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The clustering and photometric properties of faint galaxies Woods, David 1996

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T H E C L U S T E R I N G A N D P H O T O M E T R I C P R O P E R T I E S O F F A I N T G A L A X I E S By David Woods B. Sc. (Physics and Astronomy) University of Toronto M . Sc. (Astronomy) University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF D O C T O R OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES ASTRONOMY We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA October 1996 © David Woods, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Astronomy The University of British Columbia 129-2219 Main MaU Vancouver, Canada V6T 1Z4 Date: Abstract A photometric survey of faint galaxies in three high Galactic latitude fields (each ~ 49 arcmin2) with sub-arcsecond seeing is used to study the clustering properties of the faint galaxy population. Multi-colour photometry of the galaxies has been obtained to magnitude limits of V ~ 25, R ~ 25 and I ~ 24. Two approaches are utilized to examine the clustering: close pair and angular correlation analysis. The number of close pairs of galaxies observed to faint magnitude limits, when com-pared to nearby samples, determines the interaction or merger rate as a function of redshift. The prevalence of mergers at intermediate redshifts is fundamental to under-standing how galaxies evolve and the relative population of galaxy types. Mergers have been used to explain the excess of galaxies in faint blue counts above the numbers ex-pected from no-evolution models. Using the deep imaging from one of the fields, a pair fraction is determined which is consistent with the galaxies in the sample being randomly distributed with no significant excess of close pairs. This is contrary to the pair fraction of 34% ± 9% found by other authors for similar magnitude limits and using an iden-tical approach to the pair analysis. Various reasons for this discrepancy are discussed. Colours and morphologies of our close pairs are consistent with the bulk of them being random superpositions although, as indicators of interaction, these criteria are uncertain due to contamination of field galaxies and difficulty in seeing galactic structure at inter-mediate redshifts, respectively. This observed lack of close pairs is probably linked with the decreasing amplitude of the angular correlation function at faint limits. If our faint samples are comprised of galaxies which have a higher average redshift than brighter samples studied by other workers then either the merger rate has been overestimated or n there is a change in its behaviour from what is observed at brighter magnitude limits. Angular correlation analysis is applied to magnitude-limited and colour-selected samples of galaxies from the three fields, for larger angular separations than those studied with close pair analysis. General agreement is obtained with other recent studies which show that the amplitude of the angular correlation function (u>(6)) is smoothly decreasing as a function of the R limiting magnitude. This decline of u)(6) rules out the viability of merger-dominated galaxy evolution models. The <JH{0) amplitudes calculated from our fields show stronger clustering than the measurements of Brainerd et al. (1995) in the same magnitude range. Using redshift distributions extrapolated to faint magnitude hmits, galaxy clustering evolution models are calculated and compared to the observed /-band u>(6). Faint galaxies are determined to have correlation lengths and clustering evolution parameters of either TQ ~ 4 h-1 Mpc and e ~ 0 - 1; r 0 ~ 5 - 6 h~x Mpc and e > 1; or r0 ~ 2 — 3 h_1 Mpc and e ~ —1.2. The latter case is for clustering fixed in co-moving coordinates and is probably unrealistic since most local galaxies are observed to be more strongly clustered. The first of the three cases has the most reasonable rate of clustering evolution but distinguishing the correct TQ for the faint galaxies is not possible with the current data. No significant variations in the clustering amplitude as a function of colour are detected, for all the colour-selected galaxy samples considered. The validity of this result is discussed in relation to other determinations of u>(9) for galaxies selected by colour. ru Table of Contents Abstract ii List of Tables vi List of Figures viii Acknowledgement x 1 Introduction - Recent Research on Faint Galaxies 1 1.1 Overview 1 1.2 Number Counts 2 1.3 Redshift Distributions 7 1.4 Clustering Analysis 10 1.5 This Thesis 13 2 Photometric Data Reductions and Analysis 16 2.1 Observations 16 2.2 Image Pre-processing and Summation 17 2.3 Calibrations 22 2.4 Generating FOCAS Catalogs 23 2.5 Final Photometry 29 2.6 Photometry Checks 37 3 Counting Close Pairs of Galaxies in N F l 39 iv 3.1 Introduction 39 3.2 Statistical Approach 41 3.3 Galaxy Pairs 43 3.4 Discussion 57 3.5 Conclusions 61 4 Correlation Analysis of Three High-Galactic Latitude Fields 63 4.1 Introduction 63 4.2 Measuring the Angular Correlation Function 70 4.2.1 Estimator 70 4.2.2 The Clustering Model 72 4.2.3 Star Removal 74 4.2.4 Integral Constraint 75 4.2.5 Star Dilution Correction 77 4.2.6 Combining Fields, Error Analysis and Fitting 77 4.3 Angular Correlation Function Results 80 4.3.1 Magnitude-Limited Samples 80 4.3.2 Colour-Selected Samples 89 4.3.3 Comparison with Models of the Spatial Correlation Function . . . 93 4.4 Discussion and Conclusions 99 5 Conclusions 105 5.1 Thesis Summary 105 5.2 Future Work in Faint Galaxy Research 107 References 109 v List of Tables 2.1 Coordinates for the centres of the three "blank" fields and times of the observing runs 17 2.2 Total exposure times and the average seeing in V, R and / for the three fields 17 2.3 Summary of the standard star fields used for calibration for each observing run 23 2.4 Slopes of the number counts, presented in Figs. 2.7-2.9, for the three fields in V, R and 1 31 2.5 Number of objects detected in each of three fields in V, R and I. The magnitude ranges are listed in brackets 35 3.1 Pair fractions for the bright samples in V, R and I for both probability cutoffs 50 3.2 Pair fractions for the faint samples in V, R and I for both probability cutoffs. 51 3.3 Listing of probabilities of galaxy pairs in Figs. 3.5 & 3.6 55 4.1 Final angular correlation function amplitudes calculated for V, R and / , for the faintest magnitude limited samples, with different assumed power law exponents, 6. The chi-square statistics (x2) are given for each fit as well 80 v i F i n a l angular correlation function amplitudes for V, R and I , and three different magnitude limits. Stellar dilution corrections are tabulated for each field, calculated for the magnitude ranges adopted i n the three band-passes vu List of Figures 2.1 V + R + I Image of N F 1 18 2.2 V + R + I Image of N F 2 19 2.3 V + R + I Image of N F 3 20 2.4 Calibration equation solutions for the N F 1 field 24 2.5 Calibration equation solutions for the NF2 field 25 2.6 Calibration equation solutions for the N F 3 field 26 2.7 V Number Counts for N F 1 , NF2 and N F 3 32 2.8 R Number Counts for N F 1 , NF2 and N F 3 33 2.9 I Number Counts for N F 1 , NF2 and N F 3 34 2.10 Galaxy colour histograms for NF1-3 36 3.1 Concentration Parameter vs. Aperture Magnitude for N F 1 45 3.2 V , R and I Number Counts for N F 1 47 3.3 Close Pair Histograms for "Bright" Sample of Galaxies in N F 1 48 3.4 Close Pair Histograms for "Faint" Sample of Galaxies in N F 1 49 3.5 Close Galaxy Pair Images in N F 1 53 3.6 Close Galaxy Pair Images in N F 1 54 3.7 Colour Distributions of Close Pairs in N F 1 58 4.1 Galaxy-Star Separation for NF2 and N F 3 76 4.2 Angular correlation functions for magnitude-limited samples in V 82 4.3 Angular correlation functions for magnitude-hmited samples in R 83 4.4 Angular correlation functions for magnitude-limited samples in 7 84 viii 4.5 Angular correlation amplitudes for V", R and I 86 4.6 Comparison of recent angular correlation function studies as a function of l imit ing magnitude, in R 87 4.7 for colour-selected galaxy samples 91 4.8 (V — J)-selected angular correlation functions 94 4.9 Comparison of / -band w(9) and galaxy clustering evolution models. . . . 97 4.10 Further comparison of the J-band u>{6) and models 100 ix Acknowledgement "You 're like a man who moves into a room and says 'Oh this is only temporary' and doesn't unpack his trunks. Now this is all right - for a time. But if he can't find a better place, or can't make up his mind to risk moving perhaps into another town altogether, then the thing to do is to unpack his trunks and settle down whether the room's good or not. For anything's better than to live in a state of waiting." - Ludwig Wittgenstein "I'm doing my thesis on them. If you do your thesis on syphilis, you end up loving even the Spirochaeta pallida." - Casaubon in 'Foucault's Pendulum' by Umberto Eco Along the long, twisting and sometimes tortuous path towards the completion of this thesis I have been encouraged, supported and entertained by a number of people. M y sister, Ruth, has been there for me during both the lows and the highs and I am very grateful for her unwaivering love and emotional sustenance. She has been my rock through many a storm and I would not have finished this thesis without her support. M y thesis research supervisor, Greg Fahlman, generously gave a great deal of valuable input and insight, in the form of many enjoyable and stimulating conversations. I also thank Greg for his seemingly limitless patience and for all the financial support he has provided over the years without which I would not have been able to do Ph.D. research. Also, in the prof dept., I extend my thanks to Gordon Walker for being the affable guy across the hall and for putting up with my diverse vocabulary. Jaymie Matthews (Dr. Libido) has given a substantial amount of comic relief over the years, along with vast quantities of beer and movie trivia. I thank all my friends for helping keep me a relatively sane, happy individual and for providing lots of distractions from the astronomical task at hand. Special thank yous to: Jana Pika ("Take risks"), Dave & Heidi Hogg (co hospitality and Friday video nights), Corinne Reimer ("wanna workout?"), Mike Luke (the original dude), Andy & Alex Petersen (a 2 + I2), Judy & Keir Buchan (California Dreamin'), Kathy Laird-Burns (Ms. Fate), Anne van Beers (fellow cynic), Peter McGillivray (climbing, hiking and biking McDude), Pam & Jeff Eppler (the nicest people I've ever met from B .C.) , Melony Ward (for unique email), Man Hoi Lee (the Princeton/CITA/Queen's connection) and Dave and SigaH Earn (for the Cambridge sabbatical and Israel updates). At U B C , I have been fortunate enough to work and become friends with many top notch graduate students (and their S.O.'s) over the years. Ted Kennelly deserves a special mention for innumerable dinners in the Village and for keeping me updated on the extent of my beer tab. Brad Gibson provided countless email updates, movie reviews and several enlightening discussions. Thanks also to: Dan Hurley (for Whitehorse weather advisories and his talent in sarcasm), Greg Burley and Janet Mumford (congrats to the newlyweds!!), Yiman Jiang & Andrew Walker ("Yandrew"), Rodrigo Ibata and Helene Pharabod ("Coffee?"), Alex Razoumov (for putting up with his annoying officemate(s)), Remi Cabanac (for his federalist fervour), Nick Ivanans (for his enthusiasm), Chris Koenig (for many tennis games) and Scott Chapman (for practising his violin when "I was ready to get up anyways"). A nod to "old-timers". Gordon Drukier and Claudia Mendes de Oliveira for initially helping to push me up the right path. And to all the other grad students at U B C , current and former, who have added to the learning experience. Lastly, I would like to thank James Brewer for leaving the country. I'd like to acknowledge the people who have discouraged me and underestimated my abilities over the years. Without this negative feedback I probably wouldn't have been so determined and stubborn about finishing this thesis. Additional sources of inspiration include: Rush, Robertson Davies, movies, John Steinbeck, Montreal Canadiens, softball, running, coffee, the Coastal and Rocky moun-tains. Finally, I would like to thank a small piece of land in Ontario. The fields behind the house where I grew up in Grand Valley provided many hours of awestruck wonder with the boundless darkness and fluorescent stellar vistas visible there. My love of astronomy was first developed during walks at night through the "back 40" and I am grateful for these memorable hours of simple observation and appreciation. Chapter 1 Introduction - Recent Research on Faint Galaxies 1.1 Overview Faint galaxies can be considered to be an analogue, in current astronomical research, to the mysterious "nebulae" that were the focus of so much debate in the 1920s. Edwin Hubble's (1926) epochal determination of the extragalactic nature of the nebulae began a new branch of observational astronomy which has flourished in the last 70 years. In much the same way that in the early days the photometric properties of nearby galaxies could be measured with the associated distances being of great controversy, we are now struggling with similar questions related to the true nature of faint galaxies. By the designation "faint", we refer to galaxies roughly within the apparent visual magnitude range of V ~ 20 — 25. For the faintest galaxies in this range, this corresponds to objects which are approximately 100 million times fainter in brightness than what can be seen with the naked eye at a site with a dark sky. Needless to say large telescopes are indispensable tools for studying and measuring the properties of faint galaxies. Much of the debate over the nature of these objects involves our ignorance of the typical distances to these systems, along with a lack of knowledge of the relative morphological popula-tions and the evolutionary histories which link faint galaxies to the galactic systems we observe nearby. With spectroscopic surveys currently limited to / ~ 22 (or V ~ 23) no conclusive distinction between faint galaxies being nearby dwarf galaxies or distant, 1 Chapter 1. Introduction - Recent Research on Faint Galaxies 2 numerous galaxies can be made as yet. Without redshifts for faint galaxies, the inform-ation contained in photometric data must be exploited with as many different analysis techniques as possible. However, it is the primary focus of this thesis to use imaging data taken to very faint limits with a large telescope (3.6-m Canada-France-Hawaii telescope) to obtain photometric samples suitable for a detailed study of the clustering of faint galaxies. By studying the clustering of this population the aim is to learn more about the properties and nature of its constituents. In this chapter, some perspective is provided with a brief summary of recent advances in faint galaxy research. For a detailed and comprehensive review the reader is referred to Koo and Kron (1992, K K ) . The following sections emphasize work which has been done in the last few years, and which was not discussed in K K . In the final section, an outline of the work in the remaining chapters of this thesis is presented. 1.2 N u m b e r C o u n t s With widespread use of charge-coupled devices (CCDs) now commonplace in astronomy, the most recent instrumentation advances which have revolutionized imaging capabilities are large-format (2k x 2k) CCDs and C C D mosaic cameras (e.g., M O C A M and U H C A M at C F H T ) . The greater areal coverage of sky made possible by these new cameras means that CCDs will soon completely replace photographic plates, since wide-field imaging ca-pability was the last advantage the plates had over electronic detectors. The first photo-graphic surveys performed, with the goal of imaging faint galaxies, were nevertheless very impressive given the magnitude limits reached and numbers of objects obtained (Tyson and Jarvis 1979, Peterson et al. 1979 and Kron 1980). Kron (1980), in his classic paper, laid out in detail techniques necessary for measuring magnitudes and colours of faint galaxies. The Kron observations included photometry, in two bandpasses, of ~ 20, 000 Chapter 1. Introduction - Recent Research on Faint Galaxies 3 galaxies from two high-galactic latitude fields, each with an area of ~ 1080 arcmin2. These data were complete to magnitude hmits of Bj ~ 23.5 and rp ~ 22.5. To substan-tially improve upon these photometric Hmits required the introduction of CCDs. Over the last decade there have been a large number of groups doing faint galaxy count studies with CCDs and 4-m class telescopes. Number counts have been done in a variety of filters, including B (Stevenson et al. 1986; Tyson 1988, T88; Heydon-Dumbleton et al. 1989; Maddox et al. 1990; Jones et al. 1991; Metcalfe et al. 1991; Lilly et al. 1991, L C G ) , 7/ (HaU and Mackay 1984, T88, L C G , LiUy 1993) and K (Mobasher et al. 1986, Glazebrook et al. 1994, L C G , Cowie et al. 1994, Djorgovski et al. 1995, Glazebrook et al. 1995a). This list of references is far from exhaustive for the given bandpasses and does not include the work which has been done in U, V and R. Given all this expenditure of effort and telescope time, the number counts of faint galaxies as a function of magnitude are very well measured in optical and IR (K) filters. Tyson (1988, T88) was one of the first to observe an excess of counts above what would be expected for no-evolution (NE) models of galaxies. He identified the excess as being due to blue, faint galaxies which dominated the counts at the magnitude limits of his sample. Lilly et al. (1991) and Lilly (1993), using data taken with better seeing and less drastic completeness corrections than in T88, convincingly showed that the excess in the number counts is observed in the B band, while in redder filters, such as I and K, the significance of this discrepancy diminishes. The Lilly et al. study also showed that the faint galaxy population did not have as excessively blue colours as Tyson's work had originally suggested. These results from number counts are not surprising since we expect galaxies to evolve at some level over cosmological time-scales in number and/or luminosity (Tinsley 1977). Also, number count variations at longer wavelengths are expected to be partly caused by older stellar populations dominating the integrated light at longer wavelengths. The observed number counts become a real conumdrum when recent determinations of the redshift distribution Chapter 1. Introduction - Recent Research on Faint Galaxies 4 for faint galaxies, in the same magnitude range, are considered since these distributions are found to agree with NE models (Broadhurst et al. 1988, BES; Colless et al. 1990, 1993). This point is illuminated further in the next section. Lilly (1993) discusses the three standard scenarios which have been suggested to explain the obvious evolution in the counts and the lack thereof in the measured redshift distributions. First, the faint blue galaxies are explained to be proto-dwarf galaxies undergoing bursts of star formation at intermediate redshifts (z ~ 0.4) and then evolving into galaxies at the faint end of the luminosity function (L < 0.012/*, where a L* galaxy has an absolute visual magnitude of My ~ —20) by the current epoch (BES). The second model has faint galaxies being very shortlived, star-bursting objects which are subsequently disrupted or fade away in such a fashion that they aren't observed in large numbers at small redshift (Babul and Rees 1992). The final conventional model invokes merging of sub-galactic units at intermediate redshifts where current L* galaxies are the products of this process (Broadhurst et al. 1992). We will hereafter refer to these three frameworks as the "bursting dwarfs", "fading dwarfs" and "merger" models. The bulk of current, ground-based studies are concerned with measuring redshifts for the population at faint magnitude limits to test the above models. Looking for increases in the clustering of faint galaxies, using various selection criteria (small angular separation, colour etc.), is another technique which can be used to check the viability of the merger galaxy evolution model. Dwarf models can be tested by comparing the clustering behaviour of the various local and faint galaxy populations. A full outline and discussion of the clustering analysis approach is given in section 1.4 below, and in Chapters 3 and 4. More unconventional explanations for the number counts and redshift distributions evolution discrepancy include: the assumption that the local luminosity function (LLF) is not well defined and therefore one finds the optimal LLF using a least-squares tech-nique and the observed number counts, redshifts and colours of fainter galaxies (Koo, Chapter 1. Introduction - Recent Research on Faint Galaxies 5 Gronwall and Bruzual 1993, Gronwall and K o o 1995); the presence of a non-zero cosmo-logical constant (Yoshii 1993); or the existence of a significant population of low surface brightness galaxies not typically detected locally, due to observational selection effects, but found i n faint, photometric surveys (McGaugh 1994, Ferguson and M c G a u g h 1995). The first two alternative models cannot be adequately tested with the data and analysis presented here, so most of the discussion of galaxy models i n the remainder of this thesis wi l l concern the bursting and fading dwarfs, merger and low surface brightness galaxies scenarios. Further work on galaxy number counts i n the last few years has mostly concentrated on improving the K counts (Gardner et al. 1993, Cowie et al. 1994, Glazebrook et al. 1994, Soifer et al. 1994, Djorgovski et al. 1995) and pushing the ground-based, optical counts to fainter levels with 4-m class telescopes (Driver et al. 1994, Metcalfe et al. 1995) and the new 10-m Keck telescope (Smail et al. 1995). Gardner et al. (1993) surveyed the i f - b a n d number counts from 12 < K < 23 and found the slope of the counts changes at K ~ 17 from 0.67 to 0.26. The median B — K colour was also found to have a turnover at K ~ 17 and rapidly became bluer towards fainter magnitudes. This dominance of blue galaxies at faint hmits allows the K counts to be easily fitted with N E galaxy models while a large discrepancy is evident between the observed B counts and the N E models. The faintest K number counts yet obtained (Cowie et al. 1994 and Djorgovski et al. 1995) agree with theoretical models containing a small amount or no evolution for low values of Q,0, but considering model uncertainties the constraints on Q 0 are preliminary. Metcalfe et al. (1995) collected data which enabled them to measure the B number counts down to the deepest ground-based magnitude l imit yet achieved (B ~ 27.5). The primary result of this study is that a change i n the slope of the counts is detected at B ~ 25, wi th the number counts continuing to increase at fainter l imits . This flattening of the slope is interpreted as being due to z > 1 galaxies dominating the Chapter 1. Introduction - Recent Research on Faint Galaxies 6 counts. Smail et al. (1995) have also obtained deep optical galaxy counts (V, R and I ) using the Keck telescope. Beyond R ~ 24.5, (V — R) galaxy colours are observed to be redder, ( V — /) remains constant and (R — I) values continue a bluing trend. These variations are consistent with a flattening of the V counts slope which is measured for the faintest l imits. Smail et al. conclude from observed large galaxy densities and small sizes that the bulk of the faintest galaxies are consistent with being dwarf galaxies or subgalactic units. The Hubble Space Telescope (HST) has proven to be a valuable addition to the tools available for studying the faint galaxy population. Images from the M e d i u m Deep Survey ( M D S ) , a key project with H S T (Griffiths et al. 1994), have been used to determine number counts of faint field galaxies as a function of morphology (Driver et al. 1995, Glazebrook et al 1995b) down to 7. ~ 24.25 and / ~ 22 (respectively). B o t h M D S studies find counts for ellipticals and early-type spirals which are consistent with l itt le or no evolution. However, with late-type spirals and irregular galaxies the observed number counts were found to be significantly in excess of N E models. Driver et al. suggest that this excess can be explained with a local luminosity function which is rich i n dwarf galaxies or /and strong evolution occurring in a large portion of late-type galaxies. H S T has also been used to obtain what are currently the deepest images of the sky ever taken i n optical bandpasses. The public release in January, 1996 (Will iams et al 1996) of these spectacular images of the "Hubble Deep F ie ld" ( H D F ) has generated a flurry of papers on the exceedingly faint galaxy population detected i n U, B, R and I. A b r a h a m et al (1996) determined that the steeply rising number counts for "irregular/peculiar/merger" galaxies reported i n Glazebrook et al (1995b) continued out to at least I ~ 25. Also , spiral galaxies towards the fainter limits exhibited a more substantial excess over N E models while elliptical/SO galaxy counts were found to be still roughly consistent with no evolution. One caveat to these results is that the irregular/peculiar/merger class of Chapter 1. Introduction - Recent Research on Faint Galaxies 7 galaxies being counted may not be entire systems but fragments of galaxies, such as giant HII regions (Colley et al. 1996). Deep number counts wi l l continue to be an important constraint on galaxy evolution scenarios and cosmological models, especially any observed changes in the slope for a given bandpass at faint Hmits. However, the counts must be interpreted i n tandem wi th other observables such as the correlation function, redshift distributions and colours to conclusively estabHsh the nature of the faint galaxy population and its Hnk with nearby galaxies. 1.3 Redshift Distributions The acquisition of redshifts for faint galaxies requires a large amount of telescope time such that numerous spectra with sufficient signal-to-noise can be obtained. This difficulty has been mitigated with the multiplexing advantage of multi-object spectrographs which use fibers or sHtlets cut in a mask to optimize the number of objects obtainable i n one field. One of the first redshift surveys to faint Hmits with this k ind of instrumentation was done by Broadhurst et al. (1988, B E S ) . Spectra were obtained of over 200 field galaxies which occupied the magnitude range 20 < bj < 21.5. The resultant redshift distribution was consistent with predictions of no-evolution galaxy models. This was a surprising result given that the number counts, as mentioned previously, clearly exhibited evolution. Extensions of the measured redshift distribution to fainter Hmits (bj ~ 22.5) by CoUess et al. (1990, 1993) agreed with the findings of B E S , i n that N E models fit weU, suggesting that at least half of the excess population seen i n the counts should have z < 0.5. LiUy (1993) and Tresse et al. (1993) did spectroscopic surveys down to magnitude Hmits of I ~ 22, of smaU numbers of galaxies (50 and 44, respectively). B o t h of these Chapter 1. Introduction - Recent Research on Faint Galaxies 8 studies confirmed that / -band selection, which is based on old stellar populations in the galaxies, led to similar results as those obtained with jB-selected samples, wi th the resultant redshift distributions being easily modelled with no evolution. The redshift surveys of L i l ly (1993) and the Meudon group (Tresse et al. 1993) progressed into a collaboration dubbed the Canada-France Redshift Survey ( C F R S ) . The C F R S was designed to yield redshifts for a large, well defined sample of galaxies to a magnitude limit of IAB < 22.5 (/ < 22) and with a median redshift of z ~ 0.6, using the Canada-France-Hawaii telescope ( C F H T ) . In total, the C F R S (Li l ly et al. 1995b, C F R S I; Crampton et al. 1995, C F R S V ; L i l l y et al. 1995c, C F R S V I ) measured the redshifts for 591 galaxies taken from five different fields over the magnitude range of 17.5 < IAB < 22.5, with an identification success rate of 85%. This is one of the largest redshift surveys of faint galaxies to date, where special attention has been paid towards selection effects such as the surface brightness l imit of the galaxy detection and the loss of galaxy redshifts due to the Hmits of the spectral range. The primary conclusion, out of a number of interesting results, of the C F R S is that the luminosity function ( L F ) of the red galaxies i n the sample is found to change very Httle in number density or luminosity for 0 < z < 1, while the blue galaxy L F shows significant evolution for z > 0.5 ( C F R S V I ) . LiUy et al. found their blue L F appears to brighten ~ 1 mag. by 0.5 < z < 0.75 and steepen at faint levels. This blue galaxy L F evolution would appear to be consistent with the fainter H S T - M D S number counts studies, where late-type galaxies are proposed to be responsible for most of the evolution of faint galaxies. The C F R S results suggest that at low redshift (z < 0.2) galaxies with MAB(B) ~ —18 could be the descendants of the evolving faint blue galaxies observed at higher redshifts. Redshift distributions extrapolated to / ~ 24 in C F R S V I are used to generate models of the correlation function for faint galaxies in Chapter 4. Other recent work on redshift distributions includes the i f - b a n d selected surveys by Chapter 1. Introduction - Recent Research on Faint Galaxies 9 the Hawaii (Songaila et al. 1994) and Edinburgh groups (Glazebrook et al. 1995a). The Hawaii group obtain complete redshift information for a sample of 298 galaxies down to B ~ 26, I ~ 22.5 and K < 18. The redshift distribution for K < 18 is weU fit by a no luminosity evolution model. Therefore, no indication of significant positive luminosity evolution between z — 0 and 1 is found except for an evolution i n spectral features which suggests there was more star formation occurring at z — 1 than there is now. The objects selected for spectroscopy in Glazebrook et al. (1995a) are from an imaging o survey encompassing 552 arcmin2 and complete to a 5cr l imit of K ~ 17.3. Redshifts for 124 galaxies were determined. Evidence for evolution of the K - b a n d luminosity function is not found for z < 0.5 but is for redshifts higher than this, wi th M £ measured to be ~ 0.75 mag brighter at z = 1. The observed evolution is contrary to what would be expected in simple merger models of faint galaxies. A determination of the B -band galaxy L F from z ~ 0 — 0.75 was completed by Ellis et al. (1995) util izing a collection of faint galaxy redshift surveys. This compendium, referred to as the "Autofib Redshift Survey", contained a total of 1700 redshifts for galaxies with 11.5 < bj < 24.0. The principal result of this study is that the 73-band L F is found to evolve with redshift. A steepening of the faint-end slope of the L F is detected, this being a consequence of the density of star-forming galaxies decreasing by a factor of 2 since z ~ 0.5, as orginally postulated by Broadhurst et al. (1988). In contrast, wi th [Oil] emission used as a star formation indicator, the L F of "quiescent" galaxies is found to be relatively unchanged from z ~ 0.5 to the present time. A burgeoning area of research is to use the combination of spectra taken with the 10-m Keck telescope and the high resolution images provided by H S T to investigate the nature of faint galaxies to unprecedented depths (Forbes et al. 1996, Cowie et al. 1995 and K o o et al. 1996). Although samples i n these studies are stil l sparse, this approach holds much promise for comparing the local galaxy population to galaxies at Chapter 1. Introduction - Recent Research on Faint Galaxies 10 intermediate and high redshifts. Obtaining spectra and redshifts of faint galaxies is a critical step to understanding which of the three standard models mentioned i n section 1.2, or combinations thereof, is the most accurate description of how galaxy evolution proceeds with lookback time. W i t h the new generation of 8 and 10-m telescopes coming on-line as well as new multi-object spectrographs (e.g., 2dF on A A T ) being built , there is stil l much that can be learned about faint galaxies using spectroscopic techniques. 1.4 Clustering Analysis Largely due to the influential work of Peebles (1980) and the simplicity of its application, the angular correlation function (u>(0)) is the most popular galaxy clustering observable. B y measuring the redshift distribution for the magnitude range of the galaxies being studied, one can then determine the spatial correlation function (£(r); see Chapter 4 for more background on clustering). A s with number counts and redshift surveys, there has been a large amount of research done on the clustering of faint galaxies i n the last few years. Before the current plethora of C C D imaging studies to measure u>(9) there were several groups measuring galaxy clustering with photographic data down to magnitude hmits of B ~ 24 (Koo and Szalay 1984, Stevenson et al. 1985 and Pritchet and Infante 1986). The first study to measure u>(d) to faint Hmits with C C D imaging and which subsequently generated considerable interest and activity i n this area, was the work of Efstathiou et al. (1991, E B K T G ) . Using 12 deep C C D fields obtained by Tyson and Seitzer (1988) with a total sky coverage of 107 arcmin2 i n Bj, R and I, along with one field from a large-format C C D (49 arcmin2) in Bj, they determined u>(9) at an angular separation of 30" for the faint, blue galaxies discussed by T88. This faint population was observed to be very weakly clustered causing E B K T G to suggest that either they were members of a new population which faded away by the current epoch (e.g., fading Chapter 1. Introduction - Recent Research on Faint Galaxies 11 dwarfs model), that galaxy clustering growth towards the locally observed <*>(#) was much more rapid than from the simple gravitational instability picture or the universe was significantly different from the Einstein-de Sitter model. The ini t ia l study of E B K T G was quickly followed by several other papers from groups which measured u>(6) to faint limits (Neuschaefer et al. 1991, Pritchet and Infante 1992, Couch et al. 1993, Roche et al. 1993, Roukema and Peterson 1994), and de-termined clustering behaviour at intermediate magnitude limits (e.g., Bernstein et al. 1994, to Bj < 22.5). Recent work has also concentrated on measuring the clustering for infrared-selected galaxy samples (Lidman and Peterson 1996, / -band; Carlberg et al. 1996, i f -band) and calculating the amplitude of the spatial correlation function with newly obtained redshift distributions in tandem with faint galaxy photometry (Hudon and L i l l y 1996, Le Fevre et al. 1996, Shepherd et al. 1996). A number of studies have re-evaluated the oft-quoted work of Davis and Peebles (1983) on the locally observed correlation function. Loveday et al. (1992, 1995) used the S t r o m l o - A P M Redshift Sur-vey to present results on the local, spatial correlation function for their entire galaxy sample and for galaxies selected by morphology and luminosity, respectively. Redshift surveys of galaxies selected from I R A S catalogues have also been used to calculate the local correlation function (Saunders et al. 1992, Fisher et al. 1994). Measurements of the two-point spatial correlation function for nearby galaxies are important for studies of faint galaxies since they allow the extent of clustering evolution to be assessed, and i n turn help constrain large scale structure models. The deepest, ground-based images used to date for correlation analysis are those obtained by Brainerd et al. (1995, B S M ) and Metcalfe et al. (1995, M S F R ) , wi th respective magnitude limits of r < 26 and B < 27. The trend of a monotonically decreasing clustering amplitude as a function of the survey magnitude l imit continues to the limits of the B S M data. M S F R claim a flattening i n u>(6) around the limits of Chapter 1. Introduction - Recent Research on Faint Galaxies 12 their data, albeit with large random and potential systematic errors. They suggest that this observation and a change i n the number counts slope are indicative of an increasing fraction of low-luminosity galaxies at z > 1. A new approach for determining the variations i n clustering of different galaxy pop-ulations at faint magnitude hmits, where morphologies are difficult to discern, is to calculate oo(9) for colour-selected samples of galaxies. Landy, Szalay and K o o (1996, see also Bernstein et al. 1994) and Roche et al. (1996) have looked at the clustering of galaxies chosen by their colour for intermediate (20 < Bj < 23.5) and faint (B ~ 25.5, R ~ 24.5) magnitude Hmits. In section 4.3.2, the clustering of colour-selected samples of the galaxies studied in this thesis are investigated and compared to similar analyses. FinaUy, H S T data have been used to measure OJ(9) at faint Hmits. Neuschaefer et al. (1995b), wi th pre-refurbishment W F P C data obtained for the M e d i u m Deep Survey, investigated the clustering of galaxies down to I < 23 for angular separations 0.7" < 6 < V. They also examined the ampHtude of UJ(9) as a function of galaxy colour and morphological type. Clustering studies of the galaxies in the Hubble Deep Fie ld images have also been produced by CoUey et al. (1996) and ViUumsen, FreudHng and da Costa (1996, V F d C ) . As alluded to previously, the CoUey et al. work finds an increase i n the correlation ampHtude for galaxies with small sizes, blue colours and colours indicative of high redshift, suggesting that these clustered objects are actuaUy fragments of galaxies such as giant HII regions. V F d C calculate u>(9) for ~ 1700 galaxies i n the H D F down to an impressive magnitude Hmit of R ~ 29.5. The clustering ampHtudes measured at brighter magnitude Hmits are consistent with ground-based results albeit wi th substantial errors, due to the small galaxy sample. Determining the change i n the ampHtude of the two-point angular correlation function for faint populations relative to local galaxy samples is important for understanding which of the three "standard" models, if any, best describe faint galaxy evolution. For Chapter 1. Introduction - Recent Research on Faint Galaxies 13 example, i n some merger model scenarios w(0) is predicted to level off at magnitude limits where the faint galaxies dominate (Carlberg and Chariot 1992). The rate of change of clustering with magnitude l imit is also an indirect constraint for structure formation models. Studying the incidence of close pairs at small angular separations helps trace the merger rate of galaxies with lookback time. A discussion of close pair analysis and the application of it to one of the fields i n our sample is presented i n Chapter 3. Measuring the strength of the angular clustering of galaxies, with close pair and correlation analysis, at very faint magnitude limits is the primary purpose of this work. The motivation for the current work and an outline of this thesis is given i n the next section. 1.5 This Thesis Ideally redshift information is required to properly trace the clustering evolution of galax-ies from earlier epochs. However, the greatest difficulty in collecting spectral information for faint galaxies is the substantial amount of observing time needed to obtain sufficient signal-to-noise for accurate redshifts. Even with the multiplexing advantage of mult i -object spectrographs, executing large, deep redshift surveys is a very time consuming process. Although knowledge of the redshift distribution of galaxies down to a faint magnitude l imit is extremely valuable, much can stil l be learned from photometric data alone. The clearest advantage for studying faint galaxies solely with photometry is that more galaxies can be observed to significantly deeper magnitude limits than with spec-troscopy. The large galaxy samples provided by deep photometry can be util ized in a variety of ways including determining the overall number density of galaxies as a func-tion of magnitude with number counts, studying the stellar population mixtures and star formation rates within faint galaxies by measuring colours and analysing the clustering of faint galaxies relative to local galaxy samples to better understand the evolution of Chapter 1. Introduction - Recent Research on Faint Galaxies 14 galaxies and large scale structure over a range of epochs. Faint galaxy number counts and colours have been well measured by many authors. Therefore, i n the current work we concentrate on the clustering analysis of faint galaxies since the angular correlation function is stil l not accurately determined at the faintest magnitude l imits. Also , wi th the multi-colour (V, R and /) data used in this study we can calculate u(9) for colour-selected samples, an approach which only recently has begun to be exploited. In this thesis, high-resolution C C D images of three high Galactic latitude fields are analysed for clustering at small and larger angular separations by counting close pairs and measuring the correlation function, respectively. The first technique helps establish the change i n the merger rate of galaxies with magnitude l imit (or ~redshift) and it is applied to one of the three fields in Chapter 3. Combining angular correlation analysis of the three fields yields a robust measure of faint galaxy clustering as a function of magnitude and colour, as described i n Chapter 4. Both approaches for estimating clustering can be used to compare local galaxies to faint galaxy populations so that the viabil i ty of galaxy evolution models can be checked. This study is unique i n that it measures galaxy clustering to very faint magnitude Hmits (V ~ 25, R ~ 25 and I ~ 24) as weU as having colour information for the galaxies. A few groups have achieved clustering estimates at these depths but only one or two have attempted to determine the clustering ampHtudes for colour-selected, faint galaxy samples. A comparison of our results with other faint galaxy studies is given in later sections. The outHne for this thesis is as foUows. Chapter 2 gives a discussion of the pre-processing, reductions and analysis required for the photometric data presented i n this work. In particular, the novel procedures adopted to ensure uniform galaxy detection over the three fields studied are summarized i n detail. Number counts and colour distributions for each field are also shown in this chapter. Chapter 3 has a description of a close pairs analysis for one of the three fields. A fuU discussion of the technique, results and their Chapter 1. Introduction - Recent Research on Faint Galaxies 15 consequences is given. Chapter 4 contains a summary of the data analysis techniques and various corrections required for measuring the angular clustering of the faint galaxies in the three fields, along with some theoretical background. Results for both magnitude-l imited and colour-selected samples of galaxies are presented, as well as a comparison with a model of the spatial correlation function. In Chapter 5 the final conclusions from the thesis are summarized and potential future work in faint galaxy research is proposed. It should be noted that Chapter 3 and part of Chapter 2 of this thesis have been previously published as a paper i n The Astrophysical Journal (Woods et al. 1995). Also , Chapter 4 and portions of Chapter 2 comprise the content of a paper submitted to The Astrophysical Journal in October, 1996 (Woods and Fahlman 1996). C h a p t e r 2 P h o t o m e t r i c D a t a R e d u c t i o n s a n d A n a l y s i s 2.1 O b s e r v a t i o n s The V, R and I images used i n this study were obtained at the prime focus of the Canada-France-Hawaii telescope using F O C A M and the L I C K 1 and L I C K 2 large-format, 20482 C C D s from 1991 A p r i l - 1993 March . The image scale for both L I C K devices is 07207 per pixel so the ful l field-of-view of the C C D is ~ 7' on a side. These images of high Galactic latitude "blank" fields were originally obtained for a survey of Population II halo stars (see Richer and Fahlman 1992) but are also useful for studying properties of the faint galaxy population. Three fields with north Galactic latitudes were observed and are dubbed N F 1 , N F 2 and N F 3 , while data were also obtained for one field with a south Galactic latitude labelled SF1. The observing run for the SF1 field i n 1992 November was compromised considerably by poor weather, resulting in a small number of C C D frames collected on one clear night with seeing typically exceeding 1". The sparse numbers of program frames obtained were insufficient to properly pre-process the data. Therefore, we disregard the SF1 field for the remainder of this paper and concentrate our analysis on the north Galactic latitude fields, N F 1 , N F 2 and N F 3 . The right ascension and declination, and the corresponding Galactic coordinates, of the centres of these three "blank" fields are given i n Table 2.1, along with the time of the observing runs. Fields were specifically chosen to have no observable objects on the P O S S photographs and a lack of any Zwicky clusters. Seeing for the frames used i n the final summed images 16 Chapter 2. Photometric Data Reductions and Analysis 17 Blank Fie ld <*1950 ^1950 / b Dates Observed N F 1 13 / l 10 m l(K80 +43°01'06'/0 109.0° +73 .8° 1991 A p r i l 7-11 N F 2 15 f c39m01'50 +24°47'3670 39.0° +51 .9° 1992 June 4-6, 1993 M a r c h 25 N F 3 12' l29mll=70 +02°07'42"0 291.5° +64 .3° 1993 M a r c h 23-25 Table 2.1 Coordinates for the centres of the three "blank" fields and times of the observing runs. A e / / ( A ) AA (A) N F 1 N F 2 N F 3 V 5430 900 9600s 0794 9000s 0782 4800s 0772 R 6485 1280 9600s 0"89 6300s 0784 6000s 0766 I 8320 1950 12000s 0783 6000s 0776 8400s 0773 Table 2.2 Total exposure times and the average seeing in V, R and / for the three fields. is uniformly excellent, ranging from 075 to 1". Good seeing is essential for acquiring deep images in a reasonable amount of exposure time. The filter bandpasses used, and the total exposure times and average seeing for the summed frames i n each colour and field, are summarized in Table 2.2. Summed V + R + I frames which demonstrate the total multi-band exposure for each field are shown in Figs. 2.1-2.3. Note the cosmetic differences between the L I C K 1 (NF1) C C D and the L I C K 2 device ( N F 2 and N F 3 ) . 2.2 I m a g e P r e - p r o c e s s i n g a n d S u m m a t i o n A t the telescope we adopted an observational approach similar to other deep field studies. Exposures were obtained typically for 900-1200s, i n between which the telescope would be moved a few arcseconds i n alternating cardinal directions. This "dithering" was necessary i n order to use the program frames for generating a "sky-flat" i n each bandpass, for flat-fielding purposes. Various programs from the IRAF package were used to do the pre-processing of the C C D images. The individual frames were de-biased for the instrumental Chapter 2. Photometric Data Reductions and Analysis 18 Figure 2.1 V+R+I Image of N F l . Total, summed V+R+I image of the N F 1 field with each bandpass normalized to an equivalent flux level. The areas of the C C D frame with cosmetic defects, saturated stars and very bright galaxies are masked out and not considered in the image analysis. Chapter 2. Photometric Data Reductions and Analysis 19 Figure 2.2 V+R+I Image of NF2. Total, summed V+R+I image of the N F 2 field with each bandpass normalized to an equivalent flux level, as in F i g . 2.1. Chapter 2. Photometric Data Reductions and Analysis 20 Figure 2.3 V + R + I Image of N F 3 . Total, summed V+R+I image of the N F 3 field with each bandpass normalized to an equivalent flux level, as in F i g . 2.1. Chapter 2. Photometric Data Reductions and Analysis 21 dc level, as determined from the C C D bias region, using the linebias routine. A bias frame was also subtracted from each program frame to correct for pixel-to-pixel variations. Each individual frame was divided by a sky-flat which was generated by taking a median of all the program frames for the particular bandpass. Some modifications to this flat-fielding step had to be made depending on the data set being pre-processed. For the N F 1 (1991 Apr i l ) observing run the median of the program frames exhibited "hotspots", due to a bright galaxy and three saturated stars, which had to be removed. This was achieved by masking out the hotspot artifacts and then setting the mask to the adjacent background level. The smooth, final sky-flat was then used as an i l lumination correction frame for the domeflats to obtain a flat-field which could be used on the program frames. During the N F 2 (1992 June) run there was an insufficient number of program frames obtained such that a satisfactory sky-flat could be produced. To flat-field these data we fortunately were able to use some program frames from an adjacent F O C A M observing run, collected by Simon L i l l y and Dan Hudon, and kindly provided by D a v i d Schade. W i t h these additional frames a suitable sky-flat could be produced for each bandpass. The 1993 M a r c h ( N F 3 / N F 2 ) data were simply flat-fielded in the normal manner. Following the flat-fielding step, the individual frames were registered and aligned using the IRAF routine imalign and several bright stars i n each field. The alignment interpolation was performed to the nearest pixel. Combination of N F 2 data from the 1992 June and 1993 March runs introduces an additional complication into the alignment. There is a slight rotation (~ 1°) between the images obtained of N F 2 from these two runs, so, geomap and geotran were utilized to remove this discrepancy. Program frames, for each bandpass and field, were summed together using imcombine where the final, "summed" frame is actually an average of all the input frames, which are scaled according to their exposure time. C C D frames with poor seeing were not included in the final summed image to avoid any resolution degradation. F ina l combined frames were always found to Chapter 2. Photometric Data Reductions and Analysis 22 be flat to within <^  1%. Image flatness is an important feature of our processed images i n that it allows accurate faint galaxy photometry to be determined. Cosmic rays were also removed, as a final step, using the IRAF routine cosmicrays. 2.3 Calibrations Standard star fields were observed on every night of each observing run. The fields used are summarized in Table 2.3, with the pertinent references. Calibration solutions for each of the three observing runs are plotted in Figs. 2.4-2.6. Least squares fits to the measured standard stars are shown by the dot-dashed fines. The N F 1 , N F 2 and N F 3 labels i n Figs. 2.4, 2.5 and 2.6 correspond to the 1991 A p r i l , 1992 June and 1993 M a r c h observations, respectively, to denote the primary field observed during the run. Subscripts of ordi-nate variables which read "instr" indicate the instrumental magnitudes for a particular bandpass, while the capitalized variables are the standard magnitudes. Note that the N F 2 zero point (vinstr — V axis i n the top plot of F ig . 2.5) is significantly different from the two other fields due to a large change i n the gain of the L I C K 2 device, which was a result of a blue-sensitive coating being removed between the N F 2 and N F 3 observing runs. The zero points in the calibration equations were typically very stable, not varying more than ~ 0.2 mag between observing runs. Colour terms found for the calibration solutions for the three runs were accounted for by small offsets (< 0.05) assuming a mean colour for the faint galaxies or were disregarded due to the calculated coefficient being negligible. Stetson's (1987, 1990) D A O G R O W and D A O P H O T programs were used to determine accurate aperture corrections for the standard star frames. The only depar-ture from standard calibration techniques necessary was i n the case of the N F 2 V and / final frames, which were comprised of images from two different observing runs. A p -proximately 20 stars in the N F 2 field were chosen to be secondary standards. Magnitude Chapter 2. Photometric Data Reductions and Analysis 23 Observing Run Standard Star Fields 1991 April M67 (Montgomery et al. 1993, Schild 1983), N G C 4147 (Christian et al. 1985 [C85]) 1992 June N G C 4147 (C85), SA 110 (Landolt 1992 [L92]), SA 113 (L92), M92 (L. Davis, priv. comm. [D]) 1993 March SA 98 (L92), M92 (D), G12-43 (L92), RU 149 (L92) Table 2.3 Summary of the standard star fields used for calibration for each observing run. offsets between these secondary standards photometered on a single frame, from the 1992 June run, and the final, averaged frames were determined. These small offsets (< 0.05 mag) were found to be colour independent and therefore could be applied directly to the magnitudes obtained from the final averaged V and I frames for NF2, to correct for the variations between the two observing runs. 2 .4 G e n e r a t i n g F O C A S C a t a l o g s The objects in our final C C D frames were detected and analysed using F O C A S (Jarvis and Tyson 1981) routines with slight modifications to the standard analysis procedures, some of which are outlined in Valdes (1983, 1993). The two key input parameters for FOCAS are the detection threshold, given as a multiple of the sky variance (<r,fcy), and the minimum area for the objects detected, in numbers of pixels. After experimenting with these parameters and ensuring that spurious detections were minimized, we ad-opted a threshold of 2.5o~sky and a minimum area which corresponded to the seeing disk for the poorest resolved frame from the three bandpasses taken of a given field (i.e., Amin = K(HWHMpSf)/(O'!207pixel~1)2 where Amin is the minimum area obtained and FWHM — 2 * HWHM is the poorest seeing of the V, R and I images of a field; also see Steidel and Hamilton 1993). Decreasing the detection threshold or the minimum area Chapter 2. Photometric Data Reductions and Analysis 24 i — 1 — 1 — 1 i — 1 — 1 — 1 — r N F 1 -i—i—i—r-> i 1.4 1.2 1 0.8 0.6 _1 I I 1 I I I I I I I 1 I L _J I L_ -0 .2 0.2 0.4 ( V - R ) 0.6 0.8 H 1 1 1 1 1 r T -| 1 1 1 1 1 1 .| - I 1 r 0.8 J 0-6 IT I 0.4 0.2 - 0 . 2 0.2 0.4 ( V - R ) 0.6 0.8 0.4 F-0.2 0 - 0 . 2 - 0 . 4 h L _1 I u - 0 . 2 J i _j i I i i_ 0.2 0.4 ( R - I ) 0.6 0.8 Figure 2.4 Calibration plots for the N F 1 field. Calibration equation solutions are plotted for N F 1 using data from the A p r i l 1991 observing run. Magnitudes with "instr" subscripts are instrumental magnitudes while capitalized variables are standard magni-tudes. The dot-dashed lines are least squares fit to the standard star observations. Chapter 2. Photometric Data Reductions and Analysis 25 i 2 h ~i 1 r n 1 1 1 i 1 1 1 _ . + ' — i r 1.5 F-V 1 > 0.5 _l I I I 1 I L_ 0.5 1 (V -R) 1.5 I in 0.5 F-o h -0.5 u I i 1_ - i 1 1 1 r _l I I L. 0.5 1 ( R - D 1 1 r~ _j i i i_ 1.5 Figure 2.5 Calibration plots for the N F 2 field. Calibration equation solutions are plotted for NF2 using data from the June 1992 observing run, with the same format as in Fig. 2.4. Note the V i n s t r — V ordinate in the top plot is significantly different in value than Fig. 2.4 or 2.6 due to a gain difference in the CCD used (see text). Chapter 2. Photometric Data Reductions and Analysis 26 : 1 1 ' 1 1 1 1 1 1 1 1 R : N F 3 0 0.5 (R-I) Figure 2.6 Calibration plots for the NF3 field. Calibration equation solutions are plotted for N F 3 using data from the March 1993 observing run, wi th the same format as i n F i g . 2.4. Chapter 2. Photometric Data Reductions and Analysis 27 (Amin) significantly increases the number of spurious objects included in the F O C A S catalogs fainter than the magnitude Hmits. Increasing either of these two parameters results in some of the galaxies brighter than the magnitude Hmits being excluded from the final catalogs. Since we use conservative magnitude Hmits i n this study (see section 2.5) where galaxy incompleteness is negHgible and the success of the galaxy detection is checked thoroughly by eye, we are confident that the threshold and min imum area parameters chosen are appropriate. Two approaches were used for the init ial detection of the objects i n our fields. The first technique was to utiHze a total summed, master frame (V + R + I) where the average frames from each bandpass were normalized to a common flux level i n counts (Small et al. 1994). A n object Hst was generated from the master frame detections, then the galaxy magnitudes were measured off of the average images i n each respective filter. This works quite weU for fields where data have been obtained with comparable magnitude Hmits i n the three filters (NF1) . Note that V ~ R + 0.5 and R ~ I + 0.75 (see galaxy colour histograms in F i g . 2.10) for galaxies with late-type morphologies within the redshift regime that approximately corresponds to our magnitude Hmits (Frei and G u n n 1994). Exceptions to this are ellipticals which become harder to detect since the 4000A break has been redshifted beyond the V filter at z > 0.5. In fields where the faint Hmits were not roughly equivalent from filter-to-filter (NF2 and N F 3 ) , the second approach was to have the ini t ia l detection of the objects done in each individual bandpass. The objects found in each filter were then matched in master catalogs i n order to provide colour information. In particular, the i2-band data taken for N F 2 and N F 3 were found to be deeper than the V and I data. F O C A S programs are appHed to either the master frame of a field or the averaged frame in each filter (i.e., the "detection" frame) to determine an ini t ia l catalog of ob-jects, wi th a sHght modification. The detection algorithm for F O C A S wiU find different Chapter 2. Photometric Data Reductions and Analysis 28 numbers of objects, varying by a few percent, depending on the orientation of the frame. This variation i n numbers is due to the line-by-line nature of the detection algorithm and the fact that the threshold for a particular line depends on the sky history from the previous lines. We get around this problem by rotating the detection frame through 90 degree increments and matching the resultant four catalogs to produce a final catalog. A n object is included i n the final catalog if it is detected i n all the catalogs for the four orientations. It should be noted that the centroids of objects found with the detection frame in different orientations are in excellent agreement, only rarely having differences as much as one pixel in the x or y coordinates for the faintest objects. We used the "bui l t in" F O C A S filter: 1 2 3 2 1, for convolution of the image under consideration, during the detection process. Using this kernel to smooth the image for the detection algorithm helps reduce the number of spurious, noise objects in the final catalogs. The final catalog is also filtered to remove detections of objects which He within "masked" areas of the frame. Masked regions have been setup to include saturated stars, very bright galaxies, bad columns, vignetted corners and other artifacts which generate spurious detections. The masked area of each field is not a significant fraction of the total detection area, only being a few percent of the total number of pixels in the final frame. Following the application of the detection algorithm, the sky values are determined for each frame using the standard sky and skycorrect routines i n F O C A S . N o significant dependence of the measured sky values on the orientation of the frame (as was found for the number of objects detected) was found. Detections listed i n the final catalogs were split into indvidual objects and the magnitudes were evaluated using the default F O C A S programs. Splitting of multi-component groups into individual galaxies (and stars) was easily accomplished for the separations over which the angular correlation function is calculated, mostly due to the uniformly excellent seeing of the data set. Measurement of galaxy magnitudes is further discussed i n the following section. A point spread function Chapter 2. Photometric Data Reductions and Analysis 29 (PSF) was determined from ~ 15 — 20 bright stars found i n each field for each bandpass. The P S F is used in the F O C A S object classification algorithm (resolution) to separate out galaxies and stars from spurious objects i n the final master catalog. Separation of stars from galaxies i n the final samples is not done with the F O C A S classification routine but with an approach outlined in section 4.2.3. Objects listed in the final catalogs were all checked by eye to confirm their detection. A few spurious objects remained at this juncture, and were removed from further analysis, but their numbers were small relative to the final galaxy sample. 2.5 Final Photometry Total exposures for the three bandpasses, in a given field, were obtained so as to be comparable in depth to maximize the colour information for the greatest number of objects. In practice this is difficult to do at the telescope due to varying seeing, along with weather and observing time constraints. However, we obtained fairly uniform sampling of the objects allowing us to determine V, R and / magnitudes for ~ 1000 galaxies in each field, with the exceptions of the V and I data for N F 2 . Slightly lower numbers of faint objects for the N F 2 data were detected due to the data sets being collected during two observing runs which created a small loss in area from slightly mismatched fields. Only the field area which is coincident on all the data frames is included i n the final detection frames, so that uniform magnitude-limited samples are generated. Magnitudes were evaluated for each bandpass and field yielding final lists of isophotal and aperture magnitudes, and colours for all the objects. Aperture magnitudes were found to be a more reliable measure of the total brightness of the faint galaxies than isophotal magnitudes. Also , with isophotal magnitudes the magnitude l imit was found to be unrealistically faint. Ideally we are trying to measure the total magnitude for Chapter 2. Photometric Data Reductions and Analysis 30 each galaxy, but using an aperture is not appropriate for the brighter galaxies in the sample which have a significant angular extent. To circumvent these problems "hybr id" magnitudes are adopted, where apertures are used for the faint galaxies with an average radius < 3" and isophotal magnitudes are used for galaxies with characteristic sizes larger than this. Average radii were calculated using each object's average width i n the x and y coordinates and by assuming the galaxies were circularly symmetric. Isophotal and aperture magnitudes are in good agreement at the magnitude range where the transition between the two measures occurs, typically having differences <; 0.1 mag. Isophotal magnitudes are usually used for galaxies which are up to three magnitudes below the bright magnitude limit (V ~ 20 - 23, R ~ 20 - 23 and 7/ ~ 19 - 22) where galaxies are > 3" in mean diameter. Since the bulk of the galaxies in any of our samples is within two magnitudes of the faint magnitude l imit (V > 23, R > 23 and / > 22) the galaxies measured with isophotal magnitudes are a small contribution to the final sample size. The isophotal magnitudes used are not the "total" magnitudes which F O C A S generates (Valdes 1983) since we prefer to avoid doubling the isophotal area, from the init ial ly determined isophote, for the flux measurements. Considering the small numbers of galaxy magnitudes measured with isophotes this choice does not have a significant effect on the final magnitude-limited galaxy samples. Apertures of 3" were chosen since they were determined to be large enough to contain most of the flux from the majority of the faint galaxies. The adopted aperture size of 3" corresponds to a physical scale of ~ 11 fe_1kpc and ~ 13 / i _ 1 k p c for redshifts of z = 0.5 and 1.0, respectively, using H0 = 100 k m s _ 1 M p c _ 1 (h = 1) and q0 = 0.5. See §2.3 of L i l ly et al. (1991) for some further discussion of the benefits of aperture photometry for faint galaxies, as opposed to using isophotal photometry. Number counts, in V, R and J , are given for the three fields in Figs. 2.7-2.9. Note that bright stars have been removed from these counts i n the manner outlined i n section Chapter 2. Photometric Data Reductions and Analysis 31 V R I_ N F 1 0.41 ± 0 . 0 1 0.36 ± 0 . 0 1 0.32 ± 0 . 0 1 N F 2 0.42 ± 0.02 0.35 ± 0.01 0.34 ± 0.02 N F 3 0.46 ± 0.02 0.39 ± 0.02 0.33 ± 0.02 Table 2.4 Slopes of the number counts, presented i n Figs. 2.7-2.9, for the three fields in V, R and I. 4.2.3 and hybrid magnitudes are used to calculate these counts. Slopes of the galaxy counts are listed i n Table 2.4. There is good agreement between the number count slopes determined i n different fields for a given bandpass. This is an encouraging result since we want to calculate by averaging the galaxy clustering behaviour i n N F 1 - 3 , thus requiring field-to-field uniformity. From the number counts it is easily seen that conservative magnitude limits for the data are V ~ 25, R ~ 25 and I ~ 24, except the N F 1 R data which has a l imit of ~ 24.5. This latter l imit is more comparable to the V and I l imits for galaxies at intermediate redshifts (z ~ 0.5 — 0.7). Photometric errors were calculated to be typically ^ 0.1 mag. for galaxies with V < 24, R < 24 and I < 23, increasing to as much as ~ 0.3 mag. for fainter galaxies. In Table 2.5, the number of objects found within the given magnitude hmits are tabulated, along with the effective field areas. A l l of these objects comprise the magnitude l imited samples, which are used for angular correlation analysis i n Chapter 4 and for choosing galaxy samples by colour. Colour selected samples were obtained by matching the catalogs generated of a given field for the two bandpasses in the required colour. High percentages (typically > 90%) of galaxy matching between filters were achieved, particularly for the N F 1 field where the magnitude hmits in V, R and I were of similar depth. A n approximately 5 — 10% decrease in the matching success rate is observed for the faintest magnitude bins where Chapter 2. Photometric Data Reductions and Analysis 32 5.5 4.5 OB t d * 00 T3 3.5 O 2.5 - i | i | — i — | — i — | — i — | — i — | — i — | — . —ry—i— 1 — i / i — 1 — v r / / / N F l • / / i / NF2 NF3 / / / / y / / A / / V ¥ 1 / J I I ! I ! L J , L 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Total V Magnitude Figure 2.7 V Number Counts for NF1-3 . The plotted lines are least squares fits to all the points brighter than the V ~ 25 magnitude limits. Slopes are given in Table 2.4. NF3 counts have been shifted to the right and N F l counts to the left, by 2 magnitudes, for purposes of clarity. N F l , NF2 and NF3 counts are denoted by circular, triangular and square points, respectively. The coordinate on the abscissa corresponds to "hybrid" magnitudes, as discussed in the text. Chapter 2. Photometric Data Reductions and Analysis 33 5.5 4.5 (0 M TJ 3.5 M O 2.5 1 1 I 1 I 1 I 1 I ' I 1 T / - 1 I r / N F l • NF2 * NF3 • / / A / 4 4 / A IA / A V / 4 / 4 ! _ L I I I J , L J _ j L 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Total R Magnitude Figure 2.8 R Number Counts for NF1-3. The plotted lines are least squares fits to all the points brighter than the R ~ 25 magnitude l imits , except for N F l which has a magnitude l imit of R ~ 24.5. Slopes are given in Table 2.4 and the format of the plot is the same as i n F ig . 2.7. Chapter 2. Photometric Data Reductions and Analysis 34 5.5 4 .5 BO (0 QO XI 3.5 M O 3 k 2.5 i 1 r — i — 1 — i — 1 — i — r ~ 7 i — 1 — i — 1 — — i — r / r / / / / N F l • NF2 -NF3 • / / / / /* A / f I A / y % A I-A / / y / / X y 1 Z I / I ! I ! I I I ! L J , I ! I , I I L 16 17 18 19 20 21 22 23 24 2 5 26 27 28 29 Total I Magnitude Figure 2.9 I Number Counts for N F l - 3 . The plotted lines are least squares fits to all the points brighter than the I ~ 24 magnitude hmits. Slopes are given in Table 2.4 and the format of the plot is the same as in Fig. 2.7. Chapter 2. Photometric Data Reductions and Analysis 35 N F l Effective Area: ~ 0.01064 deg.2 Number of Objects (Magnitude Range) V R I 355 (20-24) 590 (20-24.5) 935 (20-25) 574 (20-24) 878 (20-24.5) 486 (19-23) 706 (19-23.5) 996 (19-24) NF2 Effective Area: ~ 0.01026 deg.2 Number of Objects (Magnitude Range) V R I 253 (20-24) 425 (20-24.5) 699 (20-25) 407 (20-24) 610 (20-24.5) 913 (20-25) 319 (19-23) 496 (19-23.5) 709 (19-24) NF3 Effective Area: ~ 0.01111 deg.2 Number of Objects (Magnitude Range) V R I 312 (20-24) 543 (20-24.5) 908 (20-25) 588 (20-24) 930 (20-24.5) 1482 (20-25) 439 (19-23), 658 (19-23.5) 992 (19-24) Table 2.5 Number of objects detected in each of three fields in V, R and I. The magnitude ranges are listed in brackets. galaxies are not detected in all three filters. Histograms of the colours of the galaxies found in the three fields are shown in Fig. 2.10. (V — R), (R — I) and (V — I) galaxy-colours are shown in the plot panels from top to bottom for N F l (solid line), NF2 (dashed line) and NF3 (dotted line). The shapes and modes of the histograms for a given colour are very similar from field-to-field, with the differences in the normalization due to the variations in the effective area of detection and in the galaxy numbers. The median colours of galaxies as a function of R magnitude were found to be roughly consistent with those measured by Smail et al. (1995), in their Fig. 3. These samples are used to measure the colour-selected angular correlation function of galaxies. Further discussion of the specific colour selection criteria is given in section 4.3.2. Chapter 2. Photometric Data Reductions and Analysis 36 u 200 6 z 100 1 1 1 1 1 1 1 1 I I I l ~ " 1 1 i i i i | i i i i | i i i i (V-R) j N F l : NF2 : NF3 : | 1 |- . 1 ! 1 | 1 1 1 1 | 1 | -1 i i 1 1 1 l i t ] 1 1 1 1 1 1 1 1 ( R - I ) : N F l -NF2 NF3 - -i IT 1 . j j . -t , i i 1 i i i i 1 1 1 1 1 1 1 1 i i 1 i i i ,.i L^oJ i 1 1 (V-I) : | 1 N F l -1 ' 1 NF2 : : NF3 -f - 1 i -i 1 i i i i 1 i r i ^ i L,— 1—i- i i i j - 1 0 1 2 3 Colour Figure 2.10 Histograms of galaxy colours for N F l , N F 2 and N F 3 . Number of galaxies observed as a function of (V — R) (top panel), (R — I) (middle) and (V — I) (bottom) colours are plotted for the three observed fields. N F l , NF2 and NF3 are demarcated by solid-line, dashed-line and dotted-line histograms, respectively. Chapter 2. Photometric Data Reductions and Analysis 37 2.6 Photometry Checks Since the N F l and N F 3 fields have overlapping regions with faint galaxy data obtained by L i l l y et al (1991, hereafter L C G ) and Griffiths et al (1994) respectively, we performed quick photometry checks on the common objects. A comparison between the N F 3 V and I photometry and the pre-refurbishment W F C -H S T data of Griffiths et al is not straightforward since the H S T filters used (F555W (V) wi th Ae// = 5397A, A A ~ 1226A and F785LP (I) with XEFF = 8617A, A A ~ 1332A) are slightly different than the C F H T filters and the H S T data have significantly better seeing, albeit wi th image deconvolution. The W F C data occupy ~ 1/5 of the N F 3 field and have a completeness l imit ~ 1 mag brighter than the C F H T data. F O C A S was applied to the N F 3 frames to generate aperture magnitudes with a radius of 3 pixels, to ease the comparison with the 1" aperture magnitudes used by Griffiths et al. Aperture magnitudes for objects with I = 19 — 24 were checked for consistency. The mean magnitude and colour differences between data sets were found to be < IMDS — INFS > — 0.12 ± 0.19 and < (V — I)MDS - (V - I ) N F S > ~ 0.23 ± 0.26. Al though the two studies are formally consistent with no large discrepancies in the photometry, the slight offsets i n both V and / are probably due to differences in the filters and seeing between data sets, and the coarser pixelization of the C F H T data. Since the above effects are non-trivial to model we take the relatively small differences in the M D S and N F 3 data as indication that the two sets of photometry are broadly consistent. About half of the SSA 13 field from L C G is contained within the N F l field, which allows a comparison of the photometry of the few (~ 20) objects in the overlapping region. Mean differences between the 3" aperture magnitudes i n the two studies are found to be < V N F 1 - V L C G >^ —0.06±0.21 and < INFi~hcG > ^ - 0 . 2 7 ± 0 . 1 9 , where V — VAB a * i d I = IAB — 0.48 have been assumed to convert the L C G magnitudes. These Chapter 2. Photometric Data Reductions and Analysis 38 slight offsets between the N F l and L C G photometric samples are not surprising since one would expect more flux in the N F l apertures where there is better seeing. As with the MDS/NF3 sample comparison, we find approximate agreement between the photometry and this is encouraging and sufficient for the purposes of the current study. Chapter 3 Counting Close Pairs of Galaxies in N F l 3.1 Introduction Physically associated pairs of galaxies have long been recognized as harbingers of star formation, A G N behaviour and, i n some cases, the inevitable merging of the init ial ly distinct galactic systems. M a n y theoretical studies have suggested an increasing rate of mergers with redshift to explain the significant evolution of the faint galaxy counts for blue bandpasses (Broadhurst, Ellis and Glazebrook 1992, Carlberg and Chariot 1992 and Col in , Schramm and Peimbert 1994, and references therein). Whether a close pair of faint galaxies is truly in the process of merging can only be determined with redshifts for both objects, and even with this information some assumptions must be made about the critical relative velocity and fraction of close pairs with physical separations larger than those projected on the sky. W i t h a photometric sample of galaxies one must make statistical arguments in order to measure the pair fraction. For faint magnitude l imited samples this approach is a necessity since current spectroscopic surveys are l imited to / ~ 22.1 (Tresse et al. 1993 and Li l ly et al. 1995a). Our galaxy sample falls into this category (/ < 24). The interactions of galaxy pairs in clusters have been shown i n recent work by Lavery and Henry (1988, 1994) and Lavery, Pierce and M c C l u r e (1992) to be a potentially important mechanism for the "Butcher-Oemler" effect at z ~ 0.2 — 0.4. For faint field galaxies the role of interactions and mergers i n galaxy evolution is not as well understood 39 Chapter 3. Counting Close Pairs of Galaxies in NFl 40 or measured. Zepf and K o o (1989, hereafter Z K ) compiled a sample of 20 close galaxy pairs from 4m plates of two regions of high galactic latitude down to a magnitude l imit of Bj < 22, for separations less than 4'.'5. Comparing to nearby pair samples they found that the frequency of close pairs increases as (1 + z ) 4 0 ± 2 , 5 . A slight excess of pairs, which may not be statistically significant, was observed for a projected separation of 3". Colless et al. (1994) obtained high-resolution imaging of 17 faint blue galaxies culled from their spectroscopic survey (Colless et al. 1993) with z ~ 0.1 — 0.7 and found that 5 exhibited fainter companions at a projected distance of less than 10fe _ 1 kpc. Recent H S T results (Burkey et al. 1994, hereafter B K W F ) show 34% of their I = 18.5 - 23 galaxies to be close pair members. They claim that this is a lower l imit to the pair fraction and find it increases with redshift as (1 + z ) 3 5 ± 0 ' 5 . Since our data are at least one magnitude deeper than that of B K W F we can check the apparently large pair fraction they find and this is our primary motivation in this study. B K W F point out that they detect very few pairs at separations less than 0'./5, although with H S T it is possible to do so. This result suggests that it is feasible to count close pairs from a good ground-based site such as C F H T although the scarcity of sub-arcsecond separation pairs in the field at intermediate redshifts should be confirmed with additional H S T data (e.g., Griffiths et al. 1994). Carlberg et al. (1994, hereafter C P I ) looked at a sample of V magnitude selected (V < 22.5) galaxy pairs with physical separations less than ~ 20ft. - 1 kpc. W i t h redshifts for 14 galaxies i n close pairs and 38 field galaxies they found no statistically significant difference between the redshift distributions for the two populations. However, they find an amplitude for the angular correlation function, OJ(9), of the field population which is higher, for separations of 6 < 6", than an extrapolation of the canonical power-law form OJ(9) oc 6~os, as well as a merger rate which goes as (1 + z ) 3 4 ± 1 0 . We measure the angular correlation function for faint galaxies in Chapter 4 using a larger galaxy sample but focus on close pairs of galaxies i n the N F l field (as a case study) i n this chapter. Griffiths et Chapter 3. Counting Close Pairs of Galaxies in NFl 41 al. (1994) use H S T M e d i u m Deep Survey ( M D S ) data of 201 galaxies with I < 25, with the caveat that the sample is not complete, to study clustering of faint galaxies and find an excess of nearest neighbours at a projected separation of ~ l'/5. Finally, Yee and Ellingson (1995, hereafter Y E ) find a similar result to C P I using a magnitude l imited sample (r < 21.5) and some redshifts initially obtained for a quasar-cluster spectroscopic survey. They estimate the fraction of close pairs, with projected separations less than 20fe _ 1 kpc, to be ~ 15%. Their merger rate of (1 + z ) 4 0 ± 1 - 5 agrees with those of C P I , B K W F and Z K (see discussion i n §5). The aim of this study is to determine the pair fraction with V , R and I photometry to fainter Hmits than previously achieved (V < 25, R < 24.5 and I < 24). Close pairs can be found with projected separations as small as ~ 1" since the summed images i n all three colours have subarcsecond seeing. The statistical approach is described in section 3.2 and the galaxy pairs found are presented i n section 3.3. A discussion of the results and conclusions are given i n sections 3.4 and 3.5, respectively. 3.2 Statistical Approach To determine whether two galaxies are closely ahgned on the sky by chance or are a physical close pair, i n Heu of redshifts, requires some statistical criteria to attempt to differentiate the two cases. We use B K W F ' s approach of calculating a statistic which de-pends on the pair separation and the surface density of galaxies as a function of Hmiting magnitude. Given a random distribution of galaxies distributed on the sky, the proba-biHty of a chance projection occurring for a companion galaxy with apparent magnitude 77i and separation 9 is: P = 2irpa exp(—irpa2)da = 1 — exp(—irp92), Jo (3-1) Chapter 3. Counting Close Pairs of Galaxies in NFl 42 with p defined as the surface density of galaxies brighter than m . The quantity within the integral sign, the nearest neighbour probability density function, is rigorously derived in an appendix of Scott and Tout (1989). The expression in equation (3.1) has a correc-tion for both u>(9) and integrals over u(0), the three-point and higher order correlation functions (White 1979). Since the amplitude of the two-point correlation function is measured to be too small to affect the pair probabilities at our faint magnitude limits (Brainerd et al. 1995), we disregard this correction. The angular two-point correlation function, UJ(0), cannot be measured accurately for small separations given the number of galaxies in our one field (~ 1000). Adopting a pair statistic, as above, over correlation analysis is a necessity but it serves our purpose of looking for a significant pair excess at small separations and allows a direct comparison with the B K W F result. For the iV objects detected i n each filter to a given magnitude l imi t , each of the N(N — l ) / 2 possible pairs have their separation calculated along with a local surface density (p) calculated by integrating the number counts to the l imit of the faintest galaxy i n the pair. A s B K W F point out, integrating to the magnitude of the fainter member is conservative i n the sense that the contribution of pair members projected by chance is overestimated. To find close pairs, we adopt a probability of chance projection of P < 0.05, as did B K W F . The probability cutoff of P < 0.10 is also used i n order to check that the value of P adopted does not have a significant effect on the number of close pairs determined. Since atmospheric seeing effects don't allow us to distinguish pairs to as small a separation as H S T we must include a minor correction i n equation (3.1). If (3 is the angular separation cutoff below which individual objects cannot be independently detected then equation 3.1 becomes: f8 P — / 2irpa exp(—irpa2)da = exp(—irp/32) — exp(—irpd2). (3-2) J/3 Chapter 3. Counting Close Pairs of Galaxies in NFl 43 Adopting a value of 8 — (X'95 decreases the probability of a chance projection (P) by, at most, 0.02 and ultimately adds a few close pairs to our list. For galaxy groups (triplets, quartets, etc.) each unique pair is checked with equation 3.2 and only the pairs which have an associated probability below the chosen cutoff are further considered. 3.3 Galaxy Pairs For the galaxies detected in N F l , a ~ 3" diameter aperture is used to measure the magnitudes unless the galaxy is found to be larger than the aperture, in which case an isophotal magnitude is determined. Some of the galaxies in pairs with small separations will have magnitudes which are overestimated due to contamination from the neighbour's flux. This boosting of magnitudes will be most important for the fainter (and smaller) galaxies which need to be at smaller angular separations to be counted as a close pair. With overestimated magnitudes a close pair is more apt to be considered real (e.g., P < 0.05) since the galaxies in the pair can have a larger angular separation. This will increase the number of close pairs found but it does not seem to be a significant effect, as shown by the results in this section. Since N F l is a high Galactic latitude field (fe ~ 74°) there is not significant contam-ination from stars in our galaxy samples. Star-galaxy separation is more of a concern for B K W F since the fine of sight to their fields is closer to the galactic plane (fe ~ 38°) although their discrimination should be fairly unambiguous. Fig. 3.1 shows the crude shape parameter we use to remove the brighter stars, in this example for the I data. The difference between the "core" magnitude, corresponding to the central 3x3 pixels of each object, and the aperture magnitude is plotted versus the aperture magnitude. A hori-zontal sequence is clearly seen which is consistent with the objects having the same point spread function. These objects are observed to have stellar appearances in the images Chapter 3. Counting Close Pairs of Galaxies in NFl 44 and therefore this sequence is identified to be stellar in origin. This stellar sequence is observed at (mcore — maper) ~ 1.1 with the shape parameter rising to larger values for bright magnitudes due to saturation. Using this crude discriminant we separate the stars and galaxies down to J = 21, V = 22 and R = 21. The identification of the stars is double checked with the more sophisticated star-galaxy separation technique employed in an independent study (Richer and Fahlman 1992) to identify Population II halo stars. Fainter than the hmits given above the stars are left in the sample since their contribu-tion is negligible at best and we want to avoid removing compact galaxies which have a stellar appearance. From the halo star study it is estimated that there are 38 stars for the magnitude range 21 < I < 24 leaving 869 galaxies with this brightness. Regardless of whether the stars are included i n the final samples or not, there is not a significant effect observed on the determined close pair fractions. Number counts determined for the three bandpasses are given in F i g . 3.2. Magnitude hmits are conservatively estimated to be V ~ 25, R ~ 24.5 and / ~ 24. These counts are used to determine the surface density of objects as a function of magnitude (p(m)) required for the pair statistic. Note that the V and I counts have been shifted along the abscissa for illustrative purposes. The dashed fine is a least squares fit to the counts, using data points with brighter magnitudes than the magnitude hmits, for each colour. Slopes calculated for the V , R and I number counts are 0.41 ± 0.01, 0.36 ± 0.01 and 0.32 ± 0.01, respectively. These values agree well wi th those published i n the literature. For the R and I counts Tyson (1988) finds a slope of 0.39 and 0.34, while L i l l y et al. (1991) obtain 0.32 for I. We define a "bright" sample with magnitudes of: 19 < V < 24, 18.5 < R < 23.5 and 18 < / < 23 with 359, 391 and 496 galaxies, respectively. Our "faint" sample goes one magnitude deeper in each bandpass (but includes al l the galaxies i n the "bright" sample), to the completeness hmits, resulting i n 938 galaxies i n V , 891 i n R and 1005 i n I. These six photometry samples were subjected to the pair statistical Chapter 3. Counting Close Pairs of Galaxies in NFl 45 Figure 3.1 Concentration Parameter vs. Aperture Magnitude for N F l . Our crude concentration parameter, mcore — maper, is plotted vs. the aperture magnitudes in the I filter. Note the stellar sequence which is apparent down to / ~ 21. The magnitude l imit is estimated to be / ~ 24. Chapter 3. Counting Close Pairs of Galaxies in NFl 46 analysis. Using the technique outlined i n section 3.2, we find all the close pairs i n the bright and faint V , R and I samples for probability cutoffs P < 0.05 and P < 0.10 and separations ranging from 1 — 11". A physical separation of 20 h~x kpc, using a Hubble constant of H0 = 100 k m s _ 1 M p c _ 1 (h = 1) and qo = 0, for redshifts of 0.2 and 0.8 corresponds to angular separations of 9" and 4", respectively. Therefore a considerable range of angular separations between pairs is studied although at small separations we are l imited at 1" (corresponds to ~ 4 kpc at z = 0.5) by seeing effects. Our results are shown i n Figs. 3.3 and 3.4 for the bright and faint samples respectively. The solid-lined bar histograms are the observed close pairs which satisfy the statistical criterion. The solid line curves are the distributions of pairs expected if the galaxies were randomly distributed i n the images. In each case, these were calculated from the average of a pair analysis of 1000 catalogues constructed by assigning the observed galaxies ran-dom coordinates in the field. The dashed lines above and below the solid line represent the 95% confidence levels bracketing the expected random pair distribution. One complication we do not consider i n the random simulations is the angular extent of the galaxies. This shouldn't be important since the galaxies in our images have small average sizes and the total area of the images covered by galaxies is a few percent of the total field. In these random simulations of galaxies distributed i n our field any "pairs" found with separations < 1" are considered to be one object since this is the small separation cutoff we use in the real data, as is evident from the histograms i n Figs. 3.3 and 3.4. In the simulations objects are not allowed to occupy the positions corresponding to the masked regions in the observed frames. Another potential problem with our image analysis could be that F O C A S has difficulty splitting objects which have small angular separations. This was manually checked by looking at all the objects detected i n the frames over the ful l magnitude range. No evidence was found for a significant number of Chapter 3. Counting Close Pairs of Galaxies in NFl 47 5.5 i 1 1 1 1 1 r 4.5 h i — 1 r v R I (0 T J 3.5 h O i — 1 — i — 1 — i — 1 — i — " ~ ~ A — ' — i — 1 — 7 ^ — i — 1 — r r ~ / 7 / / / / / / i 5 i y I' 1 / /T / / 2.5 / I i I A / / / / 1 / / ' / . i J i L J , I i I i L 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Total Magnitudes (I-2.R.V+2) Figure 3.2 V , R and I Number Counts for N F l . The lines are least squares fits to all the points brighter than the magnitude limits of I < 24, R < 24.5 and V < 25. Slopes are determined to be 0.41 ±0 .01 , 0.36 ±0 .01 and 0.32 ±0 .01 , respectively. V counts have been shifted to the right and I counts to the left, by 2 magnitudes, for purposes of clarity. Chapter 3. Counting Close Pairs of Galaxies in NFl 48 Figure 3.3 Close Pair Histograms for "Bright" Sample of Galaxies in N F l . Observed close pair histograms for the "bright" magnitude limited samples in V, R and I, for probability cutoffs of P < 0.05 and P < 0.10. The solid curves are the expected pairs for galaxies which are distributed randomly in our images. Dashed lines correspond to the 95% confidence levels above and below the average random distribution, which have all been determined from 1000 simulations. Chapter 3. Counting Close Pairs of Galaxies in NFl 49 Figure 3.4 Close Pair Histograms for "Faint" Sample of Galaxies in N F l . Same as Fig.3.3 except these histograms are for the "faint" magnitude limited samples in V, R and I. Note the different scale in the number coordinate from Fig.3.3. Chapter 3. Counting Close Pairs of Galaxies in NFl 50 Table 3.1 Pair fractions for the bright samples in V, R and I for both probability cutoffs. Probability 18 < I < 23 Random 18.5 < R < 23.5 Random 19 < V < 24 Random P < 0.05 10.1 ± 1.4% 9.4% 11.3 ± 1 . 6 % 9.2% 13.6 ± 1.8% 9.3% P < 0.10 22.0 ± 1.9% 17.8% 24.3 ± 2.2% 17.6% 27.9 ± 2.4% 17.5% close pairs being missed by the detection algorithm. In particular, for the bright sample to / ~ 23 where we make the direct comparison with B K W F , this is definitely not a problem. Note that all the observed histograms, in Figs. 3.3 and 3.4, are consistent with a random distribution of galaxies, with the possible exception of the bright samples of galaxies with P < 0.10. A slight excess of close pairs is observed in some of these cases for separations around 3 — 5", albeit at the ~ 2a level. The pair fractions determined in each sample for the separation range of 1 — 11" are listed in Tables 3.1 and 3.2, along with the expected fractions for a random distribution of galaxies determined from our Monte Carlo simulations. Errors for the pair fractions (ler) are calculated from binomial statistics, while errors for the random distribution fractions are not given since they can be made arbitrarily small, in principle, if one runs enough simulations. For the conservative probability cutoff of P < 0.05 all our pair fractions agree with the random values, for both the bright and faint samples in all three bandpasses, within ~ 2a errors. With the more liberal probability cutoff P < 0.10 the V and R samples have pair fractions which depart by more than 3<r from that expected for a purely random distribution of galaxies but only the bright V sample has a significant excess. This is contrary to what B K W F found with a sample of galaxies (18.5 < I < 23) Chapter 3. Counting Close Pairs of Galaxies in NFl 51 Table 3.2 Pair fractions for the faint samples in V , R and I for both probability cutoffs. Probability 18 < 7 < 24 Random 18.5 < R < 24.5 Random 19 < V < 25 Random P < 0.05 8.7 ± 0.9% 9.4% 10.2 ± 1.0% 9.4% 12.0 ± 1.1% 9.4% P < 0.10 19.8 ± 1.3% 18.2% 23.0 ± 1.4% 18.1% 22.7 ± 1.4% 18.1% similar to our bright I sample, using an identical approach considering pairs with P < 0.05, where they measured a pair fraction of 34% ± 9%. The pair fraction we measure is 10.1 ± 1.4% but this becomes consistent with a zero fraction of physical pairs after the non-physical pairs expected from random superpositions (9.4%) are subtracted. One of the principal reasons for the discrepancy with B K W F is that they did not correct their pair fraction for randomly distributed galaxies. Y E have also pointed this out and after making a correction for optical pairs claim that the physical pair fraction in the B K W F data is ~ 17%, which agrees with C P I and their value. If the physical pair fraction at I ~ 23 was ~ 17% we should have had no problem detecting these close pairs in our images. In the only case where a significant excess above the random pairs expectation is observed, for the bright V sample with P < 0.10, a physical pair fraction of 10.4 ± 3.1% is implied. This is not a particularly large excess considering the B K W F result and since we have doubled their probability cutoff, to 0.1, to obtain it. Therefore, our main result is that there is no significant excess of close pairs for the various magnitude limited samples we consider. Another reason why the pair fraction obtained by B K W F may have been higher than our result is that they do not use a uniform magnitude limit for their pair analysis. From their number counts they appear to be complete down to J ~ 22.3 but it is mentioned that Chapter 3. Counting Close Pairs of Galaxies in NFl 52 there are differing magnitude limits in the separate fields studied. Presumably surface brightness selection effects should roughly be the same for single and paired galaxies, as discussed by B K W F . However, to calculate accurate pair fractions within a magnitude range it is important to have complete data so the total number of galaxies is well measured. We are confident that our data is complete to the limits (/ ~ 23) that B K W F quote their pair fractions for. Also, a concern with both the B K W F and Y E studies is the choice of the fields used for finding close pairs. B K W F have fields centered on or near faint radio galaxies which they claim are not in rich or poor clusters. Y E obtain a sample of field galaxies from their quasar-cluster redshift survey by using an absolute velocity difference between the galaxies and the quasar in each C C D frame of greater than 4000 km s _ 1 . Galaxies fainter than r < 21.5 are not considered due to the success rate of determining redshifts being less than 78% below this magnitude limit. Our data is of a more pristine nature for pair studies since the N F l blank field was specifically chosen to avoid Zwicky galaxy clusters and any other evidence of clustering. To illustrate the close pairs we detect from our samples, a mosaic of images of pairs found for the sample which is comparable to that of B K W F (P < 0.05, 18 < / < 23) is presented in Figs. 3.5 and 3.6. Each tickmark on the axes corresponds to 1". Clearly the fainter galaxies must be observed at smaller separations to be considered members of close pairs. Table 3.3 lists the magnitude of each object which is in a close pair in the corresponding lettered images. The galaxies are numbered and the magnitudes are given from left to right in each image. Calculated a posteriori probabilities (P) are given for each pair in the third column. In cases where there is more than one close pair in the frame the probabilities listed are for the galaxies given in the following brackets. Morphological peculiarities such as tidal tails and distortions of the isophotes of the galaxies are typically used as indicators of an interaction or merging event. With in-creasing redshift and decreasing resolution these subtle, low surface brightness features Chapter 3. Counting Close Pairs of Galaxies in NFl 53 Figure 3.5 C l o s e G a l a x y P a i r I m a g e s i n N F l . Images of galaxy pairs a — I found for 18 < / < 23 and an a posteriori probability of P < 0.05. Magnitudes and calculated probabilities for each galaxy pair are listed in Table 3.3 for the objects shown from left to right. Each tickmark on the sides of the panels corresponds to an angular separation of 1". Chapter 3. Counting Close Pairs of Galaxies in NFl 54 m n Figure 3.6 Close Galaxy Pair Images in N F l . Images of galaxy pairs m — w found for 18 < I < 23 and an a posteriori probability of P < 0.05. Magnitudes and calculated probabilities for each galaxy pair are listed in Table 3.3 for the objects shown from left to right. Each tickmark on the sides of the panels corresponds to an angular separation of 1". Chapter 3. Counting Close Pairs of Galaxies in NFl 55 Table 3.3 Listing of probabilities of galaxy pairs in Figs. 3.5 & 3.6 Image (Figs. 3.5 fe 3.6) I Magnitudes P a 22.67, 20.88, 21.09 0.037(1,2), 0.003(2,3) b 21.06, 21.34 0.008a c 19.83, 20.44 0.031 d 20.96, 19.58 0.009 e 21.95, 22.38 0.007 f 21.73, 22.69 0.020 g 22.39, 22.74 0.035 h 21.09, 22.28 0.010 i 22.89, 21.09 0.042 j 19.83, 20.03, 19.85 0.021(1,2), 0.048(1,3), 0.010(2,3) k 22.57, 20.72 0.020 1 22.37, 22.59 0.033 m 20.99, 21.28 0.006 n 20.55, 22.14 0.025 o 20.18, 21.41, 20.90 0.041(1,2), 0.030(2,3) p 20.45, 20.83 0.033b q 22.69, 22.14 0.017 r 21.93, 22.51 0.010 s 22.49, 22.73, 21.94 0.040(1,3), 0.017(2,3) t 22.42, 22.95 0.048 u 22.55, 22.10 0.022 v 19.01, 18.43 0.006 w 18.41, 20.12 0.048 "The brighter object on the left has a radial profile consistent with that of a star. It hasn't been removed since it is fainter than the J ~ 21 cutoff below which no attempt is made to remove stellar objects. ^The galaxy on the left is actually a close pair missed by the detection algorithm. Galaxies are easily detected in pairs down to angular separations of 1", so, this failure is a rare occurrence. Chapter 3. Counting Close Pairs of Galaxies in NFl 56 are difficult to detect, let alone quantify. Using simulations of W F P C 2 H S T images of galaxy mergers at z = 0.4 and z = 1.0, Mihos (1995) demonstrates that using these morphological signatures is subject to significant uncertainty due to the rapid evolution of the interacting system once the galaxies have merged. W i t h large redshifts or poorer resolution (ground-based) images he suggests that merging systems can only be found through the presence of companions. As a;test, we looked for morphological signatures of interactions or mergers i n our deepest image for the close pairs found wi th P < 0.10 and 18 < I < 23. Only ~ 7 — 10 out of 66 close pairs showed unambiguous peculiarities indicative of t idal perturbation. The identification of these features is l imited by the seeing and pixelization effects and this identified fraction is certainly a lower l imit . Nev-ertheless, the relatively small number of objects with perturbed morphology agrees with our assertion that the majority of the close pairs found by the statistical methodology applied here are merely chance alignments. Another test of the physical nature of our pair samples is to determine their colour distributions. Galaxies which are undergoing interactions or mergers wi l l have bursts of star formation induced subsequently causing the colours to be bluer (Larson and Tinsley 1978). In F i g . 3.7 we plot the colours for our bright samples i n each bandpass. Only the bright samples are plotted since we can obtain the most accurate colours for these objects and also, due to these galaxies having the only significant pair excess above that expected for non-physical pairs. Colours are determined using 3" aperture magnitudes for all galaxies. The solid-line histograms are the colour distributions of the bright sample detected i n that filter, and the dashed and dotted lines correspond to the colours of galaxies occupying close pairs for P < 0.05 and P < 0.10, respectively. N o K -corrections are made since the mean redshifts of the entire galaxy and pair distributions should be basically the same ( C P I ) . A % 2 test shows no significant differences between the general galaxy colour distributions and that of the pairs, wi th either P < 0.05 and Chapter 3. Counting Close Pairs of Galaxies in NFl 57 P < 0.10, for all three bandpasses. Therefore, one cannot rule out the pair and total galaxy distributions being drawn from the same parent population. This also supports our conclusion that the majority of close pairs we have identified are optical pairs and not physically associated. However, as Y E comment, separating the true interacting systems by colour, from those which are either optically aligned or physical pairs wi th high relative velocities, is very difficult given the small ratio of the former to the latter. Also wi th the smaller separation pairs, where we would expect at least one galaxy i n the pair to have bluer colours than the general distribution, there is going to be more flux from the neighbouring object contaminating the aperture for which the colour measurement is made. 3.4 Discussion Whether merging increases the number of close pairs or not depends on the merger timescale and when galaxies brighten during the interaction process (Broadhurst et al. 1992). Toomre (1977) has used the occurrence of galaxy pairs wi th t idal tails i n a sample of ~ 4000 R C 2 galaxies to estimate the time for two local galaxies to merge to be ~ 0.5 G y r . W i t h the stringent requirement of observable tails this is probably a conservative number and C P I , using more elaborate arguments, derive a local merging timescale of ~ 22 ± 2.6 G y r . A t higher redshifts C P I assume the pairwise velocity dispersion evolves as (1 + z ) _ 1 yielding a merger timescale of 7.1 ± 1.4 G y r at z = 0.4. There is general agreement that the measured local pair fraction is ~ 4 — 5% (Soares et al. 1995, C P I , Y E ) . The determinations of the merging rate typically parameterize it as a power law (1 + z)m, where m has been estimated to lie within the range of ~ 2.5 — 4 with substantial errors associated with the exponent (Toomre 1977, Z K , C P I , B K W F and Y E ) . The pair fraction growth can also be expressed i n the form (1 + z)n. For this Chapter 3. Counting Close Pairs of Galaxies in NFl 58 Figure 3.7 Colour Distributions of Close Pairs in N F l . Histograms of the colour distributions of pair samples found for the "bright" samples of galaxies in each bandpass with 19 < V < 24, 18.5 < R < 23.5 and 18 < I < 23, compared to the colours of all the galaxies in these magnitude limits. The solid line histogram is the general galaxy distribution while the dashed and dotted lines correspond to the pairs found with P < 0.05 and P < 0.10, respectively. All colours are determined using 3" aperture magnitudes. Chapter 3. Counting Close Pairs of Galaxies in NFl 59 parameterization, B K W F assume that the merger rate evolves as (1 + z ) n _ 1 while CPI use dynamical reasoning to obtain a merger rate which increases as (1 + z ) n + 1 . Finally, Y E suggest that the observed pair fraction is a good estimator of the actual merging population at a given redshift so that the merging rate is (1 + z)n (i.e., m ~ n). For the I ~ 23 limit Y E find the physical B K W F close pair fraction to be ~ 17% when the optical pairs are accounted for. If this pair fraction exists at I ~ 23, or even to I ~ 24, we should be able to measure it with our photometry since we are using an identical approach to the analysis as B K W F . So, why do we not observe a strong excess in the pair fraction for our magnitude Hmits? It should be noted that the magnitude Hmits for which CPI (V ~ 22.5) and Y E (r ~ 21.5) calculate their pair fractions are much brighter than our Hmits and BKWF's . Assuming the average redshift of galaxies in our bright and faint samples are higher than that of the CPI and Y E samples, there should be an increase in the pair fraction above the ~ 10 — 15.5% they measure. The fact that we only see a comparable pair fraction for the bright V sample with P < 0.10, while the other samples have substantially lower fractions, suggests that either the exponent for the merging rate has been overestimated or there is a dropoff in mergers at our magnitude Hmits. Another possible compHcation is that there may be different merger behaviour depending on the dominant morphological type of galaxy at the specific magnitude Hmit (Glazebrook et al. 1994, Driver et al. 1994). Ideally the pair fraction evolution should be determined as a function of galaxy morphology and magnitude. We can consider the angular correlation function since it is related to the expected pair fraction, although measuring w(0) at small scales with small samples of galaxies has inherently large errors. As an additional test, we measured u>{6) for the bright and faint magnitude Hmits in the V , R and I images down to a separation of 2". The w(0) calculated for all cases were found to be consistent with randomly distributed galaxies Chapter 3. Counting Close Pairs of Galaxies in NFl 60 within the substantial error l imits . This is: what is expected from the pair fractions we determine. A better determined <JJ(9) from multiple fields with larger galaxy samples w i l l be discussed i n Chapter 4. Brainerd et al. (1995) have measured w(6) wi th a sample of ~ 5700 galaxies to r ~ 26, which corresponds to R ~ 25.5, down to separations of ~ 22". Since the amplitude of the correlation function is observed to fall off monotonically wi th the magnitude l imit of the sample, we use the Brainerd et al. fits and their F i g . 2 to estimate the amplitude of u>(9) for our bright and faint magnitude limits of R ~ 23.5 and R ~ 24.5. A form for the correlation function of u>(0) = Aw6~0% has been adopted. B y extrapolating w(9) to small scales a prediction for the fraction of "non-random" pairs (those not resulting from random superpositions), expected for a particular angular separation, can be made. For the bright l imit i n R this gives k>(l") — 0.6 and w ( l " ) ~ 0.31 for the faint sample. These values correspond to ~ 38% and ~ 24% of the pairs being non-random. If we use CPI ' s operational definition which requires a physical pair to have a separation 9 such that ^(9) > 1, then our estimated values for u>(9) do not bode well for finding "real" pairs. If the extrapolation of the Brainerd et al. u)(9) to small separations is to be believed and since our min imum pair separation is 1", it is not surprising we do not detect an excess of close pairs, at least i n our R data. So, it may be a consequence of the weak amplitude of the angular correlation function at these magnitude l imits that a stronger pair fraction is not observed i n our sample, although we emphasize that a measurement of u>(0) is not being made but an extrapolation. More accurate determinations of u>(9), using larger numbers of galaxies than previous samples, down to faint l imits and small angular separations with good angular resolution are needed. New C C D mosaic cameras wi th wide-field imaging capabilities currently becoming available would be ideal for this task and these data would further constrain merger models (see, for e.g., F i g . 5 i n Carlberg and Chariot 1992). It is interesting to note that the M D S survey, using H S T , has determined Chapter 3. Counting Close Pairs of Galaxies in NFl 61 u)(0) down to I ~ 22 for small separations (a few ") and found no significant excess of close pairs above a canonical power law slope (Neuschafer et al. 1995a). 3.5 Conclusions Our principal result is that we find no evidence for a significant excess of close pairs of galaxies for magnitude limits of I ~ 23 and I ~ 24 (and similarly for R ~ 23.5,24.5; V ~ 24,25). This result is contrary to the large pair fraction found by B K W F for I < 23. If the probability cutoff P < 0.10 is used slight pair excesses are found in our V and R samples although typically yielding substantially smaller pair fractions than those observed by other workers at brighter hmits. The lack of a large physical pair fraction hmits the usefulness of small photometric surveys as a tool for studying the merger rate among the faint galaxies. Further, if the faint galaxies studied here are representative of a more distant sample than the galaxy samples of CPI and Y E then this suggests the merger rate has either been previously overestimated or there is a change in its behaviour beyond their brighter magnitude Hmits. Of course, fainter galaxy samples (7 ~ 23 or 24) need not have higher average redshifts than bright samples since spectroscopic surveys haven't measured redshift distributions for these faint Hmits. The relative contribution of different galaxy morphologies to the number counts at a given magnitude Hmit (Driver et al. 1994) could also affect the observed merger rate for the total galaxy sample. The absence of a significant excess of galaxy pairs is consistent with an extrapolation of the angular correlation function for faint galaxies to smaller separations (Brainerd et al. 1995). This conclusion should be verified by a direct measurement of u>(6) for 0 ~ 1" - 10". Additional tests of looking for morphological signatures of interactions or differences Chapter 3. Counting Close Pairs of Galaxies in NFl 62 between the colours of close pairs and the total galaxy population agree with our con-clusion that the majority of our pairs are chance alignments, with the caveat that these criteria are subject to considerable uncertainties. Our result doesn't preclude there being any mergers at our magnitude limits (some obviously interacting galaxies are observed in our data) but suggests that there are fewer close pairs than what is observed with brighter samples. Although our C C D imaging has better than 1" seeing, additional HST data of appropriate "blank" regions with faint magnitude limits are required to study the frequency of merging systems in the field with sub-arcsecond separations. To unambigu-ously determine the physical pair fraction, and thereby the merger rate, as a function of lookback time many more redshifts of faint, isolated and paired galaxies are required. Chapter 4 Correlation Analysis of Three High-Galactic Latitude Fields 4.1 Introduction Angular correlation function analysis of large photometric samples of galaxies is a pop-ular tool for quantifying the large-scale structure of the local universe. Its power as a diagnostic of the galaxy distribution lies in the simplicity of its application, essentially requiring only a counting of pairs of galaxies at given angular separations and normaliz-ing these results with respect to the number of pairs expected for a random distribution. The angular correlation function (u>(8)) is a two-dimensional analogue of the spatial two-point correlation function (£(?*)), where the latter quantity has the important property of being the Fourier transform of the power spectrum (P(k)) of the galaxy distribution. Measurements of either £(r) or P(k) are important tests of structure formation models, such as Cold Dark Matter (CDM) scenarios (Davis et al. 1985), and are also necessary for understanding the relationship between nearby, bright galaxies and their faint, distant counterparts, along with the evolutionary processes that link the two populations. The angular and spatial two-point correlation functions for bright magnitude-limited samples of nearby galaxies have been determined, in various forms, by several authors over the last few decades (see Peebles 1980, 1993, and references therein). Numerous studies have measured £(r) to be a power-law although there is still considerable uncer-tainty concerning the exact normalization and slope due to possible systematic errors from morphological mixing, redshift distortions from clusters of galaxies and differing 63 Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 64 approaches to the correlation analysis (e.g., Bernstein et al. 1994 and Loveday et al. 1995). The spatial correlation function is observed to have the form: £(r) = (^) - 7, with 7 ~ 1.7 — 1.8 and the correlation length, r0(z = 0), estimated to be ~ 5 h~l Mpc for local galaxy populations (Davis and Peebles 1983 and Loveday et al. 1992). A potential problem for studies of nearby galaxies may be that a significant population of low surface brightness galaxies is being systematically ignored, due to high surface brightness selec-tion effects, in contrast to faint galaxy surveys where this is not a problem (McGaugh 1994). Low surface brightness galaxies as candidates for faint, blue galaxies are discussed further in section 4.4. Since the study of Davis and Geller (1976) it has been apparent that the clustering of galaxies is directly dependent on morphology. The correlation function for early-type galaxies is observed to have steeper power-law slopes and larger amplitudes than that obtained for late-type galaxies. The purported relationship of clustering with the intrinsic luminosity of galaxies is more controversial. Loveday et al. (1995, LMEP) review previous work on the dependence of galaxy clustering on both morphology and luminosity, and present results from the Stromlo-APM redshift survey. For the full sample of galaxies, on scales of 0.2 — 20/i - 1 Mpc, the spatial correlation function is fit by a power-law slope of 7 = 1.71 and a correlation length of ro = 5.1h -1 Mpc. LMEP find early-type galaxies are more clustered than late-types by a factor of 3.5 — 5.5, and the respective morphological samples have 7 = 1.85 and 7 = 1.64. Low-luminosity galaxies were observed to be more weakly clustered, by a factor of ~ 2, than L* galaxies for scales > lh~x Mpc. Also, the power-law slope found by LMEP for low-luminosity galaxies was steeper than the brighter-luminosity sample slope, yielding a clustering amplitude for the former sample which was a factor of 4 lower at IO/1-1 Mpc. Redshift surveys of IRAS-selected galaxies have also been used for local studies of the spatial correlation function (Saunders et al. 1992, Fisher et al. 1994). Both studies find smaller values for the correlation lengths Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 65 (3.79 ± 0.14 h-1 Mpc, 3.76io!23 Mpc; respectively) and shallower power laws for the correlation function (7 = 1.57 ± 0.03, 7 = 1.66+0^1), than typical values estimated from optically selected galaxy samples. This suggests that the correlation function for IRAS galaxies is shallower than the £(r) measured for optical galaxies. Pioneering photographic studies of faint galaxies done by Koo and Szalay (1984), Stevenson et al. (1985) and Pritchet and Infante (1986) determined w{6) down to magni-tude limits of B < 23 — 24. Only in the last decade has it been possible to measure u>(6) for fainter galaxy samples because of the emergence of C C D imaging cameras, in partic-ular the large-format devices. A study by Efstathiou et al. (1991, hereafter E B K T G ) has generated a flurry of interest in measuring <JJ{0) for faint galaxies. E B K T G found the faint blue galaxy population in their C C D images (for 24 < Bj < 26) to be weakly clus-tered at 30" separations relative to local galaxy populations. They concluded that there were three possible explanations: most of the faint galaxies were members of an hitherto unobserved population which had faded away by the current epoch, galaxy clustering was insufficiently described by basic models of gravitational instability or that space-time geometry departed significantly from an Einstein-de Sitter universe. However, as Koo and Kron (1992) point out, the E B K T G result implicitly assumes that galaxies with different morphologies have similar intrinsic clustering properties. This is not the case locally (see above and Giovanelli et al. 1986) and clearly this is a an effect which needs to be addressed by using various sample selection criteria for faint galaxies, in addition to magnitude limits, such as morphologies and colours. Neuschaefer et al. (1991, hereafter NWD) measured <JJ{0) down to g ~ 25 before significant incompleteness in their sample set in and found a similar amplitude as E B K T G at g ~,24.8. The monotonic decrease of the amplitude of w(9) for a given angular separation as a function of survey magnitude limit, demonstrated by N W D , has been observed in a number of other studies (Pritchet and Infante 1992, Couch et al. 1993, Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 66 Roche et al. 1993, Roukema and Peterson 1994, Brainerd et al. 1995, hereafter B S M , and Metcalfe et al. 1995, hereafter M S F R 9 5 , to name a few). This decrease of clustering with survey depth shows some indication of beginning to flatten out at the faintest hmits i n some studies (Roche et al. 1993 and M S F R 9 5 ) but the errors involved i n these measurements preclude any f irm conclusions from being made. We w i l l discuss the results from the aforementioned and other clustering studies i n more detail i n section 4.4. Other recent work has emphasized identifying the local counterparts of the weakly clustered faint galaxy population, if there are any. Bernstein et al. (1994, hereafter B T B J ) measure the angular correlation function for galaxies wi th magnitudes i n the range 20 < Bj < 22.5. Using redshift distributions measured by Broadhurst et al. (1988) and Colless et al. (1993) B T B J determined £(r = 2 5 0 ^ _ 1 kpc) and found that the clustering behaviour of their sample is similar to that of nearby IRAS-selected galaxies. W h e n the galaxies are separated into blue and red subsamples, the reddest third of their sample shows a clustering strength which is consistent with A P M local samples (Bj < 18), assuming a conventional clustering growth rate. The bluest two-thirds of their sample sti l l exhibits a clustering strength which is consistent wi th the local I R A S galaxies. This suggests that the weak clustering observed for galaxies at Bj ~ 22 could be due to changes i n the relative fractions of morphological types i n the general population. The study of the clustering of galaxies with colour-selected samples, as i n B T B J , is a technique which has only recently been applied to discriminate faint galaxies of differing morphologies (Landy, Szalay and K o o 1996, hereafter L S K ; Roche et al. 1996). L S K used 4m plate photometry i n two independent fields of a total of over 5900 galaxies wi th 20 < Bj < 23.5 to measure <JJ(6) as a function of colour. They found a strong increase i n the ampHtude (greater than a factor of 10) of the angular correlation function for the extremely blue and red subsets of galaxies i n their sample, using U — RF as the colour discriminant. L S K claim the u>(9) excess for the reddest galaxies is due to intrinsic Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 67 clustering of the sample (morphology-density relation) since most of these objects are probably E/SO galaxies and both fields contain known clusters. The increase of the clustering amplitude for the bluest objects i n the sample is explained as being caused by a faint population of galaxies with z < 0.3. Roche et al. (1996) calculated u>(9) for colour-selected samples to significantly fainter magnitude limits (B ~ 25.5 and R ~ 24.5) than L S K . For ~ 7000 galaxies, they determined the amplitude of u>(9) for the red ((B — R) > 1.5) sample to be higher than that calculated for the blue ((B — R) < 1.5) galaxies. This result led Roche et al. to suggest that the decrease i n the amplitude of u>(9) for all galaxies wi th B > 23 is caused by the same blue galaxies that are responsible for the number counts excess around this magnitude range. Using a pure luminosity evolution model they explain the correlation function colour dependence at B ~ 25 as being due to red galaxies with z < 1, i n addition to the blue galaxy sample consisting of both late-type dwarfs at low/moderate redshifts and evolving L* galaxies having redshifts from z ~ 0.5 — 3. Colour-selected samples of galaxies are analysed for variations i n clustering i n this work i n section 4.3.2. The Canada France Redshift Survey ( C F R S , L i l l y et al. 1995b) has provided the redshift information necessary for measuring the spatial correlation function of faint galaxies down to a magnitude l imit of I ~ 22 (Le Fevre et al. 1996). The ampli-tude of £(r) is found to decrease strongly with redshift resulting i n a correlation length of r0(z = 0.53) = 1.33(1.57) ± 0 . 0 9 h'1 M p c for q0 = 0.5(0). No significant difference was found for the clustering of red and blue galaxies with z > 0.5, while at lower redshifts (0.2 < z < 0.5) the blue galaxies were observed to be marginally less clustered than red galaxies. Hudon and L i l l y (1996) used a photometric survey, which covered ~ 0.33 deg2 and contained 13,000 galaxies observed to a magnitude l imit of R ~ 23.5, along with the C F R S to calculate £(r). In broad agreement with the evolutionary behaviour ob-served by Le Fevre et al. (1996), Hudon and L i l l y estimate the correlation length to be Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 68 r 0(z = 0.48) = 1.86(2.16) ± 0.43(0.49) h'1 Mpc for q0 = 0.5(0.05). Another determina-tion of the spatial correlation function from a recent redshift survey is that of Shepherd et al. (1996) which uses a sample of field galaxies gleaned from the C N O C cluster survey. With a sparse sample of 144 galaxies having magnitudes 17.0 < r < 21.7 and redshifts 0.21 < z < 0.53 the correlation length was calculated to be r0 — 2 . l i ° J h 1 Mpc for q0 = 0.5 (7*0 = 2.5to!l Mpc for qo = 0.1) for a median redshift of z m e d = 0.36, in good agreement with the two previously mentioned studies. In this study we use the /-selected CFRS (and an extrapolation thereof, see section 4.3.3) to generate different evolutionary models of £(r) for comparison with the galaxy clustering measured at faint magnitude hmits (/ < 24). The deepest angular correlation function studies to date are those by Brainerd et al. (1995, BSM) and Metcalfe et al. (1995, MSFR95). B S M measure u)(6) to a limiting magnitude of r = 26 (R ~ 25.5) with ~ 5700 galaxies detected over a 90.1 arcmin2 field. They found the ampHtude of the two-point angular correlation function smoothly decreases with magnitude Hmit, and there was no evidence for flattening. No significant difference was observed between the clustering of the red and blue galaxies, separated in colour at (g — r) = 0.3. BSM propose that the relatively weak clustering amphtudes they measure can be best explained by a modest amount of evolution and a correlation length of ~ 2.0 h~l Mpc, which is similar to the value determined locally for low surface brightness galaxies (Santiago and da Costa 1990). MSFR95 have imaged a field to the deepest Hmit (B ~ 27.5) yet achieved with a ground-based telescope. A decrease in the slope of the blue number counts is detected faintward of B ~ 25 but the counts continue to rise towards the magnitude Hmit. The flattening of the counts is interpreted as the galaxies in this magnitude range predominantly being at z > 1. MSFR95 claim they observe a flattening in the ampHtude of u>(0) at B > 25, which also impHes that a large portion of the galaxies at these magnitudes have high redshifts. They also suggest Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 69 that the change i n the clustering amplitude with magnitude rejects simple merger galaxy evolution models (Roche et al. 1993, Roukema and Yoshii 1993). L idman and Peterson (1996, L P ) provide a good comparison for the measurements of galaxy clustering made i n this work since /-selected studies are scarce. L P calculate u>(9) using galaxy samples obtained over ~ 13 deg2 to magnitude limits significantly brighter (/ ~ 22.75) than ours. Their results are i n general agreement wi th other groups i n that the amplitude of w(0) is found to monotonically decrease w i t h magnitude l imi t . The amplitudes measured i n L P fall below what no-evolution models of galaxies predict wi th modest clustering evolution. Also , a blue-selected sample ((V—I) < 1.5) of galaxies (with magnitudes 18 < / < 20) is observed to have half the clustering amplitude of that of the red sample ( ( V — /) > 1.5), albeit wi th large measurement errors. Neuschaefer et al. (1995b, N R G C I ) is another clustering study which is useful for checking our results since the observations were taken in V and /-bandpasses. Using pre-refurbished W F P C - H S T data from the M e d i u m Deep Survey N R G C I calculated u>(6) for various galaxy samples down to / ~ 23. W i t h the high resolution capability of H S T imaging they are able to measure galaxy clustering down to arcsecond scales and accurately determine galaxy morphologies to fainter limits than is possible from the ground. N R G C I find no excess of pairs at angular separations of 0.5" — 4" and a magnitude l imit of I < 22, this implying a constant merging rate for galaxies out to z m e j ~ 0.5. A s i n other studies, strong evolution of the clustering amplitude is seen relative to that observed for local galaxy populations. Stronger correlation amplitudes are found for early-type galaxy samples than for those containing mostly disk galaxies. Galaxies with the 50% reddest (V — I) colours have clustering which is ~ 4 — 8 times stronger than the blue half of the sample, but it should be noted that both the colour and morphology-selected samples have substantial errors associated with the clustering measures. B o t h of the aforementioned papers are discussed further, i n relation to our results, i n section 4.4. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 70 In this chapter, w(9) is calculated for a sample of galaxies imaged i n V, R and I to respective magnitude Hmits of 25, 25 and 24, combining data from three different high Galactic latitude fields. In Chapter 3, the close pair fraction was determined for one of the fields ( N F l ) . This nearest neighbour approach is complementary to the cor-relation analysis since it measures clustering behaviour at the smallest possible angular separations while the greater numbers of galaxies analysed i n this chapter can be used to estimate the angular correlation function over a range of larger angular separations. The galaxy samples presented here are among the deepest yet used for measurements of Also , the multi-bandpass data allow the clustering variations with colour to be measured for these faint galaxies. The adopted technique for estimating the angu-lar correlation function, along with a summary of the galaxy clustering model used, is presented i n section 4.2. Clustering results for magnitude-Hmited and colour-selected samples of galaxies are given i n section 4.3. This section also contains a comparison of the clustering detected i n the / -band to models of the spatial correlation function, which are calculated wi th extrapolated redshift distributions provided by the C F R S . Final ly, possible interpretations of the results are discussed and conclusions are drawn i n section 4.4. 4.2 Measuring the Angular Correlation Function 4.2.1 Estimator If one considers two differential elements of sofid angle on the sky, dVt\ and dQ,2, the joint probabifity (dP) that galaxies wi l l occupy the two elements with an angular separation of 9 can be written as: dP = n2(l+w(9))dn1dn2, (4.1) Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 71 where w(9) is the angular correlation function and n is the mean surface density of galaxies. A random (Poisson) distribution of galaxies yields u)(9) = 0 for all 9. Therefore, (JU(0) is simply a measure of the number of galaxy pairs observed at a given separation projected on the sky, normalized by the number of galaxy pairs expected if the galaxies are randomly distributed. The traditional estimator for OJ(9) (Peebles 1980), where equal numbers of galaxies and random points are considered, is of the form: where DD(9) and RR(9) represent the number of data-data and random-random pairs at the angular separation, 9 (hereinafter pair symbols w i l l impl ic i t ly be assumed to be functions of 9 and the hat symbol is used to denote an estimate of a function). Another variant of this estimator includes a cross-correlation of the data and random objects: (see E B K T G , Infante and Pritchet 1995, and references therein), where Nr> and NR are the number of galaxies and random objects, respectively. The estimators i n equations 4.2 and 4.3 have greater than Poissonian variance, so, to minimize the noise we adopt the estimator suggested by Landy and Szalay (1993, hereafter L S ; also see Hamil ton 1993): W = ( D D - 2 ™ + R R ) -For a given galaxy sample, 100 files of random positions, containing the same number of "galaxies" i n each as those observed, were generated, yielding ~ 0.5 — l . O x l O 5 random objects. Increasing the number of random position files from 100 to 1000 did not sig-nificantly improve the accuracy of the final u>(9) estimation but substantially increased the computing time. The random position files were created using the same detection mask which was used for the galaxy image (in all three bandpasses for a given field) to Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 72 block out saturated objects and areas of the C C D with defects and vignetting. A mean of the 100 random files cross correlated with the particular galaxy sample yielded D R i n the estimator. In addition, by averaging the files one could calculate the probabilities of obtaining a pair and triplet at a separation 9 (LS refers to these probabilities as Gp{9) and Gt{9\ respectively), quantities which are required to calculate u>(#) and the errors associated wi th the estimator. To check the clustering estimator i n equation 4.4, counts i n cells were also determined using: m = lN<Ni> -1, (4.5) with the number of galaxies i n cells i and j denoted by Ni and Nj, and the angular brackets representing an average of all the cells wi th an angular separation within the bin 9 ± 89. Images were divided into square cells, 5" on a side. Excellent agreement was obtained between the data pairs (equation 4.4) and cell counts estimators, so, we wi l l only refer to the data pairs approach hereafter. 4.2.2 The Clustering Model The standard model for the angular correlation function is as follows: w{9) = AJ~\ (4.6) where 8 has been found to range from ~ 0.6—0.8 for faint samples. The value of 8 could be dependent on the angular scales (9) probed or the magnitude hmit of the galaxy sample (Maddox et al. 1990, B T B J and Neuschaefer and Windhorst 1995). If we want to use the estimated u>{9) projected on the sky to determine the spatial two-point correlation function, a model must be adopted which includes possible clustering evolution with Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 73 redshift (z). The conventional model for the spatial correlation function is as follows: « r , z ) = ( ^ ) " 7 ( l + z ) - ( 3 + e \ (4.7) where evolution with redshift is parameterized by e, r is the proper length and ro is the correlation length for z = 0 (see Phillipps et al. 1978, Peebles 1980, Efstathiou et al. 1991, Infante and Pritchet 1995 and Hudon 1995). If the redshift distribution (^zO * s k n o w n for the magnitude Hmits for which the angular correlation function has been measured, the relationship between u>(9) and £(r, z) is determined with an integral, known as Limber's equation (see, for example, Peebles 1980 and Hudon and LiUy 1996): = Cr&- f I * . ) ' W ( l + *)-<*+" ( f ) ' * [ f ( f ) * ] " • (4.8) D(z) is the angular diameter distance defined as: r>(*\ - c q ° z + ^ ° ~ + 2 q ° z ~ l j (A. Q\ D [ z ) - H 0 q*(i + zy ' t4-yj with g(z) and C given by: and g{z)=—{{l + z)*(l + 2q0z)l)-1 (4.10) - « 0 r ( ( 7 -1)/2) c - v * r ( 7 /2 ) ' ( 4 1 1 ) with go, -Ho and T being the deceleration parameter, Hubble constant and Gamma func-tion, respectively. Note that equation 4.7 and Limber's equation (4.8) gives 8 = 7 — 1. If 7 ~ 1.8, which is the usually adopted value due to surveys of bright, nearby galaxies (Davis and Peebles 1983), this leads to clustering fixed in comoving coordinates having an e = 7 —3 = —1.2. "Stable clustering", with the clustering fixed in proper coordinates, is the result when e = 0. If e > 0 then this corresponds to a growth in the clustering with redshift, in proper coordinates. Using an extrapolated redshift distribution (^ 7) Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 74 from the CFRS (Lilly et al. 1995c) with the same faint magnitude limit as our galaxy sample (/ < 24), Limber's equation can be solved for an assumed cosmology, 7 and e. The growth of clustering (e) can then be estimated by comparing the models with the observations of u(0) (see section 4.3.3). 4.2.3 Star Removal The number counts of objects at faint magnitudes are dominated by galaxies but at bright limits the stellar component makes a significant contribution. To correct for this stellar contamination we plot a crude shape parameter vs. the aperture magnitudes for all the detected objects. This shape parameter is simply the difference between the "core" and aperture magnitude of an object, where the former is the magnitude corresponding to the flux incident on the inner 3x3 pixels. The core and aperture magnitude difference is a measure of the object's light concentration and is analogous to Kron's (1980) r_2 statistic and Petrosian's (1976) radius, which are both essentially half-light radii (Kron's statistic is proportional to the half-light radius). A plot of the core-aperture vs. aperture (3") magnitudes for V, R and I in NF2 and I in NF3, is given in Fig. 4.1 (see also Fig. 3.1 in Chapter 3, for N F l in J). The stellar sequences (see discussion in section 3.3) are demarcated with a solid-lined rectangular box and at the brightest magnitudes they sometimes exhibit increasing values for the shape parameter due to saturation. Magnitude limits for each galaxy sample are shown by dashed vertical lines. Stars brighter than V, R or I = 22 are removed from the final galaxy samples for all three fields, except in N F l where stars are removed down to R or / = 21. With these limits the star/galaxy separation is unambiguous. For fainter objects no attempt is made to further eliminate stars from the sample since compact galaxies could be mistakenly removed and stellar numbers are very small relative to the galaxies at these faint limits. Some of the data obtained for the NF3 field, particularly in V and Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 75 R, have asymmetric point-spread functions (PSFs) due to the C F H T secondary mirror experiencing windshake. Fortunately, the I data for N F 3 have a symmetric P S F such that a well-defined stellar sequence could be discerned for the field. Therefore, the stars i n N F 3 are identified only with the / data, but this isn't a concern considering the great deal of overlap between the stellar samples found with the three bandpasses i n N F l and N F 2 . Note that the "plume" of galaxies i n the N F 3 panel of Fig.4.1, at large values of the shape parameter and near the magnitude l imit of the data, is due to spurious noise objects which have not yet been cleaned from the "raw" galaxy catalog. 4.2.4 Integral Constraint A correction must be made for the integral constraint which is due to the estimation of the density of galaxies, at a given magnitude l imi t , wi th a bounded, finite sample (Peebles 1980). This bias has the effect of reducing the amplitude of w(9). Following B S M and LS we calculate the integral constraint (wn) using: with Q, representing the solid angle of the masked field. We also assume the angular correlation function has the functional form given i n equation 4.6. Integral constraints are calculated for each field and for power laws with S ranging from 0.5 — 0.9, resulting i n U/Q ~ 0.08A,, — 0.01-4.^,. For a u>(8) oc $ ~ 0 8 power law the integral constraints are determined to be u>n ~ 0.0195<AW,0.0199AW,0.0193AW for N F l , N F 2 and N F 3 , respectively. Note that the values are comparable since the field sizes and geometries are similar, but the correction is substantial compared to the small amplitudes of u>(6) measured from the faint galaxy samples (see section 4.3). (4.12) Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 76 Figure 4.1 Galaxy-Star Separation Plots for N F 2 and N F 3 Star-galaxy separa-tion diagrams for NF2 and NF3 with the difference between the "core" and aperture magnitudes plotted on the oordinate and just the latter on the abscissa. The field and bandpass are shown in the upper left corner of each panel. Magnitude Hmits are marked with dashed Hnes and the stellar sequences are demarcated by sohd Hne rectangular boxes. Note the "plume" of galaxies seen in the NF3-7 panel, at the upper right of the data near the magnitude Hmit, is caused by spurious noise objects yet to be removed from the galaxy catalog. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 77 4.2.5 Star Dilution Correction Since stars have not been removed from the photometric samples fainter than V, R or / ~ 22 (see section 4.2.3) a correction must be made to the amplitude of u>(9) to account for the stellar component present down to the magnitude l imit . The number of expected stars for a given bandpass to faint magnitude limits is not well constrained by observations and we must depend on the highly uncertain model of Bahcal l and Soneira (1980). Fortunately, the three fields i n this study were chosen to be at high Galactic latitudes where stars are relatively scarce thereby requiring small stellar dilution corrections. The amplitude of u>(6) after the correction for stellar dilution is made is given by: A SC ( Nobj where N<&j is the number of objects used to calculate u>(0), Nt is the number of stars predicted by the BS model and Aw is the "raw" amplitude of w(#) before any corrections or weighting have been applied. Stellar dilution correction terms calculated for the various magnitude l imited samples are listed i n Table 4.2. 4.2.6 Combining Fields, Error Analysis and Fitting Since using one of the three observed deep fields to obtain a determination of u>(0) is of insufficient accuracy, a strategy must be adopted to combine the data sets and to calculate the average or "f inal" <JJ{9) and the appropriate errors. We follow a similar approach to that of N R G C I , where the average u>(9) for a given b in (6) is calculated using: %„w. ^m, (4.14) with summations over the i = 1,3 fields and where 77; are the weights for each field, which include the stellar dilution correction (section 4.2.5) and the galaxy number densities. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 78 The IC superscript for the u(0) estimates for each field denotes that corrections have been made for the integral constraint. We use number densities of objects as weights since the total areas of the three fields are slightly different. Corrections for higher order correlations (e.g., three-point correlation function) are disregarded since they are negligible relative to the values of the u)fc(6) calculated in the individual fields. Although the estimator for <*>(#) which we use has been shown to give Poissonian vari-ance for uncorrelated data by LS, it does not necessarily follow that it behaves this way for correlated data, as first pointed out by Bernstein (1994). Also, Bernstein emphasized that most authors do not properly account for the interdependence between the various bins by determining a covariance matrix, to yield realistic errors when fitting the angular correlation function with the canonical model. Therefore, we calculate errors using a scheme outlined by Fisher et al. (1994) where the covariance matrix for a particular estimate of w(0) is determined with bootstrap resampling (Barrow, Bhavsar and Sonoda 1984). Alternatively, Bernstein (1994) derives an approximate analytical expression for the covariance matrix, with the model fitting procedures in either study being essentially equivalent. As in NRGCI, for a given field and magnitude-limited sample, resampled estimates of the L>f°(8) are calculated by applying equation 4.4 to a resampled list of galaxies with the same number of objects as the original, real sample. The resampled list is generated by randomly selecting galaxies from the original list with replacement, such that a galaxy can be chosen anywhere from zero up to several times. For each magnitude-limited sample, 50 bootstrap-resampled estimates of u)/,n(0) are made by av-i eraging resampled estimates ofQfc(9) calculated for the three fields. The final bootstrap errors for the different angular separation bins are simply given by the variance of the resampled estimates of £>/;„(#). Finally, a covariance matrix is generated so that the power-law model (equation 4.6) can be properly fit to the clustering observations for each magnitude-limited sample, following the technique described by Fisher et al (1994, Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 79 Appendix A ) . Since the galaxy samples are fairly small (see Table 2.5) we choose to fix the power-law exponent (8) and only let the amplitude (Au) vary when fitt ing the model (equation 4.6) to the data. W i t h one linear parameter i n the model, this makes the % 2 minimization analytic. The power-law exponent, 8, has been measured to range from ~ 0.6 — 0.9 (Neuschaefer and Windhorst 1995, and references therein) for galaxies at faint magnitude Hmits. Accordingly, each £>(#) calculated for a magnitude-Hmited sample i n this study is fit wi th power laws having 8 = 0.5 — 0.9, i n 0.1 increments. The \ 2 statistic calculated for each fit gives an idea of what the most appropriate value for 8 is, although the measurement is not that weH constrained. Our data favour larger values of 8, so we fix 8 = 0.8 to ease comparison with other studies. Table 4.1 iUustrates the relative insensitivity of the final ampfitudes to fits wi th power laws having different values for 8, i n this case for the faintest magnitude-Hmited samples in V, R and I. The % 2 values decrease as the power law approaches 8 = 0.8 — 0.9 but for 8 > 1.0 the errors i n the ampHtude fit increase dramatically. For decreasing values of the power-law exponent, 8, the integral constraint increases (see §4.2.4) and this is the primary reason the % 2 statistic increases dramatically, as Hsted i n Table 4.1. Given the Hmited statistics of our sample, fixing 8 — 0.8 seems to be the best approach but it should be emphasized that the errors given for the fitted ampfitudes using this technique are probably underestimates. See section 4.3.1 for further details of the results of the model fitting. For each magnitude-Hmited sample i n a particular field, £o(9) is calculated for angular separations ranging from 10 — 126", within 6 equally spaced logarithmic bins. The binning and angular separation range were carefuUy chosen to optimize the measurement of u>(0) given the available imaging data. The upper Hmit for 9 was chosen to be roughly one-third of the angular extent of the smallest field, thereby avoiding border effects. A lower Hmit of 10" yielded error bars for the smallest angular Reparation bin which were roughly Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 80 20 < V < 25 20 < R < 25 19 < I < 24 A*Jn X 2 A}in x2 x2 8 = 0.5 0.653 ± 0.024 49.3 0.686 ± 0.022 62.6 0.689 ± 0 . 0 2 0 67.5 6 = 0.6 0.596 ± 0.036 26.9 0.640 ± 0.033 36.7 0.641 ± 0.031 37.9 8 = 0.7 0.546 ± 0 . 0 5 3 13.7 0.605 ± 0 . 0 4 9 20.9 0.603 ± 0.045 20.2 8 = 0.8 0.508 ± 0 . 0 7 7 6.6 0.583 ± 0 . 0 7 2 11.9 0.578 ± 0 . 0 6 7 10.5 8 = 0.9 0.485 ± 0 . 1 1 0 3.1 0.581 ± 0 . 1 0 4 7.1 0.570 ± 0.097 5.4 Table 4.1 Final angular correlation function amplitudes calculated for V, R and I, for the faintest magnitude limited samples, with different assumed power law exponents, 8. The chi-square statistics (x2) are given for each fit as well. comparable to those obtained for bins with a larger 8. A discussion of close galaxy pairs with smaller angular separations (< 10") is given in Chapter 3. 4.3 Angular Correlation Function Results 4.3.1 Magnitude-Limited Samples Measurements of the angular correlation function for the magnitude-limited samples de-fined in section 2.5, for V, R and I, are presented in Figs. 4.2-4.4. Note that the correlation amplitudes generally decrease over the small range of magnitudes probed, most obviously with the R data. The solid lines in Figs. 4.2-4.4 are fits of the model, UJ(8) = AU8~0A, to the data and the errors are calculated using bootstrap resampling, as described in the previous section. Amplitudes measured from the fits to u)f{n(8) (A*Jn) for the various magnitude ranges are listed in Table 4.2, and are appropriate for angular separations (8) given in arcseconds. Table 4.2 also tabulates the stellar dilution correc-tions used for the galaxy samples gleaned from each field, for the three bandpasses (see section 4.2.5). The angular correlation functions measured in V, R and I at the separation 8 = 1' for Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 81 V 20 < V < 24 20 < V < 24.5 20 < V < 25 Ag" 0.550 ± 0 . 2 3 3 0.527 ± 0 . 1 4 1 0.508 ± 0.077 (N"0-N.Y (NF1) L 0 8 7 1 M 7 ' (NF2) 1.494 1.356 1.255 (NF3) 1.198 1.147 1.104 R 20 < R < 24 20 < R < 24.5 20 < R < 25~ Ag* 0.804 ± 0 . 1 2 0 0.676 ± 0.079 0.583 ± 0.072 ( i v ^ k ) 2 ( N F 1 ) 1-105 1.082 (NF2) 1.300 1.253 1.204 (NF3) 1.156 1.117 1.085 I 19 < I < 23 19 < J < 23.5 19 < I <24~ Af™ 0.545 ± 0 . 1 7 6 0.627 ± 0 . 1 1 6 0.578 ± 0.067 (N"°-N.y ( n f i) L°83 L°75 L°65 ' (NF2) 1.185 1.183 1.179 " (NF3) 1.072 L077 1.070 Table 4.2 F i n a l angular correlation function amplitudes for V, R and I, and three different magnitude l imits. Stellar dilution corrections are tabulated for each field, calculated for the magnitude ranges adopted i n the three bandpasses. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 82 0.1 0.05 0 -0 .05 0.1 0.05 0 -0.05 0.1 0.05 0 -0.05 C O 3^ 10 M i l l 20<V<24.5 -I 1 1 K-+-20<V<25 100 0(arcsec.) Figure 4.2 UJ(6) for V magnitude limited samples.Measurements of the angular cor-relation function using V-band data from N F l , N F 2 and N F 3 , for the listed magnitude ranges. Error bars are calculated using bootstrap resampling. The solid lines represent the standard model fit to the data assuming S — 0.8. See text for details. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 83 Figure 4.3 <JJ(9) for R magnitude limited samples.As in Fig.4.2, but for the R data. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 84 0.1 0.05 0 -0 .05 0.1 ~ 0.05 CD 0 -0.05 0.1 0.05 0 -0.05 10 H h H 1—I—h 19<I<23.5 H 1—I—H 19<I<24 100 0 ( a r csec . ) Figure 4.4 u(6) for I magnitude limited samples.As in Fig.4.2, but for the I data. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 85 a power law with 8 = 0.8 are plotted i n F i g . 4.5. Values of u>(9 — 1') from filter-to-filter are roughly comparable within the errors, wi th the .R-band data producing the strongest clustering signal. Galaxies detected in the three bandpasses are used to generate colour-selected samples which are described and analysed i n section 4.3.2. L i l l y et al. (1995c) have measured the redshift distribution for faint galaxies selected to a magnitude hmit of I ~ 22 and also make extrapolations for N(z) to l imits as faint as those obtained for the photometry i n this work. W i t h these redshift distributions and the / - b a n d angular correlation analysis by L idman and Peterson (1996) at brighter magnitude hmits, we use our co(9) measurements i n / to constrain model parameters for the spatial correlation function i n section 4.3.3. Finally, since there have been many recent correlation studies of faint galaxies i n /2-bandpasses, these provide a comparison for the clustering detected wi th /2-nTter data i n this study. In F i g . 4.6, the angular correlation function extrapolated to 9 = 1°, using the standard power law model with 8 = 0.8, is plotted as a function of the R magnitude hmit . The measurements by different groups are denoted by the various symbols shown beside the authors' initials and year of the particular paper, i n the plot reference list. The IJJ{9) amplitudes determined i n this work are given by the solid circles for R = 20 — 24,20 — 24.5,20 — 25. Magnitude transformations for B S M were made using their assumption of R ~ r — 0.55. For Couch et al. (1993, CJB93) and others the conversions given by Yoshii et al. (1993) and Roche et al. (1993) yield R magnitudes from the original VR and rp values. The only observation plotted i n F i g . 4.6 which was not taken with a red filter is that of Metcalfe et al. (1995, M S F R 9 5 ) . Since the M S F R 9 5 result is the u>(9) ampHtude with the faintest magnitude Hmit yet measured from the ground, it is interesting to include it for comparison using the approximate relationship BCCD — R + 1-Our F i g . 4.6 foUows the F i g . 2 presented i n B S M , but even with the additional results included from recent studies there stil l isn't general agreement on the precise Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 86 •1.5 M a g n i t u d e L i m i t Figure 4.5 Angular correlation function calculated for a separation of l' .The amplitudes of the angular correlation function calculated for a separation of one arcminute for V, R and I, assuming a power law with 8 = 0.8. Measurements for the V, R and /-band data are shown in the left, centre and right panels, respectively. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 87 -1 .5 • This Work • BSM95 o : EBKTG91 • MSFR95 A SSFM85 CO d -2 .5 too XI 60 o -3 .5 K HL96 -fr IP95 ° CJB93 -X- RSMF93 a RSMF96 0 i J 1 18 19 20 21 22 23 24 25 26 R Magnitude Figure 4.6 A m p l i t u d e o f w(B) a s a f u n c t i o n o f t h e R m a g n i t u d e limit.Amplitudes of the angular correlation function extrapolated to 1 degree assuming 8 = 0.8 for this work and other studies from the literature, in the R band. Details of the magnitude transformations used for the different observations are given in the text. Each symbol is listed with the initials of the authors' names and the year of the study it denotes. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 88 slope of the monotonic decrease in clustering amplitude with limiting magnitude. No clear indication of the ampHtude starting to level off at a roughly constant value for the faintest magnitudes is observed, which would be expected for some merger models of galaxy evolution (Carlberg and Chariot 1992) or if there was a magnification bias from weak gravitational lensing (ViUumsen 1996). Roche et ,al. (1993) and Roukema and Yoshii (1993) have also found merging model clustering behaviour to be inconsistent with current measurements of <JJ(0). Other observations which seem to rule out the most extreme merger galaxy evolution models (Carlberg 1996) are a collection of redshifts for very faint galaxies obtained with the Keck telescope (Koo et al. 1996). This spectroscopic sample is stiU sparse, so the results should be treated as prehminary, but the median redshifts obtained for I > 22 are contrary to what is expected for Carlberg's "maximal merging model". MSFR95 claim that the ampHtude of u>(0) flattens for B data at about the same magnitude where the slope of the number counts flattens (B ~ 25 or R ~ 24), which they attribute to an effective redshift cutoff for the galaxies. The data from various groups plotted in Fig.4.6 show that any flattening in the clustering ampHtude with magnitude is not that weU constrained as yet, especially considering the inherently large random and systematic errors which plague the measurement of OJ{6) at faint Hmits. It is notable that the u(6) values presented in this work form a smooth continuation of the previous observations made by Infante and Pritchet (1995) and Hudon and Lilly (1996) for R = 21 — 23.5, where the latter study used the same R filter as the current observations. Our data agrees reasonably weU with the Efstathiou et al. (1991) datum point and very weU with the overlapping observations of Roche et al. (1996). The largest discrepancy with this study is seen with BSM's <JJ{9) measurements where our clustering amphtudes are observed to be factors of ~ 2 — 3 larger. A possible explanation for part of this difference is that the B S M field is at low Galactic latitude (6 ~ 35°) requiring larger steUar contamination corrections than our three high Galactic latitude fields. Another Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 89 possibility is that our clustering amplitude errors are underestimates since they are the fitting errors for a $~ o s power law. Nevertheless, the clustering measurements in this study are in agreement with Roche et al. (1996), who in turn agree with the BSM results, so there is a reasonable level of consistency between studies in the R ~ 24 — 25 magnitude range. With three fields the current work is less susceptible to variations in general clustering behaviour induced by large scale structure. The fact that our V and /-band estimates of u>(0) (Fig.4.5) do not show well defined decreases with magnitude, as the Pt-filter data does, is not surprising due to the narrow magnitude range of galaxies and poorer statistics in the V and I filters (note the galaxy samples are cumulative towards fainter magnitudes, not differential). Therefore, the apparent flatness of the clustering amplitudes with magnitude for the V and / galaxy samples should not be interpreted as a strong trend but merely a clustering measure over a small range of magnitudes. Since clustering over a significant magnitude range cannot be tracked with just the data from this study, other studies must be included for a proper analysis of galaxy clustering evolution. This is done for the /-band data in section 4.3.3. 4.3.2 Colour-Selected Samples With the multi-colour deep imaging in each field, the angular correlation function can be measured with colour-selected samples down to faint magnitude limits. A few approaches for gleaning colour-selected samples were attempted to maximize the number of galaxies and thereby improve the accuracy of the u){6) measurements over a significant range in colour. However, it should be noted that having just V, R and / images to work with, and no bluer bandpasses, unfortunately leads to a limited baseline of observed colours for the faint galaxies. In Fig. 4.7 the angular correlation functions selected by (V — R), (R — I) and (V — I) colours are presented, in a plot which is analogous to Fig. 1 of Landy et al. (1996, LSK). Clustering amplitudes measured using the full range of Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 90 angular separations, 10 — 126" (yielding an effective separation of ~ 35"), are plotted as a function of the colours which the galaxies observed are either less than (top row of plots) or greater than (bottom row). Poisson error bars are shown but should be considered to be underestimates of the true errors. The final amplitudes have been corrected for the integral constraints and stellar dilution factors calculated for each field. Also , the plotted values are obtained with weighted averaging, where the weights are determined from the number of galaxies detected i n both bandpasses for a given colour. L S K used (U — RF) colours to find the u>(9) ampHtude increased over a factor of ten for the reddest and bluest galaxies taken from a bright magnitude Hmited sample (Bj < 23.5). F i g . 4.7 shows no indication of this behaviour for the current, fainter galaxy sample with ( V — R), (R — I) and (V — I) colours. W i t h i n the errors, the angular correlation ampHtude integrated over the ful l range of separations is relatively constant regardless of the colours of the galaxies being analysed. This is consistent wi th what B S M (R £ 25.5) and Infante and Pritchet (1995, bj < 24 and RF < 23) have found although one should note that these previous studies made only one division i n colour (blue/red) and did not look at the extremely blue or red galaxies, as did L S K . It is possible the lack of an increase i n the clustering ampHtude i n F i g . 4.7 is due to the V, R and I colours not discriminating the reddest and bluest galaxies weU enough (unHke the substantial colour basehne provided by U and Rp in L S K ) . Another possibiHty is that the numbers of galaxies available for this study are simply insufficient for providing an accurate measure of w(6) wi th colour. There also may be physical reasons for the non-detection of a clustering increase with extreme colour, and these are discussed below. To do a more direct comparison of the clustering observed wi th colour-selected samples i n this and another study, the correlation ampfitudes for (V — 7)-selected galaxy samples are plotted i n F i g . 4.8, for a fixed angular separation of 1'. Results from Neuschaefer et al. (1995b, N R G C I ) are given as open symbols for the 50% blue, 50% red and entire Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 91 0.05 CO II ^ 3 0.05 •H (V-R)< H 1 I (V-R)> J i I -r-H (R-I)< (R-I)> 1 , 1 , 1 V (v-i)< I I / (v-i)> 0 1 2 3 0 1 2 3 0 1 2 3 Figure 4.7 for colour-selected galaxy samples. Measurements of u>(6) for the full range of angular separations (10 — 126"; giving an effective separation of ~ 35") as a function of (V — R), {R — I) and (V — I) colours. The top panels show the clustering amplitudes as a function of colours the galaxies are less than. Similarly, the bottom panels show clustering with the abscissa giving the colours the galaxies are greater than. Note the limited range of colour the V, R and I filters provide. The Poisson error bars plotted are determined from the combination of the three fields, and are underestimates of the true errors (see text). Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 92 samples while the filled symbols are measurements for objects i n the current work with ( V — I) < 1.3, (V — I) > 1.3 and for the full sample, wi th the median I magnitude plotted on the abscissa. The value (V — I) — 1.3 was chosen as the dividing line such that the entire sample could be cut into roughly 50% blue and red galaxies. Note the u>(0) amplitudes calculated from our sample assume that 8 = 0.7 i n order to ease the comparison with N R G C I and the error bars for the (V — J)-selected samples i n this study are calculated using bootstrap resampling. A difference between the clustering of red and blue galaxies was observed by N R G C I for the brighter objects i n their 50% samples, wi th substantial errors. They also argue that there is an increase i n amplitude for the 20% colour marginals (bluest and reddest 1/5 of the galaxies) from the 50% samples, although these two samples for either blue or red objects are consistent within the errors. The ( V — 7)-selected sample i n this study, which is one magnitude deeper than that of N R G C I , shows no sign of colour segregation of the clustering amplitudes beyond I ~ 22, albeit wi th large error bars for the measurements. Also , there is no significant difference observed between the clustering of the galaxies i n the fu l l , /-selected sample and the blue and red samples. The amplitudes from our red and blue galaxy samples do not bracket those calculated for the ful l sample due to not all the galaxies being detected i n both the V and / images. Clustering amplitudes for the red and blue 20% marginals i n our sample were not determined since large errors would result from the small sample size. L i d m a n and Peterson (1996, L P ) also determined u>(9) for ( V — /)-selected samples of galaxies, in the magnitude range / = 18 — 20. They used (V — I) — 1.5 as the blue/red boundary and found a marginally significant difference between the samples wi th red galaxies exhibiting stronger clustering, similar to the N R G C I results at brighter magni-tudes. This comparison of L P , N R G C I and the current work suggests that a difference i n the clustering amplitudes for blue and red galaxies (using (V — /) selection) exists at Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 93 bright magnitudes (7 ~ 18 — 21.5) and either disappears or has not been detected at fainter magnitude Hmits (7 ~ 21.5 — 24). Interpreting these colour-selected clustering results is compHcated i n that, for /-selected samples (for e.g., the C F R S wi th I < 22), red galaxies tend to be confined to a fairly narrow range of intermediate redshifts while blue galaxies are observed to have more broadly distributed redshifts (z ~ 0 — 1), wi th a lower mean z . The results summarized i n F ig . 4.8 may be showing that significant galaxy evolution is occurring at faint magnitudes relative to brighter magnitudes (lower redshift). Red and blue galaxies are observed to cluster differently at lower z , which is simply a reflection of the morphology-density relation. Another possibifity is that the blue sample is more diluted with lower luminosity galaxies which have stronger clustering properties, making the clustering measurements for the blue and red faint galaxy samples indistinguishable. A more accurate approach for tracing clustering evolution of " typical" L* galaxies may be to select out red galaxies and measure their clustering variations with magnitude, since the luminosity function of these objects shows very Httle change over 0 < z < 1 (LiUy et al. 1995c). Obviously, larger multi-colour imaging surveys of faint galaxies are required to more accurately measure the colour-selected angular correlation function and further check the viabifity of various galaxy evolution scenarios. 4.3.3 Comparison with Models of the Spatial Correlation Function The angular correlation function is the two dimensional analogue of the spatial, two-point correlation function, £(r). Clearly, the latter function (or the power spectrum, which is the Fourier transform of £(r)) contains more information and is an essential diagnostic for testing models of large scale structure formation and evolution. To determine a viable model for the evolution of £(r ) wi th redshift (equation 4.7) from measurements of u>(6), one requires the redshift distribution (^f-) of the galaxies within the magnitude interval being considered. W i t h a realistic redshift distribution Limber's equation (4.8) can be Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 94 n 1 1 r T 1 1 r T-—i 1 r ~i 1 1 r - 1 0 O II o OS 3 o •1.5 -2 .5 t5 0 NRGCI 95 A 50% Blue • 50% Red O F u l l S a m p l e 0 This Work A (V-I)<1.3 (Blue) • (V-I)>1.3 (Red) • F u l l S a m p l e _j i i i_ 20 21 _i i i_ _j i i i_ _i i i i_ 22 23 m e d Figure 4.8 (V —7)-selected angular correlation functions.Amplitudes of the angular correlation function calculated for separations of 1 arcminute assuming 8 = 0.7, following Fig.3 of Neuschaefer et al. (1995, NRGCI) to ease the comparison with this work. The abscissa is the median I magnitude for each galaxy sample plotted. Open symbols show the "50% Blue", "50% Red" and full samples of NRGCI. Filled symbols show the red, blue and entire galaxy samples for this study, with (V — I) = 1.3 used as the blue/red dividing line. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 95 solved and relationships between the observed and inferred £(r) can be determined for different cosmologies (q0,H0) and clustering evolution (e), with the power law index for £(r) constrained from the angular clustering results (7 = 8 + 1). Unless otherwise noted we assume that q0 = 0.5 and H0 = 100 km s _ 1 M p c - 1 . Since the correlation length at z = 0, ro, corresponds to the amplitude of the locally observed a range of reasonable values for ro and e are assumed in order to generate models of the clustering evolution. These models are then compared to the observed values of u>(6). The I photometric data in this study have a magnitude limit (I ~ 24) which is a full two magnitudes fainter than the currently largest deep redshift survey (Lilly et al. 1995b). For an estimate of the redshift distribution at the limits of the photometry, the ^ measured to 7 ~ 22 can be extrapolated to fainter magnitude limits with a no-evolution assumption for the galaxies. Evolution is obviously occurring for the galaxies at some level towards fainter magnitudes but the discrepancy between the observed galaxy counts and extrapolated-^ number counts is fairly small for the two magnitude interval beyond I ~ 22 (see Figs. 8 and 9 in Lilly et al. 1995c). Using the Lilly et al. extrapolations to I ~ 24 and the observed redshift distributions for brighter magnitude limits, the variation of u>(#) with I magnitude is calculated for given values of r0, e and 7 (8). Errors are almost certainly present in the extrapolated redshift distributions used to calculate the clustering evolution models. Since the amplitude of UJ(9) calculated using Limber's equation (4.8) has a strong dependence on the shape (essentially the width) of errors will occur if this shape is poorly estimated with the extrapolation, while the effect of an inaccurate normalization will be small. The shape of ^ at our faintest magnitudes (7 ~ 22 — 24) will be incorrect if a particular galaxy population dominates at these limits but is not detected at brighter magnitudes. The calculations using Limber's equation are presented keeping this caveat in mind. Lidman and Peterson (1996, LP) have measured w(9) for a wide range of brighter Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 96 magnitudes in I and we use these results as a comparison to this study, as well as to the models. In Fig. 4.9 the logarithm of u(0) at an angular separation of 1' is plotted with the median I magnitude given on the abscissa. All of the samples obtained from the two fields (CL and FBS) in LP are included, in addition to the Efstathiou et al. (1991) point for the /-band. Models calculated using the aforementioned redshift distributions from Lilly et al. (1995c) are plotted as a series of lines for different values of ro(z = 0) (5.4 h~x Mpc solid lines, 4 h~l Mpc dashed lines and 2 h~x Mpc dotted lines) and e (—1.2,0,1,2 from top to bottom for each set of lines with a given r 0). The value of 7 is fixed at 1.8 following the discussions in sections 4.2.2 and 4.2.6, but the effects of varying it are shown below. Note that the LP results are amplitudes obtained from galaxies within narrow luminosity bins (1 or 2 magnitudes wide) spanning I — 16 — 23 while the points determined in the current study are for galaxies with / = 19 — 23,23.5,24. Our magnitude Hmits were chosen to minimize the error in the ui(9) measurements since there are a Hmited number of galaxies available (2697 with I — 19 — 24). There is fairly good agreement between the correlation function amphtudes from the three studies at the faintest magnitudes in Fig. 4.9. At Imed ~ 22 the LP data are consistent with our measurement of the clustering. For the LP point at Imed ~ 22.5 and the Efstathiou et al. result at Imed ~ 23 there is agreement within 2a of the amphtudes obtained from this study but with a substantial error for the LP measurement at their magnitude Hmit. Comparing the observations to the models in Fig. 4.9 leads to some general con-clusions. Clustering evolution which is fixed in co-moving coordinates (e = —1.2) is a viable scenario only if r0(z = 0 ) ~ 2 — 3 h~x Mpc. Values for r0 are typically not ob-served to be this small for the entire galaxy population. The correlation length usually ranges from ~ 4 h~x Mpc, calculated using IRAS-selected redshift surveys (Saunders et al. 1992, Fisher et al. 1994), to the canonical optical survey correlation length of Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 97 -0 .5 a II - 1 . 5 h O -2 h m e d Figure 4.9 Comparison of / -band u>{9) and galaxy clustering evolution models. The angular correlation function determined for separations of 1 arcminute plotted as a function of the median I magnitude. Results for the /-selected samples of Lidman and Peterson (1996, LP), Efstathiou et al. (1991, E B K T G ) and this work are shown. Four different subsamples of LP are listed with the corresponding symbols, along with the E B K T G observation and the filled squares which denote the /-filter measurements of this work. Each family of lines correspond to evolutionary models calculated for a given correlation length (solid lines: r0 — 5.4 hT1 Mpc, dashed lines: r0 = 4.0 h_1 Mpc and dotted lines: r 0 = 2.0 h~x Mpc) and e = —1.2,0,1,2 (from top to bottom, respectively). Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 98 r0(z = 0) — 5.4 h'1 M p c from Davis and Peebles (1983). B S M show that with a cor-relation length of ro ~ 2 h~l M p c and a rate of clustering growth predicted by linear theory (e ~ 0.8) they can match their clustering observations and models at faint Hmits. From this result B S M claim that low surface brightness (LSB) and/or dwarf galaxies are dominating the faint galaxy population since some local measurements of the cor-relation lengths for these objects yield r0 ~ 2.3 — 2.7 h'1 M p c (Santiago and da Costa 1990). However, the values for the L S B / d w a r f galaxy local correlation lengths are stiU controversial and may be larger (Thuan et al. 1991). A s noted earHer (section 4.3.1), the amphtudes of the B S M observations i n the i?-band are significantly lower than what is observed i n this study at similar magnitude Hmits. Given that most studies to date have found local correlation lengths with r0 ^ 4 h'1 M p c along with the assumption that faint galaxy populations evolve into locally observed galaxies, our /-f i l ter observations then suggest that e > 0, i n agreement with Hudon and LiUy (1996), Le Fevre et al. (1996) and Shepherd et al. (1996). For a non-negative value of e, two general possibifities remain for the evolution of the faint galaxy population. The first scenario is that e ~ 0 - 1 and r0(z = 0) ~ 4 h~l M p c , where the excess of faint blue galaxies is due to objects which are analogous to I R A S -selected galaxies with respect to star formation, morphology and clustering, as suggested by B T B J . The second possibifity is that e > 1, implying significant evolution i n the clustering from faint Hmits to locally observed galaxies such that a value of ~ 5 — 6 h'1 M p c is found for r0. This correlation length is i n agreement wi th most optically-selected redshift survey measurements of £ ( r ) . We note that Efstathiou et al. (1991) only considered clustering models with —1.2 ^ e ^ 0 in order to obey the standard gravitational instabifity picture. However, more recent N-body studies (Melott 1992, Yoshi i , Peterson and Takahara 1993) have found that models with e ~ 0 — 3 are indeed possible due to the continual merging of groups as the universe expands. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 99 To illustrate the sensitivity of the models for u>(0) in I to the assumed 90, 7 and e, for a given ro, the observed clustering amplitudes are plotted again in Fig.4.10 versus the I magnitude limit along with three different families of models. Note that each ordinate of the three panels covers a different range of clustering amplitudes but has a total range of 1 dex. For each r0 listed (5.4,4,2 h-1 Mpc from top to bottom) in the lower left corner, the solid line corresponds to the model calculated for q0 = 0.5, 7 = 1.8 and e = 0. Assuming a small-fi0 universe with q0 = 0.1 yields the dotted line model for each ro. Changing just the power-law index for the correlation function to the two extremes of what is observed, 7 = 1.9 and 7 = 1.6, gives the short-long dashed lines above and below the solid line, respectively. Finally, the long dashed lines are associated with changing just the value of e, as in Fig. 4.9. The long-dashed line above the solid line is for e = —1.2 while the two dashed lines below correspond to e = 1,2 for decreasing amplitude. Clearly the comparison between the observations and models does not have a strong dependence on the assumed value of qo. Not surprisingly, a variation in 7 away from the conventional value of 1.8 causes the most pronounced change from the solid-line model when r0 = 5.4 h~x Mpc. The clustering evolution parameter e provides the greatest leverage in parameter space for matching the observations to clustering models, in addition to being poorly constrained. If the inherent degeneracy of fitting clustering models to measurements of u>(0) is to be broken, better determinations of r0 and 7 for local samples of galaxies selected by morphology, luminosity and surface brightness are required. 4.4 Discussion and Conclusions Using the measurements of w(0) presented in the current work is it possible to rule out any of the galaxy evolution models outlined in Chapter 1 (section 1.2)? The lack of Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 100 Figure 4.10 Further comparison of the / -band u>(8) and models. A further com-parison of the /-band observations of a>(0) with 6 — V to a suite of evolutionary models calculated using the extrapolated CFRS redshift distributions (Lilly et al. 1995c). Note the abscissa is the / magnitude hmit not the median magnitude. The correlation length (ro) used is shown in the lower left corner of each panel. For each panel, the solid line corresponds to the model with q0 = 0.5, 7 = 1.8 and e — 0. Just a change of q0 = 0.1 yields the dotted line. Setting 7 = 1.9,1.6 moves the solid line to the short-long dashed lines above and below, respectively. Changing just e, to —1.2, gives the long dashed line above the solid line, while the dashed lines in decreasing ampHtude below are for e = 1,2. Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 101 sustained flattening in the amplitude of the angular correlation function over a significant range of magnitude, or a well defined change in the slope of the decreasing u>(9) with magnitude, is not seen with current data suggesting merger models (Guiderdoni and Rocca-Volmerange 1990, Broadhurst et al 1992, Carlberg and Chariot 1992) may not be viable descriptions of faint galaxy evolution. Since our three measurements of the /-selected u>(0) are over a narrow magnitude range (Figs. 4.9 and 4.10) no claim is made for a detection of flattening in the clustering amplitude. When combined with the LP clustering study a generally smooth decline in the amplitude of w(8) with magnitude is observed. For the slightly better statistics of the R observations (Fig.4.6), a decrease in the clustering amplitudes with magnitude limit is unambiguously seen to the faintest limits, in conjunction with other studies. Merger-dominated galaxy evolution models generally exhibit amplitudes of u(8) which are larger than those determined in the /2-band (Roukema and Yoshii 1993, Roche et al. 1993 and Metcalfe et al 1995) with an eventual change in slope or flattening of the clustering amplitude towards fainter magnitudes. It is still possible to incorporate some merging into galaxy evolution models without being inconsistent with the observed clustering and redshift distributions, but models with "maximal merging" are certainly ruled out. This can also be concluded from preliminary results of a deep redshift survey which uses the Keck telescope (Koo et al 1996). A lack of a significant amount of merging of galaxies at faint limits is in agreement with our results from Chapter 3 (Woods et al. 1995) where no substantial excess of close pairs of galaxies in N F l were found down to / < 24. Obviously the clustering measurements displayed in Fig.4.6 are still too inaccurate to reasonably constrain any detailed galaxy evolution model. Larger photometric surveys of faint galaxies are essential to achieving more precise determinations of u>(#). Assessing whether the faint galaxy population is predominantly bursting dwarfs or fading dwarfs is very difficult (if not impossible) to distinguish at this juncture using only Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 102 estimates of the clustering. Two general scenarios which bracket the range of clustering evolution (see Fig.4.9) could be occurring: (i) the majority of faint galaxies are evolving into a local population which is similar to IRAS-selected galaxies (7*0 ~ 4 h~x Mpc) with moderate clustering evolution (e ~ 0 — 1) or (ii) the clustering of the faint population is evolving at a greater rate (e > 1) yielding local galaxies with similar clustering properties to what is observed in optical surveys (r0 ~ 5 — 6 h~x Mpc). The other general possibility for the clustering evolution (e ~ —1.2) is more unpalatable since it requires the local counterparts of faint galaxies to be weakly clustered (r0 ~ 2 — 3 h~x Mpc), which is typically not measured. More local surveys for low surface brightness galaxies are required to better constrain the purported weak clustering behaviour of this population. The three approximate pairs of ro and e which match the clustering model to our J-band <JJ(6) measurements, are also in broad agreement with those found by Hudon and Lilly (1996) in the R bandpass. Clustering evolution processes (i) and (ii) could both be occurring but recent HST observations suggest that late-type and irregular galaxies dominate the number counts at faint hmits (Driver et al. 1995, Glazebrook et al. 1995b). Loveday et al. (1995) have measured the local correlation lengths for early and late-type galaxy morphologies to be 5.9 ± 0 . 7 h~x Mpc (E and SO) and 4.4 ± 0 . 1 h'1 Mpc (Sp and Irr), respectively. If spirals and irregulars are the dominant population comprising the faint counts it is possible that the primary path for faint galaxy evolution is scenario (i) above, with a moderate change in the clustering (e ~ 0 — 1). This value of e is consistent with what is predicted by linear theory and is easier to account for than larger values of e. Another point which should be made is that the large fraction of irregular galaxies found at faint magnitude hmits with HST data (Driver et al. 1995, Glazebrook et al. 1995b, Abraham et al. 1996) could possibly be morphologically "regular" galaxies at high redshift. Since galaxy morphologies at large redshifts are subject to K-corrections, Chapter 4. Correlation Analysis of Three High-Galactic Latitude Fields 103 evolutionary effects of stellar populations and a strong dependence on surface brightness (Giavalisco et al. 1996) it is not necessarily clear that strong morphological evolution is occurring at faint magnitudes. Morphology studies of local galaxies at U V wavelengths and further faint galaxy spectroscopy should address this concern. Considering the previ-ous point, it is possible that moderate galaxy clustering evolution (e ~ 0 — 1) is occurring for a majority of the faint galaxies which are normal, late-type spiral galaxies or dwarf galaxies, undergoing enhanced star formation at high redshifts and then evolving into lo-cal populations with r0 ~ 4 h~r M p c . This picture is broadly consistent wi th the C F R S luminosity function ( L F ) study of galaxies out to z ~ 1, which finds very little evolution for the L F of red galaxies but a significant amount of evolution for the blue galaxy L F . Deeper redshift surveys (Koo et al. 1996) should help identify the dominant species i n the faint galaxy population. No evidence was found i n this study for a dependence of clustering on galaxy colour (Figs.4.7 and 4.8), such as what L S K found using U — RF colours at brighter magnitude hmits. It is possible that the V, R and I filters used i n the current work were not a large enough colour baseline. Also , none of the filters may have been blue enough to discriminate extremely blue (or red) galaxies which are suggested to be more clustered than the overall population. Nevertheless, Roche et al. (1996) find a difference i n the clustering of red and blue galaxies at B ~ 25.5. They suggest that the observed clustering is consistent wi th the L S K result at B < 23.5. No significant differences i n the clustering measured for red and blue samples of galaxies were found by Efstathiou et al. (1991) and B S M . Neuschaefer et al. (1995b), as mentioned earlier, find only marginal differences i n the clustering for (V — 7)-selected samples of red and blue galaxies at faint hmits (I > 21.5) while larger discrepancies are observed for brighter magnitudes (I < 21.5). This roughly consistent clustering of red and blue galaxies at faint magnitudes is i n agreement with what is observed for the (V — 7)-selected samples of this work (Fig.4.8). Chapter 4. Correlation Analysis o f Three High-Galactic Latitude Fields 104 It remains to be seen if the difference (or consistency) of the clustering between red and blue galaxies is a function of the sample magnitude limit. The accuracy of the measurements of the colour-selected w(#) in the current work does not warrant further analysis. Larger photometric samples of faint galaxies will be required to further elucidate the clustering behaviour of galaxy samples chosen by colour. A summary of the conclusions now follows. The amplitude of u>(6) is found to de-crease with magnitude limit when our R and /-band results are combined with clustering determined by other authors. This observed monotonic decrease with magnitude rules out merger-dominated galaxy evolution models. Angular correlation function estimates (in /) of the current study and LP were compared to galaxy clustering evolution models, generated with CFRS redshift distributions which were extrapolated to the faint limits of the photometry. The observed clustering of the faint galaxies can be explained with local correlation lengths for this population of ~ 4 h'1 Mpc or ~ 5 — 6 h'1 Mpc for moderate (e ~ 0 —1) or strong (e > 1) clustering evolution, respectively. Clustering evolution which is fixed in co-moving coordinates (e = —1.2) is possible but requires smaller correlation lengths (~ 2• — 3 h~l Mpc) than what is usually observed for local galaxies. No evidence is found for variations in clustering which are dependent on galaxy colour in this study. Larger photometric surveys are required to confirm the stronger clustering amplitudes for red galaxies found by Roche et al. (1996) at faint magnitude limits. Chapter 5 Conclusions 5.1 Thesis Summary A photometric survey of faint galaxies has been performed using high-quality, V, R and I C C D images of three high Galactic latitude fields, obtained with the Canada-France-Hawaii telescope. Using image analysis techniques described in Chapter 2, faint galaxy samples have been obtained down to magnitude limits of V ~ 25, R ~ 25 and I ~ 24. The primary purpose of this thesis was to analyse the clustering of the galaxies with two complementary techniques, close pair and angular correlation analysis. Close pairs of galaxies have been detected and analysed in one of the three fields (NFl). This close pair analysis was given in Chapter 3. The number of close pairs observed at faint magnitudes, in comparison to local pair fractions, constrain the growth of the merger rate with redshift. No evidence was found in N F l for a significant excess of close pairs for angular separations of 1" — 11" to the magnitude limits in each filter. The faint pair fractions found were actually consistent with the galaxies being randomly distributed. This result is contrary to what B K W F found for I < 23 using W F P C pre-refurbishment HST images and an identical approach to the pair analysis. Colours and morphologies of the pairs are also consistent with the majority of the pairs being random superpositions although these are much less convincing indicators of interaction at faint limits. Assuming the fainter galaxies in our samples are at higher redshifts suggests that the merger rate of galaxies has been overestimated or changes significantly from what is 105 Chapter 5. Conclusions 106 observed at brighter magnitude l imits. To better study close pairs and the associated merger rate, high-resolution H S T imaging of "blank" fields and more redshift surveys of paired and isolated faint galaxies are necessary. In Chapter 4, the theory, technique and execution of angular correlation analysis was presented for the three C F H T fields. The main result was that the amplitude of the angular correlation function monotonically decreases to the faintest magnitude limits i n R, the filter with the deepest images and consequently largest numbers of galaxies (most reliable statistics) i n our sample. This decrease i n OJ(0) wi th magnitude is inconsistent with merger-dominated models of galaxy evolution. Using galaxy clustering evolution models derived with C F R S extrapolated redshift distributions, predictions were made for the identity of the dominant constituent of the faint galaxy population: (1) galax-ies with correlation lengths characteristic of IRAS-selected galaxies or late-type spirals ( r 0 ~ 4 h~x M p c ) with moderate clustering evolution (e ~ 0 — 1); (2) galaxies wi th correl-ation lengths similar to that observed for the overall local population ( r 0 ~ 5 —6 h~x M p c ) experiencing substantial clustering evolution (e > 1); or (3) weakly correlated galaxies ( r 0 ~ 2 — 3 h'1 M p c ) with clustering evolution which is fixed in co-moving coordinates (e ~ —1.2). The faint galaxy population could be comprised of any combination of the above three candidates but option (1) is the most attractive since large clustering evolu-tion and weakly correlated galaxies are not required. Finally, no variations of clustering were found for galaxy samples selected by colour. This result may be an artifact of the l imited colour discrimination of the V, R and / filters used. Larger photometric surveys than the current work wi l l be needed i n the future to more accurately measure angular clustering of galaxies for magnitude and colour-selected samples, so that the nature of faint galaxies beyond the spectroscopic l imit can be understood. Also , more redshift sur-veys of local galaxy populations and faint galaxies wi l l be needed to break the degeneracy of galaxy evolution models. Chapter 5. Conclusions 107 5.2 Future Work in Faint Galaxy Research The introduction of cameras which contain a mosaic of large-format C C D s , such as M O -C A M and U H C A M (~ 0.25deg.2 field-of-view) at C F H T , significantly improves the areal coverage for a given exposure time. This innovation i n imaging technology makes it possible to obtain significantly larger samples of photometry of faint galaxies. A larger number of galaxies is crucial i n obtaining an accurate measurement of the angular cor-relation function for both small and large separations. W i t h mosaic cameras an order of magnitude increase i n the number of galaxies being imaged can be achieved using a similar amount of telescope time as this study. For example, wi th an X-shaped con-figuration of five M O C A M fields observed to a magnitude hmit of R ~ 25.5 one could observe ~ 60,000 galaxies and measure the correlation function for separations ranging from below 1" up to scales of 14'. Using U H C A M would increase this by a factor of four yielding images of an incredible 2 .5xl0 5 galaxies. It should be emphasized, obtaining either of these impressive data sets would be possible with only 5-6 nights on C F H T . Large spectroscopic surveys of local and faint galaxies are necessary for an accurate measurement of the evolution of £(r), for galaxies with different morphologies, luminosi-ties and surface brightnesses. M a n y new spectroscopic surveys are underway or wi l l begin soon including the C N O C field galaxy survey, 2dF survey and D E E P survey. In partic-ular, the D E E P , survey which uses the 10-m Keck telescope to obtain spectra of faint galaxies (Imed ^ 22) at high redshift (zmed ~ 0.8) wi l l provide further clues towards un-derstanding the nature of faint galaxies. These redshift surveys along with the upcoming Sloan Digi ta l Sky Survey wi l l further illuminate how the clustering and luminosity func-tions of intermediate and high redshift galaxies evolve into the locally observed galaxy populations. W i t h the substantial increase i n photometric survey area provided by C C D mosaic cameras and the advent of multi-plexed spectroscopic surveys, some significant Chapter 5. Conclusions gains are going to be made i n the next few years i n faint galaxy research. "The best thing for being sad," replied Merlyn, beginning to puff and blow, "is to learn something. That is the only thing that never fails. You may grow old and trembling in your anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honour trampled in the sewers of baser minds. There is only one thing for it then - to learn. Learn why the world wags and what wags it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting." - T.H. White References A b r a h a m , R . A . et al . 1996, M N R A S , 279, L47 Babul , A . , k Rees, M . J . 1992, M N R A S , 255, 346 Bahcal l , J . N . , k Soneira, R . M . 1980, A p J S , 44, 73 Barrow, J . D . , Bhavsar, S . R , k Sonoda, D . 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