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The faint extragalactic radio sky Vernstrom, Tessa 2015

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The Faint Extragalactic Radio SkybyTessa VernstromB.Sc., The University of Minnesota, 2008M.Sc., The University of British Columbia, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Astronomy)The University of British Columbia(Vancouver)August 2015c© Tessa Vernstrom, 2015AbstractThe radio sky covers a large range of sources, from small single galaxies to largeclusters of galaxies and the space between them. These sources consist of someof the most powerful objects in the Universe, as well as diffuse weak emitters.Understanding the radio sky tells us about how galaxies have evolved over time,the different kinds of galaxy populations, the star formation history of the Universe,and the role of magnetism, as well as details of large-scale structure and clustering.Advancements in radio telescopes now allow us to push observational limits to newdepths, probing fainter galaxies and farther back in cosmic time.We use a multi-pronged approach to examine several aspects of the faint ex-tragalactic radio sky. Using new deep data from the Karl G. Jansky Very LargeArray telescope, combined with the confusion analysis technique of P(D), we ob-tain the deepest estimates of the source count of individual radio galaxies and theircontribution to the cosmic radio background temperature. Additionally, these dataare used to catalogue the individual galaxies in order to study characteristics suchas source size, spectral dependence, galaxy type, and redshift. We then examinethe contribution from extended large-scale diffuse emission to the radio sky usingdata from the Australia Telescope Compact Array. This yields constraints on thetotal emission from such sources, including galactic halos, galaxy cluster halos andrelics, and the inter- and intra-cluster medium. Finally, we investigate the radio an-gular power spectrum using interferometric data. These measurements show thefluctuations coming from the unresolved radio background as a function of angularscale.Together these studies present the deepest constraints available for the faint ra-dio sky across a range of statistical areas. The measurements obtained here provideiiconstraints on the evolving population of galaxies through their radio emission inorder to further our knowledge of galaxy evolution in general.iiiPrefaceThe text of this dissertation includes modified reprints of previously published ma-terial as listed below. Per publishing convention, throughout this thesis the word“we” is used when discussing the work performed. However, the breakdown of mycontribution to the work is as follows 1.Chapter 1 (published):• T. Vernstrom, D. Scott, J. V. Wall, Contribution to the Diffuse Radio Back-ground from Extragalactic Radio Sources, MNRAS, 415, 3641 (2011) [182]The work for this paper was performed as part of, and is also published in, myMasters Thesis. This paper presents an analysis of radio source counts, and theircontribution to the radio background temperature, across a range of frequenciesfrom a compilation of previously published data. I wrote the entire text and per-formed all of the analysis. D. Scott and J.V. Wall provided support for the work,including the initial idea, along with comments on the writing.Chapter 2.2 (published):• J.J. Condon, W.D. Cotton, E.B. Fomalont, K.I. Kellermann, N. Miller, R.A.Perley, D. Scott, T. Vernstrom, J.V. Wall. Resolving the Radio Source Back-ground: Deeper Understanding through Confusion, ApJ, 758, 23 (2012) [37]This paper presents the initial data reduction methods and analysis of new datafrom the Karl G. Jansky Very Large Array. This work was a collaboration between1 Chapters 6 and 7 describe work not yet published. I have performed the entirety of the analysisfor those chapters, with commentary contributions from D. Scott and J.V. Wall.ivmyself and advisors (D. Scott and J.V. Wall) at the University of British Columbiaand collaborators at the National Radio Astrophysical Observatory (NRAO). Thework presented was carried out, and written up, predominately by W.D. Cotton andJ.J Condon, who was the PI for the genesis of this project. I assisted in the taking ofthe observations, and provided considerable input in the writing of the paper. Thework presented in Chapter 2.2 presents a portion of the work from the publishedpaper.Chapter 3.2, Chapter 4 (published):• T. Vernstrom, D. Scott, J.V. Wall, J.J. Condon, W.D. Cotton, E.B. Fomalont,K.I. Kellermann, N. Miller, R.A. Perley. Deep 3 GHz Number Counts froma P(D) fluctuation analysis, MNRAS, 440, 2791 (2014) [183]This paper was a follow-up to the previously mentioned paper and presents myanalysis of the data described in that work. The initial data calibration and imagingwas performed by W.D. Cotton. However, I carried out the entirety of the analysisand wrote the text of the paper. Important feedback on the work and text was pro-vided by all of the co-authors.Chapter 2.1, Chapter 5 (published):• T. Vernstrom, R.P. Norris, D. Scott, J.V. Wall. The Deep Diffuse Extragalac-tic Radio Sky at 1.75 GHz, MNRAS, 447, 2243 (2015) [184]This work was a collaboration between myself and advisors at UBC and R.P.Norris at the Australia Telescope National Facility/Commonwealth Scientific andIndustrial Research Organization (ATNF/CSIRO). The idea for the project and ini-tial telescope proposal were the responsibility of R.P. Norris. I assisted with thetelescope observations at the Australia Telescope Compact Array in Australia andperformed all data calibration and imaging. I carried out the analysis and wrote thetext of the paper, with helpful commentary from my co-authors on the analysis andwriting.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Acronyms and Abbreviations . . . . . . . . . . . . . . . . . . . . xviList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Source Counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 The Radio Background . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 Source Count Contribution to Temperature . . . . . . . . 141.3.2 ARCADE 2 . . . . . . . . . . . . . . . . . . . . . . . . . 162 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1 Australia Telescope Compact Array . . . . . . . . . . . . . . . . 242.1.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 25vi2.1.2 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.3 Image Noise . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Karl G. Jansky Very Large Array . . . . . . . . . . . . . . . . . . 312.2.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.2 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2.3 Image Noise . . . . . . . . . . . . . . . . . . . . . . . . 372.2.4 The SNR-Optimized Wideband Sky Image . . . . . . . . 392.3 SKADS Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 403 Confusion and P(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1 Confusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Probability of Deflection . . . . . . . . . . . . . . . . . . . . . . 463.2.1 P(D) Simulation Tests . . . . . . . . . . . . . . . . . . . 473.2.2 Clustering and Source Sizes . . . . . . . . . . . . . . . . 504 Discrete-Source Count . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 P(D) and Image Noise . . . . . . . . . . . . . . . . . . . . . . . 554.3 Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.4 Choice of Model . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4.1 Modified Power Law . . . . . . . . . . . . . . . . . . . . 664.4.2 Node Model . . . . . . . . . . . . . . . . . . . . . . . . 674.5 Discrete Source Count Fitting Results . . . . . . . . . . . . . . . 714.5.1 Estimated Number Counts . . . . . . . . . . . . . . . . . 714.5.2 Parameter Degeneracies . . . . . . . . . . . . . . . . . . 744.5.3 Background Temperature . . . . . . . . . . . . . . . . . . 794.6 Discrete Emission Discussion . . . . . . . . . . . . . . . . . . . . 814.6.1 Image Artefacts . . . . . . . . . . . . . . . . . . . . . . . 814.6.2 Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . 854.6.3 Comparison to Other Estimates . . . . . . . . . . . . . . 864.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90vii5 Extended-Source Count . . . . . . . . . . . . . . . . . . . . . . . . . 965.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.2 P(D) and Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.3 Discrete Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3.1 Source Subtraction . . . . . . . . . . . . . . . . . . . . . 1015.3.2 Counts and Confusion . . . . . . . . . . . . . . . . . . . 1025.4 Extended Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.4.1 High Resolution Extended Emission . . . . . . . . . . . . 1055.4.2 Source Size Sensitivity . . . . . . . . . . . . . . . . . . . 1075.5 Extended Source Count Models . . . . . . . . . . . . . . . . . . 1075.5.1 Shifted Discrete Count Model . . . . . . . . . . . . . . . 1105.5.2 Parabola Model . . . . . . . . . . . . . . . . . . . . . . . 1115.5.3 Node Model . . . . . . . . . . . . . . . . . . . . . . . . 1115.6 Background Temperature . . . . . . . . . . . . . . . . . . . . . . 1135.7 Extended Emission Source Count Fitting Results . . . . . . . . . 1135.7.1 Summary of Fits . . . . . . . . . . . . . . . . . . . . . . 1135.7.2 Model Uncertainties . . . . . . . . . . . . . . . . . . . . 1145.7.3 ARCADE 2 Fits . . . . . . . . . . . . . . . . . . . . . . 1195.8 Extended Emission Discussion . . . . . . . . . . . . . . . . . . . 1205.8.1 Sources of Diffuse Emission . . . . . . . . . . . . . . . . 1205.8.2 Cluster Emission . . . . . . . . . . . . . . . . . . . . . . 1225.8.3 Dark Matter Constraints . . . . . . . . . . . . . . . . . . 1255.9 Integral Counts . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306 Discrete-Source Catalogue . . . . . . . . . . . . . . . . . . . . . . . 1326.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.1.1 Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 1346.1.2 Completeness, False Detections, and Blending . . . . . . 1416.2 Catalogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.2.1 Angular Size Distribution . . . . . . . . . . . . . . . . . 1476.2.2 Extended Sources . . . . . . . . . . . . . . . . . . . . . . 1516.3 Source Count . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153viii6.4 Spectral Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.5 Cross-Identifications . . . . . . . . . . . . . . . . . . . . . . . . 1646.5.1 Optical and IR . . . . . . . . . . . . . . . . . . . . . . . 1646.5.2 Radio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . 1677 The Radio Angular Power Spectrum . . . . . . . . . . . . . . . . . . 1737.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1737.2 Visibility Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.2.1 Primary Beam . . . . . . . . . . . . . . . . . . . . . . . 1757.2.2 Frequency Weighting . . . . . . . . . . . . . . . . . . . . 1787.2.3 Mosaicking . . . . . . . . . . . . . . . . . . . . . . . . . 1797.3 Bare Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . 1817.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1847.5 Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1887.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 1917.7 ATCA Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1947.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1998 Impact and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 2028.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2028.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.2.1 High Resolution . . . . . . . . . . . . . . . . . . . . . . 2058.2.2 Low Frequencies . . . . . . . . . . . . . . . . . . . . . . 2068.2.3 The Far Infrared to Radio Correlation . . . . . . . . . . . 2078.2.4 Polarisation . . . . . . . . . . . . . . . . . . . . . . . . . 2088.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210A C Catalogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232B CB Catalogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252ixList of TablesTable 1.1 References for the extragalactic radio count data compilation . 11Table 1.2 Best-fit parameter and χ2 values for source count polynomial fits 11Table 1.3 Background temperatures from fitting of compiled source counts 18Table 2.1 ATLAS ELAIS-S1 pointings centres . . . . . . . . . . . . . . 24Table 2.2 VLA observing Runs Summary . . . . . . . . . . . . . . . . . 32Table 2.3 Frequency and noise properties of the 16 VLA subband images 36Table 2.4 Image properties for the wide-band VLA 3 GHz data . . . . . . 40Table 4.1 Best-fitting parameters for VLa 3 GHz modified power-law model 74Table 4.2 Best-fitting parameters from VLA 3 GHz node model . . . . . 75Table 4.3 Slopes and normalizations from best-fitting VLA 3 GHz nodemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Table 4.4 Correlation matrix for best-fitting VLA 3 GHz parameters . . . 79Table 4.5 Radio background temperatures from best-fitting source countVLA 3 GHz models . . . . . . . . . . . . . . . . . . . . . . . 81Table 4.6 VLA 3 GHz best-fitting model parameters . . . . . . . . . . . 90Table 5.1 Angular and physical source sizes at different redshifts . . . . . 109Table 5.2 Best-fitting results for extended emission Model 1. . . . . . . . 114Table 5.3 Best-fitting results for extended emission Model 2. . . . . . . . 118Table 5.4 Best-fitting results for extended emission Model 3. . . . . . . . 118Table 5.5 Luminosity and redshift estimates for Model 2. . . . . . . . . 120Table 5.6 Integrated source count values of the different models scaled to1.4 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129xTable 6.1 Differential Euclidean normalized source count . . . . . . . . 160Table A.1 Discrete VLA C Catalogue . . . . . . . . . . . . . . . . . . . 233Table B.1 Discrete VLA CB Catalogue . . . . . . . . . . . . . . . . . . 253xiList of FiguresFigure 1.1 Differential and integral source counts with different normal-izations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.2 Compilation of published survey source counts and fits at mul-tiple radio frequencies . . . . . . . . . . . . . . . . . . . . . 10Figure 1.3 The Cosmic Backgrounds . . . . . . . . . . . . . . . . . . . 13Figure 1.4 Background temperature distributions from fitting publishedsource counts . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 1.5 Extragalactic background temperature values from ARCADE2 and source counts . . . . . . . . . . . . . . . . . . . . . . . 17Figure 2.1 Example Interferometer uv Coverage . . . . . . . . . . . . . 22Figure 2.2 ELAIS-S1 1.75 GHz mosaic images . . . . . . . . . . . . . . 28Figure 2.3 Instrumental noise from ATCA 1.75 GHz images . . . . . . . 29Figure 2.4 Frequency dependence of ATCA 1.75 GHz wide-band mosaicimage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Figure 2.5 Lockman Hole 3 GHz images . . . . . . . . . . . . . . . . . 38Figure 3.1 Confusion of simulated noiseless images with different beamsizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 3.2 Confusion of simulated images with different noise values . . 45Figure 3.3 Euclidean-normalised source count from SKADS simulation . 49Figure 3.4 Comparison of pixel histograms from 1.4 GHz simulation . . 50Figure 4.1 Change in VLA image noise as a function of distance . . . . . 56xiiFigure 4.2 Differences in PDFs of purely Gaussian noise and weightedvarying noise. . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 4.3 VLA 3-GHz contour images of the Lockman hole. . . . . . . 60Figure 4.4 P(D) Bin correlations and uncertainties . . . . . . . . . . . . 61Figure 4.5 Comparison of Monte Carlo fitting output for 3 GHz sourcecount with different model settings . . . . . . . . . . . . . . . 72Figure 4.6 MCMC source count fitting results of VLA 3 GHz data usingmodified power-law model . . . . . . . . . . . . . . . . . . . 73Figure 4.7 P(D)s of best-fitting source counts at 3 GHz from modifiedpower-law model . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 4.8 MCMC source count fitting results of VLA 3 GHz data usingnode model . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 4.9 P(D)s of best-fitting source counts at 3 GHz from node model 78Figure 4.10 One and two dimensional likelihood distributions for the sixVLA 3 GHz nodes . . . . . . . . . . . . . . . . . . . . . . . 80Figure 4.11 Histogram of CRB temperatures at 3 GHz from modified power-law model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 4.12 Histogram of CRB temperatures at 3 GHz from node model . 83Figure 4.13 Negative flux-density region of the VLA 3 GHz P(D) distribution 84Figure 4.14 Comparison of P(D) fitting for wide-band images with differ-ent spectral indices in the weighting . . . . . . . . . . . . . . 87Figure 4.15 Best-fit modified power-law model of VLA source count com-pared with other models and observed counts . . . . . . . . . 91Figure 4.16 Best-fit node model of VLA source count compared with othermodels and observed counts . . . . . . . . . . . . . . . . . . 92Figure 4.17 Source count bump models for ARCADE 2 temperature . . . 93Figure 5.1 Images of the synthesized beams for the 1.75 GHz data . . . . 99Figure 5.2 P(D) distributions for the ATCA 1.75 GHZ mosaic central re-gions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 5.3 Discrete source P(D) distribution at 1.75 GHz . . . . . . . . . 103Figure 5.4 Simulatied point source and extended emission images at 9 arcsecresolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108xiiiFigure 5.5 Fitted P(D)s of simulated images with different source sizes . 109Figure 5.6 Discrete source counts of AGN and starburst at 1.75 GHz withflux density shift . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 5.7 P(D) distributions from fitting for extended emission sourcecount models . . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure 5.8 Temperature histograms from MCMC fitting of the three ex-tended emission source count models . . . . . . . . . . . . . 116Figure 5.9 S2 normalized source counts at 1.75 GHz of best-fitting models 117Figure 5.10 Comparison of the radio cluster halo model from Zandanelet al. [198] with ATCA 1.75 GHz data . . . . . . . . . . . . . 124Figure 5.11 Comparison of one dark matter modelling approach with ATCA1.75 GHz data . . . . . . . . . . . . . . . . . . . . . . . . . . 126Figure 5.12 Integrated source counts at 1.4 GHz and 1.75 GHz . . . . . . 128Figure 6.1 Catalogue Simulation Uncertainties . . . . . . . . . . . . . . 135Figure 6.2 Catalogue Simulation 2D Size Ratio Distribution . . . . . . . 137Figure 6.3 Catalogue Simulation 2D Size Ratio Uncertainty Distribution 138Figure 6.4 Catalogue Simulation Corrected Uncertainties . . . . . . . . . 139Figure 6.5 Catalogue Simulation Completeness . . . . . . . . . . . . . . 142Figure 6.6 Catalogue Simulation False Detections . . . . . . . . . . . . 143Figure 6.7 Catalogue Flux Distribution . . . . . . . . . . . . . . . . . . 145Figure 6.8 Catalogue Resolution Comparison . . . . . . . . . . . . . . . 148Figure 6.9 CB Catalogue Angular Size Distribution . . . . . . . . . . . . 149Figure 6.10 C Catalogue Angular Size Distribution . . . . . . . . . . . . . 150Figure 6.11 Catalogue Mean Sizes vs Flux . . . . . . . . . . . . . . . . . 152Figure 6.12 Images for extended source at J2000 161◦.6051, 59◦.090913 . 154Figure 6.13 Images for extended source at J2000 161◦.41572, 58◦.955855 155Figure 6.14 Images for extended source at J2000 161◦.45361, 58◦.90226 . 156Figure 6.15 Images for extended source at J2000 161◦.6575, 58◦.906266 . 157Figure 6.16 3GHz Catalogue Source Count . . . . . . . . . . . . . . . . 159Figure 6.17 Catalogue Spectral Index Histograms . . . . . . . . . . . . . 162Figure 6.18 Catalogue Mean Spectral Indices vs Flux Density . . . . . . . 163Figure 6.19 Details of Catalogue Cross Matching . . . . . . . . . . . . . 165xivFigure 6.20 Colour-Colour plots of optical and NIR CB catalogue matches 168Figure 6.21 Colour-Colour plots of optical and NIR C catalogue matches . 169Figure 6.22 1.4 GHz to 3 GHz Spectral Indices and Fluxes . . . . . . . . 170Figure 7.1 ATCA Spatial Scale Responsivity . . . . . . . . . . . . . . . 177Figure 7.2 2D Log Power with ATCA uv coverage . . . . . . . . . . . . 180Figure 7.3 Power vs uv Distance Simulated Visibilities . . . . . . . . . . 183Figure 7.4 Simulation models for power spectrum . . . . . . . . . . . . 184Figure 7.5 Simulated Clustered Point Sources . . . . . . . . . . . . . . . 186Figure 7.6 Bootstrap Power Spectrum Scatter Random Positions . . . . . 189Figure 7.7 Bootstrap Power Spectrum Scatter Clustered Positions . . . . 190Figure 7.8 Power Spectra from Simulations–Clustered . . . . . . . . . . 192Figure 7.9 Power Spectra from Simulations–Extended . . . . . . . . . . 193Figure 7.10 Power Spectra from Simulations–Mosaic . . . . . . . . . . . 195Figure 7.11 Parameter Distributions from fitting Simulated 1D random powerspectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Figure 7.12 Parameter Distributions from fitting Simulated 1D clusteredpower spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 197Figure 7.13 Power Spectra from ATCA data . . . . . . . . . . . . . . . . 198Figure 7.14 C` ∆T/T Limits . . . . . . . . . . . . . . . . . . . . . . . . 200xvList of Acronyms andAbbreviationsAGN active galactic nucleiATCA Australia Telescope Compact ArrayATLAS Australia Telescope Large Area SurveyBAO baryon acoustic oscillationsCFHT Canada France Hawaii TelescopeCHIME Canadian Hydrogen Intensity Mapping ExperimentCMB cosmic microwave backgroundCRB cosmic radio backgroundDec DeclinationGMRT Giant Metrewave Radio TelescopeEMU Evolutionary Map of the Universe surveyFIR far infraredFOV field of viewFR Fanaroff-RileyFWHM full width at half maximumxviICM intracluster mediumIGM intergalactic mediumIF intermediate frequencyLIRG luminous infrared galaxyMCMC Monte Carlo Markov ChainQSO QuasarPSF point spread functionPDF probability distribution functionRA Right AscensionRFI radio frequency interferenceSKA Square Kilometre ArrayASKAP Australian Square Kilometre Array PathfinderSDSS Sloan Digital Sky SurveySED spectral energy distributionSERVS Spitzer Extragalactic Representative Volume SurveySKADS Square Kilometre Array Design StudiesSNR signal-to-noise ratioSWIRE Spitzer Wide-Area Infrared Extragalactic SurveyUKIDSS UKIRT Infrared Deep Sky SurveyULIRG ultra-luminous infrared galaxyVLA Karl G. Jansky Very Large ArrayVLASS VLA Sky SurveyxviiList of SymbolsS flux densityNdof Number of degrees of freedomDL luminosity distanceH0 Hubble constantφL luminosity function; number density as a function of luminosityΩm matter densityΩλ vacuum energy density; cosmological constantDa antenna diameterb (maximum) baseline lengthCT flux to temperature conversion factorα spectral indexΩB beam solid angleΩS source solid angleTb background temperaturekB Boltzmann constantEl left hand complex electric field amplitudexviiiEr right hand complex electric field ampltiudeρ angular offset from the pointing centredN/dS differential source countN(> S) integral source countD deflection (observed brightness)P (D) Probability of deflectionB(θ, φ) beam or point spread functionR(x) mean number of pixel values per steradian with observed intensitiesbetween x and x+ dxσc confusion noiseσn instrumental noiseσ∗n weighted instrumental noiseς histogram bin uncertaintyθM major axis FWHMθDM deconvolved major axisθm minor axis FWHMθDm deconvolved minor axisθB beam FWHMφ position angleSP peak flux densityST total integrated flux densityw(θ) 2-point angular correlation functionxixAcknowledgmentsI acknowledge the financial support of the Natural Sciences and Engineering Re-search Council (NSERC) of Canada. I thank the staff of the VLA , which is oper-ated by the National Radio Astronomy Observatory (NRAO). The National RadioAstronomy Observatory is a facility of the National Science Foundation operatedunder cooperative agreement by Associated Universities, Inc. I thank the ATCAand CSIRO staff. The Australia Telescope Compact Array is part of the AustraliaTelescope National Facility which is funded by the Commonwealth of Australiafor operation as a National Facility managed by CSIRO.I would like to personally thank my advisors Douglas Scott and Jasper Wall forall their help. And thank my parents, sister, and husband for all their support.xxChapter 1IntroductionThe radio sky can be broken up into three main components. There is the contribu-tion from our Galaxy the Milky Way (and all the sources within it), the contributionfrom the cosmic microwave background (CMB) radiation, and a contribution fromall extragalactic sources. The Milky Way contribution is by far the dominant sourceat radio wavelengths (10’s of Kelvin in temperature), whereas the other two compo-nents contribute less (roughly milliKelvins in Temperature), though it depends onthe exact frequency to say which of the CMB or extragalactic is a stronger contrib-utor. This work focus on the the extragalactic component. We break this categorydown into two parts: discrete sources and extended sources.Discrete sources are point sources, or small angular scale sources (typicallybelow a few arcsec). This is emission from individual galaxies, which at radiofrequencies are predominantly active galactic nuclei (AGN) and star forming, orstarburst, galaxies. AGN were first discovered by Bolton et al. [14] and Ryle et al.[153]. These are galaxies which host a supermassive black hole at the centre whichis actively accreting material yielding a much higher luminosity than normal atsome, if not all, wavelengths. AGN are some of the most luminous persistentsources of radiation in the Universe. At 1.4 GHz the flux densities range from mJyto 10s or hundreds of Jy’s (with Jy being the conventional unit of flux density and1 Jy= 1 × 10−26 W m−2 sr−1 Hz−1). The accreting black holes generate strongmagnetic fields resulting in powerful synchrotron emission. At radio frequenciesAGN can have several features such as relativistic jets, lobes, or hotspots.1There are several classifications of AGN, although with the current unifiedmodel it is thought these are all the same type of galaxy with the differences a resultof our particular viewing angle (Barthel [6], Orr and Browne [132], Scheuer andReadhead [158], Urry and Padovani [179]). These can be classified as low-radio-power edge-dimmed Fanaroff-Riley (FR) type 1 ([48]) which have broad emissionlines (optically featureless continua), and powerful FR type 2 (edge-brightened)with narrow lines in the optical/UV spectra sometimes known as radio quasars(FSRQs). It is generally accepted that type 2 sources are obscured versions of type1. These can be broken down into flat and steep-spectrum sources. The emissionfrom the core (stellar-like) tends to have a flat spectrum and these are known as BLLacs or Quasars (sometimes referred to as Blazars). Whereas the emission domi-nated by the extended optically thin components (the jets, lobes, or hot-spots) has asteep spectrum. In general each source has both a compact flat-spectrum core andextended steep-spectrum lobes, it just depends on our viewing angle with respectto the source.There are radio-loud and radio-quiet AGN, which refers to the type of accre-tion happening, hot-mode and cold-mode, respectively. Radio-loud objects are thebrighter sources which produce the large scale radio jets and lobes, with the kineticpower of the jets being a significant fraction of the total bolometric luminosity;whereas, the weak ejecta of the radio-quiet objects are energetically insignificant.Radio-loud objects tend to be associated with elliptical galaxies which have under-gone recent mergers, while radio-quiets generally have spiral hosts. The thermalemissions from the AGN (continua and lines from X-ray to infrared wavelengths)are quite similar in the two classes (Sanders et al. [154]), which has been arguedto mean the black hole masses and mass accretion rates are not greatly differentand that the difference is associated with the spin of the black hole [192]. Therehas been much work looking into the different AGN classes, the emission mecha-nisms, and more and for a reviews of such topics see e.g. Antonucci [3], Wilsonand Colbert [192].In terms of non-AGN radio galaxies the main kinds are star-forming and star-burst galaxies. Star-forming galaxies are considered “normal” spiral-type galaxies(sometimes referred to as main sequence galaxies). Starburst galaxies are under-going a large burst of star formation and are usually classified as luminous infrared2galaxies (LIRG) or ultra luminous infrared galaxies (ULIRG). The primary emis-sion is synchrotron emission from the star formation in the disk of the galaxies.These both tend to be less luminous than AGN (or at least than radio-loud AGN)with flux densities ≤mJy’s.The counts (or number of objects per sr as a function of flux density) of discreteradio sources and their contribution to the cosmic radio background (CRB) canbe used to constrain galaxy evolution. The history of radio source counts goesback to Mills [119] who discussed the cumulative count or ‘ogive’, giving one ofthe first published source counts, followed by Ryle and Scheuer [152], with thediscovery that the slope of the source count was steeper than expected from a staticEuclidean universe, which implied that sources must be evolving in space densityor luminosity. Since then many more surveys have been carried out to measurethe source counts at various radio frequencies, both whole-sky and limited areasurveys, which confirmed the importance of evolution [see 42, for a recent review].These counts can now be broken down into different source populations.With advancements in radio telescope capabilities we have been able to probethe source count to ever increasing depth, so that estimating the count in the µJyand sub-µJy regions is now possible. Investigating the count at these faint fluxdensities is important for understanding the evolution of sources at lower lumi-nosities and/or higher redshifts (z ≥ 2). How the count below 10µJy behaves hasbeen unknown until now: what the slope is in this region, whether the Euclidean-normalised count declines or begins to rise again (possibly indicating a new popu-lation), and whether the sources continue to obey the well-known far infrared (FIR)to radio correlation [35, 41, 95] out to higher redshifts.The brightest radio sources are known to be radio-loud AGN. However, it wasnot until the 1980s that the radio source counts at the sub-mJy level revealed anew radio population (Mitchell and Condon [120], Windhorst et al. [193]). Itis now known from various studies (e.g. Seymour et al. [161], Simpson et al.[164], Smolcˇic´ et al. [167]) that the bulk of discrete radio sources in the sub-mJyrange, for frequencies near 1.4 GHz, are star-forming galaxies (starbursts, spirals,or irregulars) and low luminosity radio AGNS (faint FR I radio galaxies, Seyfertgalaxies, radio-quiet Quasar (QSO), see e.g Mignano et al. [117], Padovani et al.[136]). Using optical/IR cross-match identifications and redshift data, their space3distributions have been modelled via their luminosity functions to determine howthey evolve with redshift. The number counts dN/dS, or the differential numberof sources per steradian per flux density interval, have now been measured wellinto the sub-mJy region [37, 134], as well as the contribution from these sources tothe isotropic diffuse radio background.However, what is less well characterized is extended large-angular-scale dif-fuse low-surface brightness sources (sizes of 10s of arcseconds to arcminutes) .There have been few surveys carried out for diffuse arcminute-scale extragalacticemission, and very few that also have high sensitivity at that scale. The most sensi-tive lower resolution survey yet published was by Subrahmanyan et al. [172], whichreached an rms of 85µJy beam−1 with a 50 arcsec beam. This angular scale en-compasses emission from individual galaxy haloes, emission from galaxy clustersand groups (e.g. Cassano et al. [23], Fabian et al. [47], Liang et al. [109], Petersonand Fabian [140], Venturi et al. [180, 181]), or emission from large-scale structure(the cosmic web), such as the intergalactic medium (IGM) or inter-cluster medium(e.g. Bagchi et al. [4], Kronberg et al. [101]).Clusters of galaxies are the largest gravitationally bound systems in the Uni-verse. They are formed by hierarchical structure formation processes, or merg-ers of smaller systems. It is believed the underlying dark matter in the Universeconsists of filaments and voids. Denser regions form a filament and clusters areformed within the filaments. Major cluster mergers are some of the most energeticevents in the Universe. During mergers, shocks are driven into the intraclustermedium (ICM) , following with the injection of turbulence. Eventually, clusterscan reach a relaxed state with a giant galaxy and hot gas in the centre. In recentyears there has been growing evidence of cluster large-scale diffuse radio sourcesof synchrotron origin associated with the ICM ([e.g. 54, 71]). Diffuse featuresdemonstrate that the thermal ICM plasma is mixed with the non-thermal compo-nents of large-scale magnetic fields with relativistic electrons in the cluster volume.These magnetic fields have been observed through studies of the Faraday rotationof polarized radio galaxies in the background or within the magnetized ICM.The diffuse sources are usually grouped into three categories: haloes, minihaloes, and relics. Haloes are hosted in clusters showing merging processes (Fer-etti [51]) and are located near the cluster centre. Relics are located in the cluster4peripheral regions of both merging and relaxed clusters. Mini haloes are hosted inrelaxed cool-core clusters and are usually centrally located near a powerful radiogalaxy. These objects all tend to have steep spectrums (≥ 1.), however, the relics,at the peripherals, tend to be more strongly polarized (Feretti [51]). For a detailedreview of these objects see Feretti et al. [53].1.1 Thesis OutlineRather than looking at a small number of radio sources in detail, this work focuseson characterizing the population of sources by looking at large numbers of them.Statistical surveys complement studies of individual sources, thereby yielding in-formation on the flux densities, numbers, and source sizes, the source count (i.e.the 1-point statistics) and the 2-point statistics (e.g power spectrum or two-pointcorrelation function). The goal of this work is to use these methods to examine thefaint radio sky and faint radio sources at both large and small angular scales. Wefirst give an introduction to source counts and the cosmic radio background in theremainder of this Chapter.In Chapter 2 we discuss the data that are used in the analysis, including theobservation, data calibration, and imaging details based on Condon et al. [37] andVernstrom et al. [183, 184], as well as the details of the main simulation data fromWilman et al. [191] that are used throughout this thesis. In Chapter 3 the mainmethod used (at least in Chapters 4 and 5) is explained. This includes exampletests of the method using the realistic simulations described in Chapter 2.Chapter 4 details the work carried out to investigate the faint discrete sourcecount using new deep radio observations, based on Vernstrom et al. [183]. Chap-ter 5 is based on the work of Vernstrom et al. [184] investigating extended radioemission. In Chapter 6 a preliminary version of the discrete-source catalogue isconstructed and described, using the data from Condon et al. [37] and Vernstromet al. [183]. Chapter 7 presents a first measurement of the radio angular power spec-trum at faint intensities. Finally, Chapter 8 gathers together the conclusions fromthe previous chapters and considers their broader impacts, as well as discussingsome avenues for future work to continue these investigations.51.2 Source CountsSource counts, namely the surface density of sources as a function of flux density(S), can be directly compiled from any complete sample, without any additionaldata. Because flux density is primarily a function of both redshift and luminosity,source counts are the first probes into the behaviour of galaxy evolution. Study-ing source counts is important for a number of reasons. They tell us the num-ber and density of galaxies in the Universe. Source counts can, in principle, alsotell us about the cosmology of the Universe, particularly the faint counts. Sourcecounts come from the luminosity function, φL, which is the number of galaxies asa function of luminosity in some volume. The volume depends on cosmologicalparameters such thatdVdz=cH0D2L(1 + z)2(1 + Ωmz)− z(2 + z)ΩΛ. (1.1)Here DL is the luminosity distance, H0 is the Hubble constant, Ωm is the matterdensity, and Ωλ is the energy density. The luminosity distance is dependent is alsoa function of the cosmological parameters,DL =cH0∫ z(L,S)0(1 + z)dz√(1 + z)2(1 + Ωmz)− z(2 + z)ΩΛ. (1.2)The flux density depends on the luminosity distance, whereS =L4piD2L. (1.3)Thus, if ΩΛ 6= 0, there is faster expansion, a larger volume, and more galaxies(though evolution of source populations is the dominant factor in the form of thesource counts, and distinguishing cosmological parameters in practice is compli-cated by issues such as survey systematics).Source counts enable us to investigate the evolution of galaxies, comparinggalaxies today with galaxies in the past. Additionally, we can quantify and con-strain the evolution in galaxy populations and discriminate the nature of the evolu-tion.6Source counts can reveal the contribution of different galaxy populations tothe overall population of galaxies. The counts at different wavelengths are dom-inated by different types of galaxies, for example X-ray are predominantly AGN,radio counts are AGN (bright flux densities) and star-forming galaxies (dominateat S ≤ 1 mJy), whereas infrared is mainly spiral star forming galaxies. Comparingdata from the different wavelengths can reveal information about the underlyingphysical processes. For example comparing radio data to X-ray data can tell usabout the connection between synchrotron emission and Inverse Compton scatter-ing ([9, 50, 83, 92]). The radio and infrared can both be tracers of star formation;thus combing data from those two wavebands can help to map the star formationhistory of the Universe.Source counts can be presented in different forms, the most basic being thecumulative integral source count (N(> S)) (Fig. 1.1, top left panel), describingthe expected number of sources per unit area above a given flux density. Becauseof its cumulative nature, consecutive data points are not statistically independent,which can be problematic, especially in the corresponding error analysis. Thedifferential form of the source count dN/dS (Fig. 1.1, top right panel), describesthe number of sources per unit area in a given flux density bin and avoids theproblem of dependence of consecutive values.Because the differential source count is generally very steep, possibly hidingsome important features, it is customary to represent it relative to the count ofuniformly distributed sources in a flat, non-evolving Euclidean Universe, wherethe flux S ∝ 1/D2 and the number density N ∝ D3. Thus the integral count hasthe formN(> S)Euc = KS−3/2. (1.4)Here the arbitrary constant K is usually taken from the number density of sourceswith flux-densities above 1 Jy. In the differential form the Euclidean count becomesS−5/2. Normalizing the source count by this factor provides the relative Euclidean-normalized differential source count, dNdS S5/2 (Fig. 1.1, bottom left panel), whichis the third form of count usually considered. The fourth form of the count shownin Fig. 1.1 (bottom right panel) is weighting the differential counts by S2. Thisalternate weighting of S2dN/dS is proportional to the source count contribution7100101102103104105106107108 N(>S) (sr−1)Integral 10-1101103105107109101110131015 dN/dS (sr−1 Jy−1 )Differential10-4 10-3 10-2 10-1 100 101 102 103 104S (mJy)10-210-1100101102 S5/2dN/dS (sr−1 Jy1.5 )Euclidean Normalized Differential10-4 10-3 10-2 10-1 100 101 102 103 104S (mJy)101102103 S2dN/dS (sr−1 Jy)S2  Normalized DifferentialFigure 1.1: Example source count (taken from Condon and Mitchell [34]and Vernstrom et al. [183]) showing the differential and integral sourcecounts with different normalizations. The top left panel shows the in-tegral source count, N(> S), the number of sources brighter than fluxdensity S per sterradian. The top right panel is dN/dS, or the numberof sources per sterradian with flux densities between S and S + dS.The bottom left panel is the differential source count normalized by theEuclidean (static universe) count of S−5/2 and the bottom right panel isthe differential source count normalized by S2 (as appropriate for esti-mating the contribution to the radio background).8to the background temperature per logarithmic interval of flux density.Gervasi et al. [69] obtained fits to the source count data for different radio-emitting galaxy populations across a range of frequencies from ν = 150 to 8440MHz. From their fits, which ranged from 1µJy to 100 Jy, they were able tointegrate the source counts to obtain an estimate of the sky brightness tempera-ture contribution at each of the frequencies. They determined a power-law skybrightness temperature dependency on frequency, with a spectral index (α) of –2.7,which is in agreement with the frequency dependence of the flux density emitted bysynchrotron-dominated steep-spectrum radio sources [31]. These estimates wereused to interpret absolute measurements of the radio sky brightness by the TRISexperiment [199].As part of my Masters thesis, I performed similar work using published sourcecount data at ν = 150 MHz, 325 MHz, 408 MHz, 610 MHz, 1.4 GHz, 4.8 GHz,and 8.4 GHz. The survey data were was compiled in de Zotti et al. [42], withall the individual surveys listed in Table 1.1. For each frequency I fit a 5th orderpolynomial of the formF (S) = A0 +A1S +A2S2 +A3S3 +A4S4 +A5S5. (1.5)The fitting was initially performed using a χ2 minimization routine. The co-efficients given by the χ2 minima were then used as starting points in a MonteCarlo Markov Chain (MCMC) approach [107], which was used to refine the fitsand obtain estimates of uncertainty. The best-fit values for all the parameters ateach of the frequency bands can be found in Table 1.2, along with χ2 values foreach fit. The data and the best-fit lines are plotted in Fig. 1.2, which shows theEuclidean-normalized data.Table 1.2 shows that the χ2 values of the fits are generally reasonable (giventhe different choices made for calibration and other corrections for synthetic effectsamong the data sets) , with all but one of the reduced χ2 values being below 2. Theexception is for the 1.4 GHz data set, with a χ2 of over 20 per degree of freedom.To obtain anything like a reasonable χ2 we would have to increase the errors bya factor of four. It is worrisome that the 1.4 GHz compilation is the one with themost available data. As can be seen in the plot, there are many data points that are9105 106 107 108S (µJy)102103104S5/2 dN/dS (sr−1 Jy−1.5)Fit to 150 MHz countsCounts ν=150 MHz102 103 104 105 106 107S (µJy)101102103104S5/2 dN/dS (sr−1 Jy−1.5)Fit to 325 MHz countsCounts ν=325 MHz104 105 106 107 108S (µJy)101102103104S5/2 dN/dS (sr−1 Jy−1.5)Fit to 408 MHz countsCounts ν=408 MHz101 102 103 104 105 106 107S (µJy)100101102103104S5/2 dN/dS (sr−1 Jy−1.5)Fit to 610 MHz countsCounts ν=610 MHz101 102 103 104 105 106 107 108S (µJy)10-1100101102103104S5/2 dN/dS (sr−1 Jy−1.5)Fit to 1.4 GHz countsCounts ν=1.4 GHz101 102 103 104 105 106 107S (µJy)10-210-1100101102103S5/2 dN/dS (sr−1 Jy−1.5)Fit to 4.8 GHz countsCounts ν=4.8 GHz101 102 103 104 105 106 107S (µJy)10-1100101102103S5/2 dN/dS (sr−1 Jy−1.5)Fit to 8.4 GHz countsCounts ν=8.4 GHz101 102 103 104 105 106 107S (µJy)10-1100101102103104S5/2 dN/dS (sr−1 Jy−1.5)Fit to 150 MHz countsFit to 325 MHz countsFit to 408 MHz countsFit to 610 MHz countsFit to 1.40 GHz countsFit to 4.80 GHz countsFit to 8.40 GHz countsFigure 1.2: Compilation of published survey source counts and fits at multi-ple radio frequencies. The published data (balck points) were compiledfrom de Zotti et al. [42] and the models (solid coloured lines) are fitsto the data using eq. 1.5. The references for the surveys used at eachfrequency are listed in Table 1.1 and the best-fit parameter values aregiven in Table 1.2. The bottom right panel shows the models from allthe frequencies on the same axes for comparison.10Table 1.1: References for the extragalactic radio count data compilationFrequency References150 MHz Hales et al. [81], McGilchrist et al. [115].325 MHz Oort et al. [131], Owen and Morrison [134], Sirothia et al. [166].408 MHz Benn et al. [8], Grueff [77], Robertson [150].610 MHz Bondi et al. [15], Garn et al. [67], Ibar et al. [94].Katgert [99], Moss et al. [123].1.4 GHz Bondi et al. [16], Bridle et al. [17], Ciliegi et al. [29], Fomalont et al. [62],Gruppioni et al. [78], Hopkins et al. [91], Ibar et al. [94], Miller et al. [118]:Mitchell and Condon [120], Owen and Morrison [134], Richards [148],Seymour et al. [161], White et al. [190].4.8 GHz Altschuler [2], Donnelly et al. [44], Fomalont et al. [59], Gregory et al. [76] ,Kuehr et al. [102], Pauliny-Toth et al. [139], Wrobel and Krause [195].8.4 GHz Fomalont et al. [61], Henkel and Partridge [85], Windhorst et al. [194].Table 1.2: Best fit parameter and χ2 values for source count polynomial fitsν χ2 Ndof A0 A1 A2 A3 A4 A5MHz150 68 45 6.58 0.36 −0.65 −0.19 0.26 0.099325 59 34 5.17 0.029 −0.11 0.36 0.17 0.20408 66 44 4.13 0.13 −0.34 −0.003 0.035 0.01610 75 59 3.02 0.71 0.97 0.91 0.28 0.0281400 4230 196 2.53 −0.052 −0.020 0.051 0.010 −0.00134800 32 47 1.95 −0.076 −0.15 0.020 0.0029 −0.000798400 41 29 0.79 −0.10 −0.23 −0.051 −0.019 −0.0029inconsistent with each other, even with the relatively “large” error bars.There are systematic differences between different surveys at 1.4 GHz, thusparticularly at the faint end (see Condon [33] for further discussion). In the µJyrange it is difficult to obtain reliable counts, since this range is close to the confu-sion limit of most radio surveys ([34, 193], though this depends on survey resolu-tion) and hence the level of incompleteness may be incorrectly estimated in somesurveys. Moreover, at the bright end there are significant and systematic sourcesof error introduced when attempting to correct for source extension and surfacebrightness limitations (see discussion in Singal et al. [165] and Subrahmanyan et al.11[172]). In addition to these effects, the total variance is enhanced by source clus-tering. We know cosmic variance can lead to differences in counts for small fields,but an order of magnitude less than the observed scatter.1.3 The Radio BackgroundThe sum of all emission at different wavelengths yields a measure of the back-ground, with different sources contributing at the different wavelengths. Measure-ments of the backgrounds as a function of wavelength is shown in Fig. 1.3. Most ofthe electromagnetic energy of the Universe is in the cosmic microwave backgroundradiation left over from the hot big bang. It has a nearly perfect 2.73 K blackbodyspectrum peaking at λ ∼ 1 mm (300 GHz) [55]. The strong UV/optical peak isprimarily thermal emission from stars, supplemented by a smaller contribution ofthermal and nonthermal emission from the active galactic nuclei (AGN) in Seyfertgalaxies and quasars. Most of the comparably strong cosmic infrared backgroundis thermal re-emission from interstellar dust heated by absorbing that UV/opticalradiation. The cosmic X-ray and gamma-ray backgrounds are mixtures of non-thermal emission (e.g., synchrotron radiation or inverse-Compton scattering) fromhigh-energy particles accelerated by AGN and thermal emission from very hot gas(e.g., gas in clusters of galaxies).By comparison, the cosmic radio-source background is extremely weak. Al-though the overall contribution from these sources is small there are some ex-tremely powerful radio sources; they are just so rare that the build up of integratedradiation is small. These powerful sources are now known to play an important rolein the formation and evolution of galaxies. Radio sources do trace most phenom-ena that are detectable in other portions of the electromagnetic spectrum, and mod-ern radio telescopes are sensitive enough to detect extremely faint radio emission.The cosmic radio background is a combination of emission from these extragalac-tic sources, a component from the CMB blackbody, and Milky Way emission. Itdepends on the frequency as to how the contributions from these components com-pare. However, at radio frequencies the Galaxy emission is dominant. In orderto study the extragalactic component it is therefore necessary to observe areas farfrom the Galactic plane. Since the CMB blackbody radiation is well measured, it12Figure 1.3: The Cosmic Backgrounds: the electromagnetic spectrum of theuniverse [figure 1 from 159]. The brightness per logarithmic frequency(or wavelength) interval is shown as a function of the logarithm ofthe wavelength, so the highest peaks correspond to the most energeticspectral ranges. These data are based on a compliation from Resselland Turner [147], with data from Smoot [168] (CMB); Hauser et al.[84], Lagache et al. [103] (FIB); Leinert et al. [106] (NIR, UV); Dwekand Arendt [45] (NIR); Pozzetti et al. [144] (optical); Gendreau et al.[68], Miyaji et al. [121] (X-ray); Kappadath et al. [98], Sreekumar et al.[169], Weidenspointner et al. [187] (γ-ray). The purple shaded regionshows the radio wavelengths which we are interested in here.13can be effectively removed, leaving the component from extragalactic radio sourcesto be studied further.1.3.1 Source Count Contribution to TemperatureWe can easily obtain an estimate of the contribution from sources to the radio back-ground temperature (Tb). The source count and the sky temperature at a frequencyν are related by the Rayleigh-Jeans approximation,∫ ∞SminSdNdSdS =Tb2kBν2c2. (1.6)In the above equation kB is the Boltzmann constant (kB), and Tb is the sky tem-perature from all the sources brighter than Smin. Equation (1.6) is also equivalentto ∫ ∞SminS2dNdSd[ln(S)] =Tb2kBν2c2. (1.7)It is for this reason that it is convenient to show the source count weighted not bythe Euclidean S5/2 but by S2. With such a plot the source count must fall off atboth ends to avoid over-predicting the background (i.e. violating Olber’s paradox1); hence the higher end must turn over at flux densities above those we have plotted.The fitting of the compiled source count data can be used to obtain estimatesof the background temperature contribution by extragalactic sources at each fre-quency. The results from integrating the best-fit models of Table 1.2 yields thebackground temperatures shown in Table 1.3. To investigate the uncertainties thor-oughly, we carried out our fits using MCMC analysis, by adopting CosmoMC [107]as a generic MCMC sampler. The χ2 function was sampled for each set using thepolynomial in eq. 1.5, which was then fed to the sampler to locate the χ2 minimum.Each of the six parameters of the polynomials was varied for each step of the chainand the chains were run with 500,000 steps. CosmoMC generates statistics for thechains, including the minimum χ2, the best fit values for each of the parameters,and their uncertainties.1Obler’s paradox states that a static, infinitely old universe with an infinite number of stars andgalaxies would yield a bright rather dark night sky. Thus to avoid violating this paradox the numberof galaxies must at some point turn over and stop increasing.1418000 18500 19000 19500 20000Tb (mK)0.00000.00050.00100.0015Probability densityµ=18847.88 mKν=150 MHz2400 2600 2800 3000 3200 3400 3600 3800Tb (mK)0.00000.00050.00100.00150.00200.00250.00300.00350.00400.0045µ=3133.27 mKν=325 MHz1450 1500 1550 1600 1650 1700Tb (mK)0.0000.0020.0040.0060.0080.0100.0120.0140.016Probability densityµ=1577.17 mKν=408 MHz600 650 700 750 800Tb (mK)0.0000.0020.0040.0060.0080.0100.0120.0140.016 µ=690.21 mKν=610 MHz110 120 130 140Tb (mK)0.000.020.040.060.080.100.12Probability densityµ=119.71 mKν=1.4 GHz2.6 2.8 3.0 3.2 3.4 3.6 3.8Tb (mK)0.00.51.01.52.02.5 µ=3.22 mKν=4.8 GHz0.45 0.50 0.55 0.60 0.65 0.70 0.75Tb (mK)024681012Probability densityµ=0.58 mKν=8.4 GHzFigure 1.4: Background temperature distributions from fitting publishedsource counts. Using Monte Carlo Markov Chains we fit the sourcecounts in Fig. 1.2 with eq. (1.5), then each step in the chain is used tocompute a background temperature using eq. (1.6).15Histograms of the chain values for the background temperature are shown inFig. 1.4. From the width of these histograms we are able to measure the uncer-tainty in our estimates for the background temperature, taken here as the 68 percentarea values, fully accounting for the correlations among the parameters in the poly-nomial fits. The 1σ uncertainties are listed in Table 1.3.Most of the histograms are fairly Gaussian, which is a reflection of the qual-ity of the data. Source counts at frequencies with well-sampled data around thepeak contribution tend to have well-constrained background temperature values,e.g. at 408 MHz. However, there is a noticeable irregularity with the 325 MHzhistogram. Because of the limited data available at 325 MHz, and with the peakarea of contribution having little to no data, the histogram at this frequency doesnot have a well defined shape, and the uncertainty is far from Gaussian.Over the years there have not been many estimates of the CRB made usingsource count data [52, 69, 110, 143, 185]. And within this list, the frequenciescovered were rather limited and uncertainties not always quoted. It is importantto see how our estimates compare with these previous estimates. Longair [110]gives a value for T178 = 23 ± 5 K. Wall [185] lists estimates of T408 = 2.6 K,T1.4 = 0.09 K, and T2.5 = 0.02 K. Our results are in agreement with these earlierestimates to within ± 2σ. The values for source contributions from Gervasi et al.[69] tend to be a little higher than the values found here, the differences beingtraceable to choices made for the limits of integration and for the parameterizedform for the fits.1.3.2 ARCADE 2Results from the Absolute Radiometer for Cosmology, Astrophysics, and DiffuseEmission, or ARCADE 2 [56, 160], revived interest in the CRB and how it maybe related to the faint2 counts. The results of the ARCADE 2 experiment havehad a significant impact on the study of the radio background.This instrument pro-vided absolute measurements of the sky temperature at 3, 8, 10, 30, and 90 GHz.These results showed measured temperatures of the radio background about 5 timesgreater than that currently determined from radio source counts, with the most no-2In this work the terms“faint” and “bright” refer to the flux density of the source or sources andnot to intrinsic brightness or luminosity.1610-1 100 101Frequency (GHz)100101102103104105Temperature (mK)Vernstrom et al.,2011ARCADE 2Figure 1.5: Extragalactic background temperatures from ARCADE 2 andsource counts at multiple frequencies. The blue points and dot-dashedline is an extrapolation to the frequencies used in Vernstrom et al. [182]using the best-fit power-laws (eq. (1.8) and eq. (1.9)) from the ARCADE2 experiment. The red points and dashed line are from the fitting of pub-lished source counts as discussed in Sections 1.2 and 1.3.1.17Table 1.3: Values of the integrated sky brightness and temperature contribu-tion from radio source counts for different frequency bands. The uncer-tainties are 1σ limits determined from Markov chain polynomial fits tothe data.ν νIν T δTMHz W m−2 sr−1 mK mK150 1.8 ×10−14 17800 300325 2.1 ×10−14 2800 600408 2.9 ×10−14 1600 30610 4.2 ×10−14 710 901400 7.5 ×10−14 110 204800 8.0 ×10−14 3.2 0.28400 9.6 ×10−14 0.59 0.05table excess of emission being detected at 3 GHz. Since most systematic effectsexplaining this emission were ruled out, we are left with the question of whether itcould be caused by some previously unknown source of extragalactic emission (orobservational error).The ARCADE 2 experiment measured a background temperature of (54 ±6) mK at 3.3 GHz. Several fits are provided to these data, which allow for scalingof the result to different frequencies. The initial fit provided in Seiffert et al. [160]isTb = (1.06± 0.11K)( ν1GHz)−2.56±0.04. (1.8)There is another fit, incorporating data from lower frequencies, given in Fixsenet al. [57] asTb = (24.1± 2.1K)( ν310MHz)−2.599±0.036. (1.9)The quantities extrapolated from the ARCADE 2 data fits and our current es-timates from source counts are shown in Fig. 1.5. Here it can be seen that theARCADE 2 absolute measurements lie far above source estimates, particularly atlower frequencies. Clearly, the excess detected around 3 GHz would correspondto a large excess at lower frequencies if the power-law behaviour continued (whichwas the main result from Vernstrom et al. [182], the outcome of my Masters thesiswork).18Singal et al. [165] suggested some possible explanations for this excess as anew population of radio sources below current detection limits; or diffuse largescale emission from galaxies or clusters, or the cosmic web (either from conven-tional synchrotron emission from cosmic ray electrons or electrons from dark mat-ter particle annihilation or decay causing synchrotron emission). All possibilitieswe set out to examine with the work in this thesis.19Chapter 2DataThe work presented in this thesis relies on data from radio interferometric tele-scopes. Before describing the data used I will provide a brief overview of radiotelescopes, interferometers, and the data calibration and imaging process (full de-tails can be found in e.g. Christiansen and Hogbom [27] and Rohlfs and Wilson[151]).Electromagnetic wave electric field oscillations can induce voltage oscillationsin a conductor. In a radio telescope this process happens at the antenna focus in adevice called the feed; the simplest sort of feed is a linear dipole. The output of afeed is a voltage representing power from the radio source. This power is measuredin kelvins and is referred to as the antenna temperature (the antenna temperature isnot the physical temperature of the antenna but the temperature a matched resistorwhose thermally generated power per unit frequency equals that produced by theantenna) .In a single dish telescope the angular resolution is given byθ =1.22λDa, (2.1)with λ the wavelength of the radiation received and Da is the diameter of thetelescope. In the optical, a 6-m dish provides approximately 0.025 arcsec resolu-tion, however at radio wavelengths a single dish can achieve at best only a fewarcminute resolution. For example, the Effelsberg telescope is one of the largest20steerable single dish radio telescopes with a diameter of 100 m and has a resolutionof ∼ 9 arcminutes at 1.4 GHz. It is too difficult and costly to build a single dishwith a large enough diameter to achieve arcsecond or sub-arcsecond resolution andthus we rely on interferometers.The key concept behind an interferometer is that one can link many single dishtelescopes together, combining the signal received at each, and effectively simu-lating a large single radio telescope dish with a diameter equivalent to the largestdistance between the smaller dishes. The distance between two dishes is knownas a baseline. The angular resolution achievable by an interferometer is 1.22λ/b,where b is the longest baseline distance. For comparison, an interferometer with alongest baseline of 30 km can have a resolution of 1.5 arcseconds at 1.4 GHz. Thefield of view (FOV) is limited by the size of the individual antennas.Each antenna measures different phases of the electric field wavefront arrivingfrom the source. The signals from each antenna are cross-correlated, the resultdepending on the path taken and the distance between the antennas; the interfer-ometer output will thus either be constructive or destructive. Adding in the Earth’srotation, one can imagine a source moving through the interferometer beam, giv-ing positive and negative output, and producing a fringe pattern. The measurementsthen become the complex visibilities from each baseline. Chapter 7 has more de-tails on this method, and for a full discussion and derivations of the mathematicsof interferometry see e.g. Burke and Graham-Smith [22] and Goldsmith [74].The beam pattern from an instantaneous sample from one pair of antennas rep-resents one point in the Fourier plane, or uv coordinates. As the source is observedfor more time the antenna pair traces out a track in uv space. The exact shapes of thetracks depends on the the number of baselines, antennae configuration, amount ofobserving time, source declination, and local hour angle. Figure 2.1 shows two ex-amples of the uv coverage from a 12 hour observation with the Karl G. Jansky VeryLarge Array (VLA) in its most compact configuration at +60◦ Declination (Dec)and the Australia Telescope Compact Array (ATCA) in a compact configuration at−40◦ Dec both at ν = 3 GHz. The uv coordinates are the Fourier conjugate ofRight Ascension (RA) and Dec, with units of inverse radians (sometimes writtenas kλ). The outer boundary shows the limit of the resolution, or rather the bound-ary of the beam for a single dish with Da = b. It is important to have this uv2110 5 0 5 10 15 20u (kλ)10505101520v (kλ)4 3 2 1 0 1 2 3 43210123Figure 2.1: Example interferometer uv coverage from two arrays. The mainpanel shows the coverage for a 12 hour observation of the Very LargeArray in its most compact configuration (27 antennas, maximum base-line 1 km) at +60◦ Dec. The inset shows the coverage for a 12 hourobservation with the Australia Telescope Compact Array (5 antennas,maximum baseline 352 m) at −40◦ Dec. The coloured lines show thebaseline tracks. The grey regions show the coverage of a single dishtelescope with an antenna diameter equal to the maximum baselinelength.22plane as filled as possible, which requires longer observations and telescopes witha large number of baselines; note how much more filled in the VLA array is withits 27 antennas compared to the ATCA with only five antennas. The more baselines(and thus more coverage) increases the sensitivity and increases the accuracy whentransforming to the image plane.The basic calibration steps include, deleting or flagging bad points, e.g. thosecorrupted due to radio frequency interference (RFI) such as from satellites. In ad-dition to observing the source of interest it is usually necessary to observe a strongsource with a well known brightness and spectrum in order to calibrate the dataamplitude and spectrum, and possibly a second source (usually more frequentlythrough out the observations) with well known phases, in order to calibrate the tar-get phases as a function of time. Sometimes other corrections are necessary forantenna positions offsets, delays, or the atmosphere.In terms of imaging, the individual visibilities are gridded (with chosen weights)and then Fourier transformed to make an image. In radio interferometry images thepoint spread function (PSF) resulting from the Fourier transform of the uv coverageand weighting functions is known as the “dirty” synthesized beam. The dirty beamgenerally contains positive and negative sidelobes. A “dirty” image is the Fouriertransform of the uv data with all pixels convolved by the “dirty” beam. The imagethen goes through a process known as cleaning. The cleaning process finds brightpeaks, stores them in a model as pixel flux densities known as the “clean” model,and subtracts them from the dirty image. These model components are then con-volved with the central Gaussian distribution of the dirty beam (which is known asthe “clean” synthesized beam) and added back to the residual image, with the finalimage known as the “clean image”. This clean beam is free of sidelobes (though thecleaned image still contains contamination from the dirty beam and its sidelobesfor fainter sources).The remainder of this chapter will describe the observations specifically madefor use in this thesis, as well as the details of the calibration and imaging processes.23Table 2.1: ATLAS ELAIS-S1 pointings.ATLAS RA Dec σnPointing (HH:MM:SS.ss) (HH:MM:SS.ss) (µJy beam−1)el1 1 00 : 32 : 03.55 −43 : 44 : 51.24 53.7± 4.64el1 5 00 : 32 : 57.67 −43 : 28 : 09.00 52.3± 2.66el1 6 00 : 33 : 50.79 −43 : 44 : 57.36 57.0± 5.31el1 7 00 : 35 : 38.02 −43 : 44 : 57.36 57.8± 3.18el1 8 00 : 34 : 44.40 −43 : 28 : 11.88 58.1± 7.28el1 16 00 : 34 : 44.40 −44 : 01 : 42.84 59.2± 6.51el1 17 00 : 32 : 57.67 −44 : 01 : 42.84 50.8± 3.612.1 Australia Telescope Compact ArrayIn order to study extended emission we were granted 12 hours of time with theATCA, which has good short baseline coverage needed for lower resolution imag-ing.We targeted a portion of the European Large Area Infrared Space Observa-tory Survey – South 1 field [ELAIS-S1, 129], an extragalactic region originally se-lected for ISO observations. This field was chosen because it has previously beensurveyed with higher resolution for the Australia Telescope Large Area Survey(ATLAS) [80, 116, 126, Franzen et. al, 2014 in preparation, Banfield et. al, 2014in preparation]. Our new observations were made with the ATCA EW352 arrayconfiguration, which has a maximum baseline of 352 m and a minimum baselineof 30.6 m. A total of 12 hours of observation time was obtained in a single sessionon November, 28 2013. We observed using the ATCA band which is centred on2.1 GHz, with 2 GHz of bandwidth. The bandwidth is separated into 2048 channelsof 1 MHz width. The resolution with this configuration ranges from 1 to 2 arcmin,depending on the image frequency. We observed seven pointings in the ELAIS-S1 field, chosen from the 20 pointings used by the ATLAS survey. The pointingcentres are listed in Table 2.1 and are shown in Fig. 2.2.242.1.1 CalibrationThe calibration, editing, and imaging were performed using the MIRIAD1 softwarepackage [155]. Following several rounds of RFI flagging, the source J1934+638was used for bandpass and flux density calibration. The source PKS 0022−423(PMN J0024−4202) was observed for 2 -minute intervals every 10 minutes andused to correct the gain phases. The task GPCAL was utilized to derive frequency-dependent gain solutions, solving for the gains of the upper and lower parts of theband separately.Observations at this frequency are highly affected by RFI, most notably at thelowest frequencies. The task MIRFLAG was used for automated RFI flagging on thephase calibrator source and the target fields. This allowed us to flag the majorityof interference, so that only a small amount of manual flagging was required. Eachof the seven pointings was flagged individually for uv points above an amplitudethreshold. The data were then split into two frequency bands (1.1 to 2.1 GHz and2.1 to 3.1 GHz), and separated into individual data sets for each pointing. The lasthour of time was not usable, since the source was setting, and for the the final fourhours Antenna 1 was lost due to shadowing. In the end about 55 per cent of thedata was flagged (i.e. not used) in the 1.1 to 2.1 GHz frequency band, and about30 per cent in the 2.1 to 3.1 GHz band.The following analysis is only carried out for the lower part of the band (1.1to 2.1 GHz), which, after flagging, ranged from 1.38 to 2.1 GHz, with a centrefrequency of 1.75 GHz. This decision was made because it more closely matchesthe image frequency of the ATLAS survey. We planned to use the ATLAS point-source models to subtract discrete emission from our data. The change in size ofthe primary beam going from the lower band to the upper band (2.1 to 3.1 GHz)is large, which makes accurate scaling of the point source models difficult and theoutput of the subtraction at the higher frequency less reliable. For this reason we donot believe the addition of the upper band would contribute additional meaningfulinformation for our analysis.1http://www.atnf.csiro.au/computing/software/miriad/252.1.2 ImagingImaging was first performed on the full uv data sets, primarily for the purposes ofself-calibration of the data. However, the ultimate goal was to perform subtrac-tion of the known point sources in the fields and re-image the source-subtracteddata for further analysis. The subtraction process is discussed in more detail inSection 5.3.1.The MIRIAD tasks INVERT, MFCLEAN, and RESTOR were used to create andclean the images. Due to the large frequency range covered, we used multi-frequencysynthesis and deconvolution, or cleaning (MFCLEAN). MFCLEAN attempts to solvefor a frequency dependent intensity, I(ν). HereI(ν) = I(ν0)(νν0)α, (2.2)and solving for the partial derivative of the intensity with frequency gives the spec-tral index,I(ν0)α = ν0∂I∂νν0. (2.3)Thus by using MFCLEAN the resulting image has two planes, the intensity at thereference frequency and the intensity times the spectral index. This allows us totake advantage of the large bandwidth and solve for the frequency dependence ofsources (though a high signal-to-noise ratio is usually required in order to producean accurate measurement). Note that this process can be complicated by the chang-ing primary beam size at the different frequencies. There should therefore be anadditional term representing the spectral dependence of the primary beam; how-ever, currently MFCLEAN only allows for fitting of one additional spectral term.Instead the primary beam frequency dependence was accounted for during the mo-saicing process.Each pointing was cleaned separately, initially down to a level of 600µJybeam−1. At this stage we performed two rounds of phase-only self-calibrationand one of amplitude and phase. The final images were cleaned down to 250µJybeam−1. The resulting PSF, or synthesized clean beam, size is 150 arcsec×60 arcsec,with a position angle of 6◦, using Briggs2 weighting [18] and a robustness factor of2Uniform weighting of the uv data usually results in a better-behaved synthesized beam, and260.5. A mosaic of the seven pointings was made using LINMOS, with each pointinghaving a primary beam full width at half maximum (FWHM) of roughly 27 arcmin;the final mosaic (Fig. 2.2) has a total area of approximately 2.46 deg2.Regarding the brightness units, radio images are often generated with bright-ness units of Jy beam−1. However, it can be useful (particularly with this type ofdiscussion) to convert these units to that of brightness temperature in K (or mK).Conversion between these units can be computed using a factorCT =λ210−26Wm−22kBΩB, (2.4)such that T = CTS in Kelvin, with S the flux density in units of Jy beam−1,and where ΩB is the beam solid angle in steradians, and kB is the Boltzmannconstant. Thus for our case of a Gaussian elliptical beam with FWHM sizes of150 arcsec×60 arcsec and frequency of 1.75 GHz, CT= 44.54. Throughout thediscussion of the ATCA data, for convenience, we present results in both units.2.1.3 Image NoiseObtaining a precise measure of the instrumental noise (σn) is difficult, becausewith the large beam size the confusion rms σc is expected to dominate over theinstrumental noise. However, for our analysis goals an accurate measurement andcharacterization of the noise is required. We employed two different techniques inorder to estimate the instrumental image noise. First we made measurements ofthe noise using the “jackknife” method. This involves taking two (approximately)equal halves of the data and creating separate images. Each of these images shouldhave noise equal√2σtotal. By taking the difference between the images and divid-ing by two, the result is noise of the combined image, with all the signal subtractedout. Since the noise in each half adds in quadrature, then after the subtraction,σ =√σ21 + σ222=√(√2σtotal)2 + (√2σtotal)22= σtotal. (2.5)smaller side lobes, but usually with higher noise. Natural weighting generally gives the best signal-to-noise ratio (though not in the confusion-limited case), but at the expense of an increased beamsize. Briggs, or “robust”, weighting allows for weighting between the two options, doing so in anoptimal sense (similar to Wiener optimization).27-44.50°-44.00°-43.50°-43.00°Dec (J2000)el1_1el1_5el1_6el1_7el1_8el1_16 el1_1707.50°08.00°08.50°09.00°09.50° RA (J2000)-44.50°-44.00°-43.50°-43.00°Dec (J2000)07.50°08.00°08.50°09.00°09.50° RA (J2000)Figure 2.2: ELAIS-S1 mosaics. The top left panel shows the full area,with the seven pointings outlined and labelled at their centres. Thetop right panel is the final 1.75 GHz mosaic image. The bottom leftpanel shows the noise across the mosaic field, with contour levels at46, 48, 78, 120, 190, 305, 480, 760, 1200, 1900, and 3000µJy beam−1.The bottom right panel shows the image after subtraction of the ATLASpoint sources.28200 0 200 400 600D (µJy beam−1 )10-410-310-2P(D) (µJy beam−1)−1Jackknife el1_5rms = 49.8 µJy/bmrms = 2.5 mKσFit = 49.8 µJy/bmσFit = 2.4 mK10 0 10 20 30DT  (mK)200 0 200 400 600D (µJy beam−1 )Jackknife el1_7rms = 56.7 µJy/bmrms = 2.5 mKσFit = 53.4 µJy/bmσFit = 2.4 mK10 0 10 20 30DT  (mK)200 0 200 400 600D (µJy beam−1 )Stokes V el1_5rms = 55.0 µJy/bmrms = 2.6 mKσFit = 55.1 µJy/bmσFit = 2.6 mK10 0 10 20 30DT  (mK)200 0 200 400 600D (µJy beam−1 )Stokes V el1_7rms = 59.2 µJy/bmrms = 2.6 mKσFit = 59.3 µJy/bmσFit = 2.6 mK10 0 10 20 30DT  (mK)10-210-1P(D) (mK)−1Figure 2.3: Measurements and estimates of the instrumental noise using thejackknife method (first two panels) and Stokes V method (second twopanels) for two of the pointings. The black solid lines are from the pixelhistograms of the images and the red dashed lines are fitted Gaussiandistributions. The quoted values are the measured rms from the imagepixel values, while σFit is the width of the fitted Gaussian distributions.It can be challenging with interferometry to create images with equal halves ofthe data. Choosing two equal time chunks can introduce issues with different uvcoverage between the two data sets. We therefore chose to create two images usingthe even and odd numbered spectral channels, which should give approximatelyhalf in each set with most of the obvious types of systematic effects common toboth. The images were cleaned in the same manner and then subtracted for eachpointing. We measured the rms in the cleaned portion of the image, as well asfitting the pixel distribution with a Gaussian to obtain a fitted rms noise σn. Thiscan be seen for two of the pointings in Fig. 2.3. The jackknife procedure yieldedmeasurements of the instrumental noise of the individual pointings of 50–65µJybeam−1, or 2.2–2.9 mK.We used a second approach as a check on this procedure. The Stokes V pa-290 10 20 30 40 50Distance (arcmin)1.401.451.501.551.601.651.701.751.80Frequency (GHz)1.481.521.561.601.641.681.721.76Figure 2.4: Frequency dependence of the final mosaic image due to the wide-band primary beam correction. The solid black line shows the effectivefrequency 〈ν〉 from the centre to the edge of the image, as a function ofradius. The inset is the full mosaic image, with the colour scale showingthe change in effective frequency.rameter measures circular polarization and is defined asV = 〈E2l 〉 − 〈E2r 〉, (2.6)where El and Er are, respectively, the left and right hand complex electric fieldamplitudes in the circular basis as measured by the antennas. The total intensity,or the Stokes I parameter, is defined asI = 〈E2l 〉+ 〈E2r 〉. (2.7)Extragalactic radio sources generally have low levels of circular polarisation [146]and so a Stokes V image should have subtracted out all the signal, leaving only30instrumental noise (similar to the jackknife, but performed in the uv plane ratherthan the image plane). We therefore made Stokes V images of all the pointingsand again measured the rms and fit Gaussian distributions to the pixel probabilitydistributions to obtain a fitted rms σn. This yielded similar estimates of 55–65µJybeam−1 ( 2.4–2.9 mK), as can also be seen in Fig. 2.3. For final values of σnwe averaged the measured and fitted values from the jackknife and Stokes V foreach pointing, and have listed them in Table 2.1. These values only account forinstrument noise and do not include any additional noise contributions from theimaging process, such as uncleaned dirty beam sidelobes, artefacts from brightsources, or from sources out in the lobes of the primary beam (of which there areseveral).For the final mosaic, each pointing had a primary beam correction applied to it,raising the noise radially. LINMOS takes in the values of σn for each pointing andcombines pixels by weighting asS(x, y) =∑iSi(x, y)(σn,i/pi(x, y))2, (2.8)where S(x, y) is the final flux density of the pixel, Si(x, y) is the flux density inpointing i, σn,i is the noise in pointing i and pi(x, y) is the primary beam correc-tion of pointing i at position (x, y). This results in non-uniform noise across thefield. The resulting instrumental noise for the full mosaic is shown with contoursin Fig. 2.2. The actual procedure used to combine the pointings is more compli-cated than eq. (2.8), since, due to the wide bandwidth, the primary beam correctionbecomes frequency dependent. LINMOS takes into account the bandwidth used,as well as the spectral index information found from MFCLEAN, to correct for thefrequency effects. This results in an effective frequency 〈ν〉 in the field that varieswith distance from the centre, going from 1.75 to 1.4 GHz, as shown in Fig. 2.4.2.2 Karl G. Jansky Very Large ArrayThe VLA has four main antenna configurations: A, B, C, and D. These configura-tions range from most to least extended, respectively, and have average resolutionsat 1.4 GHz of 1.3′′, 4.3′′, 14′′, and 46′′. There are also three hybrid configurations:31Table 2.2: Observing Runs Summary. The B∗ data were taken during BnAconfiguration and transition period to A configuration, although imageswere made at B configuration resolutionConfiguration Start Date End Date HoursC 2012 Feb 21 2012 Mar 18 57.0B∗ 2014 Feb 02 2014 Feb 17 24.0DnC, CnB, BnA. These hybrid configurations have the antennas on the east andwest arms moved out for the next configuration, and have resolutions similar to thesmaller configuration of the hybrid.Another set of observations was made with the VLA in S-band, which rangesfrom 2 to 4 GHz. There were two main observing sessions: one with the C con-figuration (maximum baseline 3.4 km), with an average of 21 antennas; and onewith higher resolution in the BnA configuration (maximum baseline ∼ 11.1 km).The BnA configuration’s resolution is comparable to that of the smaller, or B, con-figuration. The observing dates and time spent in each configuration are listed inTable 2.2.The 3-GHz VLA pointing was selected explicitly to overlap the region Owen& Morrison (2008) observed in the Lockman Hole at 1.4 GHz. The field is cen-tred on RA= 10h46m00s, Dec= +59◦01′00′′ (J2000), and was originally chosenas it is known to be a “random” (i.e. for our purposes quite crowded) field, withfew bright sources (the brightest source being 7 mJy), and no very bright radiosources nearby. It is also covered in many other wavebands (Spitzer, Chandra,Herschel, GMRT, and more) allowing for source cross-identification, investigationof AGN contribution, and study of the far-IR/radio correlation. The 3 GHz (centrefrequency) S-band was chosen rather than the 1.4 GHz L-Band, because the con-tamination from interference is less, the requirements on dynamic range are lower,the confusion is lower, and additionally S-Band has greater available bandwidth(2 GHz), all resulting in overall better sensitivity on average. This is in addition toS-band being closer to the frequency of the largest ARCADE 2 observed excessthat we wanted to investigate.322.2.1 CalibrationThis section describes the steps and details of calibrating the VLA data. The proce-dure below describes the work on the C-configuration data; however, the steps werethe same for the B-configuration data, which were added to the C-configurationdata once they were calibrated.Our 57 hours of observing time was spread over six observing nights between2012 February 21 and March 18, and 50 hours of this time was spent on the targetfield. The phase calibrator J1035+564 was monitored for 30 seconds every 30minutes. The flux density and bandpass calibrators 3C 147 and 3C 286 were eachobserved once per night.The 2−4 GHz frequency range was divided into 16 sub-bands (subband alsoknown as intermediate frequency (IF)), each with 64 channels of width 2 MHz.The primary beam attenuation pattern is very nearly that of a uniformly illuminatedcircular aperture [36]:A(ρ/θp) ≈[2J1(3.233ρ/θ)(3.233ρ/θ)]2, (2.9)where J1 is the Bessel function of the first kind and order, ρ is the angular offsetfrom the pointing centre, andθp = (43.3′ ± 0.4′)(νGHz)−1(2.10)is the measured FWHM of the S-band primary beam at frequency ν. The primaryFWHM ranges from 21.7′ at ν = 2 GHz to 10.8′ at ν = 4 GHz.We used the OBIT package [39]3 to edit and calibrate the uv data; with thesix observing sessions calibrated and edited separately. Two of the 16 subbandscontained satellite RFI strong enough to cause serious Gibbs ringing in the rawdata (sidelobes in the data at the location of a discontinuity or change in the sub-bands due to the finite number of correlator lags). Hanning smoothing (combiningadjacent spectral channels with weights 1/4, 1/2, and 1/4) suppressed this ringingand cut the number of channels per IF to 32. Spectral channels still containing3http://www.cv.nrao.edu/∼bcotton/Obit.html33strong interfering signals were flagged and removed from the data, as were a fewedge channels in each subband. Prior to calibration, the remaining data samplescontaining strong interfering signals were identified by their large deviations fromrunning medians in time and frequency, and they were flagged.Subsequent calibration and editing consisted of the following steps:1. Instrumental group delay offsets were determined from observations of allcalibrators and applied to all of the data.2. Residual variations of gain and phase with frequency were corrected bybandpass calibration based on 3C 286.3. Amplitude calibration was based on the VLA standard spectrum of 3C 286and was used to calibrate J1035+564. The more frequent observations ofJ1035+564 were then used to calibrate the amplitudes and phases of the tar-get uv data. Data from some antennas, time intervals, and frequency rangeswere still degraded by interference that had evaded earlier editing, corrupteda small fraction of our amplitude and phase solutions, and were thus flagged.4. The calibrated data were then subjected to further editing in which data withexcessive Stokes I or V amplitudes were deleted, as well as another pass atremoving narrowband interference.After this calibration and editing, the initial calibration was reset and the wholeprocess was repeated using only the data that had survived the editing process. Fi-nally, we were able to flag the small amount of the data having amplitudes sig-nificantly above the noise. About 53% of the uv data survived all of the editingsteps. The calibrated and edited uv data from all six observing sessions were thencombined for imaging. Before imaging, the data were averaged over baseline-dependent time intervals subject to the constraints that (1) the averaging should notcause time smearing within our central image and (2) the averaging time shouldnever exceed the 20 second phase self-calibration interval.2.2.2 ImagingAgain this section details the work on the C-configuration data, with the process forimaging the combined C+B-configurations being the same except for the subband34weighting used.Observations spanning the frequency range between the low frequency limit νland the high frequency limit νh have centre frequency νc = (νl + νh)/2, band-width ∆ν = (νh−νl), and fractional bandwidth ∆ν/νc. The fractional bandwidthcovered by our 2−4 GHz uv data is large: ∆ν/νc = 2/3. Using such data to makean image suitable for measuring confusion encounters three problems, namely thatthe field of view, the PSF, and the flux densities of most sources can vary signifi-cantly with frequency. To deal with these problems, we separated the uv data into16 subbands, each having a small fractional bandwidth, ∆ν/νc  1. The uv datawere tapered heavily in the higher-frequency subbands, and each subband was im-aged with an independent “robustness” [18] to force nearly identical PSFs in allsubbands. We weighted and recombined the narrowband images to produce a sen-sitive wideband image characterized by an “effective frequency” 〈ν〉, where 〈ν〉at any position in a wideband image is defined as the frequency at which the fluxdensity of a point source with a typical spectral index 〈α〉 ≡ d lnS/d ln ν = −0.7equals its flux density in the wideband sky image. The effective frequency declineswith angular distance ρ from the pointing centre because the primary beam widthis inversely proportional to frequency.Table 2.3 lists the centre frequencies νc of the 16 subbands and the rms noisevalues σn of the 16 subband images. The fractional bandwidths of these subbandsrange from 3% to 6%, so each subband image is a narrowband image. We used theOBIT task MFIMAGE (also a multi-frequency clean algorithm) to form separatedirty and residual images in each subband. We assigned each subband image aweight inversely proportional to its rms noise, generated a combined widebandimage from their weighted average, and used this sensitive combined image tolocate clean components for a joint deconvolution. The clean operation used fluxdensities from the individual subband images, but at locations selected from thecombined image. At the end of each major clean cycle, the clean components withflux densities from the individual subband images were used to create residualsubband uv data and then new residual images.In order to obtain nearly identical PSFs in all subbands, we tapered the uvdata in each subband differently and assigned individual robust weighting factors,adjusted to ensure that each synthesized dirty beam was nearly circular with major35Table 2.3: Frequency and noise properties of the 16 VLA subband images.Subband Frequency σn C σn CBnumber (GHz) (Jy) (µJy beam−1)01 2.0500 9.22 8.4602 2.1780 13.19 26.1403 2.3060 18.17 1000.904 2.4340 4.48 5.0905 2.5620 4.24 5.0006 2.6900 4.31 4.4207 2.8180 4.04 4.2608 2.9460 3.72 4.7509 3.0500 4.48 4.0110 3.1780 3.14 3.4911 3.3060 3.07 3.5612 3.4340 2.97 3.1913 3.5620 2.88 3.4714 3.6900 3.47 4.0415 3.8180 4.45 5.6616 3.9460 4.40 117.38and minor axes between 7′′ and 8′′. After CLEANing, each residual image wassmoothed by convolution with its own elliptical Gaussian distribution, tailored toyield a circular and nearly Gaussian PSF with a precisely 8 arcsec FWHM. Finally,all CLEAN components were restored with an 8′′ FWHM circular Gaussian beam(for the CB-combined data the clean beam is a circular Gaussian with FWHM2.75′′).Two iterations of phase-only self calibration were used to remove residualphase errors. Fluctuations, indistinguishable from confusion, are produced byany dirty-beam sidelobes remaining in our combined image. Fortunately, theyare small, because the combination of long observing tracks and bandwidth syn-thesis over our wide fractional bandwidth ensures excellent uv -plane coverageand keeps the dirty-beam sidelobe levels well below 1% of the peak. Conse-quently the highest dirty-beam sidelobes from sources in the residual image are< 0.01 × 10µJy beam−1 ≈ 0.1µJy beam−1, so their contribution to the imagevariance is more than two orders of magnitude below the (1µJy beam−1)2 contri-36butions of noise and confusion.2.2.3 Image NoiseWe used the AIPS task IMEAN to calculate the rms noise values σn of the CLEANedsubband images in several large areas that are well outside the main lobe of the pri-mary beam and contain no visible (Speak ≥ 6µJy beam−1) sources. This ensuresthat the contribution from source signals in these regions is negligible. The σnvalues are listed in Table 2.3. Next we assigned to each subband image a weightinversely proportional to its noise variance σ2n, generated a wideband image fromthe weighted average of the subband images, and measured the noise distribution infour large regions well outside the primary main beam and free of visible sources.The best-fit to the logarithmic noise histogram indicates a Gaussian distributionwith an rms σn ≈ 1.02µJy beam−1.The B-configuration data had several subbands that were more severely af-fected by interference issues that were not resolved through the flagging and edit-ing process; this is evident in the σn values shown in Table 2.3. For imaging andanalysis these subbands were given very small weights when combining the imagesto make a centre image.We also wanted to know the rms statistical uncertainty ∆σn in our estimate ofσn. The image PSF is a Gaussian, and the effective noise area for a Gaussian PSFis that of the Gaussian PSF squared [32, 36]. Squaring a Gaussian PSF of widthθ yields a narrower Gaussian of width θ/√2 and solid angle Ωb/2. Consequentlythere are actually 2N statistically independent noise samples in Ω = NΩb beamsolid angles, and the rms fractional uncertainty in σn is∆σnσn=(12N)1/2, (2.11)not the commonly believed (1/N)1/2; for further discussion on this see AppendixA of Condon et al. [37].The final result is σn = 1.012± 0.007µJy beam−1. In theory the rms noise isuniform across the image prior to correction for primary-beam attenuation, so weused this value to estimate the noise in confusion-limited regions near the pointing3758.858.959.059.159.259.3Dec (J2000)3210123456µJy58.858.959.059.159.259.310080604020020406080100µJy161.0161.5162.0 RA (J2000)58.858.959.059.159.259.3Dec (J2000)15102550µJy161.0161.5162.0 RA (J2000)58.858.959.059.159.259.32.102.252.402.552.702.853.00GHzFigure 2.5: Lockman Hole VLA 3 GHz images. The top left shows the C-configuration field before correction for the primary beam. The top rightpanel is the field after wide-band primary beam correction. The bot-tom left panel shows the noise change across the field after wide-bandprimary beam correction. The bottom right panel shows the effectivefrequency after the wide-band combination.38centre. The total intensity-proportional error arising from uncertainties in the flux-density calibration and primary beamwidth is not more than 3% inside the primarybeam half-power circle. For more details on the imaging process and noise mea-surements see section 2.4 of Condon et al. [37]. The noise for the combined CBdata is σn = 1.15± 0.007µJy beam−1.2.2.4 The SNR-Optimized Wideband Sky ImageOur final wideband image of the sky was made with weights designed to correct forprimary-beam attenuation and simultaneously maximize the signal-to-noise ratio(SNR) for sources having spectral indices near 〈α〉 = −0.7, the mean spectralindex of faint sources found at frequencies around 3 GHz [31]. This differs fromthe traditional weighting designed to minimize noise, which maximizes the SNRin a narrowband image, but in a wideband image only if 〈α〉 ≈ 0. The brightnessbi(ρ) of each pixel in each of the i = 1, 16 subband images was assigned a weightWi(ρ, νc) ∝[ν〈α〉cσnA(ρ, νc)]2, (2.12)where the rms noise σn and centre frequency νc of each subband image is listed inTable 2.3. Each pixel in the weighted wideband sky image was generated from theratiob(ρ) =16∑i=1[bi(ρ)Wi(ρ)]/ 16∑i=1Wi(ρ). (2.13)The final C-configuration images, before and after primary beam correction, areshown in the top panels of Fig. 2.5.Even at the pointing centre, weighting to optimize the SNR for 〈α〉 = −0.7increases the sky image noise slightly from σn = 1.012 ± 0.007µJy beam−1 toσn = 1.080± 0.007µJy beam−1. Away from the pointing centre, the frequency-dependent primary-beam attenuation correction causes the rms noise on the skyimage to grow with the radial offset ρ. Weighting also affects how the effectivefrequency 〈ν〉 at each point in the wideband sky image decreases with the offset ρfrom the pointing centre. Also, 〈ν〉 in our sky image declines monotonically from3.06 GHz at the pointing centre to 2.96 GHz at ρ = 5 arcmin. The radial changes in39Table 2.4: Image properties for the wide-band VLA data. The reported noisevalues are all after correction for the primary beam and frequency weight-ing effects, with ρ being the distance from the pointing centre. The cleanbeam size, θB, is the FWHM, and the synthesized beam solid angle, ΩB,is (θ2Bpi)/(4ln2).Quantity Value C-data Value CB-data Unit〈ν〉 in centre 3.06 3.036 GHz〈ν〉 at 5 arcmin 2.96 2.96 GHz〈ν〉 inside 5 arcmin Ring 3.02 3.02 GHzPixel size 1.252 0.5 arcsecClean beam FWHM, θB 8.00 2.75 arcsecBeam solid angle, ΩB 72.32 8.55 arcsec2σn(ρ=0) 1.08 1.15 µJy beam−1σn(ρ = 5′) 1.447 1.54 µJy beam−1σn(ρ ≤ 5′) 1.255 1.33 µJy beam−1noise and frequency are shown in the bottom panels of Fig. 2.5. The final image isconfusion limited in the sense that the rms fluctuations are everywhere larger thanthe noise levels. Table 2.4 provides a summary of the image properties.2.3 SKADS SimulationThroughout this work we rely on using realistic simulated radio data from theSquare Kilometre Array (SKA) Simulated Skies (S3) simulation4. This is a com-puter simulation of the radio and submm Universe, dedicated to the preparation ofthe Square Kilometre Array and its pathfinders. This simulation was led by the Uni-versity of Oxford as part of the Square Kilometre Array Design Studies (SKADS).While several simulations were done, we used only data from S3-SEX, which is asemi-empirical simulation of extragalactic radio continuum sources in a sky areaof 20◦ × 20◦, out to a cosmological redshift of z = 20.The simulation is based on a realisation of the linear matter power spectrumproduced by CAMB [108]. The cosmological model used is: Ωm = 0.3, Ωk = 0.0,w = −1.0, h = 0.0, fbaryon = 0.16, σ8 = 0.74, b = 1.0, and fNL = 0. This den-sity field realisation is gridded in cells of size 5 h−1 Mpc, from which galaxies are4http://s-cubed.physics.ox.ac.uk/40sampled. The galaxy bias function b(z) follows the description of Mo and White[122] with a cut-off redshift for different galaxy types. Thus cut-off is chosen sothat the bias is held constant above a given redshift to prevent exponential blow-upof the clustering. Galaxy clusters are identified by looking for regions with den-sities larger than the critical density with the use of the Press-Schecter [145] andSheth-Tormen formulations [163]. The simulated sources were drawn from fourtypes of calculated luminosity functions (radio-loud AGN of high and low lumi-nosities, radio-quiet AGN, star-forming galaxies, and starburst galaxies). Thesegalaxies were inserted into the evolving dark matter density field. This simulation,therefore, has realistic approximations of the known source counts and containsboth small and large-scale clustering. Full details of the simulation are describedin Wilman et al. [191].41Chapter 3Confusion and P(D)3.1 ConfusionThe traditional way of doing a source count, as discussed in Section 1.2, is tofit for and count all the sources in an image above some threshold, generally >5σ. The problem with this is the uncertainties in the peak finding and fitting, aswell as corrections for sizes, flux boosting, and clean bias (see Chapter 6 for morediscussion on these corrections) all create a lot of uncertainty in the source count.Another way to obtain a statistical estimate of the source count is to take advantageof something known as confusion. Confusion is a term which refers to the blendingof faint sources due to the telescope beam or PSF. There is such a thing as naturalconfusion, which is the blending of faint sources due solely to their number densitycausing overlap on the sky. However, here we focus on the confusion resulting fromthe finite resolution of the telescope.In a noiseless image with just point sources convolved with a telescope beamthe shape of the normalized histogram of the pixels, or 1D probability distributionfunction (PDF) , is dictated by the source count in the image and the shape ofthe beam. To illustrate this we made a simulated image of point sources with nonoise and a realistic source count. We then convolved it with several different sizedGaussian beams and looked at the pixel histograms, which is shown in Fig. 3.1. Thetop row shows the simulated image convolved with four different Gaussian beams.The middle row shows the beams, while the bottom row shows the PDFs. This42figure shows that as the beam size gets larger more sources are blended together,thus widening the PDF.The confusion noise (σc) is the width of the PDF due to confusion. The noise-less distribution will generally have an extended bright tail such that the methodof finding σ of a Gaussian shaped curve does not give an accurate estimate of thewidth. Instead, to find σc we find the median and D1 and D2 such thatmedian∑D1P (D) =D2∑medianP (D) = 0.34 (3.1)when normalised such that the sum of the PDF= 1 (with D being the observedbrightness or pixel value), since in the Gaussian case 68 per cent of the area isbetween ±1σ. The confusion noise is then σc = (D2 − D1)/2 (thus yielding anestimate of something resembling the σ of a Gaussian distribution).However, in the real world we have instrument noise from our telescope addedto the confusion noise, such that the total noise is σt =√σ2n + σ2c . In terms ofthe PDF the instrument noise adds more width and becomes degenerate with thecontribution from faint sources. For this reason, to attempt to measure faint countsusing confusion, it is best to have σn ≤ σc. We illustrate this by taking one ofthe noiseless images from before, at a set beam size of 8 arcsec, and add Gaussianrandom beam-convolved noise to the image while varying the σn of the noise. Thisis shown in Fig. 3.2, where the top row is the sources plus noise, the middle rowis just noise, and the bottom row shows the histograms from the noise, noiselesssources, and noisy sources. This shows that as the instrumental noise decreases,and drops below the confusion noise, the combined histogram’s shape approachesthat of the noiseless histogram, which is what is to be estimated.For data where the confusion noise is larger than the instrumental noise, whichcan be achieved by choosing the beam size and exposure times, the method ofProbability of deflection (P (D)) can be used in order to model the source countbelow the current cut-offs. This method, which forward models the source countby fitting the image histogram, is described in detail in the next section.438006004002000200400600800Sources (arcsec)ΩBeam=2.5" ΩBeam=5.0" ΩBeam=8.0" ΩBeam=12.0"10050050100Beams (arcsec)5 0 5 10 15 20D (µJy beam−1 )10-210-1100P(D) (µJybeam−1 )−1Confusion σ=0.065 0 5 10 15 20D (µJy beam−1 )Confusion σ=0.415 0 5 10 15 20D (µJy beam−1 )Confusion σ=1.295 0 5 10 15 20D (µJy beam−1 )Confusion σ=3.19Figure 3.1: The top row shows a region of simulated noiseless images eachconvolved with beams with FWHM=2.5, 5.0, 8.0, and 12 arcsec; allimages have the same colour scale. The middle row shows the peaknormalized beams. The bottom row shows the image pixel histogramsand the confusion widths as measured by eq. 5.1.448006004002000200400600800Sources+noise (arcsec)ΩBeam=8.0" σN=12.04 ΩBeam=8.0" σN=5.99 ΩBeam=8.0" σN=1.5 ΩBeam=8.0" σN=0.488006004002000200400600800Noise (arcsec)20 10 0 10 20 30D (µJy beam−1 )10-310-210-1P(D) (µJybeam−1 )−1σT=12.53σC=1.29σN=12.0420 10 0 10 20 30D (µJy beam−1 )σT=6.5σC=1.29σN=5.9920 10 0 10 20 30D (µJy beam−1 )σT=2.13σC=1.29σN=1.520 10 0 10 20 30D (µJy beam−1 )σT=1.4σC=1.29σN=0.48Figure 3.2: The top row shows simulated images with sources and noise eachconvolved with s beam of FWHM= 8.0 arcsec; all images have the samecolour scale. The middle row shows the same regions with just noiseand no sources; all images have the same colour scale. The bottom rowshows the image pixel histograms for the noiseless case (solid blacklines), just noise (dashed lines), and sources plus noise (solid colouredlines). The σ values listed are σt =total combined width, σc = noiselessconfusion, and σn = instrumental noise as measured by eq. 5.1.453.2 Probability of DeflectionThe method of P(D) was introduced by Scheuer [156] as the probability of pendeflections on a chart-recorder from a single baseline of a two-element radio inter-ferometer. The P(D) distribution of an image is the PDF of pixel intensities, or the“1-point statistics”. Condon [30] and Scheuer [157] gave analytical derivations ofP(D) for a single power-law model of a source count. The method which has beenmost often applied is to count the objects in the map brighter than some cut-off(usually about 5σn) and use P(D) analysis for the faint end of the count, constrain-ing an amplitude and a slope. A similar approach with the VLA data describedhere was carried out in Condon et al. [37], where a simple power law was fit tothe count below 10µJy. In this paper we follow the more computationally inten-sive approach of Patanchon et al. [138] to apply a histogram-fitting procedure forthe full range of image source brightnesses. This approach does not require thatthe source count model be a power law, allowing for more flexibility in accuratelymodelling the true source count. For completeness we give here a brief summaryof the statistics of P(D), providing some specific details on how we applied thisto the 3 GHz VLA data. For more detailed derivations see Condon [30], Takeuchiet al. [176], and Patanchon et al. [138].The deflection, D, at any point (pixel) is an image intensity (in units such asJy per beam solid angle) at that point. P(D) is then the probability distribution ofthose deflections in some finite region of the image. The differential number countdN(S)/dS is the number of sources per steradian with flux densities between Sand S+dS per unit flux-density interval. The relative point spread functionB(θ, φ)is the relative gain of the peak-normalised CLEAN beam at the offset of a pixelfrom the source1. The image response to a point source of flux density S at a pointin the PSF where the relative gain isB(θ, φ) is x = SB(θ, φ). The mean number ofsource responses (e.g. pixel values) per steradian with observed intensities between1An assumption with the P(D) method is that the PSF is constant across the image. With singledish observations or those done at other wavelengths, such as sub-mm or infrared, this may notalways be the case. However, with our interferometric image the synthesized beam is set beforetransformation from the Fourier plane to the image plane. Thus, with our VLA data the PSF is aconstant size and shape across the entire image46x and x+ dx is R(x)dx [see 30, for example], withR (x) dx =∫ΩdNdS(xB(θ, φ))B(θ, φ)−1 dΩ dx . (3.2)The PDF for the observed flux density in each sky area unit (in this case an im-age pixel) is the convolution of the PDFs for each flux density interval over all fluxdensities – this is P(D). The convolution in the image plane is just multiplicationin the Fourier plane of the individual characteristic functions. In this case D is thetotal flux density from all sources with the observed flux density x. Thus, p(ω) isp(ω) = exp[∫ ∞0R (x) exp (iωx) dx−∫ ∞0R (x) dx], (3.3)and P(D) is the inverse Fourier transform of this,P (D) = F−1 [p(ω)] . (3.4)The P(D) distribution in a noisy image is the convolution of the noiseless P(D)distribution with the noise intensity distribution. Convolution is equivalent to mul-tiplication in the Fourier transform plane, and the Fourier transform of a Gaussianis a Gaussian, so for Gaussian noise with rms σn,2P (D) = F−1[p(ω) exp(−σ2nω22)]. (3.5)The task then boils down to using the measured P(D) to constrain a model fordN/dS, via R(x), for a given noise and beam.3.2.1 P(D) Simulation TestsTo test our model and statistical approach we used data from the SKADS S3 sim-ulation (see Section 2.3 for details). Using these data allowed us to test not only2In the case of single dish observations, or steep-slope counts (γ > 2), the mean deflection aboveabsolute zero µ should also be subtracted off, such thatD would then represent the deflection about µrather than zero. The mean deflection can be found from µ =∫xR(x)dx. The zero point of the P(D)distribution is lost in an interferometer image, which has no “DC” response to isotropic emission, sothe zero point must be a free parameter when fitting our VLA data to model P(D) distributions.47the functionality and accuracy of our code, but also any effects that small-scale(beam-sized) clustering might have on the output, by comparing the fitted modelto the known input.We used the simulated data at 1.4 GHz, the closest frequency in the simula-tion to our VLA data. The full size of the simulation is 400 deg2, from whichwe extracted the central 1 deg2. The simulated image was constructed to have thesame beam and pixel size as our VLA data. Random (beam-convolved) Gaussiannoise was added to the simulated image, with σn = 2.14µJy beam−1 rms. Thisnoise value is slightly larger than that of our VLA image central 5 arcmin due tothe simulation being at 1.4 GHz instead of 3 GHz. The model count was set up asdescribed in Section 4.4, with six variable nodes and two fixed ones. The faintestnode was set at 10 nJy, as this was the faintest flux density simulated in the data.The second node was set at 0.1σn.The output from the MCMC fitting to the simulated data can be seen in thetop panel of Fig. 3.3 and the P(D) distributions are shown in Fig. 3.4. The plotshows the marginalised mean amplitudes from each parameter’s likelihood distri-bution for all six nodes. The values for the six nodes and the χ2 values at eachpoint in the chain can be used to compute 68 per cent confidence intervals (use-ful for examining the full likelihood surface, since there are shape changes due tothe parameter degeneracies, as discussed in Section 4.5.2). The results from thissimulated image indicate that our fitting procedure is unbiased; the input sourcecount model is always within the relevant confidence regions. There is some slightdeviation from the input count for the faintest three nodes. This is perhaps not un-expected, since this region is well below the instrumental noise. However, whilethe error bars on the faintest node are large, it is important to note that that thecount is still constrained; even the 95 per cent limits for this node do not reach thehigh and low limits given to the MCMC routine. The marginalised mean for thefaintest node is within 1 per cent of the input value, even though this is two ordersof magnitude below the noise limit. This test shows that the method and model arenot only capable of fitting the underlying source count of an image, but that thereis still information about the count well below the instrumental noise, as long asthat noise value is known well.4810-410-310-210-1100101S5/2 dN/dS (sr−1 Jy)Known 1.4 GHz countsMarg. meansInput count68% Confidence region10-2 10-1 100 101 102S (µJy)10-410-310-210-1100101S5/2 dN/dS (sr−1 Jy)Known 1.4 GHz countsMarg. means full simulation68% Limits full simMarg. means no sizes68% Limits no sizesMarg. means no sizes or clustering68% Limits no sizes or clusteringFigure 3.3: Euclidean-normalised source count from SKADS simulation.The black points are real data counts from the de Zotti et al. [42] com-pilation. Top: MCMC marginalised mean (blue line and points) nodepositions. The black dashed line is the input source count model fromWilman et al. [191]. The shaded area is the 68 per cent confidence re-gion. Bottom: The marginalised mean node positions from the simu-lated image, zoomed in on the region S ≤ 2µJy, taking into accountsource sizes and clustering (blue solid line), but with sources all un-resolved (red dashed line), and for unresolved sizes and random po-sitions (green dotted line), with 68 per cent confidence regions as thedot-dashed purple, orange, and green lines.495 0 5 10D (µJy beam−1 )10-310-210-1P(D) (µJy beam−1)−1Gaussian noise σ=2.14 µJy beam−1P(D) model fitImage histogramFigure 3.4: Comparison of pixel histograms from 1.4 GHz simulation (reddots) with the output of the marginalised means model P(D) using asinput six variable nodes (solid line) and a fixed σn = 2.14µJy beam−1noise level (dashed line).3.2.2 Clustering and Source SizesThe simulated image included sources with varying sizes, sources with multiplecomponents, and the underlying clustering information. In Wilman et al. [191]the angular two-point correlation function, w(θ), is shown for the full simulation;this is a little higher than measurements made by Blake et al. [13] at a somewhatbrighter flux density limit. We computed w(θ) for the specific 1 deg2 simulatedsample. The function w(θ) is usually approximated by a power law of the formw(θ) = Aθ−γ( or sometimes written as w(θ) = (θ/θφ)−γ). Blake and Wall[11, 12] found A = 1.0× 10−3 and γ = 0.8, using data from the the NRAO VLASky Survey [NVSS; 36]. We assumed γ = 0.8 and calculated θφ for the subset of50simulated data we used. Using all the sources down to the limit of 10 nJy we foundθφ = 1.6 × 10−5 deg or 0.06 arcsec. The sources are certainly clustered on thescale of our beam (' 8 arcsec), but very weakly, because θ◦ (the angular scale fornon-linear clustering) is so small compared with our synthesized beam size. OurP(D) calculation does not take into account any clustering correction. It also doesnot account for source sizes, but assumes that all the sources are unresolved. Whilein the case of both the VLA data and the simulated data, many of the sources aresmaller than the beam, we know that this is not the case for all of them. In thesimulated data there are roughly 700,000 sources, with a mean major axis size of1.4 arcsec and mean minor axis of 0.8 arcsec, giving a mean source solid angle ofΩS' 1.27 arcsec2 (before convolution with the beam). This is much smaller thanour beam solid angle of 72.3 arcsec2.To test what kind of effect source sizes and clustering have on the model fitting,two other images were made. The first kept the source position information, so thatany clustering would be preserved, but all source size information was neglected.Every source, single and multi-component, was set to a single delta function withflux density equal to the total source flux density, and then this was convolved withthe beam. The second image also had all the sources as delta functions, but in thiscase the positions were randomised as well, so that the sources were unclustered.The MCMC fitting was rerun on histograms from these two simulated images withall other factors being the same. The results from fitting each of the three imagesare compared in the bottom panel of Fig. 3.3.The amplitudes and error regions of the three brighter S nodes are the same ineach case. The only differences are for the faintest nodes, which are more difficultto constrain. Comparing the full image with the case where no size informationis present, we see that the full image case is higher; as anticipated given that withthe larger source sizes one might expect more blending, more bright pixels, anda slightly wider histogram. When comparing the case with randomised positionsand unresolved sources, again the results are the same down to about the noiselevel, although fainter than this does give lower values. This again is expected, dueto the lack of both source sizes and clustering. Clustering within the beam willtend to boost the pixel values after beam convolution, producing a slight wideningof the distribution [see 175]. We would expect these fainter nodes to be of lower51amplitude without clustering, as seen. When not accounting for source sizes orclustering, the largest fractional change in node amplitude from the full image is2.3 per cent at the first node, 2.2 per cent for the second node, with the others all1 per cent or less; all of the values lie within the 68 per cent confidence limits ofthe full simulation. These results make us confident that neglecting the effects ofclustering and source size when fitting our real data results in no significant bias.Regarding the issue of source sizes, it is important to note that P(D) countsare much more robust than comparably deep individual source counts. This isbecause P(D) counts use a much bigger beam. For example, the 8 arcsec VLAbeam corresponds to about one source per beam. Individual sources can be countedreliably only if there are at least 25 beams per source. This means the beam widthfor individual counts can not be much bigger than 8./√25 ∼ 1.6 arcsec, which isquite close to the mean source size in the SKADS simulation and would requirelarge corrections for partial resolution of the sources. The mean source size fromthe SKADS simulation is small enough compared to our beam that source sizeshould not impact the P(D). It should be noted, however, that the SKADS sizesare larger than estimates from high-resolution studies of faint radio sources. Forexample, Muxlow et al. [124] finds the average source size for weak radio sourcesto be closer to 0.7 arcseconds.52Chapter 4Discrete-Source Count4.1 IntroductionThere has been considerable discussion recently about possible different types ofradio sources contributing to the source count at flux densities fainter than the limitsof the current source counts. Measurements from ARCADE 2 indicate the pres-ence of an excess of radio emission over previous measurements or estimates usingsource count data. Vernstrom et al. [182], motivated by the ARCADE 2 results,presented new estimates of lower limits to the background from a compilation ofsource counts at eight frequencies and found an expected value for the backgroundtemperature almost five times lower than that of ARCADE 2 at 1.4 GHz.The ARCADE2 results suggested that this excess emission might be comingfrom a previously unrecognized population of discrete radio sources below the fluxdensity limit of existing surveys, and that this new population might be seen inradio source counts extending to lower flux density levels. This issue was furtherexamined by Singal et al. [165], who concluded that this emission could primarilybe coming from ordinary star forming galaxies at z > 1 only if the far-IR/radioratio decreases with redshift. In other words we can only explain the backgroundresults with sources if they break the far–IR/radio correlation [79].Vernstrom et al. [182] (Chapter 1) also showed that the known radio sourcecounts cannot on their own account for the ARCADE 2 excess, although the sourcecounts, at least at 1.4 GHz, are not inconsistent with a possible upturn below about5310µJy. Such a possible upturn is mainly driven by the faintest count availableat 1.4 GHz, from Owen and Morrison [134]. Owen & Morrison found that their(Euclidean-normalised) count, which extends down to 15µJy, did not decreasewith decreasing flux density (compared to static Euclidean counts), but seemed tolevel off or even show signs of increasing. It is important to note that the Euclidean-normalised count (S5/2dN/dS) does not need to level off or turn up to explain thehigh ARCADE 2 background temperature; it is sufficient that S2dN/dS levels offor turns up.New 3-GHz data from the VLA, reach down to µJy levels [37] and the result-ing map is the deepest currently available. In this previous paper we estimated thesource count from 1 to 10µJy using a technique known for historical reasons asP(D) analysis [30, 156]. This approach allows a statistical estimate of the countfrom a confusion-limited survey, extending down to flux densities below the con-fusion limit. P(D) is the probability distribution of peak flux densities in an image.This approach results in statistical estimates of the source count that are muchfainter than the faintest sources that can be counted individually (about 5 times therms noise). The count model used in Condon et al. [37] was a single power lawover a limited flux density range. However, there appeared to be evidence for abreak in the slope somewhere in this region and certainly the results did not sup-port any upturn in the count. While this previous result puts strong limits on theµJy count, it is possible that more comprehensive analysis of the P(D) distribution,with a more general count model, could reveal additional information about thetrue shape of the count, as well as constraining the count fainter than 1µJy.Here we present a more sophisticated modelling approach to the P(D) fittingprocess, motivated by Patanchon et al. [138], using a model based on multiplejoined power laws. The statistical uncertainties here are evaluated using Markovchains. We test this technique with a large-scale simulation incorporating realisticsource sizes, multi-component sources, and clustering. This method allows forexploration of the flux density limit of the P(D) approach, and the count below theconfusion noise, as well as a thorough non-parametric error analysis.Sections 4.2 and 4.3 describe the process adopted for model fitting and erroranalysis. In Section 4.4 we discuss the models used in the fitting and the detailsof their application to the VLA data. In Section 4.5 we present the results of the54fitting from two different models, a discussion of the parameter degeneracies, andthe derived radio background temperature. Section 4.6 gives a discussion of thesystematics and comparisons with previous results.For all the work in this Chapter only the VLA C-configuration data were used,as described in Section 2.2.4.2 P(D) and Image NoiseIt is important to have an accurate measure of the instrumental noise in the imagefor analysis. As mentioned in Section 2.2 we created 16 images from the differentS-band frequency sub-bands. The images created from the UV data should haveconstant instrumental noise across the images, before any primary beam correc-tions and neglecting any deconvolution artifacts or contamination from dirty-beamsidelobes (which in our case were small with the largest dirty-beam sidelobe con-tamination being ' 0.1µJy beam1). We used the AIPS task IMEAN to calculatethe rms noise values of the CLEANed sub-band images in four large areas welloutside the primary beam of each. This ensures that the contribution from sourcesignals in these regions is negligible. The 16 images were combined with weightsinversely proportional to the sub-band noise to create the 3 GHz centre image. Thenoise was then measured again in several large areas outside the 3 GHz primarybeam area of the centre image. From these measurements we obtained our noiseestimate of σn = 1.012 ± 0.007µJy beam−1, constant across the image, beforethe primary beam correction. For more details on the imaging process and noisemeasurements see section 2.4 of Condon et al. [37].For the basic P(D) calculation using eq. (3.5), it is assumed that the noise,σn, is constant across the image. However, for our VLA data this is not the case.While the true instrumental noise does not change, because of the primary beamcorrection and frequency weighting effects, the noise measured in µJy beam−1increases radially with distance, ρ, from the pointing centre. The noise for thering of pixels at a radius of 5 arcmin has already increased from 1.08µJy beam−1to 1.447µJy beam−1. However, the actual noise contributing to the P(D) is aweighted combination of the variance of the rings inside some set radius. Thus,for a circle of radius 5 arcmin the weighted effective noise from all the rings inside550 2 4 6 8 10Ring Radius (arcminutes)1.01.52.02.53.03.54.0Noise (µJy beam−1) Pixel noise at ring of radius RPixel noise inside ring of radius RFigure 4.1: Change in image noise as a function of ring radius. The linesshow how the noise at each ring changes with distance from the centre(black solid line) and the weighted noise within a ring of that radius (reddashed line).is 1.255µJy beam−1, as seen as the red dashed line in Fig. 4.1. We have to choosean area where the variation in the noise is not too large, since for a P(D) analysis wewant σn to be roughly constant. For highest accuracy we would also like σn ≤ σc,where σc is the confusion noise; and yet we want the area to be as large as possibleto provide the most samples. We chose to carry out the main P(D) calculationwithin the central 5 arcmin, where the fractional change in the noise has a broadminimum and the effective noise is ≤ σc.When binning the pixels for the histogram, weighting must be applied for thehistogram to reflect the effective width of σ∗n= 1.255µJy beam−1. To accomplish5610-610-510-410-310-210-1100P(D) (µJy beam−1)−1Mean weighted noise histogramMean Gaussian fitMean noise and source model convolutionMean Gaussian fit and source model convolution4 2 0 2 4 6S µJy beam−10.00.20.40.60.81.01.2Noise/ModelResiduals from just noiseResiduals from noise*source modelFigure 4.2: Differences in PDFs of purely Gaussian noise and weighted vary-ing noise. The top panel shows the mean results from 100 simulatednoise generations. The solid black line is the mean from the weightednoise image histograms, and the red dashed line is the mean from Gaus-sian distributions fitted to those histograms. The dot-dashed black lineline is the mean from convolving the noise image histograms with anoiseless source count P(D) and the red dotted line is the mean fromconvolving the same source count P(D) with the fitted Gaussian distri-butions. The bottom panel shows the ratios of the means. The blackline is the mean from the fitted Gaussian distributions divided by noiseimage histograms and the red dashed line is the mean from the fittedGaussian distribution convolution divided by the mean from image con-volution. This shows how the noise weighting we apply to our datacauses it to deviate from purely Gaussian noise.57this the area is split into sub-rings with radii (as measured from the mid-point radiusof the ring) increasing by 0.11 arcmin. A histogram is made for each ring and avalue equal towk =1σ4nk, (4.1)gives the pixel weight in the kth ring. The σnk is the value of the noise, after theprimary beam correction, in the kth ring (the black line of Fig. 4.1). The weights,wk, go as σ−4nk because in this case the estimator is a variance, and thus the weightsgo as the square of the variance, or the variance of the variance [see section 3 of 37,for a more detailed discussion of the noise, weighting, and choice of area]. Theseweights are applied to each ring histogram and the histograms are combined. Therings used for the central 5 arcmin can be seen in Fig. 4.3. This weighting schemetakes into account the areas of the rings but also favours the more sensitive (lowernoise) rings.The weighting also affects the uncertainties on the bins for the combined his-togram. There are 23 rings in the central area and thus 23 histograms; each of thosehistogram’s bins has Poisson uncertainties of ςi,k =√ni,k, for the ith bin of thekth histogram (or kth ring). The uncertainties of the combined histogram are thena weighted combination of these such that,ςi2 =∑kni,kw2k, (4.2)which can be seen compared with the standard√ni Poisson value in the bottompanel of Fig. 4.4. It is these bin uncertainties that are used when model fitting.Additionally, to increase the amount of data used to constrain the count weran the fitting in two other zones. The first extends from 5 to 7.5 arcmin, andthe second covers from 7.5 to 10 arcmin. The effective noise inside this secondzone is 2.005µJy beam−1 and the effective noise inside the third zone is 3.550µJybeam−1. With just the 0 to 5 arcmin zone we are sampling about 8 per cent ofthe available pixels. The use of all three zones brings that up to around 32 percent of the image pixels. While this still leaves a large fraction of the total imageunused for constraining the count, outside a 10 arcmin radius the instrumental noiseoverwhelms the confusion noise.58The frequency-dependent primary beam correction, and our weighting scheme,does mean that our image noise is not purely Gaussian, as is assumed in the P(D)calculation. We ran a simulation to see by how much our noise might be deviatingfrom Gaussian and whether this could impact our fitting. We created images ofrandom Gaussian noise, of the same size as our central 5 arcmin and convolvedthem with a Gaussian distribution the same shape and size as our beam. We thenapplied the same corrections as to our actual data and created a weighted histogramof each using the process described above. A Gaussian distribution was fit and cal-culated for each sample noise image, and then the noise image P(D) and that ofthe fitted Gaussian distributions were both convolved with a noiseless P(D) from asource count model (the specific source PDF used can be seen in Fig. 4.9). After100 trials we calculated the mean P(D) from the noise histograms, fitted Gaussiandistributions, noise histograms convolved with the source model PDF, and fittedGaussian distributions convolved with the source model PDF. These four meanscan be seen in the top panel of Fig. 4.2, with the ratio of the fitted Gaussian dis-tributions to the noise images shown in the bottom panel. We can see that for thenoise alone the true weighted histograms do deviate from Gaussian distributionsstarting at around 3σ, with the largest deviations being about a factor of 2.5 in the5σ region. However, once convolved with the source count P(D) the deviation ismuch smaller. There is then no discernible difference in the two distributions onthe positive side. On the negative side the maximum deviation from the Gaussianmodel is only about a factor of 1.25, and this is only in the 4−5σ range, where inthe images there are likely only 0−3 pixels/bin. Thus, with our current data thisshould not present any bias in the fitting.4.3 Model FittingWhen calculating the P(D) distribution we use very fine binning in flux density: 218bins with bin size = 0.04µJy beam−1. The output PDF is then interpolated ontothe bins used for the image histogram to perform the fit. We calculate and fit P(D)over the entire range of pixel values in the given image. For the image histogramwe use a bin size of 0.3µJy beam−1 below D = 10µJy beam−1. However, forpixels with flux densities above 10µJy beam−1 there would be very few pixels per5910h45m30s45s46m00s15s30s RA (J2000)+58°56'58'+59°00'02'04'06'DEC (J2000)10h45m00s30s46m00s30s47m00s RA (J2000)+58°52'56'+59°00'04'08'DEC (J2000)Zone1Zone 2Zone 310 arcmin 5 arcmin7.5 arcminFigure 4.3: VLA 3-GHz contour images of the Lockman hole. The upperpanel shows the central 5 arcmin, where the red dashed lines are ringsused for weighting the histogram for the primary beam and the bluecrosses are the pixel locations from one of the grids, with spacing be-tween the points equal to the beam FWHM. The lower panel is the sameimage out to 10 arcmin, with the red dashed lines now showing the sep-aration of the three noise zones discussed in Section 2.2.3.600.40.20.00.20.40.60.81.0Correlation Coefficient ρBin correlations no griddingBin correlations gridded-6.0 -1.7 1.0 3.4 5.8 8.2D (µJy beam−1 )51015202530Bin Uncertainty ς (No. pixels) √n  ςWeighted ςFigure 4.4: Bin correlations and uncertainties. The top panel shows the bin-to-bin correlation coefficients for one row of the correlation matrix at thepeak, computed using eq. (4.5). The black dashed line shows the corre-lation values computed from full pixel histograms of 20,000 simulatedimages. The red solid line indicates the correlation values for the samebin computed from 20,000 simulated images but with the histogramsmade from pixels separated by one beam FWHM. The bottom panelshows the uncertainties for each bin. The black dashed line is ς=√niand the red solid line is the uncertainty ς due to weighting calculatedfrom eq. (4.2).61bin, because of the small bin size as well as the lack of bright sources in the image;thus a majority would have value 0 or 1. To ensure a large enough number of pixelsper bin (to use a Gaussian approximation for fitting) we used expanded bin sizes inthe tails. The bin size above 10µJy increases to ensure a minimum of 10 pixels inall bins. A total of 65 bins were used for the central 5 arcmin region, spanning therange −7µJy beam−1 to 6900µJy beam−1.We have developed a code to fit P(D) based on a set of input model parameters.The input model need not be a simple power law, and may take on various forms, aslong as it is continuous over the chosen flux density range. To fit the parameters weforward-model and use Markov chain Monte Carlo (MCMC) sampling methods.We make use of the publicly available MCMC package COSMOMC [107]1, which,while developed for use in cosmological modelling, may be used as a generic sam-pler, if one provides data, model, and likelihood function. The MCMC code variesthe input parameters in order to minimize the chosen fit statistic. Once the chainhas past the “burn-in” phase it converges near the minimum and will then samplethe parameter space, drawing from the parameter’s proposal density to decide onthe next step in the chain. A well chosen proposal density can improve the effi-ciency of the fitting procedure. For all of our chains we first ran sample chains,with about an order of magnitude fewer steps than the final chains, and used theseto compute the covariance matrix of the parameters, which we then supplied to theMCMC code to use for the proposal density.There has been discussion about the optimal choice of statistic to use for P(D)fitting. One possibility is to use the classical χ2, as done by Friedmann and Bouchet[65] and Maloney et al. [112]. However, the weighting of 1/ni, with ni being thenumber of pixels in the ith bin, will tend to over-weight the bins when ni is small,giving more weight to the tails of the distribution Since for small numbers theuncertainty is not well modelled by√ni, this option is not ideal. Another choiceis to minimize the more correctly calculated negative log likelihood, as done byPatanchon et al. [138] and Glenn and e.a. [73]. While this method gives properweighting, the problems come when trying to interpret the goodness of the fit. For1http://cosmologist.info/cosmomc/62the P(D) model with Poisson statistics the log likelihood is defined aslogL = −∑ini log(pi)− log(N !) +∑ilog(ni!). (4.3)Here N is the total number of pixels in the image, pi is the probability in the ithbin when the PDF is normalised to sum to one, and ni is the the number of imagepixels in the ith bin. In the limit that ni  1 this approximates a χ2 distribution:χ22'12∑i(ni −Npi)2Npi+K, (4.4)where K is a normalisation factor usually taken to be K = (1/2)∑i(Npi). How-ever, when the log likelihood of eq. (4.3) does not equal the left hand side ofeq. (4.4) it can be difficult to determine K and therefore difficult to interpret thelog likelihood.Neither of these two methods takes into account the fact that the pixels in themap (and hence bins in the histogram) are correlated. Due to the sources and noisebeing convolved with the beam, values in one location will affect neighbouringpixel values within an area roughly equal to the size of the beam [or the size of thebeam area divided by 2 in the case of the noise; for further explanation see 37].Furthermore, one source, when convolved with the beam, will contribute pixels tomultiple bins. Ignoring these issues will underestimate the uncertainties of the binsand correspondingly the uncertainties of the fit parameters. 2When dealing with correlated variables the ideal solution is to use the gener-2Both Patanchon et al. [138] and Glenn and e.a. [73] discuss the related issue of the optimalsmoothing kernel for obtaining maximum signal-to-noise ratio for using P(D) to constrain counts.However, it is important to realise that the situation in interferometry is fundamentally different thanfor single dish data. In direct imaging observations the instrumental noise is (ideally) independent atthe map level, and hence it makes sense to further smooth this by a kernel of approximately the beamsize [as shown in figure 3 of 138]. However, for interferometric imaging, the noise is independent inthe Fourier plane, and when going to the image plane has already been convolved by the synthesized(dirty) beam. Chapin et al. [25] found from simulations of sub-mm data that in the very confusedregime the optimal filter is the inverse of the PSF in Fourier space, i.e., the map is de-convolved by thebeam; and in the regime dominated by instrument noise the optimal filter is the PSF. In our case, withσn ' σc our current weighting scheme may not be optimal for P(D), but is likely close. To determinethe ideal weighting and filtering scheme for our type of data would require a more thorough analysis,starting in the Fourier plane and looking at ways to optimize before transformation to the imageplane, rather than applying filters post transformation. This is beyond the scope of this investigation.63alised form of χ2, which includes the covariance matrix of the data. However, testsrun using simulated images show this correlation matrix to be highly dependent onthe underlying source count model. As we do not know in advance the true sourcecount for our data, we want to avoid biasing the results by using a covariancematrix calculated from simulations performed with only an approximated sourcecount. Additionally, using the MCMC method the covariance matrix should bethat of the model being tested. This would entail making many simulated imagesto obtain a covariance matrix from each source count model in the MCMC chain,which is much more computationally expensive.In order to remove the bin-to-bin correlations we instead sampled the image us-ing a grid of positions with spacings of one beam FWHM. This should ensure thatthe pixels are approximately independent and correspondingly that the histogrambins are also independent. Of course the optimal sampling will be a compromisebetween reducing the correlations, and not losing too much fine-scale information,so it is certainly necessary to test that sampling with 1×FWHM spacing is close tothe best choice.We tested the effectiveness of this method using simulations. We used a simplebroken power-law source count of slope −1.7 for flux densities less than 10µJyand −2.3 for sources brighter than 10µJy, to generate sources that were randomlyplaced in an image with the same number of pixels as our data image. We con-volved these sources with a beam of the same size as ours, and added them tobeam-convolved Gaussian noise with σ = 1.255 × 10−6. We simulated 20,000realisations in this way, made full histograms of each and also created histogramsusing pixels sampled from a grid with spacings of FWHM/√2, 1×FWHM, and√2×FWHM. We computed the mean number of pixels per bin from these andthen computed the corresponding correlation matrix. Each entry in the correlationmatrix was computed such that,ρi,j =120000∑k(ni,k − µi)(nj,k − µj)ςiςj, (4.5)the diagonals of which are equal to 1. The correlation coefficient, ρi,j , is equal toCi,j/ςiςj , where Ci,j is the covariance of the ith and jth bin. One row from this64matrix, near the peak of the histogram, is plotted in the top panel of Fig. 4.4. Thisshows that by taking FWHM-separated samples from the grid we remove nearlyall of the correlation between the bins; the off-diagonals of the gridded simulationare all zero within statistical error. The samples with grid spacings of FWHM/√2showed higher off-diagonal correlations. The√2×FWHM samples have roughlythe same correlations as using 1×FWHM, but with lower resolution. Thus wechose to use grids with the FWHM spacing.The images we use have a beam width of approximately six pixels. Hence theFWHM grid which samples the image could be shifted in RA and Dec, with 36different choices possible without repeating any pixels. An example of the gridscan be seen in the top panel of Fig. 4.3, where the blue mini-crosses represent thepositions of the image pixels selected for binning that grid. From the 36 histogramswe are able to compute the scatter for each bin, which can be used as a check onthe calculated bin uncertainties described in eq. (4.2).We chose to carry out the MCMC fitting by minimisingχ2 =12∑i(ni −Npi)2ς2i, (4.6)where the uncertainties used were not the usual Poisson√ni error bars, but rather(due to weighting effects from the primary beam) those from eq. (4.2). We per-formed MCMC trials on our VLA data using both eq. (4.6) and eq. (4.3) (bothusing the gridded image histograms). Comparisons of the output fit parameters forthe different methods can be seen in Fig. 4.5. Although the outputs from the twomethods are consistent, because the value of the log likelihood does not equal theχ2/2 it is difficult to interpret the goodness of the fit.Also in Fig. 4.5 we show the gridded method against the results of a trial usingall of the image pixels. The output is not significantly different for the parame-ters; however, as mentioned, the full resolution method underestimates the limits,the 68 per cent error region being roughly a factor of 1.5 to 3 times smaller inlog10 dN/dS. We know that the fits performed with the gridded data use approxi-mately independent samples. Even though we do lose some resolution we believethis method to be more statistically robust and to model more accurately the vari-65ance and correlations.4.4 Choice of ModelIn Condon et al. [37] a single power-law model was fit to the data in this field.The best fitting single power law in the range 1 < S < 10µJy was dN/dS =9000S−1.7 Jy−1 sr−1. It was noted that power law models from Condon [31],dN/dS = 9.17×104S−1.5 Jy−1 sr−1, and Wilman et al. [191] simulations, dN/dS =2.5 × 104S−1.6 Jy−1 sr−1, were both reasonably good approximations to the datain this range (assuming 〈α〉 = −0.7 to convert from 1.4 GHz to 3 GHz). However,it is the case that no single power law fits well across the whole µJy region 3 .4.4.1 Modified Power LawSince the single power-law model had already been explored, we first decided totry fitting a modified power law of the formdNdS= κSα+β log10 S+γ(log10 S)2, (4.7)in the range 0.01 < S < 60µJy. For S > 60µJy we connected the modifiedpower law to the model from Condon [31] (scaled to 3 GHz using 〈α〉 = −0.7),where this model is in good agreement with known counts. We chose the cut-off at 60µJy so that we would fit the data not just in the µJy region but also in theslightly brighter area where the count from Owen and Morrison [134] was found tobe higher than expected at 1.4 GHz. We fit for α, β, and γ, while κ was calculatedas a normalisation constant to ensure continuity at S = 60µJy. The results arepresented in Section 4.5.The modified power law is a better fit than a simple power law (with a signif-icant change in χ2 for the change in degrees of freedom). One might say that thisis an obvious result as the modified power-law model has more free parameters.3With this method of P(D) a physical model such as a luminosity function may be used. Weopted to use a non-physical model because we intend to investigate a region where no prior dataexist. Using a luminosity function would assume something about the type and number of differentpopulations in that region, which could bias the results. Thus using a non-physical model allowsus to more freely sample the shape of the counts and then, in the future, determine what type(s) ofphysical models and populations could result in a similar shape.66However, if the underlying region were simply a power-law with a nearly constantpower-law index in that region, than a model which allowed for more departurefrom that would not necessarily result in a significant improvement in the fittingsimply due to the addition of the extra parameters.One would still like to be able to constrain the shape of the count in moredetail over different intervals of flux density. With the modified power law, the fitparameters are not very sensitive to the region S ≤ σn, even though there is stillinformation in the image at these faint flux densities. This model also does notallow us to investigate the faintest limits for which constraints are still possible.Therefore, we have followed the approach of Patanchon et al. [138] and Glenn ande.a. [73] and fit a phenomenological parametric model of multiple joined powerlaws, allowing for more variation in the shape of the count. In this approach we fixthe position in log10(S) of a fixed number of nodes, and fit for the node amplitudeof log10 dN/dS. Between the nodes the count is interpolated in log space to ensurea continuous function, with the count outside the highest and lowest nodes set tozero. The node amplitudes do not actually represent the value of dN/dS at thepositions of the nodes, but rather represent an integral constraint on some regionsurrounding the node. Therefore, the best-fit position of any given node dependsnot only on the underlying source count but also on the number, or spacing, of thenodes, and also the type of interpolation used between the nodes.4.4.2 Node ModelThe choice of the number and position of the nodes is somewhat subjective. Thereneed to be enough nodes across the flux-density range to be able to account forchanges in the underlying count, and the choice is also influenced by the result-ing uncertainties on the parameters. The fits of the node positions are degenerate;neighbouring nodes will be most strongly correlated, and so, adding too manynodes will increase the correlations and parameter degeneracies. We examined tri-als using five, six, and eight nodes. We found that there was no significant changein the ∆χ2 with the total of eight nodes, and with six the results were most consis-tent over repeated trials. Comparison of the results with different number of nodescan be seen in Fig. 4.5; based on this, we decided to fit six nodes, spaced roughly67evenly in log10 S. The value of the faintest node is to be considered only as anupper limit, since the code cannot distinguish between low amplitude values andzero. Therefore, the situation is effectively that we fit five well constrained nodesand one upper limit. In the P(D) calculation we also considered two additionalbrighter nodes at fixed dN/dS values. The highest node is far above any source inour field, and it was found that changing its value during P(D) calculation had noeffect on the output. The second highest is also in a very sparsely populated fluxdensity area for our image (only one source brighter). These two node positionsare in a well-constrained range of the 1.4 GHz source count, so rather than fittingfor these nodes their values were estimated from existing 1.4 GHz source counts,scaled to 3 GHz using 〈α〉 = −0.7. Adding these extra nodes is essentially thesame as adopting a prior on the brightest count region considered.The positions for the six nodes were chosen through trial and error. We foundthat the results were not sensitive to a faintest node below −7.3, in log10(S), andthus this position was chosen for the lowest node. For the second faintest node,we found that any nodes placed in the region between the faintest and ∼ 0.25σnwere difficult to constrain and very degenerate for more than one in that region.We therefore chose to place the second node at about a quarter of the instrumentalnoise, which produces reasonably robust constraints. As far as the spacing betweenthe second and sixth nodes, the requirements are to have fairly evenly spaced nodesin log10(S), while still having at least one node in the µJy region, one near theOwen & Morrison (2008) flux density limit, and one between that and the fixednode near our brightest flux density. We ended up with four nodes (three powerlaws) encompassing the region from 0.2 to 17.2µJy, fully covering the region fit inCondon et al. [37] and the Owen & Morrison (2008) sources. Although the nodeplacement was fixed, to make sure that the precise positions did not bias the resultswe also ran chains at ±0.1 in log10 S of the centre nodes, the results of which canalso be seen in Fig. 4.5. Since no discernible difference was observed when varyingthe positions, for the rest of the analysis the centre positions were adopted.Since the source count comes from a redshift integral over luminosities, thecount must be continuous between Smin and Smax. Smax is set by the flux den-sity of the brightest node, 0.0126 Jy, which, as above, was chosen to be brighterthan any source in our image, but not so bright as to greatly increase the range (so68that our bin size could be kept as small as possible). Smin in our case is set bythe number of bins, and is thus 0.0126/218 = 0.04µJy. Since we are fitting fornodes at only a few positions, it is necessary to interpolate the count between thenodes. As well as using linear interpolation (multiple power laws), we considereda cubic spline model, with the cubic spline interpolation done in log10 dN/dS andlog10 S. We ran chains using both models while keeping other variables fixed, andcompared the output, which can be seen in Fig. 4.5. The comparison is not straight-forward, since the values at each node do not have exactly the same meaning, be-ing effectively integral constraints over different flux density regions. However,the two methods produce very similar results: the marginalised means are almostexactly the same, but the uncertainties in the fainter regions are larger for the cubicspline model. For simplicity we decided to use the power-law model for the rest ofthe analysis.Some additional constraints on the fitting parameters were applied to ensurephysically reasonable results. A prior on the background temperature from the in-tegrated count was used. It was set as a cut-off, such that any count model yieldinga temperature greater than 95 mK at 3 GHz was not considered. This was imposedto allow the count to produce (but not overproduce) the background temperatureseen by ARCADE 2 of around 70 mK. This is a very weak prior, and hence veryreasonable to impose, as it not only exceeds the ARCADE 2 value but also greatlyexceeds previous source count temperature estimates of 13 mK. It is important toset some limit on the amplitude of the faintest nodes, where the data constraintsare weakest. For the brighter nodes, a starting estimate of the count was given byapproximating known source counts around the node at 1.4 GHz scaled to 3 GHz.High and low cut-offs were placed on the nodes, limiting the region to be sampled.These were chosen based on the observed high and low count values in the regionaround the node measured using the compilation of 1.4 GHz source counts fromde Zotti et al. [42], scaled to 3 GHz. For the nodes fainter than the current cut-offas set by Owen and Morrison [134], starting estimates were based on the scaledCondon (1984) model at 1.4 GHz. Limits were placed on the sampling space byextrapolating two lines (in log-log space) from the current cut-off, one with a pos-itive slope and one a negative slope. The extreme allowed values for the last node,at ∼ 0.05µJy, were 20 and 14 (in log10[dN/dS]). This yielded a wide area to be69sampled in a region where no previous information existed.It is very important to have an accurate value for the instrumental noise in thiscalculation, because it convolves the noise-free P(D) distribution. Unless σn  σc,then small changes in σn can have a significant effect on the output, particularly inthe faint flux density regime. Our estimate for the confusion noise is roughly thesame as our estimate of the effective instrumental noise inside the 5 arcmin ring,σc = 1.2 ' σ∗n = 1.255. Since our noise estimate comes from a weighted averageof the instrumental noise of the 16 frequency sub-band images, and then a weightedaverage of the noise after primary beam correction, any errors in the measurementor calculation of those would affect our calculated noise value. To allow for thepossibility of uncertainty in our noise value we performed the MCMC P(D) fittingwith: (1) the noise fixed at the calculated values for σ∗n for each model; and (2)allowing the noise to be a free parameter. In this latter case the calculated noisevalue was given as a starting estimate for the fitting and we allowed a samplingrange of (1.255± 0.05)µJy beam−1.In the modified power-law case the marginalised mean for the noise is σ∗n =1.268 ± 0.005, while the node-based model gives a marginalised value of σ∗n =1.250±0.006. These are consistent with the original estimate of 1.255µJy beam−1.The results of fitting with the noise being variable versus fixed can be seen in thebottom left panel of Fig. 4.5. The noise parameter is strongly degenerate with thefaintest two node amplitudes. These nodes do not contribute much to the brighttail of the P(D), but mainly affect its width. This explains why, for the variablenoise case, the faintest two nodes are slightly higher than in the fixed noise case,since the fitted σ∗n is smaller. For both models the fixed and variable noise resultsare consistent within uncertainties. For the rest of the analysis only the fixed noiseresults are used.In terms of the multiple noise zones, the three zones were all fit independently;the results are shown in Fig. 4.5. We also fit all three zones simultaneously, suchthat the fit χ2 was a sum of the individual χ2s. So in this case we minimizedχ2total =∑iχ2i , (4.8)where χ2i is the χ2 of eq. (4.6) from each zone for a given set of input model70parameters. The results presented in Section 4.5 report the fitting of just the firstzone (with the lowest noise) and the three zones together ,for both the modifiedpower-law model and the node-based model.4.5 Discrete Source Count Fitting Results4.5.1 Estimated Number CountsFor all the models investigated here we report the means from the marginalisedparameter likelihood distributions for the variable parameters and any derived pa-rameters. This is done both for fitting just the first noise zone and for fitting allthree zones simultaneously. The limits listed are 68 per cent (upper and lower)confidence limits for the marginalised means, except for the first node which isonly an upper limit. We can also compare these results with the single power-lawbest-fit from Condon et al. [37] and with a compilation of known source countsfrom de Zotti et al. [42]. The confusion noise is measured from the noiseless P(D)distribution (eq. (3.4) with the noise term set to zero). Calculating the standard de-viation is not an accurate way of finding σc, since it is such a skewed distribution.Instead we found the median and D1 and D2 such thatmedian∑D1P (D) =D2∑medianP (D) = 0.34 (4.9)when normalised such that the sum of the P (D) = 1, since in the Gaussian case68 per cent of the area is between ±1σ. Then we took σc = (D2 − D1)/2. Theconfusion noise values for the different models are listed in Tables 4.1 and 4.2.The value estimated from the single power-law fit in Condon et al. [37] is 1.2µJybeam−1, in the middle of our range of 1.05 ≤ σ∗c ≤ 1.37µJy beam−1.The MCMC fitting was first run with the modified power-law model. Theresults from these runs are listed in Table 4.1, with the fits scaled to 1.4 GHz plottedin Fig. 4.6. The data and model P(D) distributions can be seen in Fig. 4.7, alongwith the noise distributions and model noiseless P(D) distributions. Above about3µJy all the fits are consistent. Below this the results from fitting the three noisezones simultaneously fall off faster than the fits from the first noise zone alone.7110-210-1100101S5/2  dN/dS (sr−1  Jy−1.5 )χ2-Log(Likelihood)Grid samplingAll pixels10-210-1100101S5/2  dN/dS (sr−1  Jy−1.5 )Linear power lawCubic spline Nodes = 6Nodes = 810-1 100 101 102 103 10410-210-1100101S5/2  dN/dS (sr−1  Jy−1.5 )-.1 Middle+.1 10-1 100 101 102 103 104σn=1.255 µJy beam−1σn=1.250 µJy beam−110-1 100 101 102 103 104S (µJy)10-210-1100101S5/2  dN/dS (sr−1  Jy−1.5 )Zone 1Zone 2Zone 3ν=3. GHz ν=3. GHzν=3. GHz ν=3. GHzν=3. GHz ν=3. GHzν=3. GHzFigure 4.5: Comparison of MCMC output for source counts with differentsettings. Points are best-fitting amplitudes and dot-dashed lines repre-sent the 68% confidence regions. The blue solid line is the same in allplots and is the best-fit for zone 1 reported in Section 4.5 (Model 1), thered-dashed and green dotted lines are comparisons (Model 2 and Model3). Top left: Model 1 is fit with χ2/2, Model 2 is fit with the logL. Topright: Model 1 is the gridded pixel histograms, and Model 2 is all pix-els. 2nd row left: Model 1 uses linear interpolation in log10(S), whileModel 2 uses cubic spline. 2nd row right: Model 1 has six nodes andModel 2 has eight. 3rd row left: Model 1 is the run the with initial po-sitions Scentre, Model 2 has positions log10(Scentre)− 0.1 and Model 3has log10(Scentre) + 0.1. 3rd row right: Model 1 is fixed noise of 1.255µJy beam−1 and Model 2 allows the noise to float. Bottom: Model 1 isnoise zone 1, Model 2 is noise zone 2, and Model 3 is noise zone 3.7210-310-210-1100101S5/2  dN/dS (sr−1 Jy−1.5)Marg. means Zone 1Marg. means Zones 1,2,3Zone 1 68% Confidence regionZones 1,2,3 68% Confidence region10-1 100 101 102 103 104S (µJy)10-1100101102103S2 dN/dS (sr−1 Jy)Marg. means Zone 1Marg. means Zones 1,2,3Zone 1 68% Confidence regionZones 1,2,3 68% Confidence regionν=3. GHzν=3. GHzFigure 4.6: Source count at 3 GHz from MCMC fitting of the modifiedpower-law model from eq. (4.7). Lines are from the marginalised meansof the parameters of eq. (4.7) (red dashed is from all three zones, i.e. outto 10 arcmin, while blue solid is from zone 1, i.e. 5 arcmin). The dottedlines is where the model was fixed to the values of the Condon (1984)model. The shaded areas are 68 per cent confidence regions. The toppanel uses the Euclidean normalisation, while the bottom panel has theS2 normalisation.73Table 4.1: Marginalised fits for the modified power law in eq. (4.7) at 3 GHz.The quoted uncertainties are 68 per cent confidence intervals. For thecombined fit we treat each zone separately, and hence the number ofdegrees of freedom is approximately 3 times higher.Noise zones 1 1, 2, 3Parameter Marginalised means Marginalised meansα −4.5+1.3−1.3 −4.7+1.2−1.2β −0.17+0.25−0.25 −0.16+0.25−0.25γ 0.012+0.017−0.017 0.016+0.016−0.016log10(κ) −4.34+1.3−1.3 −5.01+1.2−1.1σc (µJy beam−1) 1.122+0.009−0.009 1.068+0.008−0.008χ2 87.3 160.3Ndof 59 149The results for the node-based model are listed in Table 4.2. The slopes andnormalisation constants for the interpolated power laws between the nodes, of theform dN/dS = kSγ , are listed in Table 4.3. The source counts from these modelsare plotted in Fig. 4.8 and the P(D) distributions are shown in Fig. 4.9. The χ2values are lower than in the modified power-law model, though the χ2 values forall four model fits are reasonably consistent with Ndof the number of bins minusthe number of fit parameters. The models are consistent with each other, except forS ≤ 1µJy, where the node-based model falls off more slowly. The modified powerlaw has the advantage of being a single continuous function, as well as havingless fit parameters. However, the node-based model allows for a larger range ofpossibilities than the modified power law and is much more sensitive to the countbelow the noise level, as it is able to fit that region with little to no effect on thebrighter values. With this model, the count for the one-zone case is above thosefrom the three-zone case in the faint region, although the marginalised means arealmost identical.4.5.2 Parameter DegeneraciesThe values of the parameters are highly correlated, particularly between adjacentnodes where the correlation is negative. This means that the errors on the numbercount parameters will also be correlated, giving non-Gaussian shapes to some of74Table 4.2: Marginalised mean amplitudes for the six fit nodes and two fixednodes at 3 GHz, given separately for the deepest noise zone and for allthree noise zones fit simultaneously. The brightest two nodes were fixedto values estimated from known counts at 1.4 GHz and scaled to 3 GHzusing 〈α〉 = −0.7.Noise Zones 1 1, 2, 3Node Marginalised means Marginalised meansµJy log10[sr−1 Jy−1] log10[sr−1 Jy−1]0.05 16.17+1.69 15.79+1.200.20 15.06+0.56−0.56 15.05+0.45−0.430.50 14.43+0.38−0.40 14.43+0.20−0.202.93 13.45+0.09−0.09 13.48+0.03−0.0317.2 12.16+0.06−0.06 12.11+0.02−0.02100 10.27+0.11−0.11 10.35+0.02−0.02572 8.55 8.5512600 6.32 6.32σc (µJy beam−1) 1.283+0.006−0.007 1.266+0.003−0.003χ2 54.8 153.05Ndof 59 149Table 4.3: Slopes and normalisation constants for the interpolated power lawsbetween the nodes, of the form dNdS = kSγ at 3 GHz.Noise Zones 1 1, 2, 3Between Marginal fit Marginal fitNodes (µJy) γ log10 k3GHz γ log10 k3GHz0.05−0.20 −1.79 3.05 −1.19 7.060.20−0.50 −1.65 4.01 −1.55 4.690.50−2.90 −1.23 6.63 −1.25 6.572.93−17.2 −1.69 4.09 −1.78 3.6317.2−100 −2.46 0.43 −2.30 1.15100−560 −2.29 1.08 −2.40 0.75572−12600 −1.66 3.16 −1.66 3.16756 4 2 0 2 4 6 8 10D (µJy beam−1 )10-310-210-1100P(D) (µJy beam−1)−1Gaussian σ=1.255 µJy beam−1P(D) Marg. means Zone 1Noiseless P(D) Zone 1Noiseless P(D) Zones 1,2,3Image histogram15 10 5 0 5 10 15 20 25D (µJy beam−1 )10-310-210-1P(D) (µJy beam−1)−1P(D) Marg. means Zone 1Image histogram Zone 1P(D) Marg. means Zone 2Image histogram Zone 2P(D) Marg. means Zone 3Image histogram Zone 3Figure 4.7: Comparison of 3 GHz pixel histograms (red dots) with themarginalised means model P(D) for zone 1 only (top panel) and mod-els for all three zones (bottom panel) using a modified power-law inputmodel. The dashed line is Gaussian noise of σ = 1.255µJy beam−1.The noiseless P(D) for each model is shown by the blue dot-dashed linefor the one zone fit and the green dotted line for the three zone fit.the joint likelihoods of the two parameter distributions. Sources at a given fluxdensity contribute to many different P(D) pixel values when convolved with thebeam. This means that some sources could be effectively moved from one fluxdensity bin to another, still retaining the same shape for the resulting histogram.This is illustrated in the confidence regions plotted with the source counts (seeFig. 4.5). Instead of being straight power laws from one parameter’s upper limit tothe next, the confidence regions tends to “bow” inwards between the two nodes; as7610-310-210-1100101S5/2  dN/dS (sr−1 Jy−1.5)Marg. means Zone 1Zone 1 68% LimitsMarg. means Zones 1,2,3Zones 1,2,3 68% Limits10-1 100 101 102 103 104S (µJy)100101102103S2 dN/dS (sr−1 Jy)Marg. means Zone 1Zone 1 68% LimitsMarg. means Zones 1,2,3Zones 1,2,3 68% Limitsν=3. GHzν=3. GHzFigure 4.8: Source count at 3 GHz from MCMC fitting of the node-basedmodel using six free nodes and two fixed nodes. Points and correspond-ing lines are the node marginalised means, with the red dashed line be-ing from all three noise zones (out to 10 arcmin), while the blue solidline is from one zone (5 arcmin). The dot-dashed lines are 68 per centconfidence regions (purple for Zone 1, orange for all three zones). Thetop panel uses the Euclidean normalisation, while the bottom panel hasthe S2 normalisation indicative of contribution to the background tem-perature.774 2 0 2 4 6 8 10D (µJy beam−1 )10-310-210-1100P(D) (µJy beam−1)−1Gaussian σ=1.255 µJy beam−1P(D) Marg. means Zone 1Noiseless P(D) Zone 1Noiseless P(D) Zones 1,2,3Image histogram Zone 115 10 5 0 5 10 15 20 25D (µJy beam−1 )10-310-210-1P(D) (µJy beam−1)−1P(D) Marg. means Zone 1Image histogram Zone 1P(D) Marg. means Zone 2Image histogram Zone 2P(D) Marg. means Zone 3Image histogram Zone 3Figure 4.9: Comparison of 3 GHz pixel histograms (red dots) with themarginalised means model P(D) for Zone 1 only (top panel) and modelsfor all three zones (bottom panel) for the node-based input model. Thedashed line is Gaussian noise of σ = 1.255µJy beam−1. The noiselessP(D) for each model is shown by the blue dot-dashed line for the onezone fit and the green dotted line for the three zone fit.one node amplitude is raised the amplitude of the neighbours must decrease. Thisdegeneracy is strongest for the fainter flux densities, as they are not only degeneratewith neighbouring nodes, but also with the instrumental noise.The Pearson correlation matrix for the two cases is listed in Table 4.4, andthe 2D likelihood distributions are shown in Fig. 4.10. The degeneracy meansthat adding more nodes in the fainter regions does not improve the fit. We wouldrequire lower instrumental noise, as well as increased resolution, to benefit from78Table 4.4: Correlation matrix for parameters. Coefficients are computed forfitting all three zones (upper triangle) and just zone one (lower triangle),following the definition Cij =∑r pipj/√∑r p2i∑r p2j , where pi andpj are parameter numbers i and j, and r is the realization number.Node (µJy) 0.05 0.20 0.50 2.90 17.2 1000.05 1.00 −0.16 −0.17 0.02 0.03 0.010.20 −0.23 1.00 0.35 −0.28 0.15 −0.110.50 −0.26 0.66 1.00 −0.64 0.33 −0.192.93 0.01 −0.44 −0.61 1.00 −0.78 0.4417.2 0.03 0.25 0.31 −0.79 1.00 −0.72100 0.01 −0.11 −0.14 0.37 −0.68 1.00extra nodes.4.5.3 Background TemperatureUsing eq. (1.6) we are able to obtain estimates for the discrete-source contributionto the background temperature from our results. Integrating the MCMC outputat each step in the chains allows us to look at the distribution of temperatures.Fig. 4.11 shows the histograms obtained from the modified power-law fitting forboth noise zone cases at 3 GHz, as well as scaled to 1.4 GHz; the same is shown inFig. 4.12 for the node-based model. For the modified power-law fits we integratedover the flux density range 0.05 ≤ S (µJy) ≤ 60 and used the values from theCondon (1984) model for 60<S (µJy)< 109. For the node-based model the fitresults were used in the range 0.05<S (µJy)< 1.26×104 and the Condon (1984)model for 1.26× 104<S (µJy)< 109.The outputs obtained from the MCMC fitting allow us to compute 68 per centconfidence intervals for each distribution, as well as the means, medians, peaks,and values from the source counts from the marginalised means from each param-eter. These values are listed in Table 4.5. The values from the different models andnoise settings are all consistent. These yield a background temperature of around14.5 mK at 3 GHz, corresponding to 115 mK at 1.4 GHz. The distributions fromthe node-based models tend toward higher values and have more elongated tails.Because of this skewness, the 68 per cent confidence limits for these two distribu-7914.0 16.5 19.00.001.01Node-1 (0.05 µJy)13.5 14.9 16.414.016.519.013.5 14.4 15.314.016.519.013.313.4 13.614.016.519.012.0 12.1 12.214.016.519.010.2 10.3 10.414.016.519.014.0 16.5 19.013.414.916.5Node-2 (0.5 µJy)13.4 14.9 16.5 13.5 14.4 15.313.514.916.413.313.4 13.613.514.916.412.0 12.1 12.213.514.916.410.2 10.3 10.413.514.916.414.0 16.5 19.013.314.515.5Node-3 (2.9 µJy)13.4 14.9 16.513.314.515.513.3 14.5 15.5 13.313.4 13.613.514.415.312.0 12.1 12.213.514.415.310.2 10.3 10.413.514.415.314.0 16.5 19.013.213.513.7Node-4 (17.2 µJy)13.4 14.9 16.513.213.513.713.3 14.5 15.513.213.513.713.2 13.513.7 12.0 12.1 12.213.313.413.610.2 10.3 10.413.313.413.614.0 16.5 19.012.012.212.4Node-5 (100 µJy)13.4 14.9 16.512.012.212.413.3 14.5 15.512.012.212.413.2 13.513.712.012.212.412.0 12.2 10.2 10.3 10.412.012.112.214.0 16.5 19.0Node-1 (0.05 µJy)9.910.310.6Node-6 (100 µJy)13.4 14.9 16.5Node-2 (0.2 µJy)9.910.310.613.3 14.5 15.5Node-3 (0.5 µJy)9.910.310.613.2 13.513.7Node-4 (2.9 µJy)9.910.310.612.0 12.2Node-5 (17.2 µJy)10.310.69.9 10.3 10.6Node-6 (100 µJy)Figure 4.10: One and two dimensional likelihood distributions for the six fitnodes. The upper triangle 2D plots (yellow background) are for thethree noise zone fits and the lower triangle 2D plots are the one noisezone fits. The 1D plots show the marginalised likelihood distributionsfor those nodes with three noise zones (red dashed line) and one noisezone (blue solid line). For the 2D plots the contours are 68 (greensolid) and 95 per cent (purple dashed) confidence limits. The blackdots show the positions of the marginalised means. Parameter unitsfor each plot are log10[sr−1Jy−1].80Table 4.5: Radio background temperatures from integration of thesource counts using eq. (1.6).Model Node model Modified power-law modelFrequency (GHz) 3.0 3.0 1.4 1.4 3.0 3.0 1.4 1.4Zones 1 1, 2, 3 1 1, 2, 3 1 1, 2, 3 1 1, 2, 3Peak (mK) 14.6 14.6 115.8 116.3 13.1 13.4 104.7 106.1Median (mK) 14.9 14.7 118.7 117.3 13.3 13.4 106.4 106.968% Lower Limit (mK) 14.4 14.4 115.5 115.5 13.1 13.2 103.9 104.968% Upper Limit (mK) 16.4 15.3 127.7 121.5 13.9 13.8 110.4 109.9marginalised Fit (mK) 14.9 14.8 109.2 111.7 13.5 13.4 107.7 106.8tions are computed from the median instead of the mean. This skewness is simplydue to the fact that this model allows for more possible values in the faintest re-gion, letting the faintest node rise to higher amplitudes, thus affecting the integratedtemperature.4.6 Discrete Emission Discussion4.6.1 Image ArtefactsThis P(D) fitting technique assumes that the instrumental noise is Gaussian dis-tributed and well characterised. In practice our image noise is very nearly Gaus-sian, with the highest contamination from dirty beam sidelobes being only about0.1µJy beam−1. There is, however, another effect that contributes to the shapeof the histogram: it appears to have a tail of excess negative flux density pixelsand thus does not drop off in a purely Gaussian way on the negative side. Visualinspection of the image reveals that the pixels responsible (with values > 5σ, evenconsidering the primary beam correction) are all clustered around the brightest twosources, with almost all of them around the brightest source in the image which isabout 7 mJy, right at the edge of the 5 arcmin ring.It is clear that this is an artefact caused by the imaging and cleaning process, orby asymmetry in the antenna pattern. The VLA antennas use alt-az mounts whichcause the antenna pattern to rotate on the sky with parallactic angle. The supportlegs for the secondary introduce asymmetries in the antenna pattern, which, when8111.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5Temperature (mK)0.00.20.40.60.81.01.21.4ProbabilityMean=13.45Median=13.35Peak=13.14Marg. means T=13.49σ=0.5168% Confidence region11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5Temperature (mK)0.00.20.40.60.81.01.21.41.6ProbabilityMean=13.32Median=13.42Peak=13.36Marg. means T=13.41σ=0.8468% Confidence region100 105 110 115 120 125100 105 110 115 120Figure 4.11: Normalised histogram of radio background temperatures at3 GHz from integrating each step in the MCMC according to eq. (1.6)using the modified power-law model. The top panel comes from justfitting noise Zone 1, while the bottom panel is from fitting for all threenoise zones. Insets are temperature histograms at 1.4 GHz made byscaling the 3 GHz chains with 〈α〉 = −0.7.combined with the rotation with parallactic angle, cause sources away from thepointing centre to appear variable. This in turn causes areas of negative excesspixels around brighter sources. The effect gets stronger with distance from the fieldcentre, with increasing frequency, and source brightness. When examining the 16sub-band images it does seem that this effect increases in strength with frequency.However, it is difficult to say if this is truly the cause, as beyond 5 arcmin there arevery few bright sources. To be able to remove this effect we would need detailed8212 14 16 18Temperature (mK)0.00.10.20.30.40.50.60.7ProbabilityMean=15.3Median=14.9Peak=14.6Marg. means T=14.9σ=1.468% Confidence region12 14 16 18Temperature (mK)0.00.20.40.60.81.01.21.4ProbabilityMean=14.9Median=14.7Peak=14.6Marg. means T=14.8σ=0.968% Confidence region115 125115 125Figure 4.12: Normalised histogram of radio background temperatures at3 GHz from integrating each step in the MCMC according to eq. (1.6)using the node-based model. The top panel comes from just fittingnoise Zone 1, while the bottom panel is from fitting for all three noisezones. Insets are temperature histograms at 1.4 GHz made by scalingthe 3 GHz chains with 〈α〉 = −0.7.measurements of the antenna beam pattern at S-band. Such measurements havenot yet been made; we hope that future imaging will be able to correct for artefactsof this type.With the current data the presence of this negative tail seriously affects theMCMC fitting. Since there is a large deviation from the predicted P(D) calcula-tion with Gaussian noise, attempting to fit the entire histogram gives too broad adistribution, and too low a peak. The fitting procedure inflates the faintest (and8310 8 6 4 2 0D (µJy beam−1 )10-1010-910-810-710-610-510-410-310-210-1P(D) (µJy beam−1)−1P(D) model fitFull image histogramMasked image histogramFigure 4.13: Negative flux-density region of the P(D) distribution. The reddashed line is from the full histogram of the data. The black dot-dashedline is from the marginalised mean node parameters from the MCMCfitting. The blue solid line shows the image histogram after masking.possibly second faintest) node to higher amplitudes to achieve this. Without beingable to correct or model the antenna pattern we simply masked out the negativepixels around the 7 mJy source and a smaller region near a second (1 mJy) source.This decreased the total number of pixels used by about 0.2 per cent. We alsomasked the negative pixels in the second noise zone around this source, decreasingits number of pixels by 0.07 per cent.The negative side of the image P(D) can be seen in Fig. 4.13. Red pointsindicate the image values, while the black line is the P(D) model using the nodevalues from the first noise zone fitting. The blue line shows the image values aftermasking out the negative regions. Tests run on the masked and unmasked versionsclearly show that the masking has no effect on any nodes other than the first two,84which become artificially inflated in the unmasked fitting. Thus we feel justified inperforming the masking. All of the results presented in Section 4.5 were fit usingthe masked images.4.6.2 WeightingNew data reduction and imaging challenges arise from the 2 GHz bandwidth ofthe VLA at S band. Across this bandwidth there are substantial changes in thesynthesised beam size and the primary beam, as well as source flux-density changesdue to the spectral dependence. In our particular case each sub-band was imagedindependently (although cleaned simultaneously), with weighting and taper factorsapplied during cleaning to force the synthesized beams to be the same size. In thenarrow-band case changes due to the frequency bandwidth are usually small andthus weighting to produce an image at the centre frequency of the band does notusually need to include any spectral dependence. With wide-band data this type ofweighting scheme would maximize signal to noise only for sources with 〈α〉 = 0.Instead one could perform a weighted fit of the spectral dependence in each pixelof the 16 sub-band images, correct for the primary beam spectral dependence atthe distance of each pixel from the centre, and use that value to calculate the fluxdensity at the centre frequency. However, this requires having enough signal-to-noise in each pixel to obtain an accurate fit. The weighting scheme we used wasWi(ρ, νi) ∝[ν〈α〉cσniA(ρ, νc)]2, (4.10)where i labels the sub-bands, σni is the noise in each sub-band image, andA(ρ, νc)is the primary beam value at pixel distance ρ and sub-band frequency νi. Usingthese weights the 3-GHz pixel values were given byb3GHz(ρ) =16∑i=1[bi(ρ)Wi(ρ)]/ 16∑i=1Wi(ρ), (4.11)with bi being the pixel brightness in the ith sub-band. This combination is designedto maximize the signal-to-noise ratio for sources with 〈α〉 = −0.7, the averagespectral index for faint sources in this frequency range [e.g. 31].85However, it is possible that this choice of weighting scheme might have af-fected our P(D) results. To test this we created two new wide-band images withdifferent weightings applied. One image was made using 〈α〉 = −0.45 and onewith 〈α〉 = −0.95. The MCMC fitting was rerun on both of these images, leavingthe noise as a free parameter, since changing the weighting could have also affectedthe noise level. The marginalised mean values for the noise are σ∗n = 1.259µJybeam−1 for 〈α〉 = −0.45 and σ∗n = 1.245µJy beam−1 for 〈α〉 = −0.95. The re-sults of the MCMC fitting can be seen in Fig. 4.14, compared with the 〈α〉 = −0.7case with variable noise. There is very little difference in the fits. The largest frac-tional difference between the marginalised fits is still only 0.6 per cent for the thirdnode between the −0.7 and the −0.95 cases. Therefore, it does not appear that thespectral dependence of the weighting has a significant effect on the output.4.6.3 Comparison to Other EstimatesCondon et al. [37] found that the best-fit slope for a single power-law in the µJyregion was γ = −1.7. It was noted that at the fainter end a shallower slope ofγ = −1.5 or −1.6 might be better. Looking at the slopes between our fit nodesand the earlier result, we have three power-law sections that cover this region:between the second and third nodes corresponds to the region 0.2 ≤ S (µJy) ≤0.5; between the third and fourth nodes is the region 0.5 ≤ S (µJy) ≤ 2.9; andbetween the fourth and fifth nodes is the region 2.9 ≤ S (µJy) ≤ 17.2. The slopesfor these regions can be seen in Table 4.3. For the faint part of the region our slopesrange from −1.23 to −1.65, while for the brighter part they range from −1.69 to−1.78. The new results therefore agree well with a−1.7 slope for the brighter partof the µJy region and do seem to suggest a shift to a shallower slope in the µJyregime. Our χ2 value of 153.0 for Ndof = 149 over the full 10 arcmin is lowerthan those we obtain with a single-slope model of slope −1.7, where χ2 = 249.1for Ndof = 153 (or a slope of −1.6, which gives χ2 = 292.3). By using thenode-based model instead of the single power-law model the improvement in thefit yields ∆χ2 = 96.1, which is a highly significant improvement for 149 degreesof freedom.Condon [31] used the local luminosity function to constrain the epoch-dependent8610-210-1100101102S5/2  dN/dS (sr−1 Jy−1.5)α =-0.7 Marg. meansα = -0.95 Marg. meansα =-0.45 Marg. means10-1 100 101 102 103 104S (µJy)101102103S2 dN/dS (sr−1 Jy)α =-0.7 Marg. meansα = -0.95 Marg. meansα =-0.45 Marg. meansν=3. GHzν=3. GHzFigure 4.14: Comparison of MCMC P(D) fitting for wide-band images madewith different spectral indices in the weighting scaled to 3 GHz. Thetop panel is with the Euclidean normalisation, while the bottom panelis S2 normalised. Lines and points are the parameter marginalisedmeans. The red dashed line is for 〈α〉 = −0.95, the green dotted lineis for 〈α〉 = −0.45, and the blue solid line is 〈α〉 = −0.7. The dot-dashed lines are the 68 per cent confidence limits.87spectral luminosity function of extragalactic radio sources, finding a simple modelbased on luminosity, redshift, and frequency that accurately predicted the sourcecount at 1.4 GHz at that time. This model shows two peaks in the S2dN/dS sourcecount, one dominated by starburst-powered galaxies peaking at 50µJy, and theother dominated by AGN-powered galaxies peaking near 0.1 Jy. In the brighterflux density range, where there is a large amount of observational data, this modeldescribes the source count well. We have plotted this Condon (1984) model againstour fits for comparison in Fig. 4.15 and Fig. 4.16 . At the brighter end of the count,S ≥ 90µJy, there is good agreement between this model and all of our fits. Inthe region S ≤ 3µJy the model is also within the uncertainties for all the node-based fits, and lines up quite closely with the marginalised mean fits. However,in the region 3 ≤ S (µJy) ≤ 90 the Condon model is consistently below ourfits, both from the node-based model and the modified power law. In this regionthe dominant component of the Condon model is star-forming galaxies. The dis-crepancy between the model and our source count results suggests the contributionfrom these galaxies is greater than previously thought. If the star-forming compo-nent from Condon’s model is increased by roughly a factor of 2 it would matchquite closely. It is clear that any successful models should not deviate too stronglyfrom the Condon (1984) model, but may need a slightly different treatment of star-forming galaxies.We have also compared our results with the empirical model from Be´therminet al. [10]. This model is derived from the infrared luminosity functions of starforming galaxies, broken into two groups: ‘main sequence’ galaxies and ‘starburstgalaxies’, combined with new spectral energy distributions from the Herschel ob-servatory as well as source counts from a range of IR and submm wavelengths.The Be´thermin model was scaled to 1.4 GHz assuming a non-evolving IR-radiocorrelation of qTIR ≡ log(LIR3.75×1012W ×WHz−1L1.4)= 2.64 out to high redshift anda spectral index of α = 0.8. This model is plotted in Fig. 4.15 and Fig. 4.16, withour best fit results, as well as the Condon (1984) model and Condon et al. [37]power law. In contrast to the Condon (1984) model, the Be´thermin model matchesour results quite closely in the region 1 ≤ S (µJy) ≤ 50, where the star-formingcontribution is dominant. However, for 50 ≤ S (µJy) ≤ 1000 the model dropsbelow our best fits, as well as the Condon (1984) model, and is clearly under-88predicting the observed counts. From figure 3 of Be´thermin et al. [10] this is theregion where the main sequence contribution starts to decline and where the star-burst contribution peaks. If the starburst contribution is increased by a factor ofaround 3 then the model in this region more closely approximates the other esti-mates.All of our model fits, even allowing liberal uncertainties, lie below the source-count values from Owen and Morrison [134]. These points are highlighted in redin Fig. 4.17; they seem to level off, or rise, toward fainter flux densities. We do notsee any such indication for our results, all of our model fits declining in amplitudewithin this region and beyond. As discussed in Condon et al. [37] we believe thisdiscrepancy to be mainly due to incorrect source size corrections in constructingthe earlier source count estimates. Higher resolution VLA observations at 3 GHzwould resolve the size issues definitively.As to the matter of the ARCADE 2 excess emission, it seems unlikely fromthese results that it could be coming from discrete sources. All of our model fits,both node-based and modified power law, as well as the single power-law fromCondon et al. [37], imply a background temperature at 3 GHz of around 13 mK.Using the fit provided in Fixsen et al. [56] to scale the ARCADE 2 result from3.2 GHz to 3 GHz yields a temperature of 62 mK, far outside the uncertainties inour results.There is no indication in our results of any new population of sources. Fig. 4.17shows two possible bumps (representing new possible populations) that would in-tegrate up to the extra temperature necessary to account for the ARCADE 2 result.These were modelled as simple parabolas in the log10[S] − log10[S5/2dN/dS]plane, with fixed peak position. The bump peaking at around 2µJy is clearly muchhigher in amplitude than any of our fits. Any kind of new population peaking aboveabout 50 nJy can be ruled out. Of course there is still the possibility that a new pop-ulation could exist that is even fainter than our current limits, peaking somewherebelow 50 nJy. The fainter bump shown in Fig. 4.17 is one such example. How-ever, the source density required for such sources to contribute significantly to thebackground is extreme.Between the faintest two nodes, 0.05µJy and 0.20µJy, and particularly nearthe faintest node, the count is not well constrained and the uncertainties do allow89Table 4.6: Marginalised parameter means for the best fit (three-zone) nodemodel and modified power law model scaled to 1.4 GHz using 〈α〉 =−0.7. The form for the modified power law is given in eq. (4.7)Node 1.4 GHz Marginalised meansµJy log10[sr−1 Jy−1]0.08 15.55+1.200.34 14.82+0.45−0.430.86 14.20+0.20−0.205.02 13.24+0.03−0.0329.3 11.87+0.02−0.02171. 10.11+0.02−0.02963 8.3121600 6.08Parameter 1.4 GHz Marginalised means0.08 ≤ S (µJy) ≤ 100.α −4.7+1.2−1.2β −0.16+0.25−0.25γ 0.016+0.016−0.016log10(κ) −4.68+1.2−1.1for a rise in the count. We therefore cannot rule out bumps with peaks fainterthan 10 nJy. However, as the peak goes to fainter flux densities the bump needs toincrease in width or height to produce the required background temperature. Notethat any such population would far exceed the total number of known galaxies, aswell as requiring a complete departure from the radio/far-IR correlation (assumingthe sources are not AGN and are star-forming or starburst galaxies).4.7 ConclusionsOur VLA image [CO12] is the deepest currently available, with an instrumentalnoise of 1µJy beam−1. To do justice to these data we have developed a novel andthorough P(D) analysis that has revealed the structure of the 3-GHz source countdown to 0.1µJy.The novel features are the following.1. We have modelled the source count by a series of nodes joined by short9010-1100101102103S2dN/dS(Jysr−1)Modified Power-Law P(D) ModelPublished 1.4 GHz CountsOwen & Morrison 1.4 GHzModified Power-Law P(D) ModelSKADS RQQSKADS FRISKADS FRIISKADS SFSKADS SBSKADS All10-1 100 101 102 103 104S1.4 GHz (µJy)10-1100101102103S2dN/dS(Jysr−1)Modified Power-Law P(D) ModelBethermin et al 2012 AllBethermin et al 2012 MSBethermin et al 2012 SBBethermin et al 2012 AGN10-1 100 101 102 103 104S1.4 GHz (µJy)Modified Power-Law P(D) ModelCondon 1984 Model AllCondon 1984 Model SBCondon 1984 Model AGNCondon et al 2012 P(D)Condon et al 2012 P(D) 68% ConfidenceFigure 4.15: Source counts at 1.4 GHz of models and observed counts. Inall four plots the solid red line is the modified power-law model fromfitting of the 3 noise zones scaled to 1.4 GHz using α = −0.7. Theorange regions are the 68 per cent confidence regions. Top left: Blackpoints are known counts compiled from de Zotti et al. [42], with theyellow stars showing the Owen & Morrison 2008 counts. Top Right:The solid black line is the source count from the SKADS simula-tion. The green dashed and dot-dashed lines are the star-forming andstarburst populations, respectively. The blue dot-dashed, dotted, anddashed lines are FRI AGN, FRII AGN, and radio quiet AGN, respec-tively. Bottom left: The purple solid line is the evolutionary modelfrom Be´thermin et al. [10], with the dashed, dot-dashed, and dottedlines being main sequence, starburst, and AGN galaxies. Bottom right:The green solid line is the evolutionary model from Condon [31], withthe dashed and dot-dashed lines being the starburst and AGN popula-tions. The solid light blue line is the P(D) result from Condon et al.[37], with surrounding error box.9110-1100101102103S2dN/dS(Jysr−1)Node P(D)Published 1.4 GHz CountsOwen & Morrison 1.4 GHzNode P(D)SKADS RQQSKADS FRISKADS FRIISKADS SFSKADS SBSKADS All10-1 100 101 102 103 104S1.4 GHz (µJy)10-1100101102103S2dN/dS(Jysr−1)Node P(D)Bethermin et al 2012 AllBethermin et al 2012 MSBethermin et al 2012 SBBethermin et al 2012 AGN10-1 100 101 102 103 104S1.4 GHz (µJy)Node P(D)Condon 1984 Model AllCondon 1984 Model SBCondon 1984 Model AGNCondon et al 2012 P(D)Condon et al 2012 P(D) 68% ConfidenceFigure 4.16: Source counts at 1.4 GHz of models and observed counts. Inall four plots the solid red line is the node model from fitting of the 3noise zones scaled to 1.4 GHz using α = −0.7. The orange regions arethe 68 per cent confidence regions. Top left: Black points are knowncounts compiled from de Zotti et al. [42], with the yellow stars showingthe Owen & Morrison 2008 counts. Top Right: The solid black lineis the source count from the SKADS simulation. The green dashedand dot-dashed lines are the star-forming and starburst populations,respectively. The blue dot-dashed, dotted, and dashed lines are FRIAGN, FRII AGN, and radio quiet AGN, respectively. Bottom left: Thepurple solid line is the evolutionary model from Be´thermin et al. [10],with the dashed, dot-dashed, and dotted lines being main sequence,starburst, and AGN galaxies. Bottom right: The green solid line isthe evolutionary model from Condon [31], with the dashed and dot-dashed lines being the starburst and AGN populations. The solid lightblue line is the P(D) result from Condon et al. [37], with surroundingerror box.9210-510-410-310-210-1100101102S5/2  dN/dS (sr−1 Jy−1.5)Known 1.4 GHz countsCondon 1984 modelARCADE 2 bump peak=2µJyARCADE 2 bump peak=5nJyMarg. means Zone 1CO12 Power law γ = -1.7Owen counts68% Confidence regionCO12 Error region10-3 10-2 10-1 100 101 102S (µJy)10-1100101102103104S2 dN/dS (sr−1 Jy)Known 1.4 GHz countsCondon 1984 modelARCADE 2 bump peak=2µJyARCADE 2 bump peak=5nJyMarg. means Zone 1CO12 Power law γ = -1.7Owen counts68% Confidence regionCO12 Error regionν=1.4 GHzν=1.4 GHzFigure 4.17: Faint end of the 1.4 GHz source count. The top panel is withthe Euclidean normalisation, while the bottom panel is S2 normalised.The bumps are two examples of counts that, when integrated, producethe extra background temperature necessary to match the ARCADE 2emission, with the red dot-dashed line peaking at 2µJy and the purpledashed bump peaking at 5 nJy. The red star points show the sourcecount of Owen and Morrison [134].93sections of power-law form [138]. In this way, there is no prescription, as-sumption or constraint on the form the count might follow. The parametersin our model then simply become the node values.2. We have used Markov chain Monte Carlo sampling throughout to provideunbiased determinations of the parameters and accurate estimates of param-eter uncertainties. This demonstrates with clarity the dependence on fluxdensity, how the inter-parameter dependencies increase with decreasing fluxdensity, and the faintest limits to which P(D) is sensitive.From the use of these novel techniques we have drawn the following conclu-sions.1. The MCMC approach shows that the uncertainties are dominated by samplevariance rather than systematic effects, at least at the high end of the count.Hence a wider image at the same depth would lead to an improved estimateof the source count.2. Our results are broadly consistent with the single power-law slope of −1.7found by CO12, although differing slightly in detail. They show that theerror estimate of CO12 is somewhat generous. They also show with greaterconviction the change to a shallower slope below 3µJy suggested by CO12.3. The consistency with previous estimates persists even when we take intoaccount changes in the instrumental noise with frequency and position withinthe primary beam, different weightings of the wide-band bandpass data, andnon-Gaussian features in the noise.4. We have shown that the method allows extraction of count information fromthese data to flux densities an order of magnitude below the limit traditionallyset by noise plus confusion, and far below the 5σ noise limit of around 5µJyset by direct source-counting.5. Using a realistic large-scale simulation from Wilman et al. [191], we haveverified our approach and shown that it is unbiased. This simulation enabledus to quantify the effects of clustering and source sizes on the P(D) distri-bution, both of which we found to be insignificant. While simulated P(D)94from a model sky is not new [e.g. 186], never before has a comprehensivesimulation been combined with a comprehensive count-fitting technique.6. Our source count estimates rule out any new populations that could be in-voked to account for the ARCADE 2 excess temperature, down to a levelof about 50 nJy. The count is closely represented by existing models ofevolving luminosity functions, including the contributions of star-formingand starburst galaxies and radio-quiet AGN at the faintest flux densities ob-served; this suggests that we have a substantially robust accounting of thegalaxies that contribute to the radio sky.Here we presented a brief summary of the conclusions from this chapter. For adiscussion of these results in a broader context see Chapter 8.95Chapter 5Extended-Source Count5.1 IntroductionExtended low-surface-brightness radio emission can be difficult to survey. Galactic-and cluster-scale emission can extend up to several arcminutes. Single-dish tele-scopes at radio frequencies have beams on much larger scales and are limited intheir continuum sensitivity by systematic errors, while most interferometers are notideal for measuring low-surface-brightness extended objects. The surface bright-ness sensitivity of an interferometer is limited by its spatial frequency coverage inthe image domain, which is the Fourier transform of its coverage of the apertureplane, often referred to as its ‘uv coverage’. For example, if an interferometerconsists of antennas of diameter Da, and the length of the shortest baseline is b,then the interferometer is generally insensitive to objects in the sky with angularsize greater than λ/(b − Da) radians. An interferometer with Da = 25 m andb = 1000 m observing at 20 cm is therefore insensitive to astronomical objectswith scale sizes greater than 0.7 arcmin. Mosaicing can recover spatial informa-tion > λ/Da in size but nothing can recover information between > λ/Da and< λ/(b − Da), as that has not been measured by the interferometer. Thus, notmany deep extended emission surveys have been carried out at radio frequencies.It is unknown how much this large-scale emission may contribute to the cosmicradio background (CRB) temperature. This background at radio frequencies (Tb)is composed of emission from the cosmic microwave background (CMB, TCMB),96the contribution from the Milky Way (TGal), and the contribution from extragalac-tic sources (Tsource); thus Tb = TCMB + TGal + Tsource. The CMB contribu-tion has been measured to high accuracy and corresponds to a blackbody withT = 2.7255 K [55]. Recent estimates from the deep survey by Condon et al.[37] and Vernstrom et al. [183, hereafter V14] were made of the contribution fromextragalactic sources using the Karl G. Jansky Very Large Array (VLA) at 3 GHz.They found the contribution from compact sources to be Tsource = 14 mK at 3 GHzand 120 mK when scaling this result to 1.4 GHz. However, the synthesized beamsize from the VLA at 3 GHz was 8 arcsec and the image was constructed from uvweighting that filtered out scales much larger than the beam. Thus that surveywould not have been sensitive to emission on larger scales.The issue of large-scale emission and the CRB has been of greater interestin the last few years, following the results of ARCADE 2. This balloon-borneexperiment observed the sky at several radio frequencies, ranging from 3.3 to100 GHz. It measured a background temperature at 3.3 GHz that is much higherthan current estimates from extragalactic sources, (54± 6) mK compared with theTsource ' 14 mK of Chapter 4. Singal et al. [165] proposed that the excess couldbe due to a new population of faint distant star-forming galaxies. Chapter 4 ruledout any new populations of discrete compact sources having peaks in the sourcecount above 50 nJy.For compact sources to be causing the excess emission seen by ARCADE 2,the additional sources would need to have number-count peaks at very faint fluxdensities. This could raise a problem with the far-IR to radio correlation if thesources are not some form of AGN (unless this correlation evolves with redshift),and conflict with limits on the overall number of galaxies.However, the cause of the ARCADE 2 excess could be larger-scale emission(scales ranging from around 0.5 arcmin up to the 12◦ primary beam size of theARCADE 2 experiment). It has been proposed that the emission could be causedby dark matter annihilation [63, 64, 89, 196], in which case it would trace the darkmatter distribution of clusters of galaxies, with a characteristic scale size of arcmin.Other emission processes could include those normally seen from clusters, such asradio relics and haloes, or with diffuse synchrotron emission from the cosmic web[20]. Such emission processes do not directly correlate with star formation and97therefore could evade constraints from the far-IR radio correlation.In this chapter we use deep low-resolution radio observations from the Aus-tralia Telescope Compact Array (ATCA) to investigate the emission that might bepresent at larger angular scales and constrain how it might contribute to the CRB.Section 5.2 describes the technique used to examine the data. In Section 5.3 wediscuss our treatment of discrete point sources, our subtraction method, and thecontribution from faint un-subtracted sources. We discuss issues of detecting ex-tended emission at both high and low resolutions in Section 5.4. Section 5.5 detailsthe models we use for investigating the extended or diffuse emission. Section 5.6discusses the conversion from source count to background temperature, as well asthe predicted background temperatures from ARCADE 2 at our image frequency.Section 5.7.1 presents the results of fitting our extended emission source countmodels to our new data and their contribution to the CRB, and discusses modelsfit to the ARCADE 2 results. In Section 5.8 we discuss our findings, in particularwhat the results might mean in terms of astrophysical sources. We examine modelsof cluster halo emission as well as a source count models from dark matter. Finally,in Section 5.9 we present our current estimates of integral source counts for bothdiscrete and extended source count models.5.2 P(D) and BeamsThe technique described in Section 3.2 was used in this analysis as well.In order to fit an accurate P(D) with a source-count model in this way, theshape of the beam and the image noise must be well understood. Ordinarily onewould use a Gaussian model of the synthesized clean beam in the calculation of themodel P(D), under the assumption that it is not significantly different from the dirtysynthesized beam. However, in our case, the dirty beam has fairly large sidelobes,and is not well approximated by the clean beam. This is shown in Fig. 5.1, withthe full-sized beams and with a close-up of the regions near the peaks. The peaksidelobes are at about the±0.1 level. However, there are pronounced streaks in theouter regions, of amplitude around±0.02, which, when convolved with a source ofS ≈ 100µJy, would create many pixel values in the µJy region. If only the cleanbeam were used in the calculation then a source count model with a large number98-2200 -1100 0 1100 2200Arcsec-2200-1100011002200Arcsec0.050 0.025 0.000 0.025 0.050Peak Normalized Dirty Beam Power-2200 -1100 0 1100 2200Arcsec0.2 0.4 0.6 0.8Peak Normalized Clean Beam Power-300 -150 0 150 300Arcsec0.0 0.2 0.4 0.6Peak Normalized Dirty Beam Power-300 -150 0 150 300Arcsec-300-1500150300Arcsec0.2 0.4 0.6 0.8 1.0Peak Normalized Clean Beam PowerFigure 5.1: Images of the synthesized beams for the 1.75 GHz data. The firsttwo panels are the full ‘dirty’ and ‘clean’ synthesized beams. The thirdand fourth panels show close ups of the region around the peaks of thebeams (dirty and then clean). All beams have been peak-normalized tounity.of µJy sources would be required to achieve a decent fit, even if no such populationof sources truly existed. Thus in all following P(D) calculations we used the dirtybeam for all sources below our cleaning limit of S < 150µJy, while for sourceswith S > 150µJy the clean beam values were used.5.3 Discrete SourcesThe discrete source count is now known quite well, and has been shown to providea very much lower background temperature than the one seen by ARCADE 2,down to at least 50 nJy Chapter 4. In this paper we are therefore interested inmore diffuse extended emission, which would be resolved out at higher resolution.By discrete sources we are referring to sources which are point sources in our150 arcsec×60 arcsec beam, or sources with Ωsource  Ωbeam. In order to focus onthis emission we first need to subtract out the known contribution from point sourceemission. We are only able to subtract out sources down to a certain flux density;therefore we must also consider any discrete emission that was not subtracted out.9910-510-410-3P(D) (µJybeam−1 )−1Sources 50% Powerσrms = 225.5 µJy/bmσrms = 10.0 mKσFit = 179.8 µJy/bmσFit = 8.0 mK20 0 20 40 60 80DT (mK)10-410-310-2P(D) (mK)−1500 0 500 1000 1500 2000D (µJy beam−1 )10-510-410-3P(D) (µJybeam−1 )−1No Sources 50% Powerσrms = 155.2σrms = 6.9 mKσFit = 149.3σFit = 6.6 mk10-410-310-2P(D) (mK)−1Figure 5.2: P(D) distributions for the mosaic image central regions, wherethe increase in noise due to the primary beam is 1.5 times the minimumnoise or lower, an area of roughly 0.61 deg2. The top panel shows thepixel histogram for the mosaic before point source subtraction. The bot-tom panel shows the distribution for the mosaic after subtraction of theATLAS point sources. The solid black lines are the image distributions,while the red dashed lines are fitted Gaussians.1005.3.1 Source SubtractionWe used the clean component models from the ATLAS survey third data release(Franzen et. al, 2014 in preparation, Banfield et. al, 2014 in preparation) as pointsource models for subtraction, since the ATLAS resolution is significantly higherthan our data, at around 10 arcsec. It is not entirely clear what the median sourcesize might be and how it changes with flux density, but we expect a value between1 and 3 arcsec for typical galaxies in evolutionary models [e.g. 191]. Thus theATLAS resolution should be sufficient to measure all of the discrete or point sourceemission. The ATLAS point source models were split into two frequency bands:the lower frequencies from 1.30 to 1.48 GHz; and the higher frequencies from1.63 to 1.80 GHz. For the subtraction we split our seven uv-data sets (for eachpointing) into two equal frequency bands as well: 1.30 to 1.70 GHz; and 1.70 to2.10 GHz. The ATLAS images were made using multi-frequency deconvolutionand thus contain estimates of the spectral indices of the clean components, whichcan be used to scale the flux density to different frequencies during subtraction.The task UVMODEL was used to subtract the appropriate pointing and frequencycoverage clean model from each corresponding uv-data set; then the uv-data foreach pointing were concatenated using the task UVGLUE (combining the lower andupper frequency parts for each pointing). An independent image was constructedfrom each pointing with a mosaic constructed subsequently.The ATLAS survey has an rms sensitivity of 15 to 25µJy beam−1 (dependingon the individual pointing) and the models were cleaned down to a level of 150µJybeam−1. Thus all point sources with S > 150µJy should have some fraction oftheir discrete emission subtracted out. There is some residual emission apparentaround the brightest sources, which is visible in the bottom right panel of Fig. 2.2.We cannot say if this is due to some slight calibration or subtraction error, possibletime variability of AGN sources, or if this represents a portion of the sources’diffuse emission. Looking at the peak positions of the well defined objects in eachof the images, the average residual is only 5 per cent of the peaks. The P(D)s forthe central region of the mosaic images before and after source subtraction arepresented in Fig. 5.2; this shows a clear decrease in the size of the positive sourcetail for the subtracted image.101When comparing our data to P(D) predictions from source-count models weuse the P(D) of the source-subtracted mosaic image, including only pixels from re-gions where the noise due to the primary beam correction is not more than 1.5 timesthe lowest noise value. This is because the P(D) calculation from a source-countmodel assumes a constant value for image noise. The noise is certainly inhomo-geneous in our data. However, simulations have shown that the effect on the P(D)calculation is small if we limit ourselves to a region where the change in the noise issmall and create a noise-weighted histogram. Using a weighting scheme describedin Chapter 4, we calculate a mean noise in this area (approximately 0.61 deg2) ofσn = (52± 5)µJy beam−1, or (2.3± 0.2 mK).5.3.2 Counts and ConfusionIt is necessary to estimate the contribution of discrete emission from sources thatwere not subtracted out. For sources below the clean threshold of the ATLASmodels we took the discrete source count of Chapter 4, including sources up toS = 150µJy, which is measured via confusion analysis down to S ' 0.05µJy at3 GHz. We scaled this to 1.75 GHz according to S ∝ να, with α = −0.70± 0.05being the mean spectral index of star-forming galaxies [31]. We found that slightvariation in this spectral index produces no significant effect on the output.For the bright residuals left over from the subtraction process the issue is not asstraightforward. Even neglecting any errors in calibration or subtraction, the cleanprocess which generated the models is highly non-linear. The clean componentsmay only represent a fraction of the true flux density, which can vary by peak fluxdensity and from pointing to pointing. We do not believe there to be extendedemission brighter than approximately 150µJy beam−1 (as discussed in more detailin Section 5.4.1). To account for unresolved residuals brighter than this we countedall the peaks in the source-subtracted image brighter than 150µJy beam−1 that areassociated with point sources in the image with no subtraction, and calculated apower law index for their differential source count of −2.50.Our model for the unsubtracted point-source contribution is then the scaledChapter 4 source count up to 150µJy with a power law of slope −2.50 attachedfor sources with 150µJy < S < 3 mJy (3 mJy being the brightest residual in the102200 400 600 800 1000D (µJy beam−1 )10-410-3P(D) (µJybeam−1 )−1θbeam = 150x60 arcsecσconfusion = 124.7 µJy bm−1σconfusion = 5.5 mKMedian = 500.0 µJy bm−1Median = 22.2 mK10 15 20 25 30 35 40 45DT (mK)10-2P(D) (mK)−1400 200 0 200 400 600 800D (µJy beam−1 )10-510-410-3P(D) (µJybeam−1 )−1σn=52.0 µJy beam−1σn=2.3 mKNoise ⊗ Source modelImage P(D)10 0 10 20 30 40DT (mK)10-310-210-1P(D) (mK)−1Figure 5.3: Source confusion distribution for discrete sources (point sourcesin our 150 arcsec×60 arcsec beam). The top panel shows the noise-less P(D) from the source count of Chapter 4, scaled from 3 GHz to1.75 GHz using α = −0.7, including only sources up to a flux densityof S = 150µJy, with a differential source count logarithmic slope of−2.5 for 150 ≤ S ≤ 3000µJy. The measured confusion rms from thisdistribution is σc = (125 ± 10)µJy beam−1, or (5.5 ± 0.44) mK, withthe dashed lines showing the median and the shaded regiom showingthe ±1σ values. The bottom panel shows this distribution convolvedwith a Gaussian of width σn = 52µJy beam−1. The Gaussian is theblue dotted line, the convolution is the red solid line, and the P(D) fromthe inner region of the source-subtracted mosaic image is shown as theblack dashed line.103fitting area). We computed the P(D) from this count and convolved this P(D) with aGaussian noise distribution of width σn = (52±5)µJy beam−1, or (2.3±0.2) mK.The noiseless and convolved P(D) distributions are shown in Fig. 5.3. We measuredthe confusion noise σc, or width of the distribution, by first finding D1 and D2,median∑D1P (D) =D2∑medianP (D) = 0.34, (5.1)when normalised such that the sum over the P (D) is 1. Then we take σc = (D2 −D1)/2. We do this since, in the Gaussian case, 68 per cent of the area is between±1σ, and since, in the more realistic case, the long positive tail makes the varianceof the full distribution a poor estimate of the width if the peak. The estimated widthof the source-subtracted image P(D) is σ = 155µJy beam−1 ( 6.9 mK) with anuncertainty of ±5µJy beam−1 (±0.22 mK) measured from bootstrap resampling.For the discrete source model P(D) we find a value of σc = 125µJy beam−1( 5.5 mK). The P(D) of this model convolved with Gaussian noise thus has an rmsof σc⊗n = 135µJy beam−1 ( 6.0 mK).This discrete model estimate should be treated with some caution. The re-sult is dependent on the exact value of the noise used in the calculation and theexact shape of the unsubtracted discrete count contribution. The unsubtracted dis-crete count is based on a model which is dependent on the maximum flux densityvalue for the point sources with no subtraction, as well as the power law usedfor the brighter sources. Taking these points into consideration we adopt an un-certainty of ±10 µJy beam−1, or ±0.44 mK, on the measure of σc = 125µJybeam−1= 5.5 mK, yielding a measurement and uncertainty for the width of thenoise convolved distribution of σc⊗n = (135±12)µJy beam−1, or (6.0±0.53) mK.We want to know how different the model of unsubtracted discrete source emis-sion is from the data. To do this we performed a bootstrap significance test. We ran-domly selected a subset of half the image pixels, generated random numbers fromthe noiseless model distribution and added varying amounts of Gaussian noise (toaccount for the uncertainty in the model). We then combined the real and modeldata into one set and drew two new subsets at random from the combined distribu-tion. We compared the binned real data to the binned model data, and the binned104combined random sets to each other. We repeated this procedure 5000 times. Thisyields a distribution of the test statistic from the combined random samples of thenull hypothesis (that the observed and model data come from the same population)and a distribution of the test statistic when comparing the ordered sets (the ob-served and model sets not combined). We computed three different test statistics:the Euclidean distance (the root-mean-square distance between the histograms);the Jeffries-Matusita distance (similar to the Euclidean distance but more sensitiveto differences in small number bins); and a simple χ2.The results of the bootstrap test show an average excess width of (76±23)µJybeam−1, (3.4 ± 1.0) mK, with the value of 76 coming from√σ2 − σ2c⊗n, whichwith the measured values is√1552 − 1352. The exact significance of this excessdepends on the test statistic . However, regardless of which test statistic is usedthe data and model are statistically different, with a minimum of 99.5 per cent con-fidence. This excess cannot be converted directly into a background temperaturesince the conversion depends on the underlying source-count model responsiblefor the width (see Section 5.6 for more discussion on the temperature conversion).Based on these tests, we conclude that there is more emission present than thatfrom compact galaxies alone at the roughly 3σ level. However, due to the un-certainty in the source subtraction process, this excess and any resulting extendedemission models are here considered as upper limits on the extended emissionpresent.5.4 Extended Sources5.4.1 High Resolution Extended EmissionBefore attempting to model any extended emission in the ATCA data we considerhow extended emission is detected at higher resolutions, comparing the VLA dataused by Chapter 4 and the ATLAS ATCA high resolution images. The VLA 3-GHz beam used in Chapter 4 had a FWHM of 8 arcsec, while the ATLAS beamwas roughly 10 arcsec. We would like to know how emission on arcmin scalesappears with these types of observations, since we know that some emission willbe resolved out at higher resolution.105This was tested using sources from 1 deg2 of the SKADS simulation [191] at1.4 GHz. This simulation was shown in Chapter 4 to be a close approximation toobserved source counts. Using the flux densities provided, we made one imagecontaining only point sources. Then assuming each point source has an extendedhalo with total flux set to Sdis/10, with Sdis being the point source (discrete) flux,we made two images, assuming all the haloes were Gaussians with FWHM of 30or 60 arcsec. We added the point sources to these and convolved the images with a9 arcsec beam (the average size of the VLA and ATCA resolutions).The confusion noise of each of the noiseless images are 1.53, 1.95, and 1.78µJybeam−1 for the discrete, discrete+30 arcsec, and discrete+60 arcsec data sets at1.4 GHz. The 30 arcsec haloes add a width of σ30 =√1.952 − 1.532 = 1.21µJybeam−1, and the 60 arcsec haloes add σ60 =√1.782 − 1.532 = 0.91µJy beam−1.The P(D)s for the images with point sources plus extended emission are shownin the bottom panel of Fig. 5.4. The smaller the extended objects the greater theincrease in the width of the distribution. For images with the same total flux den-sity the distribution for the larger sources would have its DC level shifted to higherflux densities; however interferometers are not sensitive to the DC level (or lowestspatial frequency) and thus do not measure total flux densities.The measured confusion rms from the 3 GHz VLA data is approximately (1.2±0.07)µJy beam−1 (depending on the source-count model). Scaling the simulatedvalues to 3 GHz, the addition of the 30 arcsec extended emission to the VLA pointsource model would yield a width of 1.38µJy beam−1, with the 60 arcsec haloesyielding 1.31µJy beam−1. Although in this case these exceed the estimated un-certainty, the simulated confusion widths depend on the exact source count usedand the assumption of how the extended emission depends on the point-source fluxdensity. Thus these particular extended emission models produce excess widthsin the P(D) distributions that are large enough to have been detected in deep highresolution images. However, these simulations show that there likely exist modelswith either fainter or larger-scale extended emission that would have been unde-tected in the VLA P(D) experiment of Chapter 4.From the simulated extended-size images we see that sources with total haloflux densities greater than approximately 150µJy would be visible in the VLA orATLAS images. The top panel of Fig. 5.4 shows a cut-out of the simulated images;106when the point-source flux density is faint (≤ 200µJy), the extended emission isnot visible in the image. However, for brighter point sources (with brighter haloemission), the extended haloes are visible in both the 30 and 60 arcsec images.Since nothing of this nature is seen in either the VLA or ATLAS images, we con-clude that any extended emission in the current low resolution ATCA data shouldalso have total flux density less than about 150µJy, or else has to be very rare.5.4.2 Source Size SensitivityThe P(D) calculation does not use any size information and assumes only unre-solved sources. Therefore, it is important to understand how resolution affects theP(D) fitting. To test this we used the simulated halo flux densities for the extendedemission described in Section 5.4.1 and made four separate images for sourcestreated simply as point sources and as Gaussians with FWHM of 60, 90, and 300arcsec. These give a range of sizes in relation to the ATCA beam. We then raneach image through the fitting routine for source-count amplitudes at specific fluxdensities, i.e. a set of connected power laws [e.g. 138, 183].The results show that there is no significant change in the fitting results be-tween the point source and 60 arcsec size images. However, the count amplitudesfor the 90 arcsec sizes are lower than the true count at both the faintest and bright-est flux densities, while the 300 arcsec size results are significantly lower at all fluxdensities. The results of this test are presented in Fig. 5.5. This shows that the P(D)fitting procedure is reliable for sources on the order of the beam size or smaller; Ta-ble 5.1 shows the linear sizes for the angular scales to which we are sensitive givena range of redshifts, assuming standard ΛCDM cosmology withH0 = 67.8 km s−1Mpc−1, Ωm = 0.308, and ΩΛ = 0.692 [141].5.5 Extended Source Count ModelsWe have shown that there is a significant excess in the width of the observed dis-tribution over that estimated from noise and discrete point sources, suggesting thepresence of diffuse or extended sources. This emission could be low surface bright-ness diffuse emission around individual galaxies, diffuse cluster emission, or some-thing more exotic, such as emission from dark matter annihilation in haloes. We107Point Sources Point + 30" Point + 60" 10 5 0 5 10 15 20 25D (µ Jy beam−1 )10-310-210-1P(D) (µJybeam−1 )−1 Point SourcePS+30" haloesPs+60" haloes20 10 0 10 20 30 40DT (mK)10-310-2P(D) (mK)−1Figure 5.4: Simulation showing point source and extended emission at higherresolution. The top panels show cut-outs of the simulation with justpoint source emission (left), point sources plus haloes of 30 arcsec di-ameter (middle), and point sources plus haloes of 60 arcsec diameter(right), all convolved with a 9 arcsec beam and with Gaussian noiseof 2µJy beam−1. The total flux density of each halo is taken as thepoint source flux density divided by 10. The bottom panel shows theP(D) distributions from the three images, with the solid black line beingfor point sources only, the red dashed line point sources plus 30 arcsechaloes, and the blue dot-dashed line point sources plus 60 arcsec haloes.10810-310-210-1100dN/dSS5/2  (sr−1Jy−1.5)Real countsize 0"size 60"size 90"size 300"100 101 102 103S (µJy)103104105106107108109N(>S) (sr−1)Real countsize 0size 60"size 90"size 300"Figure 5.5: Results of P(D) fitting of simulated images with different sourcesizes. The top panel shows the Euclidean-normalized differential sourcecount of the input count (solid black line) and best-fitting results ofthe point source image (red diamonds), the 60 arcsec size image (bluesquares), the 90 arcsec size image (green stars), and the 300 arcsec sizeimage (magenta pentagons). The bottom panel shows the same infor-mation, but plotted as integrated source counts.Table 5.1: Angular and physical source sizes at different redshifts.Angular Size Physical Sizez = 0.25 z = 0.5 z = 1 z = 2(arcsec) (Mpc) (Mpc) (Mpc) (Mpc)30 0.12 0.19 0.25 0.2660 0.24 0.38 0.49 0.51100 0.40 0.63 0.82 0.86150 0.60 0.94 1.23 1.29109then used three source count models to investigate the possible excess (extended)emission. We follow the fitting procedure described in detail in Chapter 4. Weuse Monte Carlo Markov Chains (MCMC), employing the software package COS-MOMC [107]1, to minimize χ2 for each model. The three most negative bins(−500 ≤ D(µJy beam−1) ≤ −250) from the image histogram were neglectedin the calculation of χ2. This is because the data have a clearly non-Gaussiannegative tail, due in part to the non-constant noise but also due to the areas of over-subtraction, which produce an excess of negative points (see the bottom panel ofFig. 5.2). Tests on subsets of the data, and using different detailed approaches forsubtracting bright sources, showed that these effects were restricted to the mostnegative bins, with the rest of the histogram being quite stable.5.5.1 Shifted Discrete Count ModelUsing evolutionary models [e.g. 31, 90] the source count can be broken into contri-butions from two populations, namely AGN and star-forming galaxies, as shown inFig. 5.6. The simplest extended-emission model assumes that each of these popu-lations has a radio-emitting halo on arcmin scales, proportional to some fraction ofthe discrete flux density (or Sdiscrete × C), separately for the two populations.Theorigin of these haloes would be either cosmic ray electrons (or dark matter annihi-lation products in the haloes) interacting with the galaxy’s magnetic fields. Thesewould have to be quite faint, or diffuse, to have not already been observed, and isnot believed to be a likely source of much emission, though still a possibility. Thecounts associated with this extended emission must then retain the shape of thediscrete counts for each population, but can be shifted in flux density. To estimatethe extended counts that are consistent with our data we took the discrete countsfor each population and simply applied a shift in log10[S] separately. Thus,dN(Sext)AGNdSext=dN([SdisC1])AGNd[SdisC1],dN(Sext)SBdSext=dN([SdisC2])SBd[SdisC2],(5.2)1http://cosmologist.info/cosmomc/110where C1 and C2 are constants that are varied to fit the counts. When combinedwith the unsubtracted discrete count and Gaussian noise, we can find the values thatbest fit the observed P(D) distribution of our source-subtracted image. Figure 5.6shows an example of this model with the two populations of discrete counts, eachwith shifts applied. We plot the results with the usual S2 normalization and with nonormalization (so that the horizontal shifts can be seen).This model will be referredto as Model 1.5.5.2 Parabola ModelWe also wished to investigate the possibility of the extra emission being fit by asingle new population. To do this we introduce a new population as a parabola inlog10[S2dN/dS] of the formS2dN(S)extdS= A(x− h)2 + k. (5.3)Here x = log10[S] and A, h, and k are all free parameters. The parameter h is thepeak position in log10[S], k is the amplitude or height of the peak, and A (alongwith k) controls the width. We chose this model because it allows for a smoothcurve, and since the discrete count populations are themselves crudely approxi-mated by parabolas in log10[S2dN/dS]. This model will be referred to as Model2.5.5.3 Node ModelThere may be several types of sources or populations contributing to the extendedemission counts, including individual galaxies, clusters, dark matter, intra-clustermedium, etc. Without having physical models for these different populations, wewould require too many parameters to fit separate models for each. Therefore, wehave chosen also to fit a model of connected power laws. This model allows for theshape of the source count to vary over a particular flux density range, rather thanhaving a fixed shape based on a few parameters. It therefore has the potential to besensitive to contributions from different populations at different flux densities.The model consists of fitting for the amplitude of log10[dN/dS] at specific flux11110-210-1100101102103104105S2 dN/dS (sr−1Jy)AGNdiscreteStarburstdiscreteAGNdiscrete log[C1 ]=-1.5Starburstdiscrete log[C2 ]=-.2510-1 100 101 102 103 104 105 106 107S1.75 GHz (µJy)10-310-11011031051071091011101310151017dN/dS (sr−1Jy−1 )AGNdiscreteStarburstdiscreteAGNdiscrete log[C1 ]=-1.5Starburstdiscrete log[C2 ]=-.25Figure 5.6: Discrete and shifted source counts of AGN and starburst. The toppanel shows the discrete AGN and starburst source counts (black dottedline and black dashed line) using S2 normalization. The red lines areexample of the shifting model described in Section 5.5.1, where S forthe AGN count has been shifted by log10[C1] = −1.5 and the starburstcount is shifted by log10[C2] = −0.25. The bottom panel shows thesame lines with no normalization on the source counts. This demon-strates how applying only a horizontal shift in log10[S] will appear as acombination of vertical or amplitude shift when the S2 normalization isapplied.112densities, or nodes, and interpolating linearly (in log space) between the nodes –for more details on this model see Chapter 4. We specifically use five nodes spacedevenly in log10[S], covering the range of 0.5 ≤ S ≤ 1000µJy. This model will bereferred to as Model 3.5.6 Background TemperatureThe discrete source count used by Chapter 4 integrates (up to S = 900 Jy) to abackground temperature at 1.75 GHz of Tdis(1.75 GHz) = 63 mK, where Tdis isthe temperature from the discrete source contribution, using eq. 1.6.The ARCADE 2 experiment measured a background temperature of (54 ±6) mK at 3.3 GHz. Using both of the fits from eq. 1.8 and eq. 1.9 we calculatedthe estimated background temperature at 1.75 GHz by taking the average from thetwo equations, and an uncertainty using the highest and lowest values from theuncertainties in the equation parameters. This yields TAR2(1.75 GHz) = (265 ±45) mK, which corresponds to a total flux density, given our beam size, of 5600µJybeam−1.In addition to fitting the data with no constraints, we also fit the models to seewhat kind of count shapes would be necessary to achieve the ARCADE 2 temper-ature. We fit the models as described above, only this time adding a prior requiringthat the integrated temperature be in the range of 150 to 300 mK. This should showif there is any such source count model consistent with both ARCADE 2 and ourdata.These models are referred to as Model 1A (shifts), Model 2A (parabola), andModel 3A (nodes).5.7 Extended Emission Source Count Fitting Results5.7.1 Summary of FitsUsing the three models from Section 5.5 we (a) examined what model parame-ters best fit our new ATCA data, (b) calculated the resulting contribution to thebackground brightness temperature, and (c) modelled what would be necessary toachieve a background temperature consistent with ARCADE 2.113Table 5.2: Best-fitting results for Model 1. The temperature, Text, is the con-tribution to the background (using eq. 1.6), for the extended source countcontribution only.Model 1 Model 1A(unconstrained) (constrained)log10[C1] −1.91± 0.11 −2.0± 0.15log10[C2] −0.61± 0.16 0.39± 0.03Text (mK) 1.65+1.85−0.75 201.2± 40σc ( µJy beam−1) 62.63 480.1σc (mK) 2.78 21.36χ2 (Ndof = 42) 109.6 45200The results from Model 1 and Model 1A are in Table 5.2, results from fittingModel 2 and 2A are in Table 5.3, and results from Model 3 and Model 3A arelisted in Table 5.4. Each of these extended counts was added to the unsubtracteddiscrete count model (discrete source count fainter than the subtraction limit plus apower law for subtraction residuals, as discussed in Section 5.3.2) to compute theP(D) for each model. The P(D) models, convolved with Gaussian noise of 52µJybeam−1, are shown in Fig. 5.7, along with the P(D) for the central region of oursource-subtracted mosaic image.Each step in the MCMC chains is another source count model. For each modelwe used the MCMC results and calculated the background temperature distribu-tions, using eq. (1.6), which are plotted in Fig. 5.8. The temperature distributionsimply a mean temperature of (10± 7) mK. The resulting source count models arepresented in Fig. 5.9, broken down by population and shown along with the discretecounts at 1.75 GHz.5.7.2 Model UncertaintiesWe tried variations in the fitting method by first changing the fit statistic used (χ2vs. log likelihood), which produced little change in the output; and second bytrying different models. Instead of the parabola we tried a Gaussian in S2dN/dS.The Gaussian model produces a peak in roughly the same spot as the parabola,though the parameters are not as well constrained. All models tried resulted in11410-610-510-410-3P(D) (µJybeam−1 )−1Model 1AModel 2AModel 3ASubtracted Mosaic P(D)20 10 0 10 20 30 40 50DT (mK)10-410-310-2P(D) (mK)−1500 0 500 1000D (µJy beam−1 )10-610-510-410-3P(D) (µJybeam−1 )−1Model 1Model 2Model 3Subtracted Mosaic P(D)10-410-310-2P(D) (mK)−1Figure 5.7: P(D) distributions for various extended-emission source counts.The top panel shows the P(D) distributions for the best-fitting modelsof extended-emission counts with a prior for the ARCADE 2 tempera-ture for Model 1A (blue solid line), Model 2A (green dot-dashed line), and Model 3A (red dashed line). The bottom panel shows the resultsof fitting the same models, but without the temperature requirement.All models have been convolved with Gaussian noise of σn = 52µJybeam−1 and the unsubtracted discrete source count contribution. Theblack points are the source-subtracted mosaic histogram (as seen in thebottom of Fig. 5.2).1150 1 2 3 4 5 6Temperature (mK)050100150200Steps in MCMC chain µT = 2.4 mKModel 10 10 20 30 40 50 60Temperature (mK)050100150200250300350Steps in MCMC chain µT = 13.1 mKModel 20 10 20 30 40 50Temperature (mK)0100200300400500Steps in MCMC chain µT = 5.7 mKModel 3Figure 5.8: Histograms of the contribution to the background temperaturefrom MCMC fitting of the three source count models. Temperaturesare for extended emission counts only, with the discrete source countbeing Tdis = 63 mK. The top panel shows the background temperaturesfrom fitting Model 1, the middle panel is the histogram from Model 2,and the bottom panel from Model 3. The solid vertical lines are themedians for each distribution, with the grey shaded regions showing the68 per cent confidence regions.116100 101 102 103 104 10510-210-1100101102103104S2 dN/dS (sr−1Jy)No prior Model 1 Discrete counts - Tb=63.6 mKModel 1 - Tb=1.6 mKShifted AGNShifted starburstARCADE 2 prior Model 1A Discrete counts - Tb=63.6 mKModel 1A - Tb=201.2 mKShifted AGNShifted starburst10-1100101102103104S2 dN/dS (sr−1Jy)No prior Model 2 Discrete counts - Tb=63.6 mKDiscrete AGNDiscrete starburstModel 2 - Tb=12.3 mKARCADE 2 prior Model 2A Discrete counts - Tb=63.6 mKDiscrete AGNDiscrete starburstModel 2A - Tb=171.3mK100 101 102 103 104 105S1.75 GHz (µJy)10-1100101102103104S2 dN/dS (sr−1Jy)No prior Model 3 Discrete counts - Tb=63.6 mKDiscrete AGNDiscrete starburstModel 3 - Tb=7.2 mK100 101 102 103 104 105S1.75 GHz (µJy)ARCADE 2 prior Model 3A Discrete counts - Tb=63.6 mKDiscrete AGNDiscrete starburstModel 3A - Tb= 159.6Figure 5.9: S2 normalized source counts at 1.75 GHz. The black lines are thesame in all plots and are counts of discrete sources from the estimatesin Chapter 4 at 3 GHz, scaled to 1.75 GHz using α = −0.7, while thecoloured lines are the extended-emission counts from models. The dis-crete count is broken into two populations, AGN and starbursts, basedon evolutionary models, shown as the dotted and dashed lines, respec-tively, with the solid lines being the sum of both components. The leftpanels show the models with no priors and the right panels show mod-els with ARCADE2 priors. The top panels are Model 1 and 1A (blueand red lines). The middle panels show Model 2 and 2A (magenta andgreen lines). The bottom panels show Model 3 and 3A (orange and cyanlines). The grey regions are the 68 per cent confidence intervals.117Table 5.3: Best-fitting parameter results for Model 2 and Model 2A. The tem-perature, Text, is the contribution to the background (using eq. 1.6), forthe extended source count contribution only.Parameter Model 2 Model 2A(unconstrained) (constrained)A −0.79± 0.29 −2.04± 0.22h −5.55± 0.40 −6.19± 0.04k 2.58± 0.49 3.93± 0.05Text (mK) 12.3+22.8−7.90 171.3+16.2−13.3σc (µJy beam−1) 62.81 63.10σc (mK) 2.79 2.81χ2 (Ndof = 41) 76.1 111.1Table 5.4: Best-fitting parameter results for Model 3 and Model 3A. The tem-perature, Text, is the contribution to the background (using eq. 1.6), forthe extended source count contribution only.Parameter Model 3 Model 3A(unconstrained) (constrained)log10[SJy ] log10[dN/dSsr−1 Jy−1] log10[dN/dSsr−1 Jy−1]−6.25 15.01± 1.26 16.97± 0.04−5.43 13.06± 0.74 13.77± 0.07−4.62 11.04± 0.62 10.67± 0.13−3.81 8.50± 0.76 7.73± 0.55−3.00 6.04± 0.92 7.05± 0.17Text (mK) 7.2+14.0−5.20 159.6+9.50−12.6σc (µJy beam−1) 62.73 78.12σc (mK) 2.79 3.47χ2 (Ndof = 39) 75.3 251.6118best-fit parameters that yielded background temperature estimates for the extendedemission in the range of (10± 7) mK.We tested whether an incorrect estimate of the instrumental noise of (52 ±5)µJy beam−1 could affect the results by re-fitting the models while allowing thenoise to vary between 40 and 70µJy beam−1. This has little effect, except in Model3, where the faintest node is degenerate with the noise. Thus a higher noise woulddecrease the amplitude of the faintest node. Nevertheless, we conclude that ournoise estimate cannot be far enough off to explain the excess P(D) width.5.7.3 ARCADE 2 FitsWe refitted the models to explore what source counts would be necessary to yieldbackground temperatures in the range predicted by ARCADE 2, and to assess howwell those source counts fit our data. It is clear from Fig. 5.7 that shifting the twopopulations with the ARCADE 2 prior (Model 1A) is strongly inconsistent withour data. With Model 2A or Model 3A it is possible to obtain source count tem-peratures in the ARCADE 2 range and find a reasonable fit to our data. Figure 5.9shows that in doing so, such a population would need to be extremely faint andnumerous. The typical flux density of the peak of the parabola is three orders ofmagnitude below our instrumental and confusion noise limits. That region of thesource count is nearly impossible to constrain with existing data. With Model 3A,the fitting routine makes the faintest node higher in amplitude, since changes tothe counts that far below the noise result in very little change in the predicted P(D)shape.The two models are also difficult to interpret in terms of physical objects. Sincethese extended, faint, numerous objects would completely overlap on the sky, mod-elling them as discrete objects fails. Future work will examine whether a faintdiffuse cosmic web structure could produce this emission.We conclude that there are no source count models, to a depth of 1µJy, thatare consistent with both our data and the ARCADE 2 background temperature.Scaling our best-fitting discrete and extended source-count temperature (70 mK)to the ARCADE 2 frequency of 3.3 GHz via a spectral index of −0.7 gives only13 mK, compared with the nearly 55 mK result from ARCADE 2.119Table 5.5: Luminosity and redshift estimates for Model 2.Speak S−50% S+50% zpk = 0.25 zpk = 0.5 zpk = 1 zpk = 2log[L1.4] ∆z log[L1.4] ∆z log[L1.4] ∆z log[L1.4] ∆z(µJy) (µJy) (µJy)2.8 0.7 11.0 19.8 0.39 20.6 0.84 21.5 1.83 22.5 4.08Chapter 4 ruled out a new discrete population peaking brighter than 50 nJy.Combining that with our constraint on an extended population peaking above 1µJyindicates that the ARCADE 2 result is highly unlikely to be due to extragalac-tic emission. Residual emission from subtraction of the Galactic component thusseems a more likely explanation for the excess seen by the ARCADE 2 experi-ment. The contribution from extragalactic sources from ARCADE 2 depends onthe model used for the contribution of the Galactic component. Subrahmanyan andCowsik [170] showed that using a more realistic model of the Galaxy, as opposedto the plane parallel slab used by ARCADE 2 [100], obviates the need for anyfurther contribution from the extragalactic sources beyond that predicted from thecounts.5.8 Extended Emission DiscussionWhen unconstrained by the ARCADE results, we find that Model 2 and Model 3 fitour data significantly better than Model 1 (an improved ∆χ2 per degree of freedomof around 34). Though the χ2s for Model 2 and Model 3 are still somewhat high,either model is a reasonable approximation to the data, at least when compared withthe unsubtracted discrete model on its own, which has a χ2 = 335 for 44 degreesof freedom. We now consider potential astrophysical sources of this emission.5.8.1 Sources of Diffuse EmissionModel 1, consisting of only shifts in the discrete counts, is considered here as anapproximate representation of individual galaxy haloes. The best-fit results fromthis model should be considered only as upper limits for galactic haloes. Thismodel on its own does not optimally fit the data. If there are other sources con-tributing to the measured P(D), the fitting process would push the shifts artificiallyhigh in an attempt to make the model as consistent with the data as possible. Also,120this model falls apart when considering nearby galaxies that have been observedwith single dish telescopes. According to the model, a 1 Jy nearby galaxy wouldhave an extended halo of 250 mJy. This type of emission has not been seen aroundsuch sources. This implies that if all galaxies have some form of diffuse halo thenthe flux density of that halo cannot simply be a fraction of the discrete flux den-sity. Models using the luminosity functions of the separate populations, where thehalos may be a fraction of the point source luminosity, and or may have differentevolution with redshift, would likely produce more consistent results.For any of the models, in order to be consistent with known constraints onthe cosmic infrared background (CIB) the emission process(es) must not be linkeddirectly to star formation rates. Moreover, as noted in Section 5.4.2, this techniquecan only constrain sources that are roughly 2 arcmin or smaller. Thus, these modelsare valid only for objects in that size range.Additionally, since we assume the sources in these models are powered bysynchrotron emission, we must also consider the associated X-ray emission andhow that compares with the known cosmic X-ray background (CXB). The electronsthat generate the synchrotron emission can inverse-Compton (IC) scatter off ofCMB photons to generate X-ray emission (e.g. [9, 50, 83]), the brightness of whichwe can estimate as follows.The synchrotron and IC power are related by,LICLsync=U0(1 + z)4UB(5.4)[see e.g. 125]. Here Lsync and LIC are the synchrotron and IC luminosities, UB isthe magnetic field energy density, and U0 = 4.2× 10−14 J m−3 is the CMB energydensity at z = 0. Using the simplification that all of the sources are at the sameredshift, we can calculate the IC flux density for a range of redshift and magneticfield values. We used redshifts of 0.01 ≤ z ≤ 4 and magnetic field values inthe range of 0.01µG ≤ B ≤ 10µG [based on B values for nearby clusters beingaround 1µG, 53].When integrating the new source count (dN /dSIC) we obtain a range of valuesof the CXB for energies of 100s of keV. Churazov et al. [28] presented observationsof the CXB spectrum measured by INTEGRAL of E2dN/dE ≤ 15 keV2 s−1121cm−2 keV−1 sr−1 for E ≥ 100 keV. In these units our models yield values of10−13 to 5 for E2dN/dE, depending on the assumed redshift and magnetic fieldstrength (larger values for lower B and higher z). This shows that such modelsshould not have a large impact on the X-ray background.5.8.2 Cluster EmissionIf we are to assume that Model 2 (middle right panel of Fig. 5.9) represents as-trophysical sources, we need to determine how they compare to known objects.Making some simple assumptions, we can calculate possible luminosities and red-shifts. We chose several redshifts for the peak of the parabola and calculated theK-corrected 1.4 -GHz luminosity, assuming a spectral index of α = −1.1. Then,assuming the objects all have the same intrinsic luminosity, we calculated the red-shifts at which the counts have fallen to 50 per-cent of the peak. We did this forpeak redshifts of z = 0.25, 0.5, 1, and 2; the results are listed in Table 5.5.It seems unlikely that this population could represent cluster emission fromradio haloes or relics. The luminosity values for such objects, given in Feretti et al.[53], are in the range of 23 ≤ log10[L1.4] ≤ 26, several orders of magnitude largerthan seen here. To date, we know of less than 100 clusters that host giant or miniradio haloes [53]. Extended radio emission in clusters has only been observed inhigh mass clusters (≥ 1014M) at low redshift, and all with total 1.4 GHz fluxdensities in the 10s to 100s of mJy.There could of course be similar objects (relics, haloes, etc.) in smaller massgroups at higher redshifts that are contributing. Nurmi et al. [128], using data fromthe SDSS survey, found that the majority of galaxies reside in intermediate massgroups, as opposed to large clusters. Stacking of subsamples of luminous X-rayclusters by Brown et al. [19] found a signal of diffuse radio emission below theradio upper limits on individual clusters. It is possible that there are clusters orgroups that are more ‘radio quiet’, below current detection thresholds [e.g. 21, 24].Zandanel et al. [198] used a cosmological mock galaxy cluster catalogue, builtfrom the MultiDark simulation [197], to investigate radio loud and radio quiethalo populations. Their model, which assumes 10 per cent of clusters to have ra-dio loud haloes, is a good fit [see figure 5 of 198] to the observed radio cluster122data from the NVSS survey [72]. The luminosity limit for the observed NVSSdata is log10[L1.4 (W Hz−1)] ' 23.5, while the simulation continues to a limit oflog10[L1.4 (W Hz−1)] ' 20.It is instructive to compare these simulated haloes with our ATCA data. Us-ing the online database to access the simulation [149]2, we used the 1.4-GHz halosimulation snapshots for z = 0, 0.1, 0.5, and 1, scaling the luminosities for eachredshift snapshot to give the flux density that would be observed at 1.75 GHz. Wecomputed a source count from these data, combined it with the unsubtracted dis-crete emission model and Gaussian noise to obtain a predicted P(D). The sourcecount and P(D) are shown in Fig. 5.10. The source count from this model onlyadds an additional 1.5 mK to the radio background temperature.The fit to the image P(D) is not unreasonable, with this model adding onlya modest excess width to the distribution compared with the unsubtracted discretemodel on its own. The source count would likely not decrease as significantly in thesub-mJy region if the simulation included data from redshifts higher than z = 1,and this would likely improve the fit. The χ2 is high mainly due to this modelhaving a slightly higher number of bright objects and thus over-predicting the tailof the distribution. However, some of these brighter haloes would be relativelynearby, hence larger on the sky and so potentially resolved out in our data (seeTable 5.1).The halo model has similar count amplitude to our best-fit for Model 3 around1 mJy. This halo model then begins to fall off, whereas the node model rises;this again could be due to the lack of high redshift objects. This type of halomodel is therefore not necessarily inconsistent with our phenomenological model.Assuming the model from Zandanel et al. [198] is a realistic extension of radiohaloes to fainter luminosities, then it is possible for such haloes to exist givenour data. However, more deep observations of clusters are necessary to test theaccuracy of this model.One thing to keep in mind is that the model from Zandanel et al. [198] dealswith the issue of the origin of radio haloes, i.e. haloes being generated from re-acceleration or hadronic-induced emission. This model assumes a fraction of the2http://www.cosmosim.org12310-1 100 101 102 103 104S1.75 GHz (µ Jy)10-210-1100101102S2 dN/dS (sr−1Jy)Discrete countsZandanel et al. 2014 Halo counts500 0 500 1000D (µ Jy beam−1 )10-610-510-410-3P(D) (µJybeam−1 )−1Zandanel et al. 2014 Halo countsMosaic P(D)20 10 0 10 20 30 40 50DT (mK)10-410-310-2P(D) (mK)−1Figure 5.10: Comparison of the radio cluster halo model from Zandanel et al.[198] with current data. The top panel shows the S2 normalized sourcecount derived from taking the halo radio luminosity values at redshiftsnapshots z = 0, 0.1, 0.5, and 1, and converting to 1.75 GHz (bluedashed line), compared with the discrete radio source count (blacksolid line). The bottom panel shows the output P(D) for the halo modelplus the unsubtracted point source contribution, convolved with Gaus-sian noise of 52µJy beam−1 (black points).124observed radio emission is of hadronic origin. However, if the hadronic contribu-tion is negligible, acting only at radio-quiet levels, the predicted counts would bedramatically lower at all masses.5.8.3 Dark Matter ConstraintsIt has been proposed that radio emission may originate from WIMP dark matterparticles. Dark-matter particle annihilation in haloes releases energy as chargedparticles, which emit synchrotron radiation due to the magnetic field of the sur-rounding galaxy or galaxies. The predicted emission depends on the mass of thedark matter particle and halo mass or density profile, as well as the strength of themagnetic field.Fornengo et al. [63] presented one dark matter model with two source-countpredictions, the first assuming all the halo substructures are resolved and the sec-ond assuming all the substructures are unresolved. The predicted source counts,shifted to 1.75 GHz, is shown in the top panel of Fig. 5.11 along with the discreteradio source count. Their best-fit model has a dark matter mass of 10 GeV, assum-ing annihilation or decay into leptons. We computed the predicted P(D) for bothmodels, plus the unsubtracted discrete source contribution convolved with Gaus-sian noise of 52µJy beam−1. The model P(D)s are shown in the bottom panel ofFig. 5.11 along with our radio image P(D).Clearly these particular models are not consistent with our current radio data.Any other dark matter models would need reduced amplitude of the counts forflux densities greater than about 10µJy. Models with the dark matter count am-plitude as high as or higher than that from known radio sources for these brighterflux densities would overproduce the emission seen and are therefore ruled out.Dark matter models consistent with our data and responsible for the ARCADE 2emission would need to produce a large portion of the emission from the sub-µJyregion, a region not constrained by our data. However, the required number countswould render such predictions unrelated to galactic haloes (i.e. the number sourcesthat faint required to account for that amount of emission would be too very largecompared with the number if individual galaxy count models/predictions).12510-1 100 101 102 103 104S1.75 GHz (µ Jy)101102103S2 dN/dS (sr−1Jy)Discrete countsDark Matter resolvedDark Matter unresolved200 0 200 400 600 800D (µ Jy beam−1 )10-410-3P(D) (µJybeam−1 )−1Dark Matter resolvedDark Matter unresolvedMosaic P(D)10 0 10 20 30 40DT (mK)10-2P(D) (mK)−1Figure 5.11: Comparison of one particular dark matter model with current ra-dio data. The top panel shows the two predicted source count models[see figure 3 in 63],shifted to 1.75 GHz, for a 10 GeV dark matter par-ticle mass, assuming all the structures are resolved (blue dashed line)and unresolved (red dot-dashed line), together with the discrete radiosource count (black solid line), with S2 normalization. The bottompanel shows the output P(D)s for the two models plus the unsubtractedpoint source contribution, convolved with Gaussian noise of 52µJybeam−1 (black points).1265.9 Integral CountsNow that we have closed the loophole of extended emission, we can revisit sourcecount constraints in general. It is important for future deep survey designs to havean accurate estimate of the expected number of source detections. To estimate thiswe can derive the integral source counts N(> S), or the total number of sourceswith flux density greater than S per unit area. Deep and accurate estimates ofN(> S) can provide useful information for surveys at a range of frequencies,with proper scaling; in the synchrotron-dominated regime we should be able toextrapolate by a factor of a ∼ ±2 in frequency.This is of particular relevance to the SKA, and its pathfinders, Australian SquareKilometre Array Pathfinder (ASKAP) and MeerKAT, as well as the new planneddeep VLA survey. The VLA Sky Survey (VLASS) is aiming to map an area10 deg2 to a depth of 1.5µJy at 1.4 GHz with a resolution of roughly 1 arcsec[97]. The Evolutionary Map of the Universe survey (EMU) continuum survey[127] planned for ASKAP will cover the entire sky south of Dec +30◦ with a res-olution of 10 arcsec at 1.4 GHz, and will also be sensitive to diffuse emission witha sensitivity at 1 arcmin scale similar to that reached in this paper. The deep surveywith MeerKAT [MIGHTEE, 96], will reach an rms of 1µJy over 35 deg2 with arc-sec resolution. In the following decade, the SKA will conduct an all-sky survey toan rms of 1µJy, and a smaller survey to an rms of 100 nJy. It would be helpful inplanning to know what source densities are expected in these surveys.We can obtain the integral source counts fromN(> S) =∫ ∞SdNdSdS. (5.5)We have derived the integrated source counts from the discrete model in Chapter 4,as well as that discrete model plus the best-fits from the extended emission models.These are shown in Fig. 5.12, with values listed in Table 5.6. Also shown on theplot are the expected SKA and SKA Pathfinder survey limits.The SKA and Pathfinders should not be limited by any natural source confusionfor discrete sources. The natural confusion limit is the confusion caused by thefinite source sizes, as opposed to confusion caused by the telescope beam size.12710-1 100 101 102 103 104S1.4 GHz (µJy)101102103104105 N(>S) (deg−2 )V14 Discrete CountsV14 Disc+Model 1V14 Disc+Model 2V14 Disc+Model 310-1 100 101S1.4 GHz (µJy)104105SKA Deep SurveyMeerKat MIGHTEE/SKA WideASKAP EMU10-1 100 101 102 103 104S1.75 GHz (µJy)101102103104105106107 N(>S) (deg−2 )V14 Disc+Model 1AV14 Disc+Model 2AV14 Disc+Model 3AFigure 5.12: Integrated source counts at 1.4 GHz (top) and 1.75 GHz (bot-tom). The solid black lines are the discrete source count (DS) fromChapter 4, scaled from 3 GHz using α = −0.7. The green dottedlines are DS + Model 1, the red dot-dashed lines are DS + Model 2,and the blue dashed lines are DS+ Model 3. The shaded grey areasrepresent 68 per cent confidence regions of the discrete count derivedfrom Chapter 4. The upper right-hand panel shows a close up of theregion marked by the solid rectangle in the upper left panel. The threepoints show the expected number of sources per square degree for theupcoming SKA and SKA Pathfinder surveys based on their expecteddepths (the circle is SKA, the square is MIGHTEE and the all skySKA, and the star is the EMU survey). The bottom panel (1.75 GHz)shows Model 1A, Model 2A, and Model 3A as orange dotted, ma-genta dot-dashed, and cyan dashed lines, respectively (models fit withthe ARCADE 2 prior). 128Table 5.6: Integrated source count values of the different models scaled to1.4 GHz.log10[SJy ] Discrete Dis+Mod1 Dis+Mod 2 Dis+Mod 3(No. deg−2) (No. deg−2) (No. deg−2) No. deg−2)−7.0 2.4× 105 2.8× 105 4.9× 105 3.6× 105−6.5 1.5× 105 1.7× 105 3.5× 105 2.7× 105−6.0 8.8× 104 9.8× 104 2.1× 105 1.7× 105−5.5 4.7× 104 5.1× 104 8.3× 104 6.3× 104−5.0 1.8× 104 1.9× 104 2.3× 104 2.1× 104−4.5 5.4× 103 5.6× 103 5.7× 103 5.8× 103−4.0 1.2× 103 1.2× 103 1.2× 103 1.2× 103−3.5 2.9× 102 2.9× 102 2.9× 102 2.9× 102For discrete sources with an average source size of approximately 1 arcsec2 forfaint sources, the natural confusion limit would be less than 10 nJy. However,extended objects of 2 arcmin diameter, for example, would begin to heavily overlapabove 1000 sources per deg2 which corresponds to a flux density at 1.4 GHz ofapproximately 100µJy.To highlight some numbers (ignoring extended emission now) the discretemodel predicts 1 × 109 sources over the whole sky brighter than 23µJy, and10 sources per arcmin2 brighter than 4.6µJy. At a limit of 1µJy we estimate88,500 sources per square degree at 1.4 GHz. For relatively modest extrapolationsin flux density and frequency, the cumulative counts for 0.1 ≤ S ≤ 5µJy can bewell described byN(> S) ' 84, 800(S1µJy)−0.48 ( ν1.4GHz)−0.33deg−2, (5.6)and for 5 < S ≤ 500µJyN(> S) ' 296, 700(S1µJy)−1.20 ( ν1.4GHz)−0.33deg−2. (5.7)1295.10 ConclusionsOur ATCA image is the deepest available with a mean frequency of 1.75 GHz and aFWHM resolution of 150 arcsec× 60 arcsec. The image is confusion-limited withan rms of (155±5)µJy beam−1 = (6.9±0.2) mK and average instrumental noiseσn = (52± 5)µJy beam−1 = (2.3± 0.2) mK. Using this data we were able to testtechniques and constrain models.The novel techniques are the following.1. We have used wide-band high resolution data to perform source subtractionin our low resolution data while accounting for frequency change and un-subtracted sources.2. We tested simulations at both low and high angular resolution in order todetermine the how large scale emission appears on small scales.3. With this data with have performed a P(D) analysis and bootstrap tests toconfirm a excess emission over that from point sources at the 3σ level.4. We have used Markov chain Monte Carlo sampling with P(D) analysis totest a variety of source count models for this excess extended emission.Using these techniques we have drawn the following conclusions.1. The P(D) approach is a viable test for data of this larger resolution as longas sources are still not much larger than the beam size.2. It is possible for there to be a large amount of emission from extendedsources that would be missed, or resolved out, with higher resolution im-ages.3. Our extended source count models rule out emission or populations thatcould account for the ARCADE 2 emission down to the 1µJy level.4. Cluster emission and dark matter annihilation emission source count modelcan be constrained by this technique, with the tightest constraints for fluxdensities ≥ 0.5 mJy.1305. Faint large scale emission may only be detectable by such techniques as theconfusion level at large angular scales is too great to allow for imaging ofindividual sources.Here we presented a brief summary of the conclusions from this chapter. For adiscussion of these results in a broader context see Chapter 8.131Chapter 6Discrete-Source CatalogueThere is far more to be gleamed from a radio survey than just the source count.In order to fully explore the sources and their properties a catalogue is necessary.Catalogues yield information on source sizes, spectral dependence, and when com-bined with catalogues from surveys at other wavebands, information on galaxypopulation types, redshifts, the star formation rates, evolution and more.We decided to catalogue both the C-configuration data and the CB-configurationdata separately (with the C data being the data from just the VLA C antenna con-figuration and CB being the data from the C-configuration and the B-configurationtogether, see Chapter 2.2). The CB data have a higher resolution, which shouldprovide more positional accuracy in the fitting. However, it also has higher noisefor the resolution. This is because, even though it is a combination of the C andB data, and thus should have lower noise given more data, the majority of the datais from the lower resolution and so to achieve higher resolution, weighting is ap-plied to the data before transforming to an image. This down-weights the lowerresolution and up-weights the high resolution samples where there are fewer data,essentially decreasing the total amount of data and increasing the image noise.We also want to catalogue both resolution data sets in order to see how theycompare for future surveys. The proposed VLASS survey [97] would have a res-olution closer to our CB data (or higher) whereas SKA Pathfinder surveys, suchas EMU [127], will have lower resolution similar to the C data. Cataloguing thetwo resolutions of the same field, calibrated and transformed to images in the same132way, should inform us about what kind of differences can be expected between thesurveys.This work has not yet been published and the catalogue presented here maynot be the published version. In Section 6.1 we describe the general process ofsource finding and fitting and how we used simulations to test for uncertainties aswell as other effects such as completeness. Section 6.2 details the catalogues fromthe actual VLA data, including a comparison between the two resolutions and ourinvestigation of extended sources. Section 6.3 presents the source counts fromthe two catalogue versions. Section 6.4 discusses how we can use the wide VLAbandwidth to look at the spectral dependence of the catalogue sources. Finally, inSection 6.5, we present preliminary cross-matches of our catalogues with othersfrom the optical, IR, and radio.6.1 SimulationsFor source fitting we used the OBIT task FNDSOU. The images are searched forpeaks down to the 3σ level. Each peak area is then fit with a 2D elliptical Gaussianwith the fit parameters being the peak flux density (SP), the centre RA and centreDec positions, the major axis FWHM (θM) and minor axis FWHM (θm) and theposition angle (φ). The fitting was constrained such that the major and minor axescould not be less than the image beam FWHM (θB). The total integrated fluxdensity (ST) is computed asST = SpeakθMθmθ2B. (6.1).To understand the fit uncertainties associated with our images, noise properties,and fitting software, we generated simulated images and run them through ourfitting procedure. We are then able to compare the fit output with the known input.We performed several sets of simulations.First we ran the simplest case of inserting point sources with set flux densitiesof 3, 4, 5, 8, 12, and 100 µJy. These were inserted at points chosen from a gridover the image, such that the distance between the sources was  θB. These133point sources were convolved with the beams from our C and CB images, thenadded to backgrounds of random Gaussian beam-convolved noise with σ = 1.05and 1.15µJy beam−1 for the two resolutions. This initial case was meant simply todetermine the uncertainties of fitting in the presence of correlated noise. We createdfour simulated images (each made at both the high and low resolutions), each imagehaving 400 sources in it. These images were then fit using FNDSOU. The outputof FNDSOU was cross-matched with the true positions to within a 5 arcsec radius.Then the data from all four cases were collated to look at the results.However, the real world case is more complicated. We next wanted to carry outtests with as realistic a setup as possible. We used the source catalogues from eightseparate 0.5◦ × 0.5◦ areas of the SKADS S3 simulation. We know already that thesource count from the simulation matches fairly closely to published source counts,and it includes some clustering as well as a fairly realistic source size distribution.We scaled the flux densities from 1.4 GHz to 3 GHz using α = −0.7. We thencut out any sources with S > 1 mJy to more closely approximate our field, wherethe brightest source is around 3 mJy. We also cut out AGN objects with extendedlobes or jets, because in real images the few such sources would not be fit withGaussians and we are solely concerned here with single component sources. Theimages were generated and the 3 GHz VLA primary beam was applied to eachsimulated image. Then the images were convolved with the CB and C beams tocreate high and low resolution versions of each. Finally, beam-convolved noisewas added to each image. These eight images were then fit using FNDSOU, withthe output of FNDSOU cross-matched with the known positions to within a 5 arcsecradius. Finally the data from all cases were collated.6.1.1 UncertaintiesIn Condon [32] the errors for 2D elliptical Gaussian fits are discussed. The squareof the overall signal-to-noise ratio p2 isp2 =pi8 ln 2θMθmS2Ph2µ2, (6.2)where h is the pixel size, and µ is the image noise. The errors on each of the fitparameters as related to p2 (from Condon [32]) are134101 102 103Fit SPeak (µJy beam−1 )0.60.70.80.91.01.11.2(True/Fit) S PeakCB Data C Data 101 102 103Fit SPeak (µJy beam−1 )0.30.40.50.60.70.80.91.01.1(True/Fit) S TotalCB Data C Data 101 102 103Fit SPeak (µJy beam−1 )0.50.60.70.80.91.01.1(True/Fit) Major FWHMCB Data C Data 101 102 103Fit SPeak (µJy beam−1 )0.800.850.900.951.001.051.101.151.20(True/Fit) Minor FWHMCB Data C Data 101 102 103Fit SPeak (µJy beam−1 )1.51.00.50.00.51.01.5∆ RA (arcsec)CB Data C Data 101 102 103Fit SPeak (µJy beam−1 )1.51.00.50.00.51.01.5∆ Dec (arcsec)CB Data C Data Figure 6.1: The mean of either the ratio of the true to fitted values or differ-ence from the realistic simulations in bins of peak flux density. The Cvalues (blue circles) were made with C data image resolution and CBvalues (red squares) made with CB data image resolution. The linesshow the 1σ uncertainties (red solid lines for CB data and blue dashedlines for C data). From left to right and top to bottom the panels areSpeak ratios, Stotal ratios, major axis size ratios, minor axis size ratios,∆RA, and ∆Dec.1352p2'µ2(SP)S2P=µ2(ST)S2T=8 ln 2µ2(x0)θ2M= 8 ln 2µ2(y0)θ2m=µ2(θM)θ2M=µ2(θm)θ2m=µ2(φ)2(θ2M − θ2mθMθm).(6.3)Here x0, and y0 are the centre coordinates.However, these equations do not factor in the presence of correlated noise. Thesimple simulation case should then test whether these equations hold up when thenoise has also been convolved with the beam. Taking the fitted and matched dataand binning by true flux density we computed the mean and standard deviation forthe fitted peak flux density, total flux density, positions, axis sizes, and positionangle, and observe that the resulting parameter uncertainties are consistent withthose predicted by eq. 6.3.We took all the sources from the realistic simulations that were found fromthe fitting routine and cross-matched that catalogue with the true, or input, sourcecatalogue. Using the matches, we computed the ratios of the true to fitted valuesfor the peaks, major axis size, minor axis size, total flux density, and the differencein RA and Dec positions and binned them in bins of peak flux density. Figure 6.1shows the mean ratios and standard deviations for these bins for both image resolu-tions. We can see from this that the error bars decrease as expected with peak fluxdensity. Also noticeable here is that the fitting overestimates the major axis sizeand peak flux density for faint flux density sources. This is a known complicationwith most fitting routines when dealing with low signal to noise sources.The goal of the realistic simulations was to use the results in order to estimateuncertainties on the fit results for the real data. To do this we took all the matchedsimulated sources and binned them by fitted peak flux density and fitted decon-volved major axis size, where the deconvolved major axis (θDM) and deconvolvedminor axis (θDm) were computed as θDM =√θ2M − θ2B and θDm =√θ2m − θ2B.1360.0 0.025 1.25 2.25 3.75 5.0 9.0 20.0Fitted θDM (arcsec)0.60.80.91.11.41.92.23.1log10[Fitted Speak] (µJy beam−1)0.600.650.700.750.800.850.900.951.00Mean (True θ DM)/(Fit θDM)Figure 6.2: Mean ratio of true deconvolved major axis size to fitted decon-volved major axis size of sources from the realistic simulation in binsof peak fitted flux density and fitted major axis FWHM. The white binsmean there are no sources within those size and peak ranges. The left-most bins are unresolved sources (θM = θB).1370.0 0.025 1.25 2.25 3.75 5.0 9.0 20.0Fitted θDM (arcsec)0.60.80.91.11.41.92.23.1log10[Fitted Speak] (µJy beam−1)0.000.020.040.060.080.100.120.140.16σ (True θDM)/(Fit θDM)Figure 6.3: Standard deviation of the ratio of true deconvolved major axissize to fitted deconvolved major axis size of sources from the realis-tic simulation in bins of peak fitted flux density and fitted major axisFWHM. The white bins mean there are no sources within those size andpeak ranges. The leftmost bins are unresolved sources (θM = θB).138101 102 103Fit SPeak (µJy beam−1 )0.60.70.80.91.01.11.21.3(True/Fit) SPeakCB Data CB Data corrected101 102 103Fit SPeak (µJy beam−1 )0.60.70.80.91.01.11.21.3(True/Fit) SPeakC Data C Data corrected101 102 103Fit SPeak (µJy beam−1 )0.50.60.70.80.91.01.11.21.3(True/Fit) Major FWHMCB Data CB Data corrected101 102 103Fit SPeak (µJy beam−1 )0.50.60.70.80.91.01.1(True/Fit) Major FWHMC Data C Data correctedFigure 6.4: Mean of the ratio of the true to fitted values for peaks (top row)and major axis size (bottom row) from the realistic simulations in bins ofpeak flux density. The C values were made with C data image resolution(left column) and CB values made with CB data image resolution (rightcoloumn). The corrected values are obtained after using the method of2D binning and interpolation to get correction factors and uncertaintiesfor individual sources.139We computed the mean and standard deviations of the ratios of the true values tothe fitted values, i.e. PeakTrue/PeakFit in two dimensional bins; an example ofthe mean ratios is shown in Fig. 6.2 and the standard deviation ratios in Fig. 6.3for major axis size. These show similar trends as fig. 6.1, however, the additionof binning by size as well shows that the effect of over estimation in a parameteris largest for those not just with the faintest peaks but also with the largest sizes.These figures show the 2D arrays for correction of the major axis size, however,corresponding arrays and corrections were computed for the fitted peaks and minoraxis size, as well as uncertainties for the RA and Dec.The bin setups were chosen to ensure a minimum of 20 sources in a bin (orwhen genuinely zero). We then interpolated the values of ratios and ratio standarddeviations for peak flux density, total flux density, major and minor axes decon-volved sizes, and mean and standard deviation for ∆RA and ∆Dec, for all of thesources fit. We tested this correction method on the simulated sources. Figure 6.4shows how the mean ratios and uncertainties change after interpolating new valuesfor the simulated fit values.This method should enable us to account for the effect of flux boosting in ourcatalogue. This well known effect means that, particularly for sources near thenoise limit, the fitted fluxes densities are more likely to be overestimated thanunderestimated, due to both instrumental and confusion noise. There are manypapers that discuss correcting for flux boosting using Bayesian methods [e.g. 25].However, many of those deal with submm or infrared data, where θB  θS andtherefore are only fitting for a total flux density rather than a peak together withsize and shape. Submm and infrared images are also not affected by the primarybeam, which complicates the Bayesian method used to compute probabilities di-rectly from source counts. Therefore, considering our more complicated case, wechose to use the realistic simulations and this 2D interpolation to account for thiseffect by computing the corrections for the both size axes and the peak flux densi-ties.1406.1.2 Completeness, False Detections, and BlendingThe simulation catalogues can also be used to examine how well the fitting softwaredoes at finding all of the known sources, how often it includes a false detection, andhow often if mistakes multiple sources for one source.For each simulation realisation there are approximately 1000 sources withinthe primary beam area that are “detectable” by the limits set. Among those 1000for the low resolution images, there are on the order of 50 to 100 sources that are fitas one source when there are in fact multiple (fainter) sources in or around the fittedsource position. In these “blended” cases the fitting software tends to fit large (12to 20 arcsec convolved) sizes for one or both of the axes and the total flux densityfor the one source is on average larger than the sum of the totals of the individualsit encompasses. This will have an impact on the source count of a catalogue, obvi-ously lowering the count of fainter sources and increasing the count of the brightersources. One possible way to improve this, or at least investigate it, would be to fitthe same image multiple times, starting with the sizes being fixed to the beam sizeand then increasing the maximum allowed axis sizes. Comparing the results andresiduals from this could help shed light on whether there is some fitting constraintprocess that would minimize blending or else some some limits on the parameterspace to identify blended objects. This issue is under active investigation.The issues of completeness and false detections are somewhat complicated bywhether one is interested in uncorrected or primary beam-corrected peak or totalfluxes and whether we look at the fitted values or the true values of the matchedsources. Examples of this for completeness are shown in Fig. 6.5. These plots showthe number of sources, in bins of flux density, that were found by the fitting routineand had a known match within 5 arcsec, divided by the total number of knowninserted objects in that flux density bin. This shows us that the completeness levelsare lower if we consider the total flux density values rather than the peak fluxdensities. Also, the fitted flux densities values, both peaks and totals, seem to over-estimate the number of bright sources. The figure also shows the difference thatthe primary beam and source size makes. For example the CB image for totalflux densities at the faintest fluxes, when considering those that could have beendetected (i.e. peaks > 4σ, bottom middle panel) is roughly 85%, whereas it drops14165707580859095100105Completeness %Speak No PB correctionCB True match/TrueC True match/TrueCB Fit match/TrueC Fit match/TrueSpeak PB corrected Speak>4σCB True match/TrueC True match/TrueCB Fit match/TrueC Fit match/TrueSpeak PB corrected all SCB True match/TrueC True match/TrueCB Fit match/TrueC Fit match/True101 102 103S (µJy)60708090100Completeness %Stotal No PB correctionCB True match/TrueC True match/TrueCB Fit match/TrueC Fit match/True101 102 103S (µJy)Stotal PB corrected Speak>4σCB True match/TrueC True match/TrueCB Fit match/TrueC Fit match/True101 102 103S (µJy)Stotal PB corrected all SCB True match/TrueC True match/TrueCB Fit match/TrueC Fit match/TrueFigure 6.5: Completeness of sources from simulation. The plots show thenumber of sources binned by flux density of the sources with matchesdivided by the true total number of inserted sources. The top row showsthe sources binned by peak flux density while the bottom row showsbinned by total flux density. The left most plots show the sources binnedby flux density uncorrected for the primary beam. The middle columnis primary beam corrected flux densities but only those sources whoseuncorrected peaks are> 4σ, or those that could have been detected. Theright column is the primary beam corrected flux density for all sources.The red solid and green dashed lines are for the CB images true matchedvalues and fitted matched values respectively. The blue dot-dashed lineand magenta dotted lines are for the C images true matched values andfitted matched values.142020406080100False Detections % Speak No PB correctionCB match/allC match/allSpeak PB corrected Speak>4σCB match/allC match/allSpeak PB corrected all SCB match/allC match/all101 102 103S (µJy)020406080100False Detections % Stotal No PB correctionCB match/allC match/all101 102 103S (µJy)Stotal PB corrected Speak>4σCB match/allC match/all101 102 103S (µJy)Stotal PB corrected all SCB match/allC match/allFigure 6.6: Percentage of false detections of sources from simulations. Theplots show one minus the ratio of matched to all fit sources binned byflux density. The top row shows the sources binned by peak flux densitywhile the bottom row shows them binned by total flux density. Theleftmost plots show the sources binned by flux density uncorrected forthe primary beam. The middle column is primary beam-corrected fluxdensities, but only those sources whose uncorrected peaks are > 4σ,or those that could have been detected. The right column shows theprimary beam-corrected flux density for all sources. The green solidlines are for the CB images matched values, while the magenta dottedlines are for the C images matched values.143closer to 65% when considering all of the known sources whose total flux densityis above 4σ (including ones whose peak is below 4σ, bottom right panel).Examples of the false detection rate in the simulations are shown in Fig. 6.6.The plots show (1 minus) the number of sources binned by flux density for thesources with matches divided by the total number of sources detected. Again wecan see how the false detection rate drops quickly to zero when considering non-primary beam-corrected flux densities, but levels off much more when looking atthe corrected fluxes. There is an overall lower number of false detections with theC images, likely due to its lower noise.When considering the completeness and false detection rate together thereshould be some flux density level which balances the number of possible falsedetections with a high completeness percent that can be used to set some kind ofdetection threshold. These constraints are usually dependent on the type of sci-ence to be carried out with the catalogue. Such results from these simulations forcorrections to our catalogues is still under active investigation.6.2 CatalogueThe catalogues were made in the same manner as described for the simulations.We fit the C and CB images separately before any primary beam corrections. Anysources with peaks < 4σ were removed, with the CB data having σ = 1.15µJybeam−1 and the C data having σ = 1.02µJy beam−1. Fitted Gaussians for sourceswith structure were also cut from the initial catalogue and were treated separately(see below). The total flux densities were computed according to eq. 6.1. The fluxdensities are corrected for the primary beam based on the primary beam value atthe fitted location. Full versions of the catalogues are given in Appendix A and B.The method of 2D interpolation from simulated data (described in Sec. 6.1.1)was used for each catalogue and yielded uncertainty estimates for each source pa-rameter. Residual images were made using the original fitted parameter values andthe corrected parameter values. Where the χ2 inside a box with size 2θB × 2θBwas smaller with the corrected values, those values were adopted as the new valuesin the catalogue. Spectral index estimates for each source are included, with themethod described further in Sec. 6.4. The flux density distributions for each cata-144100 101 102 103 104Stotal (µJy)020406080100120140Number of SourcesCB DataC Data0 5 10 15 20 25 30 35Stotal (µJy)020406080100120140160Number of SourcesC DataCB DataFigure 6.7: Histograms of total flux densities from both the CB and C cat-alogues. The top panel show the full flux density distributions for theCB data (red solid line) and C data (blue dashed line). The bottom panelshows the distribution for just the fainter flux densities, with the CB databeing red and the C data being blue.145logue are shown in Fig. 6.7, which shows that the catalogues are roughly completedown to 10 to 20µJy.We cross-matched the two resolution catalogues with each other, using a match-ing radius of 5 arcsec. Roughly 80 per-cent of the sources in the CB catalogue werefound in the C catalogue, with about 60 percent of the C sources matched. One rea-son for only 80 per-cent of the CB sources being matched (besides possible falsedetections or faint sources near the detection limit), even though the C data hashigher sensitivity, is source blending in the C catalogue. There are a number ofsources were two (or possibly more) individual sources were fit in the CB imagewhere only one blended source was fit in the C image. Thus during the cross-matching only the CB source closest to the C blended source was included. Thenon-blended sources without a counterpart (in either catalogue) are all near thenoise threshold. This means they are either false detections or, due to the localnoise distribution, the source is above the noise threshold in one image and belowin the other. For the final version of the catalogue a source-by-source comparisonwill be necessary to accurately match the two resolution catalogues.The C catalogue has about 1.3 times as many sources, mainly due to the lowernoise threshold. Figure 6.8 shows a comparison of some of the parameters for thematched sources. The top left panel of this figure shows the total fluxes from eachsource from each catalogue. There is little scatter from a one-to-one relation forsources with S ≥ 100µJy. As well for these brighter sources the separation incentre positions is small (bottom left panel). There is certainly much more scatterbetween the two catalogues for fainter sources, which is not surprising. There isvery little correlation in the major axis sizes (top right panel), which could be dueto the poorer size sensitivity of the C data, incorrect matches, or source blendingin the C data.In some cases it is necessary to correct for the effects of time and bandwidthsmearing. Bandwidth smearing is an effect of the limited spectral resolution andwill radially broaden the synthesized beam by convolving it with a rectangle ofangular width ∆θ∆ν/ν, where ∆θ is the radial offset from the pointing centre.With our VLA data, even after the spectral averaging that was done, at the FWHMof the primary beam the bandwidth smearing is roughly 1.0 arcsec at 3 GHz, andhence is not a large concern for us. Similarly, time averaging can also produce146a radial smearing. The amount of smearing due to the time resolution goes as∆θ∆t/f , where f is the Earth’s sidereal rotation period of 86164 s/2pi = 1.37 ×104 s. With our VLA data, after the time averaging for imaging, the maximumsmearing from this effect at is only ∼ 0.4 arcsec at the FWHM of the primarybeam.6.2.1 Angular Size DistributionThere are several conventions when it comes to dealing with source sizes. If itis clear that a source has extended structure and is not well fit by a Gaussian weconsider it to be extended. If both fitted axes have their lower limits larger thanthe beam these objects are fully resolved. If θM = θm = θB then these objectsare unresolved. If for the major or minor axis θM,m −∆θM,m < θB < θM,m thenthe axis is partially resolved. For some previous catalogues the procedure has beenthat when an axis is partially resolved than it is just set to the beam size [e.g. 118].We have chosen not to do this when we report sizes and use sizes to calculate totalflux densities. Instead we leave the axes at the fitted values and report a size flagto indicate the status. These flags are: (0) extended; (1) resolved; (2) major axisresolved minor partially resolved; (3) major axis resolved minor axis unresolved;(4) both major and minor axes partially resolved; (5) major axis partially resolvedand minor axis unresolved; and (6) both axes unresolved.The issue of the source angular size distribution as a function of flux densityis very important (e.g. for P(D) studies and source counts) yet the distribution isstill not well known. It is not known how the size distribution changes with fluxdensity at faint flux densities, and how if it varies by galaxy type, at least not withgreat accuracy. The source size distribution has an effect on the source count aswell, and it is believed that differences in corrections regarding source sizes arelikely to blame for the large amount of scatter in the counts from different surveysin the sub-mJy region. A deep survey with high resolution will miss larger sourcesnear the survey limit, and this has sometimes been ignored, but at other times over-corrected.The C image resolution is not ideal for investigating this, since the larger beamdecreases the accuracy when measuring deconvolved source sizes. Because of147100 101 102 103 104Total SC (µJy)100101102103104Total SCB (µJy)0 2 4 6 8 10 12 14Major axis FWHM C (arcsec)0246810Major axis FWHM CB (arcsec)3 2 1 0 1 2 3αIM C3210123α IM CB3 2 1 0 1 2 3αIF C2.52.01.51.00.50.00.51.01.52.0α IF CB100 101 102 103 104Peak SCB (µJy)012345Separation (arcsec)4 2 0 2 4∆ RA (arcsec)42024∆ Dec (arcsec)Figure 6.8: Comparison of matched sources from the C and CB catalogues.The top left panel is the total flux density of the sources from each cat-alogue. The top right panel is the deconvolved major axis sizes for thematched sources. The middle panels show the spectral indices for αIM(left) and αIF (right). The bottom left panel is the matched separationvs. the fitted CB peak flux density. The bottom right panel is the differ-ence in RA vs the difference in Dec.1480 2 4 6 8 10 12θDM (arcsec),4.0<=Stotal/µJy <7.60.00.10.20.30.40.50.60.7Probability (arcsec−1)CB Corrected (No. of sources=9)CB All (No. of sources=25)CB Corrected (No. unresolved=39)CB All (No. unresolved=8)0 2 4 6 8 10 12θDM (arcsec),7.6<=Stotal/µJy <14.5CB Corrected (No. of sources=103)CB All (No. of sources=143)CB Corrected (No. unresolved=66)CB All (No. unresolved=19)0 2 4 6 8 10 12θDM (arcsec),14.5<=Stotal/µJy <27.60.00.10.20.30.40.5Probability (arcsec−1)CB Corrected (No. of sources=161)CB All (No. of sources=195)CB Corrected (No. unresolved=56)CB All (No. unresolved=13)0 2 4 6 8 10 12θDM (arcsec),27.6<=Stotal/µJy <52.5CB Corrected (No. of sources=114)CB All (No. of sources=133)CB Corrected (No. unresolved=37)CB All (No. unresolved=12)0 2 4 6 8 10 12θDM (arcsec),52.5<=Stotal/µJy <100.00.00.10.20.30.40.5Probability (arcsec−1)CB Corrected (No. of sources=52)CB All (No. of sources=60)CB Corrected (No. unresolved=18)CB All (No. unresolved=8)0 2 4 6 8 10 12θDM (arcsec),100.0<=Stotal/µJy <900.0CB Corrected (No. of sources=34)CB All (No. of sources=41)CB Corrected (No. unresolved=14)CB All (No. unresolved=6)Figure 6.9: Deconvolved major axis size distributions from the CB cataloguein bins of total flux density. The sizes have been “corrected” if thepartially resolved sources are set to the beam size.1490 2 4 6 8 10 12θDM (arcsec),4.0<=Stotal/µJy <7.60.00.10.20.30.40.50.60.7Probability (arcsec−1)C Corrected (No. of sources=14)C All (No. of sources=115)C Corrected (No. unresolved=115)C All (No. unresolved=9)0 2 4 6 8 10 12θDM (arcsec),7.6<=Stotal/µJy <14.5C Corrected (No. of sources=84)C All (No. of sources=241)C Corrected (No. unresolved=192)C All (No. unresolved=33)0 2 4 6 8 10 12θDM (arcsec),14.5<=Stotal/µJy <27.60.00.10.20.30.40.5Probability (arcsec−1)C Corrected (No. of sources=104)C All (No. of sources=196)C Corrected (No. unresolved=131)C All (No. unresolved=42)0 2 4 6 8 10 12θDM (arcsec),27.6<=Stotal/µJy <52.5C Corrected (No. of sources=79)C All (No. of sources=120)C Corrected (No. unresolved=82)C All (No. unresolved=39)0 2 4 6 8 10 12θDM (arcsec),52.5<=Stotal/µJy <100.00.00.10.20.30.40.5Probability (arcsec−1)C Corrected (No. of sources=40)C All (No. of sources=52)C Corrected (No. unresolved=28)C All (No. unresolved=16)0 2 4 6 8 10 12θDM (arcsec),100.0<=Stotal/µJy <900.0C Corrected (No. of sources=25)C All (No. of sources=26)C Corrected (No. unresolved=21)C All (No. unresolved=20)Figure 6.10: Deconvolved major axis size distributions from the C cataloguein bins of total flux density. The sizes have been “corrected” if thepartially resolved sources have been set to the beam size.150this we expect the C catalogue to be missing smaller sources at the faintest fluxdensities. However, the CB catalogue will likely be missing larger sources at thefaintest flux densities, since their sizes will make it such that their peaks are belowdetection. This is indeed what we see, as is shown in Figs. 6.9 and 6.10. Theseplots show the size distributions in bins of total flux density for the two catalogues.The values labelled “Corrected” have the sizes set to the beam size if the axis ispartially resolved. There is a lack of large sources (θDM > 8 arcsec for the CB dataregardless of flux density. This may mean there are no sources of this size in ourdata; however, it is more likely that these sources are missed in the CB data as thelarge size makes the peak flux below detection. The CB plot also shows there tobe a peak in the size distributions around θDM ' 2 arcsec in all of the flux densitybins. In the C data distributions, however, the distributions are much more evenlyspread out with no discernible (common) peaks.Figure 6.11 shows the mean sizes from these distributions in bins of flux den-sity. It appears from this plot that the mean source sizes are much larger thanthe ∼ 0.7′′, estimated from high-resolution imaging (Muxlow et al. [124]). How-ever, the means shown here do not take into account sources that are unresolved orpartially resolved, which, particularly at the 8′′ resolution, will be the majority ofsources. Many of the sizes 1′′ or less cannot be measured with the fit uncertainties.This plot indicates a trend in the average source size as a function of flux densityregardless of catalogue resolution or corrections. But it is unclear how this mightbe biased by incompleteness, which is something that is still under investigation.6.2.2 Extended SourcesThere are four sources that by our examination (as well as in the catalogue fromOwen and Morrison [134]) are not well fit by Gaussians and are categorized asextended, or having extended structure. For each of these four sources the totalflux densities were found by summing the flux density values inside lines guidedby the contours but edited to not include nearby sources. The reported positions arefound by cross-matching with optical and infrared catalogues as well as the Owenand Morrison [134] catalogue. Figures 6.12, 6.13, 6.14, and 6.15 show these foursources for the C image contours against the CB image, along with the spectral151100 101 102 103Stotal (µJy)123456<θDM> (arcsec)C data AllC data CorrectedCB data AllCB data CorrectedFigure 6.11: Mean deconvolved major axis size in bins of total flux density.The sizes have been “corrected” if the partially resolved sources havebeen set to the beam size. The blue solid line and purple dashed lineare for the C data with original size values and corrected size values,respectively. The red dot-dahed line and green dotted line are the CBdata with original size values and corrected size values.152index image, optical image from the Canada France Hawaii Telescope (CFHT),and infrared Spitzer IRAC image.The source in Fig. 6.12 is a known Quasar classified as a double-lobed FR typeII object [58]. It also has associated X-ray as seen with Chandra (X-ray ID CXOXJ104623.9+590522). The spectral index image in Fig. 6.12 shows that the core hasa flatter spectrum (α ' −0.6), while the lobes have steeper spectrums (α ' −1.3),as is commonly seen with AGN lobes. The other three extended objects did notcross match with any known AGN or QSO objects in catalogues.It is clear looking at the optical and IR images that there could be emissionfrom multiple sources contributing in the radio images. As stated above the sourcein Fig. 6.12 has a clear optical match near the centre that is responsible for themajority of the radio emission seen in the centre and the two lobes. The sourcein Fig. 6.13 has a bright optical/IR counter part near the centre. The extendedtrailing emission seen in the radio images is likely associated with that source, withpossible contributions from some of the other nearby optical/IR sources seen, butagain the radio emission of the two non-central parts of this source is quite bright(hundreds of µJys), making it likely to be mainly associated with the bright centralsource rather than solely from multiple smaller fainter nearby sources (also thetrailing radio emission does not line up directly with any of the optical/IR sources).The source in Fig. 6.14 again has multiple possible optical/IR counterparts.None of these line up well with the C image contours that show emission to thesides and below the source. Similarly for the source in Fig. 6.15, with multiple pos-sible optical/ID counterparts for the emission seen to the left, but with no obviouscounterparts with the emission on the right. For both of these sources, it is possiblethe radio emission is some combination of the multiple optical/IR sources, or alsopossible that none of the optical/IR sources are true counterparts. Either way, noneof these sources are fit well by single or even multiple Gaussian models, thereforewe classify them as extended.6.3 Source CountA source count was made of all the best-fit total primary-beam-corrected flux den-sities for both the C and CB catalogues. The differential source count, dN/dS, for153Figure 6.12: Extended source image for source position J2000 161◦.6051,59◦.090913. The top left panel is the CB-data 3 GHz image. Thetop right is the C-data spectral index image. The bottom left imageis the CFHT g-band image. The bottom right is the Spitzer 3.6µmimage. The overlaid contours are from the lower resolution C-data.The contours levels are 2, 5, 12, 30, 75, 200, 500, 1200, and 3000 µJy.154Figure 6.13: Extended source images for source position is J2000161◦.41572, 58◦.955855. The top left panel is the CB-data 3 GHzimage. The top right is the C-data spectral index image. The bottomleft image is the CFHT g-band image. The bottom right is the Spitzer3.6µm image. The overlaid contours are from the lower resolution C-data. The contours levels are 2, 4, 8, 15, 30, 50, 100, 200, and 700µJy.155Figure 6.14: Extended source images for source position J2000 161◦.45361,58◦.90226. The top left panel is the CB-data 3 GHz image. The topright is the C-data spectral index image. The bottom left image is theCFHT g-band image. The bottom right is the Spitzer 3.6µm image.The overlaid contours are from the lower resolution C-data. The con-tours levels are 2, 3.5, 6, 10, 20, 30, 50, 90, and 150 µJy.156Figure 6.15: Extended source images for source position J2000 161◦.6575,58◦.906266. The top left panel is the CB-data 3 GHz image. Thetop right is the C-data spectral index image. The bottom left imageis the CFHT g-band image. The bottom right is the Spitzer 3.6µmimage. The overlaid contours are from the lower resolution C-data.The contours levels are 2, 4, 6, 12, 20, 35, 65, and 120 µJy.157bin a was computed asdNdS a=NaAa∆Sa. (6.4)Here Na is the total number of sources in bin a, ∆Sa = Sa,high − Sa,low, and Aais the area over which the sources with mean flux 〈Sa〉 could be detected given theprimary beam, noise, and detection limit. The angle over which a source can bedetected isΘa =√lnPaln 2ΩFWHM2, (6.5)where ΩFWHM is the FWHM of the primary beam in radians and Pa = 〈Sa〉/(4σ),which is signal-to-detection limit ratio rather than just the signal-to-noise ratio.Then the area (for small Θa) isΩa = piΘ2a. (6.6)This is the area over which a point source could be detected with a 4σ peak de-tection cutoff. However, for non-point sources Pa changes to Pa = 〈Sa〉/(4σN ),where N is the size correction or N = (θMθm)/θ2B.We computed the source count for our catalogues by computing N for all thesources, binning the sources by N , computing dN/dS for each N bin and thensumming over all the N counts. The final source count is thendNdS=∑i(dNdS)Ni. (6.7)The N bins used are N=[1, 1.13, 1.375, 2, 4.25]. The mean flux density for thesummed source count is the average of the mean flux densities for each size binin each flux density bin. This is shown in Fig. 6.16, along with the P(D) fits fromVernstrom et al. [183] and Condon et al. [37]. Table 6.1 shows the counts fromusing all the sources. The counts from both catalogues are in good agreement witheach other and with the P(D) models, being slightly lower than the P(D) modelaround 100µJy.The correction for flux boosting has been performed on the individual sourceflux densities via the 2D interpolation corrections, rather than a correction to the158100 101 102 103 104S (µJy)10-210-1100101102S5/2 dN/dS (sr−1 Jy−1.5)Condon 2012 Power Law Slope -1.7Vernstrom 2014 P(D) FitCB DataC DataFigure 6.16: 3 GHz source count. The black dashed line is the P(D) countfrom Vernstrom et al. [183]. The black solid line is the power-law P(D)from Condon et al. [37]. The red circles are the count using the best-fittotal flux densities from the CB image, with the blue diamonds beingthe count from the C image. The counts from the catalogues are binnedcounts (with the points plotted at the mean flux densities), whereas theP(D) counts are for a power-law model (or multiple connected powerlaws).159Table 6.1: Differential Euclidean normalized source countSlow Shigh 〈SCB〉 〈SC〉 dN/dSCB S5/2 dN/dSC S5/2(µJy) (µJy) (µJy) (µJy) (Jy1.5 sr−1) (Jy1.5 sr−1)4 12 8.8 9.1 0.9 ±0.1 0.9 ±0.112 36 22 23 1.9 ±0.1 1.8 ±0.136 130 61 64 1.5 ±0.1 1.5 ±0.1130 320 210 200 1.9 ±0.4 1.6 ±0.4320 1500 810 800 4.3 ±1.4 4.6 ±1.41500 8700 5100 5100 13. ±9.5 13. ±9.3source count. Corrections for completeness, false detections, and “clean bias”should all be looked at for these catalogues. The clean bias is a bias resulting fromthe cleaning process. This involves flux being removed from fainter sources, whichare affected by the sidelobes of brighter sources. To correct for this, simulationsare required, which entails inserting fake sources into the uv data and repeating thecleaning process as before and examining the flux of the simulated sources. Wedo not expect a large clean bias for our data as the sidelobes of our beams are notlarge and there are not many bright sources; however, this still needs to be exam-ined. The completeness and false detections have been briefly examined using thesimulation data. However, the corrections are not expected to be large at the fluxdensity thresholds we selected (with the peak completeness from the simulationsaround 80% at the threshold and false detection rate around 30%), and hence acomprehensive analysis of those effects is postponed to a future study.6.4 Spectral IndicesThe 2–GHz bandwidth of the VLA correlator allows us to obtain information onthe spectral dependence of the sources. We first retrieved the flux density values ofeach source at the positions of the fitted peaks in each of the 16 sub-band images.We applied a primary beam correction to each value, based on the primary beam foreach sub-band frequency and source position. We then performed a least squaresfit for the spectral index αi and the corresponding fitted 3-GHz flux density, F , of160each source by minimizingχ2 =16∑i=1(Si − ((νi/3.0)αF ))2wi. (6.8)Here wi is the normalized weight from the sub-band divided by the primary beamcorrection at the source position:wi =1σ∗i2∑i1σ∗i2 . (6.9)If the source is outside the 10 per cent power region for a particular sub-band’sprimary beam it was not included the fitting. This method works well for sourceswith large signal-to-noise, but for the fainter sources it is difficult to obtain a well-constrained fit, since each sub-band has noise that is 3 to 18 times the noise levelin the combined 3-GHz image.In an attempt to deal with the noiser sources we also performed a separatemethod to estimate spectral indices. We made two new images at 2.5, and 3.5 GHz,each using 1 GHz of bandwidth, resulting in noise values of 1.75 and 3.5µJybeam−1. These were again constructed to have nearly circular 8 arcsec beams.We took the primary beam-corrected images and made a spectral index image bytaking log10 of the images and dividing one image by the other. This yielded a spec-tral index image with α2.5−3.5. Some parts of this image can be seen in Figs. 6.12,6.13, 6.14, and 6.15 for the extended sources. The spectral index for each cataloguesource was then read off from this image at the pixel position of the fitted peak.A separate spectral index image was made for both the CB and C data. Thespectral indices are labelled as αIF when fitting the 16 sub-bands and αIM from thespectral index image. Values outside the range −2.25 ≤ α ≥ 2.25 were set to zeroin the catalogue. Histograms showing the different spectral indices are shown inFig. 6.17 and plots of the mean α in bins of flux density are shown in Fig. 6.18.All versions of the spectral index distribution have means near α = −0.7, aswould be expected if the majority of sources are star-forming galaxies; the onlyexception is αIF for the CB data. This is likely due to several of the sub bands inthe CB data being more severely affected by RFI than in the C data alone (note1613 2 1 0 1 2 3αIM0102030405060708090Number of sourcesµ =-0.69median =-0.83CB Data3 2 1 0 1 2 3αIF020406080100120140Number of sourcesµ =-0.52median =-0.71CB Data3 2 1 0 1 2 3αIM020406080100120140Number of sourcesµ =-0.71median =-0.9C Data3 2 1 0 1 2 3αIF020406080100120140160180Number of sourcesµ =-0.64median =-0.77C DataFigure 6.17: Catalogue spectral index histograms. The top panels show thespectral index distributions for the CB data and the bottom panels arefor the C data. The left panels are indices obtained from the spectralindex images, while the right panels show indices obtained from sub-band fitting.162101 102 103STotal (µJy)1.51.00.50.00.5mean αCB Data αIMCB Data αIF101 102 103STotal (µJy)1.51.00.50.00.5mean αC Data αIMC Data αIFFigure 6.18: Catalogue mean spectral indices in bins of flux density. The toppanel shows the mean αIM (red solid line) and αIF (green dashed line)for the CB data. The bottom panel shows the mean αIM (blue solidline) and αIF (purple dashed line) for the C data. The uncertaintiesshow the inter-quantile range for each bin.163these sub bands were not used in the fitting at all). The image method is moreconsistent between the two catalogues, as seen by the tighter correlation in themiddle left panel of Fig. 6.8, which is likely due to the higher SNR.There has been some recent discussion of the mean spectral index for faintersources being closer to −0.5 rather than −0.7. Looking at Fig. 6.18, there doesseem to be some indication for this. However, for any method of estimation it ismore difficult to obtain an accurate estimate with a smaller SNR; thus we shouldtreat the faintest flux density spectral index estimates with some caution.The differences in spectral indices dependent on the type of source, as wellas the redshift (and frequency). To say the mean spectral index expected is α =−0.7 depends on observing frequency as synchrotron self absorption can lead toflattening of the spectrum if observed at lower frequencies or if the source is at ahigh redshift.6.5 Cross-IdentificationsThe 3 GHz catalogue best fit positions were used to cross match the sources with26 other catalogues, though some of those come from the same instrument or sur-vey, but with different filters. The source positions were matched with a searchradius of 2σB, or 2.25 arcsec and 6.75 arcsec for the CB and C catalogues, respec-tively. Details of the matching can be seen in Fig. 6.19. A more detailed analysisis required to determine true counterparts from these catalogues. This will entaillooking at the likelihood ratio of the matches using information on the positionaloffset uncertainties, and the counterpart distributions rather than just using the clos-est matches. This was deferred to a future study of the properties of the cataloguesources.6.5.1 Optical and IRFor optical and near infrared data our catalogue was cross-matched with severalcatalogues. There is the Sloan Digital Sky Survey (SDSS) data release 9 [1], whichincludes u, g, i, r, and z bands. There is deeper optical data from CFHT, in theu, g, i, r, and z bands. There is Spitzer/IRAC data from the Spitzer Wide-AreaInfrared Extragalactic Survey (SWIRE) survey at 3.6, 4.5, 5.8, and 8.5µm [111,1640 5 10 15 20Number of Catalog Matches101102103S Peak (µJy beam−1 )CB data0 100 200 300 400 500 600Number of Catalog Matches0.00.51.01.52.02.53.03.54.0Number of SourcesSDSS90cm610MHzWISEChandraUKIDSS2MASSGALEXCFHT_GCFHT_ICFHT_RCFHT_UCFHT_z20cmz-SWIREz-photoSERVSIRACSPITZER_24 Spire0100200300400500600700800Number of sources matchedTotal=7070 5 10 15 20Number of Catalog Matches101102103S Peak (µJy beam−1 )C data0 200 400 600 800 1000Number of Catalog Matches0.00.51.01.52.02.53.0Number of SourcesSDSS90cm610MHzWISEChandraUKIDSS2MASSGALEXCFHT_GCFHT_ICFHT_RCFHT_UCFHT_z20cmz-SWIREz-photoSERVSIRACSPITZER_24 Spire0100200300400500600700800Number of sources matchedTotal=932Figure 6.19: Details of catalogue cross-matching. The panels on the right arefor the CB data catalogue, while the panels on the left are for the Cdata catalogue.165173], as well as “warm” Spitzer data from the Spitzer Extragalactic RepresentativeVolume Survey (SERVS) at 3.6 and 4.5 µm. There are also data from the UKIRTInfrared Deep Sky Survey (UKIDSS) data release 9 survey [104, 105] in K and Jbands. We computed colour cuts for each of our two catalogues using the cross-matched data, which can be seen in Figs. 6.20 and 6.21.There are distinct features in the colour-colour plots. These can tell us aboutthe redshifts of the sources, as well as the separate populations (i.e. AGN vs star-forming galaxies). We have highlighted some features. The bottom panel of thefigures shows IRAC [5.8]− [8.0]µm vs [3.6]− [4.5]µm colours. The yellow regionshows the divide between z < 1.2 and z > 1.2 from Marsden et al. [114], with theequation for the dividing line being([3.6]− [4.5]) = 0.0682× ([5.8]− [8.0])− 0.075. (6.10)Also on this panel the AGN selection criteria from Donley et al. [43] is shown asthe grey shaded region. These criteria are define as follows, where ∧ is the logical“AND” operator:x = log10(f5.8f3.6), y = log10(f8.0f5.8)(6.11)x ≥ 0.08 ∧ y ≥ 0.15∧y ≥ (1.21× x)− 0.27 ∧ y ≤ (1.21× x) + 0.27∧ f4.5 > f3.6 ∧ f5.8 > f4.5 ∧ f8.0 > f5.8.(6.12)These criteria were then converted from this flux colour space to the magnitudecolour space shown in the figure. Additionally, we have shown four galaxy red-shift tracks (starting at z = 0 going to z = 4 and assuming no evolution) usingrest-frame galaxy spectral energy distribution (SED) templates from Polletta et al.[142]. We chose one galaxy from each category of elliptical, spiral, starburst orultra-luminous infrared galaxy (ULIRG), and quasar.The optical and IR colours, combined with these tracks, show that the majorityof sources are likely spiral or star-forming type at intermediate redshift. This isnot unexpected. From the source count and luminosity functions, it is believed166that star-forming galaxies dominate in the sub-mJy regime. These criteria, whencombined with firm optical matches, spectral indices and available redshifts, willallow us to categorize and investigate the properties of the different galaxy types inthe catalogues.6.5.2 RadioWe cross matched our VLA catalogue with the 1.4 GHz VLA catalogue from Owenand Morrison [134]. The majority of the 1.4 GHz sources that did not have matchesin our catalogue were near the Owen signal-to-noise limit and a large majoritywere out near the edge of our cataloguing area, where the sensitivity is weak fromthe primary beam (the primary beam is smaller at 3 GHz than at 1.4 GHz). The1.4 GHz catalogue reports the total flux density corrected for the primary beam, aswell as bandwidth and time smearing effects.We computed the spectral indices of the matched sources for both versions ofour catalogues, with histograms and flux-flux plots shown in Fig. 6.22. The av-erages of the spectral indices range from −0.9 to −1.04, which is clearly steeperthan the expected −0.7. The exact cause for this has not yet been determined.The average deconvolved size for the 1.4 GHz sources is smaller than the averagedeconvolved size at 3 GHz; thus to obtain a steep spectral index with our sourcesizes being larger would mean either we our underestimating our flux densities onaverage or Owen & Morrison over estimated the 1.4 GHz flux densities. We be-lieve that it is more likely that the 1.4 GHz flux densities have been overestimated,since the source count produced by Owen & Morrison is higher than our sourcecounts for the fainter flux densities (both the catalogue counts as well as the P(D)count); this would not make the Owen & Morrison count less believable exceptthat (as discussed in earlier Chapters) the counts are expected to show a downturnat fainter flux densities rather than the levelling off (or upturn, depending on thenormalization) shown by Owen & Morrison .6.6 Conclusions and Future WorkAs has been mentioned, what is presented here are only the preliminary versionsof these catalogues and a brief investigation of the source properties. We plan to1670.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5u−g0.50.00.51.01.52.02.5g−rCFHTSDSSStarburstEllipticalSpiralQSO0.5 0.0 0.5 1.0 1.5 2.0g−r0.50.00.51.01.52.02.5r−iCFHTSDSSStarburstEllipticalSpiralQSO0.5 0.0 0.5 1.0 1.5 2.0 2.5r−i0.50.00.51.01.5i−zCFHTSDSSStarburstEllipticalSpiralQSO1.0 0.5 0.0 0.5 1.0 1.5[3.6]AB−[4.5]AB µm0.50.00.51.01.52.02.5K mag−[3.6] ABµmSWIRESERVSStarburstEllipticalSpiralQSO1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5[5.8]AB−[8.0]AB µm0.60.40.20.00.20.40.60.81.0[3.6] AB−[4.5] ABµmz<1.2z>1.2SWIREStarburstEllipticalSpiralQSOFigure 6.20: Colour-Colour plots of optical and NIR catalogue matches forthe CB catalogue. The top and middle left panels include data fromthe SDSS DR9 and CFHT. The middle right panel is Spitzer IRAC andUKIDSS data, while the bottom panel is Spitzer IRAC. The colouredlines are redshift tracks (for 0 < z < 4 and assuming no evolution)using galaxy templates from Polletta et al. [142]. The yellow shadedregion in the bottom panel shows z < 1.2 from eq. 6.10 (Marsden et al.[114]). The grey shaded region is the AGN criteria from Donley et al.[43]. 1680.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5u−g0.50.00.51.01.52.02.5g−rCFHTSDSSStarburstEllipticalSpiralQSO0.5 0.0 0.5 1.0 1.5 2.0g−r0.50.00.51.01.52.02.5r−iCFHTSDSSStarburstEllipticalSpiralQSO0.5 0.0 0.5 1.0 1.5 2.0 2.5r−i0.50.00.51.01.5i−zCFHTSDSSStarburstEllipticalSpiralQSO1.0 0.5 0.0 0.5 1.0 1.5[3.6]AB−[4.5]AB µm0.50.00.51.01.52.02.5K mag−[3.6] ABµmSWIRESERVSStarburstEllipticalSpiralQSO1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5[5.8]AB−[8.0]AB µm0.60.40.20.00.20.40.60.81.0[3.6] AB−[4.5] ABµmz<1.2z>1.2SWIREStarburstEllipticalSpiralQSOFigure 6.21: Colour-Colour plots of optical and NIR catalogue matches forthe C catalogue. The top and middle left panels include data from theSDSS DR9 and CFHT. The middle right panel is Spitzer IRAC andUKIDSS data, while the bottom panel is Spitzer IRAC. The colouredlines are redshift tracks (for 0 < z < 4 and assuming no evolution) us-ing four galaxy templates from Polletta et al. [142]. The yellow shadedregion in the bottom panel shows z < 1.2 from eq. 6.10 (Marsden et al.[114]). The grey shaded region is the AGN criteria from Donley et al.[43]. 1693 2 1 0 1α1.4−3020406080100120Number of sourcesµ =-0.92median =-0.97CB Data101 102 103 104S3 (µJy)101102103104S 1.4 (µJy)CB Dataα=−0.73 2 1 0 1α1.4−3020406080100120140160180Number of sourcesµ =-1.0median =-1.04C Data101 102 103 104S3 (µJy)101102103104S 1.4 (µJy)C Dataα=−0.7Figure 6.22: The left panels show the histograms of spectral indices fromthe cross-matched sources at 1.4 GHz of Owen and Morrison [134]with our 3 GHz catalogues. The right panels show the 1.4 GHz fluxdensities vs the 3 GHz flux densities for each cross-matches source.The top panels are the CB data while the bottom panels are the C data.The black diagonal lines show the predicted flux densities for an α =−0.7.170build on this analysis by looking more deeply at the completeness and false detec-tions, the clean bias, the redshift distribution, the source size distribution, the truecounterparts and population investigation. This will include counterpart matchingat the presented counterpart wavelengths, as well as those not discussed in detailhere (infrared, submm, X-ray, and lower frequency radio). However, we have al-ready made detailed simulations and used them for analysis of the uncertainties inour catalogues. These early catalogues are likely not very different from their finalversions. We have shown that VLA wideband data can be used to obtain spectralindex estimates and that these agree with previous estimates for the mean spectralindex of 〈α〉 = −0.7, with the possibility of shallower spectral indices for fainterflux densities. We have shown there are many counterparts in other wavebands andthat their distributions show distinct features. We were able to match a majority ofour catalogue sources with those from the Owen and Morrison [134] catalogue, andhave shown again that the previous flux densities seem to be overestimated (bothfrom comparing source counts and spectral indices). Our source count from bothcatalogues agrees well with the P(D) work from Vernstrom et al. [183] and Con-don et al. [37], as well as being in line with most previous source counts from othersurveys (with the exception of the low flux density counts of Owen and Morrison[134]).When comparing the two catalogues of different resolution, they seem to per-form comparably well when it comes to finding the same sources and their param-eters. From our simulations, with the CB resolution 77% of the input sources hada match in the fit results. The mean ∆RA/Dec was 0.003 ± .0.75, the mean ratioof true to fitted total flux density was 0.83 ± 0.3 and the mean ratio for the majoraxis was 0.87±0.24. For the C resolution, 81% of the input sources were matchedin the fit results. The mean ∆RA and Dec was 0.03 ± 1.25. The mean ratio forthe total flux density was .078 ± 0.65 and the mean ratio for the major axis was0.81± 0.19. These numbers are for all sources, regardless of flux density.When thinking in the context of future surveys, such as VLASS (higher res-olution) vs EMU (lower resolution), we would suggest that the higher resolutiondata has the distinct advantage of providing more accurate positions and avoids theproblem of blending. On the other hand, the low resolution data have the advan-tage of being able to find more of the extended sources near the noise cutoff. Both171resolution choices, as long as the noise is comparably deep, should find roughlythe same number and the same sources and provide accurate estimates of theirparameters.172Chapter 7The Radio Angular PowerSpectrum7.1 IntroductionThe clustering of radio emission on the sky is represented by the power spectrumwhich is the distribution of power C` in each mode as a function of angular scaleor spherical mode `. There currently exist measurements of the angular powerspectrum of resolved radio sources (Blake et al. [13]) through the 2-point angularcorrelation function (w(θ)) but there exist only a few upper limits on the fluctua-tions from the unresolved radio background. Given the nature of radio emission,produced from magnetic fields between galaxies and clusters, models predict thatthe radio power spectrum should trace large-scale clustering or the cosmic web.Whether or not this emission is strong enough to be currently detectable is unclear.However, most previous studies have focused on trying to detect specific structures,such as filaments between clusters. Detecting the overall structure in a statisticalsense, as we attempt here, should be easier.Searches for CMB anisotropies at low frequencies provide strong constraintson clustering of the radio background. Some previous studies providing upper lim-its on anisotropies have been performed, such as Fomalont et al. [60] using theVLA at 4.8 GHz, Partridge et al. [137] with the VLA at 8.4 GHz , and Subrah-manyan et al. [171] with the ATCA at 8.7 GHz (see figure 1 from Holder [88]).173These limits on CMB clustering can also be used to constrain CRB clustering.This is complicated by the fact that the value of the CRB temperature (i.e. the DClevel of monopole of the sky) is unknown.The study of the C` of the radio sky extends our 1-point statistical studies(Chapters 4 and 5) to 2-point statistics. This is another means by which we canstudy and constrain models of cluster emission, as well as dark matter particlemodels.The power spectrum is a natural quantity to investigate here since the powerspectrum is measured in Fourier space and radio interferometry data are also mea-sured in Fourier space – this is in contrast to data which start in the image plane andmust be transformed to the Fourier domain. Nevertheless, this is not a trivial pro-cess, since it requires a firm understanding of all the corrections that must be made,particularly uv -weighting, primary beam correction and calibration. Similar inter-ferometric studies of CMB anisotropies have been carried out at higher frequen-cies (e.g. Hobson and Maisinger [86], Sutter et al. [174], White et al. [188, 189]).There has also been much related recent work using radio interferometer data andredshifted 21 cm emission to try to measure baryon acoustic oscillations (BAO)or the re-ionization signal. There are currently projects underway with the GiantMetrewave Radio Telescope (GMRT), the LOFAR telescope, the Canadian Hy-drogen Intensity Mapping Experiment (CHIME), and others (e.g. Bandura et al.[5], Ghosh et al. [70], Harker et al. [82], Trott et al. [178]). This involves mea-suring the 3D power spectrum as a function of angular scale as well as redshift orfrequency, rather than the 2D power as a function of multipole, but these worksprovide valuable references for the treatment of interferometric data.In this Chapter we will discuss our (preliminary) attempt to measure the angu-lar power spectrum using our existing ATCA data. Section 7.2 discusses the visibil-ity data that are used, with subsections describing the telescope primary beam, theissues of wideband frequency coverage, and of mosaicked data. Section 7.3 goesover the current method used for estimating C` from visibility data. Section 7.4 de-tails our simulations used to demonstrate and investigate these effects as well as theissues of clustering and extended emission, while Section 7.5 describes our modelfitting procedures. Section 7.6 compares early results from these simulations andour real ATCA data, along with some discussion of the results.1747.2 Visibility DataHere we discuss the relation between the visibility correlation and the angularpower spectrum of the intensity I(~θ, ν), or equivalently the brightness temperatureT (~θ, ν) distribution on the sky under the flat-sky approximation. The quantity ~θ isthe two dimensional vector on the plane of the sky with the origin at the pointingcentre.In radio interferometric observations, every pair of antennas in the interferom-eter measures a complex visibility V(u, ν) at a given point in the Fourier plane,where u is the vector of u, v coordinates in the Fourier plane, with dimensions ofinverse angle measured in wavelengths, and is the variable conjugate to ~θ. Sinceinterferometers are missing the zero-spacing they are insensitive to the DC levelin any region mapped and are thus sensitive only to angular fluctuations on scalessmaller than the primary beam and bigger than scales probed by the longest avail-able baseline.7.2.1 Primary BeamThe complex visibility is a measure of the sky signal plus instrument noise,V(uj , ν) = S(uj , ν) +N (uj , ν). (7.1)where S(uj , ν) is the contribution from the sky signal andN (uj) is the instrumen-tal noise on the jth visibility (which we expected to be approximately uncorrelatedand Gaussian). The visibility contribution from the sky signal is the Fourier trans-form of the product of the sky brightness I(~θ, ν) and the antenna power pattern, orprimary beam, A(~θ, ν),S(uj , ν) =∫d~θA(~θ, ν)I(~θ, ν)e2piiu·~θ. (7.2)The field of view is set by the primary beam size. The power pattern is very nearlya uniformly illuminated circular aperture [36]:A(~θ, ν) = A(ρ, ν) =[2λpiρDJ1(piρDλ)]2, (7.3)175where D is the diameter of the antenna, ρ = |~θ| is the angular offset from thepointing centre, and J1 is the Bessel function of the first kind and order. However,in practice the primary beam can be closely approximated by a Gaussian functionwith FWHM θF = 1.02λ/D:A(ρ, ν)G = e−ρ2θ20 . (7.4)In this case θ0 = θF√2/2.3548 ' 0.6θF. Each pair of antennas is separated bya baseline of length d metres projected on the plane perpendicular to the sky inunits of the observing wavelength λ. Each antenna pair is sensitive to all baselinesbetween d−D and d+D, with the responsivity determined by the antenna primarybeam. This means that the array is insensitive to any angular scales correspondingto baselines smaller than dmin −D or larger than dmax +D. This is shown for theATCA setup in Fig. 7.1, which has D = 22 m, shortest baseline of dmin = 30 m,and longest baseline of dmax = 352 m.The visibilities are convolved by the Fourier transform of A(ρ),A(kθ) =∫A(ρ, ν)e2piikθ·ρdρ, (7.5)where kθ is the uv distance or |u| (kθ has units of inverse radians, not to be confusedwith the spatial scale k with units of inverse Mpc) and in the small sky approxima-tion ` = 2pikθ. This can also be approximated by a GaussianA(kθ)G =1piU0e−k2θ/U20 , (7.6)where U0 = 1/piθ0 = 0.53θF.The two dimensional power spectrum P (u, ν) is〈∆I˜(u, ν)∆I˜∗(u, ν)〉 = δ2(u− u′)P (u, ν), (7.7)where ∆I˜(u, ν) is the Fourier transform of δI(~θ, ν) and is S(uj , ν) without a pri-mary beam convolution, and δ2(u−u′) is a two dimensional Dirac delta function.The angular brackets denote an ensemble average over different realizations of the1760.0 0.5 1.0 1.5 2.0k (kλ) ν=1.7GHz0.00.20.40.60.81.0Responsivity0 50 100 150 200 250 300 350 400Baseline (m)Figure 7.1: Responsivity of the ATCA interferometer setup as a function ofthe uv distance kθ at a frequency of 1.75 GHz and baseline length (upperaxis). The different colours represent the 10 different baselines, withthe shape of each curve determined by the individual antenna powerpatterns, or primary beam.177sky intensity fluctuations. We know that P (u, ν) is related to C` byC`(ν) =(∂B∂T)−2P (`/2pi, ν). (7.8)In this(∂B∂T)−2=(2kBλ2)−2is used to convert from Jy sr−1 to temperature. Fromeq. 7.7 and eq. 7.1 we can see that the visibility correlation function is related toC`,〈ViV∗j 〉 = V0e−|∆ui,j |2/σ0C`i + δij2σ2n, (7.9)where σ0 =√2U0 = 0.76/θF. The factor V0 is is defined asV0 =piθ202(∂B∂T)2=piθ202(2kBλ2)2. (7.10)The effect of the primary beam convolution on C` is to lower the mean ampli-tude by a factor fPb,fPb =1∫A(ρ)2dρ. (7.11)Equation 7.11 holds for the mean of many realisations, or if the flux distribution isfairly uniformly distributed in ρ. The exact value of fPb for an individual realisa-tion of C` varies depending on the exact flux density distribution as a function ofρ. By this we mean that if there are outlier sources with S  〈S〉 far out in theprimary beam then fPb will be greater than eq. 7.11 predicts, and likewise if outliersources were located near the pointing centre then fPb would be less than predicted.The value of fPb for our ATCA data, at the mean frequency, is fPb = 8.05.7.2.2 Frequency WeightingThe upgraded correlator on the ATCA telescope provides for wide frequency band-width coverage. This allows for more data and thus higher signal-to-noise, as wellas the possibility of measuring spectral changes across the band. It is less clear,however, what is the best way in which to handle the large frequency coveragefor the power spectrum. The power spectrum could be measured individually atmany different frequencies across the band. This decreases the data available formeasurements and thus decreases the SNR, but this would be recovered by averag-178ing the estimates together. However, splitting into many frequencies also results insparser coverage of the uv plane at each frequency. This is shown in Fig. 7.2, wherethe top panel gives the uv coverage of one ATCA pointing for all frequencies andthe bottom panel gives the uv coverage at a single frequency.There is also the option of measuring all of the visibilities together for a centralor mean frequency. In this case the change in amplitude from the spectral indexmust be accounted for. If α is known, then the visibilities can be corrected byV(uj , 〈ν〉) = V(uj , νi)(〈ν〉νi)α. (7.12)While the exact value of α may vary between sources, using simulations we havefound that the mean expected α is a good approximation.The spectral index of the primary beam αPb(ρ) is a function of distance fromthe pointing centre and is simply the derivative of eq. 7.3 with respect to frequency.When calculating C` at a single frequency using multi-frequency data, fPb be-comes frequency dependent, such thatfPb(νi) = fPb(〈ν〉)θF(〈ν〉)θF(νi). (7.13)7.2.3 MosaickingThe field of view, and correspondingly the uv resolution, are set by the size ofthe primary beam. To obtain high resolution in `-space the Fourier transform ofthe primary beam needs to be narrow, implying small antennas. However, highsensitivity requires a large collecting area for each antenna. This problem can besomewhat circumvented by the technique of mosaicking [38, 46, 87]. Mosaickingentails observing overlapping fields on the sky, which increases the effective sur-vey size and thus improves the `-resolution. Mosaicking takes advantage of thefact that a telescope samples a whole superposition of visibilities along a baseline.Unfortunately, overlapping observations also require that we take into account thecorrelations between the fields. The fields, and their primary beams, are correlated179Figure 7.2: Logarithm of the 2D power of one simulated sky model with theATCA uv coverage. The top panel shows the full uv coverage for onepointing with all frequencies. The bottom panel shows the same imagewith the uv coverage from one pointing at just one frequency.180by a phase shift relative to the global phase centre, such thatS(uj,l, ν) = S(uj,0, ν)e2piiu·∆θ˜. (7.14)Here S(uj,l, ν) is the sky contribution to the complex visibilities from the lth point-ing, S(uj,0, ν) is the sky contribution at the phase centre, and ∆θ˜ is the distance ofthe lth pointing centre to the phase centre.There are two routes to analyzing mosaicked data. The first is to treat the vis-ibilities from each pointing as separate data, highly correlated in a calculable way,with apparently low resolution, but a lot of information in the correlations betweenvisibilities. The other route is to statistically weight the visibilities from the differ-ent pointings to form a synthesized data set with fewer visibilities and correlationsand intrinsically higher resolution. This is is discussed in some detail for CMB in-terferometric measurements in White et al. [189] and Hobson and Maisinger [86].7.3 Bare EstimatorWe employ the Bare Estimator method as described in Choudhuri et al. [26]. Thismethod uses the individual visibilities rather than gridding them into uv bins. Eachvisibility corresponds to a Fourier mode of the sky signal, and the visibility squared| VV∗ | gives the angular power spectrum. This simple estimator, however, has thesevere drawback that the noise contribution 2σ2n is usually much larger than thesky signal. This can be mitigated by only including contributions where the noiseis uncorrelated, or not including the correlations of any visibility with itself. Thevisibilities at two different baselines are correlated only if the separation betweenthem is small, meaning | ∆uij |2≤ σ0.The Bare Estimator EˆB(a) is defined asEˆB(b) =∑i,j wij Vi V∗j∑i,j wije−|∆uij |2/σ20V0, (7.15)for bin b, where all kθ are in the range kθ(b) − dkθ/2. ≤ kθ ≤ kθ(b) + dkθ/2.The weights wij are chosen to: (1) maximize the SNR; and (2) go to zero when avisibility is correlated with itself, in order to avoid the correlated noise contribution.181The simplest case, ignoring any issues arising from calibration errors, is wij =(1− δij)e−|∆uij |2/σ20 , which is proportional to its contribution to EˆB(b).The Bare Estimator is a measure of the average angular power spectrum C¯`b atthe mean ` for bin b ¯`b, i.e.〈EˆB(b)〉 =∑i,j wij 〈Vi V∗j 〉∑i,j wijV0, e−|∆uij |2/σ20=∑i,j wijV0, e−|∆uij |2/σ20 C`i∑i,j wijV0, e−|∆uij |2/σ20. (7.16)If one is using multifrequency data to estimate the primary beam-corrected C`,then 〈Vi V∗j 〉 become the corrected visibilities〈VCi V∗Cj 〉 = 〈Vi V∗j 〉fPb(νk)(〈ν〉νk)α. (7.17)Figure 7.3 shows the effect of these corrections on simulated noiseless visibilitiesof randomly distributed point sources. Because of the different sizes of the primarybeams the different frequency visibilities are calculated separately for 〈EˆB(b)〉,〈EˆB(b)C〉 =∑k,i,j wkij 〈VCi (νk)V∗Cj (νk)〉∑k,i,j wkijV0, e−|∆ukij |2/σ0(νk)2. (7.18)The variance of the Bare estimator for bin b, σ2Eb , isσ2Eb = 〈EˆB(b)2〉 − 〈EˆB(b)〉2, (7.19)assuming the visibilities are Gaussian random variables. Then the average angularmultiple for bin b, ¯`b, is¯`b =∑i,j wij e−|∆uij |2/σ20`i∑i,j wije−|∆uij |2/σ20. (7.20)The choice of the bin size ∆` is driven by the trade-off between the desired narrowwidths for localizing features in the power spectrum and the correlations betweenbins introduced by the transform of the primary beam, where the minimum resolu-tion is set by the size of the primary beam.We note that this method can potentially be very computationally expensive,182Figure 7.3: Power vs uv distance for the visibilities of one simulated dataset (where the model includes no clustering or extended sources). Thecolours represent the different frequencies, ranging from blue at the lowend to red at the high end. The top left panel shows the noiseless visibil-ity power without any corrections. The top right panel shows the samevisibilities with a correction for: (1) frequency scaling applied accord-ing to eq. 7.12; (2) the primary beam spectral scaling of eq. 7.13; and(3) the primary beam amplitude of eq. 7.11. The bottom panel showsthe same as the top right, but with Gaussian noise of σ = 0.1 Jy addedto the visibilities.18330 20 10 0 10 20 30arcsec3020100102030arcsec0.00 0.05 0.10 0.15µJy30 20 10 0 10 20 30arcsec0.00 0.05 0.10 0.15µJy30 20 10 0 10 20 30arcsec0.012 0.025 0.037µJyFigure 7.4: Images of simulated sources from the SKADS simulation usedas models for simulated ATCA observations. The left panel shows thesingle frequency point sources, while the middle panel shows the pointsources plus extended emission halos, and the right panel being only theextended halos.depending on the size of the data set. It scales as N2, with N the total number ofvisibilities, which depends on the telescope setup (number of baselines, number offrequency channels, and integration time). For our ATCA data, which uses only 12hours of total time, approximately 44 unique frequency channels (after flagging),and 10 baselines, there are on the order of 70,000 visibilities. However, for a setuplike our VLA data the number of visibilities is over 1,000,000. A data set like thatwould likely require a method involving gridding of the visibilities (such as is donebefore imaging).Finally, we note that the baseline visibilities contain two uv points consideringthe property V(u) = V∗(−u). Thus in data analysis only one half of the visibilitiesneed be used, as long as the this property is taken into consideration. This alsomeans that even when performing a cross-correlation the imaginary part of thesum cancels out.7.4 SimulationsWe first examined simulated data sets in order to test many of the issues previouslymentioned (visibility weighting, frequency scaling, primary beam correction, etc.).We used the source catalogue in 1 deg2 of the SKADS simulation at 1.4 GHz. This184simulated area roughly matches the size of the ATCA primary beam. The catalogueincludes positions, flux densities, and sizes for about 600, 000 sources with S >1 nJy. With α = −0.7 we scaled all the fluxes to a frequency of 1.70 GHz to moreclosely match the ATCA data. We only included sources with S ≤ 5 mJy, to moreclosely match the source-subtracted ATCA data, as well as to ensure the Poissonsignal from a few bright sources does not overwhelm fainter signals. We did notinclude any source sizes here, setting all objects to point sources or delta functionsin the image.We created several cases to test different features. These test images includethe following features:1. Point sources; random positions.2. Extended Gaussian halos; random positions .3. Point sources + extended Gaussian halos; random positions.4. Point sources; clustered positions.5. Extended Gaussian halos; clustered positions.6. Point sources + extended Gaussian halos; clustered positions.7. Point sources; random positions: 7 pointing mosaic of 2 deg2.These cases allow us to examine the power spectrum from point sources, orPoisson noise, from large-scale clustering, and from extended emission, as well asto test the effects of the frequency correction, mosaicking, the primary beam, andthe particular uv coverage. The models with random position point sources, pointsources plus halos, and just halos are shown in Fig. 7.4.The extended halos were created as Gaussian distributions centred around pointsources. The total halo flux was set to Sps divided by some factor ranging from 3to 8, with brighter sources having a smaller factor, and Sps being the flux densityof the point source. The size of the haloes ranged from 15 to 90 arcsec, scaled withSps such that the brightest point sources would have the largest haloes.The SKADS catalogue positions were used for the point source only mosaicmodels, which should include some level of clustering. In Wilman et al. [191] it18530 20 10 0 10 20 30arcminute3020100102030arcminute30 20 10 0 10 20 30arcminute30 20 10 0 10 20 30arcminute0.50 0.25 0.00 0.25 0.50Log10[Number Density] 9 8 7 6Log10[µJy] 9 8 7 6Log10[µJy]Figure 7.5: Simulated clustered and random skies. The left panel shows thedensity field (degree of over or under density). The middle panel showsthe clustered positions with the brightest flux densities being set to thepixels with the highest over-densities, thus a strong clustering bias. Theright panel shows the same clustered positions but with randomized fluxdensities.is said the simulation has different clustering for (i) radio-quiet AGN, (ii) radio-loud AGN and (iii) star-forming galaxies. The radio-quiet AGN clustering wasderived to match observations from the 2dFQZ survey ([40]); while the radio-loudclustering comes from the NVSS and FIRST surveys ([133]). The star-formingcluster was taken from Overzier et al. [133] and using IR data from Farrah et al.[49]. This is likely as good of an approximation to the actual clustering spectrumas may be possible with current data and simulations.However, even though the SKADS catalogue positions should already have thisclustering information encoded we do not know the exact shape of the clusteringspectrum for those simulations and it appears to be quite weak compared to thePoisson level. For these simulations we would like to clearly test the method andpossible effects on a clearly known and visible input clustering spectrum. Thus, wedecided to take the catalogue fluxes (since we know that the source count is fairlyrealistic) and generate positions with a known clustering signal, relatively strong186compared to the Poisson level and with a distinguishable shape. The clustered po-sitions were generated assuming a Gaussian source distribution in redshift N(z)with mean=0 and standard deviation=0.5, between z = 0.5 and z = 1.5. Sourceshad 3D clustering ξ(r) = (r0/r)γ . We used r0 of 5 Mpc h−1 and γ = 1.8. Thisproduces an angular correlation function w(θ) = 0.0021θ−0.8deg , matching that mea-sured by Blake et al. [13]. The clustered positions and this method were providedto us by Chris Blake.This process resulted in realistically clustered positions; however, it is not soclear how to assign flux densities to those positions, since the brightness-dependenceof clustering is more of a mystery. We certainly know there should be a Poissoncomponent from the source distribution. The Poisson power term can be computedfrom the source countPshot =∫ Scut0S2dNdSdS. (7.21)The fluxes can then either be randomly assigned or, to strengthen the clusteringsignal, higher fluxes can be assigned to the positions in higher over-densities. Fig-ure 7.5 shows the over and under-densities of the clustered positions, as well as onecase of assigning the fluxes with a bias and one with random assignment.Thus we have simulated images with SKADS clustering (the point source mo-saics) and the w(θ) of Blake et al. [13] (point sources and extended single point-ings) along with single pointings with randomised positions. Thus we should havesimulations that may more closely resemble that true radio source spectrum andothers that allow for clearer testing of the method effects on a spectrum with clus-tering.To create simulated radio interferometer observations from these sky modelswe used the MIRIAD task UVMODEL. UVMODEL allows you to use a model imageand a real visibility data set and then replace the real visibilities with model visibil-ities. We used the central pointing of the ATCA observations as the input, exceptfor the mosaic simulation, which used all seven ATCA pointings. This ensures thatthe simulated data have the same observational setup as our real data. Each of thesimulated images was used as a model for this task, with primary beams applied (tomimic a clean component model image) and a second plane describing the spec-tral index of each source (α = −0.7) plus the spectral index due to the primary187beam at each position. This allows the task to scale the single frequency image toall the observing frequencies. This created noiseless visibility sets. We averagedfrom the full 1024 spectral channels to 64, though only about 44 frequencies hadnon-zero amplitude visibilities, due to the flagging in the real data; this seemed agood compromise between having a large number of visibilities and still testing forthe effects of wide-bandwidth.The power spectrum was then computed using these noiseless visibilities viaeq. 7.18, as well as a computation with Gaussian random noise of σ = 0.03 Jyadded to the visibilities (consistent with the rms per visibility for our ATCA ob-servations). These were then compared with the auto-power spectra from the inputmodel images.In order to investigate the effects of the binning, cosmic variance, and the pri-mary beam, we performed a bootstrap analysis. First, we took the SKADS fluxdensity catalogue and used it to generate 1000 realisations of the source count withrandom positions each time. We applied a primary beam to each realisation. Wethen computed the 1D power spectrum (using the auto-correlation) of each modelimage, with and without the primary beam, and using both linear and logarith-mic binning in `, with the number of bins ranging from 4 to 12 in the range of600 < ` < 16500. We then repeated the procedure, except that this time we usedclustered positions with randomized flux density assignment. The mean, uncertain-ties, and 100 realisations for 7 bins are shown in Fig. 7.6 and Fig. 7.7. From theunclustered power spectra we can see that we recover the input well, with perhapsa slight bias at the lowest `s after performing the beam correction. Similarly, theclustered simulations show that we can recover approximately the correct powerspectrum. These bootstraps can also be used to look at the correlations or covari-ance between the bins.7.5 Model FittingWhen it comes to fitting a model, the most straightforward case is to perform aleast squares fit to the 1D power spectrum. It may be worthwhile to use the fullcovariance matrix of the bins in the fitting rather than just the bin uncertainties.This covariance matrix can be obtained from a bootstrap analysis such as described1882000 4000 6000 8000 10000120001400016000`60708090100110120C `Linear bins no primary beam2000 4000 6000 8000 10000120001400016000`6810121416182022C `Linear bins primary beam103 104`050100150200250C `Log bins no primary beam103 104`010203040506070C `Log bins primary beamFigure 7.6: Results from bootstrap tests of random Poisson (i.e. uniformly-distributed) sources, using both linear and logarithmic bins (top and bot-tom rows) and with and without a primary beam correction (left andright columns) for 7 bins. The solid black line is the mean from 1000realizations. The coloured lines are 100 different individual realizations.The results are clearly consistent with the Poisson (i.e. flat power) in-puts.1892000 4000 6000 8000 10000120001400016000`30405060708090100110C `Linear bins no primary beam2000 4000 6000 8000 10000120001400016000`2468101214161820C `Linear bins primary beam103 104`050100150200250300C `Log bins no primary beam103 104`020406080100120C `Log bins primary beamFigure 7.7: Results from bootstrap tests of clustered sources (with randomlydistributed flux densities) using both linear and logarithmic bins (topand bottom rows) and with and without a primary beam correction (leftand right columns) for 7 bins. The solid black line is the mean from1000 realizations. The coloured lines are 100 different individual real-izations.190above. However, it is unknown what effect (particularly any bias) the model usedto generate the bootstrap may have on the output and thus we have not used thismethod here. There are many other possible ways of model fitting, some of whichwe plan to explore at a later date.For the choice of model, the simplest approach is to simply fit for the Poissonlevel (Model 1),C` = P. (7.22)If we assume there is some clustering which follows a power law, we can fit for thePoisson level plus the clustering power law (Model 2)C` = D(``0)β+ P, (7.23)after picking some value for `0. Optionally, if no correction for the primary beamis applied during the power spectrum calculation, the value of fPb(〈ν〉) can be fitas a free parameter, since it is not exact for an individual realisation of the sky. Asfor a model to include an extended emission component, this is more difficult. Itis not known how much extended emission contributes to the power spectrum orwhat shape it will take. Models using Gaussian type halos could be fit by a simplemodel; however, it is unknown how realistic that type of model is. At this time wehave therefore not fit any models for extended emission.It is necessary to apply some priors on the parameters to constrain the fits. Wehave left these fairly relaxed at the moment, constraining the Poisson level and theclustering amplitude to be greater than zero and less than the maximum amplitudefrom the input. The β value for the clustering model is constrained to be less thanzero and greater than −6. When fitting the primary beam-correction amplitude asa free parameter, a Gaussian prior is set with a mean of the expected value fromeq. 7.11.The model fitting is performed using MCMC analysis.7.6 Simulation ResultsThe C`s from all of the simulations are shown in Fig. 7.8 (clustering), Fig. 7.9(extended emission), and Fig. 7.10 (mosaics). We have also run some early trials of191103 104`10-210-1100C ` (Jy2sr−1 )Random All SModel no PBModel PBSim No NoiseSim Noise103 104`10-1100 Cluster All SModel no PBModel PBSim No NoiseSim Noise103 104`10-1100C ` (Jy2sr−1 )Cluster S>1 µJyModel no PBModel PBSim No NoiseSim Noise103 104`10-510-410-310-2Cluster S<1 µJyModel no PBModel PBSim No NoiseSim NoiseFigure 7.8: C` measurements from simulations that contain clustering. Theblack solid lines are the input noiseless model images, while the blackdotted lines are the input models with a primary beam applied. Theblue dashed and red dot-dashed lines are the simulated visibilities withno noise and with Gaussian random noise of σ = 0.02, respectively.All simulations here include only point sources. The top left panel hasrandomized positions for all S (Smax = 0.5 mJy). The top right panel isclustered positions for all S. The bottom left panel is clustered, includ-ing only sources with S > 1µJy. The bottom right panel is clustered forsources with S < 1µJy. The shaded regions are the ±1σ uncertaintiesfrom the visibilities.192103 104`10-1C ` (Jy2sr−1 )PS+EX RandomModel no PBModel PBSim No NoiseSim Noise103 104`10-1100PS+EX ClusteredModel no PBModel PBSim No NoiseSim Noise103 104`10-410-310-210-1C ` (Jy2sr−1 )EX RandomModel no PBModel PBSim No NoiseSim Noise103 104`10-410-310-210-1EX ClusteredModel no PBModel PBSim No NoiseSim NoiseFigure 7.9: C` measurements from simulations with extended emission. Theblack solid lines are the input noiseless model images, while the blackdotted lines are the input models with a primary beam applied. Theblue dashed and red dot-dashed lines are the simulated visibilities withno noise and with Gaussian random noise of σ = 0.02, respectively.The top panels are point sources plus extended halos for randomizedpositions (left) and for clustered positions (right). The bottom panelsare extended halos only, for random positions (left) and for clusteredpositions (right). The shaded regions are the ±1σ uncertainties fromthe visibilities.193model fitting on the simulated data sets. Parameter distributions (1D and 2D) fromfitting Model 2 (cluster power law plus Poisson) for the simulated noisy data areshown in Figs. 7.11 (random positions no clustering) and 7.12 (clustered positions).These were fit without any additional priors on the known or estimated Poissonlevel, which should be 0.1 Jy2 sr−1. We can see that for the data with no clusteringthe fitted Poisson level is approximately right (median value of 0.097 Jy2 sr−1), andthat the routine had difficulty fitting the clustering level and index, as it should. Onthe other hand, when fitting the data that contains clustering, the true index shouldbe β ' −1.2 ,and we can see that the peak of the distribution is near this value (the median is −1.08). The Poisson level is still close to the input value, althoughperhaps a bit low (median value of 0.07 Jy2 sr−1).We can see it is certainly possible to detect the Poisson level of an image even,with the existing noise level. The clustering level and index strongly depend on theamount of noise, amount of clustering, and the particular realisation of the sky. Ifthe uncertainties from cosmic variance large compared to the clustering amplitudethen the clustering is much harder to constrain. The particular clustering setup wehave used in the simulations seems optimistic, at least for the clustering amplitude,compared to the Poisson amplitude. While the clustered positions are set up tomatch previous observations, the assignment of flux-density values is likely not asrealistic. In reality the clustering amplitude is likely to be lower, since we have notseen something with such a distinctive power-law addition to the flat Poisson levelin the real data. Perhaps with the higher multipole resolution from mosaicking,lower noise, or a better treatment of the primary beam effect the clustering levelscould be fit with smaller uncertainty.7.7 ATCA ResultsThe C`s from our real ATCA data individual pointings are shown in Fig. 7.13.These data have had the bright (S ≥ 150µJy) point sources subtracted, as describedin Section 5.3.1, using the ATLAS models. Using the source count from Chapter 4scaled to 1.75 GHz and eq. 7.21 with Scut = 150µJy, the expected Poisson levelis 3.8 Jy2 sr−1, with the mean from all seven pointings being 3.36 Jy2 sr−1.We have not yet attempted any model fitting on these data, since there are still194103 104`10-1100101C ` (Jy2sr−1 )Model all 2degModel PT=1Model PB PT=1PT=1103 104`10-1100101Model all 2degModel PT=5Model PB PT=5PT=5103 104`10-1100101Model all 2degModel PT=6Model PB PT=6PT=6103 104`10-1100101C ` (Jy2sr−1 )Model all 2degModel PT=7Model PB PT=7PT=7103 104`10-1100101Model all 2degModel PT=8Model PB PT=8PT=8103 104`10-1100101102 Model all 2degModel PT=16Model PB PT=16PT=16103 104`10-1100101C ` (Jy2sr−1 )Model all 2degModel PT=17Model PB PT=17PT=17Figure 7.10: C` measurements from simulations with mosaicked data. Theblack solid lines are from the input noiseless model image of the wholefield, the black dot-dashed lines are the input models for the individualpointings, and the black dashed lines are the individual input modelswith a primary beam applied. The solid coloured lines are the simu-lated noiseless visibilities from the individual pointings.1950.000.050.100.150.20D(Jy2sr−1 )0.0900.0950.1000.105P (Jy2 sr−1 )642β0.000.050.100.150.20D (Jy2 sr−1 )6 4 2βFigure 7.11: Parameter distribution (1D and 2D) results from MCMC fittingof Model 2 on the simulated noisy data with random positions andrandom flux assignment. The solid blue lines indicate the true inputvalues for each parameter and and back dashed vertical lines show themedians for the each parameter distribution.1960.120.180.240.300.36D(Jy2sr−1 )0.020.040.060.08P (Jy2 sr−1 )2.41.60.8β0.120.180.240.300.36D (Jy2 sr−1 )2.4 1.6 0.8βFigure 7.12: Parameter distribution (1D and 2D) results from MCMC fittingof Model 2 on the simulated noisy data, with clustered positions andrandom flux assignment. The solid blue lines indicate the true inputvalues for each parameter and and back dashed vertical lines show themedians for the each parameter distribution.197103 104`100101C ` (Jy2sr−1 )Expected Poisson levelStokes I PT=1103 104`100101Expected Poisson levelStokes I PT=5103 104`100101Expected Poisson levelStokes I PT=6103 104`100101C ` (Jy2sr−1 )Expected Poisson levelStokes I PT=7103 104`100101Expected Poisson levelStokes I PT=8103 104`100101Expected Poisson levelStokes I PT=16103 104`100101C ` (Jy2sr−1 )Expected Poisson levelStokes I PT=17103 104`100101Expected Poisson levelStokes I averageFigure 7.13: C` measurements from individual ATCA pointings with all cor-rections applied and with ATLAS point source models (S ≥ 150µJy)subtracted. The black solid line represents the expected Poisson am-plitude using the source count from Chapter 4 scaled to 1.75 GHz andeq. 7.21 with Scut = 150µJy.198issues being resolved with the use of the simulations. The weighting for these datais likely different than that used in the simulations as the data include phase andcalibration errors, as well as time dependent noise. The time dependent noise is aresult of the source approaching the horizon during the final few hours of the 12hour observation and some antennas being shadowed. This likely needs to be takeninto account in the visibility weights; we are currently investigating the optimalweighting in this case.It is clear from visual inspection that if there is clustering signal present, itsamplitude is weaker than the Poisson amplitude, and weaker than that adopted inour simulations. This does not mean there is no clustering signal, simply that itcould be entirely swamped by the Poisson contribution.Previous upper limits on the contribution to the unresolved background comefrom Fomalont et al. [60], Partridge et al. [137] and Subrahmanyan et al. [171]at 4.86, 8.4, and 8.7 GHz, respectively. We can compare results by looking atthe fractional rms or ∆T/T , the question being what T to use (i.e. the tem-perature predicted by ARCADE 2, predicted from source counts, or somethingelse). These limits are shown in Fig. 7.14, along with the ranges from the sevenATCA pointings. Work continues on the proper mosaicking treatment to makeone measurement from the seven. We show three different cases. The first usesT = TCMB = 2.7255 K. The second normalizes the C`s by the ARCADE 2 tem-perature at each frequency TAR2, computed using the fits from eq. 1.8 and eq. 1.9.The third shows the C`s normalized by the temperature predicted from integratingthe source count at each frequency Tcount.7.8 ConclusionsMeasuring the radio angular power spectrum is challenging using data from tradi-tional interferometers, which are effectively optimized for the detection of compactsources. Using realistic physical assumptions we created simulated skies with arange of features. Using these we created simulated interferometric observationswith the same setup as for our ATCA data. We have used these to investigate anddetermine the effects from (and corrections for) wide frequency coverage, uv sam-pling, the primary beam, mosaicking, and more. We have shown that by using the199103 104 105`10-510-410-310-210-1[`(`+1)C `/2pi]1/2(∆T/T CMB)Fomalont 88, 4.86 GHzPartridge 97, 8.4 GHzSubrahmanyan 00, 8.7 GHzPointing Min/Max, 1.75 GHzPointing Min/Max σ, 1.75 GHz103 104 105`10-410-310-210-1100[`(`+1)C `/2pi]1/2(∆T/T AR2)Fomalont 88, 4.86 GHzPartridge 97, 8.4 GHzSubrahmanyan 00, 8.7 GHzPointing Min/Max, 1.75 GHzPointing Min/Max σ, 1.75 GHz103 104 105`10-310-210-1100101[`(`+1)C `/2pi]1/2(∆T/T count)Fomalont 88, 4.86 GHzPartridge 97, 8.4 GHzSubrahmanyan 00, 8.7 GHzPointing Min/Max, 1.75 GHzPointing Min/Max σ, 1.75 GHzFigure 7.14: Normalized C` measurements to show ∆T/T from ATCA datawith all corrections applied and ATLAS models subtracted (S ≥150µJy), with upper limits from Fomalont et al. [60] (green dashedline), Partridge et al. [137] (purple solid line), and Subrahmanyan et al.[171] (blue triangle). The red regions are the maximum and minimumrange from the 7 pointings, while the grey regions are the range ofthe 1σ uncertainties. The top left panel shows the C`s normalized byTCMB = 2.7255 K. The top right panel is normalized by the ARCADE2 temperature at each frequency (TAR2), while the bottom panel usesthe temperature from the source count at each frequency, Tcount.200Bare Estimator to calculate the average C`s from the visibilities, we are able torecover the input power spectrum. Early attempts at model fitting seem promisingfor fitting the Poisson level, which is something that has not been measured in thisway at radio frequencies before. Moreover, there are indications that it may bepossible to detect the clustering signal as well.Work remaining on this project includes determining the optimal visibilityweighting and the correct measure of the uncertainty, considering things like phasesor calibration errors. We also need to understand how to properly combine andmeasure mosaicked data, the optimal fitting procedure, and how to fit for extendedemission. Once these issues have been explored, applying this technique fully toactual ATCA data and model fitting will be possible.It is clear that this type of experiment would benefit from a different observa-tional strategy. Observations covering a larger sky area would be optimal, sincewe are interested in clustering and large angular scale emission. Plus, getting moreshort baseline data will increase sensitivity to large scales. Also, observations ata lower frequency would be beneficial, since the majority of the emission is pro-duced via synchrotron processes, which become stronger at lower frequencies, thusincreasing the signal to noise. However, determining the optimal observing strat-egy for future measurements of radio clustering will require further study withsmaller-scale data sets, such as those we already have in hand.201Chapter 8Impact and Conclusions8.1 SummaryMy research lies primarily in investigating the faint (µJy) extragalactic radio sky. Inbroad terms this work involves statistically characterizing faint radio galaxies: howmany there are; what kind of galaxies they are; how large are these galaxies etc.Characterizing these rarely studied galaxies on both large and small angular scalescan tell us much about galaxy evolution, dynamics in galaxy clusters, differenttypes of galaxy populations, as well as the star formation history of the Universe.The radio-galaxy phase of galaxy evolution is a critical one because the feedbackfrom AGN significantly effects galaxy star formation thereby governing the growthand evolution of the galaxies. Studying the characteristics of radio emission tonew depths will help to better understand this crucial phase, and may also help toconstrain models of dark matter particles. There are also more specific issues tostudy such as the contribution of faint galaxies to the cosmic radio background:how the size of radio galaxies scales with brightness; how the far-infrared to radiocorrelation evolves (or otherwise) with redshift; and what we can learn from theradio power spectrum. In this thesis I have detailed some of the ways in which Ihave already started to investigate these areas.We have presented several different estimates of the radio source count ex-tracted directly from P(D) analysis at 3 GHz, and the count extrapolated to 1.4 GHzusing new deep data from the VLA. These involve different assumptions about the202count shape, e.g. the number of nodes, and different choices of data, namely thevarious noise zones. We have provided an estimate of the extragalactic discretesource contribution to the CRB at 3 GHz of (15 ± 1) mK and (120 ± 3) mK at1.4 GHz. We have also found an upper limit to the peak of any new discrete popu-lation of 50 nJy.Comparing our source count to known luminosity function models shows agood match in some flux density ranges whilst a poor fit in other. There is certainlyroom for improvement in the physical models. The Be´thermin et al. [10] modeltuned by considering both infrared and radio counts comes up too low in the ∼ 50to several hundred µJy region. This could indicate some kind of evolution betweenthe radio to infrared correlation (see below for more discussion on this). Furthermodelling of the luminosity function would be important for connecting the sourcecount to galaxy population evolution.Using the ATCA telescope we have found, after discrete source subtraction, aroughly 3σ excess in the 1.75 GHz image PDF, which could be caused by extendedemission. After fitting several models we find an upper limit on the contributionto CRB at this frequency from extended emission to be (10 ± 7) mK. We found alimit to any possible extended population that could cause the ARCADE 2 excessof 1µJy.The models used represent upper limits on the extended emission, and arevalid for sources with angular size of approximately 2 arcmin or less. Assum-ing the excess is truly from extended emission, rather than data artefacts, we dis-cussed some possible sources for the extended emission. These include individualgalaxy haloes from starburst or AGN galaxies, haloes from another population suchas dwarf spheroidals (or something unknown), and some contribution from clus-ters (or smaller mass groups) through emission structures such as radio relics andhaloes. We compared one model of cluster halo emission and found it is not ruledout by our data, although there is room for some additional emission besides thatprovided by the model. The amount of faint cluster emission is not well knownand depends on many factors (e.g. origin of the halo, dissipation timescales). Thistype of technique will be important for studying this issue, since for sources of thissize the confusion level is quite high, making imaging of individual faint objectsdifficult.203Modelling is required to see if any known objects could produce similar sourcecount shapes, either by means of ‘normal’ emission, i.e. synchrotron emissionfrom magnetic fields, or from something more exotic such as via dark matter anni-hilation. We showed an example of dark matter models from Fornengo et al. [63]and found them to be inconsistent with our data. It is clear that the resulting sourcecount for any WIMP dark matter model, for source sizes up to 2 arcmin, must liebelow the source count of current radio galaxies, at least for flux densities greaterthan around 10µJy.Using the deep VLA data at two different resolutions, we presented discretesource catalogues from these data along with an analysis of the source fitting anduncertainties using simulations. We have shown that, as long as the data are c