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Searching for hemispheric asymmetry and parity violation with the cosmic microwave background Contreras, Dagoberto 2018

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Searching for hemispheric asymmetryand parity violation with the cosmicmicrowave backgroundbyDagoberto ContrerasB.Sc., The University of Waterloo, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2018c© Dagoberto Contreras 2018The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for external examination,the dissertation entitled:Searching for hemispheric asymmetry and parity violation with the cos-mic microwave backgroundSubmitted by Dagoberto Contreras in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in physics.Committee:Douglas ScottSupervisorJeremy HeylSupervisory Committee MemberGary HinshawSupervisory Committee MemberScott OserSupervisory Committee MemberiiAbstractThe current standard model of cosmology is an extremely successful theorythat describes all available data with only half a dozen or so parameters.Nevertheless there are aspects of the model that remain mysterious. Inthis thesis I test two assumptions with the cosmic microwave background(CMB) as measured by the Planck satellite mission. The first is statisti-cal isotropy, motivated by hints in the temperature anisotropies that poweron large scales exhibits a dipolar asymmetry. I confront this claim withdata and formulate a mechanism to predict the corresponding asymmetryin different modes given a specific model. I apply this to temperature, CMBlensing, and polarization specifically. I find that while lensing is not con-straining enough to help in distinguishing models, cosmic-variance-limitedpolarization will prove very helpful in doing so. I forecast that if the asym-metry signal is correct, then Planck polarization is quite unlikely to detectit, while a cosmic-variance-limited polarization experiment will increase theprobability of a detection greatly. Furthermore via their over production oftotal power to the CMB, I rule out a class of models that try to explainthe asymmetry such as an asymmetry in tensors or isocurvature. The sec-ond assumed symmetry is parity. The CMB is sensitive to parity-violationin the electromagnetic sector via correlations of temperature and E modeswith B modes. The parity-violation is parameterized by an angle α, de-fined on the sky. I use polarization data to constrain a uniform α, settingthe current best limits on this angle α = 0.◦35 ± 0.◦05 (stat.) ± 0.◦28 (syst.).I demonstrate that this measurement is now dominated by systematic ef-fects and thus unlikely to be improved upon in the near future. I alsoset the current best constraints on large-scale anisotropies of α via a scale-invariant power spectrum L(L+1)CL/2π < [2.2 (stat.)±0.7 (syst.)]×10−5 =[0.07 (stat.) ± 0.02 (syst.)] deg2. Furthermore I constrain power on L = 1,√3C1/4π = 0.◦32 ± 0.◦10 (stat.) ± 0.◦08 (syst.) and L = 2 modes α20 =0.◦02± 0.◦21. The CMB is therefore consistent with no parity violations.iiiLay SummaryIs the Universe lop-sided and/or would it care if you looked at itthrough a mirror?The standard model of cosmology is an extremely successful theory thatdescribes all available cosmological data with only half a dozen or so param-eters. Nevertheless there are aspects of the model that remain mysterious.In this thesis I test two assumptions with the cosmic microwave background,which could provide clues into some of the basic questions about the cosmo-logical model. The first is the cosmological principle, which states that thereare no preferred directions in the Universe. The second is parity violationin electromagnetism (theory of light), which has to do with how somethinglooks when viewed through a mirror. I confront these assumptions withcosmological data and find them to be consistent with the standard model.This, while not yet providing new insight, will narrow down the realm ofpossibilities for which answers can be found.ivPrefaceThe introduction to Part I was adapted from text I wrote as part of discus-sions with the Planck collaboration that was published in Planck Collabo-ration XVI (2016).Chapter 1 is based on the work published in Zibin and Contreras (2017); Iwas responsible for the derivation of the estimator, the data analysis, andmuch of the writing. J.P. Zibin was responsible for developing the formalismused and manuscript editing.Chapter 2 is based on work published in Contreras et al. (2017a); I was thelead author and responsible for concept formulation, the data analysis, andmost of the manuscript composition. J.P. Zibin was involved in discussingwhich models to look at specifically and manuscript editing. D. Scott,A.J. Banday, and K.M. Go´rski were the supervisory authors involved inmanuscript composition.Chapter 3 is based on work published in Contreras et al. (2018); I wasresponsible for concept formulation of the latest version of the paper, andall of the data analysis and manuscript composition. J. Hutchinson, A. Moss,and J.P. Zibin were involved in concept formulation of an early version ofthe paper. J.P. Zibin was also heavily involved in concept formulation andmanuscript editing throughout. D. Scott was the supervisory author andinvolved in manuscript composition.Chapter 4 is adapted from the work published in Planck Collaboration XLIX(2016); I did all of the work in that chapter, with the exception of producingthe final figure. The final figure was produced by A. Gruppuso.vPrefaceChapter 5 is based on work published in Contreras et al. (2017b); I was thelead author and was responsible for concept formulation, did all the analysis,and the writing for this work, though the idea was developed along with anundergraduate student (Paula Boubel) I was helping to supervise at thattime. Paula Boubel was involved in concept formulation of an early versionof the paper, and manuscript editing throughout. Douglas Scott was thesupervisory author and involved in manuscript composition.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1I Dipolar asymmetry . . . . . . . . . . . . . . . . . . . . . . . 251 CMB Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2 Primordial adiabatic k-space modulation . . . . . . . . . . . 331.3 Effect on CMB temperature anisotropies . . . . . . . . . . . 361.3.1 Multipole covariance . . . . . . . . . . . . . . . . . . 361.3.2 Connection to general asymmetry form . . . . . . . . 411.4 Effect on lensing potential . . . . . . . . . . . . . . . . . . . 421.5 Fitting the k-space modulation to temperature data . . . . . 441.5.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . 441.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 471.6 Predictions for CMB lensing . . . . . . . . . . . . . . . . . . 501.6.1 Modulation power spectrum . . . . . . . . . . . . . . 501.6.2 Detectability for ideal lensing map . . . . . . . . . . . 52viiTable of Contents1.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 CMB Polarization . . . . . . . . . . . . . . . . . . . . . . . . . 562.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.2 Modulation approach . . . . . . . . . . . . . . . . . . . . . . 582.2.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . 582.2.2 Adiabatic modulation . . . . . . . . . . . . . . . . . . 622.2.3 Tensor modulation . . . . . . . . . . . . . . . . . . . 632.2.4 Isocurvature modulation . . . . . . . . . . . . . . . . 642.3 Dipole asymmetry estimator . . . . . . . . . . . . . . . . . . 642.3.1 Connection to previous approaches . . . . . . . . . . 642.3.2 Full-sky, noise-free case . . . . . . . . . . . . . . . . . 652.3.3 Realistic skies . . . . . . . . . . . . . . . . . . . . . . 662.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.4.1 Temperature only . . . . . . . . . . . . . . . . . . . . 672.4.2 Including polarization . . . . . . . . . . . . . . . . . . 682.4.3 Distinguishing modulated from isotropic polarization 792.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 CMB temperature . . . . . . . . . . . . . . . . . . . . . . . . . 863.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.4 Modulation estimator . . . . . . . . . . . . . . . . . . . . . . 903.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95II Parity symmetry . . . . . . . . . . . . . . . . . . . . . . . . 984 Isotropic Birefringence . . . . . . . . . . . . . . . . . . . . . . 1024.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.2 Impact of birefringence on the CMB polarization spectra . . 1034.3 Data and simulations . . . . . . . . . . . . . . . . . . . . . . 1044.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.4.1 Transformed Stokes parameters . . . . . . . . . . . . 1074.4.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . 1084.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.6 Systematic effects . . . . . . . . . . . . . . . . . . . . . . . . 1144.6.1 Noise properties of polarization . . . . . . . . . . . . 116viiiTable of Contents4.6.2 Beam effects . . . . . . . . . . . . . . . . . . . . . . . 1174.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185 Anisotropic Birefringence . . . . . . . . . . . . . . . . . . . . . 1205.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.2 Impact of birefringence on the CMB . . . . . . . . . . . . . . 1225.3 Data and simulations . . . . . . . . . . . . . . . . . . . . . . 1245.4 Measuring local rotations . . . . . . . . . . . . . . . . . . . . 1265.4.1 Low multipoles . . . . . . . . . . . . . . . . . . . . . . 1275.4.2 High multipoles . . . . . . . . . . . . . . . . . . . . . 1285.5 Tests of the method . . . . . . . . . . . . . . . . . . . . . . . 1305.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.6.1 Maps and power spectrum . . . . . . . . . . . . . . . 1315.6.2 Constraints on a scale-invariant power spectrum . . . 1325.6.3 The dipole . . . . . . . . . . . . . . . . . . . . . . . . 1355.6.4 The M = 0 quadrupole . . . . . . . . . . . . . . . . . 1365.7 Systematic effects . . . . . . . . . . . . . . . . . . . . . . . . 1375.7.1 Foregrounds . . . . . . . . . . . . . . . . . . . . . . . 1375.7.2 Point sources . . . . . . . . . . . . . . . . . . . . . . . 1395.7.3 Relative uncertainty on the PSB orientations . . . . . 1405.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145AppendicesA Effect of modulation on small-scale T anisotropies . . . . . 165B Dipolar modulation of polarization . . . . . . . . . . . . . . . 168C Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172D Simulating modulation parameters . . . . . . . . . . . . . . . 173D.1 Isotropic estimates . . . . . . . . . . . . . . . . . . . . . . . . 173D.2 Anisotropic estimates . . . . . . . . . . . . . . . . . . . . . . 173E Detection and removal of aberration . . . . . . . . . . . . . . 175ixTable of ContentsF Stacking on E-mode peaks . . . . . . . . . . . . . . . . . . . . 176xList of Tables1.1 Marginalized posterior mean values and their 68% uncertain-ties for the modulation parameters of the model of Eq. (1.9),along with their corresponding maximum-likelihood values.The angles l and b are the Galactic longitude and latitude,respectively, calculated via Eqs. (1.51) and (1.52). The finalrow is the projected combined constraint including an idealCMB lensing experiment assuming a modulation with am-plitude AR = 0.122 and the remaining temperature meanvalues. The addition of lensing does not appreciably help toconstrain the model. . . . . . . . . . . . . . . . . . . . . . . . 512.1 Best-fit modulation parameters for the Planck temperaturedata, given the models described in Sect. 2.2. . . . . . . . . . 702.2 Probability of a “2σ”-detection (as defined in Sect. 2.4.3) of areal modulation as described by the model in the first columngiven modulated Planck or cosmic-variance-limited polariza-tion. The “best fit” columns refer to modulating polariza-tion with the best-fit values from the temperature data (seeTable 2.1). The “sampling” columns refer to modulating po-larization using parameters chosen by randomly sampling thefull likelihood of the temperature data. The latter values area more conservative approach to how the polarization mightbe modulated and thus give smaller probabilities of detection. 843.1 Data sets used for the isotropic constraints. BKP refers to theBICEP2/Keck Array-Planck joint analysis (BICEP2/KeckCollaboration et al. 2015). . . . . . . . . . . . . . . . . . . . . 93xiList of Tables3.2 Percentage of the posterior for which the amplitude A exceeds3σX , i.e., P>3σ, as well as A/σX for the maximum-likelihoodparameters, for different combinations of data. These quan-tify whether the model can source significant asymmetry giventhe data, a necessary but not sufficient condition for prefer-ring the model over ΛCDM. The asterisk denotes the additionof a fully anti-correlated isocurvature component. . . . . . . . 943.3 95% CL (or upper limits) for the parameters r0.002 and α0.002for various tensor and isocurvature models and data combi-nations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.1 Mean values and (1σ) statistical uncertainties for α (in de-grees) derived from the stacking analysis for all component-separation methods, coming from hot spots, cold spots, andall extrema. We include the fit from each component-separationmethod’s half-mission half-difference (HMHD)Q and U maps,as an indication of the expectation for noise. NILC and SMICAhave smaller uncertainties compared with Commander and SEVEM,which follows from the naive expectation of the rms in the po-larization maps (see Table 1 of Planck Collaboration IX 2016).1134.2 χ2 values for the model with α = 0, derived from the stackinganalysis for all component-separation methods. The ∆χ2 isthe reduction of χ2 given the values of α in the correspondingentry in Table 4.1. The number of degrees of freedom is 2500coming from a 5◦ × 5◦ patch with 0.1◦ pixel size. . . . . . . . 1155.1 95% CL upper limits on the amplitude A of a scale-invariantpower spectrum. Here “All L” refers to the combination ofour uniform weighting low-L and high-L likelihoods. The lastcolumn comes from BICEP2 Collaboration et al. (2017), usingpolarization data from the BICEP2/Keck Array. . . . . . . . 1345.2 Mean posterior values and 68% uncertainty levels for the am-plitude and direction of the dipole in α. The correspond-ing 68% radial positional uncertainty around the best-fit di-rection is about 25◦, with a corresponding p-value of 1.4%.The difference between both methods is attributable to resid-ual foregrounds (which are more apparent for the hits-mapweighting, see Section 5.7.1), as well as a significant system-atic effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135xiiList of Tables5.3 Probability to exceed the χ2 obtained from the cross-correlationof our α maps (or just the dipole) with the corresponding αfmap (or just dipole) from each foreground of the data. Wefind a marginally significant correlation with polarized dust,due to residual dust in the data that is not accounted for inour simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 138xiiiList of Figures1 Mollweide projection in Galactic coordinates of the temper-ature anisotropies and Q and U polarization as measured bythe Planck mission (top to bottom, respectively). The unitsare in µK and the greyed-out area is the mask applied toremove residual foregrounds, mainly from our Galaxy. . . . . 122 A demonstration showing that rotating a polarization per-turbation field about 2π means you have actually rotated thefield twice, hence it is a “spin-2” field. . . . . . . . . . . . . . 133 Polarization patterns for a pure E and B mode, in red andblue, respectively. Looking at these two patterns in a mirror,the E-mode pattern would look the same, while the B-mode“wind-mill” pattern would change handedness. . . . . . . . . 154 The power spectrum of temperature anisotropies DTTℓ = ℓ(ℓ+1)CTTℓ /2π as measured by the Planck satellite (Planck Col-laboration I 2016). The red curve is the best-fitting ΛCDMtheory curve, and the lower panel shows the residuals relativeto this model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 The power spectrum of E-mode polarization as measured bythe Planck satellite (Planck Collaboration I 2016). The redcurve is the best-fitting ΛCDM theory curve. The green curveindicates a residual systematic effect coming from the imper-fect separation between temperature and polarization. . . . . 186 The power spectrum of temperature E mode cross-correlationDTEℓ = ℓ(ℓ+ 1)CTEℓ /2π, as measured by the Planck satellite(Planck Collaboration I 2016). The red curve is the best-fitting ΛCDM theory curve. The green curve indicates aresidual systematic effect coming from the imperfect sepa-ration between temperature and polarization. . . . . . . . . . 197 The CMB temperature anisotropies for ℓ ≤ 64, centred onthe direction of the dipolar power asymmetry, left, and theanti-direction, right. . . . . . . . . . . . . . . . . . . . . . . . 26xivList of Figures1.1 Limber approximation kernels (in arbitrary units) for vari-ous cosmological observations (contours) out to last scattering(r = rLS). The horizontal axis represents the magnitude of a3D k-mode, while the vertical axis shows the comoving dis-tance to last scattering (we are located at r = 0, and z = 0).The grey box indicates very roughly the reach of the plannedEuclid survey (Laureijs 2009). Dotted magenta curves corre-spond to fixed multipole scales, with the red hatched regiongeometrically inaccessible (ℓ < 1). The vertical cyan linecorresponds approximately to the scale ℓ = 65 in the primaryCMB (narrow green box at the top); scales roughly to the leftof it exhibit dipolar asymmetry in the CMB. If only modesto the left of the cyan line are modulated then only probessourced in this region will be able to test modulation models.Adapted from Zibin and Moss (2014). . . . . . . . . . . . . . 311.2 Marginalized posteriors for the parameter set {pi,∆X,∆Y,∆Z};dark and light blue (solid) contours enclose 68% and 95% ofthe likelihood, respectively. The black and grey (dashed) con-tours and curves represent the theoretical distributions of theparameters coming solely from cosmic variance in statisticallyisotropic skies. The values (kc,∆ ln k) = (5× 10−3Mpc−1, 0)would correspond roughly to the often-considered ℓ-space mod-ulation to ℓ ≃ 65. . . . . . . . . . . . . . . . . . . . . . . . . . 491.3 Marginalized posteriors for the parameter set {pi, AR}; darkand light blue (solid) contours enclose 68% and 95% of thelikelihood, respectively. The black and grey (dashed) con-tours and curve represent the theoretical distributions of theparameters coming solely from cosmic variance in statisticallyisotropic skies. . . . . . . . . . . . . . . . . . . . . . . . . . . 501.4 Temperature anisotropy isotropic power spectrum, CΛCDMℓ(solid black curve), anisotropic power spectrum, C loℓ (dashedred curve), and power spectrum increment from equator topole, ∆Cℓ (dot-dashed green curve), for the case of the maximum-likelihood modulation from Table 1.1, which fits the observedtemperature asymmetry. The modulation in amplitude is ata level of roughly 7% to ℓ ≃ 50. . . . . . . . . . . . . . . . . . 52xvList of Figures1.5 Lensing potential isotropic power spectrum, C lensℓ (black curve),predicted anisotropic power spectrum, δC lensℓℓ (red curve), andpredicted power spectrum increment from equator to pole,∆C lensℓ (green curve), for the case of the maximum-likelihoodmodulation from Table 1.1, which fits the observed temper-ature asymmetry. The modulation in amplitude is at a levelof roughly 1.5% or less to ℓ ≃ 50. . . . . . . . . . . . . . . . . 532.1 Marginalized posteriors for the adiabatic power-law (top lefttriangle plot), ns gradient (top right), and tanh (bottom)models, using Planck temperature data only. Dark and lightblue regions enclose 68% and 95% of the likelihood, respec-tively. The black dashed curves represent the theoretical dis-tributions of the parameters coming solely from cosmic vari-ance in statistically isotropic skies. . . . . . . . . . . . . . . . 692.2 ΛCDM power spectra for TT , TE, and EE (top to bottompanels) compared to the best-fit asymmetry spectra, AC loℓ , tothe Planck temperature data (see Table 2.1), for the variousmodels. The purple curve in the bottom panel is the noisepower spectrum for a single FFP8 noise realization. The best-fit TT asymmetry spectra give several-percent-level asymme-try for ℓ . 100, as expected. Here Dℓ ≡ ℓ(ℓ+ 1)Cℓ/(2π). . . . 712.3 Cosmic variance for a measurement of the amplitude of mod-ulation for the tanh model (with ∆ ln k = 0.01) for TT (blackcurve) and EE (red). Polarization does considerably betterthan the naive ℓ-space expectation of identical cosmic vari-ance for TT and EE. . . . . . . . . . . . . . . . . . . . . . . . 722.4 Asymmetry spectra C loℓ for temperature (black curves) andE-mode polarization (red) for the tanh model of Sect. 2.2.2,with ∆ ln k = 0.01 and kc = 7.45× 10−3Mpc−1 (solid curves)and kc = 2 × 10−2Mpc−1 (dashed). The same physical k-space modulation produces substantially more modulation ofE than of T for the lower kc value, and conversely for thehigher kc value. . . . . . . . . . . . . . . . . . . . . . . . . . . 73xviList of Figures2.5 Improvement in the error bar for a measurement of the am-plitude of modulation for the tanh, adiabatic power-law, andns gradient models (top to bottom panels), assuming that themodulation direction is known, when Planck (blue curves) orcosmic-variance-limited (orange) simulated polarization data(to ℓmax = 1000) are added to Planck temperature. The de-pendence on kc for the Planck case tanh model follows thepeak structure of the EE power spectra relative to the noise(see Fig. 2.2, bottom) and can exceed the naive expectationof√2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.6 Posteriors for kc, ∆ ln k, and AR (top to bottom panels) forthe tanh model for temperature alone (black curves) and tem-perature with isotropic (blue) and modulated (orange) po-larization simulations for the model parameters in Table 2.1.The posteriors using polarization have been averaged over 500polarization realizations. Solid curves refer to Planck polar-ization, while dashed curves refer to cosmic-variance-limitedpolarization. The parameter kc is typically somewhat moreconstrained in the modulated than in the isotropic polariza-tion case, but cosmic variance is still significant. . . . . . . . . 762.7 Posteriors for nlos and AR (top and bottom panels, respec-tively) for the adiabatic power-law model for temperaturealone (black curves) and temperature with isotropic (blue)and modulated (orange) polarization simulations for the pa-rameters given in Table 2.1. The posteriors using polarizationhave been averaged over 500 polarization realizations. Solidcurves refer to Planck polarization, while dashed curves referto cosmic-variance-limited polarization. . . . . . . . . . . . . 772.8 Posteriors for k∗ and ∆ns (top and bottom panels, respec-tively) for the ns gradient model for temperature alone (blackcurves) and temperature with isotropic (blue) and modulated(orange) polarization simulations, for the parameters given inTable 2.1. The posteriors using polarization have been aver-aged over 500 polarization realizations. Solid curves referto Planck polarization while dashed curves refer to cosmic-variance-limited polarization. . . . . . . . . . . . . . . . . . . 78xviiList of Figures2.9 Histogram of the logarithm of Oˆj0 [defined by Eq.(2.29)] forthe tanh model using the Planck temperature data with 500realizations of statistically isotropic (red outlines) or mod-ulated (black outlines and green filled) polarization as de-scribed in the text. The top panel uses Planck polarization,while the bottom uses cosmic-variance-limited polarization.Large values of Oˆj0 relative to the isotropic histograms in-dicate that the modulation model should be preferred overΛCDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.10 As in Fig. 2.9 except for the adiabatic power-law model. . . . 812.11 As in Fig. 2.9 except for the ns gradient model. . . . . . . . . 823.1 ΛCDM temperature spectrum compared to the best-fit asym-metry spectra, C loℓ , for the various models. The best fitscorrespond roughly to a 5–10% asymmetry for ℓ . 100, asexpected, with the exception of the ISW modulation, whosemaximum amplitude (and shape) is fixed by ΛCDM. . . . . . 923.2 Posteriors for α0.002 or r0.002 and tilt of the isocurvature (toppanel) and tensor (bottom) models. Contours enclose 68%and 95% of the posteriors. We have conservatively assumedmaximal modulation, so that the vertical axes are also a mea-sure of the level of modulation relative to the isotropic ΛCDMspectrum. We can see that the modulation allowed by theasymmetry constraints is reduced substantially when addingthe isotropic constraints. . . . . . . . . . . . . . . . . . . . . . 974.1 Stacked images of the transformed Stokes parametersQr (left)and Ur (right) for Commander temperature hot spots. The ro-tation of the plane of polarization will act to leak the signalfrom Qr into Ur. Note that the right plot uses a differentcolour scale to enhance any weak features. Finer resolutionstacked images can be seen in Figure 40 of Planck Collabora-tion XVI (2016). . . . . . . . . . . . . . . . . . . . . . . . . . 110xviiiList of Figures4.2 Stacked images of the transformed Stokes parametersQr (left)and Ur (right) for SMICA E-mode hot spots. The rotation ofthe plane of polarization will act to leak the signal from Qrinto Ur. The quadrupole pattern in the right plot is relatedto “sub-pixel” effects (Planck Collaboration XV 2014; PlanckCollaboration XI 2016); fortunately, our results are insensi-tive to this feature, because it disappears in an azimuthalaverage (see Sect. 4.5). . . . . . . . . . . . . . . . . . . . . . . 1114.3 Profiles of Ur from stacking on temperature (left) and E-mode(right) extrema for the four component-separation methods.The best-fit curves for each component-separation methodare also shown, with α values given in the fourth column ofTable 4.1. Note that we have included both hot and coldspots in this figure, i.e., we have co-added the negative of theprofile from cold spots to the profile of the hot spots. Errorbars correspond to 68% confidence regions. . . . . . . . . . . 1144.4 Constraints on α coming from published analyses of severalsets of CMB experimental data sets (shown in grey) as re-viewed in Kaufman et al. (2016), compared with what is foundin the present chapter (in blue). For each experiment the lefterror bars are for statistical uncertainties at 68% CL, whileright error bars (when displayed) are obtained by summinglinearly the statistical and systematic uncertainties. The er-ror bar of BOOM03 already contains a contribution from sys-tematic effects, and Bicep2 did not consider systematic un-certainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.1 Effective beam for the high-L analysis described in Section 5.4.2,induced by the smoothing procedure. . . . . . . . . . . . . . . 1295.2 Top: Low-L (1 ≤ L ≤ 30) α-maps for an input α realization(left) and for reconstruction by our method, as described inSection 5.4.1 (right). Bottom: High-resolution α-maps for aninput α realization (left), along with the Weiner-filtered out-put of our high-L reconstruction, as described in Section 5.4.2(right). The induced beam (Fig. 5.1) is applied to the inputmap for comparison purposes. The input and output mapsare clearly correlated, although the output has considerablymore scatter on small scales due to the significant noise in thepolarization maps. . . . . . . . . . . . . . . . . . . . . . . . . 131xixList of Figures5.3 Recovery of a scale-invariant α power spectrum with A =10−4/2π from a suite of simulations. The blue points arethe mean recovered power spectrum from simulations, whilethe bars denote the standard deviations from the same set ofsimulations. The input power spectrum is shown in orange.The overestimation (blue points consistently above the or-ange curve) comes from the reconstruction bias of the method,which is accounted for by the B parameter in Eq. (5.21) in allsubsequent power spectrum plots. . . . . . . . . . . . . . . . . 1325.4 Top: Low resolution (Lmax = 30) data maps of α weightedby the hits map (left) or using uniform weighting (right).Bottom: Smoothed hits map used for the wp = Hp analysis(left), together with high-resolution data map (right). . . . . 1335.5 Left: Power spectrum for α. The vertical dashed grey linedenotes the boundary between our low L and high L recon-structions; note the differing y-scale for low-L compared tohigh-L. Uncertainties shown are standard deviations of ourset of null simulations; at low-L the CααL are not Gaussianor symmetric, which is accounted for in our likelihood (seeSection 5.4.1). The power spectrum here justifies our use ofthe small-angle approximation. Right: Posteriors for the am-plitude (A) of a scale-invariant power spectrum defined byEq. (5.18). The constraint is mainly driven by the lowestLs, which is the reason for the non-symmetric shape of theposterior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.6 Best-fit dipole in α from the full-mission data using wp = Hp(left), compared to the dipole from uniform weighting (right).The dipoles are consistent, although the amplitude clearlydecreases when using uniform weighting. . . . . . . . . . . . . 136xxAcknowledgementsFirst and foremost I would like to thank my friends and family for alwaysbeing supportive of me even if they never quite understood what I was doing.In particular I would like to thank my mom, Yanira Contreras, for reasonstoo long and personal to list here.Next I would like to thank my supervisor Douglas Scott. He was ex-tremely present and supportive when I needed it. I always learned some-thing in our discussions, though not always about science! I would also liketo acknowledge and thank Jim Zibin, who is also my next most frequent col-laborator. His way of asking questions had a way of sharpening the rigourof my arguments. Furthermore if the writing in this thesis is any good, it isbecause of Douglas and Jim. I also benefited immensely by being a memberof the Planck collaboration.I would like to thank the grad students I have met during my PhD, pri-marily here at UBC, but also in other institutions around the world. Thereare too many individuals to name here, but the friendship and solidaritywithin the graduate student community has certainly helped to keep mesane over the past five years!xxiIntroductionWork work workwork work work. . .—RihannaThe standard model of cosmology, given the name ΛCDM, is a set of sim-plifying assumptions, along with about half a dozen free parameters, de-scribing the content and large-scale structure of the Universe (Lahav andLiddle 2014). ΛCDM describes the Universe remarkably well and derivesits name from the two dominating forms of energy in the Universe today.“Λ”, refers to the cosmological constant, a source of energy that behaves likevacuum energy. It is the cause of the current accelerated expansion of theUniverse and accounts for about 69.2% (Planck Collaboration XIII 2016) ofthe total energy budget of the Universe. “CDM”, refers to cold dark matter,an unknown form of (currently non-relativistic) matter that has little to nointeractions (other than gravitational) with regular matter. It accounts forroughly 25.8% of the total energy budget of the Universe. The remainingingredients of the Universe are “baryonic” matter as cosmologists call allmatter made from or containing particles from the Standard Model of par-ticle physics (with the exception of neutrinos, and radiation-like particles).This nomenclature is not accurate, since for example, electrons, which areleptons and not baryons, are included in the “baryonic” budget by cosmol-ogists. Nevertheless this nomenclature has become standard, and baryonsaccount for the remaining 4.8% of the energy of the Universe. The finalingredients are electromagnetic-radiation (photons) and neutrinos, whichaccount for small fractions of a percent of the total energy of the Universe.Included in the ΛCDM paradigm are a set of initial conditions that de-scribe how primordial perturbations were laid out in the early Universe, themechanism for which, likely an inflationary epoch, is yet unknown (Dodel-son 2003; Mukhanov 2005; Durrer 2005). The early Universe was endowedwith nearly scale-invariant fluctuations at the level of 1 part in 109 in power.These perturbations appear as temperature deviations at the level of 1 partin 105 (Peebles and Yu 1970; Bond and Efstathiou 1984) at the time when1Introductionthe Universe became transparent to photons. The perturbations are essen-tially frozen in the left-over radiation of this era and make up the cosmicmicrowave background (CMB) anisotropies. These perturbations were alsothe source of gravitational instabilities where dark matter halos formed,housing galaxies and ultimately giving rise to all the structure in the Uni-verse that we see around us today.Implicit in ΛCDM are a set of basic assumptions. Aside from the un-known ingredients mentioned above, no new physics is required. Specificallythe theory describing gravity is simply General Relativity. The global cur-vature of the Universe is assumed to be flat, and no non-standard topologies(e.g., torodial) are assumed, as well as many other proposed kinds of physics.Furthermore, the cosmological principle is assumed out of simplicity, whichstates that there are no special locations in the Universe. An addition tothis principle is that there are no special directions either, i.e. statisticalisotropy. That is, statistically, any place in the Universe is indistinguishablefrom any other place in the Universe. Another basic assumption is thatthe laws of electromagnetism are invariant under parity transformations (orparity inversions x → −x). In this thesis I will explore some of the con-sequences of relaxing the latter two assumptions mentioned, in Part I andII respectively, using the CMB data, which will be introduced later in thischapter.Units, Notation, and ConventionBefore diving in I will outline some basic discussion on the units and notationused throughout this thesis (unless otherwise specified), and particularly inthis chapter. I will use two fundamental constants, c, ~ (the speed of light ina vacuum, and Planck’s constant respectively), to define a set of “natural”units that are often helpful at revealing the underlying physics at a glance.In SI units they have the valuesc = 299 792 000 m s−1,~ = 1.0545718× 10−34 J s.= 1.0545718× 10−34 kgm2 s−1.I will now use units in which c = ~ = 1. Starting with c = 1, this meansthat time can be measured in units of length, or length can be measured inunits of time. By setting ~ = 1 we see that we can measure energy (with SIunit J) in units of inverse time. More common in cosmology is to measuremomentum in units of Mpc−1. A Mpc (106 pc) stands for megaparsec and2Introductionis a unit of length:1 Mpc = 3.262× 106 light years = 3.086× 1022 m. (1)Since momentum has units of mass times length over time, therefore in natu-ral units momentum can be thought of as having units of inverse length. Forcomparison of scales 1 Mpc is roughly 30 times the size of our Milky-WayGalaxy, and slightly larger than the distance to our next nearest neighbour-ing galaxy Andromeda.Greek indices (α, β, µ, ν, λ, . . . ) will run over all space-time coordinatesfrom 0 to 3. Throughout the thesis, Einstein summation convention will beused, which means that, unless a summation sign is explicitly used in anequation, repeated up and down indices are summed over. As an examplethe expression vλuλ is short hand for v0u0 + v1u1 + v2u2 + v3u3. Theserepeated indices are often referred to as “dummy” indices.I will use the (−,+,+,+) signature for the metric, gµν . Specifically thismeans that there always exists a local frame in which the metric takes theformgµν =−1 0 0 00 1 0 00 0 1 00 0 0 1 .Also when discussing perturbations to the metric, unless otherwise specified,the conformal Newtonian gauge (to be shown later) will be used.Indices after commas indicate partial derivatives with respect to therespective coordinates. That is, for an arbitrary tensor, Aµ1...µNν1...νN , we haveAµ1...µN ,λν1...νN ≡ ∂Aµ1...µNν1...νN /∂xλ,Aµ1...µNν1...νN ,λ ≡ ∂Aµ1...µNν1...νN/∂xλ.Indices after semicolons indicate covariant derivatives with respect to therespective coordinatesAµ1...µNν1...νN ;λ ≡ Aµ1...µNν1...νN ,λ+ Γµ1αλAαµ2...µNν1...νN+ · · ·+ ΓµNαλAµ1...µN−1αν1...νN+ Γαν1λAµ1...µNαν2...νN+ · · ·+ ΓανNλAµ1...µNν1...νN−1α.The Γ coefficients are called Christoffel symbols and are functions of themetric to be defined later on in this section.3IntroductionI will also use ′ and ˙ symbols to denote special time derivatives. Foran arbitrary function, f of time I definef˙ ≡ dfdt,f ′ ≡ dfdη,where t and η refer to cosmic and conformal time, respectively (and will bedefined later in this chapter).Linear perturbation theory is all that is necessary in the early Universe(with few exceptions) to describe the anisotropies of the CMB. Because ofthis, it is often convenient to express quantities in Fourier space instead ofreal space. The conjugate variable to the position vector x, with magnitudex, is the wave number (or momentum) k, with magnitude k.For analysis on the 2D sphere I will use the spherical harmonic functionsYℓm. These are complex orthonormal functions with the property∫dΩYℓmY∗ℓ′m′ = δℓℓ′δmm′ . (2)The index ℓ runs from 0 to infinity (in principle), whilem runs from−ℓ to +ℓ.Larger ℓ corresponds to structure on smaller angular scales θ, approximatedcrudely by the relation θ ∼ π/ℓ. I will often make use of the followingformula∫dΩ [s1Yj1m1(nˆ)][s2Yj2m2(nˆ)][s3Yj3m3(nˆ)]=√(2j1 + 1)(2j2 + 1)(2j3 + 1)4π×(j1 j2 j3m1 m2 m3)(j1 j2 j3−s1 −s2 −s3).(3)Here the sYℓm are spin-weighted spherical harmonic functions, which are theanalogues of Yℓm for higher spin quantities. The bracketed terms are calledWigner 3-j symbols, and are related to Clebsch-Gordan coefficients(j1 j2 j3−s1 −s2 −s3)=(−1)j1−j2−m3√2j3 + 1〈j1m1j2m2|j3(−m3)〉 . (4)The j and m entires are angular momentum quantum numbers, so theClebsch-Gordon coefficients denote how a combination of two objects withquantum numbers j1,m1, and j2,m2, form a new object with quantum num-bers j3, and m3 (encoded in the inner product on the right hand side). The4Introductionfinal related definition needed is the Wigner D-matrix, which can be writtenin terms of Euler angles and spherical harmonic functionsDℓ−ms(α, β,−γ) = (−1)m√4π2ℓ+ 1sY∗ℓm(β, α)eisγ . (5)The homogeneous UniverseThe governing equation of General Relativity equates two symmetric 4× 4tensors to each other:Rµν − 12Rgµν = 8πGTµν − Λgµν . (6)The left hand side is a second-order (non-linear) differential equation ofthe metric, gµν , which encodes the geometry of the space-time. The righthand side is Newton’s gravitational constant (G) multiplying a stress energytensor of all matter fields in the space-time, and Λ is simply an arbitraryconstant. Equation 6 hides much of the complexity of the field equations.More explicitly we can expand the left-hand side asR ≡ Rµµ, (7)Rµν ≡ Γαµν,α − Γαµα,ν + ΓαβαΓβµν − ΓαβνΓβµα, (8)Γαµν =12gαβ (gβµ,ν + gβν,µ − gµν,β) . (9)In this form it is slightly more clear how horrendously complicated an ar-bitrary solution for the metric could be. In cosmology we are helped bythe “cosmological principle”, which states that there is a range of lengthscales for which deviations from homogeneity and isotropy are small. Beforedefining exactly what “small” means, it is worth pointing out that thereare obvious scales for which deviations from homogeneity and isotropy can-not be considered small. For humans interacting on Earth the directions“up” and “down” have distinct meanings that are special and different from“forward” and “backward” (along with “left” and “right”). In the Solarsystem the Sun is a special place compared to all others, and our MilkyWay galaxy has a specific structure with spiral arms in a given orientation,etc. If the cosmological principle applies at all, it therefore has to be validon scales larger than galactic scales. It turns out that the magic scale isroughly 100Mpc; today, on scales of 100Mpc or larger the Universe looksvery homogeneous and isotropic. This scale varies, since the non-linearitiesof gravity grow with time, so earlier in the Universe the scale of homogeneitywas smaller.5IntroductionOn scales larger than about 100Mpc, the metric takes the form (assum-ing spatial flatness)ds2 = gµνdxµdxν = = −dt2 + a2(t) (dx2 + dy2 + dz2) , (10)= −dt2 + a2(t)dx2 (11)The time coordinate here is often referred to as “cosmic time”, but it canjust be thought of as our regular notion of time. The spatial coordinates hereare called “comoving” coordinates, since they do not change as a(t) varies,rather “physical” coordinates defined by dxp = a(t)dx do. This metricdescribes a universe that has no preferred directions, since the function a2(t)is applied to all spatial coordinates equally. It also describes an expandinguniverse where the distance between any two points in physical coordinates(xp) changes in time by a factor of a(t). This function is given the name“scale factor” for this reason.The form of the isotropic metric is extremely restrictive and simplifiesthe right-hand side of Eq. (6) immensely. In fact it dictates the form that thematter fields will take on the right-hand side (as it must). For convenience,we often (and arbitrarily) set the scale factor today (t0) to be unity, a(t0) ≡a0 = 1. We will see that the evolution of the Universe is such that a wassmaller at earlier times. That is, in these coordinates, the physical distancebetween any two points was smaller by a factor of a at an earlier time. Oftenit is convenient to introduce a new definition of time, by factoring out thescale factor:ds2 = gµνdxµdxν = a2(η)(−dη2 + dx2) , (12)where dη ≡ dta. (13)Here the time coordinate η is called “conformal time”. Later I will considerperturbations to the above metric. Under the assumption that only scalarperturbations are present we can write down the perturbed metric in termsof two scalar fields:ds2 = a2(η)[−(1− 2ψ)dη2 + (1 + 2φ)dx2] . (14)The φ field has the interpretation of being the Newtonian potential. In theabsence of anisotropic stresses (a very good approximation) we have ψ = φ.The notion that perturbations should be “small” now has the meaning that|ψ| ≪ 1 and |φ| ≪ 1.We will now perform an inventory of the types of fluids that Eq. (6) willcontain.6Introduction1. A scalar field, Φ (not to be confused with the gravitational potentialφ): generally speaking this field is invoked to perform the process ofinflation in the very early Universe. Inflation could be caused by somemore complex phenomenon, but a single scalar field is a simple scenariothat appears to work. The total energy of the scalar field is given byΦ˙2/2 + V (Φ), where V (Φ) is a potential that the field evolves in andultimately determines how inflation proceeds and how it ends.2. Radiation: any massless or nearly massless particles.3. Matter: this contains any form of matter with appreciable mass; forthe moment it will not be necessary to break this into dark and bary-onic matter.4. Dark energy: this is the Λ term in Eq. (6), which can in principle begeneralized (but we will not be concerned with such possibilities here).We further assume that all fluids in the Universe are perfect fluids, whichis a good approximation in the absence of anisotropic stresses (which aretypically small). In this case the stress energy tensor takes the form,Tµν =−ρ 0 0 00 P 0 00 0 P 00 0 0 P , (15)in the rest frame of the fluid, where ρ and P are the energy density andpressure of the fluid, respectively.The stress-energy tensor must be covariantly conserved (Tµν;µ = 0, this isthe relativistic notion of energy and momentum conservation), which leadsto the fluid equationρ˙+ 3a˙a(ρ+ P ) = 0, (16)dictating how the energy of the fluid evolves in time. In order to fully closethese equations one has to specify how ρ and P are related, which is donewith the equation of state relation,P = wρ, (17)w =Φ˙2/2−V (Φ)Φ˙2/2+V (Φ), scalar field,−1, cosmological constant,0, non-relativistic matter,13 , relativistic matter.(18)7IntroductionThe scalar field Φ is defined by its kinetic energy Φ˙2/2 and its potentialV (Φ). In order to source inflation the scalar field should be dominated byits potential, such that w ≃ −1, this condition leads to nearly scale-invariantperturbations (to be discussed in greater detail below). Inflation ends whenthe field eventually decays into regular Standard Model particles, via a stillunknown process. The second relation, w = −1 can be seen directly fromEq. (6) by interpreting the cosmological constant as a fluid with a stress-energy tensor. We can make the following definition:T cosm. const.µν ≡ −Λ8πGgµν ; (19)Tµ,cosm. const.ν = −Λ8πG1 0 0 00 1 0 00 0 1 00 0 0 1 . (20)Identifying this with Eq. (15) demonstrates that ρ = −P = Λ/8πG, andtherefore w = −1. The term “non-relativistic matter” implies that theenergy is dominated by its rest energy. In this case the pressure term is givenby kinetic energy, P ∼ mv2, while, ρ ∼ mc2, therefore w = P/ρ ≃ v2/c2 ≃ 0.At the other extreme is relativistic matter (including radiation species),which maximize the contribution of pressure to P = ρ/3. For massivefluids, 0 < w < 1/3, where w usually depends on the mass and the epoch.Equation (16) is solvable by plugging in Eq. (17):ρ˙ρ= −3 a˙a(1 + w), (21)⇒ ρ ∝ a−3(1+w), (22)ρΛ ∝ constant, (23)ρm ∝ a−3, (24)ρr ∝ a−4. (25)Therefore Λ has a constant energy density as the Universe expands, whilethe energy density of matter decreases with the volume of space as one mightexpect. The energy density of radiation decreases with an extra factor of acompared to matter due to the expansion of its wavelength. Therefore wecan see that the early Universe (but after inflation happened) was radiationdominated, then transitioned to a matter-dominated phase, and the finalfate of the Universe will be a state of Λ domination. In the early Universetemperatures are high and so w ≃ 1/3; as the Universe expands and cools8Introductiondown to of order the mass of the fluid particle w transitions to lower valuesuntil the temperature becomes much lower than the mass and the fluidthermalizes and w ≃ 0. Then even later, Λ takes over and we approach atotal equation of state w ≃ −1.We are now able to consider how the scale factor evolves in time byconsidering the time-time component of Einstein’s equations (Eq. 6). Astraightforward but tedious calculation gives the Friedmann equation (seethe review article Tamvakis 2005, for example),H2 =8πG3ρ+Λ3, (26)with H ≡ a˙a. (27)Therefore, the evolution of the homogeneous Universe is entirely dictatedby the types of species in it and their amounts:H2 = H20(Ωra−4 +Ωma−3 +ΩΛ). (28)Here H0 is H evaluated today and is often called the Hubble constant,whereas H itself is the Hubble parameter. The Ω quantities are the energydensities of radiation, matter, and Λ with respect to the critical energydensity ρcritical ≡ 3H20/8πG evaluated today. Defined in this way ρcriticalis the energy density required to keep the curvature of the Universe flat(something I will take for granted throughout), which implies that Ωr +Ωm +ΩΛ = 1.Note that it is an observational fact that the Universe is currently ex-panding (Hubble and Humason 1931; Perlmutter et al. 1999), i.e., a˙|t0 > 0.Another observational fact is that Λ > 0 (Riess et al. 1998; Perlmutter et al.1999; Planck Collaboration XIII 2016). Furthermore the only way for a˙ tochange sign is for the right-hand side of Eq. (26) to be zero. All forms ofmatter discussed here share the property that ρ > 0 and therefore a˙ cannever change sign. Simply put, the fact that the Universe is expanding nowand filled with ordinary types of matter, means that the Universe was al-ways expanding and therefore a(t) is a monotonically increasing function.This fact also allows us to tell time in several different ways: one can simplygive the value of time, t, which increases monotonically, or the scale factor,a, which I just showed also increases monotonically.The Cosmic Microwave BackgroundWe have already seen that physical distances were closer together in the earlyUniverse than they are today. This means that the Universe was necessar-9Introductionily hotter and denser than it was today. If we assume thermal equilibriumduring radiation domination then H ∼ √GT 2 (e.g., Challinor 2005). Theexpansion does not disturb the thermal equilibrium, and therefore if someof the radiation at early times is still present today it should have the sameblackbody spectrum with a much cooler temperature, related to how muchthe Universe has expanded since then. This leftover radiation is the cosmicmicrowave background (CMB), and was first observed in 1964 by A. Penziasand R. Wilson (Penzias and Wilson 1965), confirming the hot big bang sce-nario (Dicke et al. 1965) as the simplest explanation. This observation givesus one of the fundamental parameters of the standard model of cosmologyand that is the temperature of the radiation today, T0 = 2.7255± 0.0006K(Fixsen 2009). The fact that the CMB is a near perfect blackbody impliesthat the Universe was in thermal equilibrium at some point in the past.Here a is the scale-factor at some early time during radiation domination,and a0 is the scale factor today (conventionally taken to be unity). If lightis emitted with wavelength λE and later received with wavelength λR, thenthe relative shift in the wavelength is given by z = (λR − λE)/λE. Thewavelength of light scales with the scale factor and thus increases as theUniverse expands; this relation is given byz = aR/aE − 1, (29)a =11 + z. (30)The second line follows from the first upon setting aR = a0 = 1 and allowingaE to be arbitrary. We have now seen a third way to tell time in the Uni-verse, which is through the redshift z, since it is a monotonically decreasingfunction of a and thus time. A fourth way is through the temperature of theCMB itself, T , which is a monotonically decreasing function of time since itscales with 1/a.The CMB last-scattered after this hot radiation-dominated era, whenthe Universe cools down enough for neutral atoms (dominated by hydrogen)to form. This transition means that the radiation no longer scatters off freeelectrons, and this “surface of last scattering” occurs is at z ≃ 1100 (PlanckCollaboration XIII 2016), when T ≃ 3000K. The bulk of the information inthe CMB, however, resides in its anisotropies rather than the shape of itsfrequency spectrum. That is, in its variations in temperature on the sky, aswell as its state of polarization. The deviations from isotropy are accuratelymeasured (first in Smoot et al. 1992), are at the roughly 300µK level andare consistent with adiabatic initial conditions (see Fig. 1, top).10IntroductionHowever, this near uniformity of the background causes a problem calledthe horizon problem. The blackbody nature of the CMB spectrum impliesthat the CMB was in thermal equilibrium at some early time, but given thefiniteness of the age of the Universe there simply is not enough time for theradiation on opposite sides of the sky to have ever been in causal contactif the Universe was radiation dominated at early times. The maximumcomoving distance that light can propagate from some early time ti to lastscattering is (e.g., Tamvakis 2005)χp =∫ tlstidta(t)=∫ alsaidaa2H. (31)During radiation domination H ∝ a−2 and thus the integrand decreases fordecreasing a; therefore even in the limit ai → 0, χp remains finite. Inflationoffers a convenient solution to this problem by creating an earlier period ofexponential expansion (H ∝ constant), which means the integrand increaseswith decreasing a (Guth 1981). The maximum comoving distance becomesχp ∝ 1/ai − 1/als. This can be made arbitrarily large by decreasing aisufficiently, thus allowing more time for the radiation to come into causalcontact and achieve thermal equilibrium.The spatial anisotropy field on the sky δT (nˆ), can be decomposed as (Goudaet al. 1991)δT (nˆ) =∑ℓmaTℓmYℓm(nˆ), (32)where Yℓm are the usual spherical harmonic functions. Increasing ℓ corre-sponds to providing detail on decreasing angular scales. The CMB is alsolinearly polarized, which is often expressed in terms of Stokes parameters Q,and U (Hu and White 1997b). Q is the intensity of radiation polarized in alocal North/South orientation, minus the intensity of radiation polarized ina local East/West orientation on the sky. U , on the other hand is the dif-ference in the intensities of radiation polarized in a South-west/North-eastversus a North-west/South-east orientations (the Q orientations rotated by45deg).The temperature anisotropies and Q and U polarization for our CMBsky, as seen by the Planck satellite (described later in this chapter) areshown in Fig. 1. Note that Q and U are spin-2 quantities (Hu and White1997b), which means that under a rotation by an arbitrary angle, α, thepolarization states rotate by an angle 2α. More explicitly, if Q′ and U ′ arethe rotated polarization states, then they relate to the original polarization11Introduction-300 300-15 15-15 15Figure 1: Mollweide projection in Galactic coordinates of the temperatureanisotropies and Q and U polarization as measured by the Planck mission(top to bottom, respectively). The units are in µK and the greyed-out areais the mask applied to remove residual foregrounds, mainly from our Galaxy.12Introduction0 π/2 π 3π/2 2πAngle of RotationFigure 2: A demonstration showing that rotating a polarization perturbationfield about 2π means you have actually rotated the field twice, hence it is a“spin-2” field.in the following way:Q′ ± iU ′ = (Q± iU)e∓2iα. (33)A visualization of this can be seen in Fig. 2. In this figure the red and blue“sticks” are the intensity of polarized radiation along the North/South orEast/West directions. Initially the incident radiation has a higher inten-sity of light polarized in the North/South direction compared to East/West,therefore Q = Q0 > 0. As the polarization is rotated by π/2 the inten-sity of light polarized in the East/West direction is now greater than theNorth/South direction and thus Q = −Q0 < 0. After only half a rotation(π) the polarization state is back in its original orientation, Q = Q0 > 0,and after a full rotation (2π) the polarization state has done two completerotations.Polarization can be decomposed into a set of spherical harmonics similarto Eq. (32) in the following way (Zaldarriaga and Seljak 1997):Q(nˆ)± iU(nˆ) = −∑ℓm(aEℓm ± iaBℓm)±2Yℓm. (34)Here the ±2Yℓm are spin-2 spherical harmonic functions (ℓ > 1) and theaEℓm, aBℓm are coefficients representing the two degrees of freedom for the13Introductionspin-2 field. The two coefficients are referred to as E and B modes, in anal-ogy with the electric ( ~E) and magnetic ( ~B) fields from electromagnetism. Emodes are curl-free modes, while B modes are divergence free. For electro-magnetic fields the previous statement is encoded in the relations ∇× ~E = 0,and ∇· ~B = 0, and equivalent relations exist for polarization, defined on thesurface of a sphere. An intuitive picture of the difference between E- and B-mode polarization can be seen in Fig. 3. A pure E-mode (> 0) polarizationpattern is seen in the top panel, where there appears to be no “curl” in thepattern. A negative E-mode pattern would correspond to each polarizationstick being rotated by 90deg, such that they point radially to the centre. Apure B-mode (> 0) polarization pattern is shown in the bottom panel, andclearly has no “divergence”. A negative B-mode pattern would have eachpolarization stick rotated by 90deg, such that the “wind-mill” would pointin the other direction.Another equivalent way to uniquely distinguish E- and B-mode polariza-tion is to focus on their parity transformation properties. As mentioned ear-lier, a parity transformation is a discrete coordinate transformation x → −x.This definition requires there to be central point, which for us is simply thecentre of the sphere defined by the CMB, or where we are today. Withthis definition, consider a small patch of sky on the sphere, small enoughfor the flat-sky approximation to hold. In this approximation a parity-transformation is equivalent to looking at that part of the sky in a mirrorparallel to the plane of the sky. If we apply this to Figs. 3, we see thatthe E-mode pattern is unchanged, while the B-mode pattern flips sign (thewind-mill has the opposite handedness). In other words E- and B-modepolarization transform oppositely under parity transformations. More gen-erally, the aℓms, under a parity transformation transform asaTℓm → (−1)ℓaTℓm, (35)aEℓm → (−1)ℓaEℓm, (36)aBℓm → (−1)ℓ+1aBℓm. (37)Rotational symmetry of the background implies that the spherical har-monic coefficients are not correlated with each other, i.e.〈aXℓmaY ∗ℓ′m′〉= CXYℓ δℓℓ′δmm′ , (38)with X,Y = T,E,B. The angular brackets here denote an average over anensemble of realizations, and the CXYℓ are referred to as “power spectra”.There are nine distinct power spectra available, however three of them are re-dundant by symmetry (CTEℓ = CETℓ , CTBℓ = CBTℓ , CEBℓ = CBEℓ ). Therefore14IntroductionFigure 3: Polarization patterns for a pure E and B mode, in red and blue,respectively. Looking at these two patterns in a mirror, the E-mode patternwould look the same, while the B-mode “wind-mill” pattern would changehandedness.the remaining six distinct power spectra are CTTℓ , CEEℓ , CBBℓ , CTEℓ , CTBℓ ,and CEBℓ . Furthermore, using the parity relations from Eqs. (35)–(37) wefind that CTBℓ = CEBℓ = 0.For a single realization of aℓms, an unbiased estimate of the power spectrais (Knox 1995) ∑maXℓmaY ∗ℓm2ℓ+ 1= C˜XYℓ . (39)This is useful because ΛCDM predicts the ensemble-averaged power spec-tra (not the individual aℓms, which are mean zero), but it also means thatour uncertainty on the underlying power spectra of our Universe is funda-mentally limited by the number of aℓms we can observe, which is neces-15Introductionsarily finite. The variance of the power spectra are (ignoring incompletesky coverage and instrumental noise) given by (Zaldarriaga and Seljak 1997;Kamionkowski et al. 1997)Var(C˜XYℓ ) =22ℓ+ 1(CXYℓ )2. (40)This variance is referred to as “cosmic variance” and is largest for low ℓ(large scales).It is also expected that the a{T,E,B}ℓm s themselves are Gaussian complexrandom numbers, with mean zero and covariance, C, given byC = CTTℓ CTEℓ CTBℓCETℓ CEEℓ CEBℓCBTℓ CBEℓ CBBℓ (41)= CTTℓ CTEℓ 0CTEℓ CEEℓ 00 0 CBBℓ . (42)Here symmetry and parity considerations allow the second line to followfrom the first. In ΛCDM the aℓms are empirically found to be consistentwith being Gaussian (with small corrections due to lensing) and thereforeessentially all the cosmological information comes from the CMB covariance,or the four non-zero power spectra. An equivalent statement is that all theinformation in the CMB comes from the 2-point function (Eq. 38), with allhigher-order moments being expressible in terms of the power spectra (Scottet al. 2016).The power spectra (as measured by Planck in Planck Collaboration I2016) are shown in Figs. 4–6. Absent from this set is the CBBℓ power spec-trum. Primary B modes, those coming from the surface of last scattering,have so far not been detected (although lensing-induced B modes, gener-ated from E modes at low redshifts, have been). Such primary B modesare not sourced by standard scalar perturbations since they require an extradegree of freedom to excite the magnetic-parity mode. ΛCDM does howeverpredict the existence of B modes from two sources: the first is tensor (orgravitational wave) perturbations, and the second comes from the second-order effect of gravitational lensing. The former is parameterized by theratio of tensor power to scalar power, r, and has yet to be detected (Adeet al. 2016), while the latter has been detected with Planck and several otherprobes (to be discussed later).On large angular scales (ℓ . 60) we can readily write down the contri-butions to the CMB anisotropies. These scales are particularly important16Introduction0100020003000400050006000DTT`[µK2]30 500 1000 1500 2000 2500`-60-3003060∆DTT`2 10-600-3000300600Figure 4: The power spectrum of temperature anisotropies DTTℓ = ℓ(ℓ +1)CTTℓ /2π as measured by the Planck satellite (Planck Collaboration I2016). The red curve is the best-fitting ΛCDM theory curve, and the lowerpanel shows the residuals relative to this model.because this is roughly equal to the Hubble scale (aH) at the time of lastscattering, which means these perturbations were frozen in place since in-flation took place. Therefore these modes offer a very clean look into whatthe perturbations caused by inflation looked like, or in other words, we aredirectly seeing the initial conditions. Approximating the last scattering sur-face as sharp (which is a good approximation for large angular scales) andignoring the quadrupole anisotropy, we can write down how the anisotropiesat last scattering (ls) evolve as they reach our detectors d (Kodama andSasaki 1986):[δT (nˆ)T0+ ψ]d=[δTT0+ ψ]ls− nˆ · vb|ls +∫ dls(ψ′ + φ′)dη. (43)The interpretation of the (non-derivative) potential terms are simply thereis a contribution to the anisotropies coming from the difference in the poten-tial at last scattering and here today; this is referred to as the “Sachs-Wolfe”term (Sachs and Wolfe 1967). The Sachs-Wolfe term is responsible for theplateau at low ℓ in the temperature power spectrum (see Fig. 4). The veloc-ity term in Eq. (43) comes from the Doppler shift, arising from scattering17Introduction020406080100CEE`[10−5µK2]30 500 1000 1500 2000`-404∆CEE`Figure 5: The power spectrum of E-mode polarization as measured by thePlanck satellite (Planck Collaboration I 2016). The red curve is the best-fitting ΛCDM theory curve. The green curve indicates a residual systematiceffect coming from the imperfect separation between temperature and po-larization.off the moving electrons (with velocity vb). The integral term is called the“integrated Sachs-Wolfe” (ISW) effect, is a sub-dominant term, and is onlypresent when the potentials are evolving in time. During matter domina-tion the potentials are constant (Dodelson 2003), and therefore there aretwo ISW contributions to the CMB. The first comes from early times whenthe Universe evolves from radiation to matter domination, and the secondcomes from late times as the Universe becomes Λ dominated. This lattercase has a larger effect and is often called the “late time ISW” effect; how-ever I will often simply refer to it as the ISW effect. During Λ dominationpotentials decay and therefore a photon entering a potential leaves the po-tential having gained some energy since it escapes a shallower potential thanwhen it entered it. This effect is strongest on large scales and its effect onthe CMB will be shown later in Part I. Note that there is no Sachs-Wolfecontribution to polarization, since the gravitational potential can only alterthe energy state of the radiation and not the relative states of differentlypolarized radiation.18Introduction-140-70070140DTE`[µK2]30 500 1000 1500 2000`-10010∆DTE`Figure 6: The power spectrum of temperature E mode cross-correlationDTEℓ = ℓ(ℓ + 1)CTEℓ /2π, as measured by the Planck satellite (Planck Col-laboration I 2016). The red curve is the best-fitting ΛCDM theory curve.The green curve indicates a residual systematic effect coming from the im-perfect separation between temperature and polarization.Initial conditionsBefore the Universe becomes neutral, the photons are tightly coupled tothe baryons. The mean free path for photons is ∼ 5 × 104(Ωbh2)−1(1 +z)−2Mpc (Challinor 2005), and the photon-baryon fluid has a (comoving)sound speed given by (Hu and Sugiyama 1995)c2s(η) =13(1 +R), (44)with R ≡ 3ρb4ργ. (45)Note that the sound speed of photons is only modified from the value 1/√3due to the presence of baryons and not dark matter; this is simply becausethe dark matter does not interact with the photons (by definition). Thisfact allows the CMB to distinguish between baryonic and dark matter.19IntroductionThe initial conditions and baryonic physics can be encoded in a transferfunction, ∆ℓ(k), useful for computing the angular power spectrum. Thepower spectra can then be written asCXYℓ =∫dk∆Xℓ (k)∆∗Yℓ (k)P(k). (46)Here P(k) is the primordial power spectrum. The transfer functions de-termine how power in 3D Fourier space gets distributed to 2D harmonicspace. When perturbations are sourced by some other mechanism (tensorsor isocurvature) then the transfer functions differ. Thus, the same primor-dial power spectrum will give rise to different angular power spectrum ifsourced by isocurvature or tensor perturbations. The transfer functions arealso often called “kernels”. One useful approximation (on small scales) isthe Limber kernel (Limber 1953), which states that each angular mode getscontribution from a single k-mode, i.e., k = ℓ/r (where r is the comovingdistance to the source, in this case the surface of last scattering). Here I willonly use the Limber approximation for illustrative purposes in Chapter 1.Under adiabatic initial conditions all species start with the same over-density and velocity, provided they are tightly coupled. The initial powerspectrum is predicted to be nearly scale-invariant (independent of wave num-ber k) and given by the potential evaluated during inflation, however it isoften parameterized asPΛCDMR (k) = As(kk0)ns−1. (47)Here As is the amplitude of perturbations, which is of order 10−9, and nsis slightly less than unity (predicted by inflationary models). The pivotscale k0 is arbitrary, but the scales 0.05Mpc−1 and 0.002Mpc−1 are oftenused. The power can also be written in terms of the potential V (Φ) andits first derivative evaluated during inflation (Lyth 1997), and therefore aninflationary model with a distinct potential will give distinct predictions forAs and ns. These conditions imply that photons in an initial over-density arestationary and move out to a distance given by the sound speed integratedout to the time of last scattering (η∗), called the sound horizon,rs ≡∫ η∗0dη′1√3(1 +R). (48)This provides a characteristic physical scale (rs ≃ 150Mpc) that is seen inthe CMB power spectra as a characteristic angle θs (θs = rs/DA ≃ 0.◦61).20IntroductionHere DA is the angular diameter distance to the last-scattering surface (≃14Gpc). The first peak of DTTℓ corresponds to the first oscillation mode ofthe photon-baryon fluid and is therefore given by ℓs ≃ (π − π/4)/θs ≃ 220(see Fig. 4 and Planck Collaboration XVI 2016). The extra π/4 phasefactor comes from the fact that more 3D k-modes contribute to a 2D ℓ-mode (i.e., a break down of the Limber approximation). The positions of theother peaks and troughs are roughly multiples of ℓs reflecting the harmonicseries of oscillations that occurred during the period before the CMB lastscattered (Dodelson 2003; Challinor 2005; Mukhanov 2005). In other words,the single scale rs in 3D real space corresponds to a harmonic series in 2Dℓ-space showing up in the CMB power spectra.Other initial conditions for different kinds of perturbation are possible,but have not been observed in the data. One of these possibilities are isocur-vature modes. Isocurvature perturbations are perturbations that preservecurvature, the most relevant being cold dark matter (CDM) isocurvatureperturbations. These are such that the CDM and rest of the matter oscil-late out of phase with each other in order to keep the curvature the same.These perturbations have a different acoustic peak structure from regularadiabatic perturbations and contribute much less power on small scales com-pared to adiabatic perturbations. The effect of isocurvature on the powerspectra will be shown later on in Part I.TensorsTensor perturbations (or gravitational waves) are transverse-traceless wavesand are spin-2 objects, that is they share the same properties as polarizationunder rotations. Assuming that they propagate in the z direction the tensorperturbations show up in the spatial part of the metric asgij = a2 1 + h+ h× 0h× 1− h+ 00 0 1 , (49)where i, j run over the spatial indices only, and the h+ and h× quantitiesare directly analogous to Q and U . The power spectrum for h+ and h× can,similarly to the scalar perturbations (but different in detail), be written interms of the potential V (Φ) directly (Lyth 1997; Mukhanov 2005; Durrer2005). Therefore since the scalar perturbations are very well constrained,the ratio of tensor power to scalar power, r, is directly proportional to the21Introductionenergy scale at which inflation occurred (Lyth 1997):V1/4inf ≃( r0.07)1/4× 1.8× 1016 GeV. (50)Thus its detection would be another step in vindicating the inflationarypicture of the early Universe. One of the consequences of linear theoryis that the tensor perturbations are completely uncorrelated with scalarperturbations. The same holds true for vector perturbations, which areanother class of perturbations not discussed here, since they decay with timeand are thought to be cosmologically unimportant. Tensor perturbationsare able to excite the magnetic-parity mode of polarization, due to havingthe same degrees of freedom, and thus it is a cosmological source of B-modepolarization (Hu and White 1997b; Durrer 2005). Thus far only upper limitson B-modes from tensor perturbations exist (Ade et al. 2016). The effect ofthe tensor contribution to the CMB power spectra will be shown later on inPart I.CMB secondary anisotropiesA number of effects influence the CMB after z ≃ 1100 and the photonspropagate to us; these are sometimes called CMB secondary anisotropies orline-of-sight effects. One of these effects comes from reionization, which isthe era when the first stars start to form and reionize the neutral gas in theUniverse. The Universe is almost completely reionized by z ≃ 6, determinedby measuring the suppression of emission from quasar spectra at wavelengthssmaller than the Lyman-alpha emission line (Gunn and Peterson 1965), andis parameterized by the optical depth, τ . The main effect of reionizationon the temperature anisotropies is to act as another scattering surface andpartially smooth out perturbations. That is, on small scales Cℓ is reducedby a factor of e−2τ (Dodelson 2003). On large scales, however, (ℓ . 10),the era of reionization acts as another source for polarization and causesan enhancement to CEEℓ and CTEℓ proportional to τ . CMB photons haveanother chance at scattering off hot gas via the Sunyaev-Zeldovich effect(Sunyaev and Zeldovich 1980), which is comes from scattering in the ionizedgas within hot galaxy clusters. This is a measured effect in the directionsof massive clusters, but is very small when integrated over the sky and willnot affect any of the analysis performed here.One of the most important CMB secondary anisotropies comes fromlensing of the CMB. As the light from the CMB propagates from the surfaceof last scattering to us it must pass through all the structure of the Universe22Introductionin between. This intervening matter deflects the path of the light and causesa slight distortion to the CMB. This distortion is another cosmological sourceof B modes since they can be produced from lensed E modes (Zaldarriagaand Seljak 1998) It is worth noting that while the effect of lensing is asecond-order perturbation on the CMB anisotropies, the lensing potentialitself is perfectly well described by linear theory,ψlens(nˆ) = −2∫ rls0drrls − rrlsrψ(t(r), rnˆ). (51)Here the integral runs from us (r = 0) to the last scattering surface (r =rls). Therefore ψlens is a mean-zero quantity with covariance given by itspower spectrum Cψψℓ . Light propagating from the CMB is deflected by theangle −∇ψlens, with typical magnitude of 2′, where the ∇ operator is thegradient restricted to the surface of a sphere. Aside from mixing E andB modes, the main effect of lensing is to introduce correlations betweendifferent multipoles, such that Eq. (38) no longer holds (Okamoto and Hu2003). This induces non-standard 4 point correlations in the CMB, theexploitation of which is how this effect is measured.One last secondary effect is the ISW effect, which was already discussedin Eq. (43).PlanckThe European Space Agency’s Planck satellite, dedicated to studying theearly Universe and its subsequent evolution, was launched in 14 May 2009and scanned the microwave and submillimetre sky continuously between 12August 2009 and 23 October 2013 (Planck Collaboration I 2016). It ob-served temperature fluctuations in 30-, 44-, 70-, 100-, 143-, 217-, 353-, 545-,and 857-GHz frequency bands out to angular scales of 30′ to 5′ (dependingon frequency). It also observed polarization in all but the highest two fre-quency channels. The large frequency range was partly chosen to be ableto remove non-cosmological signals in the measurement of the CMB. As-trophysical foregrounds coming mostly from dust and synchrotron emissionfrom our MilkyWay Galaxy also contribute to the microwave sky and are dis-tinguished from the CMB signal via their spatial distributions and their dis-tinct frequency dependences. The Planck mission is one of the latest CMBexperiments, improving on its predecessor satellites COBE and the Wilkin-son Microwave Anisotropy Probe (WMAP) (Bennett et al. 2013) in terms ofhigher angular resolution and lower detector noise levels. Ground-based ex-periments are also used, and are able to observe much smaller angular scales23Introductionwith ever-improving sensitivity. The most important are the Atacama Cos-mology Telescope (ACT) (Sievers et al. 2013) and the South Pole Telescope(SPT) (Story et al. 2013). Then there are dedicated B-mode experimentslike the BICEP2/Keck array (BICEP2 Collaboration et al. 2014), which aimto look for B modes on intermediate scales (30 . ℓ . 150), where the con-tribution from tensor perturbations is the largest. In this thesis I only usePlanck data, with one exception in Chapter 3 where BICEP2/Keck data arealso used.24Part IDipolar asymmetry25They don’t think it be like it is.But it do.—Oscar GambleHere I consider one important signal that comes from a family of “CMBanomalies” that was originally detected in the WMAP sky maps, and laterconfirmed in the analyses described in Planck Collaboration XXIII (2014).The term “anomaly” in this context refers to a mildly significant (usually inthe 2–3σ range) departure from the expectations of ΛCDM. These appearon large angular scales, ℓ . 100.The analysis presented in this part of my thesis investigates an apparenthemispheric asymmetry of power on large scales, which is demonstrated inFig. 7. There is more power on the left hemisphere than the right, i.e., thered and blue spots are redder and bluer on one half of the sky compared tothe other half. This is often referred to as the “hemispheric asymmetry”, or“dipolar power asymmetry” in the literature.Figure 7: The CMB temperature anisotropies for ℓ ≤ 64, centred on thedirection of the dipolar power asymmetry, left, and the anti-direction, right.The microwave sky is intrinsically statistically anisotropic due to ourmotion with respect to the CMB rest frame. This is often called the “Dopplerboosting effect” and leads to a strong dipole signal, as well as modulationwas first detected in the 2013 Planck data (Planck Collaboration XXVII2014). In almost all analysis below we use CMB simulations produced bythe Planck collaboration but (due to a bug in the code) some of the effectsof Doppler boosting were not included properly in the 2015 simulation set.In each chapter I explain how this error is corrected for.26Before presenting the results, I need to bring up the issue of a posterioricorrection, which particle physicists refer to as correcting for the “look-elsewhere effect”, and statisticians refer to as the “multiplicity of tests”.Since there are many tests that can be performed on the data to look fora violation of statistical isotropy, we expect some to indicate detections at,for example, roughly 3σ levels, since even a statistically isotropic CMBsky is a realization of an underlying statistical process corresponding tomany independent random variables. However, in the absence of an existingtheoretical framework (i.e., a physical model) to predict such anomalies, it isdifficult to interpret their significance; without a prior prediction, one needsto consider the possibility of a wide set of potential anomalies. It is thennecessary, and equally challenging, to address the question of how oftensuch detections would be found for statistically isotropic Gaussian skies.Unfortunately, it is not always clear how to answer this question.There will always be a degree of subjectivity when deciding exactly howto assess the significance of these types of features in the data. As anexample, one could perhaps argue that the large-scale dipole modulationsignal we see is coming specifically from super-Hubble modes (i.e., thosebigger than the Hubble scale at the time of last-scattering), in which caseperforming an a posteriori correction for dipole modulation that could havebeen seen on small scales (ℓ & 100) would not make sense. Models for sucha super-Hubble modulation exist and an example was examined in PlanckCollaboration XX (2016), the conclusion being that the model could onlyexplain part of the dipole modulation and that the allowed part was perfectlyconsistent with cosmic variance.Although many of the observed effects described in this and the nextsection may elude theoretical prediction today, we continue to highlight themsince there is a real possibility that the significance of one or more mightincrease at a later date, perhaps when polarization data are included in theanalysis, and lead to new insights into early Universe physics. Alternatively,such observations may directly motivate the construction of models that canmake predictions for features that can be sought in new data sets. This isparticularly the case for anomalies on the largest angular scales, which mayhave a specific connection to inflation.27Chapter 1CMB Lensing1.1 IntroductionThe standard cosmological model, known as Λ cold dark matter (ΛCDM),describes the large-scale and early Universe remarkably well, with only ahandful of parameters (see, e.g., Planck Collaboration XIII 2016). Veryfew hints of departures or tensions with ΛCDM exist in the present cos-mological data. Of these, considerable attention has been paid to variousso-called “anomalies” in measurements of the cosmic microwave background(CMB) radiation (see, e.g., Bennett et al. 2011; Planck Collaboration XVI2016; Schwarz et al. 2016). In some cases, the anomalies are known tobecome statistically insignificant when correcting for the line-of-sight inte-grated Sachs-Wolfe (ISW) contribution (see, e.g., Efstathiou et al. 2010;Rassat et al. 2013; Rassat and Starck 2013). In these cases, due to theweak correlation between the ISW and primary anisotropies, the anomaliesare unlikely to be due to some physical mechanism and hence are almostcertainly statistical flukes. However, in all cases the anomalies are of onlyweak to moderate statistical significance, which typically is reduced furtherwhen correcting for a posteriori selection effects (also known as the “lookelsewhere effect” Bennett et al. 2011; Planck Collaboration XVI 2016).One intriguing feature of the CMB temperature (T ) anisotropies is aroughly dipolar power asymmetry (Eriksen et al. 2004). Measurements withthe Planck mission (Planck Collaboration XVI 2016) indicate a roughly 6%amplitude of asymmetry up to multipole ℓ ≃ 65, with a significance (as mea-sured by a p value) of roughly 1%. Equivalently, the measured amplitude isonly about 2–2.5 times the expected level of asymmetry due to cosmic vari-ance in statistically isotropic skies (Planck Collaboration XVI 2016). Thesignificance of the asymmetry becomes lower out to higher ℓ (Flender andHotchkiss 2013; Quartin and Notari 2015; Aiola et al. 2015; Planck Collab-oration XVI 2016), and is reduced to of order 10% if we do not consider thescale ℓ ≃ 65 as special, and correct for a posteriori effects (Bennett et al.2011; Planck Collaboration XVI 2016).However, despite its underwhelming statistical significance the dipolar281.1. Introductionpower asymmetry remains interesting because of its large-scale character.The asymmetry involves scales that are roughly super-Hubble at last scat-tering, and a number of early-Universe or inflationary mechanisms mightconceivably affect these scales preferentially. For example, CDM isocurva-ture fluctuations naturally imprint on scales ℓ . 100. However, a particularmodulated isocurvature model (Erickcek et al. 2009) was recently tested inPlanck Collaboration XX (2016) and found to not be preferred to ΛCDM.More generally, it appears to be very difficult to construct a physicalmechanism for generating a scale-dependent dipolar modulation (see Byrneset al. 2016a, for a thorough discussion and summary of previous attempts).This contrasts with the relative ease in producing a quadrupolar modulation(see, e.g., Dimastrogiovanni et al. 2010; Soda 2012; Naruko et al. 2015).The crucial difference is that a quadrupolar asymmetry on the sky can beproduced via a quadrupolar statistical anisotropy in k space, associated,e.g., with a homogeneous vector field. However, despite some claims to thecontrary (Schmidt and Hui 2013), a k-space anisotropy cannot lead to adipolar asymmetry on the sky: the fact that the fluctuations must be realnumbers implies that the k-space power spectrum must have even parity(see, e.g., Abramo and Pereira 2010). Instead, a dipolar asymmetry mustbe the result of statistical inhomogeneity, perhaps due to modulation with along-wavelength mode. Note that this distinction holds more generally forany odd compared with any even type of asymmetry. (Parity violation maycircumvent this argument; see, e.g., Ashoorioon and Koivisto 2016).It is clear that the important question of whether the observed dipolarasymmetry in the CMB temperature fluctuations is due to a statistical flukeor to a real, physical modulation of the primordial fluctuations will not beresolved through further study of the temperature fluctuations. This is sim-ply because the large-scale T data are already cosmic-variance limited, sothere will be no significant reduction of noise by remeasuring them. Whatare needed are observations that can probe independent fluctuation modesfrom those which source temperature fluctuations. The most obvious suchobservations are of the CMB polarization. Although E-mode polarization ispartially correlated with temperature, it is largely sourced by independentmodes. Polarization has long been recognized as useful for providing inde-pendent checks of “anomalies” found in the T data (see, e.g., Dvorkin et al.2008; Paci et al. 2010; Copi et al. 2013; Ghosh et al. 2016).It is worthwhile considering whether observations other than polariza-tion might also be able to address this question. The essential difficulty isthat the scales at which the T asymmetry is observed are extremely large.To illustrate this, I plot in Fig. 1.1 the Limber approximation kernels for291.1. Introductionvarious cosmological observations in the k-r plane. (See Zibin and Moss2014, for details on the calculations involved.) The vertical line indicatesthe k scale corresponding approximately to multipole ℓ = 65 in the primaryCMB. We can see that only the ISW effect and CMB lensing are currentlycapable of reaching the required large scales (nevertheless, limits on dipolarasymmetry in the quasar distribution on much smaller scales were placedin Hirata 2009). However, the ISW effect is mainly sourced at low red-shifts. Therefore, for a primordial fluctuation modulation linear in position,the modulation amplitude would be expected to be very small for the ISWeffect (we will see this explicitly for the case of lensing in Sect. 1.4). In ad-dition the ISW contribution mainly appears at the very smallest multipoles,so will be heavily affected by cosmic variance.Therefore it appears that, after polarization, CMB lensing offers thebest chance at testing the asymmetry. However, it should be apparent fromFig. 1.1 that, as with the ISW effect, lensing is sourced considerably closer tous than the primary CMB, and hence, for a spatially linear modulation, weexpect a lower modulation amplitude. In addition, the k scales modulated inthe CMB will appear at larger angular scales, i.e. we expect the asymmetryto appear to lower maximum multipole, in lensing. Thus we expect fewermodulated modes for lensing than for temperature. For these reasons weexpect the significance of detection achievable with lensing to be lower thanthat from temperature. On the other hand, the modes sourcing lensing willbe essentially completely uncorrelated with the primary CMB temperature,whereas CMB polarization shares significant correlation with temperature.While most previous studies of the CMB large-scale asymmetry havebeen restricted to ℓ or map space, if we observe some amplitude of asym-metry out to some multipole scale in temperature we do not expect a CMBlensing modulation of the same amplitude and scales, as just explained. Thissame point will also apply to polarization, due to the different kernels fromk space to multipole space for these observations. Therefore, in order toobtain predictions for lensing or polarization we must proceed via a k-space(or position-space) modulation model.In this chapter I have two main goals. The first is to present a formal-ism for fitting a k-space modulation to CMB T data. This involves firstdescribing a spatially linear modulation in k space, and then deriving itseffect on the T fluctuations. I show that this effect can be calculated ac-curately in a very simple way. I then fit the modulation to Planck T datausing Bayesian parameter estimation. Our next goal is to determine whatthe k-space model predicts for CMB lensing. To do this I must introduce aformalism for calculating the effect of a k-space modulation on lensing.301.1. Introduction0.10.10.30.31.01.03.00.10.1.30.31 01.03.00.10.10.10.10.30.30.31.01.01.03.03.010 - 4 10 - 3 10 - 2 10 - 1 100[Mpc - 1]0.00.20.40.60.81.0/LS100 101 102 103 104LS0.01.03.010.0100.01000.0ISWCMB lensinggalaxy lensing=3=30=300=3000Figure 1.1: Limber approximation kernels (in arbitrary units) for variouscosmological observations (contours) out to last scattering (r = rLS). Thehorizontal axis represents the magnitude of a 3D k-mode, while the verticalaxis shows the comoving distance to last scattering (we are located at r = 0,and z = 0). The grey box indicates very roughly the reach of the plannedEuclid survey (Laureijs 2009). Dotted magenta curves correspond to fixedmultipole scales, with the red hatched region geometrically inaccessible (ℓ <1). The vertical cyan line corresponds approximately to the scale ℓ = 65 inthe primary CMB (narrow green box at the top); scales roughly to the leftof it exhibit dipolar asymmetry in the CMB. If only modes to the left of thecyan line are modulated then only probes sourced in this region will be ableto test modulation models. Adapted from Zibin and Moss (2014).311.1. IntroductionOur approach will also be applicable to predicting the signal of modu-lation in CMB polarization based on the T observations. However, besidesproviding such predictions, our rigorous approach to fitting is importantin its own right. While grounding the study of the asymmetry firmly ink space, we find that temperature data alone are not constraining enoughto clearly define a k-space modulation. In particular, the often-quoted 6%modulation out to ℓ ≃ 65 does not stand out in the data.In previous related work, Dai et al. (2013) predicted the polarizationasymmetry given a simplified procedure for fitting to the T data, for mod-ulations of various cosmological parameters. Namjoo et al. (2015) consid-ered what the T asymmetry predicts for polarization asymmetry via mod-ulated primordial spectra, using a similar fitting procedure. Importantly,they found that the polarization predictions are strongly dependent on thek-space model. Rath et al. (2015); Kothari (2015) performed more carefulfitting, but restricted their models. None of these groups considered lensing.Flender and Hotchkiss (2013) looked for a power asymmetry in the Plancklensing map, finding no significant signal in the low-ℓ T asymmetry direc-tion. Additionally, Mukherjee and Souradeep (2016) claimed that lensing Bmodes could confirm a physical modulation at high significance, due to themode mixing that takes low-ℓ lensing modes to high-ℓ B modes. However,this paper treated the statistics of the lensed B field as Gaussian, whereasit is known that non-Gaussianity reduces the total signal-to-noise ratio ofthe lensing B power spectrum by a large factor (see, e.g., Li et al. 2007).Also, Mukherjee and Souradeep (2016) did not consider a physical lensingmodulation mechanism and simply took the expected lensing modulationamplitude to be 7% to ℓ = 70.In this chapter we approach this topic in a much more rigorous way.In the first few sections we lay out our modulation formalism. Section 1.2describes our treatment of the k-space modulation, while Secs. 1.3 and 1.4derive the effects of the k-space modulation on CMB temperature anisotro-pies and the lensing potential, respectively. The following sections presentour approach to fitting the k-space modulation to the CMB temperaturedata (Sect. 1.5), and describe the predicted effect of the modulation on theCMB lensing (Sect. 1.6).Throughout this chapter we use the set of ΛCDM cosmological param-eters chosen for the Planck Collaboration Full Focal Plane (FFP8) simu-lations (Planck Collaboration XII 2016); namely, we set Hubble parameterH0 = 100h km s−1Mpc−1, with h = 0.6712, baryon density Ωbh2 = 0.0222,CDM density Ωch2 = 0.1203, neutrino density Ωνh2 = 0.00064, cosmolog-ical constant density parameter ΩΛ = 0.6823, primordial comoving curva-321.2. Primordial adiabatic k-space modulationture perturbation power spectrum amplitude As = 2.09 × 10−9 at pivotscale k0 = 0.05Mpc−1 and tilt ns = 0.96, and optical depth to reionizationτ = 0.065. However, we expect our results to be only very weakly dependenton these parameters.1.2 Primordial adiabatic k-space modulationOur basic premise is to ask: If the large-scale CMB temperature dipolarasymmetry is due to a real, physical modulation of the primordial fluctu-ations, then what would this predict for CMB lensing (or polarization)?As discussed in the Introduction, a T asymmetry of, say, 6% to ℓ ≃ 65will not correspond to a lensing (or polarization) modulation of the sameamplitude and angular scales. To proceed we must specify a form for a pri-mordial modulation in position or k space. This could take the form of amodulation of the large-scale adiabatic fluctuations, or alternatively a CDMisocurvature or tensor modulation. The latter two are motivated by the factthat they naturally give a contribution only on large scales. Tensor modes,however, are expected to produce only tiny gradient-type lensing (Kaiserand Jaffe 1997). CDM isocurvature modes produce considerably less lens-ing than adiabatic modes, for comparable large-scale CMB T contributions.Therefore, we will restrict our analysis here to the modulation of adiabaticmodes. However, when considering the predictions for polarization, it willbe important to consider these other fluctuation types as well (Namjoo et al.2015).It is clear that there is no significant scale-independent dipolar asymme-try in the CMB temperature fluctuations (see, e.g., Planck CollaborationXVI 2016). Studies indicate an asymmetry amplitude of roughly 6% outto multipoles ℓ ≃ 65, with decreasing amplitude to larger ℓ (Planck Collab-oration XVI 2016). This apparent scale dependence motivates us to treatthe primordial adiabatic fluctuations as the sum of a large-scale dipole-modulated part and a small-scale statistically isotropic part. The scale de-pendence of the large-scale part will be free, although the total statisticallyisotropic power will agree with ΛCDM. In the following we will indicatemodulated fields by a tilde, while statistically isotropic fields will have notilde. We therefore write the total primordial (and hence time-independent)comoving curvature perturbation, R˜(x), asR˜(x) = R˜lo(x) +Rhi(x), (1.1)331.2. Primordial adiabatic k-space modulationwhere the high-k part is statistically isotropic,〈Rhi(k)Rhi∗(k′)〉 = 2π2k3PhiR(k)δ3(k − k′). (1.2)On the other hand, the low-k part is taken to be linearly modulated:R˜lo(x) = Rlo(x)(1 +ARrrLScos θ)(1.3)= Rlo(x)(1 +ARzrLS), (1.4)where rLS is the comoving radius to last scattering, AR is a constant, the“modulation amplitude”, and θ is the angle from the modulation direction,which we here define to coincide with the zˆ direction. Rlo satisfies〈Rlo(k)Rlo∗(k′)〉 = 2π2k3P loR(k)δ3(k − k′). (1.5)Finally, we take Rlo and Rhi to be uncorrelated,〈Rlo(k)Rhi∗(k′)〉 = 0, (1.6)so that the total statistically isotropic fluctuations, R(k) ≡ Rlo(k)+Rhi(k),must have the usual ΛCDM power spectrum,〈R(k)R∗(k′)〉 = 2π2k3PΛCDMR (k)δ3(k − k′), (1.7)wherePΛCDMR (k) = P loR(k) + PhiR(k). (1.8)In words, the full-sky “average” (or “equatorial”) power spectrum will agreewith that of ΛCDM (at least to lowest order in AR). As we explain below,treating the fields as two uncorrelated components does not restrict thegenerality of our approach.Also, note that we have in mind that Rhi contributes only negligibly tothe largest scales, so that we expect AR ≃ 0.06 when P loR(k) extends only toscales corresponding to ℓ ≃ 65, according to the observed T asymmetry. Inthis study we will take the low-k modulated component to have the spectrumP loR(k) =12As(kk0)ns−1 [1− tanh(ln k − ln kc∆ ln k)]. (1.9)341.2. Primordial adiabatic k-space modulationThis spectrum approaches the standard ΛCDM spectrum for small k andapproaches zero for large k, with cutoff scale kc and width of cutoff ∆ ln k.Recall that the total (isotropic) power spectrum is still constrained to havethe standard power-law form via Eq. (1.8). This particular tanh scale de-pendence is not intended to model any particular mechanism for the mod-ulation of fluctuations. But it can capture some interesting cases. For largekc, the modulation becomes scale-invariant. By decreasing kc we can repre-sent a modulation only on large scales, e.g. scales that are super-Hubble atlast scattering, which may be related to some early-Universe process. Forkc ≃ 5×10−3Mpc−1 and ∆ ln k → 0 in particular, we produce a modulationon the commonly quoted angular scales of ℓ . 65 (keeping in mind that thek–ℓ kernels imply that there is no one-to-one correspondence between k andℓ values).The form of the modulation in Eq. (1.4), i.e. that of a spatially linearlymodulated primordial field, is an important assumption here. We regardit as the simplest form that would lead to a dipolar asymmetry. A linearmodulation can be considered the lowest-order term in an expansion, forgeneral modulations varying slowly on our Hubble scale. Other choices addcomplexity and require more parameters to specify, e.g., generalizing thelinear form to quadratic or higher order spatial dependence, or taking thefluctuation spectrum to jump like a step function across a “wall”. Thesemore complicated scenarios could be tested, since they would predict asym-metry beyond dipolar, but considering the low signal-to-noise ratio of the Tasymmetry we restrict this study to the simplest possibility. Crucially, thelinear modulation means that CMB lensing, which is mainly sourced at lowredshifts, is expected to be modulated with considerably lower amplitudethan the observed T amplitude of roughly 6%. This conclusion will clearlybe strongly dependent on the assumed form of the k-space modulation. Also,note that we take the linear modulation to act on the primordial field, R.This is what would be expected in most proposed models where the modula-tion originates in some very early physics, e.g. during inflation. Also, it leadsto the linear dependence on comoving distance in Eq. (1.4). Conversely, itseems very unlikely that a late-time field (e.g. the zero-shear gauge fluctu-ation ψσ; see below) would be directly modulated. Such a scenario couldinvolve an anisotropic dark energy, which would be subject to strong con-straints at the background level. Nevertheless, we will show that, insofar asCMB T and lensing are concerned, to a good approximation we can equallyconsider either the early- or late-time fields to be linearly modulated.351.3. Effect on CMB temperature anisotropiesIn k space the modulation of Eq. (1.4) becomesR˜lo(k) = Rlo(k) + iARrLS∂∂kzRlo(k). (1.10)This implies that the total R˜(k) covariance (to first order in AR) is givenby〈R˜(k)R˜∗(k′)〉 = 2π2k3PΛCDMR (k)δ3(k − k′)+ 2π2iARrLS[P loR(k)k3+P loR(k′)k′3]× δ2(k⊥ − k′⊥)δ′(kz − k′z), (1.11)where k⊥ is the projection of k orthogonal to zˆ and the prime on the Diracdelta denotes a derivative with respect to the argument. Note importantlythat, for a Gaussian field R˜, Eq. (1.11) is a complete statistical descrip-tion. This means that the details of our implementation, i.e. in terms of thecomponents Rlo and Rhi, are irrelevant: in the end we obtain a covariancecorresponding to a standard isotropic part (the diagonal part of Eq. (1.11)),plus a dipole-modulated part with arbitrary scale dependence, as determinedby P loR(k) (the off-diagonal, imaginary part of Eq. (1.11)). In particular, ourapproach does not restrict us to some early-Universe mechanism which pro-duces two uncorrelated components, Rlo and Rhi. The separation into thosetwo components is purely a convenient calculational device which will makethe analytical work considerably simpler, as we will see next. We remainagnostic as to the physical modulation mechanism. Note that Eq. (1.11)describes statistically inhomogeneous fluctuations, whereas the effect in ℓ ormap space will be statistical anisotropy.1.3 Effect on CMB temperature anisotropies1.3.1 Multipole covarianceIn general, the effect of the modulation, Eq. (1.4), on the CMB anisotropieswould be very difficult to calculate (see Kothari 2015, for such a generalapproach). However, we will show that, to a very good approximation,the effect will be simply to introduce an ℓ to ℓ ± 1 coupling with spectrumdetermined by P loR(k), as one might intuitively expect for scales much smallerthan the length scale of variation of the modulation.361.3. Effect on CMB temperature anisotropiesWe begin by demonstrating this on the largest scales, for which we cananalytically write down the T anisotropies. Since the observed modulation ison large scales, this is a relevant regime. The large-scale approximation usedhere will begin to break down on scales ℓ ∼ 50, although in this case a simpleargument will allow us to write down the multipole covariance immediately.Nevertheless, we will provide a detailed examination of the small-scale casein Appendix A.On the largest scales, it is a good approximation to treat the plasma astightly coupled prior to an instantaneous recombination. In this approxi-mation, the T anisotropies are determined entirely by the zero-shear (longi-tudinal) gauge metric perturbation, ψσ, which is related to the primordialcomoving curvature perturbation, R, viaψσ(k) = −35T (k)R(k), (1.12)where T (k) is the transfer function that captures the effect of radiationdomination (see, e.g., Zibin and Scott 2008). Since here we are consideringonly the largest scales, we will ignore the component Rhi in this subsectionand drop the superscript “lo” for brevity.Note that in general a linear modulation of R will not imply a linearmodulation of ψσ, i.e. the operations of linear modulation and filtering viaT (k) will not commute. An easy way to see this is to consider the ex-treme case of a very narrow filtering around some scale k¯, T (k) ≃ δ(k − k¯).Then applying T (k) to the linearly modulated R will simply give a nearlymonospatial-frequency ψσ, which will not be spatially modulated, as op-posed to the case of modulating the field filtered with T (k). Therefore, ingeneral, a linear primordial modulation does not lead to a correspondinglinear modulation of ψσ, which is the field that determines the T anisotro-pies. In practice, this will mean that the calculation of the T anisotropieswill be very difficult. On the other hand, for constant T (k), the operationsof modulation and filtering clearly commute. So as long as T (k) is suffi-ciently slowly varying, we will be able to assume commutativity to goodapproximation.To determine the quantitative effect of the non-commutativity, Eq. (1.10)impliesT (k)R˜(k) =[T (k)− iARrLSkzkT ′(k)]R(k)+ iARrLS∂∂kz[T (k)R(k)] . (1.13)371.3. Effect on CMB temperature anisotropiesComparing with Eq. (1.12), this tells us that if∣∣∣∣i 1rLS kzk T ′(k)∣∣∣∣≪ T (k), (1.14)i.e., if ∣∣∣∣ 1T (k) dT (k)dkrLS∣∣∣∣≪ 1, (1.15)then the operations of modulation and filtering will essentially commute,so that we can write the total ψσ fluctuations to a good approximation aslinearly modulated according toψ˜σ(x) = ψσ(x)(1 +ARrrLScos θ). (1.16)For ΛCDM, we find numerically that T−1(k)dT (k)/d(krLS) . 3×10−3 on allscales, so that indeed it will be a very good approximation to use Eq. (1.16),which will simplify the calculations tremendously.Equation (1.16) makes it very easy to determine the effect of the modu-lation on large-scale anisotropies. Those anisotropies take the formδ˜T (nˆ)T= S˜(tLS, rLSnˆ), (1.17)for direction nˆ and where tLS is the time of last scattering, and the sourcefunction S˜(tLS, rLSnˆ) is determined fully by ψ˜σ and its first and secondderivatives (see, e.g., Zibin and Scott 2008). We have just shown that thelinear modulation of R corresponds to very good approximation to the lin-ear modulation of the ψσ part of S(tLS, rLSnˆ). Next we will examine eachderivative term. The first spatial derivative takes the form of a radial deriva-tive:1aLSHLS∂∂rψ˜σ(x) =1aLSHLS∂ψσ(x)∂r(1 +ARrrLScos θ)+ ψσ(x)AR1aLSHLSrLScos θ. (1.18)The second term on the right-hand side of this expression shows, interest-ingly, that the derivative of the modulation gives a term degenerate with themodulation of ψσ itself. However, for ΛCDM we have aLSHLSrLS = 66.4,so that this degenerate term can be ignored (for sources near rLS) and thefirst derivative of the linearly modulated field ψ˜σ can be well approximatedby the linear modulation of the derivative of ψσ.381.3. Effect on CMB temperature anisotropiesThe second spatial derivative takes the form of a Laplacian. In this case,it is trivial that the Laplacian commutes with the modulation in Eq. (1.16),due to the assumed linear nature of the modulation. The same is true for thetime derivatives, since the modulation is taken to be time independent, asdiscussed in Sect. 1.2. Therefore, the temperature anisotropies, Eq. (1.17),become to a good approximationδ˜T (nˆ)T= S(tLS, rLSnˆ) (1 +AR cos θ) (1.19)=δT (nˆ)T(1 +AR cos θ) . (1.20)In words, the modulated anisotropies are simply given by the anisotropiescalculated from the statistically isotropic (“equatorial”) fields, i.e. S(tLS, rLSnˆ),modulated.This leads directly to the simple temperature multipole covariance of theform studied in Moss et al. (2011), i.e. an ℓ to ℓ ± 1 coupling. ExpandingEq. (1.20) into spherical harmonic multipoles we finda˜ℓm = aℓm +AR∑ℓ′m′aℓ′m′ξ0ℓmℓ′m′ . (1.21)Here ξ0ℓmℓ′m′ is the polar component of the coupling coefficients ξMℓmℓ′m′ de-fined byξMℓmℓ′m′ ≡√4π3∫Y ∗ℓm(nˆ)Yℓ′m′(nˆ)Y1M (nˆ)dΩnˆ. (1.22)Explicitly,ξ0ℓmℓ′m′ = δm′m (δℓ′ℓ−1Aℓ−1m + δℓ′ℓ+1Aℓm) , (1.23)ξ±1ℓmℓ′m′ = δm′m∓1 (δℓ′ℓ−1Bℓ−1±m−1 − δℓ′ℓ+1Bℓ∓m) , (1.24)whereAℓm =√(ℓ+ 1)2 −m2(2ℓ+ 1)(2ℓ+ 3), (1.25)Bℓm =√(ℓ+m+ 1)(ℓ+m+ 2)2(2ℓ+ 1)(2ℓ+ 3). (1.26)Equation (1.21) gives a multipole covariance〈a˜ℓma˜∗ℓ′m′〉 = Cℓδℓ′ℓδm′m +AR (Cℓ + Cℓ′) ξ0ℓmℓ′m′ (1.27)391.3. Effect on CMB temperature anisotropiesto linear order in AR, where Cℓ is the power spectrum calculated fromP loR(k). This covariance is a complete statistical description of the modulatedtemperature anisotropies on large scales.When the scale of the fluctuations sourcing the anisotropies is muchsmaller than the length scale of variation of the modulation, i.e. rLS, thenwe would expect the effect of the spatial variation of the modulation tobe small (see, e.g., Hanson et al. 2009). In other words, we expect the Tanisotropies sourced by P loR(k) to be modulated to a good approximationaccording to Eq. (1.20). Nevertheless, it will be worthwhile to be morequantitative about this expectation, so we examine small scales in detail inAppendix A.The simple behaviour for small scales (and the detailed calculationsin Appendix A) indicate that to very good approximation the modulatedtemperature fluctuations on all scales are given by the generalization ofEq. (1.20):δ˜T (nˆ)T≃ δTlo(nˆ)T(1 +AR cos θ) +δT hi(nˆ)T. (1.28)Eq. (1.6) then implies the final result for the multipole covariance:〈a˜ℓma˜∗ℓ′m′〉 = CΛCDMℓ δℓ′ℓδm′m +AR(C loℓ + C loℓ′ )ξ0ℓmℓ′m′ (1.29)to first order in AR, where CΛCDMℓ is the power spectrum calculated fromPΛCDMR (k) and C loℓ is the spectrum calculated in the same way but usingP loR(k).Notice that the statistical anisotropy in Eq. (1.29) can be easily calcu-lated using software such as CAMB (Lewis et al. 2000) with the primordialspectrum P loR(k). This compares with the approach of Kothari (2015) whodo not make the approximations we have made and hence must calculatesome new integrals involving derivatives of internal CAMB variables, whichis considerably more work. Importantly, note that the form of Eq. (1.29)is completely general, in that we have the necessary standard ΛCDM formfor the statistically isotropic component, and we have a dipole-modulatedpart with a scale dependence that is as free as possible, given that it mustoriginate from a k-space function (in this case P loR(k)). This shows againthat our approach of splitting the primordial fluctuations into uncorrelatedlow- and high-k parts, while facilitating the calculations, is not restrictivein any way.We have ignored the ISW effect in this calculation. With the linearmodulation model, the modulation amplitude at the redshifts at which the401.3. Effect on CMB temperature anisotropiesISW effect is sourced is predicted to be considerably smaller (by a factorrISW/rLS ∼ 1/5) than the roughly 6% for the primary CMB. Consideringalso that the ISW signal affects mainly the very largest scales, and thatsmaller scales are generated at closer distances (recall Fig. 1.1), it shouldbe a very good approximation to ignore the ISW effect entirely for theasymmetry. That is, the spectrum C loℓ can be calculated without the ISWcomponent. Note also that the effect of the modulated lensing field on themodulated CMB can also be ignored because it is a second-order effect inAR.1.3.2 Connection to general asymmetry formUsing the notation of Moss et al. (2011), the general form for the multipolemoment covariance given a polar (m = 0) modulation can be written〈a˜ℓma˜∗ℓ′m′〉 = Cℓδℓ′ℓδm′m +12δCℓℓ′∆X0ξ0ℓmℓ′m′ . (1.30)The origin of this notation lies in the assumption that the anisotropy powerspectrum depends linearly on some parameter, X, in which case the modu-lation spectrum, δCℓℓ′ , satisfiesδCℓℓ′ =dCℓdX+dCℓ′dX. (1.31)This means that we can formally write down the increment in power betweenthe modulation equator and the poles as∆Cℓ =12δCℓℓ∆X0, (1.32)where ∆X0 is the change in the parameter X from modulation equator topole. We will refer to δCℓℓ′ as the statistically anisotropic or modulationpower spectrum.Comparing Eq. (1.30) to our final result, Eq. (1.29), we can identify∆X0 = AR (1.33)andδCℓℓ′ = 2(C loℓ + Cloℓ′). (1.34)Equation (1.32) then allows us to write an effective increment in powerbetween the modulation equator and the poles as∆Cℓ = 2ARCloℓ . (1.35)411.4. Effect on lensing potentialThis is exactly what we would expect, since a fractional modulation of thefluctuation amplitude by AR should result in a modulation of power by2AR. This also justifies the approach for calculating the modulated ℓ-spacespectra of Namjoo et al. (2015).Note that if there is a significant contribution of PhiR(k) to the lowest ℓ’s,then according to Eq. (1.35) the actual predicted asymmetry, ∆Cℓ/(Cloℓ +Chiℓ ), will be smaller than 2AR. This is why we said we had in mind thatPhiR(k) would have a negligible contribution to the largest scales: when thisis the case our parameter 2AR will agree well with the actual large-scaleasymmetry, ∆Cℓ/(Cloℓ + Chiℓ ).1.4 Effect on lensing potentialIn this section we calculate the effect of a linear modulation of the primordialfluctuations, R, on the lensing potential. The (modulated) lensing potentialis determined by a line of sight integral,ψ˜lens(nˆ) = −2∫ rLS0drrLS − rrLSrψ˜σ(t(r), rnˆ) (1.36)(see, e.g., Lewis and Challinor 2006). Inserting Eq. (1.16), which we haveshown to be an extremely good approximation for the form of the modulatedzero-shear gauge fluctuations, and using Eq. (1.12), an expansion in sphericalharmonics and Bessel functions givesψ˜lens(nˆ) =65√2π∫ rLS0drrLS − rrLSrg(t(r))∫ ∞0dk kT (k)×∑ℓm[Rloℓm(k)(1 +ARrrLScos θ)+Rhiℓm(k)]jℓ(kr)Yℓm(nˆ),(1.37)where g(t) is the growth suppression factor due to late-time dark energy andRℓm(k) ≡ iℓk∫dΩkR(k)Y ∗ℓm(kˆ). (1.38)Therefore the lensing potential multipole moments areψlensℓm =65√2π∫ rLS0drrLS − rrLSrg(t(r))∫ ∞0dk kT (k)Rℓm(k)jℓ(kr)+65√2πAR∫ rLS0drrLS − rrLSrg(t(r))rrLS∫ ∞0dk kT (k)∑ℓ′m′Rloℓ′m′(k)jℓ′(kr)ξ0ℓmℓ′m′ .(1.39)421.4. Effect on lensing potentialNote the anisotropic part of Eq. (1.39), which contains the r/rLS weightingfactor. Finally, we can write the lensing multipole covariance to O(AR),〈ψlensℓm ψlens∗ℓ′m′ 〉 =144π25∫ ∞0dkkT 2(k)PΛCDMR (k)[∫ rLS0drrLS − rrLSrg(t(r))jℓ(kr)]2δℓ′ℓδm′m+[144π25AR∫ ∞0dkkT 2(k)P loR(k)∫ rLS0drrLS − rrLSrg(t(r))jℓ(kr)×∫ rLS0dr′rLS − r′r2LSg(t(r′))jℓ(kr′) + (ℓ↔ ℓ′)]ξ0ℓmℓ′m′ . (1.40)Note that (ℓ ↔ ℓ′) denotes repeating the term in the surrounding bracketswith ℓ and ℓ′ interchanged. Using the general definition for the multipolemoment covariance given a polar (m = 0) modulation, Eq. (1.30), we canidentify the statistically isotropic part to beCℓ = Clensℓ =144π25∫ ∞0dkkT 2(k)PΛCDMR (k)[∫ rLS0drrLS − rrLSrg(t(r))jℓ(kr)]2,(1.41)while the statistically anisotropic part isδC lensℓℓ′ =288π25∫ ∞0dkkT 2(k)P loR(k)∫ rLS0drrLS − rrLSrg(t(r))jℓ(kr)×∫ rLS0dr′rLS − r′r2LSg(t(r′))jℓ(kr′) + (ℓ↔ ℓ′), (1.42)with ∆X0 again given by Eq. (1.33). The isotropic part, Clensℓ , agrees withthe standard result (Lewis and Challinor 2006), while the anisotropic part,δC lensℓℓ′ , is new. It can be easily calculated numerically for ΛCDM trans-fer function T (k) and growth function, g(t), given a modulation spectrumP loR(k). Note that unlike the case of the primary CMB anisotropies, forlensing the anisotropic part is not simply the usual lensing spectrum calcu-lated with P loR(k). The fact that lensing is sourced all along the line of sightmeans that, instead, the last integral in Eq. (1.42) is weighted by a factorof r/rLS, which reflects the linear nature of the assumed modulation. Asanticipated, this reduces the amplitude of the lensing asymmetry relative tothat of the primary CMB. The shift to larger angular scales expected forthe more closely sourced lensing potential is also encoded in Eq. (1.42).We stress that this lensing calculation is considerably simpler than thatfor the temperature fluctuations, due to the simpler relevant transfer func-tion and simpler dependence of the lensing potential on the primordial fluc-tuations. Indeed, the only approximation made here is that of Eq. (1.16),which we have shown to be extremely accurate.431.5. Fitting the k-space modulation to temperature dataTo complete our description of lensing modulation, we can again formallywrite down the increment in power between the modulation equator and thepoles as∆C lensℓ =12ARδClensℓℓ . (1.43)1.5 Fitting the k-space modulation totemperature data1.5.1 FormalismNext we describe how we fit the k-space modulation spectrum P loR(k), whichwe have assumed to take the tanh form of Eq. (1.9), to CMB temperaturedata. The spectrum depends on two free parameters: kc determines whichscales are modulated, and ∆ ln k determines the sharpness of the transitionfrom modulated to statistically isotropic scales. We denote these parametersby pi = {kc,∆ ln k}, for brevity. We begin with the likelihood function forthe CMB temperature multipoles given the modulation parameters,L(d|∆XM , pi) ∝ 1√|C| exp(−12d†C−1d). (1.44)Here d is the vector of multipole moments and the dependence on the modelparameters (∆XM , pi) is contained in the multipole covariance matrix C.Previously we had taken the modulation direction to coincide with the zˆdirection, but now we must keep the direction free and fit for it. Hence thecovariance matrix, Eq. (1.30), becomes (Planck Collaboration XVI 2016)Cℓmℓ′m′ ≡ 〈a˜ℓma˜∗ℓ′m′〉 (1.45)= Cℓδℓ′ℓδm′m +12δCℓℓ′∑M∆XMξMℓmℓ′m′ . (1.46)The three model parameters ∆XM determine the amplitude and directionof the modulation (they are the spherical harmonic transforms of the mod-ulating dipole term, see Eqs. (1.50)–(1.52) below), while the two modula-tion parameters pi determine the scale dependence of the modulation viaEq. (1.34), and so we have in total five parameters which describe the sta-tistical anisotropy (we hold the main cosmological parameters fixed).For fixed pi, we can find the ∆XM which maximize the likelihood fromEq. (1.44) to first order in AR. Specifically, for dipole modulation, we use441.5. Fitting the k-space modulation to temperature datathe estimator from Planck Collaboration XVI (2016), which generalizes thatof Moss et al. (2011) (see Hanson et al. 2009, for related optimal estimators):∆X˜0 =6f10∑ℓm δCℓℓ+1AℓmSℓm ℓ+1m∑ℓ δC2ℓℓ+1(ℓ+ 1)FℓFℓ+1, (1.47)∆X˜1 =6f11∑ℓm δCℓℓ+1BℓmSℓm ℓ+1m+1∑ℓ δC2ℓℓ+1(ℓ+ 1)FℓFℓ+1, (1.48)and ∆X˜−1 = −∆X˜∗1 . HereSℓmℓ′m′ ≡ T ∗ℓmTℓ′m′ − 〈T ∗ℓmTℓ′m′〉, (1.49)where the Tℓm are C-inverse filtered temperature multipoles and Fℓ is themean power spectrum of the Tℓm. The expectation value in Eq. (1.49) isan average over a set of realistic simulations, which provides a mean-fieldcorrection (described in great detail in Planck Collaboration XVII 2014;Planck Collaboration XV 2016; Planck Collaboration XVI 2016). The f1Mfactor corrects for normalization errors introduced by masking (its explicitform can be seen in Planck Collaboration XVI 2016). The C-inverse filteris identical to that used in Planck Collaboration XVII (2014), Planck Col-laboration XV (2016), and Planck Collaboration XVI (2016), and optimallyaccounts for masking effects. In practice, we bin the estimator, Eqs. (1.47)and (1.48), into bins of width ∆ℓ = 1, which means that the corrections tothe data described above only need to be calculated once. This gives exactlythe same result as if the estimators were computed for each set of pi fromscratch; however, it allows us to dramatically speed up the exploration of theparameter space (this technique was also employed in Planck CollaborationXX 2016, for the same reasons). In the following subsection we describethe data and corresponding simulations used for obtaining these estimators.Given these estimates of the ∆XM , we can write the best-fit amplitude anddirection asA˜R =√∆X˜20 + 2|∆X˜1|2, (1.50)θ˜ = cos−1(∆X˜0A˜R), (1.51)φ˜ = − tan−1[Im(∆X˜1)Re(∆X˜1)]. (1.52)The central limit theorem suggests that the ∆XM will be Gaussian dis-tributed (this has been verified explicitly with the use of simulations), and451.5. Fitting the k-space modulation to temperature dataspecifically for statistically isotropic skies they will have mean zero. Theirvariances can be calculated exactly from Eqs. (1.47) and (1.48) to beσ2X(pi) ≡〈∣∣∆X2M ∣∣〉 = 12∑ℓ(ℓ+ 1)δC2ℓℓ+1C−1ℓ C−1ℓ+1. (1.53)The posterior for the ∆XM parameters for a fixed pi = p¯i is then given byP (∆XM , p¯i|d) = 1(2π)3/2σ3X× exp[−∑M |∆XM −∆X˜M |22σ2X]. (1.54)Using these relations, we can evaluate the log-likelihood function at themaximum-likelihood values ∆X˜M to belnL(d|∆X˜M , pi) =∑M|∆X˜M |22σ2X, (1.55)to first order inAR and ignoring terms independent of the statistical anisotropy.This tells us that the expectation of the log-likelihood in statistically isotropicskies is independent of pi. It also says that the expected increase of the log-likelihood coming from the introduction of the ∆XM parameters is 3, asexpected. Note also that this relation means that the likelihood will be verysimple to evaluate numerically.Bayes’ theorem allows us to write the posterior for the model parametersasP (∆XM , pi|d) = L(d|∆XM , pi)P (∆XM , pi), (1.56)with prior P (∆XM , pi) on the model parameters, up to an overall normal-ization. We can calculate the posterior marginalized over the ∆XM s, withthe resultP (pi|d) ∝ σ3XL(d|∆X˜M , pi)P (pi). (1.57)Inserting Eqs. (1.55) into Eqs. (1.57) then tells us that a natural choice forthe prior on the pi isP (pi) ∝ σ−3X , (1.58)which yields a posterior agnostic to σX (determined by the scales beingmodulated), that is the posterior will be driven by Eq. (1.55). Hence this isthe prior we choose. We also choose a flat prior in the ∆XM s, as is usuallydone.461.5. Fitting the k-space modulation to temperature dataOnce the best-fit modulation spectrum parameters (kc,∆ ln k) are found,it will be a simple matter to evaluate the lensing asymmetry using themethod laid out in Sect. 1.4, and, in the future, the polarization asymmetryas well.1.5.2 ResultsThe results presented here are based on the component-separated temper-ature maps provided by the Planck Collaboration (Planck Collaboration I2016). Namely, we use the Commander, NILC, SEVEM, and SMICA 2015 tem-perature maps (Planck Collaboration IX 2016) at a HEALPix (Go´rski et al.2005) resolution of Nside = 2048 (for brevity we only quote the results forSMICA; however, we have checked that the other maps do not give substan-tially different results). We also use the UT78 mask provided by the PlanckCollaboration, referred to as the “common” mask. We use a set of 1000FFP8 simulations1 (Planck Collaboration XII 2016), corresponding to eachcomponent separation method, in order to make mean-field and normal-ization corrections to the data, as was done in Planck Collaboration XVI(2016).Using the relations of Eqs. (1.50)–(1.52), we can perform a one-to-onelinear transformation from the ∆XM to Cartesian modulation components,{∆X, ∆Y, ∆Z}. The {∆X, ∆Y, ∆Z} are simply the components of thedipole modulation vector in Cartesian Galactic coordinates. In what fol-lows we will present results in this coordinate system for convenience. Wescan the model space over the following parameter ranges: ln(kc [Mpc−1]) ∈[−7.2,−3.2], ∆ ln k ∈ [0.01, 0.5], and |∆X|, |∆Y |, |∆Z| ≤ 1. The lower limiton kc is placed to ensure that we only look for modulation on scales thatare observable, while the lower limit on ∆ ln k corresponds essentially to anabrupt cutoff in k space. The upper limits on kc and ∆ ln k are somewhatarbitrary: in multipole space they correspond approximately to limiting themodulation to ℓ < 1000. We are primarily interested in large-scale modula-tions, and previous ℓ-space results (Flender and Hotchkiss 2013; Quartin andNotari 2015; Aiola et al. 2015; Planck Collaboration XVI 2016) indicated noevidence for modulation on scales smaller than this limit. For AR > 1 thefluctuations in Eq. (1.4) will go to zero somewhere within our last scatteringsurface, and the details of the modulation in this case will depend on thespecific modulation mechanism. For the tanh model we do not approach1Available on NERSC, at http://crd.lbl.gov/departments/computational-science/c3/c3-research/cosmic-microwave-background/cmb-data-at-nersc/471.5. Fitting the k-space modulation to temperature datathis regime: the limits on the modulation amplitude components turn outto be generous. We explore the parameter space using a simple grid ap-proach, which is adequate since the parameter space is effectively only twodimensional via Eq. (1.54).Results are summarized as the posterior distribution of the full param-eter set {kc, ∆ ln k, ∆X, ∆Y, ∆Z} in Fig. 1.2. We also present results fora condensed version of the parameter space, i.e. the set {kc, ∆ ln k, AR},where the angular variables have been marginalized over, in Fig. 1.3. Wecan see from the distributions that no parameter is constrained very well. Inparticular, ∆ ln k is completely unconstrained by the data, which suggeststhat there is no well-defined transition in the data between modulated andunmodulated scales. This is not surprising, since we have opened up the pa-rameter space in our formalism with respect to most previous studies, whichconsidered a sharp cutoff in ℓ space and only found apparently significantmodulation when that cutoff was fixed. In Table 1.1 we quote the meanvalue parameters and their uncertainties, which we take as the mean of themarginalized posteriors and the area that encloses 68% of the likelihood.We also quote the maximum-likelihood parameters. For comparison, whentesting for an ℓ-space modulation to ℓ = 65, (Planck Collaboration XVI2016) found AR = 0.062+0.026−0.013 in the direction (l, b) = (213◦,−26◦) ± 28◦,where the error bar is circular around the best-fit direction.The temperature anisotropy modulation spectrum C loℓ and effective powerspectrum difference from modulation equator to pole, ∆Cℓ, for the maximum-likelihood modulation parameters from Table 1.1, are plotted in Fig. 1.4.These were calculated using CAMB with the corresponding best-fit primordialspectrum P loR(k) and setting the ISW source to zero for redshifts z < 30.(Negligible differences were found when the ISW effect was included in theanisotropic spectrum.) The modulation in amplitude is at a level of roughly7% to ℓ ≃ 50. Importantly, while this agrees crudely with the often-quotedlevel of 6–7% to ℓ ≃ 65, we stress that the temperature asymmetry is poorlyconstrained: the kc posterior in Figs. 1.2 and 1.3 has significant weight overa large range of values, corresponding to ℓ ≃ 50–250. As the kc-AR panelin Fig. 1.3 shows, these two parameters are anticorrelated, with a larger kcimplying a smaller AR. Furthermore, as the dashed contours in that panelshow, this anticorrelation follows the trend expected from cosmic variance,which arises simply because larger kc implies more modes and hence lowercosmic variance. This poorly-defined character of the asymmetry may besurprising, but has previously been found in ℓ space (see in particular thepeaks at ℓ ≃ 200–300 in Figure 30 of Planck Collaboration XVI (2016) andFigure 15 of Bennett et al. (2011), which have similar significance to the481.5. Fitting the k-space modulation to temperature data0.20.4∆ln(k)kc [Mpc−1]-40410×∆X0.10.20.30.40.5∆ln(k)ln(kc)-40410×∆Y0.10.20.30.40.5∆ln(k)0.40.20.00.20.410-310-2kc [Mpc−1]-40410×∆Z0.2 0.4∆ln(k)-4 0 410×∆X-4 0 410×∆Y-4 0 410×∆ZFigure 1.2: Marginalized posteriors for the parameter set {pi,∆X,∆Y,∆Z};dark and light blue (solid) contours enclose 68% and 95% of the likeli-hood, respectively. The black and grey (dashed) contours and curves rep-resent the theoretical distributions of the parameters coming solely fromcosmic variance in statistically isotropic skies. The values (kc,∆ ln k) =(5 × 10−3Mpc−1, 0) would correspond roughly to the often-considered ℓ-space modulation to ℓ ≃ 65.peaks at ℓ ≃ 65).Finally, note that Fig. 1.4 shows that an origin to the T asymmetry as amodulation of the ISW effect alone (perhaps via an anisotropic sound speed491.6. Predictions for CMB lensing0.20.4∆ln(k)10-310-2kc [Mpc−1]0.20.4AR0.2 0.4∆ln(k)0.2 0.4ARFigure 1.3: Marginalized posteriors for the parameter set {pi, AR}; darkand light blue (solid) contours enclose 68% and 95% of the likelihood, re-spectively. The black and grey (dashed) contours and curve represent thetheoretical distributions of the parameters coming solely from cosmic vari-ance in statistically isotropic skies.for dark energy) is unlikely to produce a good fit to the data, since the ISWcontribution is extremely weak for ℓ & 50.1.6 Predictions for CMB lensing1.6.1 Modulation power spectrumHaving used the CMB temperature data to fit the k-space modulation spec-trum, P loR(k), in the last section, we are now ready to present the predictionfor the CMB lensing asymmetry. Once the fitting has been done, we knowthe modulation direction via the ∆X˜M , and so we can write the multipolecovariance as Eq. (1.30) with the polar direction along the modulation di-rection and amplitude A˜R.501.6. Predictions for CMB lensingParameter Mean value Max likelihood103kc [Mpc−1] 7.08+12.56−2.34 7.83∆ ln k unconstrained 0.5∆X −0.060+0.054−0.018 −0.0610∆Y −0.063+0.069−0.010 −0.0414∆Z −0.056+0.062−0.004 −0.0347AR 0.122+0.014−0.112 0.0871l [◦] 224+43−44 214b [◦] −31+31−16 −25AR 0.095+0.026−0.080 · · ·Table 1.1: Marginalized posterior mean values and their 68% uncertaintiesfor the modulation parameters of the model of Eq. (1.9), along with theircorresponding maximum-likelihood values. The angles l and b are the Galac-tic longitude and latitude, respectively, calculated via Eqs. (1.51) and (1.52).The final row is the projected combined constraint including an ideal CMBlensing experiment assuming a modulation with amplitude AR = 0.122 andthe remaining temperature mean values. The addition of lensing does notappreciably help to constrain the model.Using the maximum-likelihood k-space modulation spectrum parame-ters, kc, ∆ ln k, and ∆X˜M , from Table 1.1, we calculated the statisticallyanisotropic lensing spectrum, δC lensℓℓ , using Eq. (1.42). The result is plottedin Fig. 1.5. The lensing spectrum is modulated at a level of about 3% andless in power (about 1.5% and less in amplitude), out to scales as smallas ℓ ≃ 50. As predicted in Sect. 1.1 on geometrical grounds, the lensingpotential is modulated to a larger minimum angular scale and by a smalleramplitude than the corresponding temperature best fit presented in Fig. 1.4.This directly leads to a low modulation detection significance for lensing, aswe will see in the next subsection.Note that the anisotropic spectrum grows relative to the isotropic spec-trum at large to intermediate scales. This can be understood with the helpof Fig. 1.1, where it is apparent that larger lensing multipoles are typicallysourced at greater distances. For our assumed linear modulation form, largerdistances, and hence larger multipoles, will be modulated with larger am-plitude. Compared with this lensing case, the corresponding temperatureanisotropy spectrum in Fig. 1.4 exhibits a much more similar shape to theisotropic spectrum, up to the cutoff kc and allowing for the lack of the ISW511.6. Predictions for CMB lensing1 10 100 100010-1210-1110-1010-9(+1)/(2)Figure 1.4: Temperature anisotropy isotropic power spectrum, CΛCDMℓ (solidblack curve), anisotropic power spectrum, C loℓ (dashed red curve), and powerspectrum increment from equator to pole, ∆Cℓ (dot-dashed green curve), forthe case of the maximum-likelihood modulation from Table 1.1, which fitsthe observed temperature asymmetry. The modulation in amplitude is at alevel of roughly 7% to ℓ ≃ 50.contribution in the anisotropic spectrum. This is simply due to the fact thatthe primary CMB is sourced at essentially a single distance, rLS, and henceis modulated at a single amplitude for our linear model.1.6.2 Detectability for ideal lensing mapIn order to ascertain the detectability of the predicted lensing potentialmodulation, we must compare the prediction to the expected uncertainty inthe measurement. The expected variance of a CMB lensing measurement ofasymmetry will be determined by cosmic variance (of the lensing potentialmodes) and lensing reconstruction noise. It turns out that for a lensing re-construction based on cosmic-variance-limited temperature and polarizationanisotropy measurements, the reconstruction noise is small compared to thelensing potential cosmic variance, at least over the relevant scales (Lesgour-gues et al. 2006).2 Therefore, an ideal lensing measurement can be consid-ered essentially cosmic-variance limited. Realistic lensing experiments willhave higher noise, which will necessarily reduce our ability to detect a mod-2Note that a lensing reconstruction will be valid even for the statistical anisotropiesconsidered here.521.6. Predictions for CMB lensing1 10 10010-1010-910-810-7[(+1)]2lens/(2)Figure 1.5: Lensing potential isotropic power spectrum, C lensℓ (black curve),predicted anisotropic power spectrum, δC lensℓℓ (red curve), and predictedpower spectrum increment from equator to pole, ∆C lensℓ (green curve), forthe case of the maximum-likelihood modulation from Table 1.1, which fitsthe observed temperature asymmetry. The modulation in amplitude is at alevel of roughly 1.5% or less to ℓ ≃ 50.ulation. Hence our conclusions with respect to the effectiveness of usinglensing will be conservative.We can easily evaluate the cosmic variance of the modulation amplitude∆X0 given a lensing modulation spectrum, δClensℓℓ′ , using Eq. (1.53). UsingEqs. (1.33), (1.41), and (1.42) for the case of the maximum-likelihood param-eters, we find√〈A2R〉= 0.111. This means that the maximum-likelihoodmodulation amplitude determined from the T anisotropies, AR = 0.0871, is0.0871/0.111 = 0.8 standard deviations from zero for a lensing measurementalong the known T modulation direction. For the mean value modulationparameters from Table 1.1, we find an expected measurement of 0.9σ. How-ever, in this case the highly non-Gaussian posterior (recall Fig. 1.3) meansthat the mean value parameters are biased towards high significance. Infact, given the likelihood from temperature we can determine that the meandetection significance for AR by lensing is 0.7σ and the probability of ob-taining a greater than 1σ detection of AR in lensing is of order 10%. Theprobability is of order 0.1% for finding a greater than 1.5σ detection of ARand decreases quite rapidly for higher detection limits. Therefore, even inthis case of an ideal, cosmic-variance-limited lensing map, lensing will tellus very little about whether the asymmetry is real or not.531.7. DiscussionWe can illustrate the weakness of CMB lensing for testing a physicalmodulation in another way. The last row of Table 1.1 lists the result ofcombining the constraint on AR from the T anisotropies with the expectedconstraint for lensing, assuming a modulation with parameters given by themean values of Table 1.1, which were determined by the T likelihood. It isapparent that lensing does not improve the constraint on AR significantly. Ifwe consider relaxing the condition here that the pi be fixed, we can see thatCMB lensing will not be able to constrain the modulation model significantlybetter than CMB T alone.1.7 DiscussionIn this chapter I have presented a rigorous formalism for describing a lin-early modulated primordial fluctuation field in k space with arbitrary scaledependence, and have calculated its effects on CMB temperature fluctua-tions as well as the lensing potential, which probes independent modes fromthe primary CMB. We performed a Bayesian parameter estimation for thek-space modulation spectrum, fitting to Planck temperature data. We thenpredicted the corresponding CMB lensing modulation, and found that evenan ideal lensing experiment would expect to see the modulation at onlyabout 0.7σ. Hence it appears that CMB lensing will never tell us muchabout whether the observed T modulation is a statistical fluke or is due toa real, physical modulation of the primordial fluctuations. Also, this meansthat the null result for asymmetry in the Planck lensing map (Flender andHotchkiss 2013) is completely unsurprising, given that the Planck lensingmap contains substantially more noise than an ideal map would.In principle, correlating CMB lensing with other probes should improvethe attainable significance of the expected modulation. However, recall fromFig. 1.1 that current galaxy surveys have weak sensitivity at the requiredextremely large scales. In addition, such surveys reach to relatively lowredshifts, and hence we would expect a low modulation amplitude, at leastfor a linear modulation. Nevertheless, it may be worth considering thismore carefully, given our result that the upper limit for the cutoff is nearkc ≃ 0.02Mpc−1. The ISW contribution reaches to sufficiently large scales,but is sourced so close to us that, again, its modulation amplitude is expectedto be very small.It is important to point out that, although it appears that we cannotusefully probe the asymmetry with CMB lensing, it will still be importantto examine lensing maps for departures from statistical isotropy. Lensing541.7. Discussionprobes a large fraction of our observable volume that is inaccessible by othermeans. Hence it provides a unique opportunity to test the simplest modelsof fluctuations (Zibin and Moss 2014).Our results also highlight a seldom-stressed aspect of the temperatureasymmetry. We found that no well-defined k-space modulation exists, andinstead that the modulation cutoff scale, kc, is only weakly constrained. Inparticular, there is no reason to single out an approximately 6% modulationto ℓ ≃ 65. However, this poor constraint means that our results should beonly weakly sensitive to our choice for P loR(k), i.e. to departures from thetanh form.It is clear that polarization will be our best opportunity in the near termto test for a physical modulation. However, even though polarization cansample about as many independent modes as temperature, Namjoo et al.(2015) finds strong k-space model dependence for the predictions of polar-ization. It will be important to examine this with our fitting procedure. Inparticular, we will need to generalize our approach to incorporate isocurva-ture and tensor mode modulations.In the distant future 21-cm surveys may have the ability to reach tolarge distances and very large scales. They will have, in principle, vastlymany more modes within reach via three-dimensional mapping than do thetwo-dimensional CMB or lensing measurements. Hence they should finallyresolve the status of the power asymmetry and other anomalies.Note added.—After this work was nearly complete, a related study ap-peared (Hassani et al. 2016), which examines the effect of dipole modulationon lensing. That study seems to predict considerably larger CMB lensingmodulation amplitudes than we find. However, it appears that Hassaniet al. (2016) ignore the spatial dependence of the modulation, replacing myEq. (1.3) withR˜lo(x) = Rlo(x) (1 +AR cos θ) . (1.59)Hence they do not see the large reduction in modulation amplitude due tothe sourcing of lensing at relatively low redshifts. In addition, Hassani et al.(2016) do not fit a modulation to T data nor do they predict the detectabilityfor lensing.55Chapter 2CMB Polarization2.1 IntroductionThe largest scales of our cosmic microwave background (CMB) tempera-ture sky exhibit a dipolar asymmetry, with an amplitude at the severalpercent level (Eriksen et al. 2004; Hanson and Lewis 2009; Bennett et al.2011; Planck Collaboration XXIII 2014; Planck Collaboration XVI 2016).While the cosmological origin of this signal is not in dispute (existing inboth WMAP and Planck data with very different systematic effects andfrequency coverage), its statistical significance is debated (see Bennett et al.2011; Planck Collaboration XVI 2016, and references therein). Withoutcorrection for the effects of a posteriori selection, this large-scale signal hasa significance at approximately the 3σ level and an amplitude of roughly6–7%, when restricted to a multipole range of ℓ . 65. Applying a look-elsewhere correction on the maximum multipole reduces the correspondingp-value to of order 10% (Bennett et al. 2011; Planck Collaboration XVI2016). However, the asymmetry is present on scales that are roughly super-Hubble at last scattering, suggesting a possible physical origin related tovery-early-Universe physics. If this were the case, the former significanceestimate might be more relevant, given a model that predicted it.The dipolar asymmetry in temperature is now characterized as well asit ever will be, because the measurements on these scales are limited bycosmic variance and hence essentially all the cosmological information hasalready been extracted. The only way to unambiguously determine if thissignal is due to a physical modulation of fluctuations or is just a randomstatistical fluctuation is to acquire new information (i.e., measure new andindependent modes Scott et al. 2016). This can be achieved with the addi-tion of large-scale polarization data. Polarization data are available from thePlanck satellite; however, residual systematics (particularly at large angularscales) have so far prevented a full investigation of the polarization dipoleasymmetry signal. Other probes of new modes that have been examinedin this context include large-scale structure (Hirata 2009; Ferna´ndez-Coboset al. 2014; Yoon et al. 2014; Baghram et al. 2014), CMB lensing (Zibin and562.1. IntroductionContreras 2017; Hassani et al. 2016), 21-cm measurements (Shiraishi et al.2016), and spectral distortions (Dai et al. 2013).The use of polarization in the investigation of dipole asymmetry has beenexamined previously in Dvorkin et al. (2008), Paci et al. (2010), Chang andWang (2013), Paci et al. (2013), Dai et al. (2013), Ghosh et al. (2016),Kothari et al. (2016), Kothari (2015), Namjoo et al. (2015), and Bunn et al.(2016). In particular, it was appreciated (Dvorkin et al. 2008; Namjoo et al.2015) that a modulation model must be constructed in position (or k) spaceand propagated to map (or spherical harmonic) space in order to consis-tently describe both temperature and polarization modes. In addition, itwas found that the predicted signature in polarization is quite dependenton the assumed model for the k-space modulation (Namjoo et al. 2015).Here we will use the temperature data to predict the polarization signaturegiven a k-space modulation model, following the formalism developed inChapter 1.Naively, one might expect that the temperature signal, together withthe assumption of a physical modulation of the large-scale three-dimensionalfluctuation field, would predict a roughly 6–7% asymmetry in polarizationin the ℓ range of 2 to 65. However, this is not necessarily the case, since themapping (transfer functions) from k to ℓ space differs between temperatureand polarization. Therefore, in the absence of a detailed physical modu-lation model we cannot use polarization to test for such a physical origin.In addition, as stressed in Chapter 1, the signal in the temperature dataalone is not strong enough to pick out a well-defined modulation scale de-pendence. In this chapter we will explore how these issues are modified withthe inclusion of polarization data in more detail.One well-known concern when considering polarization is the correla-tion between temperature and gradient- (or E-) mode polarization. Morespecifically, modes we can measure from polarization are not completely in-dependent of temperature. This correlation may alter a possible polarizationasymmetry signal or mimic such a signal in its absence. We deal with thecorrelation by calibrating our estimator on Planck Full Focal Plane (FFP8)temperature and polarization simulations (Planck Collaboration XII 2016),which are appropriately correlated. A further concern is the spurious en-hancement of a modulation signal when including polarization. That is,simply due to cosmic variance and noise, adding polarization to tempera-ture might sometimes increase the significance of a signal even when thereis no true underlying modulation. This is especially true when the originaltemperature signal is of low to moderate significance, as is the case withour real CMB sky. We address this concern by quantifying the expected572.2. Modulation approacheffect of adding polarization both with and without an underlying physicalmodulation.Our main goals are to determine how likely it is that a physical origin fora modulation could be confirmed or refuted, and to quantify the expectedimprovement in constraints on the k-space modulation model parameters,with the addition of polarization data. We will present projections for Planckas well as a cosmic-variance-limited polarization measurement. We will notperform a blind multipole-space dipole asymmetry search as has been donewith temperature, though this can be done with the estimator we employhere and will be important to perform once the data are available.For this study we use the FFP8 cosmological parameters, with Hub-ble parameter H0 = 100h km s−1Mpc−1, where h = 0.6712, baryon densityΩbh2 = 0.0222, cold dark matter (CDM) density Ωch2 = 0.1203, neu-trino density Ωνh2 = 0.00064, cosmological constant density parameterΩΛ = 0.6823, primordial comoving curvature perturbation power spectrumamplitude As = 2.09 × 10−9 (at pivot scale k0 = 0.05Mpc−1) and tiltns = 0.96, and optical depth to reionization τ = 0.065.Note that at high ℓ our results would be biased due to the effect of aber-ration (Planck Collaboration XXVII 2014), which is not present in the FFP8simulations (Planck Collaboration XII 2016). In Appendix E we describehow we detect aberration in the temperature data and remove it so as notto bias our results.2.2 Modulation approach2.2.1 FormalismOur goal is to construct position- or k-space models that generate scale-dependent dipolar asymmetry, while remaining agnostic as to the detailedorigin (presumably inflationary) of the modulation. Based on the tempera-ture signal, we would like to modulate the largest scales while maintainingconsistency with the usual isotropic ΛCDM power spectra. We considerthree different types of model: the first is a modulated adiabatic mode,which comprises a part of the total primordial power spectrum;3 the secondis a modulated CDM isocurvature mode; and the third is a modulated ten-sor mode.4 For the adiabatic scalar case, we must modulate a large-scale3One particularly motivated model of this form would be a modulated integrated Sachs-Wolfe component, which will be examined in Chapter 3.4We note that a modulation of the optical depth to reionization (Dai et al. 2013) islikely extremely disfavoured, since it would imply temperature asymmetry to the smallest582.2. Modulation approachpart of the spectrum. The contributions from CDM isocurvature and ten-sor modes, however, are naturally restricted to scales ℓ . 100 (at least fornear-scale-invariant spectra). Therefore in these cases we only need to applya scale-invariant modulation to the tensor or isocurvature component. Wewill analyze each of these models with a slight generalization of the approachin Chapter 1, which considered only the adiabatic scalar case, and we referto that chapter for full details. Our approach will be readily applicable toother modulation models.We begin by decomposing the primordial fluctuations into two compo-nents. The first, Q˜lo(x), is spatially linearly modulated, and hence its inter-section with our last-scattering surface will be dipole modulated. It takesthe formQ˜lo(x) = Qlo(x)(1 +Ax · dˆrLS), (2.1)where Qlo(x) is statistically isotropic with power spectrum P lo(k), A isthe modulation amplitude, dˆ is the direction of modulation, and rLS is thecomoving distance to last scattering. The second, unmodulated component,Qhi(x), is statistically isotropic with power spectrum Phi(k). The two fieldsare uncorrelated, i.e., 〈Qlo(k)Qhi∗(k′)〉= 0. (2.2)The field Q˜lo(x) will correspond to the isocurvature, tensor, or modulatedadiabatic component, while Qhi(x) will be the remaining, unmodulated adi-abatic component. The superscripts “lo” and “hi” refer to the fact thatgenerally these components will dominate at low and high k, respectively.Strictly, we should consider only amplitudes A ≤ 1, since for larger A thefluctuations in Eq. (2.1) will vanish somewhere inside the last scatteringsurface and the details in this case may depend on the specific (presumablyinflationary) realization of the model.As shown in Chapter 1, the total temperature anisotropies will be givento very good approximation by the sum of the uncorrelated contributionsfrom the modulated and unmodulated fluctuations, i.e.,δT (nˆ)T0=δT lo(nˆ)T0(1 +Anˆ · dˆ)+δT hi(nˆ)T0. (2.3)scales, contrary to the observations (Hansen et al. 2009; Hanson and Lewis 2009; Flenderand Hotchkiss 2013; Planck Collaboration XVI 2016; Quartin and Notari 2015; Aiola et al.2015).592.2. Modulation approachThe anisotropies δT lo/T0, with power spectrum CT,loℓ , are generated by theperturbations with power spectrum P lo(k), while δT hi/T0, with spectrumCT,hiℓ , are generated by the uncorrelated perturbations with power spectrumPhi(k). The form of Eq. (2.3) is easy to understand in the limit where theanisotropies are much smaller than the length scale of modulation (i.e., rLS).The large-scale case is less obvious, but Eq. (2.3) still holds to very goodapproximation (recall Chapter 1). We have ignored the ISW effect here,since it would introduce a negligible modification (see Chapter 1).For the E-mode polarization anisotropies, we similarly useδE(nˆ)T0=δElo(nˆ)T0(1 +Anˆ · dˆ)+δEhi(nˆ)T0. (2.4)Due to the effects of reionization and the non-local definition of E modes,Eq. (2.4) becomes inaccurate on the very largest scales, ℓ . 10. In Ap-pendix B, we derive the effect of a spatially linear modulation on the E(and B) modes, taking the non-locality into effect. We find that omittingthis correction results in a bias to our recovered modulation amplitudes ofroughly 3% in the worst case (the most red-tilted power law model we con-sider). Hence we do not apply this correction here (but plan to implementit in future work). In Appendix B, we also show that the correct treatmentresults in novel couplings between B modes and E or T modes for a lin-ear modulation. The very-large-scale nature of the corrections implies weakstatistical weight, but these unusual couplings could in principle be used toconstrain large-scale signals.In terms of spherical harmonic coefficients, the total fluctuations inEqs. (2.3) or (2.4) can be written asaℓm = aloℓm + ahiℓm +∑M∆XM∑ℓ′m′aloℓ′m′ξMℓmℓ′m′ , (2.5)where aloℓm are the statistically isotropic modes, and the ∆XM are the spher-ical harmonic decomposition of Anˆ · dˆ (the dipolar nature ensures thatM = −1, 0, 1). The ξMℓmℓ′m′ are coupling coefficients defined byξMℓmℓ′m′ ≡√4π3∫Yℓ′m′(nˆ)Y1M (nˆ)Y∗ℓm(nˆ)dΩnˆ, (2.6)and given explicitly byξ0ℓmℓ′m′ = δm′m (δℓ′ℓ−1Aℓ−1m + δℓ′ℓ+1Aℓm) , (2.7)ξ±1ℓmℓ′m′ = δm′m∓1 (δℓ′ℓ−1Bℓ−1±m−1 − δℓ′ℓ+1Bℓ∓m) , (2.8)602.2. Modulation approachwhereAℓm =√(ℓ+ 1)2 −m2(2ℓ+ 1)(2ℓ+ 3), (2.9)Bℓm =√(ℓ+m+ 1)(ℓ+m+ 2)2(2ℓ+ 1)(2ℓ+ 3). (2.10)From Eq. (2.5) we can find the covariance of the total temperature orpolarization anisotropy multipoles to first order in the modulation amplitude∆X ≡√∑M |∆XM |2 = A:Cℓmℓ′m′ ≡ 〈aℓma∗ℓ′m′〉 (2.11)= Cℓδℓℓ′δmm′ +δCℓℓ′2∑M∆XMξMℓmℓ′m′ , (2.12)where δCℓℓ′ ≡ 2(C loℓ + C loℓ′ ). The above equation explicitly shows that adipole modulation will lead to coupling of ℓ to ℓ± 1 modes in the multipolecovariance (Prunet et al. 2005). In Eq. (2.12) Cℓ is the the total isotropicpower spectrum, which, since the two fluctuation components are uncorre-lated, is given to linear order in the asymmetry byCTℓ = CT,loℓ + CT,hiℓ , (2.13)and similarly for polarization. Clearly, CTℓ must be consistent with mea-surements of the isotropic power. For the adiabatic cases, we takeP loR(k) + PhiR(k) = PΛCDMR (k), (2.14)wherePΛCDMR (k) = As(kk0)ns−1(2.15)is the comoving curvature power spectrum for ΛCDM, so that the isotropicpower constraints are automatically satisfied. However, constraints on isocur-vature and tensor isotropic contributions (Planck Collaboration XX 2016;BICEP2/Keck Collaboration et al. 2015; Ade et al. 2016) should be incor-porated in addition to the constraints from the temperature asymmetry inorder to obtain the tightest constraints for those models (see Chapter 3).612.2. Modulation approach2.2.2 Adiabatic modulationtanh spectrumFor scalar adiabatic modes (recall Chapter 1) the fluctuation fields Qlo(x)and Qhi(x) both correspond to the comoving curvature perturbation, R.Our first specific model is intended to capture a large-scale modulation witha small number of parameters. We choose a modulated component spectrumof the formP loR(k) =12PΛCDMR (k)[1− tanh(ln k − ln kc∆ ln k)]. (2.16)This smooth step function in k ensures that mainly the largest scales, k .kc, are modulated. The quantity ∆ ln k determines the sharpness of thetransition from modulated to unmodulated scales. The other parametersof the model are the amplitude of modulation (A = AR) and its direction(l, b) in Galactic coordinates. The unmodulated contribution is fixed byEq. (2.14). It must be noted that the temperature data alone are not strongenough to constrain all five modulation parameters (kc, ∆ ln k, AR, l, b); seeChapter 1. The shapes of the best-fit asymmetry spectra for the tanh model(and all of the others) will be illustrated in Sect. 2.4.1.As shown in detail in Chapter 1, the fact that we have split the adiabaticfluctuations into two uncorrelated parts does not restrict the modulationmechanism (presumably inflationary) in any way. Instead this is a conve-nient way to describe an adiabatic spectrum modulated with arbitrary scaledependence.Power-law spectrumThe tanh modulation model in Eq. (2.16) is not explicitly motivated by anyearly-Universe model. Perhaps better motivated would be a simple power-law modulation, i.e.,P loR(k) = PΛCDMR (klo0 )(kklo0)nlos −1, (2.17)where nlos is the tilt and klo0 is the pivot scale of the modulated componentof fluctuations. This model will be abbreviated “ad.-PL”. We again imposeEq. (2.14) in order to define the unmodulated PhiR(k). We consider only redtilts with nlos ≤ ns, and choose klo0 = 1.5 × 10−4Mpc−1, which correspondsroughly to quadrupolar angular scales. Larger klo0 would contradict the622.2. Modulation approachpositivity of PhiR(k) on the largest observable scales, while smaller klo0 wouldbe degenerate with the modulation amplitude. Again we modulate P loR(k)with amplitude AR, so the total modulation fraction approaches AR on largeangular scales. This model also should be taken with some caution, since forlarge departures from scale invariance (i.e., 1 − nlos 6≪ 1), we might expecthigher-order terms to become relevant in the modulation spectrum.Modulated scalar spectral indexNext we consider a single-component adiabatic model with a linear gradientin the tilt, ns, of the primordial power spectrum (Moss et al. 2011). Weabbreviate this model “ns-grad”. In this case we do not strictly follow thetwo-component formalism of Sect. 2.2.1, but instead can directly write theasymmetry spectrum as (Moss et al. 2011)C loℓ = −12dCΛCDMℓdns. (2.18)Here we have used a linear approximation for the effect of the gradient, whichwill be well justified by our results. We allow for free modulation amplitude,∆ns, and let the tilt pivot scale, k∗, vary. Note that this treatment isdegenerate with fixed pivot k∗ and additional modulation of the primordialamplitude, As. Since a modulation of tilt produces extra power on largescales in the −dˆ direction, we have included a minus sign in Eq. (2.18) sothat the best-fit modulation directions will be directly comparable to thoseof the other models.2.2.3 Tensor modulationThe possibility that a modulated tensor component is present is partic-ularly well motivated observationally (Dai et al. 2013; Scott and Frolop2014; Chluba et al. 2014), since the contribution of (near-scale-invariantor red-tilted) tensors is negligible at small scales. Tight constraints on thetensor-to-scalar ratio r will make it difficult to achieve sufficient modulation,however, via the isotropic power constraint (Chluba et al. 2014; Contreraset al. 2018).In this case we take the unmodulated component Qhi(x) to be adiabaticfluctuations with the standard ΛCDM form, Eq. (2.15). The modulatedcomponent Qlo(x) will be a scale-invariant tensor contribution, i.e.,Pt(k) = r0.05PΛCDMR (k0), (2.19)632.3. Dipole asymmetry estimatorwith r0.05 = 0.07 (this is the 95% upper limit from Ade et al. 2016) atpivot scale k0 = 0.05Mpc−1. The tensor modes are uncorrelated with theadiabatic fluctuations (note that even in the presence of such correlations,the scalar and tensor anisotropy power spectra would still be additive Zibin2014) and modulated in a scale-invariant way with amplitude A = At. Asbefore, the tensors produce anisotropy power CT,loℓ (and similarly for E).2.2.4 Isocurvature modulationAlso well motivated as a modulation model is the CDM isocurvature spec-trum, since it naturally contributes mainly at large angular scales for near-scale-invariant (or red-tilted) isocurvature modes (Erickcek et al. 2009). Inthis case we assume that the unmodulated Qhi(x) is an adiabatic contri-bution that takes the standard, ΛCDM form, Eq. (2.15). The modulatedpart Qlo(x) will be the isocurvature component, which we take to be scale-invariant, i.e.,PI(k) = α1− αPΛCDMR (k0), (2.20)where we use the same pivot scale k0 as for the tensors. The isocurvaturemodes are taken to be uncorrelated with the adiabatic fluctuations and mod-ulated in a scale-invariant way with amplitude A = AI . This isocurvaturespectrum then determines the modulated component of the CMB fluctu-ations, C loℓ . The isocurvature fraction, α, should properly be constrainedby the isotropic Planck likelihood (as will be done in Chapter 3), but herewe simply choose the Planck upper limit for uncorrelated, scale-invariantisocurvature, α = 0.04 (Planck Collaboration XX 2016).2.3 Dipole asymmetry estimatorIn this section we describe the estimator that we use to extract modulationparameters from data or simulations given a modulation model. This es-timator is applied in harmonic space, exploiting the fact that (to leadingorder in the anisotropy) dipole modulation is equivalent to the coupling ofℓ with ℓ± 1 modes, as we saw in Sect. 2.2.1.2.3.1 Connection to previous approachesThe estimator that we use was originally developed for temperature data (Mosset al. 2011; Planck Collaboration XVI 2016; Zibin and Contreras 2017), butcan equally be applied to polarization (subject to the caveats discussed in642.3. Dipole asymmetry estimatorSect. 2.2.1). Our current implementation includes new improvements to thetreatment compared with Planck Collaboration XVI (2016) and Chapter 1 inorder to account for the expectation of noisy polarization data. Differencesin the implementation of the estimator between this work and Chapter 1are outlined in Appendix C, while the consequent differences in tempera-ture results can be seen by comparing Table 2.1 with Table 1.1 (though thedifferences are not significant). Here we present a condensed description ofour estimator (the full details are in Appendix C of Planck CollaborationXVI 2016).We note that our estimator is essentially identical to that of Hansonand Lewis (2009) and can also be rewritten in terms of so-called BiPoSHcoefficients (Hajian and Souradeep 2003, 2006); however, the physical mo-tivation for this approach comes from Moss et al. (2011), where generalcosmological parameter modulations were explored. Our implementation todeal with masking and noise uses an inverse-variance filter on the spheri-cal harmonic coefficients that optimally account for the masking, as usedin Planck Collaboration XVII (2014), Planck Collaboration XV (2016), andPlanck Collaboration XVI (2016).2.3.2 Full-sky, noise-free caseFrom Eq. (2.12) it is clear that the multipole covariance can be decomposedinto an isotropic part and a small anisotropic part proportional to ∆XM .In other words, we can make the identification Cℓmℓ′m′ = CI + CA, withCI being the first term on the right-hand side of Eq. (2.12) and CA beingthe second term. The inverse covariance matrix can then be written asC−1I − C−1I CAC−1I (to linear order in the anisotropy). The best-fit ∆XMvalues are given by∆XM =d†C−1I (∂CA/∂∆XM )C−1I dTr[C−1I (∂CA/∂∆XM )C−1I (∂CA/∂∆XM )] , (2.21)for multipole vector d. These can be written more explicitly as∆X0 =6∑ℓm δCℓℓ+1C−1ℓ C−1ℓ+1Aℓma∗ℓmaℓ+1m∑ℓ δC2ℓℓ+1C−1ℓ C−1ℓ+1(ℓ+ 1), (2.22)∆X+1 =6∑ℓm δCℓℓ+1C−1ℓ C−1ℓ+1Bℓma∗ℓmaℓ+1m+1∑ℓ δC2ℓℓ+1C−1ℓ C−1ℓ+1(ℓ+ 1), (2.23)and ∆X−1 = −∆X∗+1. We note that Eq. (2.21) is completely general andcan be used to examine modulation of any kind (i.e., beyond simple dipolar652.3. Dipole asymmetry estimatormodulation). The cosmic variance of the modulation amplitude estimatoris given byσ2X ≡〈|∆XM |2〉 = 12(∑ℓ(ℓ+ 1)δC2ℓℓ+1CℓCℓ+1)−1. (2.24)As we mentioned in Sect. 2.2.1, we strictly only consider values of themodulation amplitude (AR, ∆ns, At, or AI) less than unity. Note, however,that the O(∆X2) terms which were dropped in the multipole covariance,Eq. (2.12), couple ℓ with ℓ and ℓ ± 2. Therefore the estimator, Eqs. (2.22)and (2.23), will be insensitive to them. In other words, our approach willrecover modulation amplitudes as large as unity without bias: we have noneed to restrict to small ∆X. However, the results will not be optimal, andcould be improved for large ∆X by incorporating these extra couplings intothe estimator.2.3.3 Realistic skiesThe estimators presented in Sect. 2.3.2 are only adequate for full sky cover-age with no noise. In this subsection we show how we include the effects ofmasking and noise in the data.The combination of C−1I d in Eq. (2.21) is suggestive that we should ap-ply an inverse-covariance filter to the data. This is exactly what we do.The effects of masking are readily dealt with by employing inverse-variancefiltering to the data, as described in Planck Collaboration XVII (2014),and Planck Collaboration XV (2016). Pixels within the mask are giveninfinite variance and thus are given zero weight, which optimally accountsfor masking effects. The effects of inhomogeneous noise could also readilybe handled by including its variance contribution in the inverse-variancefilter. However, we have not included these effects, which means we willhave a slightly suboptimal though still unbiased estimate. Residual effectsnot captured by our approach (e.g., inhomogeneous noise) are handled bysubtracting a mean field term, derived from simulations with the same fore-ground and noise properties as the data. We note that this same approachhas been used for lensing estimators in Planck Collaboration XVII (2014),and Planck Collaboration XV (2016) and specifically for temperature dipolemodulation in Planck Collaboration XVI (2016) and Chapter 1. Cut-skyaberration effects (Jeong et al. 2014) are taken into account by removingaberration from the data as described in Appendix E.662.4. ResultsThe estimator, Eqs. (2.22)–(2.23), then becomesX˜WZ0 =6∑ℓm δCWZℓℓ+1AℓmS(WZ)ℓm ℓ+1m+M∑ℓ(δCWZℓℓ+1)2(ℓ+ 1)F(Wℓ FZ)ℓ+1, (2.25)X˜WZ+1 =6∑ℓm δCWZℓℓ+1BℓmS(WZ)ℓm ℓ+1m+M∑ℓ(δCWZℓℓ+1)2(ℓ+ 1)F(Wℓ FZ)ℓ+1, (2.26)withSWZℓmℓ′m′ ≡W ∗ℓmZℓ′m′ − 〈W ∗ℓmZℓ′m′〉 . (2.27)Here,WZ = TT, TE,EE;Wℓm and Zℓm are inverse-covariance filtered data;FWℓ ≃ 〈WℓmW ∗ℓm〉; and the last term on the right-hand-side of Eq. (2.27)denotes the mean-field correction (details including the precise form of FWℓcan be found in Appendix A.1 of Planck Collaboration XV (2016) and Ap-pendix C of this thesis). The parentheses in the superscripts indicate sym-metrization over the enclosed variables.The estimators of Eqs. (2.25)–(2.26) can be combined with inverse-variance weighting over all data combinations (TT , TE, EE) to obtaina combined minimum-variance estimator, given by∆X˜M =∑WZ ∆X˜WZM(σWZX)−2∑WZ(σWZX)−2 . (2.28)Here we calculate the variance from the scatter of simulations, althoughthey agree closely with the Fisher errors given in Planck Collaboration XVI(2016).2.4 Results2.4.1 Temperature onlyIn this section we present constraints using Planck temperature data only.Specifically, we use the SMICA 2015 temperature solution, one of four Planckcomponent-separation methods (Planck Collaboration IX 2016), all of whichproduce very similar results (Planck Collaboration XVI 2016). The tem-perature data are evaluated up to a maximum multipole ℓmax = 1000 (nosignificant difference was found when extending to higher multipoles), withthe exception of the modulated ns model, which uses ℓmax = 2000.672.4. ResultsFig. 2.1 shows the posteriors of the modulation parameters for our adi-abatic models. Results for the tanh model have already been commentedon in Chapter 1; here we simply reiterate that the model is (unsurprisingly)not constrained well with temperature alone. The pile-up of the posteriorat low modulation amplitudes (present for all models, though most notablyfor the adiabatic power-law model) is a volume effect that arises due to ourchoice of prior, which expects uniform posteriors for the k-space parameters(ln kc,∆ ln k, ln k∗, nlos ) for statistically isotropic data.The best-fit modulation parameters for all models are presented in Ta-ble 2.1. We see that the modulation amplitudes required for the tensorand isocurvature models exceed unity, i.e., the best fits have At, AI > 1.This suggests that, for the case of tensor and isocurvature isotropic contri-butions at the current upper limits (r0.05 = 0.07 and α = 0.04), maximalmodulation (i.e., modulation amplitude unity) is insufficient to produce theobserved temperature asymmetry. For this reason we do not discuss thesemodels further here. This result will be addressed more quantitatively inChapter 3. Note that this conclusion for a specific isocurvature model wasoriginally made in Planck Collaboration XX (2016) and the difficulty of pro-viding sufficient modulation for tensor models was discussed in Dai et al.(2013); Scott and Frolop (2014); Chluba et al. (2014). Furthermore, thebest-fit value of ∆ns = 0.014 for the ns-grad model is tighter than the cor-responding value of ∆ns = 0.03 in Dai et al. (2013). This illustrates theconstraining power of the small-scale anisotropies in our analysis.In Fig. 2.2 we compare the ΛCDM power spectra to the asymmetryspectra, AC loℓ , for the temperature best-fit modulation parameters given inTable 2.1, for each of our models. Note that the best-fit TT asymmetryspectra correspond very roughly to 5% modulation in amplitude out to ℓ ≃60, as expected from the previous ℓ-space analyses of the asymmetry.It is worth reiterating that none of these models are currently favouredover base ΛCDM: the goodness of fit of these models to the asymmetry (andisotropic) data is discussed in detail in Chapter 3. Hence the interest herein pursuing polarization data to improve constraints and test for a physicalorigin to the asymmetry.2.4.2 Including polarizationE vs. T asymmetry spectraWe claimed in Sect. 2.1 that it was not reasonable to take the observedmultipole-space temperature asymmetry, e.g., 6% asymmetry to ℓ = 65,682.4. Results1.0 0.5 0.0 0.5nlos0123A0 1 2 3A102101100k* [Mpc1]1100×ns1100× ns0.20.4ln(k)103102kc [Mpc1]0.2A0.2ln(k)0.2AFigure 2.1: Marginalized posteriors for the adiabatic power-law (top lefttriangle plot), ns gradient (top right), and tanh (bottom) models, usingPlanck temperature data only. Dark and light blue regions enclose 68%and 95% of the likelihood, respectively. The black dashed curves representthe theoretical distributions of the parameters coming solely from cosmicvariance in statistically isotropic skies.and predict a 6% asymmetry in E to ℓ = 65. We demonstrate this explicitlyin this subsection, using the tanh model as an example. First, in Fig. 2.3 weillustrate the cosmic variance [calculated via Eq. (2.24)] for a measurementof the amplitude of tanh modulation versus cutoff scale kc, for ∆ ln k = 0.01(which corresponds closely to a step function in k-space, with modulation692.4. ResultsParameter tanh ad.-PL ns-grad tensors isocurvature103kc [Mpc−1] 7.45 · · · · · · · · · · · ·∆ ln k 0.5 · · · · · · · · · · · ·nlos · · · −0.09 · · · · · · · · ·k∗ [Mpc−1] · · · · · · 0.10 · · · · · ·∆X −0.065 −0.457 −0.011 −2.2 −1.2∆Y −0.044 −0.566 −0.008 −2.0 −1.0∆Z −0.035 −0.499 −0.004 −1.3 −0.8AR 0.086 0.882 · · · · · · · · ·∆ns · · · · · · 0.014 · · · · · ·At · · · · · · · · · 3.3 · · ·AI · · · · · · · · · · · · 1.8l [◦] 214 231 214 221 221b [◦] −24 −34 −17 −24 −28Table 2.1: Best-fit modulation parameters for the Planck temperature data,given the models described in Sect. 2.2.only for k < kc). It is clear that for most values of kc, the cosmic variance isconsiderably smaller for an EE measurement than for a TT measurement.In particular, this applies around the best-fit value, kc = 7.45×10−3 Mpc−1,where the EE standard deviation is smaller by a factor of nearly two thanthe TT value. This is in stark contrast to the naive ℓ-space expectation,where identical modulation cutoffs for TT and EE leads to identical cosmicvariance. However, we can also see that there are some values of kc for whichthe EE cosmic variance is comparable to or even larger than the TT value.To help understand these differences between TT and EE, we plot inFig. 2.4 asymmetry spectra CT,loℓ and CE,loℓ for the tanh model with ∆ ln k =0.01. For the case kc = 7.45×10−3Mpc−1 (i.e., the best-fit value), we can seethat the asymmetry spectra differ substantially between T and E (note thatthis effect is also visible in Figure 1 of Namjoo et al. 2015). In particular, wepredict substantially larger modulation, and hence lower cosmic variance, inE than in T , since CE,loℓ > CT,loℓ for all ℓ. The reason is that the transferfunctions from k- to ℓ-space are narrower for E than for T , and hence thestep in k-space is better resolved in E. On the other hand, for the casekc = 2 × 10−2Mpc−1 some fine k-space structure (a dip in this case) isresolved by polarization and leads to lower asymmetry power for EE andhence larger cosmic variance. This illustrates the necessity of working in702.4. Results101102103102101100101102103104TT [K2]CDMtanhad.-PLns-gradtensorsisocurvature101102103104103102101100101102TE [K2]101102103105104103102101100101102EE [K2]Figure 2.2: ΛCDM power spectra for TT , TE, and EE (top to bottompanels) compared to the best-fit asymmetry spectra, AC loℓ , to the Plancktemperature data (see Table 2.1), for the various models. The purple curvein the bottom panel is the noise power spectrum for a single FFP8 noiserealization. The best-fit TT asymmetry spectra give several-percent-levelasymmetry for ℓ . 100, as expected. Here Dℓ ≡ ℓ(ℓ+ 1)Cℓ/(2π).712.4. Results10-3 10-2[Mpc - 1]10-310-210-1Figure 2.3: Cosmic variance for a measurement of the amplitude of modu-lation for the tanh model (with ∆ ln k = 0.01) for TT (black curve) and EE(red). Polarization does considerably better than the naive ℓ-space expec-tation of identical cosmic variance for TT and EE.k-space rather than ℓ-space when testing physical models for modulation.Importantly, these results imply that an ideal polarization measurementcan improve on the TT modulation amplitude measurement error consider-ably better than the naive ℓ-space expectation of√2. In Fig. 2.5 we showthe expected improvement to the error bar on the amplitude of modulationin a known direction (σX) when adding Planck or cosmic-variance-limitedpolarization to Planck temperature, as a function of the modulation param-eters. (Recall that for temperature we use ℓmax = 2000 for the ns gradientmodel and ℓmax = 1000 for all others.) For the tanh model the dependenceon ∆ ln k is quite weak and so we have averaged over it and only showthe dependence on kc. We can see that even with Planck polarization (bluecurves), there are parameter values for which the addition of polarization de-creases the error bar by more than the naive expectation of√2. This, again,is due to the difference in the k-to-ℓ transfer functions between polarizationand temperature. That is, for the same P lo(k) modulation, polarizationmodes are more strongly modulated (on many scales) than temperature, as722.4. Results101 1020.00.20.40.60.81.0lo/Figure 2.4: Asymmetry spectra C loℓ for temperature (black curves) and E-mode polarization (red) for the tanh model of Sect. 2.2.2, with ∆ ln k =0.01 and kc = 7.45 × 10−3Mpc−1 (solid curves) and kc = 2 × 10−2Mpc−1(dashed). The same physical k-space modulation produces substantiallymore modulation of E than of T for the lower kc value, and conversely forthe higher kc value.we saw in Fig. 2.4. It is also worth noting that the variations in improvementfollow the peak structure of the EE power spectra, i.e., the first three peaksthat are above the noise level in Fig. 2.2. This is most clearly evident withthe tanh model. The expected improvement when adding cosmic-variance-limited polarization exceeds a factor of two for some models and parameterranges, substantially exceeding the naive value of√2.ConstraintsWe cross-check and validate our method through the use of FFP8 component-separated CMB+ noise simulations (Planck Collaboration XII 2016), maskedappropriately using the Planck 2015 common mask for polarization, withunmasked fraction fsky = 0.75. For our cosmic-variance-limited results weremove the noise in the polarization simulations and apply no mask. For allmodels we examine our polarization simulations up to a maximum multipole732.4. Results103102kc [Mpc1]1.21.41.61.82.0TTX/TTTEEEX0.4 0.2 0.0 0.2 0.4 0.6 0.8nlos1.301.351.401.451.501.551.601.65TTX/TTTEEEX102101100k* [Mpc1]1.01.21.41.61.82.02.2TTX/TTTEEEXFigure 2.5: Improvement in the error bar for a measurement of the amplitudeof modulation for the tanh, adiabatic power-law, and ns gradient models(top to bottom panels), assuming that the modulation direction is known,when Planck (blue curves) or cosmic-variance-limited (orange) simulatedpolarization data (to ℓmax = 1000) are added to Planck temperature. Thedependence on kc for the Planck case tanh model follows the peak structureof the EE power spectra relative to the noise (see Fig. 2.2, bottom) and canexceed the naive expectation of√2.742.4. Resultsℓmax = 1000. Throughout we will consider adding statistically isotropic oranisotropic polarization to Planck temperature data. In both scenarios wewill use the same set of FFP8 simulations, modified to either include the ap-propriate temperature-polarization correlation with the given temperaturedata (for the case of statistically isotropic polarization, see Appendix D.1),or to include the appropriate modulation for the specific model and param-eters considered (see Appendix D.2).First we consider the case that the asymmetry does not have a physicalorigin and is simply the result of fluctuations due to cosmic variance. In thiscase the polarization must be treated as statistically isotropic (apart fromthe necessary T -E correlation). We apply Bayesian parameter fitting tothe combination of temperature data and polarization simulations (treatedas if they were data). In Figs. 2.6–2.8 we show the marginalized poste-riors of the modulation parameters for temperature data alone (in black)and when adding statistically isotropic polarization data averaged over 500simulations and shown by the blue solid and dashed curves for Planck andcosmic-variance-limited polarization, respectively. In general the addition ofisotropic polarization data will act to spread out the posteriors with respectto the temperature-only constraints. However, we find that the addition ofstatistically isotropic polarization data increases the significance of a ≥ 3σtemperature result (here we mean with respect to the amplitude parameteronly) roughly 30% or 20% of the time for Planck or cosmic-variance-limitedpolarization, respectively (for the tanh model). This is mainly due to theinitial weakness of the temperature signal. Therefore, we urge caution wheninterpreting the addition of polarization data to temperature.Next we consider the case that the asymmetry is due to a real modula-tion, so the polarization will be statistically anisotropic with a precise formdetermined by the k-space modulation model. We generate modulated po-larization simulations (modulated with the temperature best-fit parametersgiven in Table 2.1) to combine with the temperature data to forecast thetype of constraints we expect to see in Planck and cosmic-variance-limiteddata if the modulation is real. We modify the existing FFP8 simulationsfollowing the procedure in Appendix D.2.Results are also summarized in Figs. 2.6–2.8, where the addition of mod-ulated polarization is shown with the orange solid and dashed curves forPlanck and cosmic-variance-limited polarization, respectively. The curvesplotted are the mean posteriors averaged over 500 simulations. We see thatin general the addition of polarization makes the data more constraining.However, these figures do not show how often we should expect to be able todistinguish modulated polarization from statistically isotropic polarization,752.4. Results103102kc [Mpc1]0.00.20.40.60.81.0P/Pmax0.1 0.2 0.3 0.4 0.5ln(k)0.00.20.40.60.81.0P/Pmax0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200A0.00.20.40.60.81.0P/PmaxFigure 2.6: Posteriors for kc, ∆ ln k, and AR (top to bottom panels) forthe tanh model for temperature alone (black curves) and temperature withisotropic (blue) and modulated (orange) polarization simulations for themodel parameters in Table 2.1. The posteriors using polarization have beenaveraged over 500 polarization realizations. Solid curves refer to Planck po-larization, while dashed curves refer to cosmic-variance-limited polarization.The parameter kc is typically somewhat more constrained in the modulatedthan in the isotropic polarization case, but cosmic variance is still significant.762.4. Results1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75nlos0.00.20.40.60.81.0P/Pmax0.0 0.2 0.4 0.6 0.8 1.0A0.00.20.40.60.81.0P/PmaxFigure 2.7: Posteriors for nlos and AR (top and bottom panels, respectively)for the adiabatic power-law model for temperature alone (black curves) andtemperature with isotropic (blue) and modulated (orange) polarization sim-ulations for the parameters given in Table 2.1. The posteriors using polariza-tion have been averaged over 500 polarization realizations. Solid curves referto Planck polarization, while dashed curves refer to cosmic-variance-limitedpolarization.772.4. Results102101100k* [Mpc1]0.00.20.40.60.81.0P/Pmax0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5100× ns0.00.20.40.60.81.0P/PmaxFigure 2.8: Posteriors for k∗ and ∆ns (top and bottom panels, respectively)for the ns gradient model for temperature alone (black curves) and tempera-ture with isotropic (blue) and modulated (orange) polarization simulations,for the parameters given in Table 2.1. The posteriors using polarizationhave been averaged over 500 polarization realizations. Solid curves referto Planck polarization while dashed curves refer to cosmic-variance-limitedpolarization.782.4. Resultswhich will necessarily be model dependent. We will be more quantitativeabout this in the following subsection.2.4.3 Distinguishing modulated from isotropic polarizationIf the signal seen in temperature has a physical cause then it is clearlytoo low in signal-to-noise to definitively distinguish from cosmic variancefluctuations. With the addition of polarization, however, we can assess howwell one could distinguish statistically anisotropic from statistically isotropicdata. With this in mind we define the quantity Oˆjk as the ratio of themaximum likelihood of model j to that of model k. For definiteness we willorder the models in the following way:0. ΛCDM;1. tanh model;2. adiabatic power-law model;3. ns gradient.For a modulation model j, the maximum likelihood is proportional to exp[A2/(2σ2X)]max,whereas for ΛCDM, it is proportional to exp[−A2/(2σ2X)]max. Therefore wecan compute Oˆj0 asOˆj0 ={exp[A2/(2σ2X)]}max{exp[−A2/ (2σ2X)]}max . (2.29)Note that Eq. (2.29) is related to the often used odds ratio for Bayesianmodel comparison, but without the Occam penalty factor (Gregory 2005).For this reason the meaning of the absolute value of Oˆj0 is irrelevant, thoughthe relative value between different data (modulated or statistically isotropic)is valuable. We will thus use Oˆj0 as a proxy for distinguishing modulateddata from statistically isotropic data by comparing to statistically isotropicor modulated simulations.For the data we will choose to add either statistically isotropic or mod-ulated polarization to the existing Planck temperature data. The resultsare shown in Figs. 2.9–2.11, where the relation between the colours and thetype of simulations are as follows:• red– statistically isotropic polarization added to temperature data;792.4. Results2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5ln[O10]10 20 30 40 50ln[O10]Figure 2.9: Histogram of the logarithm of Oˆj0 [defined by Eq.(2.29)] forthe tanh model using the Planck temperature data with 500 realizations ofstatistically isotropic (red outlines) or modulated (black outlines and greenfilled) polarization as described in the text. The top panel uses Planck polar-ization, while the bottom uses cosmic-variance-limited polarization. Largevalues of Oˆj0 relative to the isotropic histograms indicate that the modula-tion model should be preferred over ΛCDM.802.4. Results0 5 10 15 20 25ln[O20]0 5 10 15 20 25 30 35 40ln[O20]Figure 2.10: As in Fig. 2.9 except for the adiabatic power-law model.812.4. Results0 5 10 15 20 25ln[O30]0 10 20 30 40 50ln[O30]Figure 2.11: As in Fig. 2.9 except for the ns gradient model.822.4. Results• green– modulated polarization added to temperature data, where themodulation parameters are determined by randomly sampling the fulllikelihood of the temperature data;• black– modulated polarization added to temperature data using thebest-fit modulation parameters from Table 2.1.The figures show the histogram of the logarithm of Oˆj0. Large values ofOˆj0 relative to the isotropic-polarization simulations indicate that the mod-ulation model should be preferred over ΛCDM, and indeed the modulatedsimulations are clearly shifted to the right of the statistically isotropic sim-ulations, with the black histograms further to the right than the green. Alarge overlap in the distributions would indicate that it will be difficult todistinguish modulated from statistically isotropic data using polarization.We would like to assess the probability of a detection of modulationassuming that the polarization data are modulated as predicted by tem-perature. In Table 2.2 we indicate for each model the probability that Oˆj0is greater than that of 95% of isotropic (red) simulations—we will looselyrefer to this as a “2σ”-detection. The “best-fit” values refer to polariza-tion modulated with the temperature best-fit parameters from Table 2.1. Inthis case the probability of a 2σ detection with Planck polarization rangesfrom 27% for the adiabatic power-law model to 75% for the ns gradientmodel. For cosmic-variance-limited polarization these probabilities increasesubstantially, reaching 99% for the ns gradient model. However, as the“sampling” columns in Table 2.2 show, when we sample the modulation pa-rameters from the full temperature posteriors, the probabilities are reduced.We see that even in the optimistic scenario that the true modulationparameters are given by the best-fit values of the temperature likelihood,Planck has a low probability of distinguishing this from statistically isotropicdata. The exception is the case of the ns gradient model, which has a largetail out to very large Oˆ30 values even when simulated Planck polarizationdata are used. The situation improves considerably when cosmic-variance-limited polarization is used, with the tanh and ns-grad models being almostguaranteed to be detected in the scenario that the true modulation model isgiven by the best-fit temperature parameters. These large probabilities arediminished when we consider the case that polarization is instead modulatedwith parameters randomly sampling the likelihood of the temperature data,which is not constrained well. However, these “sampling” probabilities arethe best statements we can make about detectability of modulation giventhe temperature signal, and should be considered as our most conservative(with respect to detecting a modulation model) results.832.5. DiscussionModelPlanck CV-limitedbest fit sampling best fit samplingtanh 37% 23% 94% 75%ad.-PL 27% 20% 63% 45%ns-grad 75% 58% 99% 84%Table 2.2: Probability of a “2σ”-detection (as defined in Sect. 2.4.3) of a realmodulation as described by the model in the first column given modulatedPlanck or cosmic-variance-limited polarization. The “best fit” columns re-fer to modulating polarization with the best-fit values from the temperaturedata (see Table 2.1). The “sampling” columns refer to modulating polariza-tion using parameters chosen by randomly sampling the full likelihood of thetemperature data. The latter values are a more conservative approach tohow the polarization might be modulated and thus give smaller probabilitiesof detection.2.5 DiscussionIn this chapter we have applied to CMB polarization the formalism of Chap-ter 1, which describes the effect of a spatially linear modulation with arbi-trary scale dependence. We have used the statistically isotropic tempera-ture and polarization simulations provided by the Planck collaboration toestimate the decrease in uncertainty in the modulation parameters whenpolarization is added. We have also generated asymmetric polarization sim-ulations to see how well we could test the possibility that the modulationis a real, physical effect. We have characterized the probability of a “2σ”detection of our dipolar modulation models (introduced in Sect. 2.2) whenadding Planck or cosmic-variance-limited polarization data under the fol-lowing important assumptions: 1) the modulation model is correct (i.e., theprimordial fluctuations are actually modulated according to the model inquestion); 2) the polarization data are free of any relevant systematic ef-fects; and 3) the effective sky coverage is similar to what is available forthe Planck temperature data (though our noiseless polarization simulationsuse the full sky). We found that, for the case of Planck polarization, weexpect a probability of anywhere from 20% to 75% for such a 2σ detec-tion. For cosmic-variance-limited polarization, the probability increases tothe range 45% to 99%. We have shown that these results are considerablystronger than a simple ℓ-space analysis would predict, due to the ability ofpolarization to resolve k-space detail more sharply than temperature.842.5. DiscussionOur results are clearly strongly model dependent, with the ns gradientmodel being the most likely to be ruled out or confirmed. This is due tothe distinct scale dependence for this model, with substantial modulationat very small scales. This suggests that extending the polarization data toℓmax = 2000 would provide a decisive test of this model.5Furthermore, we found that the probability that adding statisticallyisotropic polarization spuriously increases the significance of a ≥ 3σ sig-nal (with respect to the amplitude) is 30% or 20% for Planck or cosmic-variance-limited polarization, respectively. This probability is large due tothe moderate strength of the modulation signal in temperature. Thereforecaution is warranted if polarization is found to increase the significance ofthe temperature signal.For a spatially linear modulation (as with our models) the addition ofpolarization is our best short-term hope at detecting such a signal. This isbecause in practice the surface of last scattering is the furthest distance wehave access to and thus perturbations sourced there would be modulatedwith a higher amplitude than observables such as lensing (see Chapter 1) orintegrated Sachs-Wolfe effect (as in Chapter 3), for example. In spite of thepoor ability of Planck polarization to address modulation, all is not lost, as aCMB-S4 (Abazajian et al. 2016) project or CORE (Delabrouille et al. 2016)should reach noise levels such that E-mode polarization is essentially cosmic-variance limited. This should provide a strong indication of the true natureof the dipole asymmetry signal, at least for some models (recall the final twocolumns of Table 2.2). Farther into the future, 21-cm measurements of thedark ages (z & 30) may be able to help in constraining the dipole modulationmodels considered here, due to the vastly larger number of modes accessibleto three-dimensional probes.5In addition, asymmetry constraints from large-scale structure surveys (Gibelyou andHuterer 2012; Ferna´ndez-Cobos et al. 2014; Baghram et al. 2014; Shiraishi et al. 2017; Zhaiand Blanton 2017) should be important for ns-grad. E.g., for our best-fit parameters, wefind a modulation amplitude of about 1.6% at k = 1Mpc−1, which is already in strongtension with the 95% upper limit based on quasar data in (Hirata 2009).85Chapter 3CMB temperature3.1 IntroductionThe standard six-parameter Λ cold dark matter (ΛCDM) cosmological modeldescribes the temperature fluctuations in the cosmic microwave background(CMB) radiation spectacularly well, as demonstrated by the WMAP satel-lite (Bennett et al. 2013), the Atacama Cosmology Telescope (Sievers et al.2013), the South Pole Telescope (Story et al. 2013), and, especially, thePlanck satellite (Planck Collaboration I 2016). Central assumptions in theΛCDM model are that the fluctuations are Gaussian and statistically ho-mogeneous and isotropic. Despite the success of the standard model, sev-eral “anomalies” have been noticed in the CMB, which apparently violatethese assumptions (for reviews, see Bennett et al. 2011; Planck Collabo-ration XXIII 2014; Planck Collaboration XVI 2016; Schwarz et al. 2016).The statistical significance of these anomalies is not very high, and is weak-ened substantially with a posteriori (look elsewhere) corrections (Zhang andHuterer 2010; Bennett et al. 2011; Bunn 2010; Planck Collaboration XVI2016) when those are well defined.Probably the most intriguing of the anomalies is a roughly 6% dipolar orhemispherical asymmetry in the large-scale CMB temperature fluctuationpower, first noted in the WMAP one-year data (Eriksen et al. 2004). Lateranalyses showed that the asymmetry is substantially reduced on multipolescales ℓ & 100 (Hansen et al. 2009; Hanson and Lewis 2009; Flender andHotchkiss 2013; Planck Collaboration XVI 2016; Quartin and Notari 2015;Aiola et al. 2015). The significance of the asymmetry is only at the 3σ level,and is sensitive to a posteriori choices in the maximum multipole scale (Ben-nett et al. 2011; Planck Collaboration XVI 2016), so it should perhaps notbe considered a great surprise. Nevertheless, an origin to the asymmetry asa physical modulation of the primordial fluctuations would clearly be of fun-damental importance for cosmology, and in particular might provide infor-mation about inflation, given the large-scale nature of the effect. Thereforeit is worthwhile to investigate possible physical explanations. Since concreteinflationary models for modulation are difficult to construct (Byrnes et al.863.2. Formalism2016a), we consider phenomenological models in this study.While most studies of the dipolar asymmetry have been performed inangular multipole space, any physical model will necessarily be describedbest in position (or k) space. In Chapter 1 we developed a formalism fordescribing a spatial modulation and its effect on CMB temperature anisotro-pies, and for performing Bayesian estimation of the modulation parameters.This formalism was crucial for answering an important question: what doesa modulation that fits the temperature data predict for other observations,such as CMB lensing and polarization (see Chapters 1 and 2 respectively)?Given the inconclusive significance level of the asymmetry, probes of modesindependent from CMB temperature may be essential in order to confirmor refute a physical origin to the asymmetry. Our formalism is an extensionof an approach to describe the effects in the CMB of gradients in cosmolog-ical parameters (Moss et al. 2011). The effects of various such parametergradients were discussed in Dai et al. (2013).In this chapter we apply our formalism from Chapters 1 and 2 for thefirst time to determine whether any models for modulation can already beruled out. To do this we point out that some models necessarily increasethe statistically isotropic temperature power over ΛCDM, and so the or-dinary power spectra can be considered as “independent probes” to test aphysical origin for the asymmetry. We consider purely phenomenologicalmodulations of the ordinary adiabatic fluctuations, as well as a gradient ofthe scalar spectrum tilt and modulations of tensor and isocurvature con-tributions, in doing so testing several of the models discussed in Dai et al.(2013). Modulated isocurvature modes were studied in Hanson and Lewis(2009), and, in the context of a particular inflationary model Erickcek et al.(2009), in Planck Collaboration XX (2016). In the process we also provideconstraints on unmodulated tilted tensor and isocurvature modes using thelatest data.3.2 FormalismOur goal is to construct physical, position-space models for a temperaturedipolar asymmetry, which is confined mostly to large scales. We apply theformalism developed in Chapters 1 and 2, which captures scale dependenceby employing two fluctuation components. The first, Q˜lo(x), is restrictedmainly to large scales (low k) and is maximally linearly spatially modulated:873.2. FormalismQ˜lo(x) = Qlo(x)(1 +x · dˆrLS), (3.1)where Qlo(x) is statistically isotropic with power spectrum P lo(k), dˆ is thedirection of modulation, and rLS is the comoving distance to last scatter-ing. The second component, Qhi(x), is statistically isotropic with spectrumPhi(k). The two fields are taken to be uncorrelated, i.e., 〈Qlo(k)Qhi∗(k′)〉 =0. We attempt to be agnostic as to the origin of the modulation; the isotropicQhi component is adiabatic, while for the modulated component, Qlo, weconsider adiabatic, CDM isocurvature, and tensor fluctuations.Note, importantly, that while we take the first fluctuation component tobe maximally modulated according to Eq. (3.1), the amount of modulationin the total sky will be determined by a free amplitude parameter, A, insidethe definition of P lo(k). This convention differs from that used in Chapters 1and 2, in which the modulation amplitude parameter A appeared explicitlymultiplying the dipole term in Eq. (3.1). Nevertheless, the two conventionsare equivalent in terms of observable quantities.The total temperature anisotropies due to these two fields will be to avery good approximation (recall Chapter 1)δT (nˆ) = δT lo(nˆ)(1 + nˆ · dˆ)+ δT hi(nˆ), (3.2)where δT lo, with power spectrum C loℓ (called the “asymmetry spectrum”), isproduced by P lo(k), while δT hi, with spectrum Chiℓ , is produced by Phi(k).These anisotropies lead to the lowest-order spherical harmonic multipolecovariance (Prunet et al. 2005; Moss et al. 2011; Planck Collaboration XVI2016; Zibin and Contreras 2017)〈aℓma∗ℓ′m′〉 = Cℓδℓℓ′δmm′ +δCℓℓ′2∑MdMξMℓmℓ′m′ , (3.3)where δCℓℓ′ ≡ 2(C loℓ +C loℓ′ ) and dM is the spherical harmonic decompositionof nˆ · dˆ. The coefficients ξMℓmℓ′m′ couple modes ℓ to ℓ± 1:ξMℓmℓ′m′ ≡√4π3∫Yℓ′m′(nˆ)Y1M (nˆ)Y∗ℓm(nˆ)dΩ. (3.4)Crucially, the modulated component will also contribute to the totalisotropic power, viaCℓ = Cloℓ + Chiℓ . (3.5)883.3. ModelsTherefore a model that produces sufficient asymmetry to fit the temper-ature data may overproduce isotropic power at large scales and hence beinconsistent with experiments such as Planck.3.3 ModelsWe employ the same models as described in Chapter 2 to describe a large-scale modulation. First, we consider the adiabatic tanh model, with k-spaceasymmetry spectrumP lo(k) = Atanh2PΛCDM(k)[1− tanh(ln k − ln kc∆ ln k)], (3.6)wherePΛCDM(k) = As(kk0)ns−1(3.7)describes the usual ΛCDM power-law primoridal comoving curvature per-turbation spectrum. The parameters ∆ ln k and kc describe the width andposition of a small-scale cutoff and Atanh ≤ 1 is the amplitude of the mod-ulation. Next we consider an adiabatic power-law model (abbreviated “ad.-PL”):P lo(k) = APLPΛCDM(klo0 )(kklo0)nlos −1, (3.8)where nlos and APL ≤ 1 are the modulation tilt and amplitude, and klo0 =1.5× 10−4Mpc−1 is a pivot scale. For both of these adiabatic models we fixPhi(k) via the constraintP lo(k) + Phi(k) = PΛCDM(k) (3.9)(and hence C loℓ + Chiℓ = CΛCDMℓ ), so that the isotropic power is automati-cally consistent with standard ΛCDM. This constraint will be satisfied, forexample, in the scenario of Byrnes et al. (2016b).Next we consider a single-component adiabatic model with a linear gra-dient in the tilt, ns, of the primordial power spectrum (“ns-grad” for short).In this case we can directly write the asymmetry spectrum asC loℓ = −∆ns2dCΛCDMℓdns. (3.10)Here we have used a linear approximation for the effect of the gradient,which will be well justified by our results. The modulation amplitude is893.4. Modulation estimatorspecified by the increment in tilt, ∆ns, from modulation equator to pole.Note that this modulation will depend implicitly on the pivot scale for As.Finally we consider three models that naturally produce contributionson large scales. The first is a modulation of the standard ΛCDM inte-grated Sachs-Wolfe (ISW) contibution with amplitude AISW ≤ 1.6 Thisphenomenological model automatically satisfies isotropic CMB constraintsand C loℓ is simply the contribution of the ISW effect to the total power Cℓ.The second is a modulated CDM density isocurvature component,P lo(k) = αk∗1− αk∗PΛCDM(k∗)(kk∗)nI−1, (3.11)and the third is a modulated tensor component,P lo(k) = rk∗PΛCDM(k∗)(kk∗)nt. (3.12)In these latter two cases the models are described by two parameters, a pri-mordial power ratio (αk∗ or rk∗ , evaluated at scale k∗ = 0.002Mpc−1) anda tilt (nI or nt). Since these components are maximally modulated, thesepower ratios also determine the modulation amplitudes. For both isocurva-ture and tensor models we set Phi(k) = PΛCDM(k), so that, with respect toisotropic power, we simply have ΛCDM plus isocurvature or tensor modes.This will give us an extra constraint for these models over the adiabaticcases. This is reasonable since it would require a very contrived adiabaticscalar large-scale power deficit that, when combined with the isocurvatureor tensor spectrum, resulted in the usual ΛCDM spectrum. For the ten-sor model we also consider an unmodulated isocurvature component that isfully (anti-)correlated with the adiabatic scalars. Anti-correlated isocurva-ture modes would decrease power on large scales, potentially allowing fora larger contribution of modulated tensors. This inclusion adds one extraparameter, which is simply the amplitude of perturbations for the new mode.3.4 Modulation estimatorFor a full-sky, noise-free measurement of the temperature multipoles, wecan write down an estimator for the modulation amplitude ∆XM ≡ AdM6Note that the ISW effect is sourced over a wide range of distances, so it is unlikelythat a position-space modulation could result in maximal ISW modulation, i.e., AISW = 1.Therefore our results will be conservative, in that a realistic ISW modulation would likelyproduce less asymmetry.903.5. Resultsas (Moss et al. 2011; Planck Collaboration XVI 2016; Zibin and Contreras2017)∆XˆM =14Aσ2X∑ℓmℓ′m′δCℓℓ′CℓCℓ′ξMℓmℓ′m′a∗ℓmaℓ′m′ , (3.13)where A = Atanh, APL, ∆ns, AISW, αk∗/(1− αk∗), or rk∗ , depending on themodel, and where the cosmic variance of the estimator is given byσ2X = 12A2(∑ℓ(ℓ+ 1)δC2ℓℓ+1CℓCℓ+1)−1. (3.14)The presence of noise and incomplete sky coverage modifies the above re-lations. We use a C-inverse filter approach that accounts for noise, and,optimally, for the mask (as described in Planck Collaboration XVII 2014;Planck Collaboration XV 2016). Masking and residuals in the data will in-duce a mean-field value for ∆XM that can be estimated with simulations.Further details of the full estimator we use can be found in Appendix C ofPlanck Collaboration XVI (2016).For fixed modulation parameters the maximum likelihood islnL =∑M∆Xˆ2M2σ2X. (3.15)We can then build the rest of the likelihood by sampling on a grid of valuesfor the k-space parameters (recall Chapter 1). For the tensor and isocurva-ture models we assign a uniform prior on A, in order to obtain consistencywith the isotropic likelihood results. For all other models we use a prioruniform in the individual ∆XM .3.5 ResultsOur dipole asymmetry constraints come from Planck TT data using theSMICA solution (Planck Collaboration IX 2016). The best-fit asymmetryspectra for all of our models are illustrated in Fig. 3.1, where we see theexpected large-scale character of the asymmetry. The corresponding fullposteriors for α0.002 and r0.002 and their tilts are shown in Fig. 3.2 (orangecontours), where we can see that large values of α0.002 or r0.002 are neededto explain the asymmetry. (Recall that the power ratios α0.002 and r0.002also fix the modulation amplitude for the case of maximal modulation inEq. (3.1).)913.5. Results101102103102101100101102103104(+1)CTT/2 [K2]CDMtanhad.-PLns-gradtensorsisocurvatureISWFigure 3.1: ΛCDM temperature spectrum compared to the best-fit asymme-try spectra, C loℓ , for the various models. The best fits correspond roughly toa 5–10% asymmetry for ℓ . 100, as expected, with the exception of the ISWmodulation, whose maximum amplitude (and shape) is fixed by ΛCDM.For the isocurvature and tensor models we can also obtain constraintsfrom the isotropic power spectra described in Table 3.1; we will refer to theseas isotropic constraints. These were obtained with a version of CosmoMC (Lewisand Bridle 2002) modified to accomodate uncorrelated isocurvature modes.For the isotropic constraints all six of the ΛCDM parameters were varied, inaddition to the isocurvature or tensor fractions. For these models Fig. 3.2also shows the isotropic posteriors for α0.002 and r0.002 and their tilts (bluecontours), as well as the joint constraints, with the assumption that theisotropic and asymmetry likelihoods are independent (recall that they arisefrom diagonal and off-diagonal elements of the multipole covariance, respec-tively). Fig. 3.2 shows that, for both isocurvature and tensor modulation,the joint constraints are inconsistent with the level of modulation preferredby the asymmetry data. In other words, the addition of the independentisotropy data has substantially reduced the “signal” seen in the asymmetrydata.923.5. ResultsModel Data setisocurvature Planck TT,TE,EE+lowPtensors Planck TT,TE,EE+lowP+lensing+BKPTable 3.1: Data sets used for the isotropic constraints. BKP refers to the BI-CEP2/Keck Array-Planck joint analysis (BICEP2/Keck Collaboration et al.2015).Note that in Fig. 3.2 we have assumed that the isocurvature and ten-sor contributions are maximally modulated, via Eq. (3.1). This allows usto directly compare the asymmetry and isotropic posteriors, but is also aconservative choice, because for less than full modulation the correspondingr and α values preferred by the asymmetry constraints would necessarilybe larger with larger uncertainties. This would increase the tension we findbetween asymmetry and isotropic constraints and increase the dominanceof the isotropic data in the joint constraints.In order to express the above graphical results quantitatively, and deter-mine which models are viable for explaining the original asymmetry signal,we will consider two quantities for each model. The first is the probability,P>3σ, that the data allow a modulation amplitude A that is at least 3 timeslarger than the cosmic variance σX . Note that the choice of the value 3 isarbitrary; however, if P>3σ is small then the model cannot source significantmodulation and can be ruled out, even if P>3σ being large is an insufficientcondition to prefer a modulation model over ΛCDM. The second quantitywe use is the maximum-likelihood amplitude of modulation compared tothe cosmic-variance value, A/σX . For both quantities σX is calculated forasymmetry only [via Eq. (3.14)].We present these quantities for the various model and data combinationsin Table 3.2. For the asymmetry data, both quantities are large (except forthe ISW model), which simply tells us that the models can produce theconsiderable asymmetry present in the data. However, in all cases the val-ues drop substantially when adding the isotropic data. This implies thateven maximally modulated tensor or isocurvature modes cannot source thelarge asymmetry signal (or can, but with very small probability) due totheir respective isotropic constraints (consistent with the result in Chlubaet al. 2014, for tensor modulation). If we attempt to hide the isotropic ten-sor temperature power by including an anti-correlated isocurvature modethe conclusions remain the same (see the row marked nt = 0∗ in Table 3.2).933.5. ResultsThis is due to the different shapes of the tensor and anti-correlated isocurva-ture power spectra, and not, for instance, to the nondetection of primordialB-modes in the BICEP2/Keck Array-Planck data. Therefore we expect thatin general, a modulation model for which (like the tensor and isocurvaturemodels) isotropic power is added will be unable to explain the dipolar asym-metry signal. The tanh, ad.-PL, and ns-grad models are of course unaffectedby the isotropic constraint and are thus still viable modulation models as faras CMB temperature is concerned. For the ISW model both P>3σ and A/σXare small: even for maximal modulation the standard ΛCDM ISW contri-bution cannot explain the observed asymmetry. Note that, via Eq. (3.15),the ratio A/σX is essentially the best-fit χ value, which shows that the tanhmodel (which has the most free parameters) gives the best fit.ModelAsymmetry Isotropic JointP>3σ[%] A/σX P>3σ[%] A/σX P>3σ[%] A/σXtanh 63.1 3.3 – –ad.-PL 32.4 2.5 – –ns-grad 36.3 2.7 – –ISW 0.0 1.2 – –nI free 32.2 3.2 1.5 0.06 0.5 0.03nI = 1 37.4 3.1 0.33 0.10 1.0 0.10nI = ns 39.6 3.1 0.073 0.09 0.24 0.03nt free 29.9 3.1 0.003 0.03 0.001 0.02nt = 0 37.4 3.1 0.000 0.48 0.000 0.63nt = 0∗ 37.4 3.1 0.000 0.31 0.000 0.49nt < 0 32.1 3.1 0.008 0.48 0.003 0.00Table 3.2: Percentage of the posterior for which the amplitude A exceeds3σX , i.e., P>3σ, as well as A/σX for the maximum-likelihood parameters,for different combinations of data. These quantify whether the model cansource significant asymmetry given the data, a necessary but not sufficientcondition for preferring the model over ΛCDM. The asterisk denotes theaddition of a fully anti-correlated isocurvature component.For our best-fit parameters, the ns-grad model induces a modulationamplitude of roughly 1.6% at k = 1Mpc−1. On such small scales thismodel should be vulnerable to constraints from large-scale structure sur-veys (Gibelyou and Huterer 2012; Ferna´ndez-Cobos et al. 2014; Baghramet al. 2014; Shiraishi et al. 2017; Zhai and Blanton 2017). Indeed, this mod-943.6. DiscussionModel Asymmetry Isotropic JointnI free α ≤ 0.092 α ≤ 0.031 α ≤ 0.038nI = 1 0.007 ≤ α ≤ 0.083 α ≤ 0.038 α ≤ 0.044nI = ns 0.008 ≤ α ≤ 0.086 α ≤ 0.038 α ≤ 0.046nt free r ≤ 0.28 r ≤ 0.08 r ≤ 0.09nt = 0 0.02 ≤ r ≤ 0.28 r ≤ 0.07 r ≤ 0.10nt = 0∗ 0.02 ≤ r ≤ 0.28 r ≤ 0.08 r ≤ 0.09nt ≤ 0 r ≤ 0.30 r ≤ 0.09 r ≤ 0.09Table 3.3: 95% CL (or upper limits) for the parameters r0.002 and α0.002 forvarious tensor and isocurvature models and data combinations.ulation amplitude is close to (or in excess of) the 95% upper limit based onquasar data in Hirata (2009), and so a rigorous joint analysis may alreadyrule this model out.In order to determine quantitatively the level of modulation allowed bythe full data we look at constraints on the r and α parameters for theisocurvature and tensor models (where we are able to use power spectra toprovide tighter constraints). In Table 3.3 we show the 95% CLs (or upperlimits where relevant) for r0.002 and α0.002 for the different combinations ofdata. While the general tensor and isocurvature models (where the tiltsare free to vary) show no strong detection with the asymmetry constraintsalone (in the sense that we can only quote upper limits), we see that theaddition of power spectrum data strongly constrains the amount of modula-tion allowed by the data. For models where the tilt is fixed and not allowedto vary, the modulation signal is more apparent; however, the addition ofisotropic constraints removes the signal to a similar degree. Note that theasymmetry constraints in Table 3.3 allow much larger values of r than α.This is due simply to the fact that identical primordial ratios of tensor- andisocurvature-to-adiabatic scalar fluctuations produce much larger isocurva-ture temperature fluctuations.3.6 DiscussionThe models we have examined fall into two general classes. In the first,the total statistically isotropic temperature power was constrained to matchthat of standard ΛCDM. Therefore the degree of modulation could be variedwithout spoiling the success of ΛCDM. In the second class, the modulated953.6. Discussioncomponent contributed extra power to the isotropic spectra. Our main con-clusion is that models in this latter class fail to provide sufficient modulationto explain the dipole asymmetry without producing too much large-scalestatistically isotropic power. Hence these models, which include modulatedtensor and uncorrelated isocurvature, can be ruled out as the source of thelarge-scale dipolar asymmetry.Models in the first class, however, can fit the asymmetry while maintain-ing the success of the ΛCDM isotropic spectra, and hence some cannot yetbe ruled out. One exception is a modulated ISW contribution, which cannotsource enough asymmetry to explain the signal in temperature. The scalartilt gradient model produces substantial modulation on small scales, and sois at risk from survey data. The surviving models are the phenomenolog-ical adiabatic modulation models. Of course the contrived nature of suchmodels should mean that ΛCDM is still preferred: they essentially addparameters to fit features in the data that may simply be random noise.Unfortunately a Bayesian model selection procedure would not provide anunambiguous Bayes factor for these models, since the modulation modelevidence is strongly driven by the parameter prior ranges, which are com-pletely undetermined. It will only be possible to confirm or refute thesemodels by comparing future observations with their predictions for probes(such as CMB polarization) which are sensitive to independent fluctuationmodes from CMB temperature (recall Chapter 2).963.6. Discussion0 1isotropicasymmetryjoint0 1n0.000.250.500.751.001.251.5010×0.0020.0 0.5 1.0 1.510×0.0021 0 1isotropicasymmetryjoint1 0 1nt0.00.10.20.30.40.5r0.0020.0 0.2 0.4r0.002Figure 3.2: Posteriors for α0.002 or r0.002 and tilt of the isocurvature (toppanel) and tensor (bottom) models. Contours enclose 68% and 95% of theposteriors. We have conservatively assumed maximal modulation, so thatthe vertical axes are also a measure of the level of modulation relative to theisotropic ΛCDM spectrum. We can see that the modulation allowed by theasymmetry constraints is reduced substantially when adding the isotropicconstraints.97Part IIParity symmetry98When discussing symmetries in physics, there are three in particular thatstand out. These are the discrete symmetries of charge (C) conjugation,parity (P) transformations, and time (T) reversals. These are in contrast tothe continuous symmetry discussed in Part I. Charge conjugation flips thecharges on all particles, while leaving neutral particles unchanged. Paritytransformations reverse the sign of coordinates around some point of origin,i.e., x → −x; when restricted to a plane this is likened to looking in amirror. Time reversal simply sets time running in the opposite direction,t→ −t. The CPT theorem states that under very minimal assumptions thecombination of CPT is an exact symmetry of nature. There is however, norestriction that nature cannot violate any symmetry individually and indeedparity violation has been found to be maximally violated in the Weak sector.It is worth noting that these symmetries (when relevant) need only besymmetries of the underlying physics and need not be respected when look-ing at the outcome of some specific experiment. To be more concrete, con-sider the example of gas collisions in the room you are sitting in. If youcould zoom in and observe the collision of any two gas molecules, you wouldhave a hard time telling if time was moving forward or backward. In factthe physics that underlies gas dynamics is invariant under time reversal andthus it would be impossible to tell the direction of time by looking at singlecollisions. Nevertheless if all the gas in the room were to all of a suddenmove to a compact corner of the room, you might suspect that the roomstarted out with all the gas in that corner and over time the gas expandedto fill the room but someone managed to switch the direction of time foryou! Indeed the tendency for gases to expand and fill their container is justthat, a tendency, an overwhelmingly probable event compared to gas in aroom moving to a specific corner. The distinction between a symmetry re-spected in the underlying physics versus being respected in the realizationof a particular experiment is important and can often be confused.This part of the thesis concerns parity transformations, which have along history in physics.It is sometimes convenient to consider the parity operator Pˆ, whichsymbolizes the act of performing a parity inversion. If we consider somearbitrary coordinate vector x, then the following relations applyPˆx = −x, (3.16)PˆPˆx = Pˆ2x = x. (3.17)The last relation simply states that applying the parity operator twice leavesthe vector unchanged (does nothing). If we decompose Pˆ into its eigen-99values then it is clear that there are at most two values, +1,−1. Objects thattransform with eigen-value +1 are said to have even-parity, while objectsthat transform with eigen-value −1 are said to have odd-parity. We havealready encountered some of these objects in the Introduction, namely thetemperature, E-, and B-modes. Reiterating Eq. (35)–(37), we havePˆaTℓm = (−1)ℓaTℓm, (3.18)PˆaEℓm = (−1)ℓaEℓm, (3.19)PˆaBℓm = (−1)ℓ+1aBℓm. (3.20)We see that T and E-modes are even parity for even ℓ and odd parity forodd ℓ, but more importantly is that corresponding ℓ modes are of oppositeparity to B-modes. This fact will be exploited in constructing an estimatorto look for parity violation.Electromagnetism is described by the Faraday tensorFµν = ∂µAν − ∂νAµ0 −Ex −Ey −EzEx 0 −Bz ByEy Bz 0 −BxEz −By Bx 0 . (3.21)The standard Lagrangian, a scalar quantity that can be used to generateequations of motion, for a free photon theory isLEM = FµνFµν = 2(B2 − E2). (3.22)Since the Lagrangian only depends on the magnitudes of the ~E and ~B-fieldsit is invariant under parity transformations, i.e., PˆLEM = LEM. Howeveran extra term can be added to this Lagrangian by considering the dualtensor F˜µν ≡ ǫµνρσFρσ, where ǫµνρσ is a perfectly antisymmetric object inits indices, with ǫ0123 = −1. If we consider the quantityFµνF˜µν = −4 ~B · ~E, (3.23)we obtain a pseudo-scalar object which is parity odd. This is because, inanalogy with E- and B-modes, the ~E and ~B-fields transform oppositelyunder parity flips. The term in Eq. (3.23) (mediated by some other field)is what will be explored in this part of the thesis, as I will show belowthat the CMB is sensitive to it. Furthermore it has motivations for beingcosmologically relevant for certain models, which I further describe below.100Explicitly we will be looking at effects of the term−φFµνF˜µν ≃ 2∂µφAνF˜µν . (3.24)Here the ≃ symbol denotes equality up to a total derivative. If ∂µφ isreplaced with a vector then this is called a Chern-Simons term. Below I willexplain how this term becomes cosmologically relevant if φ varies with timeduring the propagation of photons from last scattering to us. A φ varyingon very short time scales (constrained very well in laboratory experiments)then has little to no effect on cosmology. Thus, the constraints presentedhere are complementary to laboratory experiments on the same term.101Chapter 4Isotropic Birefringence4.1 IntroductionMeasuring the in vacuo rotation of the plane of polarization of photons isa way to test fundamental physics in the Universe. Such a rotation is sen-sitive to parity-violating interactions in the electromagnetic sector that arefound in extensions of the Standard Model of particle physics (Carroll 1998;Lue et al. 1999; Feng et al. 2005; Li et al. 2009). For example, extendingthe Maxwell Lagrangian with a coupling (scalar, Chern-Simons, etc.) toAνF˜µν ,7 impacts right- and left-handed photons asymmetrically. Thereforea photon at the last-scattering surface with linear polarization in one orien-tation will arrive at our detectors with its plane of polarization rotated dueto this coupling term. The amount of rotation, usually denoted α, is oftenreferred to as the cosmic birefringence angle. This rotation naturally mixesE- and B-modes of CMB polarization8 and generates T–B and E–B corre-lations that would be zero in the absence of parity violations. The cosmicmicrowave background (CMB) polarization is particularly useful for mea-suring such an effect, because even if the coupling is small, CMB photonshave travelled a large comoving distance from the last-scattering surface (al-most) completely unimpeded and thus the rotation could accumulate into ameasurable signal.This effect has previously been investigated using data from many CMBexperiments (Feng et al. 2006; Wu et al. 2009; Brown et al. 2009; Paganoet al. 2009; Komatsu et al. 2011; Hinshaw et al. 2013; Polarbear Collabo-ration et al. 2014; BICEP2 Collaboration et al. 2014; Kaufman et al. 2014;Naess et al. 2014; di Serego Alighieri et al. 2014; Zhao et al. 2015; Mei et al.2015; Gruppuso et al. 2015; Contaldi 2017; Molinari et al. 2016), and also bylooking at radio galaxy data (Carroll et al. 1990; Cimatti et al. 1993, 1994;7Here Aν is the photon field, and F˜µν is the dual of the Faraday tensor, defined to beF˜µν ≡ (1/2)ǫµνρσFρσ.8We use the customary convention used by the CMB community for the Q and UStokes parameters, see e.g. http://wiki.cosmos.esa.int/planckpla2015/index.php/Sky_temperature_maps.1024.2. Impact of birefringence on the CMB polarization spectraWardle et al. 1997; Leahy 1997; Carroll 1998; di Serego Alighieri et al. 2010;Kamionkowski 2010). Thus far all the constraints are compatible with nocosmic birefringence (see discussion in Sect. 4.7).In this chapter we employ Planck 2015 CMB data to estimate an isotropicα. The birefringence angle has already been constrained with Planck datain Gruppuso et al. (2015), using the publicly available 2015 Planck Likeli-hood (Planck Collaboration XI 2016). However, that work did not use T–Band E–B data, which are essential for determining the sign of α and forincreasing the constraining power. We include here T–B and E–B cross-correlations by considering an approach, based on pixel-space maps thatemploys stacked images of the (radial) Qr and Ur Stokes parameters.The chapter is organized as follows. In Sect. 4.2 we describe the effectthat cosmological birefringence has on the angular power spectra of theCMB. In Sect. 4.3 we provide details of the data and simulations that areconsidered in our analysis, which is described in Sect. 4.4. Results for ourtwo different methodologies are summarized and compared in Sect. 4.5.Section 4.6 contains a discussion of the systematic effects that are mostimportant for the observables considered. Finally, conclusions are drawn inSect. 4.7.4.2 Impact of birefringence on the CMBpolarization spectraBirefringence rotates the six CMB angular power spectra in the followingway (see Lue et al. 1999; Feng et al. 2006, for more details):C ′TTℓ = CTTℓ ; (4.1)C ′EEℓ = CEEℓ cos2 (2α) + CBBℓ sin2 (2α); (4.2)C ′BBℓ = CEEℓ sin2 (2α) + CBBℓ cos2 (2α); (4.3)C ′TEℓ = CTEℓ cos (2α); (4.4)C ′TBℓ = CTEℓ sin (2α); (4.5)C ′EBℓ =12(CEEℓ − CBBℓ ) sin (4α). (4.6)Here α is assumed to be constant (see Liu et al. 2006; Finelli and Galaverni2009; Li and Zhang 2008, and Chapter 5 for generalizations). In this chapterwe will consider only the above parametrization, where the primed C ′ℓ are theobserved spectra and the unprimed Cℓ are the spectra one would measure1034.3. Data and simulationsin the absence of parity violations. In principle the rotation angle α coulddepend on direction (with details dictated by the specific model considered),and one could measure the anisotropies of α. We do not employ this typeof analysis here, but focus on the simple case of an isotropic α (or the αmonopole).Isotropic birefringence is indistinguishable from a systematic, unknownmismatch of the global orientation of the polarimeters. However, specificbirefringence models may predict some angular dependence in α. Further-more, large angular scale polarization in α is sourced in the reionizationepoch, as opposed to the small scales which are formed at recombination(Komatsu et al. 2011; Gruppuso et al. 2015). This will inevitably producesome angular dependency in α (assuming that the birefringence angle isproportional to the CMB photon path) and this effect could in principlebe used to disentangle instrumental systematic effects (since photons thatscattered at the reionization epoch would have traveled less than the oth-ers). However, we focus here on smaller scale data, where the reionizationeffects are not important and therefore such a distinction is not possible. ForPlanck there is an estimate of the uncertainty of the possible instrument po-larization angle using measurements performed on the ground (Rosset et al.2010), as discussed further in Sect. 4.6. Unfortunately, in-flight calibrationis complicated by the scarcity of linearly polarized sources that are brightenough, with the Crab Nebula being a primary calibration source (PlanckCollaboration VIII 2016).Eqs. (4.1)-(4.6) include all the secondary anisotropies but the weak-lensing effect. Due the current precision of data (see the discussion in Gu-bitosi et al. 2014) we safely ignore the weak-lensing effect as it contributesa negligible error.4.3 Data and simulationsWe use the full-mission Planck (Planck Collaboration I 2016) component-separated temperature and high-pass-filtered polarization maps at HEALPix9(Go´rski et al. 2005) resolution Nside = 1024; i.e., we take the Commander,NILC, SEVEM, and SMICA solutions for T , Q, U , and E, fully described inPlanck Collaboration IX (2016) and Planck Collaboration X (2016), andavailable on the Planck Legacy Archive.10 The E-mode maps are calculatedusing the method of (Bielewicz et al. 2012, see also Kim (2011)). We use9http://healpix.sourceforge.net/10http://www.cosmos.esa.int/web/planck/pla1044.3. Data and simulationsthe common temperature and polarization masks at Nside = 1024, namelyUT102476 and UPB77, respectively. For the harmonic analysis we also use half-mission data provided by the SMICA component-separation pipeline, in orderto build our DEB-estimator from cross-correlations (Planck CollaborationIX 2016). No further smoothing is applied to any of the maps (althoughthis version of the data already includes 10′ smoothing in both temperatureand polarization).We note that there are known systematic effects in the polarization mapsreleased by Planck that have not been fully remedied in the 2015 release (seeSect. 4.6 for a full discussion on the main systematic effects relevant for thisanalysis). These issues include various sources of large angular scale arte-facts, temperature-to-polarization leakage (Planck Collaboration VII 2016;Planck Collaboration VIII 2016; Planck Collaboration XI 2016), and a mis-match in noise properties between the data and simulations (Planck Col-laboration XII 2016). In order to mitigate any large-angle artefacts, we useonly the high-pass-filtered version of the polarization data. We note that ne-glecting the large scales has little to no impact on our constraining power forα. We have also checked that temperature-to-polarization leakage (PlanckCollaboration XI 2016) has very little effect on our analysis (see Sect. 4.6.2);similar conclusions are reached in Planck Collaboration Int. XLVI (2016).We pay particular attention to the mis-characterization of the noise inthe polarization data. Given the recommendation in Planck CollaborationIX (2016) we restrict our analysis to cross-correlation and stacking methods,which are less sensitive to such noise issues (Planck Collaboration VIII 2016).The analysis does not require the use of simulations,11 since we only needa relatively crude noise estimate on the scales we work at and we use aweighting approach when stacking that is only dependent on the data (wehave also checked that our results are quite insensitive to the noise level ofthe data, see Sect. 4.6.1).We use realistic full focal plane (FFP8.1) simulations described in de-tail in Planck Collaboration XII (2016). These simulations are processedthrough the four Planck component-separation pipelines, namely Commander,NILC, SEVEM, and SMICA (Planck Collaboration IX 2016), using the sameweights as derived from the Planck full mission data. The CMB outputmaps are used to build the harmonic space estimators used in this work.For our harmonic space EB estimator we use the half-mission simulations11We explicitly checked that using simulations does not change the results, which issimply a consequence of the fact that the process of stacking means we are not verysensitive to the noise properties of the data.1054.4. Analysisprovided by the SMICA pipeline.The FFP8.1 fiducial cosmology corresponds to the cosmological param-eters ωb = 0.0222, ωc = 0.1203, ων = 0.00064, ΩΛ = 0.6823, h = 0.6712,ns = 0.96, As = 2.09 × 10−9, and τ = 0.065 (where ωx ≡ Ωxh2). Note thatwe perform the analysis for the birefringence angle by fixing the other cos-mological parameters to the values reported above. This seems to be a safeassumption, since in Gruppuso et al. (2015) it was shown that α is quitedecoupled from the other parameters, at least as long as CTTℓ , CTEℓ , andCEEℓ are considered; ΛCDM parameters are not expected to be constrainedmuch from CTBℓ and CEBℓ , contrary to models that explicitly break paritysymmetry.4.4 AnalysisWe follow the stacking approach first introduced in Komatsu et al. (2011),where they were able to constrain α by stacking polarization on temperatureextrema. Here we perform the same analysis, but also stack on E-modeextrema. Our analysis is performed in map space (although we must brieflygo to harmonic space for stacking on E-modes, as described in Sect. 4.6.1)and we show that stacking polarization on temperature extrema is sensitiveto the T–E and T–B correlations, while stacking on E-mode extrema issensitive to the E–E and E–B correlations.The recommendation on the use of polarization data from Planck Collab-oration IX (2016) is that only results with weak dependence on noise are tobe considered completely reliable. For the purposes of stacking on tempera-ture peaks only cross-correlation information is used, and thus understand-ing the detailed noise properties of polarization is unnecessary. Stacking onE-mode peaks the results do depend on the noise properties of the map;this is because the expected angular profiles of the stacks depend on thefull power spectrum of the map. In Sect. 4.6.1 we demonstrate that evena strong miscalculation of the noise would result in shifts at below the 1σlevel (and more reasonable miscalculations of the noise will bias results atan essentially negligible level).1064.4. Analysis4.4.1 Transformed Stokes parametersWe use the transformed Stokes parameters Qr, and Ur, first introduced inKamionkowski et al. (1997):Qr(θ) = −Q(θ) cos (2φ)− U(θ) sin (2φ); (4.7)Ur(θ) = Q(θ) sin (2φ)− U(θ) cos (2φ). (4.8)Here φ is defined as the angle from a local “east” (where “north” alwayspoints towards the Galactic north pole) direction in the coordinate systemdefined by centring on the hot or cold spot, and θ is a radial vector. Thetransformed Stokes parameters denote whether the polarization field is ra-dial or tangential (Qr) around a point, or if it is at 45◦ (Ur). The stackingprocedure tends to produce images with azimuthal symmetry, and hence thepredictions will only depend on θ. The theoretical angular profiles for stack-ing on temperature hot spots are derived in Appendix F (see also Komatsuet al. 2011; Planck Collaboration XVI 2016) and are explicitly given by〈QTr〉(θ) = −∫ℓdℓ2πW Tℓ WPℓ(b¯ν + b¯ζℓ2)CTEℓ J2(ℓθ), (4.9)〈UTr〉(θ) = −∫ℓdℓ2πW Tℓ WPℓ(b¯ν + b¯ζℓ2)CTBℓ J2(ℓθ). (4.10)The quantities W T,Pℓ are combinations of the beam (10′ smoothing) andpixel window functions (at Nside = 1024) for temperature and polarization.Below we will use WEℓ to denote the same quantity for E modes; however,the E modes are produced at the same resolution as temperature and soWEℓ =WTℓ . The bracketed term in each of Eqs. (4.9) and (4.10) incorporatesthe scale-dependent bias when converting the underlying density field totemperature or E modes (thus, they will differ if the stacking is performedon temperature or E-mode extrema). The function J2 is the second-orderBessel function of the first kind. Angular profiles derived from stacking on E-mode hot spots can easily be generalized from the above formulae by simplynoting that E modes share the same statistical properties as temperatureand thus we only need to change the power spectra in the above formulae.Thus the angular profiles for stacking on E-mode hot spots are given by〈QEr〉(θ) = −∫ℓdℓ2πWEℓ WPℓ(b¯ν + b¯ζℓ2)(CEEℓ +NEEℓ)J2(ℓθ), (4.11)〈UEr〉(θ) = −∫ℓdℓ2πWEℓ WPℓ(b¯ν + b¯ζℓ2)CEBℓ J2(ℓθ). (4.12)1074.4. AnalysisHere NEEℓ is the power spectrum of the noise in the E-mode map, and mustbe estimated from the data. The specific forms of bν (the scale-independentpart) and bζ (which is proportional to second derivatives that define thepeak) are given in Desjacques (2008) and Appendix F. The ΛCDM pre-diction for〈UT,Er〉is identically zero and thus we will find that the vastmajority of the constraining power comes from these profiles. We also showexplicitly in Eqs. (4.9)–(4.12) that Qr and Ur are sensitive to the T–E andT–B correlation when stacking on temperature extrema or the E–E andE–B correlation when stacking on E-mode extrema. Determination of thebias parameters depends on the power spectrum of the map where the ex-trema are determined (see Komatsu et al. 2011, and Appendix F), thus theydepend on the noise properties of the map, as well as the underlying powerspectrum. Section 4.6.1 will examine to what extent misunderstanding thenoise might bias the results.For our main results we have selected extrema using a threshold of ν = 0,which mean we consider all positive hot spots (or negative cold spots);however, we have checked other choices of threshold and found consistency,provided that we do not choose such a high a threshold such that the overallsignal becomes too weak. We do not claim that our analysis is optimal,and it may be that a better weighting exists for different levels of threshold;however, tests have shown that, in terms of minimizing the uncertainty on α,the choice of ν = 0 and use of averaged bias parameters is close to optimal.For the Planck temperature data we calculate the bias parameters tobe b¯ν = 3.829 × 10−3 µK−1 and b¯ζ = 1.049 × 10−7 µK−1. For the PlanckE-mode data we calculate b¯ν = (3.622, 3.384, 2.957, 3.332)×10−2 µK−1 andb¯ζ = (1.727, 3.036, 1.874, 3.039) × 10−7 µK−1 for Commander, NILC, SEVEM,and SMICA, respectively. The derivation of Eqs. (4.9)–(4.10) and a discussionof how to calculate all relevant quantities are given in appendix B of Komatsuet al. (2011), while the derivation of Eqs. (4.11)–(4.12) is given in Appendix Fof this thesis. The reader is referred to Komatsu et al. (2011) and alsoSection 8 of of Planck Collaboration XVI (2016) for a complete descriptionof the physics behind the features in the predicted stacked profiles.4.4.2 ProcedureWe begin by locating all local extrema12 of the temperature (or E-mode)data outside the region defined by the mask, i.e., either the union of temper-12As previously mentioned, we use all positive (negative) local maxima (minima) forhot (cold) spots. Extrema are defined by comparing each pixel to its nearest neighbours.1084.4. Analysisature and polarization common masks for stacking on temperature extremaor simply the polarization common masks when stacking on E-mode ex-trema. These masks remove the Galactic plane, as well as the brighter pointsources. We define a 5◦×5◦ grid, with the size of each pixel being 0.1◦and the number of pixels being 2500. When adding Q and U images, weweight each pixel by the number of unmasked Nside = 1024 pixels that lie ineach re-gridded pixel (which is not uniform, because of the re-gridding andmasking; this weighting is used in the estimation of the covariance matrixof the stacked images). Therefore the pixels near the centre generally havesomewhat lower noise in the final stacked image. We then generate QT,Erand UT,Er images using Eqs. (4.7) and (4.8).The predictions for Qr and Ur are found by combining Eqs. (4.9)–(4.12)with Eqs. (4.2)–(4.6):〈QTr〉(θ) = − cos (2α)∫ℓdℓ2πW Tℓ WPℓ(b¯ν + b¯ζℓ2)CTEℓ J2(ℓθ); (4.13)〈UTr〉(θ) = − sin (2α)∫ℓdℓ2πW Tℓ WPℓ(b¯ν + b¯ζℓ2)CTEℓ J2(ℓθ). (4.14)For stacking on E-modes we have〈QEr〉(θ) = −∫ℓdℓ2πWEℓ WPℓ(b¯ν + b¯ζℓ2)(CEEℓ cos2 (2α) +NEEℓ)J2(ℓθ), (4.15)〈UEr〉(θ) = −12sin (4α)∫ℓdℓ2πWEℓ WPℓ(b¯ν + b¯ζℓ2)CEEℓ J2(ℓθ), (4.16)where we have assumed that CBBℓ = 0, which is consistent with our data,since Planck does not have a direct detection of B-modes. We use a uniformprior on α, P (α), when sampling the likelihood, i.e.,P (α|d) ∝ P (d|α)P (α), (4.17)withP (d|α) = 1√2π|C|e− 12{d−(Qr,Ur)(α)}TC−1{d−(Qr,Ur)(α)}. (4.18)1094.4. AnalysisHere d represents the data, consisting of the stacked Qr and Ur images, and(Qr, Ur)(α) are the predictions as a function of α (see Eqs. 4.13–4.16). Thequantity C is the covariance matrix, which is a combination of the noise inthe data and the cosmic variance due to the limited number of hot (or cold)spots in the sky. We have estimated the covariance matrix by determiningan rms level from the pixelization scheme chosen and then weighting thiswith the inverse of the total number of pixels used in each re-gridded pixel;we have also assumed that the covariance is diagonal in pixel space.For the purposes of evaluating the likelihood, we have fixed CTEℓ andCEEℓ to the theoretical power spectra, based on the best-fit Planck param-eters (Planck Collaboration XIII 2016), and simply evaluate the likelihoodin a fine grid of α values. The choice of fixing the angular power spectrais reasonable because the usual cosmological parameters are determined byCTTℓ , CTEℓ , and CEEℓ , which are minimally affected by α (no dependence,quadratic, and still quadratic dependence on α, respectively). See also com-ments at the end of Sect. 4.3.Finally, we quote the mean of the posterior on α and the width of theposterior containing 68% of the likelihood as the best-fit and statisticaluncertainty, respectively. The posterior for α is sufficiently Gaussian thatthese two values contain all necessary information about the posterior.2 1 0 1 2[deg]21012[deg]QTr [µK]0.40.30.20.10.00.10.20.30.42 1 0 1 2[deg]21012[deg]UTr [µK]0.080.060.040.020.000.020.040.060.08Figure 4.1: Stacked images of the transformed Stokes parameters Qr (left)and Ur (right) for Commander temperature hot spots. The rotation of theplane of polarization will act to leak the signal fromQr into Ur. Note that theright plot uses a different colour scale to enhance any weak features. Finerresolution stacked images can be seen in Figure 40 of Planck CollaborationXVI (2016).1104.5. Results2 1 0 1 2[deg]21012[deg]QEr [µK]2.001.751.501.251.000.750.500.250.002 1 0 1 2[deg]21012[deg]UEr [µK]0.120.090.060.030.000.030.060.090.12Figure 4.2: Stacked images of the transformed Stokes parameters Qr (left)and Ur (right) for SMICA E-mode hot spots. The rotation of the plane ofpolarization will act to leak the signal from Qr into Ur. The quadrupolepattern in the right plot is related to “sub-pixel” effects (Planck Collabora-tion XV 2014; Planck Collaboration XI 2016); fortunately, our results areinsensitive to this feature, because it disappears in an azimuthal average(see Sect. 4.5).4.5 ResultsIn the following subsections we present our constraints on α, described inthe previous section. We will quote our best-fit α values and uncertainties(statistical only, leaving consideration of systematic effects to Sect. 4.6). Wewill show specifically that the E–B correlation is more constraining thanthe T–B correlation. This is expected and can be demonstrated directly bycomputing the variance of Eqs. (4.5) and (4.6). For small α, the variance ofα based on T–B and E–B information alone is(2ℓ+ 1)fsky(σTBℓ )2 ≃ 14CTTℓ CBBℓ(CTEℓ )2&14CBBℓCEEℓ, (4.19)(2ℓ+ 1)fsky(σEBℓ )2 ≃ 14CEEℓ CBBℓ(CEEℓ − CBBℓ )2≃ 14CBBℓCEEℓ, (4.20)respectively. This can be derived from the Fisher information matrix, wherethe covariance is a simple 1 × 1 matrix containing the variance of T–B orE–B. Thus, as suggested by the above relations, our results based on E–Bare generally more constraining than our T–B results (the presence of noise,however, will modify these relations).1114.5. ResultsFirstly we note that as a basic check we have verified that Fig. 4.1 closelyreproduces the stacked images shown in Planck Collaboration XVI (2016).We also show the Qr and Ur images stacked on E-mode extrema in Fig. 4.2.The visually striking quadrupole pattern in UEr appears to be an artefactof the pixelization scheme and is related to the so-called “sub-pixel” effectsdescribed in Planck Collaboration XV (2014) and Planck Collaboration XI(2016). This happens because the pixels of the stacked Q image are im-perfectly separated near the centre of the map (the stacked U image doesnot exhibit this imperfect mixing, because the pixel boundaries align per-fectly with where the profile changes sign). The pixelization errors are moreevident in the UEr image than the UTr because the individual QE and UEimages are strongly peaked near the centre of the image, and thus whengenerating the Ur stack imperfect subtraction leads to features in the centreof Fig. 4.2 (bottom). This effect has a non-diagonal influence on the powerspectra and thus has a negligible effect on parameters (Planck CollaborationXV 2014) and this analysis. Alternatively, since constraints on α come onlyfrom the radial part of the stacked images, the pixelization pattern seen inthe centre of Fig. 4.2, which cancels out in the azimuthal average, will notbias our α results (though it will contribute to the statistical uncertainty).Fig. 4.3 shows the binned Ur profiles for the four component-separationmethods. The binning here is chosen to pick out ranges with the same sign inthe predicted curve for α 6= 0◦ (with the ΛCDM prediction being identicallyzero); this choice is for visualization purposes only, since the statistical fit isperformed on the original stacked images, i.e., Figs. 4.1 and 4.2.Results are summarized in Table 4.1 and 4.2 for Commander, NILC, SEVEM,and SMICA. Table 4.1 contains the constraint on α based on the high-pass-filteredQ and U maps and their half-mission half differences (HMHD), whichgive a useful measure of the noise in the data. We present results based onstacking on temperature and E-mode extrema, both separately and com-bined. We have estimated that the correlation of the temperature and E-mode stacks are at the sub-percent level by looking at the amount of overlapin the positions of the peaks; thus we can safely neglect correlations in thecombined fit. We report 5–7σ detections for α (with respect to statisticaluncertainty only), however, this can be completely explained by a systematicrotation of our polarization-sensitive bolometers (PSBs) which we discuss inSect. 4.6. Null-test estimates all give α within 1σ of 0◦, with the exceptionof Commander results stacked on temperature, which are slightly above 1σ(see the second and eighteenth rows of Table 4.1). We have also checkedthat there is very weak dependence on our results coming from the differentchoices for the thresholds used to define the peaks.1124.5. ResultsMethod Hot Cold AllT–BCommander 0.36± 0.12 0.34± 0.11 0.35± 0.08HMHD −0.13± 0.12 −0.20± 0.11 −0.16± 0.08NILC 0.23± 0.10 0.36± 0.10 0.30± 0.07HMHD −0.08± 0.10 0.02± 0.10 −0.03± 0.07SEVEM 0.37± 0.12 0.18± 0.12 0.28± 0.08HMHD 0.07± 0.12 0.07± 0.12 0.07± 0.08SMICA 0.42± 0.10 0.36± 0.10 0.39± 0.07HMHD −0.04± 0.10 −0.04± 0.10 −0.04± 0.07E–BCommander 0.41± 0.11 0.44± 0.11 0.43± 0.08HMHD 0.03± 0.11 −0.07± 0.11 −0.02± 0.08NILC 0.33± 0.09 0.38± 0.08 0.35± 0.06HMHD −0.10± 0.09 0.01± 0.08 −0.05± 0.06SEVEM 0.28± 0.12 0.32± 0.12 0.30± 0.09HMHD 0.04± 0.12 0.04± 0.12 0.04± 0.09SMICA 0.25± 0.09 0.37± 0.09 0.31± 0.06HMHD −0.11± 0.09 0.01± 0.09 −0.05± 0.06CombinedCommander 0.38± 0.08 0.40± 0.08 0.39± 0.06HMHD −0.05± 0.08 −0.12± 0.08 −0.09± 0.06NILC 0.28± 0.06 0.37± 0.06 0.33± 0.05HMHD −0.10± 0.06 0.01± 0.06 −0.04± 0.05SEVEM 0.32± 0.08 0.25± 0.08 0.29± 0.06HMHD 0.05± 0.09 0.05± 0.08 0.05± 0.06SMICA 0.32± 0.07 0.37± 0.06 0.35± 0.05HMHD −0.08± 0.07 −0.01± 0.06 −0.04± 0.05Table 4.1: Mean values and (1σ) statistical uncertainties for α (in degrees)derived from the stacking analysis for all component-separation methods,coming from hot spots, cold spots, and all extrema. We include the fit fromeach component-separation method’s half-mission half-difference (HMHD)Q and U maps, as an indication of the expectation for noise. NILC and SMICAhave smaller uncertainties compared with Commander and SEVEM, which fol-lows from the naive expectation of the rms in the polarization maps (seeTable 1 of Planck Collaboration IX 2016).1134.6. Systematic effects0 1 2 3θ [deg]–0.50.00.51.01.5UT r[×10−2µK]CommanderNILCSEVEMSMICA0 1 2 3θ [deg]–1.0–0.50.0UE r[×10−2µK]CommanderNILCSEVEMSMICAFigure 4.3: Profiles of Ur from stacking on temperature (left) and E-mode(right) extrema for the four component-separation methods. The best-fitcurves for each component-separation method are also shown, with α valuesgiven in the fourth column of Table 4.1. Note that we have included bothhot and cold spots in this figure, i.e., we have co-added the negative of theprofile from cold spots to the profile of the hot spots. Error bars correspondto 68% confidence regions.A stacking analysis similar to ours has been attempted in Contaldi (2017)also using Planck data. Our results based on E–B data are consistent withthose of Contaldi (2017), but with smaller statistical uncertainties; however,we disagree with Contaldi (2017) regarding the constraints coming fromT–B data, which are claimed to be too noisy to be used. We show herethat both T–B and E–B data can be successfully exploited to constrain thebirefringence angle.4.6 Systematic effectsThe main systematic effect that is completely degenerate with the signalfrom isotropic cosmological birefringence is uncertainty in the orientation ofthe PSBs used for mapmaking (Pagano et al. 2009). The nature of this er-ror is characterized in the PCCS2 paper (Planck Collaboration XXVI 2016),as well as HFI (Planck Collaboration VII 2016; Planck Collaboration VIII2016) and LFI (Leahy et al. 2010; Planck Collaboration III 2014; PlanckCollaboration IV 2014) systematics papers, and also described in Planck1144.6. Systematic effectsMethod Hot: χ2 ∆χ2 Cold: χ2 ∆χ2T–BCommander 2453.7 −9.2 2769.8 −9.3NILC 2525.8 −5.2 2641.3 −13.3SEVEM 2552.6 −9.7 2718.9 −2.3SMICA 2567.7 −17.2 2610.1 −13.0E–BCommander 2548.8 −12.1 2542.7 −15.9NILC 2554.4 −13.3 2555.9 −19.3SEVEM 2551.5 −4.6 2552.0 −6.7SMICA 2556.8 −7.9 2559.1 −17.2Table 4.2: χ2 values for the model with α = 0, derived from the stackinganalysis for all component-separation methods. The ∆χ2 is the reductionof χ2 given the values of α in the corresponding entry in Table 4.1. Thenumber of degrees of freedom is 2500 coming from a 5◦× 5◦ patch with 0.1◦pixel size.Collaboration Int. XLVI (2016). The present upper limit in any globalrotation of the HFI detectors is estimated to be better than 0.3◦; how-ever, the relative upper limit between separate PSBs is 0.9◦ (Rosset et al.2010). After converting the above numbers into standard deviations (as-suming they are approximately uniform distributions, and noting that therelative uncertainty can be averaged over the eight PSBs used by Planck) weconservatively quote the total (global and relative) 1σ uncertainty as 0.28◦.This final error is not exactly Gaussian, although it is close (68% and 95%CLs are ±0.28◦ and ±0.55◦, respectively). Given that we detect a rotationof around 0.3◦, we are, therefore, unable to disentangle the signal found inthe data from the possible presence of this systematic effect. It remains tobe seen whether or not this can be improved in a future Planck release.It might be expected that Commander would perform best in polariza-tion in terms of noise and handling of systematics (based on the angularscales probed here, see Planck Collaboration IX 2016, for details). However,given that Commander uses a slightly different set of data than the othercomponent-separation methods and given that they all use different algo-rithms, we cannot make any definitive claims as to which gives the mostaccurate constraint. We are also unable to account for the apparent dis-1154.6. Systematic effectscrepancy at the roughly 2σ level (given the large number of comparisonsperformed here, this could simply be a statistical fluke) that the Commandernoise estimate yields for α when stacking on temperature (see Table 4.1).That being said, it is reassuring that all component-separation methodsagree at the ≃ 1σ level in their constraints on α.In the following subsections we mention some other possible systematiceffects that might be present, but that we believe contribute negligibly tothe polarization rotation signal.4.6.1 Noise properties of polarizationThe recommendation from the Planck collaboration is that any analysis per-formed on polarization data should not be very sensitive to mis-characterizationof the noise. To this end cross-spectra, cross-correlation, and stacking anal-yses are examples of such approaches. Our temperature tests fulfil this cri-terion explicitly. It is less obvious that stacking on E-mode extrema shouldonly weakly depend on the noise properties; however, we find this to be thecase. This is because the statistics of the E-mode map are encoded in thebias parameters (b¯ν , b¯ζ), which depend on the total power in the map (seeAppendix F for details). Therefore the bias parameters will be accurate tothe level that the statistics of the polarization data can be determined byits two-point function. Nevertheless we will now describe to what extent amiscalculation of the bias parameters will affect our results.We use the MASTER method (to correct for masking, Hivon et al. 2002) toestimate the total power spectrum of the E-mode map in order to calculatethe bias parameters (b¯ν , b¯ζ). The noise term in Eq. (4.11) is then givenby subtracting the theoretical power spectrum (CEEℓ ) from the total powerspectrum. The main effect of noise, however, comes from the determinationof the bias terms only, since most of the discriminatory power on α comesfrom the Ur stacks (Eqs. 4.12 and 4.16 do not explicitly depend on the noiseterm).The largest difference in our noise estimation when comparing betweendifferent component-separation methods comes from SEVEM and SMICA. Forthese maps b¯µ differs by approximately 20%, and b¯ζ by 40%. Using theSEVEM bias parameter values on the SMICA data (for example) leads to aroughly 1σ shift in the posterior of α (from E-modes). Such a discrepancyestimate is overly conservative however, because each component-separationtechnique will generally produce maps with different noise levels. If insteadwe scale our noise estimate by as much as 10% (for any of the individualcomponent-separation methods) we find that α shifts by less than 0.25σ. We1164.6. Systematic effectstherefore conclude that for our analysis, mis-characterization of the noise inpolarization has little to no effect.4.6.2 Beam effectsBecause of the differential nature of polarization measurements, any beammismatch or uncertainties can induce temperature-to-polarization leakage(Hu et al. 2003; Leahy et al. 2010). Here we are interested in beam uncer-tainties that can potentially lead to T–B and E–B correlations that mightmimic a non-zero α signal. Due to circular symmetry, effects from differen-tial beam sizes or differential relative gains will not tend to produce T–Bor E–B correlations, whereas effects from differential pointing and differen-tial ellipticity will. Differences in the noise level will also in general causetemperature-to-polarization leakage.We check for these effects following the approach described in PlanckCollaboration XI (2016) and Planck Collaboration XIII (2016). Note thattemperature-to-polarization leakage estimates due to bandpass mismatchesbetween detectors have been removed from the component-separated maps(see Planck Collaboration IV 2016; Planck Collaboration VI 2016; PlanckCollaboration VIII 2016; Planck Collaboration IX 2016, for details); weperform a crude scan of the parameter space in the following temperature-to-polarization leakage model (see also appendix A.6 of Planck CollaborationInt. XLVI 2016):CTEℓ → CTEℓ + ǫCTTℓ ; (4.21)CTBℓ → βCTTℓ ; (4.22)CEEℓ → CEEℓ + ǫ2CTTℓ + 2ǫCTEℓ ; (4.23)CEBℓ → ǫβℓCTTℓ + βCTEℓ . (4.24)The ǫ and β terms are expected to be dominated by m = 2 and m = 4modes (assuming the mismatch comes from differential ellipticity) and canbe written asǫ = ǫ2ℓ2 + ǫ4ℓ4, (4.25)β = β2ℓ2 + β4ℓ4. (4.26)Varying (ǫ2, β2), and (ǫ4, β4) in the range given by σ2 = 1.25 × 10−8, andσ4 = 2.7 × 10−15 (Planck Collaboration XI 2016), we find that α is stableto < 0.1σ (this is the case for both temperature and E-mode stacks).1174.7. ConclusionsWe must stress, however, that the above temperature-to-polarizationleakage model is not completely satisfactory (see section 3.4.3 and AppendixC.3.5 in Planck Collaboration XI 2016, for full details). Nevertheless it isadequate for our purposes, since we only wish to demonstrate that our resultsremain stable to most forms of beam mismatch.4.7 ConclusionsWe have estimated the rotation, α, of the plane of polarization of CMBphotons by using Planck 2015 data. We find values of 0.35◦, for the an-gle α (using SMICA data), with statistical uncertainty of 0.05◦ (68% CL),and subject to the systematic error of 0.28◦ (68% CL) due to the uncer-tainty in the global and relative orientations of the PSBs. Our results arecompatible with no rotation, i.e., no parity violation, within the total er-ror budget. We have demonstrated that our findings are robust againstdifferent component-separation methods, choices in peak thresholds, andtemperature-to-polarization leakage, at better than the 1σ statistical level.We have also carefully chosen our analyses to be insensitive to detailedknowledge of the noise properties of the polarization data.In Fig. 4.4 we show a comparison of our estimate with the birefringenceangle estimates provided by analysis on other CMB data in which, wherepossible, the total error budget is decomposed in these two parts, i.e. sta-tistical (left point of a pair) and systematic (right). The total error budgetof our estimate is dominated by the systematic uncertainty, which is a fac-tor of 6 larger than the statistical one. It is clear, therefore, that futureCMB polarization experiments (or a future Planck release) will require amuch better understating of their polarimeter orientations, since that is thecurrent limiting factor of this investigation. With a coordination of carefulground-based measurements and improved in-flight calibration on polarizedsources we may be able to further probe possible parity violations in theUniverse.1184.7. Conclusions-10-8-6-4-2024BOOM03QUaD100QUaD150WMAP9Bicep1POLARBEARBicep2ACTPolPlanck(SMICA)_[deg]Figure 4.4: Constraints on α coming from published analyses of several setsof CMB experimental data sets (shown in grey) as reviewed in Kaufmanet al. (2016), compared with what is found in the present chapter (in blue).For each experiment the left error bars are for statistical uncertainties at 68%CL, while right error bars (when displayed) are obtained by summing linearlythe statistical and systematic uncertainties. The error bar of BOOM03already contains a contribution from systematic effects, and Bicep2 did notconsider systematic uncertainties.119Chapter 5Anisotropic Birefringence5.1 IntroductionIt is well known that our Universe violates parity via weak-sector inter-actions. It is natural then to look for violations of parity in other sec-tors. Here we use the cosmic microwave background (CMB) to constraina Chern-Simons type parity violation in the electromagnetic sector (Carrollet al. 1990). In particular we focus on the effect of cosmic birefringence,which is the in vacuo rotation of the plane of polarization of photons. Suchan effect would occur in modifications to electromagnetism or from higher-dimensional operators in effective field theories, such as the axion-photoncoupling or in some quintessence models (Carroll 1998; Lue et al. 1999),or in models with new scalar degrees that are not quintessence (Xia 2012;Li et al. 2009; Pospelov et al. 2009; Finelli and Galaverni 2009; Caldwellet al. 2011), or models driven by a modified gravitational interaction (Fenget al. 2005). In these modifications the addition of a term to the standardLagrangian, which couples a new pseudo-scalar field (or vector field) to theelectromagnetic term FµνF˜µν (or AµF˜µν), would affect left-handed photonsand right-handed photons asymmetrically. This introduces a phase shift dif-ference between orthogonal polarization states that would manifest itself asa rotation of the total linear polarization. A spatially-varying pseudo-scalarfield would result in variations of this rotation angle, denoted by α, acrossthe sky. If such an effect were to exist, then it would be lost in any search forisotropic α. Hence it is interesting to map out these potential fluctuationsin α.A search for anisotropic birefringence is motivated because if, for exam-ple, the scalar field is dynamical, then it should have spatial fluctuationsthat would propagate as spatial variations in α (Pospelov et al. 2009; Li andZhang 2008; Caldwell et al. 2011). There are also models for which a uniformrotation α vanishes, in which case these models would only be detectablein a search for the anisotropies of α. If detected, the power spectrum of αwould give considerable insight on the nature of the source of birefringenceand new physics. In particular the sourcing field could in principle contain a1205.1. Introductionspecial direction (similar to birefringent crystals such as calcite or sapphire),which would impart a signal in a large-scale map of α. A final reason forsuch a study is a more practical one, namely that a uniform α is degeneratewith a systematic uncertainty in the orientation of the detectors, particularlypolarization-sensitive-bolometers (PSBs) used in CMB experiments (Leahy1997; Hu et al. 2003). Currently, measurements of a uniform rotation aresystematics dominated at the |α| . 0.◦3 level (recall Chapter 4). Searchesfor anisotropic α can therefore in principle improve on this constraint, sincea systematic uniform rotation would cancel out, along with the monopole inα.The CMB is particularly suited for measuring cosmic birefringence be-cause it is polarized and because CMB photons propagate over cosmolog-ical distances essentially unimpeded, where such a rotation could accumu-late into a detectable signal. CMB polarization is sourced by local densityquadrupoles (Hu and White 1997a) at the surface of last scattering, produc-ing linear Q- and U -type polarization. These quantities can be decomposedby their geometric properties into E- and B-mode polarization components,which are the gradient (electric parity) and curl (magnetic parity) modes ofthe polarization field on the sky (Zaldarriaga and Seljak 1997; Kamionkowskiet al. 1997; Hu and White 1997b). Under the assumption of parity conserva-tion the T–B and E–B correlations must be null, so that their measurementinforms us of parity violations.Cosmic birefringence has previously been constrained using CMB aniso-tropies from several experiments, most recently with Planck data, under theassumption of a uniform rotation. These results were found to be consis-tent with our expectation of no cosmic birefringence (recall Chapter 4 andreferences therein). In this chapter, we use the Planck 2015 (PR2) data toconsider anisotropies in α. While the possibility of an anisotropic α has pre-viously been addressed using data from other experiments (Gluscevic et al.2012; Gruppuso et al. 2012; Li and Yu 2013; Li et al. 2015; Lee et al. 2015;Ade et al. 2015; BICEP2 Collaboration et al. 2017), Planck data can pro-vide more stringent constraints, particularly on the largest angular scales.Along with the α power spectrum, we therefore constrain special directionsin α taking the form of a dipole or an M = 0 quadrupole in an arbitrarycoordinate system.This chapter is organized as follows. In Section 5.2 we explain the effectcosmic birefringence has on the CMB angular power spectra. In Section 5.3we describe the data and simulations used. In Section 5.4 we describe ournew map-space method used to estimate the angle α locally as a function ofdirection on the sky. In Section 5.5 we demonstrate the effectiveness of our1215.2. Impact of birefringence on the CMBestimator on a known input signal. In Section 5.6 we present the results forour baseline analysis pipeline. In Section 5.7 we search for possible sourcesof systematic effects that might affect our results, and we finally concludein Section 5.8.5.2 Impact of birefringence on the CMBA model of cosmic birefringence can be generated by including the followingterm in the electromagnetic Lagrangian:L = − β4MφFµνF˜µν − V (φ) ≃ β2M∂µφAνF˜µν − V (φ), (5.1)where Fµν is the electromagnetic field strength tensor and F˜µν its dual, β isa dimensionless coupling constant, M is a suppressing mass (usually takento be the Planck scale), and the potential V (φ) depends on the details ofthe model. The ≃ symbol here denotes equality up to a total derivativethat has no effect on dynamics. The interaction in Eq. (5.1) is exactly theform of the axion-photon coupling, while, for V = constant. the symmetryφ → φ + constant suppresses couplings to other Standard Model particles(Carroll 1998). The coupling of φ to FµνF˜µν treats left- and right-handedphotons asymmetrically, leading to a rotation in the plane of polarization asphotons propagate in vacuo. The amount of rotation is determined by thetotal change of the field ∆φ along the photon travel path and is given byα =β4M∆φ. (5.2)The existence of an angle α that is non-zero would be reflected in theStokes Q and U polarization parameters, which would be modified asQ′ ± iU ′ = e±2iα(Q± iU). (5.3)This effect induces T–B and E–B correlations that are otherwise expectedto be zero, along with smaller modifications to the parity-conserving corre-lations. Specifically, the observed power (primed quantities) in these cross-1225.2. Impact of birefringence on the CMBcorrelations would beC ′TTℓ = CTTℓ , (5.4)C ′EEℓ = CEEℓ cos2(2α) + CBBℓ sin2(2α), (5.5)C ′BBℓ = CEEℓ sin2(2α) + CBBℓ cos2(2α), (5.6)C ′TEℓ = CTEℓ cos(2α), (5.7)C ′TBℓ = CTEℓ sin(2α), (5.8)C ′EBℓ =12(CEEℓ − CBBℓ ) sin(4α), (5.9)where the unprimed CXYℓ are the spectra that would be measured in thecase of no cosmic birefringence. We will assume that CBBℓ is negligible,since Planck has no direct detection of B modes. Employing the small-angleapproximation13 (in α) we see that the only modifications to the CMB powerspectra appear as non-zero C ′TBℓ and C′EBℓ :C ′TBℓ (nˆ) = 2α(nˆ)CTEℓ , (5.10)C ′EBℓ (nˆ) = 2α(nˆ)CEEℓ , (5.11)where we have now allowed α to depend on direction, which forces thepower spectra on the left to depend on direction. As shown in Chapter 4,constraints on birefringence from Planck are driven by the E–B correlation,so we primarily focus on Eq. (5.11). This relation suggests that we can uselocal measurements of the E–B correlation to determine α as a function ofdirection on the sky.The correlations of Eqs. (5.4)–(5.9) can be searched for either in har-monic space or pixel space. In the spatial domain, one can look at thecorrelations between temperature extrema and polarization to reveal T–Bcross-correlations. Similarly, correlations between E-mode extrema and po-larization reveal E–B correlations. Both approaches were used to constrainan isotropic α in Planck Collaboration XLIX (2016). To analyse polariza-tion data in the neighbourhood of extrema, or “peaks,” the modified Stokesparameters, Qr and Ur are used. This involves a transformation to radialand tangential components centred on each peak as the origin. Specifically,the value of Qr at an angular distance θ from a peak is the radial (< 0) andtangential (> 0) component of the polarization with respect to the peak.13Our power spectrum results will demonstrate that this is a good approximation. Inthe event that this approximation breaks down, one should interpret the power spectrumto be for the quantity sin (4α)/4 (as explained in Gluscevic et al. 2012).1235.3. Data and simulationsThe Ur component is non-zero if the polarization is rotated by 45◦ withrespect to these directions. The specific transformations areQr(θ) = −Q(θ) cos(2φ)− U(θ) sin(2φ), (5.12)Ur(θ) = Q(θ) sin(2φ)− U(θ) cos(2φ). (5.13)The transformed Stokes parameters are calculated in the neighbourhoodsof each peak. The patterns are expected to have azimuthal symmetry, sothe data can be compared to the following theoretical predictions (derivedin Komatsu et al. 2011; Planck Collaboration XLIX 2016):〈UTr 〉(θ) = −2α∫ℓdℓ2πB2ℓ p2ℓ (bTν + bTζ ℓ2)CTEℓ J2(ℓθ); (5.14)〈UEr 〉(θ) = −2α∫ℓdℓ2πB2ℓ p2ℓ (bEν + bEζ ℓ2)CEEℓ J2(ℓθ). (5.15)Here θ is a radial vector, with θ its magnitude, Bℓ is a 10′ beam applied to thedata, and pℓ is the pixel window function at HEALPix14 (Go´rski et al. 2005)Nside = 1024 resolution. The quantity J2 is the second-order Bessel functionof the first kind, and bT,Eν,ζ are bias parameters that arise from the selection ofpeaks from a Gaussian field (Bond and Efstathiou 1987; Desjacques 2008),which are discussed and calculated in Komatsu et al. (2011) and PlanckCollaboration XLIX (2016).5.3 Data and simulationsOur baseline results use the full-mission and half-mission Planck (PlanckCollaboration I 2016) data splits for the E, Q, and U polarization maps,15specifically using the SMICA component-separation procedure (Planck Col-laboration X 2016; Planck Collaboration IX 2016), chosen for its relativelylow noise level in polarization; however, we use Commander, NILC, and SEVEMmaps to check for consistency. The maps are provided at a HEALPix Nside =1024 resolution, smoothed with a 10′ beam. Along with these maps we usethe common polarization mask UPB77 in union with a mask that coversmissing pixels specific to the half-mission data split (Planck CollaborationIX 2016).16 For our high-multipole likelihood (described in Section 5.4.2) we14http://www.healpix.sourcefourge.net/15Available at http://www.cosmoc.esa.int/web/planck/pla16Though the areas affected by missing pixels are inadequate for measuring α, theynevertheless have very little effect on our results.1245.3. Data and simulationseventually degrade our maps to an Nside = 256 resolution. We then define anew conservative mask that is simply the original mask degraded, with allpixels that contain a masked pixel in the original mask being set to zero.It is worth recalling that in 2015 the Planck collaboration released polar-ization data with some known systematic effects still present. Specifically,there are large angular artefacts in the data that have yet to be remedied(Planck Collaboration I 2016), temperature-to-polarization leakage effects atsmaller scales (Planck Collaboration VII 2016; Planck Collaboration VIII2016; Planck Collaboration IX 2016), and a noise mismatch between thedata and simulations (Planck Collaboration XII 2016). We account for thelarge-scale artefacts by using high-pass-filtered versions of the data (andsimulations), explicitly a cosine filter that nulls scales ℓ ≤ 20 and transitionsto unity at ℓ = 40 (Planck Collaboration IX 2016). We note that the best-fit temperature-to-polarization leakage model was removed from the data(Planck Collaboration IX 2016) and had negligible effect on the uniform αconstraints (Planck Collaboration XLIX 2016), though we do not explicitlytest for its impact here. The noise mismatch does not greatly affect our re-sults, since our data come from the E–B cross-correlation; nevertheless wedo use auto-correlations of α (which are dominated by the E–E and B–Bcorrelations Gluscevic et al. 2012) on large scales in order to have a welldefined likelihood (see Section 5.4.1) and have checked that results obtainedare consistent with the cross-correlation of α determined by the half-missiondata.We use a suite of simulations for which the power in α is null, in order toestimate uncertainties for our power spectrum results. We generate polariza-tion simulations using the following fiducial cosmology, which is consistentwith the data (Planck Collaboration XIII 2016): ωb = 0.0222; ωc = 0.1203;ων = 0.00064; ΩΛ = 0.6823; h = 0.6712; ns = 0.96; As = 2.09 × 10−9; andτ = 0.065. Here ωx ≡ Ωxh2 are the physical densities. We add noise powerto our simulations in order to match the total power in our data maps; this,however, does not include a correlated noise component or non-Gaussianforeground residuals. For our birefringence analysis the cosmological pa-rameters are fixed to the values reported above. This seems to be a safeassumption, since α has no effect on CTTℓ and only affects CTEℓ , and CEEℓ(and thus parameters) at second or higher order (see Eqs. 5.4–5.9). Forsmall α, the effect of lensing is orthogonal to an anisotropic birefringence(Gluscevic et al. 2012; Yadav et al. 2009), however a bias appears at thepower spectrum level (Namikawa 2017). This bias on the power spectrumis, however, sub-percent at Planck noise levels (Namikawa 2017) and there-fore we ignore its effects here. We further generate a suite of simulations1255.4. Measuring local rotationswith a particular scale-invariant α power spectrum, described in Section 5.5,to demonstrate the effectiveness of our α reconstruction and to determinethe corresponding reconstruction bias.5.4 Measuring local rotationsWe use Eq. (5.15) to define an unbiased estimator for α at the location ofevery peak p (αˆp):U˜r(θp) = −2∫ℓdℓ2πB2ℓ p2ℓ (bν + bζℓ2)CEEℓ J2(ℓθ); (5.16)αˆp =∑p Uˆr(θp)U˜r(θp)∑p U˜r(θp)U˜r(θp). (5.17)Here Uˆr(θp) is the value of the data, and θp is a radial vector centred at thelocation of peak p. The peak positions are determined by the full-missionE-mode map, while the above fit is performed on the full or half-missionQ and U maps. Equation (5.17) is a simple linear least-squares fit to U˜rwith the identity as the covariance matrix for α and the sum is performedover all unmasked pixels within a 2◦ radius (chosen since the U˜r profilevanishes at distances > 1.◦5, as demonstrated in Komatsu et al. 2011; PlanckCollaboration XVI 2011; Planck Collaboration XLIX 2016). We also removethe monopole in α, to suppress any leakage to higher multipoles, since themonopole is systematics dominated and large (Planck Collaboration XLIX2016); however, we check that this step has no significant effect on ourresults.We fit for a scale-invariant power spectrum, which takes the formL(L+ 1)2πCL ≡ A, (5.18)for a constant A. By convention we refer to α multipoles as “L” to dis-tinguish them from ℓ multipoles (for the temperature and polarization ani-sotropies). This spectrum would be realized in a model containing nearlymassless pseudo-scalar degrees of freedom coupled to photons, such as in(Caldwell et al. 2011), for L . 100. A model like this is constrained ex-tremely well at low L compared to high L, which puts Planck data ata distinct advantage compared to smaller coverage (although with highersensitivity) ground-based experiments. For this reason we pay particularattention to recovering the low Ls accurately (see Section 5.4.1).1265.4. Measuring local rotationsPrevious estimators of anisotropic α in the literature compute the con-tribution to the 4-point function (of TB or EB) in harmonic space usingstandard quadratic maximum likelihood techniques (Gluscevic et al. 2012)(similar to CMB lensing techniques), or using the 2-point correlation func-tion (Li and Yu 2013; Li et al. 2015). Our approach reconstructs the 4-pointfunction by simply measuring the variation of the 2-point function locallyin the data. In the following subsections we describe how we take the localmeasurements of α and compute maps at low and high resolution. For powerspectra it is worth recalling that the auto-spectrum of an α-map determinedby the E–B correlation is not the power spectrum of α. This is because theauto-spectrum necessarily contains contributions from the E–E and B–Bcorrelations that are non-zero even if α = 0 (Gluscevic et al. 2012). One canobtain the true power spectrum by subtracting the mean power spectrumfrom simulations with a null α spectrum or by using cross-correlations; weemploy both of these methods below.5.4.1 Low multipolesWe begin by taking the αp values found above and using a pixel fit to recoverthe spherical harmonics αLM ,α˜LM =∑pwpα˜pY∗LM (θp, φp)∑pwp|YLM (θp, φp)|2. (5.19)Here the weights wp are uniform per pixel or chosen to incorporate theuneven hits distribution from the Planck scanning strategy. For our baselinewe use a uniform weighting scheme wp = 1; however, we also consider aweighting scheme where wp is given by a smoothed version of the 217-GHzhits map17 (denoted Hp, which we plot later). Note that our simulationsdo not contain the effects of the Planck scan strategy and therefore onlyuniform weighting is used for them. This method accounts for the mask bysimply not using any masked pixels, i.e., all αp values come from unmaskedareas. We then compute the power spectrum of α either by taking thecross-correlation of α1LM and α2LM , determined with the half-mission 1 and2 maps, or by taking the auto-spectrum of αLM determined by the full-mission data and subtracting the mean of the αLM auto-spectra calculatedfrom null simulations. The latter method is in principle more sensitive tothe noise properties of the data, but we find it to be consistent with the17The map is smoothed with a 2◦ top-hat beam, chosen to match our method of Sec-tion 5.4 that fits for α over pixels within 2◦ of each peak.1275.4. Measuring local rotationsformer method, and it has the advantage of allowing us to form a Gaussianlikelihood for the αLM s when fitting for a model. For this reason we onlydisplay these auto-correlation spectra in our low L results that follow.A direct calculation of Eq. (5.19) is computationally expensive for largeLmax, so we limit this approach to Lmax = 30 and consider higher multipolesonly in the following subsection. The low-L likelihood then takes the form−2 logL({α˜LM}30L=1|A) = log |V|+L=30∑LML′M ′α˜LM V−1LML′M ′ α˜∗L′M ′ + constant,(5.20)VLML′M ′ = 〈CL〉 δLL′δMM ′ + BA 2πL(L+ 1)δLL′δMM ′ . (5.21)The average is taken over a suite of simulations with A = 0. B is a normal-izing factor to ensure that our estimator is unbiased, and is determined bycross-correlating the input α realizations with their corresponding measuredvalue from a reference power spectrum. Bayes’ theorem, along with a flatprior on A (specifying A ≥ 0), allows us to turn this into a likelihood forA given the data and hence to obtain the posterior for the amplitude of ascale-invariant spectrum.5.4.2 High multipolesTo produce our high-L map we need to apply a smoothing to our α˜p values.The mean separation between peaks is about 0.◦2, and therefore we definean Nside = 256 map (denoted α˜256p ) populated with the mean of α˜p over allpixels within a radius of 0.◦25:α˜256p =∑p′∈|p−p′|≤0.25◦ α˜p′∑p′∈|p−p′|≤0.25◦. (5.22)This procedure induces a beam function (BL), which we show in Fig. 5.1.Our high-L power spectrum is the cross-correlation of Eq. (5.22) betweenhalf-mission 1 and half-mission 2 data, with a correction for the beam andthe cut sky, i.e.,a˜LM =∫dΩα˜256p M(Ω)Y∗LM , (5.23)C˜α1α2L =〈a˜1LM a˜∗2LM〉 ≈ fskyB2Lp2LCα1α2L , (5.24)where fsky is the fraction of the unmasked sky, M(Ω) is the applied mask,BL is the effective beam function (see Fig. 5.1) induced by the smoothing1285.4. Measuring local rotationsprocedure, and pL is the pixel window function specific to an Nside = 256resolution map. The approximation in Eq. (5.24) is the exact MASTER (Hivonet al. 2002) correction due to the masking for an L-independent power spec-trum for which the data and simulations are consistent. We further bin thepower spectrum in bins of size ∆L = 50 (with Lmin = 31 so as not to doublecount the low-L data), to minimize correlations induced by the mask fromneighbouring L modes.0 100 200 300 400 500 600 700L0.00.20.40.60.81.0BLFigure 5.1: Effective beam for the high-L analysis described in Section 5.4.2,induced by the smoothing procedure.To fit for a scale-invariant power spectrum (Eq. 5.18) we form χ2 as afunction of A:χ2 =∑bb′(Cα1α2b − CAb)G−1bb′(Cα1α2b′ − CAb′), (5.25)where G−1bb′ is the binned inverse covariance matrix derived from simulations(generated with A = 0) and CAb is the model power spectrum (Eq. 5.18)binned in the same way as the data and simulations. We have verified withour simulations that each bin is close to Gaussian and so we form a likelihoodasL({Cα1α2}b|A) ∝ exp (−χ2/2). (5.26)Once again we can transform this into a likelihood for A given the data,with a flat prior on A.1295.5. Tests of the methodFinally we combine our low-L and high-L likelihoods to form a jointconstraint on A by simply taking the product of the likelihoods in Eqs. (5.21)and (5.26). This assumes that the data at low L are uncorrelated with thedata at high L. This is known to not be strictly true, since the mask willinduce correlations, particularly for L = 30 with the first bin of the high-Ldata (which uses Lmin = 31). However, this small correlation has very littleimpact on our results and the difference in the results is minimal if Lmin isincreased to reduce the correlation.5.5 Tests of the methodWe first test the method described above in Section 5.4 by generating a singlesimulation with a known realization of α from an input power spectrum. Todo this we generate a Q and U realization from the cosmology described inSection 5.3, and we then modify the Q and U maps by the relation Eq. (5.3).In this test the α map is a realization of a scale-invariant power spectrumwith A = 10−2/2π (chosen for visualization purposes), and is shown inFig. 5.2 (left panels). We further input a noise realization to our simulation,so that the total power in each of the Q and U maps is consistent with thedata.Upon applying our method of recovering α, we obtain the panels onthe right-hand side of Fig. 5.2, for both low and high resolution. At thelevel of the α maps the output of our analysis is quite consistent with theinput; however, there is considerably more noise in our output maps (par-ticularly at high resolution) due to the addition of significant noise power.At the power spectrum level we also obtain very good agreement with ourinput spectrum, with the scatter attributable to the noise in the simula-tion. In Fig. 5.3 we show the mean recovered power spectrum on a suiteof simulations with A = 10−4/2π, along with the theoretical curve and thecorresponding uncertainties. We find that our reconstruction slightly over-estimates the true power spectrum at the 30% level, which we correct forin the B parameter in Eq. (5.21).5.6 ResultsFirst we confirm that we recover the results for the α-monopole from Chap-ter 4. However, as already noted, in the main analysis of this chapter weremove the monopole (which is dominated by systematic effects) so that itdoes not leak into higher multipoles.1305.6. Results-0.25 0.25 -0.25 0.25Figure 5.2: Top: Low-L (1 ≤ L ≤ 30) α-maps for an input α realization(left) and for reconstruction by our method, as described in Section 5.4.1(right). Bottom: High-resolution α-maps for an input α realization (left),along with the Weiner-filtered output of our high-L reconstruction, as de-scribed in Section 5.4.2 (right). The induced beam (Fig. 5.1) is applied tothe input map for comparison purposes. The input and output maps areclearly correlated, although the output has considerably more scatter onsmall scales due to the significant noise in the polarization maps.5.6.1 Maps and power spectrumOur low-L and high-L maps for the data are shown in the top row and bot-tom right panels of Fig. 5.4, respectively. Our two low-L maps are clearlystrongly correlated; however, using the weighting given by the hits map inFig. 5.4 (bottom left) we see large-scale features near the Ecliptic poles andGalactic plane. While, visually striking, these features appear to have verylittle effect on our power spectrum results (to be discussed in Section 5.7.1).They nevertheless point to systematic effects associated with residual fore-grounds not accounted for in our simulations.The power spectrum at low L and at high L (binned) are shown togetherin the left panel of Fig. 5.5. Recall that at low L the spectrum is the meanof the auto-spectrum of our simulations subtracted from the auto-spectrumof our low-L α-map. We find that this estimate is consistent with the cross-correlation of α from half-mission 1 and half-mission 2 data. The blue1315.6. Results5 10 15 20 25 30L0246810CL [×105]inputmeasuredFigure 5.3: Recovery of a scale-invariant α power spectrum with A =10−4/2π from a suite of simulations. The blue points are the mean recov-ered power spectrum from simulations, while the bars denote the standarddeviations from the same set of simulations. The input power spectrum isshown in orange. The overestimation (blue points consistently above theorange curve) comes from the reconstruction bias of the method, which isaccounted for by the B parameter in Eq. (5.21) in all subsequent powerspectrum plots.points in this figure use the uniform weighting scheme, while the orangepoints use the hits-map weighting. At high L the spectrum is derived usingcross-correlations only. We find good agreement with the expectation of anull power spectrum over all L (with the possible exception of the L = 1mode, the discussion of which is left for Section 5.6.3), and the smallnessof the power spectrum justifies our use of the small-angle approximation inEq. (5.11).5.6.2 Constraints on a scale-invariant power spectrumThe posterior for the amplitude of a scale-invariant power spectrum is shownin Fig. 5.5 (right). The high-L likelihood prefers a positive A at less thanthe 2σ level, while the low-L data are consistent with A = 0. However, theyare both quite consistent with each other, as can be seen by comparing the95% CL values for the low-L likelihoods and the full likelihood in Table 5.1.1325.6. Results-0.15 0.15 -0.15 0.15754.935 23142.7-2 2Figure 5.4: Top: Low resolution (Lmax = 30) data maps of α weighted bythe hits map (left) or using uniform weighting (right). Bottom: Smoothedhits map used for the wp = Hp analysis (left), together with high-resolutiondata map (right).The full-L constraint comes from the combination of the low-L and high-Llikelihoods, assuming no correlation between the two. Due to the L depen-dence of the model spectrum, the likelihood is dominated by the lowest Ls (abluer spectrum would be more constrained by high L than the scale-invariantone). Note that the posterior is not very Gaussian, since the CLs follow aχ22L+1 distribution. We find the constraint A < 2.2 × 10−5 (0.07 [deg2]) at95% CL. This constraint is at a similar level to the systematic uncertaintyof the α-monopole measurements, namely 0.◦3. This suggests that, in theabsence of an improved absolute calibration scheme (see Nati et al. 2017, foran example of efforts in this direction), constraints on cosmic birefringencefrom the next generation of CMB measurements will likely be focused onsearches for anisotropic α.While the noise level of Planck polarization data is large compared tothe most recent ground-based experiments, it does have the distinct advan-tage of measuring the largest scales. Our new constraints improve upon themost stringent constraints available (see Table 5.1) and are an order magni-tude smaller than previous results (Gluscevic et al. 2012; Ade et al. 2015).1335.6. Results10 20 30-20-15-10-505101520CL [×102deg2]wp=1wp=Hp200 400 600L-0.3-0.2-0.10.00.10.20.30.00 0.05 0.10 0.15 0.20 0.25A [deg2]0.00.20.40.60.81.0Figure 5.5: Left: Power spectrum for α. The vertical dashed grey linedenotes the boundary between our low L and high L reconstructions; notethe differing y-scale for low-L compared to high-L. Uncertainties shown arestandard deviations of our set of null simulations; at low-L the CααL arenot Gaussian or symmetric, which is accounted for in our likelihood (seeSection 5.4.1). The power spectrum here justifies our use of the small-angleapproximation. Right: Posteriors for the amplitude (A) of a scale-invariantpower spectrum defined by Eq. (5.18). The constraint is mainly drivenby the lowest Ls, which is the reason for the non-symmetric shape of theposterior.Somewhat tighter constraints could be found using a joint low-L and high-Lanalysis of data from Planck combined with the BICEP2/Keck Array.Low L (wp = Hp) Low L (wp = 1) All L B2KA ≤ 2.4× 10−5 A ≤ 1.9× 10−5 A ≤ 2.2× 10−5 A ≤ 3.3× 10−5Table 5.1: 95% CL upper limits on the amplitude A of a scale-invariantpower spectrum. Here “All L” refers to the combination of our uniformweighting low-L and high-L likelihoods. The last column comes from BI-CEP2 Collaboration et al. (2017), using polarization data from the BI-CEP2/Keck Array.The difference between our low-L likelihoods using two different weight-ing schemes is attributable to residual foregrounds (explored in Section 5.7.1)that are not accounted for in our simulations. The hits-map weighting ismore sensitive to these foregrounds compared to the uniform weighting. In1345.6. ResultsSection 5.7.1 we derive a systematic error for our amplitude, which primarilycomes from residual polarized dust.5.6.3 The dipoleFrom the power spectrum, Fig. 5.5 (left panel), we see that the dipole devi-ates the most from the expectation of a null power spectrum. We quantifythis by comparing C1 for the data to the values in our simulations. We findthat only about 1.4% of the simulations have a larger dipole than the data.18This is the case for both the auto-correlation and cross-correlation of thefull-mission or half-mission data sets. It is therefore worth investigating thissignal further.Method Amp. (A1 =√3C1/4π) [deg] Dir. (l, b) [deg]Uniform weighting 0.32± 0.10 (295, 17)± (22, 17)Hits-map weighting 0.40± 0.10 (280, 1)± (15, 12)Table 5.2: Mean posterior values and 68% uncertainty levels for the am-plitude and direction of the dipole in α. The corresponding 68% radialpositional uncertainty around the best-fit direction is about 25◦, with acorresponding p-value of 1.4%. The difference between both methods is at-tributable to residual foregrounds (which are more apparent for the hits-mapweighting, see Section 5.7.1), as well as a significant systematic effect.In the absence of a model, the values of α1M are Gaussian distributedwith mean zero and variance given by 〈C1〉, where the average is taken oversimulations with a null α dipole. We can convert this to an amplitude anddirection with a uniform prior on the α1M s, parameterizing the dipole asα(nˆ) = A1 cos θ. (5.27)Here θ is defined as the angle with respect to the best-fit direction and A1 ≡√3C1/4π. In Table 5.2 we quote the mean values of the posteriors and theircorresponding 68% uncertainties. We show the best-fit dipoles in Fig. 5.6 forour baseline results (right panel) and using hits-map weighting (left panel).We explore in Section 5.7.1 the effect of residual foregrounds (not presentin our simulations) on the dipole and find that these significantly affect thedirection of the dipole.18For these results we include α obtained by our T–B estimator as well, although thishas only a marginal influence on our results.1355.6. Results-0.006 0.006 -0.006 0.006Figure 5.6: Best-fit dipole in α from the full-mission data using wp = Hp(left), compared to the dipole from uniform weighting (right). The dipolesare consistent, although the amplitude clearly decreases when using uniformweighting.If the dipole in α were to be physical (and not simply a statistical fluc-tuation), then the overall signal could be fit by a sufficiently red spectrum,which would have significant implications on the nature of the sourcingpseudo-scalar (or vector) field. Alternatively, there could be a genuinelypreferred direction for the cosmic birefringence (i.e., a dipole that is uncon-nected to a power spectrum). Either way, new (preferably) all-sky polar-ization data with reduced noise levels compared to Planck are required todetermine whether or not the dipole signal is cosmological.5.6.4 The M = 0 quadrupoleWhile there is no evidence for a significantly large quadrupole in the data(see Fig. 5.5, left), there could be a special direction in the α map that wouldshow up solely in the M = 0 mode. It is therefore worth considering thismode specifically.An M = 0 quadrupole in an arbitrary direction would be related to ourcoordinate α2M values by a rotation with a Wigner D-matrix byα2M = α′20D2M0(l, b, 0). (5.28)Here α′20 is an M = 0 quadrupole in a coordinate system pointing in thedirection (l, b). With our simulations we generate a covariance, Q from oursimulations, which defines a likelihood of the formL ∝ exp[−12(αˆ2M − α2M (α′20, l, b))Q−1(αˆ2M − α2M (α′20, l, b))†]. (5.29)1365.7. Systematic effectsWe then sample the likelihood using a Markov chain Monte Carlo (MCMC)algorithm to obtain posteriors for the parameters α′20, l, and b. Marginaliz-ing over the direction we find α′20 = 0.◦02±0.◦21, which is very consistent withno special direction in the quadrupole. For this same reason the directionis unconstrained.5.7 Systematic effectsIn Section 5.3 we mentioned several kinds of systematic effect that arepresent in the Planck polarization data that we have taken efforts to avoidbeing sensitive to. These are: an uncertainty in the global orientation of thePSBs; large-scale artefacts in the data; and an un-modelled correlated noisecomponent in the data. The first effect contributes a bias to a uniform αand would thus cancel out in our search for anisotropic α. The second ismitigated by the use of high-pass-filtered data. The last effect is minimizedby using cross-correlations (between half-mission 1 and half-mission 2 data)wherever possible. Our low-L likelihood uses auto-correlations, althoughwe compared our power spectra to the corresponding cross-correlation andfound good consistency. However, this does not definitively show that cor-related noise is not a significant bias for our results, and so we now furtherinvestigate other potential sources of systematic effects.5.7.1 ForegroundsOur estimator looks for parity violations in the CMB data in the hopes ofconstraining a cosmological signal. Foregrounds contaminate this by beinglarge additive signals that do not source polarization in a way that is invari-ant under parity transformations about our location, and hence can corruptthe α signature.We first test for this contamination by enlarging our baseline mask tocover more of the Galactic plane. We take the UPB77 mask (Planck Collab-oration IX 2016), smoothed with a 200′ beam, then set all points below 0.9to zero and all others to 1, and finally multiply by the UPB77 mask againso as not to miss any small masked areas. This decreases the sky fractionavailable from UPB77 from fsky = 0.77 to 0.69. This roughly 10% decreasein sky coverage leads to a roughly 40% increase in the 95% CL in the ampli-tude of a scale-invariant power spectrum, due to increased sample variance.Since the likelihood is dominated by low L and is skewed to higher values(see Fig. 5.5, right panel), this result is still quite consistent with our base-line result. However, we cannot entirely rule out that foregrounds might be1375.7. Systematic effectsa significant systematic effect here. With that in mind, it is neverthelessstill the case that the increase in the 95% CL limit is consistent with nodetection of anisotropic α.As an additional test we can define an a posteriori mask to be zero ev-erywhere that the absolute value of the low-L map in Fig. 5.4 (top left) isgreater than 0.15, masking the visually striking features. We find that ourpower spectrum results remain consistent with the expectation of the cor-responding increased sample variance. In particular the dipole is consistentin amplitude and direction with our baseline results from Table 5.2.Some foreground contaminants, such as dust, can produce B modes thatmight induce a cross-correlation signal between lensing and α. We test forthis using the Planck 2015 lensing maps (Planck Collaboration XV 2016)and perform the cross-correlation with our low-L data and simulations. Notethat the Planck lensing maps contain no information for L < 8 (PlanckCollaboration XV 2016), so this test tells us nothing about the nature ofthe dipole in α. Nevertheless, we obtain a probability to exceed (PTE) theχ2 obtained with the data (derived from simulations), of 25%, consistentwith no detection of a cross-correlation.Foreground Low-L PTE [%] Dipole PTE [%]Dust 0.4 14Synchrotron 20 97Free-free 13 96Table 5.3: Probability to exceed the χ2 obtained from the cross-correlationof our α maps (or just the dipole) with the corresponding αf map (or justdipole) from each foreground of the data. We find a marginally significantcorrelation with polarized dust, due to residual dust in the data that is notaccounted for in our simulations.We also test for the presence of correlated polarized synchrotron, free-free, and dust emission directly, by obtaining the Planck Q, and U fore-ground maps (Planck Collaboration X 2016) and propagating them throughour analysis to obtain maps of α (denoted αfLM ). That is, we use Eq. (5.17)to fit for α in the foreground maps at the location of peaks in the CMBE-mode map. These maps contain an unnormalized estimate of the contri-bution from each foreground to α that is correlated with the CMB. We thendetermine a PTE for the χ2 obtained with the cross-correlation of α withαf , shown in Table 5.3. We find marginally significant correlations withdust only. In order to estimate the bias incurred from these foregrounds we1385.7. Systematic effectsmodel the contamination asαLM = α˜LM +∑fCfαLCffLαfLM . (5.30)Here αLM are the measured data, while α˜LM are the expected data comingjust from the CMB, CfαL is the cross-correlation of α obtained from the CMBand foreground, and CffL is the auto-correlation of α obtained from the fore-ground (with these quantities being estimated from the data). Propagatingα˜LM to obtain 95% CL values for the scale-invariant amplitude leads to a de-crease of 0.7×10−5 (still consistent with no detection). Note that this is thesame amount of shift between our low-L likelihoods using hits-map weight-ing compared to uniform weighting; we thus assign this value as a systematicerror in our result, present because our simulations do not contain residualforegrounds. When applied to the L = 1 mode specifically, we find thatthe dipole moves away from the Galactic plane by the same amount as thedifference between the hits-map weighted and uniform weighted dipole (seeTable 5.2), consistent with the presence of residual correlated foregrounds;however, the amplitude remains largely unchanged. We therefore assign aconservative systematic error of 0.◦08 to the amplitude and (5◦, 15◦) to thedirection, substantially impacting its significance.5.7.2 Point sourcesAlthough point sources in general add a source of bias to the 4-point function(Osborne et al. 2014), they are not expected to contaminate the signal weare looking for (Gluscevic et al. 2012), since they only contribute a parity-even signal. Nevertheless we test the level of contamination by including apoint-source mask.We consider the union of the Planck point source masks for polarizationfrom 100 to 353GHz. It turns out the vast majority of pixels masked bythe point source mask are already masked by UPB77. After degrading to acommon resolution of Nside = 1024, there are 16 remaining pixels (out ofa potential 24,000) that are not also masked by UPB77. It should come asno surprise then that they therefore have a negligible effect on our resultsand we thus consider point sources to be an unimportant systematic for ouranalysis.1395.8. Conclusions5.7.3 Relative uncertainty on the PSB orientationsAlthough we are insensitive to a global rotation of the HFI detectors, wecould still in principle be sensitive to a relative angular separation betweenindividual PSBs. This is because, in the component separation process,different frequencies (and thus PSBs) are used anisotropically, and thus arelative difference in orientation of PSBs at different frequencies would ap-pear as anisotropic birefringence.The relative upper limit on the PSB orientations is 0.◦9 (Rosset et al.2010), however the dispersion of α, as measured using E–B correlations be-tween the HFI frequencies, is around 0.◦2 (Planck Collaboration Int. XLVI2016). We expect that such an anisotropy would appear as a large-scalefeature characterizing the use of different frequencies in the component-separation process. Thus we do not expect that this would affect the searchfor a scale-invariant spectrum. Crudely speaking the use of different fre-quencies varies mostly with latitude (though this depends upon the methodPlanck Collaboration IX 2016), which is a pattern we do not see. In par-ticular the dipole is seen mainly in the Galactic plane, as opposed to theGalactic poles. Therefore it seems unlikely that the excess in the dipole wesee is due to the relative uncertainty in PSB orientations compared to fore-grounds; however, a full characterization of such effects will be importantfor future studies.5.8 ConclusionsWe have estimated the anisotropy in the cosmological birefringence an-gle, α, with a novel map-space based method, using Planck 2015 polariza-tion data. Our results are consistent with no evidence for parity-violatingphysics. We provide the most stringent constraints on the anisotropy atlarge angular scales and have constrained a scale-invariant amplitude tobe A < [2.2 (stat.) ± 0.7 (syst.)] × 10−5 at 95% CL.19 Here the systematicerror comes from estimating residual foregrounds (primarily dust) in thedata. This implies a constraint on dipolar and quadrupolar amplitudes tobe√C1/4π . 0.◦2 and√C2/4π . 0.◦1, respectively. These constraints are,along with the newest results from the BICEP2/Keck Array, the tightestlimits on a scale-invariant power spectrum (see Table 5.1 for a direct com-parison). We also search for special directions in α, finding that an M = 0quadrupole is constrained to be α20 = 0.◦02 ± 0.◦21, consistent with the19Conservatively, one should take the full 95% limit to be 2.9× 10−5 = 0.09 deg2.1405.8. Conclusionsnull hypothesis. Our results are consistent across four different component-separation methods and do not appear to be significantly contaminated bypoint sources. We also find no significant cross-correlation signal betweenour α maps and the Planck 2015 lensing map.One possible exception to the above conclusion is the dipole in α (whosebest-fit amplitude and direction can be found in Table 5.2, correspondingto a radial 68% uncertainty on the direction of 25◦), which is somewhatlarge compared to null simulations, with an associated p-value of 1.4%. Wefind that the significance is insensitive to the use of the auto-correlationof full-mission data or the cross-correlation of the half-mission data. Wedo find that foreground contamination, coming primarily from dust, biasesthe dipole in a significant way, pulling the direction toward the Galacticplane, accounting for part of the signal. If, on the other hand, some of thedipole is genuinely due to cosmic birefringence then this would have signifi-cant implications for the form of the field and the source of its fluctuations,necessitating a red spectrum or a specifically direction-dependent birefrin-gence. The model-space is vast and the significance is low, and clearlypartially contaminated by residual foregrounds, so we do not speculate onwhat the physical source could be here. More sensitive polarization data atlarge angular scales are required to settle the issue.In Chapter 4 it was determined that searches for a uniform angle α arenow dominated by systematic effects at the 0.◦3 level. Here we find thatconstraints on the direction dependence of α are also at about the 0.◦3 level,with no apparent dominant systematic effects limiting the search in thenear future. Therefore in the absence of an improved calibration scheme fordetermining the orientation of the PSBs, future searches for parity-violatingphysics of the form discussed here will likely be driven by the pursuit ofanisotropic cosmic birefringence.141Chapter 6ConclusionsIn this thesis I tested two fundamental assumptions of the ΛCDM model ofcosmology: statistical isotropy, and symmetry under parity transformations.In Part I I tested the degree to which there is a dipolar asymmetry inpower for the CMB. Such a signal exists in the temperature anisotropies toa mildly significant degree, but only on large-scales. The large scale natureof this signal is suggestive of an inflationary or other early-Universe expla-nation, though the moderate significance means that it should be viewedwith scepticism until confirmed with additional data. The measurement ofTT -power on these scales are also limited by cosmic variance and thus onlynew modes or a new observable feature will be able to tell us anything moreabout the suspected signal. I proposed a formalism by which asymmetrymodels can be tested with new modes and applied this to CMB lensing andpolarization. In Chapter 1 I demonstrated that the signal-to-noise ratio forCMB lensing is too small to be of much use even in the cosmic variancelimited case. For polarization I quantified the probability of a 2σ detec-tion of an asymmetry model for the expected Planck polarization data andfor cosmic-variance-limited data in Chapter 2. The result is model depen-dent and pessimistic for Planck polarization, largely due to the weaknessof the original signal and the need for an extra parameter to determine thescale of the asymmetry. In the final Chapter of Part I I introduced a newclass of model where the source of modulation also contributes to the total(isotropic) power spectra. Specifically these could be from an asymmetrictensor or isocurvature component. I confronted this class of models withtemperature data alone and found that they can be ruled out based on theirover-production of power. This is even the case when an anti-correlatedisocurvature mode is introduced in an attempt to reduce the contribution ofpower. The models left are strictly of the phenomenological variety wherethe asymmetry comes from a portion of the adiabatic perturbations, whichis generated by some yet unknown mechanism.In Part II I tested whether or not parity violation terms exist in theelectromagnetic sector. The effect of this type of parity violation is thein vacuo rotation of the plane of polarization as radiation propagates. This142Chapter 6. Conclusionsrotation, or “cosmic birefringence”, correlates B modes with T and E modes.In Chapter 4 I considered a uniform rotation and constrained this rotationto be of order the systematic error, which is driven by the uncertainty inthe orientation of the polarization-sensitive detectors. The statistical erroris a factor of approximately 6 smaller than this systematic and thereforeit is unlikely that this limit will be improved upon in the near future; Anew method of calibration will likely be required before this happens. InChapter 5 I also looked for an anisotropic birefringence, which is insensitiveto this particular systematic and common in axion and quintessence models.I set the currently strictest limits on a scale-invariant power spectrum forthe birefringence. I also looked for particular large-scale birefringent modes,specifically a dipole and an M = 0 quadrupole. The results of these testsshowed that the data are so far consistent with no parity violation in theelectromagnetic sector. The anisotropic birefringence appears to have nosignificant systematic errors and thus could be improved upon in the nearterm with upcoming CMB data.At the outset of this thesis I explained the extremely successful ΛCDMmodel, emphasizing however, its reliance on unknown ingredients like infla-tion and dark matter. I have explored two basic assumptions of this modelin the hopes that their deviation from the expected values might give someinsight into inflation or other new particle content in the Universe. In this(perhaps overly) ambitious goal I have failed. What I have done instead isnarrow down the set of possibilities where insights into new physics could belurking. There are many avenues in which clues could be hiding; this the-sis closes the door on some of those avenues but cannot be the final word.There is, however, never any guarantee that new mysteries of nature will berevealed to us – it is only our job as scientists to keep trying.One crucial point, however, is that we are not done trying. There ismuch of the Universe yet to map out (recall Fig. 1.1). The detection ofCMB lensing is not as matured as the CMB itself. I talked about usingCMB lensing to follow up an observed “anomaly” in temperature, but ananomaly could be hiding in the CMB lensing data. The same holds true forCMB polarization, for which the large scales, in particular, have not beenmeasured to cosmic-variance limited levels yet. The CMB may still havesome tricks up its sleeve! Venturing out even further we could one day hopeto measure gravitational waves from inflation, as opposed to the effects ofgravitational waves on the CMB. This would give us independent informa-tion on large scales and (slightly) further away from us than the surface oflast scattering (beyond the upper line in Fig. 1.1). Though certainly chal-lenging, their detection would be the earliest picture we will ever have of143Chapter 6. Conclusionsthe Universe. If the CMB is a baby picture then the primordial gravita-tional wave signal could be the Universe’s sonogram. On the other handwe could see evidence for new physics on much smaller scales. Observationsof the stochastic gravitational wave background from all merging sourcescould provide a completely new window into the Universe, probing modessomewhere in Fig. 1.1. If we could model the baryonic signals well enoughwe could start to ask questions about how dark matter (or maybe even darkenergy) radiate gravitational waves.There are many questions left unanswered and luckily there are manymodes of the Universe left to measure. 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Phys.Rev., D78:123529, 2008. doi: 10.1103/PhysRevD.78.123529.164Appendix AEffect of modulation onsmall-scale T anisotropiesAs we mentioned in Sect. 1.3.1, when the scale of the perturbations sourcingthe anisotropies is much smaller than the length scale of variation of themodulation, we expect the effect of the spatial variation of the modulationto be small. Nevertheless, it will be useful to be more quantitative aboutthis expectation.There are three main changes to the temperature anisotropies calcu-lated in Sect. 1.3.1 when sources on smaller scales are considered. First,the relevant transfer functions become oscillatory in k due to the acous-tic oscillations. Next, relaxing the tight-coupling approximation meansthat anisotropic stress must be included. Finally, relaxing the sudden-recombination approximation means that the anisotropies are sourced overa range of redshifts, rather than just at zLS. We will consider each of theseeffects in turn.To a good approximation, the part of the anisotropy proportional to R(sometimes referred to as the “monopole”) takes on a term of order (see,e.g., Lyth and Liddle (2009))cos (krs)R(k) ≡ T1(k)R(k), (A.1)where rs is the sound horizon. This term necessarily approaches T (k)R(k) inthe large-scale limit. Similarly, the part proportional to the radial derivativeof R (the “dipole”) becomes of ordersin (krs)R(k) ≡ T2(k)R(k), (A.2)which again must approach the large-scale limit T (k)k/(aLSHLS)R(k). Re-call that we can consider the linear modulation to commute with the transferfunction filtering if Eq. (1.15) is satisfied. Here we have∣∣∣∣ 1T1(k) dT1(k)dkrLS∣∣∣∣ = rsrLS tan(krs) ≃ 0.01 tan(krs) (A.3)165Appendix A. Effect of modulation on small-scale T anisotropiesand ∣∣∣∣ 1T2(k) dT2(k)dkrLS∣∣∣∣ ≃ 0.01 cot(krs). (A.4)Therefore for most k scales the condition for commutativity is met. For thedipole term, the cot dependence may suggest a problem as k → 0. However,we showed explicitly in Sect. 1.3.1 that the dipole term does, in fact, com-mute to a good approximation with modulation on large scales. Similarly,the periodic divergences in tan(krs) and cot(krs) at larger k values may sug-gest that commutativity breaks down at these scales. To examine the effectof these divergences, consider the covariance of T (k)R˜(k) calculated usingEq. (1.13) for T (k) = cos(krs). In addition to the expected statisticallyisotropic term proportional to cos2(krs)PR(k), we find an extra isotropicterm proportional to cos(krs) sin(krs)PR(k)rs/rLS. Very close to the zerosof cos(krs) this extra term will dominate. However, its absolute contributionis weighted by the small factor rs/rLS. The relatively broad kernel that takesus from k to ℓ space will mean that the extra term will alter the acousticpeak structure only by a small amount, in proportion to the factor rs/rLS.This tells us that the modulation commutes to good approximation with theacoustic oscillation processing.The next small-scale effect is the presence of anisotropic stress, i.e. thequadrupole Boltzmann terms. These terms are suppressed by factors k/|τ˙ | ∼10−3krLS, where τ is the optical depth (see, e.g., Hu and White 1997b). Theanisotropic stress is sourced within distances of the order of the mean freepath from the observed point on the last scattering surface, which is muchsmaller than rLS, and is determined by gradients of the primordial field.Hence, as we showed for the case of the derivative terms in Sect. 1.3.1, forthese contributions modulation will commute to a very good approximationwith filtering. Importantly, as polarization is sourced entirely by anisotropicstress, this will mean that we will be able to describe in a similar way theeffect of modulation on polarization.The final small-scale effect is the sourcing over a range of redshifts,weighted by the visibility function. Including also the high-k part of thefluctuations, the anisotropy of Eq. (1.17) becomes in this case the line-of-sight integralδ˜T (nˆ)T=∫ ∞0dr[S˜lo(t(r), rnˆ) + Shi(t(r), rnˆ)]. (A.5)The previous arguments tell us that, to a good approximation, we can write166Appendix A. Effect of modulation on small-scale T anisotropiesS˜lo(t(r), rnˆ) ≃ Slo(t(r), rnˆ)(1 +ARrrLScos θ). (A.6)Therefore, writing r = rLS + δr, the anisotropy becomesδ˜T (nˆ)T≃∫ ∞0dr[Slo(t(r), rnˆ) (1 +AR cos θ)+ Shi(t(r), rnˆ)]+O(δr/rLS). (A.7)Since the primary anisotropies are sourced over a range of distances δr/rLS ∼10−3, we have to very good approximationδ˜T (nˆ)T≃ δTlo(nˆ)T(1 +AR cos θ) +δT hi(nˆ)T. (A.8)Using Eq. (1.6), this leads immediately, as in Sect. 1.3.1, to the final resultfor the multipole covariance:〈a˜ℓma˜∗ℓ′m′〉 = CΛCDMℓ δℓ′ℓδm′m +AR(C loℓ + C loℓ′ )ξ0ℓmℓ′m′ (A.9)to first order in AR, where CΛCDMℓ is the power spectrum calculated fromPΛCDMR (k) and C loℓ is the spectrum calculated in the same way but usingP loR(k).167Appendix BDipolar modulation ofpolarizationIn this Appendix we calculate the effect of a spatially linear modulation ofthe primordial fluctuations on polarization E and B modes. We will onlyconsider the “lo” component, i.e., the modulated fluctuations. The modu-lation of the E and B modes will differ in detail from that of temperature[derived in Chapter 1, and expressed in Eqs. (3.2) and (2.5)] due to the non-local relation between E and B and the Stokes Q and U parameters, which,as we will see, are modulated analogously to temperature.The polarization we see in direction nˆ can be written as a line of sightintegral in terms of the temperature quadrupole, aT2m(rnˆ), seen at scattererposition rnˆ:Q(nˆ)± iU(nˆ) = −√610∫drdτdre−τ(r)∑maT2m(rnˆ)±2Y2m(nˆ) (B.1)(see, e.g., Hu 2000). Here ±2Yℓm(nˆ) are the spin-2 spherical harmonicsand τ is the optical depth. Importantly, for our purposes we will not needto explicitly calculate the temperature quadrupole; instead, we will onlyneed to know how it is modulated across the sky in the presence of a lineargradient in the primordial fluctuations. The quadrupole at the scatterer willbe sourced by the temperature anisotropies at the scatterer’s last-scatteringsurface. This means that different parts of that source will be modulatedby different amounts. However, since the last-scattering surface is muchthinner than the distance to last scattering, to very good approximation wecan take the quadrupole aT2m(rnˆ) to be modulated by the amplitude given bythe linear gradient evaluated at point rnˆ. Errors due to this approximationwill be of order δr/rLS ∼ 10−3, where δr is a characteristic thickness of lastscattering. Again, due to the thinness of the last-scattering surface, thisapproximation will hold independently of the radius r of the quadrupole,and so for our purposes we can place the scatterers at one radius (rLS) and168Appendix B. Dipolar modulation of polarizationwrite the polarization in Eq. (B.1) asQ(nˆ)± iU(nˆ) ∝∑maT2m(rLSnˆ)±2Y2m(nˆ), (B.2)where the temperature quadrupole is modulated according toaT2m(rLSnˆ) = aT,i2m(rLSnˆ)(1 +Anˆ · dˆ). (B.3)In this Appendix, superscript i will indicate statistically isotropic and ho-mogeneous fields.Note that for the reionization bump at the very largest angular scales,ℓ . 10, we have rre/rLS ≃ 0.7. Therefore for polarization sourced at reion-ization, the error due to the variation of modulation, δr/rLS, will be largerthan for polarization from last scattering. Also, for a spatially linear mod-ulation the amplitude of modulation at reionization will be reduced by thefactor rre/rLS compared with that in Eq. (B.3). However, given the low sta-tistical weight for these few very-largest-scale modes, we ignore this effecthere. Therefore our approach will slightly overestimate the modulation ofthe reionization bump.Combining Eqs. (B.2) and (B.3) givesQ(nˆ)± iU(nˆ) = [Qi(nˆ)± iU i(nˆ)] (1 +Anˆ · dˆ) (B.4)=[Qi(nˆ)± iU i(nˆ)]×(1 +√4π3∑M∆XMY1M (nˆ)). (B.5)In words, the Stokes parameters are simply dipole modulated, just as tem-perature was in Eq. (3.2). Again, this approximation will be good for ourpurposes, i.e., for the sake of quantifying the effect of the modulation onpolarization. Note that Eq. (B.4) was taken as the starting point for polar-ization modulation in Ghosh et al. (2016).The E- and B-mode multipole moments are defined byQ(nˆ)± iU(nˆ) = −∑ℓm(aEℓm ± iaBℓm)±2Yℓm(nˆ), (B.6)which impliesaEℓm ± iaBℓm = −∫[Q(nˆ)± iU(nˆ)]±2Y ∗ℓm(nˆ)dΩnˆ. (B.7)169Appendix B. Dipolar modulation of polarizationCombining the previous three expressions gives, for the modulated E andB multipoles,aEℓm ± iaBℓm = aE,iℓm ± iaB,iℓm +∑M∆XM∑ℓ′m′(aE,iℓ′m′ ± iaB,iℓ′m′)±2ξMℓmℓ′m′ , (B.8)where±2ξMℓmℓ′m′ ≡√4π3∫P2Y lm(nˆ)Y1M (nˆ)±2Y ∗ℓm(nˆ)dΩnˆ (B.9)generalizes the usual ξMℓmℓ′m′ matrix in Eq. (2.6) to spin-2 fields.To evaluate the ±2ξMℓmℓ′m′ coefficients we can write the spherical harmon-ics in terms of the rotation matrices (Edmonds 1974), with the result±2ξMℓmℓ′m′ = (−1)m√(2ℓ+ 1)(2ℓ′ + 1)(ℓ′ 1 ℓm′ M −m)(ℓ′ 1 ℓ∓2 0 ±2). (B.10)For the case M = 0 (i.e., a polar modulation) the non-zero coefficients cantherefore be evaluated to be±2ξ0ℓmℓ+1m =1ℓ+ 1√(ℓ+m+ 1)(ℓ−m+ 1)(ℓ+ 3)(ℓ− 1)(2ℓ+ 3)(2ℓ+ 1), (B.11)±2ξ0ℓmℓm = ∓2m(ℓ+ 1)ℓ, (B.12)±2ξ0ℓmℓ−1m =1ℓ√(ℓ+m)(ℓ−m)(ℓ+ 2)(ℓ− 2)(2ℓ+ 1)(2ℓ− 1) . (B.13)These expressions agree with those in Ghosh et al. (2016). Note that±2ξ0ℓmℓm 6= 0 (for m 6= 0), so we expect ℓ, ℓ coupling in E and B modes.Also note that ±2ξ0ℓmℓ±1m are symmetric and ±2ξ0ℓmℓm is antisymmetric withrespect to a sign change of the spin index.Using these symmetry properties we can now evaluate Eq. (B.8) for thecase of modulation along the polar axis, taking sums and differences toisolate the E and B modes. The result isaEℓm = aE,iℓm +∆X0(aE,iℓ+1m 2ξ0ℓmℓ+1m + aE,iℓ−1m 2ξ0ℓmℓ−1m + iaB,iℓm 2ξ0ℓmℓm),(B.14)aBℓm = aB,iℓm +∆X0(aB,iℓ+1m 2ξ0ℓmℓ+1m + aB,iℓ−1m 2ξ0ℓmℓ−1m − iaE,iℓm 2ξ0ℓmℓm).(B.15)170Appendix B. Dipolar modulation of polarizationThese imply〈aEℓmaE∗ℓ′m′〉= CEℓ δℓ′ℓδm′m +∆X0(CEℓ + CEℓ′)2ξ0ℓmℓ′m (δℓ′ℓ−1 + δℓ′ℓ+1) δm′m,(B.16)〈aBℓmaB∗ℓ′m′〉= CBℓ δℓ′ℓδm′m +∆X0(CBℓ + CBℓ′)2ξ0ℓmℓ′m (δℓ′ℓ−1 + δℓ′ℓ+1) δm′m,(B.17)〈aEℓmaB∗ℓ′m′〉= i∆X0(CEℓ + CBℓ)2ξ0ℓmℓmδℓ′ℓδm′m, (B.18)to first order in ∆X0. The power spectra here are for the isotropic (un-modulated) fields, i.e., CEℓ ≡〈aE,iℓmaE,i∗ℓm〉etc., but agree with those of themodulated fields to first order in ∆X0. Using Eq. (2.5) for the modulatedtemperature modes, we find〈aTℓmaE∗ℓ′m′〉 = CTEℓ δℓ′ℓδm′m+∆X0[CTEℓ(2ξ0ℓmℓ−1mδℓ′ℓ−1 + 2ξ0ℓmℓ+1mδℓ′ℓ+1)δm′m+CTEℓ′ ξ0ℓmℓ′m′], (B.19)〈aTℓmaB∗ℓ′m′〉 = i∆X0CTEℓ 2ξ0ℓmℓmδℓ′ℓδm′m, (B.20)also to lowest order in ∆X0. Note crucially that we find coupling betweenB modes and E or T modes, which of course vanishes in the statisticallyisotropic case. However, we have∑m〈aEℓmaB∗ℓm〉=∑m〈aTℓmaB∗ℓm〉 = 0, (B.21)since ±2ξ0ℓmℓm is antisymmetric with respect to m.171Appendix CFilteringIn Chapter 1 a simplified noise model was used when treating the tempera-ture data. The model did not account for variations in the noise level dueto the Planck scanning strategy, or scale dependence of the noise power, orforeground signal. Not accounting for these effects was deemed adequatedue to the noise power being subdominant on the scales probed. However,for polarization this is not the case. Here we describe our approach to ac-count for the scale dependence of the noise and foreground power (as usedsimilarly in Planck Collaboration XV 2016). The corrections below are ap-plied after the filtering process and are the cause of the small differencesbetween the temperature results here and those in Chapter 1.Our starting point will be filtered data as in Chapter 1, denoted here assX, ℓm, where X = T or E. We multiply these by a quality factor QXℓ toobtain filtered data that are closer to optimal, definingXℓm = QXℓ sX, ℓm. (C.1)The choice of QXℓ is determined by the following two requirements:FXℓ =QXℓCXXℓ +NXXℓ; (C.2)FXℓ =f−1sky2ℓ+ 1∑m|Xℓm|2. (C.3)Here fsky =∑pMp/Npix, where Mp is the map in pixel space and Npix isthe total number of pixels. Further details can be found in Appendix A.1 ofPlanck Collaboration XV (2016).172Appendix DSimulating modulationparametersD.1 Isotropic estimatesThe FFP8 simulations require modification in order to combine isotropicpolarization data with temperature data. This is because the polarizationsimulations are not correlated with temperature data in the way that thetrue polarization data are. While this is a small correction, we describebelow how we perform it.For each polarization simulation to be included with the temperaturedata we modify the modulation estimator (X˜WZM ) in the following way:X˜WZ corM = X˜WZM + X˜TT, dataMCor(X˜WZM , X˜TTM)Var(X˜TTM) . (D.1)The correlation and variance are estimated with the statistically isotropicFFP8 simulations. Note that this procedure only modifies the values of theestimators (the X˜M s) and not the CMB simulations themselves. This pro-vides a significant computational speed-up when analyzing a large numberof simulations.If the X˜M s are Gaussian (which we verify to be true to high accuracy withour simulations) then this approach is exact and amounts to simply shiftingthe mean of X˜TEM and X˜EEM by an amount given by the fixed temperaturedata (Bunn et al. 2016).D.2 Anisotropic estimatesIn the appendix of Hanson and Lewis (2009) it was demonstrated how to gen-erate anisotropic maps from isotropic ones. Such an algorithm is convenient,but can be computationally expensive when scanning over many differentanisotropic models. In this Appendix we will demonstrate our strategy for173D.2. Anisotropic estimatesquickly generating modulated estimates (X˜M s) by using isotropic estimates(thus skipping the step of generating maps, filtering them, and computingestimates from them).For simplicity we will assume that we want to generate modulation in the+zˆ-direction; however, a general direction can be implemented by simplybreaking the direction into components. The following will make use ofbinned versions of the estimators of Eqs. (2.25)–(2.26). These can be writtenasX˜WZ0,ℓ =6∑mAℓmS(WZ)ℓm ℓ+1mδCWZℓℓ+1(ℓ+ 1)F(Wℓ FZ)ℓ+1. (D.2)Thus we see that at each multipole an estimate of the amplitude (and di-rection) can be made. If we want to generate an estimate of an anisotropicsimulation we can modify an estimate of an isotropic simulation in the fol-lowing way. First we compute Eq. (D.2) for an isotropic simulation at thedesired modulation parameters (e.g., p˜i = {k˜c, ∆ ln k˜} for the tanh model),which implies a particular anisotropic power spectrum δ˜CWZℓℓ+1. Then ananisotropic binned estimator can be obtained asX˜WZ ani0,ℓ∣∣∣p˜i= X˜WZ iso0,ℓ∣∣∣p˜i+ A˜, (D.3)where A˜ is the desired amplitude of modulation. The full estimator, Eqs. (2.25)–(2.26), for a general modulation model (δCℓℓ+1) can be recovered byX˜WZ ani0 =∑ℓ X˜WZ ani0,ℓ∣∣∣p˜i˜δCWZℓℓ+1δCWZℓℓ+1(ℓ+ 1)F(Wℓ FZ)ℓ+1∑ℓ(δCWZℓℓ+1)2(ℓ+ 1)F(Wℓ FZ)ℓ+1. (D.4)174Appendix EDetection and removal ofaberrationAberration due to our velocity relative to the CMB frame adds a term to theCMB temperature multipole covariance given by Challinor and van Leeuwen(2002) 〈aℓma∗ℓ+1m〉= −βAℓm [(ℓ+ 2)Cℓ+1 − ℓCℓ] . (E.1)Here the aℓms are defined in a coordinate system where the dipole direction,(l, b) = (264◦, 48◦), is aligned with the polar direction, and β = 1.23× 10−3is the magnitude of the temperature dipole Planck Collaboration VII (2016).In our notation this implies an asymmetry spectrum of the formδCℓℓ+1 = −2 [(ℓ+ 2)Cℓ+1 − ℓCℓ] . (E.2)Using this in our estimator gives us a constraint on β. With ℓmax = 2000we obtain β = (1.5± 0.5)× 10−3 in direction (l, b) = (281◦, 57◦)± 22◦, i.e.,a roughly 3σ detection of aberration, consistent with the observed CMBdipole and the results of Planck Collaboration XXVII (2014).We are then able to remove this signal from the temperature data usingthe method outlined in Appendix D.2. Specifically we use Eqs. (D.3) and(D.4) with A˜ = −β and δ˜CWZℓℓ+1 given by Eq. (E.2). Note that the high-ℓand oscillatory nature of Eq. (E.2) means that this procedure only noticeablyaffects the results for the ns gradient model.175Appendix FStacking on E-mode peaksHere we will discuss the new procedure of stacking on E-mode extrema. Weremind readers that the full implementation and derivation of all relevantparameters are discussed in great detail in Appendix B of Komatsu et al.(2011), and a similar derivation is given in Contaldi (2017) for stacking on Qand U extrema. Here we simply explain the few details required to extendthe formalism for stacking on E modes.Selecting the peaks of an underlying Gaussian field (like T or E) leadsto a biased selection of that field. Such a bias is scale-dependent and hasthe formδpk(nˆ) =[bν − bζ∇2]E(nˆ). (F.1)The scale-dependent term (bζ) arises because peaks are defined by a vanish-ing first derivative and the sign of the second derivative Desjacques (2008).The bias parameters depend entirely on rms values of derivatives of theGaussian field, σ0, σ1, and σ2 (they also depend on special functions involvedin translating a 3-dimensional Gaussian random field to the 2-dimensionalcase, as discussed in Bond and Efstathiou 1987). These are defined asσ2j ≡14π∫dnˆ(∇2)j E2(nˆ) (F.2)=14π∑ℓ(2ℓ+ 1)[ℓ(ℓ+ 1)]j(CEEℓ +NEEℓ )(WEℓ )2. (F.3)This is the only expression that contains the noise term NEEℓ , which iswhy understanding the noise properties of the E-mode map is potentiallyconsidered to be a relevant systematic effect (see Sect. 4.6.1).When we stack Qr or Ur on the location of E-mode peaks we are explic-itly computing the cross-correlation 〈δpk(nˆ)Qr(nˆ + θ)〉 or 〈δpk(nˆ)Ur(nˆ + θ)〉.Recalling that both Qr and Ur can be written in terms of E and B-modecontributions (Zaldarriaga and Seljak 1997; Kamionkowski et al. 1997) and176Appendix F. Stacking on E-mode peaksrewriting Eq. (F.1) in the flat-sky approximation we arrive at20〈δpk(nˆ)Qr(nˆ + θ)〉 =∫d2ℓ(2π)2WEℓ WPℓ (b¯ν + b¯ζℓ2){CEEℓ cos [2(φ− ψ)] + CEBℓ sin [2(φ− ψ)]}eiℓ·θ , (F.4)〈δpk(nˆ)Ur(nˆ + θ)〉 =∫d2ℓ(2π)2WEℓ WPℓ (b¯ν + b¯ζℓ2){CEBℓ cos [2(φ− ψ)]− CEEℓ sin [2(φ− ψ)]}eiℓ·θ . (F.5)Here we have used the coordinate convention of Komatsu et al. (2011), thusℓ = (ℓ cosψ, ℓ sinψ), and θ = (θ cosφ, θ sinφ). We can perform the internalintegration over ψ using properties of Bessel functions to finally arrive atEqs. (4.11)–(4.12), i.e.,〈δpk(nˆ)Qr(nˆ + θ)〉 = −∫ℓdℓ2πW Tℓ WPℓ(b¯ν + b¯ζℓ2)CEEℓ J2(ℓθ), (F.6)〈δpk(nˆ)Ur(nˆ + θ)〉 = −∫ℓdℓ2πW Tℓ WPℓ(b¯ν + b¯ζℓ2)CEBℓ J2(ℓθ). (F.7)These angular profiles could have been derived in a more heuristic wayby realizing that an E-mode map has the same statistical properties asa temperature map, differing only in its power spectrum. Thus we couldhave gone from Eqs. (4.9)–(4.10) to Eqs. (4.11)–(4.12) by simply makingthe replacement T → E.20For brevity we henceforth drop the noise term in the expression for CEEℓ .177

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