Secondary Signals in Cosmology

Home , Hidden sector, Matter power spectrum, Newtonian gauge, Particle horizon

Zhen Pan B.S. (University of Science and Technology of China) 2010 M.S. (University of Science and Technology of China) 2013

Dissertation

Submitted in partial satisfaction of the requirements for the degree of

Doctor of Philosophy

Physics

in the

Office of Graduate Studies

of the

University of California

Davis

Approved:

Chair Lloyd Knox

Andreas Albrecht

Christopher Fassnacht

Committee in Charge

2018

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� � � ProQuest 10823014 � Published by ProQuest LLC ( 2018 ). Copyright of the Dissertation is held by the Author. � � All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. � � ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 Copyright c 2018 by � Zhen Pan All rights reserved. To the Best of Times.

-ii- Contents

List of Figures ...... vi List of Tables ...... xi Abstract ...... xii Acknowledgments ...... xiii

1 Introduction 1 1.1 History of Thermal Big Bang: background ...... 2 1.1.1 Inﬂation and Reheating ...... 2 1.1.2 Annihilation, Recombination and Reionization ...... 3 1.1.3 Dark Energy ...... 4 1.2 History of Big Bang: perturbation ...... 5 1.2.1 Initial Conditions ...... 6 1.2.2 Baryon Acoustic Oscillation and CMB ...... 7 1.2.3 Dark Matter Collapse and Matter Power Spectrum ...... 7 1.3 Overview of My Research ...... 8

2 Dependence of the Cosmic Microwave Background Lensing Power Spec- trum on the Matter Density 11 2.1 Introduction ...... 11

2.2 The spectrum below� = 1000 andω m ...... 12 2.3 Introduction to the lensing power spectrum ...... 13 2.4 The dependence of lensing power spectrum on matter density ...... 16 2.4.1 Qualitative Analysis ...... 16 2.4.2 Quantitative Analysis ...... 18 2.5 Discussion ...... 20

3 Constraints on Neutrino Mass from Cosmic Microwave Background and Large Scale Structure 22 3.1 Introduction ...... 22

-iii- 3.2 Signatures of non-zero neutrino mass ...... 23 3.3 Influence of massive neutrinos on galaxy survey observables ...... 25 3.4 Influence of massive neutrinos on the CMB lensing power spectrum . . . 28 3.4.1 Introduction to the lensing power spectrum ...... 28 3.4.2 Influence of massive neutrinos on the lensing power spectrum: results 30 3.5 Forecast of constraints on the total neutrino mass from different data sets 33 3.5.1 CMB-S4 and DESI BAO ...... 33 3.5.2 Beyond DESI BAO ...... 34 3.6 Conclusion ...... 36

4 Cosmic Microwave Background Acoustic Peak Locations 38 4.1 Introduction ...... 38 4.2 Baseline Model ...... 40 4.2.1 Before Recombination ...... 40 4.2.2 After Recombination ...... 42 4.3 Evolution of phase shifts in the photon perturbations ...... 44

4.3.1 Decoupling:φ dcp ...... 45

4.3.2 Transient:φ gr,γ ...... 46

4.3.3 Neutrinos:φ gr,ν ...... 46 4.4 Phase shifts in photon perturbations at the LSS ...... 47 4.5 Projection ...... 49 4.5.1 A Rigorous Treatment of Projection ...... 50 4.5.2 Corrections to the Baseline Model ...... 51 4.6 Comparison of predicted and measured peak locations ...... 57 4.7 Conclusions ...... 59 4.8 Appendix ...... 60

4.8.1φ dcp ...... 60

4.8.2φ gr,γ ...... 62

4.8.3φ gr,ν ...... 63

-iv- 5 Searching for Signatures of Dark Matter-Dark Radiation Interaction in Observations of Large-scale Structure 65 5.1 Introduction ...... 65 5.2 Canonical DM-DRF Interaction Model ...... 67 5.2.1 LSS ...... 68 5.2.2 CMB ...... 70 5.3 Parameter constraints from LSS data ...... 73 5.3.1 Previous Analyses ...... 73 5.3.2 Anaysis with SZ data: the impact of the mass bias parameter . . 75 5.3.3 Analysis with only CMB Lensing and DES data ...... 77 5.4 Lyman-α forest data ...... 78 5.5 Generalized dm-drf models ...... 80 5.6 Summary ...... 82

-v- List of Figures

2.1 The lensing power spectrum calculated from CAMB (solid line), calculated with Limber approximation (dashed line) and calculated with Limber approximation and settingg(a) = 1 (dash-dotted line)...... 15 2.2 Factors in the integrand of the lensing power spectrum for large� = 800 (upper panel) and small� = 20 (lower panel)...... 17 2.3 The dependence of the lensing power spectrum on the matter density, �4Cφφ (ω )n. The solid line is the numerical result from CAMB, the � ∼ m dashed line is the result of the analytic scaling law derived in the text (n=0.75(m + 1) + 0.58) and the dash-dotted line is the result of the Limber approximation settingg(a)=1...... 19 2.4 The contribution of each multipole of the CMB lensing power spectrum,

φφ TT C� to the lensing of the CMB temperature power spectrum,C L , assuming ourﬁducial cosmology (WMAP9 best-ﬁtΛCDM model), for two values ofL near acoustic peaks (1455 and 2075), and two values near troughs (1300 and 2245)...... 20

3.1 The dependence of expansion rateH(z) and comoving angular diameter

2 distanceD A(z) onM ν, where we minimize theχ (Θ,Mν) by adjusting the

6ΛCDM parametersΘ when increasingM ν from 0 to 50, 100, 200 meV. 26

3.2 The dependence of structure growth rate onM ν, where we minimize the 2 χ (Θ,Mν) by adjusting the 6ΛCDM parametersΘ when increasingM ν from 0 to 20 meV...... 26 3.3 The lensing power spectrum calculated from CLASS (solid line), calculated with Limber approximation (dashed line) and calculated with Limber approximation and settingg(a) = 1 (dashed-dotted line)...... 30

-vi- 3.4 The dependence of the lensing power spectrum on total neutrino mass, (Cφφ C φφ )/Cφφ =R (M /eV). The black line is the numerical result � − �,fid �,fid � × ν from CLASS, the red line is the result of Limber approximation setting χ = 1.4 10 4 Mpc, the blue line is the result of Limber approximation � × withg(a) = 1, the magenta line is the result of Limber approximationfixing

bothω m andg(a), and green line is the result of Limber approximation

withA s, ns,ω m,ω b andg(a)ﬁxed...... 31 3.5 Forecasted 1σ and 2σ constraints in theM ω plane, where the CMB-S4 ν − m experiment results in aσ(M ν) = 38 meV constraint, the combination of

CMB-S4 and DESI BAO yield aσ(M ν) = 15 meV constraint. and adding measurements of the structure growth rate by DESI RSD further improves

the constraint toσ(M ν) = 9 meV...... 34

3.6 Same as Fig. 3.1, but with suppressed errorbars ofD A(z) andH(z) coming from CMB-S4 and a cosmic-variance-limited BAO experiment...... 35 3.7 The uncertainties in relative distances from CMB-S4 + DESI BAO. Note that the uncertainties is multiplied by a factor of 104 in the plot...... 36

4.1 Comparison of the spectra of theﬁducial cosmology (solid curves) and the peak locations predicted by the baseline model (vertical dashed lines). . 44

4.2 The intervalsΔ(kr s) of neighboring peak-trough of [Θ0 +Φ](k,η) for mode 1 k=0.5 Mpc − . Dots are numerical results of peak-trough intervals. Solid line is the analytic result of high-order correction to the tight coupling

approximationφ dcp. Dashed line is the result of corrections from both

late-time high-order correctionφ dcp and early-time gravitational driving

φgr,γ sourced by photon perturbations...... 45 4.3 Phase shifts of sources induced by 3.046 neutrinos and measured at the LSS. 47

4.4 The phase shifts of [Θ0 +Φ] (left panel) and ofΠ (right panel) induced by diﬀerent physical eﬀects measured at the LSS...... 48

-vii- 4.5 Illustration of the projection from three to two dimensions. The round circle is the LSS, the vertical solid(dashed) lines are the peaks(troughs) of

mode �k atη �. The wiggling curve around the LSS is a Legendre polynomial P (µ) with�=k(η η ), whereµ= kˆ ˆγ, is the cosine of the angle � 0 − � · subtended by the wavevector �k and the direction of observation ˆγ. In the µ = 0 direction, the peak-trough separation of the �k mode matches that

ofP �m (µ) �m=k(η0 η�) (shown). In theµ = 1 direction, the peak-trough | − � separation of the k mode matches that ofP �m (µ) �m k(η0 η�) (not shown). 51 | � − 4.6 The comparison between the asymmetric visibility functiong(η) and two

Gaussian functions both withσ = 20 Mpc, peaking atη � = 281 Mpc and ¯η = 293 Mpc, respectively, where ¯η is the mean decoupling time deﬁned by ¯η= g(η)ηdη/ g(η)dη...... 55

TT 4.7 The contribution� to� the unlensedD � from each component: dominant monopole, subdominant dipole and early ISW, and negligible polarization and late ISW...... 56

5.1 Upper Panel: the evolution of dark matter over-densityδ dm suppression for diﬀerentk modes. Middle Panel: the “self-similar” behavior of the over-density suppression, where we displace the suppression of modesk= 1 1 0.3 Mpc− andk=1.0 Mpc − by 0.05 and 0.096 respectively. Lower Panel: the dm-drf interaction induced matter power spectrum suppression today, where the dashed line is an analyticﬁt in the form of Equation (5.5). . . 71

-viii- 5.2 Comparison of TT and EE spectra of the three models listed in Table 5.1, where in the upper panel we plot the TT spectra with damping eﬀect largely removed by multiplying a factorK = exp 2 (�/1267) 1.18 , in the � { × } lower panel we plot the EE spectra, and in the two inset plots, we normalize the spectra amplitudes to allow one to see the impact of the very small shift in peak locations induced by free-streaming species as done in [1]. In these insets the red curve (the model with additional freestreaming neutrinos) is slightly shifted to the left relative to the dashed line and blue line which overlap each other...... 72

5.3 Theσ 8 tension in theΛCDM cosmology, where the redfilled contours (1σ and 2σ) are derived from Planck 2015 temperature and polarization, and the unfilled contours corresponding to the three LSS measurements are given at 2σ level, where the SZ contour is the constraintfixing the mass bias parameter as the baseline value 1 b=0.8...... 74 − 5.4 The results of the two MCMC runs (red vs. blue) for the canonical dm- drf model using joint dataset PlanckTP+PlanckSZ. Upper Left Panel: the comparison of the CCCP prior (blue/dashed) and the resulting posterior (blue/solid) with the baseline value 1 b=0.8 (red/dot-dashed). Lower − Left Panel: the posterior contour of 1 b vs.Γ . Lower Right Panel: − 0 the marginalized posteriors ofΓ 0 with the bias parameterfixed (red/dot- dashed) or varying (blue/solid)...... 76 5.5 The tension between Lyα and CMB inΛCDM cosmology with neutrino

2 massﬁxed as 0.06 eV (ΛCDM) or varying (νΛCDM) whereΔ L andn eﬀ are the amplitude and the slope of the linear matter power spectra at 1 k h Mpc − and atz=3...... 79 �

-ix- 2 5.6 Comparison of the Lyα constraints on the amplitudeΔ L and the slopen eff of the matter power spectrum with those derived fromΛCDM (bluefilled contours) and the dm-drf interaction model (red unfilled contours) using

PlanckTP+Lensing+DES, where the color points denote diﬀerentΓ 0 values 7 1 in unit of 10− Mpc− ...... 79 5.7 Left Panel: the matter power spectrum suppression from diﬀerent dm-drf

β 1 5 7 11 interaction ratesΓ=Γ 0(T/T0) (withΓ 0/Mpc− = 10− , 10− , 10− for

β=1,2, 3, respectively), where the red band denotes the modesσ 8 is sensitive to and the blue band denotes the modes Lyα measurement is sensitive to. Right Panel: the comparison between the Hubble expansion rate and the dm-drf interaction rates...... 80

2 5.8 Comparison of the Lyα constraint on the amplitudeΔ L and the slopen eff of the matter power spectrum with those derived fromΛCDM and dm- drf interaction models using PlanckTP+Lensing+DES, where the bluefilled contours are the results of theΛCDM cosmology, the red unfilled contours

are the results of the dm-drf models, and the color bars denoteσ 8 for dm- drf withβ=1, 3, respectively. Two right panels show the sameβ=3 model but with diﬀerent ranges on the x-axis...... 81

-x- List of Tables

TT 4.1 The shift of thep-th peak in the temperature power spectrumD � deﬁned

byδ� p 302p � p(TT), consists ofδ� monopole,δ� non monopole andδ� lensing, ≡ − − where the former can be decomposed asδ� monopole =δ�[φ gr,ν] +δ�[φ gr,γ] + 2 δ�[φdcp] +δ�[k P(k)] +δ�[j �] +δ�[g(η)]...... 52 EE 4.2 The shift of thep-th peak in the polarization power spectrumD � deﬁned by δ� 302(p 0.5) � (EE), consists of δ� and δ� , where the p ≡ − − p Π lensing 2 former is decomposed asδ� Π =δ�[φ gr,ν] +δ�[φ gr,γ] +δ�[φ dcp] +δ�[k P(k)] +

δ�[j�] +δ�[g(η)]...... 52 4.3 The shift of thep-th peak in the power spectrumD TE is defined byδ� � p ≡ 151(p+0.5) � (TE), and notations used here are similar to those in Table − p 4.1 ...... 52 4.4 Locations of the peaks in the power spectra. The peak locations measured from the Planck 2015 data are listed in the 3rd column [Table E.2. in 2], and the peak locations predicated by thefiducial cosmology are listed in the 2nd column (Note that these peak locations are determined by the fitting procedure used on the data, therefore are different from the literal peak locations of theoretical power spectra)...... 58

5.1 Three models used for clarifying the impact of dark radiationfluid on cosmological observables, with different parameters N ,N ,Γ and same { ν drf 0} 9 ΛCDM parametersω = 0.02253,ω = 0.1122,A = 2.42 10 − , n = b dm s × s 0.967,τ=0.0845,H 0 = 70.4 km/s/Mpc...... 69 5.2 Constraints on the dm-drf model parameters using datasets PlanckTP, PlanckSZ, Lensing and DES, where the uncertainties are 1σ values, and the upper limits are given at 2σ confidence level...... 76

-xi- Abstract

Secondary Signals in Cosmology

In this thesis, I summarize my graduate student research of the secondary signals in cosmology, including effects to which pre-Planck cosmic microwave background (CMB) observations are not sensitive, but which are becoming increasingly important for constraining theΛCDM cosmology in the Planck era and beyond, and extensions to the standard ΛCDM. It is of particular importance to understand how the cosmological signals depend on the cosmological parameters via these physical effects, since such understanding is not only useful for constructing clear pictures of underlying physics, but also valuable for guiding future experimental designs aiming to detect some anticipated cosmological signals. To better understand effects of increasing importance in the Planck era and beyond, I have worked out an analytic understanding of the CMB lensing sensitivity to the matter densityω m, detected the neutrino induced phase shift in the Planck 2013 TT power spectrum in Chapter 2; I have examined in detail the physical effects of neutrino mass M on cosmological observables, and in particular demonstrate how to break theM ν ν − ωm degeneracy by combining different datasets in Chapter 3; I have also quantitatively explored the physical effects determining the peak locations in the CMB power spectra in Chapter 4. I have also worked to develop and constrain extensions to theΛCDM model, as well as other theoretical frameworks relevant to cosmology in Chapter 5. Motivated by solving theσ 8 tension problem, I have constrained a dark sector model with self-interacting dark matter and dark radiation, using the latest CMB and Large-scale Structure measurements. We found that prior claims of a detection of this interaction rely on a use of the SZ cluster mass function that ignores uncertainty in the mass-observable relationship. Including this uncertainty wefind that the inferred level of interaction remains consistent with the data, but so does zero interaction; i.e., there is no longer a detection.

-xii- Acknowledgments

During my graduate career at Davis, I have received numerous help from members of the Physics Department, the Statistics Department and of other universities. At the moment of receiving my long desired Ph.D, I want to thank them all. First, I would like to thank my advisor Prof. Lloyd Knox, APS Fellow, who is not only my mentor in academic research but also helps me a lot in daily life. Aboutfive years ago when I moved to Davis, it was myfirst time to leave my home country and unsurprisingly many difficulties emerged because most of my previous survival skills in China ceased to work here. Lloyd reached out to me, drove me and my roommate to a furniture store in the west Sac and the furniture I purchased on that dayfilled my empty apartment, which greatly facilitated my quick settlement in a completely new environment. In academic research, Lloyd has been guiding me to the broad and fantastic world of cosmology, instead of dictating me to do research on a given path. Lloyd was also rather open-minded even sometimes I was distracted by non-cosmology topics, e.g., black hole physics and planet meteorology. With Lloyd’s support and encouragement, I have not only enjoyed much aesthetic pleasure in cosmology study, but also developed some “explorative” interests beyond cosmology in broader regime of physics, which was one of the major reasons that Perimeter Institute (PI) decided to offer me the position of Zel’dovich Fellow, according to Prof. Neil Turok, the director of PI. Thank you, Lloyd! I am also very grateful to Prof. Andreas Albrecht, Associate Prof. Ethan Anderes, Prof. Manoj Kaplinghat (of University of California, Irvine), Prof. Cong Yu (of Sun Yat-Sen University), Associate Prof. Lei Huang (of Shanghai Astronomical Observatory) and two previous members of Lloyd’s research group, Dr. Brent Follin and Dr. Marius Millea. They have helped me a lot in various aspects, especially in my research and postdoc application. I am also thankful to my qualifying exam committee members, Prof. Chris Fassnacht, Prof. Marusa Bradac, and Prof. Bob Svoboda. Without your support, it is impossible for me to go this far. Thank you!

-xiii- Chapter 1

Introduction

Precision measurements of the Cosmic Microwave Background (CMB), e.g. by Planck [2, 3], have led to cosmological parameter constraints with percent level uncertainties. As the observations have progressed, various physical eﬀects to which pre-Planck CMB observations are not sensitive, are becoming increasingly important for constraining the ΛCDM cosmology in the Planck era and beyond. It is of particular importance to understand how the precision CMB signals depend on the cosmological parameters via these physical eﬀects, since such understanding is not only useful for constructing clear pictures of underlying physics, but also valuable for guiding future experimental designs aiming to detect some anticipated cosmological signals. Complementing the precision CMB measurements, recent Large-scale Structure (LSS) surveys [4–10] have also achieved similar precision in constraining cosmological param-

1 eters, such asσ 8, the amplitude of matterfluctuations on 8h − Mpc scale. Since the LSS surveys and the CMB measurements are sensitive to physics of different scales and (highly) different redshifts, it is enlightening to compare the two: if the two are consistent in the framework of the standardΛCDM model, then the model validity would be verified in a much larger scope; if the two yield some unreconciled tension, then it would be an indication of potential unrecognized systematics in the data analysis or more excitingly signals of new physics. I give a brief introduction to the standardΛCDM model in this chapter, then summarize my graduate student research about secondary cosmological signals in the remaining

1 chapters. Throughout this chapter, I use the conventionc = 1.

1.1 History of Thermal Big Bang: background 1.1.1 Inflation and Reheating To a good precision, our universe isflat, homogeneous and isotropic on large scale today [2]. To explain these remarkable features, the inflation paradigm was proposed and has been well accepted [11–14]. According to the inflation paradigm, our universe went through a stage of accelerated expansion at very early time. During the accelerated expansion, the curvature of the universe is exponentially suppressed, and a initially small causally connected patch exponentially expands to a large homogeneous patch which con- tains the whole observable universe. The expanding,flat, homogeneous and isotropic universe is described by the Friedman- Robertson-Walker (FRW) metric

ds2 = dt 2 +a 2(t) dx2 + dy2 + dz2 , (1.1) − � � wherea(t) is the time-dependent scale-factor, and we usually define the universe’s expan- t sion rate asH(t) =˙a/a (da/dt)/a and the conformal time asη= dt�/a(t�). Applying ≡ 0 Einstein’sfield equations on the FRW metric, we obtain the evolution� equations 8πG H2 = ρ, 3 (1.2) ä 4πG = (ρ+3p), a − 3 which shows that the acceleration of the universe is proportional to (ρ+3p), withρ and − p being the average energy density and pressure, respectively. For a universefilled with normal matter and radiation with positive energy density and pressure, the universe must be decelerating (ä< 0). For an accelerating universe, e.g. our universe in the inflation stage, the dominant energy form must have some abnormal composition satisfying the energy conditionρ+3p < 0, which can be realized by a slowly evolving scalarfield with potentialV(φ): 1 1 ρ= φ˙2 +V(φ), p= φ˙2 V(φ), (1.3) 2 2 −

2 where 1 φ˙2 V(φ) andp ρ. This is the basic picture of (slow-roll) inflation. 2 � �− In the end of inflation, the coupling between the inflatonfieldφ and matter, including standard model (SM) matter, dark matter (DM) and other possible beyond SM (BSM) matter, becomes important and it converts the inflaton energy into matter [15]. The details of this process are far from clear, due to the crucial dependence on the underlying particle physics theory beyond the SM. But as long as the matter particles are produced, they quickly reach local thermal equilibrium as a result of mutual scattering. We usually call the matter production from the inflationfield decay as reheating and call the produced matter as primordial plasma.

1.1.2 Annihilation, Recombination and Reionization As the universe expands, the temperature of the primordial plasmaT drops. For a heavy particle with massm i and in thermal equilibrium with other particles, its abundance m /T is suppressed by a factore − i . The heavy particle abundance freezes out when the annihilation rate to lighter particlesn σv drops far below the expansion rateH, where i� rel� ni,σ andv rel are the number density, annihilation cross section and relative velocity of the heavy particles, respectively. In the weakly interacting massive particle (WIMP) scenario [16], DM particles with σ v of order 1039 cm2 successfully explains the DM � i rel� relic abundanceΩ 0.3, which makes WIMP one of the most plausible DM candidates. DM ∼ When the temperatureT drops to a few MeV, the universe isﬁlled with neutrinos, electron/positron pairs, protons, neutrons and photons (νi, e±, p, n,γ) in thermal equilibrium and thermally decoupled DM. Neutrinos also decouple from the rest of the plasma as the temperature further drops to 1.5 MeV, but the temperature of neutrinosT stay � ν the same as the photon temperatureT , since both of them drop as 1/a. The equality γ ∝ breaks down aroundT m 0.5 MeV, when electron/positron pairs quickly annihilate ∼ e � + into photonse − +e 2γ, which depletes almost alle ± since the large annihilation cross → section keepse ±,γ in chemical equilibrium untilT m when the abundance ofe ± is � e me/T exponentially suppressede − . The entropy conservation during the annihilation gives

3 1/3 1/3 Tν gγ 4 = 7 = , (1.4) T + 11 γ � gγ + 8 (ge− +g e ) �after e± annihilation � � � � where we have used� the degree of freedomg =g =g + = 2, and the property of the � γ e− e � entropy of relativistic speciess T 3. The CMB temperature today was well measured ∝ by Wilkinson Microwave Anisotropy Probe (WMAP) to beT γ(a = 1) = 2.726 K with 0.1 percent level uncertainty [17], from which we also obtain the cosmic neutrino background

temperatureT ν(a = 1) to the same precision.

After thee ± annihilation, photons are tightly coupled with the residual electrons via Compton scattering (n σ H, whereσ is Thompson scattering cross section) and e T � T electrons to protons via Coulomb scattering. As the temperature drops to 1 eV, the ∼ reactionp+e − H+γ remains in equilibrium and all hydrogen atoms are ionized. As ↔ the temperature drops to 0.3 eV, the charged particles quickly combine into neutral ∼ atomsp+e − H+γ which depletes all free electrons and protons [18, 19] (similar to → the electron pair annihilation). After recombination, almost all electrons are conﬁned within neutral hydrogen and photons freely stream around. The situation changes when theﬁrst objects in the universe form and radiate ionizing photons which revert the neutral hydrogen into plasma at redshiftz 6 ([20, 21]). However, the free electrons only scatter photons mildly,n σ ∼ e T � H, since the electrons are vastly diluted by the expansion of the universe. We call this process as reionization and usually quantify the photon scattering contributed by the

reionization electrons by the reionization optical depthτ re, which is much smaller than unity.

1.1.3 Dark Energy Two-thirds of the energy in the universe today is dark energy, which is an energy form with negative pressure and which drives the accelerating expansion of the universe. In the dark energy dominated universe, the luminosity distance for objects at high redshifts is larger than that in a matter dominated universe. Therefore objects ofﬁxed intrinsic brightness appear fainter in the dark energy dominated universe, which is exactly what was discovered by two supernovae observing groups [22, 23]. In the standardΛCDM cos-

4 mology, we assume a dark energy with equation of statep = ρ , which is equivalent Λ − Λ to the cosmological constant.

To summarize, in the standardﬂatΛCDM cosmology, the background evolution of the universe is completely speciﬁed by the following 4 parameters: the Hubble expansion rate

todayH 0, the energy density of each composition: baryonρ b and DMρ dm (the photon

energy densityρ γ and neutrino energy densityρ ν are known to high precision, and the 2 dark energy densityρ Λ determined by the total energy densityρ tot = 3H0 /8πG), and the

reionization optical depthτ re.

1.2 History of Big Bang: perturbation

As stated above, the homogeneous, isotropic and expanding universe is described by the FRW metric, which is characterized by the scale factora(t). The perturbed universe requires two more functions,Ψ(�x,t) andΦ(�x,t),

ds2 = (1 +Ψ(�x,t)) dt2 +a 2(t) (1 +Φ(�x,t)) dx2 + dy2 + dz2 , (1.5) − � � where the Newtonian potentialΨ and the spatial curvature perturbationΦ are small quantities.1 To track the perturbation evolution of the universe, we need to specify reasonable conditions for the metric perturbationsΨ andΦ, and perturbation of each composi-

tion: baryon energy over-densityδ b(�x,t) = δρb/ρb, baryon velocityv b(�x,t), DM energy

over-densityδ dm(�x,t) = δρdm/ρdm and DM velocityv dm, photon perturbation variable Θ(�x,ˆp,t) =δT/T , and neutrino perturbation variable (�x,ˆp,t) =δT /T , where the γ γ N ν ν perturbation variablesΘ and depend not only on spacetime position but also on N the propagation direction of the relativistic particle. It is more convenient to work in the Fourier space, i.e.δ (k, t), v (k, t),δ (k, t), v (k, t),Θ(k, µ, t) and (k, µ, t), where b b dm dm N 1In general, the metric perturbations can be categorized into three types: scalar, vector and tensor perturbations. In this thesis, we focus on the scalar perturbation and work in the Newtonian Gauge for convenience.

5 µ=ˆp kˆ. Furthermore, we usually deﬁne multipoles · 1 1 Θl(k, t) = l l(µ)Θ(k, µ, t), (1.6) ( i) 1 P − �− and the same for , where is the Legendre polynomial of orderl. Now we proceed to N l P l specify initial conditions for these over-densities, velocities and multipoles.

1.2.1 Initial Conditions The inflation paradigm not only elegantly solves the horizon and theflatness problem, but also explains the origin of the primordial perturbations which seed the formation of large-scale structure (LSS). According to the inflation paradigm, the primordial perturbations originate from the quantumfluctuations of the inflationfield [24–28]. The quantum fluctuation of the inflationfield δφ is dictated by the uncertainty principle, δφ √�/λ, ≈ whereλ is the wavelength. This relation clearly shows quantumfluctuations are substan- tial only on the Planck scale and are strongly suppressed on macroscopic scales. With the expanding universe, thefluctuationδφ decays with the expanding wavelengthλ. However, thefluctuation is frozen out asλ becomes larger than the horizon when the crests and the troughs of the wave cease to be in causal contact. Therefore the quantumfluctuations on Planck scales are stretched to macroscopic scales with amplitudes nearly unchanged. The inflation paradigm predicts a nearly scale invariant primordial perturbation power spectrum ns 1 k3 k − P (k)=A (with 0<1 n 1), (1.7) 2π2 Φ s k − s � � ∗ � whereA s, ns andk are the scalar amplitude, spectral index andfiducial scale (e.g. 0.05 ∗ 1 Mpc− ), respectively. All other perturbation variables are connected toΦ via the Boltzmann and Einstein equations, which are complicated in general but simplify greatly in limit of early time when the perturbation modek extends well beyond the horizonk/(aH) 1 [29]: � 1 Θ = = Φ, 0 N 0 2 3 δ =δ = Φ, (1.8) b dm 2 Ψ= Φ, −

6 and all other variables are vanishing, where the vanishing “velocities”v , v ,Θ and b dm 1 N 1 show that the primordial perturbations of diﬀerent modes are in phase. For simplicity, we have kept only the leading order terms and neglect terms of order k/aH, see Ref. [30] for a more systematic analysis.

To summarize, in the standardﬂatΛCDM cosmology, the initial conditions of the primordial (scalar) perturbations are completely speciﬁed by the 2 parameters,A s and ns.

1.2.2 Baryon Acoustic Oscillation and CMB Before recombination, photons are tightly coupled with baryons in the plasma state. The photon-baryon plasma conducts acoustic oscillations due to the photon pressure. Combing with the in-phase initial conditions, we expect the oscillations to behave likeΘ (k,η) 0 ∼ cos(kcsη), wherec s is sound speed of photon-baryon plasma. After recombination, all free electrons are conﬁned within neutral atoms and the photons freely stream. As a result, the acoustic oscillation pattern is frozen at recombination cos(kc sη ), whereη ∼ ∗ ∗ is the conformal time of the last-scattering surface of CMB photons. In the following chapters, we will see that the simple acoustic oscillation picture roughly explains the peaks and troughs in the CMB power spectra. Many other physical processes, including resonant driving of gravitational potential decay in the radiation dominated epoch, the asymmetry in the acoustic oscillation arising from the baryon loading, the oscillation amplitude damping from photon diﬀusion, and the phase shift in the oscillation induced by free-streaming species, are also critical in shaping the CMB power spectra.

1.2.3 Dark Matter Collapse and Matter Power Spectrum In the standardΛCDM model, we assume DM is cold and has no non-gravitational interaction, therefore DM collapses under gravity as long as the perturbation modek enters the horizon (kη� 1). While the DM matter overdensityδ dm grows logarithmically with the scale factora in the radiation dominated epoch,δ dm grows linearly with the scale factor a in the matter dominated epoch. Combing with the initial conditions of the primordial

7 perturbations, we can robustly compute the matter power spectrum today.

Both the CMB and the matter power spectrum contain rich information about the primordial perturbations and the evolution history of the universe, and we will explore how to extract this information (speciﬁed by 6 parameters ρ ,ρ ,H ,τ ,A , n in the stan- { b dm 0 re s s} dardΛCDM model) from cosmological observables in detail in the following chapters.

1.3 Overview of My Research

In this thesis, I focus on the secondary signals in cosmology, including physical effects that are subdominant in pre-Planck era but become increasingly important in constraining model parameters in the Planck era due to higher experimental sensitivity, and signals of new physics that might resolve the tensions between different datasets. In Chapter 2 we examine the dependence of the lensing of CMB anisotropies on the matter density, which provides increasing leverage as CMB anisotropies of smaller angular scales have been well measured by progressing observations [31]. We provide an analytic explanation offinding

that the CMB lensing dependence on the matter density (ωm) mainly comes from the shape of the matter power spectrum set by the decay of small-scale potentials between horizon crossing and matter/radiation equality.

The tightest upper limit on the sum of neutrino mass eigenvaluesM ν comes from cosmological observations that will improve substantially in the near future, enabling a detection. In Chapter 3 we examine in detail the physical effects of neutrino mass on cosmological observables that make these constraints possible [32]. We found that increasing neutrino mass changes the expansion history due to an increase in energy density at all redshifts. To keep the likelihood given CMB data from decreasing dramatically, we need to keep the distance to the last scattering surfacefixed, which can be realized by decreasing the dark energy densityΛ. Therefore increasing neutrino mass leads to changes in H(z) with a particular shape: a mild increase at high redshifts, a larger decrease in low redshifts, with a transition near the onset ofΛ-domination atz 1. Since increasing the � matter density has the same effect in altering the expansion history as increasing neutrino

8 massM ν,ω m andM ν are negatively correlated from baryon acoustic oscillation (BAO) measurements. We also found the CMB-Stage IV (CMB-S4) sensitivity to the neutrino mass comes via the impact of increased expansion rates on structure growth. At scales below the neutrino free-streaming length, this increased expansion rate suppresses the growth of structure. Above the free-streaming length, the ability of massive neutrinos to cluster compensates for the increased expansion and there is no net suppression. Because increasing matter density increases lensing power amplitude as we explored in [31], the CMB lensing-derived constraints on neutrino mass have uncertainties positively correlated with the matter density uncertainties. Therefore the combination of CMB and BAO leads to improvements in the determination of both quantities. The peak structure of CMB power spectra itself has drawn special attention. About two decades ago, when only thefirst TT peak was readily measured [33–48], it was found to be consistent with the standardΛCDM model with adiabatic initial conditions, and imposed tight constraints on other competing models, for exampleΛCDM model with isocuravture initial conditions and topological defect models [e.g. 49–52]. With more peaks measured in recent years [53–63], the topological defect models were ruled out [64], and the constraints on the isocurvature modes have been improved to unprecedented precision [e.g. 65–74]. As found by [75], the propagation of neutrino (or any other free- streaming species) perturbations at speed faster than the speed of sound in the photon- baryon plasma alters the time-dependent gravitational driving of the acoustic oscillations and induce a (nearly) constant phase shift to the plasma acoustic oscillation, which is a unique signature of free-streaming species on CMB power spectra and hasfirst been detected by Follin et al. [1]. Despite the importance of the peak structure of the CMB power spectra, an analytic understanding of the peak locations had been lacking since the first CMB TT peak discovery by Cosmic Background Explorer (COBE). It is timely to provide such an analytic understanding in the Planck era, where the Planck CMB data are accurate enough to determine the peak locations of the power spectra to δ� 1 for � �� 1400 [2]. Many analytic studies have been done to explain the structure of these spectra [49, 76–

9 79]. However, there are large diﬀerences in expectations for extrema locations from simple analytic models vs. numerical calculations. In Chapter 4, we quantitatively explore the origin of these diﬀerences in gravitational potential transients, neutrino free-streaming, the breakdown of tight coupling, the shape of the primordial power spectrum, details of the geometric projection from three to two dimensions, and the thickness of the last scattering surface [1, 80]. Both the Planck CMB data and recent LSS surveys are accurate enough to constrain

σ8 with percent level uncertainties, but the CMB-derivedσ 8 constraint assumingΛCDM cosmology is 3σ lower than those derived from the LSS surveys. Motivated by theσ ∼ 8 tension, we conduct in Chapter 5 a search in the latest LSS measurements for signatures of the dark matter-dark radiation interaction proposed by Buen-Abad et al. [81] . We show that prior claims of a detection of this interaction rely on a use of the SZ cluster mass function that ignores uncertainty in the mass-observable relationship. Including this uncertainty wefind that the inferred level of interaction remains consistent with the data, but so does zero interaction; i.e., there is no longer a detection. We also point out that inference of the shape and amplitude of the matter power spectrum from Lyman-α forest measurements is highly inconsistent with the predictions of theΛCDM model conditioned on Planck CMB temperature, polarization, and lensing power spectra, and that the dark matter-dark radiation model can restore that consistency. We also phenomenologically generalize the model of to allow for interaction rates with different scalings with temperature, andfind that the original scaling is preferred by the data [82].

10 Chapter 2

Dependence of the Cosmic Microwave Background Lensing Power Spectrum on the Matter Density

This chapter originally appeared as an article in MNRAS [31].

2.1 Introduction

Observations of the cosmic microwave background (CMB) temperature power spectrum have provided us with highly precise determinations of cosmological parameters [3, 83]. These determinations are widely used to aid in the interpretation of other cosmological observables and to search for possible failures of the standard cosmological model that may point us toward new physics. Highly precise inferences are possible due to the way the rich features in the power spectrum depend on the underlying model parameters, the ability to compute the spectrum with high accuracy [84] and recent high-precision measurements of the anisotropy [3]. Given the widespread use of these parameter inferences, it is important that we acquire a physical understanding of how the parameter constraints arise. Fortunately, the anisotropies arise from relatively simple physical processes so such an understanding is possible [85]. As emphasized by Hou et al. [86], gravitational lensing of the CMB temperature

11 anisotropy power spectrum allows for constraints on the matter density from measurements of the small-scale (high�) portion of the spectrum. Such precise measurements are now becoming available thanks to the Planck [87], South Pole Telescope [SPT; 88] and Atacama Cosmology Telescope [ACT; 89] collaborations. However, a physical understanding of the dependence of the CMB lensing power spectrum on the matter density has so far been missing. In this article we provide an understanding of this dependence, within the context of theΛCDM model. We willﬁnd that it is the decay of the potentials after horizon crossing due to the presence of radiation which drives the majority of the dependence and give a simple scaling relation which captures this dependence. In Section 2.2 we review the origin of the sensitivity of large-angular scale CMB temperature measurements to the physical matter density,ω Ω h2. In Section 2.3 we m ≡ m introduce the lensing potential power spectrum and the basic equations we will use. With those preliminaries, we go on in Section 2.4 to demonstrate qualitatively and quantitatively the origin of the sensitivity of the CMB lensing potential power spectrum toω m. We present our major conclusions in Section 2.5.

2.2 The spectrum below� = 1000 andω m The rich structure in thefirst few acoustic peaks allows us to simultaneously constrain a number of cosmological parameters, including the matter density. Here we briefly review the physics behind these constraints [see e.g. 29, 90, for textbook discussions]. The peaks in the CMB spectrum arise from gravity-driven acoustic oscillations in the primordial, baryon-photon plasma before recombination. For nearly scale-invariant, adiabaticfluctuations the baryon-photon momentum density ratio,R, causes thefluid to oscillate about an offset value, leading to a modulation in the heights of the power spectrum features with enhanced compression (odd) and diminished rarefaction (even) peaks. Physically, the baryons provide a “weight” in the baryon-photonfluid, making it easier to fall into potential wells but harder to climb out. In contrast the effects of photon self-gravity cause an enhancement of thefluctuations for�� 10 2. This effect is most easily understood by considering the evolution of an

12 overdensity in a potential well as it enters the (sound) horizon and begins to collapse. Since thefluid has pressure, supplied by the photons, it is hard to compress and the infall into the potential is slower than free-fall. Since the overdensity is growing slowly the potential begins to decay due to the expansion of the Universe. The time for the photons to reach their state of maximum compression is the same time scale for the decay of the potential, and the compressed photons do not have a strong potential to work against as they climb back out. Thus the potential decay provides a near-resonant driving and leads to a large1 increase in power [77]. The boost from the in-phase potential decay is larger the more the self-energy of the baryon-photonfluid contributes to the total potentials, i.e. the earlier in time and the smaller the (stabilizing) dark matter contribution to the potentials. The boost thus imprints a dependence on the matter density2, in a similar way to the peak in the matter power spectrum except that thefluctuations in the CMB increase to smaller scales. The matter density can thus be constrained by the heights of the higher order peaks once enough peaks are seen to disentangle the confounding effects of baryon loading and tilt of the initial spectrum. Since much of the constraining power of the amplitude comes from measurements of

2 the anisotropy with�� 10 , the dependence of these multipoles onω m atﬁxedA s implies that inferences ofA s atﬁxedC � also depend onω m, though in the opposite manner.

2.3 Introduction to the lensing power spectrum

We begin with a brief review of gravitational lensing of the CMB. For details see e.g. the review by Lewis and Challinor [91]. Gradients in the gravitational potential,Φ, distort the trajectories of photons traveling to us from the last scattering surface. The deflection angles, in the Born approximation, ared= φ, where the lensing potential,φ, is a ∇ weighted radial projection ofΦ. Among other effects, these deflections alter the CMB

1In the absence of neutrinos, and if the self-gravity of the baryon-photonﬂuid dominates the potentials, the amplitude of the oscillation is enhanced by 2Ψ on top of the Ψ/3 plateau at large scales. In popular models the increase of a factor of 5 (from Ψ/3 to 5Ψ/3) is slightly− tempered by the inclusion of neutrinos and further diminished by the stabilizing− presence of dark matter. 2Or epoch of equality. We shall assume that the radiation density is known, so that equality depends primarily onω m.

13 temperature power spectrum. Regions that are magnified have power shifted to lower�. Regions that are demagnified have power shifted to higher�. The net effect of averaging over these regions is a smoothing out of the peaks and troughs, and a softening of the exponential fall off of the unlensed damping tail to a power law (set by the projected potential power spectrum and the rms of the temperature gradient). The key quantity for calculating the impact of lensing on the temperature power φφ spectrum is the angular power spectrum of the projected potential,C � , which we also call the lensing power spectrum. Taking advantage of the Limber approximation [92], it can be written as a radial integral over the three dimensional gravitational potential power spectrumP Φ

χ� � χ 2 �4Cφφ 4 dχ(k 4P ) ;a 1 , (2.1) � � Φ χ − χ �0 � � � � � whereχ is the comoving distance from the observer,a=a(χ), a� subscript indicates the last scattering surface, (1 χ/χ )2 is the lensing kernel, and the power spectrumP is − � Φ deﬁned as � (D) Φ(k;a)Φ (k�;a) =P (k;a)δ (k k �). (2.2) � � Φ − p To calculateP Φ we assume a power-law primordial power spectrumP Φ(k),

ns 1 k3 k − P p(k)=A , (2.3) 2π2 Φ s k � 0 � wherek 0 is an arbitrary pivot point,A s andn s are the primordial amplitude and power law index respectively. There are two conventions for the choice of the pivot pointk 0: 1 3 1 4 0.002 Mpc− (constraints from WMAP ) and 0.05 Mpc− (CAMB ). With our assumption of a power-law spectrum the two amplitudes are related by

ns 1 A (CAMB) =A (WMAP9) 2.5 − . (2.4) s s ×

We adopt the CAMB convention in this paper and henceforth writeA s(CAMB) asA s.

For many of our formulae we shall additionally approximaten s by 1, thus rendering the distinction moot. 3http://lambda.gsfc.nasa.gov 4http://camb.info

14 ��

��

Figure 2.1. The lensing power spectrum calculated from CAMB (solid line), calculated with Limber approximation (dashed line) and calculated with Limber approximation and settingg(a) = 1 (dash-dotted line).

The gravitational potential at late times,Φ(k, a), is related to the primordial potential Φp(k) by [e.g. 93] 9 Φ(k, a) = Φp(k)T(k)g(a), (2.5) 10 where the potential on very large scales is suppressed by a factor 9/10 through the transition from radiation domination to matter domination. For modes which enter the horizon during radiation domination when the dominant component has signiﬁcant pressure (p ρ/3) the amplitude of perturbations cannot grow and the expansion of the Universe ≈ forces the potentials to decay. For modes which enter the horizon after matter-radiation equality (but before dark energy domination) the potentials remain constant. The transfer functionT(k) takes this into account, being unity for very large scale modes and falling

2 approximately ask − for small scales. Once the cosmological constant starts to become important the potentials on all scales begin to decay. This eﬀect is captured by the growth function,g(a), which is unity during matter domination. With these deﬁnitions

ns 1 81π2 k − (k4P )(k;a) = A g2(a)kT 2(k) . (2.6) Φ 50 s k � 0 � With these pieces in place, we now examine the accuracy of the Limber approximation. Fig. 2.1 shows the lensing power spectrum calculated from CAMB compared to the lensing

15 power spectrum calculated within the Limber approximation. We see the agreement is

very good for�� 20, since the width of the kernel is much larger than the wavelength of the modes which dominate the signal on these scales. In order to understand the inﬂuence of the growth functiong(a), we also calculated the lensing power spectrum by settingg(a) 1. The growth function makes a diﬀerence only for� 50. This is because ≡ � φφ for large� the major contribution toC � comes from midway between the last scattering surface and the observer, which is well within the matter dominated era when the growth functiong(a) is close to unity. We are now ready to understand the impact of the matter φφ density onC � .

2.4 The dependence of lensing power spectrum on matter density

φφ Eqs. (2.1, 2.6) show that there are several ways thatω m impactsC � : through the pri-

mordial amplitude,A s, the transfer function,T(k), and the growth function,g(a). We will see that most of the dependence comes from the transfer function, i.e. from the decay of the potentials between horizon crossing and matter-radiation equality.

2.4.1 Qualitative Analysis

Deﬁningx=χ/χ �, the lensing power spectrum can be written as

1 � �4Cφφ A χ dx kT 2 (1 x) 2g2(a), (2.7) � � s � xχ − �0 � � �

wherea=a(x), we have dropped all constant factors and we have adoptedn s = 1. We can now gain some insight by plotting in Fig. 2.2 the various factors in this integrand for two values of�, one on each side of the� 4Cφφ peak (� 50). Let’s consider the � � � = 800 caseﬁrst. Recallingχ 104 Mpc we see the power spectrum (the kT 2 factor, � ∼ 2 1 which peaks atk k 10− Mpc− ) is monotonically rising as the comoving distance, ∼ eq ∼ x, increases and we probe power at longer wavelengths (smallerk). Since the lensing kernel is dropping the net result is a broad integrand with a peak at about 40% of the way back to last scattering. Very little power comes from times when the growth function

is signiﬁcantly diﬀerent from unity. All dependence of the lensing kernel onω m has been

16 ��

��

� ��

��

Figure 2.2. Factors in the integrand of the lensing power spectrum for large� = 800 (upper panel) and small� = 20 (lower panel).

17 pulled out into the prefactor of the integrand which, as we shall see, has only a weak dependence onω m. The bulk of the dependence then comes from the transfer function. In contrast, for the� = 20 case we can see that the turnover of the matter power spectrum (the drop in power atk

2.4.2 Quantitative Analysis

φφ To provide a quantitative test of our understanding we calculate the dependence ofC � onω m. We begin by assuming a power-law matter power spectrum and set the growth function to 1. These assumptions allow us to calculate the dependence analytically, and they are a good approximation for�>� =k χ 140. We shall then include the eq eq � � physical transfer function andﬁnally the growth function.

We begin by determining the dependence of the prefactor in Eq. (2.7),A sχ , onω m. ∗ A sample ofΛCDM models from WMAP9 chains gives the following scaling relations

0.58 0.25 A ω ,χ θ ω − (2.8) s ∝ m � � ∝ m whereθ � is the angular size of the sound horizon at recombination, which is well determined by the WMAP9 data and is almost independent ofω m. We alsoﬁnd the power law indexn s is close to uncorrelated withω m, so it is safe to setn s = 1 as we have been doing.

The scaling ofA s withω m comes directly from the “potential envelope” eﬀect described in Section 2.2. For the acoustic peak region the potential envelope of Hu and White [94]

0.58 0.58 is wellﬁt byω − soA ω keeps the power in the acoustic peaks consistent with m s ∝ m the data.

The scaling ofχ � withω m arises from the dependence of the sound horizon on the expansion rate near last scattering. If we assumez � isﬁxed, and if we were to neglect the contribution of radiation to the expansion rate, then the sound horizon (rs) would scale 1/2 0.25 asω m− . The presence of radiation softens this dependency toω m− [95]. Withθ � so well determined from the data,χ � =r s/θ� has the same scaling asr s.

18 ��

��

Figure 2.3. The dependence of the lensing power spectrum on the matter density, �4Cφφ (ω )n. The solid line is the numerical result from CAMB, the dashed line is � ∼ m the result of the analytic scaling law derived in the text (n=0.75(m + 1) + 0.58) and the dash-dotted line is the result of the Limber approximation settingg(a) = 1.

The matter power spectrumP (k) kT 2(k) which we shall approximate locally as a δ ∼ power law k m+1 km+1 P (k) kT 2(k) k eq = eq . (2.9) δ ∼ ∼ k km � � The scaling withk eq is due to the suppression of the potential that occurs between horizon crossing and the onset of matter domination. φφ We can now derive the dependence ofC � onω m:

m+1 1 4 φφ (keqχ�) m 2 2 � C� A s m dx x (1 x) g (a) ∼ � 0 − � 1 0.75(m+1)+0.58 m m 2 2 ω m �− dx x (1 x) g (a) ∼ 0 − 0.75(m+1)+0.58 m � =ω m �− f(ω m), (2.10)

where we have denoted theﬁnal integral asf(ω m) and theω m dependence comes from

the growth function,g. Since the dependence ofg(a) onω m is strong only in the late

universe where, as we have seen, the integrand is very small the dependence off(ω m) on the matter density is weak and weﬁnally obtain the scaling law

4 φφ 0.75(m+1)+0.58 m � C ω �− . (2.11) � ∼ m

19 ��

��

� ��

��

Figure 2.4. The contribution of each multipole of the CMB lensing power spectrum, φφ TT C� to the lensing of the CMB temperature power spectrum,C L , assuming our ﬁducial cosmology (WMAP9 best-ﬁtΛCDM model), for two values ofL near acoustic peaks (1455 and 2075), and two values near troughs (1300 and 2245).

Of course the power spectrum is not well approximated by a single power-law. We compute the local power-law index,m, numerically by matching the numerically-determined

4 φφ m 4 φφ � C� to� − . Using thism our analytic result for the scaling of� C� withω m is very

accurate for�>� eq, as we show in Fig. 2.3. We now demonstrate that the failure of the above prediction forn(�) at�� 50 is mostly due to neglecting the growth factor. We do so by showing how the agreement

with the numerical result improves dramatically at�� 50 if we compare to the numerical

result withg(a) set to unity. Reducingω m leads to an increase inΩ Λ in order to keep the angular size of the sound horizonﬁxed, which in turn leads to a decrease ing(a). Thus including the growth factor increasesn(�) over the range in� where the growth factor is important.

2.5 Discussion

Precise measurements of anisotropies on small angular scales allow tight constraints on the matter density due to its impact on the amplitude of gravitational lensing of the angular power spectrum of temperature anisotropies [86]. Within the context ofΛCDM models we found that this dependence on the matter density arises mainly from the dependence

20 ofP Φ(k) onω m.

The origin of the dependence ofP Φ(k) onω m is well understood [29, 90, 93], and arises from the decay of the potential that occurs after horizon crossing but before matter- radiation equality. Increasingω m increasesz eq and thereby decreases the amount of this decay.

Other sources of dependence (the lensing kernel, the correlation betweenA s andω m and the growth function) contribute in a subdominant manner. The growth factor is only important at�� 50. On such large angular scales the turnover of the matter power spectrum suppresses contributions from earlier times, thereby making nearby contributions relatively more important. This brings up the question, how much does�< 50 contribute to the lensing of the temperature power spectrum? To answer it we defineX(�) via (C TT CTT )/CTT = L − L unlensed L unlensed X(�)d ln� (using Eq.(4.12) of Lewis and Challinor [91]) and plot the integrand in Fig. 2.4 �for several values ofL. For ourfiducialΛCDM cosmology, we see that�< 50 contributes φφ a subdominant portion to these integrals. SettingC � to zero at�< 50 changes the lensedC TT by less than 0.7% at all� with peak effect at� 2240. For comparison, � ≈ φφ TT settingC � to zero at all� (i.e., turning off lensing) changes the lensedC � by about 11% at� 2240. Thus the lensing of the CMB temperature power spectrum has some ≈ sensitivity to the response ofg(a) to changes inω m, but this effect is subdominant to the main effect of the response ofP Φ(k) to changes inω m. TT We note that recent measurements ofC � at high� have led to small, but important, shifts in cosmological parameters [3]. Assuming the minimal, six-parameterΛCDM model, estimates of the matter density have gone up (and estimates ofH 0 have gone down) with the inclusion of high� data from Planck. When the Planck data are restricted to the angular scales well-measured by WMAP these shifts largely disappear [3] suggesting that the parameter shifts are due to the influence of the new data at small angular scales. As Hou et al. [86] point out, the impact of lensing on the CMB temperature power spectrum provides some of the sensitivity toω m for the high� data. With this work we now understand the physical origin of that dependence onω m.

21 Chapter 3

Constraints on Neutrino Mass from Cosmic Microwave Background and Large Scale Structure

This chapter originally appeared as an article in MNRAS [32].

3.1 Introduction

Basic questions about the neutrino mass matrix remain unanswered, such as whether the CP-violating phase is non-zero, whether the neutrinos are Majorana or Dirac, and whether the hierarchy of masses is normal or inverted [96–98]. Signiﬁcant experimental and observational eﬀorts are underway and being planned to answer these questions. Doing so may shed light on possible extensions beyond the standard model of particle physics. The question of the type of mass hierarchy may be settled by cosmological observations.

The best lower limit on the sum of the mass eigenstate masses,M ν, comes from analysis of solar and atmospheric neutrino oscillation data [99]. If cosmological determinations of this quantity tighten up near this lower bound, then the inverted hierarchy will be ruled out.

Current data lead to 0.058 eV�M ν �0.21 eV, where the upper limit comes from cosmic microwave background (CMB) and baryon acoustic oscillation (BAO) data [2]. Bringing this upper limit down is a major science goal of a Stage-IV (S4) cosmic mi-

22 crowave background project, CMB-S4, and also of the galaxy survey project Dark Energy

Spectroscopic Instrument (DESI). These projects are forecasted to determineM ν with a one-standard deviation of 45 meV (CMB-S4 alone) and 16 meV (CMB-S4 combined with DESI BAO) [100]. These uncertainties are small enough to guarantee a detection of M = 0 at a 3σ or greater confidence level. ν � Of course any conclusions from such cosmological data will be model dependent. How convincing will these data be that we are indeed seeing the impact of neutrino mass, and not misinterpreting some other signals? The forecasted precision is also not quite as strong as one would like; is there a way to guarantee at least a 5σ detection ofM = 0? ν � Here we address these questions. To address thefirst question we examine the particular signatures of neutrino mass that lead to the above forecasts. To address the second we look at what additional types of data can further tighten the expected uncertainties. The chapter is organized as follows. In Section 3.2, we briefly introduce the cosmological signatures of massive neutrinos. In Section 3.3, we focus on changes in the cosmic expansion rate and structure growth rate due to massive neutrinos. In Section 3.4, we analyze the influence of massive neutrinos on the CMB lensing potential power spectrum. Forecasts on the constraints of total neutrino mass from CMB and Large Scale Structure (LSS) measurements are given in Section 3.5 and conclusions are presented in Section 3.6.

3.2 Signatures of non-zero neutrino mass

The cosmological signatures of massive neutrinos have been investigated since decades ago, e.g. [101–103]. We can more broadly view the cosmological neutrino program as a study of the dark radiation that we know exists as a thermal relic of the big bang. By dark radiation here we mean anything, other than photons, thermally produced in the early universe that is relativistic at least through decoupling. We know that such a background of nearly massless non-photon radiation exists with high conﬁdence from light element abundances and the cosmic microwave background damping tail. Both are sensitive to the history of the expansion rate, which depends on the mean density via the Friedman equation. Combining Helium abundance and CMB data constrains the eﬀective number

23 of relativistic species to beN = 2.99 0.39 [2]. eﬀ ± Is this background entirely that of the 3 active neutrino species? Is any part of it from something else? Could there be a signiﬁcant excess of neutrinos over anti-neutrinos? These are interesting questions, also to be addressed by future CMB observations that

will significantly tighten up constraints onN eff . Our confidence that the dark radiation is indeed that of cosmological neutrinos with phase-space distributions as expected from

the standard thermal history will be greatly increased if the constraints onN eﬀ tighten

up toσ(N eﬀ ) = 0.02, as forecasted, consistent with the expected value of 3.046. For the purposes of this paper, we will assume this is what will happen. If we assume that the dark radiation background is entirely the active neutrinos with the expected phase space distributions, the assumption of non-zero neutrino mass leads to very speciﬁc predictions for cosmological observables. First we consider the expansion rate as a function of redshift. The rest-mass energy of the neutrinos begins to slow down the decline of energy density with expansion as they become non-relativistic, leading to an

increase inH(z) relative to theM ν = 0 expectation. This increase would persist toz=0 if we were holding the other contents of the low-redshift universe constant. However, for our purposes of exploring observable consequences ofM = 0 it makes much more sense ν � to hold the angular size of the sound horizon on the last-scattering surface constant, since this quantity is so well-determined from CMB observations [2, 83]. To do so one must decrease the density of dark energy. Assuming the dark energy is a cosmological constant, the shape ofΔH(z) has a very particular form, as shown in Fig. 3.1, changing sign very nearz = 1, with the onset of dark energy domination.

Were we able to trace out this departure ofH(z) from theM ν = 0 shape, it would contribute to our conﬁdence we are seeing the impact of non-zero neutrino mass. However, as we will see, the DESI determinations ofH(z) will be insuﬃcient to resolve this very small signal across redshift. That is not to say the signal is altogether observably invisible.

These changes toH(z)aﬀect comoving angular diameter distanceD A(z) in ways that are detectable by DESI. It is just that it will be diﬃcult, if not impossible, to make the case that there is the sign change in theH(z) correction nearz 1. The changes toH(z) also �

24 directly impact the growth of structure, with observable consequences for the redshift- space distortion (RSD) and the CMB lensing potential power spectrum, which we will discuss in Section 3.3 and 3.4 respectively.

3.3 Inﬂuence of massive neutrinos on galaxy survey observables

To quantify the influence of massive neutrinos, we compare afiducial cosmology with massless neutrinos and a cosmology with massive neutrinos. Thefiducial cosmology is aflatΛCDM universe with the Planck bestfit parameters [3], i.e.ω b = 0.022032,ω m = 9 0.14305,A = 2.215 10 − , n = 0.9619,τ=0.0925,H = 67.04 km/s/Mpc,M = 0 s × s 0 ν meV. The set ofM = 0 cosmologies have parametersθ=(Θ,M ), whereΘ areΛCDM ν � ν 2 parameters. Given a specificM ν, we chooseΘ by minimizingχ (Θ,Mν), with

2 χ (Θ,Mν) =F αβλαλβ

2 =F ijλiλj + 2FiνλiMν +F ννMν , (3.1) whereF is the Fisher matrix for the CMB observations,λ = (Θ Θ ) withi index- i − ﬁd i ing the 6ΛCDM parameters and summation over repeated indexesα,β, i, j is implied.

2 Minimizingχ (Θ,Mν) requires

2 1 0 =∂χ /∂λ λ = (G − ) F M , (3.2) i → i − ij jν ν whereG is a subset of the Fisher matrixF,G F . ij ≡ ij In Fig. 3.1, we show the inﬂuence ofM ν = 50, 100 and 200 meV on expansion rate

H(z) and comoving angular diameter distanceD A(z). We see thatH(z� 1) decreases andH(z� 1) increases compared to theﬁducial cosmology, and the comving distance

DA(z) increases accordingly. Though the departure ofH(z) from theM ν = 0 shape is undetectable by DESI BAO, the changes inD A(z) are readily detectable [104].

One of the observable consequences of these changes toH(z) is the impact on the structure growth rate. How one describes this impact depends on what one is using

25 ��

��

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��

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Figure 3.1. The dependence of expansion rateH(z) and comoving angular diameter 2 distanceD A(z) onM ν, where we minimize theχ (Θ,Mν) by adjusting the 6ΛCDM parametersΘ when increasingM ν from 0 to 50, 100, 200 meV.

��

� ��

��

Figure 3.2. The dependence of structure growth rate onM ν, where we minimize the 2 χ (Θ,Mν) by adjusting the 6ΛCDM parametersΘ when increasingM ν from 0 to 20 meV.

for a comparison model. We use, as a comparison model, a cosmology with massless neutrinos in place of the massive ones. One could also use as a comparison model one with additional cold dark matter in place of the neutrinos. We use the former, consistent with our underlying assumption that we have 3 neutrino species with phase-space distributions as expected from the standard thermal history.

The structure growth rate that is usually quantiﬁed byd lnσ 8(a)/d lna, can be deter-

mined by RSD from galaxy surveys, whereσ 8(a) is the amplitude of massﬂuctuationsσ R

26 1 1 on scale of 8h− Mpc, i.e.,σ (a) σ(R=8h − Mpc;a). Here 8 ≡ 3 ∞ k σ2 (a) P (k;a)W 2(kR)d lnk, (3.3) R ≡ 2π2 δ �0 � (3) whereP (k, a) is the matter power spectrum defined by, δ(k;a)δ (k�;a) =P (k;a)δ (k δ � � δ − k�), andW(kR)=3j 1(kR)/kR is the window function. Introducing the growth function D(a), δ(a) σ (a) D(a) = 8 , (3.4) ≡ δ(a = 1) σ8(a = 1) whereδ(a) is the matter overdensity at redshiftz=1/a 1, and defining perturbation − growth ratef(a) d lnD(a)/d lna [105], we may rewrite the structure growth rate as ≡ d lnσ 8(a)/d lna=f(a)σ 8(a). In the above definition, the growth functionD(a) and the structure growth rate

f(a)σ 8(a) depend onσ 8(a = 1) which varies for ourﬁducialM ν = 0 model and the model

with massive neutrinos. For easier comparison, we introduce the growth functionD e(a)

normalized at early time, say ata e = 1/1100 when massive neutrinos are relativistic,

δ(a) σ8(a) De(a) = , (3.5) ≡ δ(ae) σ8(ae)

and the structure growth rate can be rewritten as

dD (a) f(a)σ (a) = e σ (a ), (3.6) 8 d lna 8 e

whereσ 8(ae) is the same for the model with massive neutrinos and theﬁducial model. So

the variation off(a)σ 8(a) only depends on the growth rate dDe(a)/da. In Fig. 3.2, we show the impact of massive neutrinos with mass 20 meV on the structure growth rate. The structure growth rate is decreased at high redshift because of the enhanced expansion rate (Fig. 3.1). Note though that the transition of structure growth rate from suppressed to enhanced is delayed toz 0.5 compared toz 1 � � when the expansion rate transitions. The reason for this delay is that the growth rate

dDe(a)/da is determined by two factors: the expansion rate and gravitational attraction. The slower growth atz 1 leads to weaker gravitational potentials. Aroundz 1, the � � expansion rate is the same for the two models, but the gravitational potential is weaker for

27 theM = 0 model. Therefore the growth rate remains suppressed,until some later time ν � z 0.5 when the weaker gravitational potential is compensated by even slower expansion. � DESI will provide a comprehensive survey of spectroscopic galaxies and quasars cov- ering redshifts 0.1

σfσ 8(a)/fσ 8(a) = 0.35% from 0.4

Mν = 20 meV on the structure growth ratef(a)σ 8(a), which is detectable by DESI RSD.

3.4 Inﬂuence of massive neutrinos on the CMB lensing power spectrum 3.4.1 Introduction to the lensing power spectrum We begin with a brief review of gravitational lensing of the CMB. For details see e.g. the review by Lewis and Challinor [91]. Gradients in the gravitational potential,Φ, distort the trajectories of photons traveling to us from the last scattering surface. The deﬂection angles, in the Born approximation, ared= φ, where the lensing potential,φ, is a ∇ weighted radial projection ofΦ. The key quantity for calculating the impact of lensing on the temperature power spectrum is the angular power spectrum of the projected potential, φφ C� , which we also call the lensing power spectrum. Taking advantage of the Limber approximation [92], it can be written as a radial integral over the three dimensional

gravitational potential power spectrumP Φ χ� � χ 2 �4Cφφ 4 dχ(k 4P ) ;a 1 , (3.7) � � Φ χ − χ �0 � � � � � whereχ is the comoving distance from the observer,a=a(χ), a� subscript indicates the last scattering surface, (1 χ/χ )2 is the lensing kernel, and the power spectrumP is − � Φ deﬁned as � (3) Φ(k;a)Φ (k�;a) =P (k;a)δ (k k �). (3.8) � � Φ − p To calculateP Φ we assume a power-law primordial power spectrumP Φ(k),

ns 1 k3 k − P p(k)=A , (3.9) 2π2 Φ s k � 0 � wherek 0 is an arbitrary pivot point,A s andn s are the primordial amplitude and power law index respectively.

28 The gravitational potential at late times,Φ(k, a), is related to the primordial potential Φp(k) by [e.g. 93] 9 Φ(k, a) = Φp(k)T(k)s(k;a)g(a), (3.10) 10 where the potential on very large scales is suppressed by a factor 9/10 through the transition from radiation domination to matter domination. For modes which enter the horizon during radiation domination when the dominant component has signiﬁcant pressure (p ρ/3) the amplitude of perturbations cannot grow and the expansion of the Universe ≈ forces the potentials to decay. In a cosmology with massless neutrinos, for modes which enter the horizon after matter-radiation equality (but before dark energy domination) the potentials remain constant. The transfer functionT(k) takes this into account, being

2 unity for very large scale modes and falling approximately ask − for small scales. The transfer functionT(k) is independent of scale factora because in a cosmology with massless neutrinos, potentials on all scales keep constant in matter domination. The impact of massive neutrinos on the potentials is described by the function,s(k;a) , which is unity on scales above the free-streaming scale and decreases with time on scales below. Once the cosmological constant starts to become important the potentials on all scales begin to decay. This eﬀect is captured by the growth function,g(a), which is unity during matter domination. With these deﬁnitions we have

ns 1 81 k − (k4P )(k;a) = π2g2(a)s 2(k;a)kT 2(k)A . (3.11) Φ 50 s k � 0 � With these pieces in place, we now examine the accuracy of the Limber approximation. According to Loverde and Afshordi [107], it is a better approximation to replacek=�/χ with �(� + 1)/χ (�+0.5)/χ in the original Limber approximation Eq.(3.7). With the � replacement� and deﬁningx=χ/χ �, the lensing power spectrum can be written as

1 �+0.5 �2(� + 1)2Cφφ 4χ dx(k 4P ) ;a (1 x) 2, (3.12) � � � Φ xχ − �0 � � � wherea=a(x). Fig. 3.3 shows the lensing power spectrum calculated from CLASS [108, 109] compared to the lensing power spectrum calculated from the Limber approximation. We see the Limber approximation reproduces the accurate numerical result even for small

29 ��

� ��

��

Figure 3.3. The lensing power spectrum calculated from CLASS (solid line), calculated with Limber approximation (dashed line) and calculated with Limber approximation and settingg(a) = 1 (dashed-dotted line).

�. In order to understand the the influence of the growth function, we also calculated the lensing power spectrum by settingg(a) 1. The growth function makes a difference for ≡ φφ �� 50 [31]. Now we are to understand the impact of massive neutrinos onC � . 3.4.2 Influence of massive neutrinos on the lensing power spectrum: results

φφ According to Eq.(3.11) and Eq.(3.12), it is clear thatC � is determined by the primordial

perturbationA s, ns, the transfer functionT(k), the impact of massive neutrinoss(k;a),

the growth factorg(a), and the comoving distance to the last scattering surfaceχ �.

In order to quantify the dependence of the lensing power on total neutrino massM ν,

we take samples from a PlanckΛCDM +M ν chain, andfit the linear relation Cφφ C φφ � − �,fid φφ =R � M ν(eV), (3.13) C�,fid × The linearfitting result is shown as the thin solid line in Fig. 3.4. To understand the

contribution of the various parameter variations and eﬀects, we also plotR � in Fig. 3.4 for the cases shown. We now work our way towards an understanding of the full response, the thin solid curve, in stages. We begin with the case where weﬁx most of the parameters other

thanM ν, and turn oﬀ the impact of dark energy on the growth factor,g(a),ﬁxing it to

30 ��

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� ��

��

Figure 3.4. The dependence of the lensing power spectrum on total neutrino mass, (Cφφ C φφ )/Cφφ =R (M /eV). The black line is the numerical result from � − �,fid �,fid � × ν CLASS, the red line is the result of Limber approximation settingχ = 1.4 10 4 Mpc, � × the blue line is the result of Limber approximation withg(a) = 1, the magenta line is the result of Limber approximationfixing bothω m andg(a), and green line is the result of Limber approximation withA s, ns,ω m,ω b andg(a)fixed.

unity. In this case, increasingM ν has two effects: 1) it decreases the free-streaming length (T /M )/H(z) [110] , and 2) increases the expansion rate once the neutrinos start to ∼ ν ν become non-relativistic. The increased expansion rate acts to suppress the growth of structure on all length scales. However, the decreased free-streaming length acts to boost structure growth on scales above the free-streaming length, nearly exactly canceling the suppression. The result is the bottom-most curve of Fig. 3.4: nearly no effect at low�, a constant suppression of power at high�, and a smooth transition between these two regimes. The difference between the bottom-most curve and the dot-dashed curve (the one

labeled, ‘Limber withg(a) = 1’) is due to how other parameters adjust asM ν varies. We can isolate these changes as almost all due (at least at�� 50) to a correlation between

Mν andω m, as demonstrated by the following: if weﬁxω m, and letA s,n s, andω b vary, we get the second curve from the bottom which diﬀers very little from the bottom-most

curve. Once we letω m vary as well, we get the dot-dashed curve. Lettingω m vary leads

to variation inn s andA s as well, but these changes allﬂow from the correlation between

ωm andM ν.

31 Once we letg(a) vary as well we get a curve that is indistinguishable from the thin solid curve, which is boosted everywhere, and especially at�� 50. The contributions to these large angular scales come predominantly from modes to the left of the peak in the matter power spectrum. Atfixed (large) angular scale, structures that are nearer by, and therefore on smaller length scales, are closer to the peak of the matter power spectrum. Thus the large angular scales are weighted toward later times, and therefore more influenced byg(a) than the smaller angular scales. The growth factor increases with increasingM ν because to keep the angular size of the sound horizonfixedΩ Λ must decrease. The lensing kernel’s dependence on cosmological parameters comes entirely via its dependence onχ . To see how much of the variation in the lensing power spectrum is due ∗ to the lensing kernel, wefixχ = 1.4 10 4 Mpc (top-most dashed curve). By examining ∗ × the difference between the top-most dashed curve, for whichχ isfixed, and the thin solid ∗ curve, which is the full numerical result, one can see this effect is very small.1 To summarize, there are three main effects of massive neutrinos on the lensing power: 1) increased expansion rate suppresses power, 2) decreased free-streaming length compensates for the suppressed power at scales above the free-streaming length, 3) other

TT parameter variations due to partial degeneracies inC l (most notably an increase inω m) boost the power on all scales. The net result is increased power at large scales and a decrease in power at small scales. One might potentially include the growth factor here as the fourth-most important eﬀect, somewhat increasing the power at large angular scales.

TT The origin of the degeneracy inC l betweenω m andM ν is actually due to lensing itself. Planck Collaboration XVI [3] demonstrated that the dominant eﬀect leading to the constraint of neutrino mass from the CMB temperature anisotropy power spectrum is gravitational lensing. As shown in Fig. 3.4, increasingM ν suppresses the lensing power, while increasingω m increases the lensing power. The lensing power suppression by massive neutrinos can be compensated by the enhancement from increasingω m, so uncertainties

1This result is in contrast to the case of tomographic cosmic shear as a probe of dark energy. In this case the sensitivity of the data to variations in the dark energy equation-of-state parameter largely arises from the lensing kernel [111, 112].

32 inM ν andω m are expected to be positively correlated [113].

3.5 Forecast of constraints on the total neutrino mass from diﬀerent data sets

We use the Fisher matrix formalism to forecast constraints on neutrino mass from future CMB and LSS experiments. Theﬁducial cosmology used here is the same as the one used

in Section 3.3 except with a diﬀerent value of total neutrino mass,M ν = 85 meV.

3.5.1 CMB-S4 and DESI BAO Following Wu et al. [114] and Dodelson and Scott [29], the Fisher matrix for cosmological parameters constrained by CMB spectra is written as

�max 2�+1 1 ∂ � 1 ∂ � Fαβ = fskyTr �− C �− C , (3.14) 2 C ∂θα C ∂θβ �� 1 and it is related to the expected uncertainty of a parameterθ α byσ(θ α) = (F − )αα, where � TT TT TE T d C� +N � C� C� TE EE EE � =  C C +N 0  , (3.15) C � � �  T d dd dd   C� 0C � +N �    dd   andC � is the angular power spectrum of the deflectionfieldd, which is related to the dd φφ XX lensing power spectrum byC � =�(� + 1)C � . The Gaussian noiseN � is defined as θ2 N XX =Δ 2 exp �(� + 1) FWHM , (3.16) � X 8 log 2 � �

whereΔ X (X=T,E,B) is the pixel noise level of the experiment andθ FWHM is the full-width-half-maximum beam size in radians [115, 116]. The noise power spectrum of

dd deflectionfieldN � is calculated assuming a lensing reconstruction that uses the quadratic EB estimator[117]. We use the iterative method proposed by Smith et al. [118], which performs significantly better than the uniterated quadratic estimators.

For the CMB-S4 experiment, we assume the temperature noise levelΔ T = 1.5µK-

arcmin, the polarization noise levelΔ E =Δ B = √2Δ T , the fraction of covered sky

fsky = 0.5 and the beam sizeθ FWHM = 1�. With these given experiment sensitivities, we

33 ��

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Figure 3.5. Forecasted 1σ and 2σ constraints in theM ω plane, where the CMB-S4 ν − m experiment results in aσ(M ν) = 38 meV constraint, the combination of CMB-S4 and DESI BAO yield aσ(M ν) = 15 meV constraint. and adding measurements of the structure growth rate by DESI RSD further improves the constraint toσ(M ν) = 9 meV.

obtain a constraint from CMB withσ(M ν) = 38 meV. The 1σ and 2σ constraints are shown in Fig. 3.5. According to the analysis in Section 3.3 and 3.4 , DESI BAO are helpful to break

the degeneracy betweenM ν andω m. BAO uncertainties are independent from CMB experiments, so the total Fisher matrix is simply given by addition

FCMB+BAO =F CMB +F BAO, (3.17)

where the DESI sensitivities of BAO signal can be found in [104] and shown in Fig. 3.1. It

is found that adding the DESI BAO data greatly improves the constraint toσ(M ν) = 15 meV (similar forecasts were also conducted by [100, 114]).

3.5.2 Beyond DESI BAO The largest signal of massive neutrinos onH(z) andD(z) is found at low redshifts (see Fig. 3.1), where BAO has inevitably large noise because of small amount of survey volume and large cosmic variance. Other than DESI BAO, we also investigate other low-redshift tracers ofH(z) andD(z) which are possible to tighten the uncertainty of total neutrino mass. DESI RSD: similar to BAO, RSD uncertainties are also independent from those of

34 ��

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Figure 3.6. Same as Fig. 3.1, but with suppressed errorbars ofD A(z) andH(z) coming from CMB-S4 and a cosmic-variance-limited BAO experiment.

CMB observations, so the total Fisher matrix of CMB+BAO+RSD is also approximately given by addition

FCMB+BAO+RSD =F CMB +F BAO +F RSD, (3.18)

where we use the RSD sensitivities from DESI survey which can be found in [106] and shown in Fig. 3.2. Here we use the approximation that uncertainties in BAO and RSD are uncorrelated, due to they are sensitive to diﬀerent aspects of the matter power spectrum:

BAO is sensitive to its characteristic length scaler s while RSD is sensitive to its amplitude. In fact, our result is insensitive to the approximation because weﬁnd that both CMB-

S4+DESI BAO+DESI RSD and CMB-S4+DESI RSD yield the sameσ(M ν) = 9 meV uncertainty. Better BAO: The DESI survey will cover 14, 000 squared degrees (about 1/3 of the whole sky). We explore a future BAO experiment which covers the whole sky and in which cosmic variance dominates over shot noise in the redshift range 0

onD A(z) andH(z) from this BAO experiment are shown in Fig. 3.6. It is found that CMB-S4 and the cosmic variance limited BAO constrain the total neutrino mass with

uncertaintyσ(M ν) = 11 meV. So we conclude that 11 meV is a lower limit ofσ(M ν) we could measure from CMB-S4+BAO, where the limit mainly comes from noise level of the CMB lensing signal. Supernovae: The constraining power of BAO is limited by its large cosmic variance at

35 �

� � � � ��

��

� � ��

� ��

Figure 3.7. The uncertainties in relative distances from CMB-S4 + DESI BAO. Note that the uncertainties is multiplied by a factor of 104 in the plot.

low redshifts (Fig. 3.6), so supernovae distance measurements which do not suﬀer from the cosmic variance problem may be eﬀective complements if their systematic errors are well controlled. Supernovae perform better in relative distance measurements than in absolute

distance measurements. However for theΛCDM +M ν model, the uncertainties in relative distances from CMB-S4 + DESI BAO are very small (see Fig. 3.7) . We conclude that supernova observations must result in relative distance determinations with systematic errors less than about 0.05% if they are to tighten the constraints on neutrino mass. Compared to systematic errors from current supernova observations (e.g., Suzuki et al. [119]) this would be a reduction by a factor of 20 . ∼ 3.6 Conclusion

This paper is motivated by our desire to better understand the origin of current and forecasted cosmological constraints on the sum of neutrino masses. We took as a given that

determination ofN eﬀ will solidify the predicted value of 3.046, increasing our conﬁdence that the phase-space distribution of the cosmic neutrino background is what we expect based on the standard thermal history. With that as a given, the most important aspect of increased neutrino mass (relative to some reference model) is an increased neutrino energy density. If the model with increased mass is to remain consistent with CMB observations, the distance to last-scattering must be preserved and so the total energy

36 density, and therefore the expansion rate, cannot increase at all redshifts. To compensate for the increase in neutrino energy density, the cosmological constant must decrease in value. Thus varying neutrino mass leads to changes inH(z) with a very particular shape: a mild increase at high redshifts, a larger decrease in low redshifts, with a transition near the onset ofΛ domination atz 1. Unfortunately this very specific prediction of the − � shape forH(z) is difficult to verify in detail because of how small the departures from the reference model are atz> 1. We see that this difficulty persists even for a cosmic-variance limited all-sky (z < 4) BAO experiment (see Fig. 3.6). The sensitivity of CMB-S4 to neutrino mass comes via the impact of this increased expansion rate on the growth of structure. At scales below the neutrino free-streaming length, this increased expansion rate suppresses the growth of structure. Above the free- streaming length, the ability of massive neutrinos to cluster compensates for the increased expansion and there is no net suppression. Because increasing matter density increases lensing power amplitude, the CMB lensing-derived constraints on neutrino mass have uncertainties positively correlated with the matter density uncertainties. This correlation with dark matter density leads to secondary correlations of neutrino mass uncertainty with uncertainties inn s andA s. We disentangled all these various effects in Fig. 3.4.

The correlation betweenM ν andω m has the opposite sign as that from BAO, since increasing neutrino mass and increasingω m both increase the expansion rate atz> 1, and lead to a compensating decrease atz< 1. Thus the combination of CMB-S4 and DESI-BAO leads to improvements in the determination of both quantities. Finally, we brieﬂy investigated how constraints might be improved beyond the ∼ 3σ to 4σ detection expected from CMB-S4 + DESI BAO in the case of the lowest ∼ possible neutrino mass of 58 meV. The larger signals are at low redshift, so we considered supernovae as relative distance indicators and redshift-space distortions, as well as cosmic- variance-limited BAO. We found requirements on supernova precision that are probably prohibitively stringent. But better BAO, and RSD, both have the potential to improve the detection to 5σ or greater. ∼

37 Chapter 4

Cosmic Microwave Background Acoustic Peak Locations

This chapter originally appeared as an article in MNRAS [80].

4.1 Introduction

With thefirst release of Planck polarization data [2, 120] we have for thefirst time a suffi- cient measurement of the polarization spectra (both the temperature-E-mode polarization cross power spectrum (TE) and the E-mode auto power spectrum (EE)) to clearly see multiple acoustic peaks with well-defined locations. These locations provide a beautiful confirmation of expectations for the response of the primordial plasma to small initial adiabatic departures from complete homogeneity. One can work out these expectations by solving the Einstein-Boltzmann equations for evolution of the phase space distribution function of the various components [e.g. 30, 108, 121–123]. But these numerical calculations, on their own, are not entirely satisfying. In addition to knowing the answer, we want understanding. This desire has led to many papers aimed at an analytic understanding of the model power spectra [e.g. 49, 76–79, 124– 134]. In this article, motivated by the recentfirst measurements of TE and EE extrema locations, we develop a detailed analytic understanding of the locations of the peaks in TT and EE, and the extrema in TE in the context ofΛCDM model. The peak structure itself has drawn special attention. Back to about 15 years ago,

38 when only thefirst TT peak was readily measured [33–48] it was found to be consistent with the standardΛCDM model with adiabatic initial conditions, and imposed tight constraints on other competing models for example,ΛCDM model with isocuravture initial conditions and topological defect models [e.g. 49–52]. With more peaks measured in recent years [53–63], the topological defect models were ruled out [64], and the constraints on the isocurvature modes have been improved to unprecedented precision [e.g. 65–74]. Part of the beauty of the TT, TE and EE measurements is that a very simple analytic model provides us with a qualitative understanding of the observed features. In the next section we will define this model and use it to produce ‘baseline’ predictions for the peak locations. It works especially well for the relative locations of the peaks. For example, the temperature anisotropies are predominantly sourced by temperaturefluctuations at the last scattering surface (LSS). We expect that the standing-wave modes that have hit an extremum in temperature contrast right at the epoch of last scattering, will be at a null in their peculiar velocities. Further, since gradients in peculiar velocities are the dominant source of polarization anisotropy, peaks in TT should correspond to minima in EE. This is roughly what we observe. But the above picture is discrepant, in detail, with observations and with the expectations of theΛCDM model. To achieve an understanding that is quantitatively correct, at a level consistent with the precision of current measurements, we have to take into account a number of effects. We have found that all these factors are important: time-varying gravitational potentials that are still non-zero at last scattering, neutrino free-streaming, the failure of the tight-coupling approximation, the shape of the primordial power spectrum, details of the projection from three dimensions to two, and thefinite width of the LSS. We work out, sometimes analytically, mostly by numerical methods, the contribution of each one of these effects to the shifting of each of the peaks from their locations in the baseline model. This chapter is organized as follows. In Section 4.2, we introduce a baseline model interpreting the evolution of photon perturbations and the power spectra based on the tight coupling approximation and simplified projection. In Section 4.3, wefirst analyti-

39 cally derive the phase shifts of photon perturbations induced by decoupling, gravitational potential transient and free-streaming neutrinos, then numerically test the analytic results by examining the evolution of a singlek mode. In Section 4.4, we numerically measure the phase shift of the photon perturbations at the LSS, single out the contribution from each eﬀect, and analytically interpret them. In Section 4.5, we investigate the impact of projection on the peak locations in details. We compare the peak locations determined from Planck measurements to expectations under theΛCDM model in Section 4.6, and conclude in Section 4.7. In this chapter, we will work in the conformal Newtonian gauge

ds2 =a 2(η) (1 + 2Φ)dη2 + (1 2Ψ)δ dxidxj , (4.1) − − ij � � whereη is the conformal time, and the scalar perturbationΨ andΦ are related to the convention of Dodelson and Scott [29] byΦ=Ψ ,Ψ= Φ . Thefiducial Dodelson − Dodelson cosmology used in the paper is the bestfittingflatΛCDM cosmology from Planck TT+low P+lensing [2].

4.2 Baseline Model

In this section, we will construct a simple analytic model, our baseline model, that predicts the peak locations. This simple model neglects many important eﬀects. Much of the rest of the paper is then devoted to explaining the diﬀerences between the approximate predictions of this baseline model, and the numerically calculated, essentially exact predictions.

4.2.1 Before Recombination The evolution of a photon-baryon plasma is governed by the Einstein-Boltzmann equations, e.g., Eqs.(4.100 - 4.107) of Dodelson [135], Π Θ˙ Ψ˙ + ikµ(Θ+Φ) Θ Θ µV + P (µ) = − , (4.2) − 0 − b 2 2 ˙τ Π Θ˙ + ikµΘ Θ (1 P (µ)) = p p , (4.3) p − 2 − 2 ˙τ ˙ ˙a Rb Vb + a Vb + ikΦ V + 3iΘ = , (4.4) b 1 � ˙τ �

40 whereV b is the bulk velocity of baryons,Θ p is the strength of the polarizationﬁeld,

Π=Θ 2 +Θ p2 +Θ p0,τ(η) is the optical depth for a photon emitted at timeη and received

at todayη 0,R b(= 3ρb/4ργ) is roughly the ratio of baryon density over photon density, the dot denotes the derivative with respect to the conformal timeη, andµ= kˆ ˆp is the · cosine of the angle subtended by the wavevector �k and the photon propagation direction �p. To be clear, we adopte the most commonly used convention of Legendre multipoles,

� Θ(µ) =Σ ∞ ( i) (2� + 1)Θ P (µ) in Eq. (4.4), and we neglecte the small corrections �=0 − � � induced by the nonzero sound speed of bayronsc 2 T /µ , whereT is the temperature b ∼ b b b of baryons andµ b is the mean molecular weight [see e.g. 30, for details]. In the tight coupling limit, theﬁrst few multipoles can be obtained by perturbative expansion with respect to k/˙τ, which is expected to be much smaller than unity before decoupling. Expanding Eqs. (4.2-4.4) toO(k/˙τ), we get [also see 49, 76–78, 122, 129] 5 8 k i Π= Θ ,Θ = Θ ,Θ = Θ˙ Ψ˙ , (4.5) 2 2 2 − 15 ˙τ 1 1 k 0 − � � and the monopole satisﬁes d2 k2 +k 2c2 [Θ Ψ] = (Φ+Ψ), (4.6) dη2 s 0 − − 3 � �

wherec s = 1/ 3(1 +R b) is the sound speed of the photon-baryon plasma and we have dropped a small� correction R in the above equation. The monopole is actually a ∼ b simple harmonic oscillator forced by gravitational driving. PotentialsΦ andΨ decay rapidly inside horizon during radiation domination, and keep constant during matter domination. For simplicity we drop, for now, the Ψ¨ term on the left side of the above equation and we have [136] d2 +k 2c2 [Θ +Φ] = 0, (4.7) dη2 s 0 � � where we have used the facts thatΦ=Ψ in the absence of photon anisotropic stress, c2 1/3, and Ψ¨ term is small after potentials decay. Assuming adiabatic initial conditions, s � expected from the simplest inﬂationary models, [Θ˙ + Φ˙ ](η = 0) = 0, we obtain k [Θ +Φ] cos(kr ),Θ sin(kr ),Π sin(kr ), (4.8) 0 ∝ s 1 ∝ s ∝ ˙τ s η wherer s(η)= 0 csdη is the sound horizon at timeη. � 41 4.2.2 After Recombination After recombination, photons freely stream. Hence the temperature anisotropies we observed today are largely determined by the photon perturbation at the LSS,Θ(�x= 0,ˆγ,η=η ) [Θ +Φ](�x=ˆγ(η η ),ˆγ,η=η ), 1 where ˆγ is the observation di- 0 � 0 0 − � � rection,η 0 is the conformal time today andη � is the conformal time of the LSS. To study the statistical property of the anisotropies, we usually expand theﬁeld in terms of spherical harmonics

a�m = dΩY �m(ˆγ)Θ(�x=0,ˆγ,η=η 0), (4.9) � TT and deﬁne the temperature power spectrumC a a∗ . With some geometric � ≡ � �m �m� transforms [e.g. 135], the power spectrum is explicitly expressed as

TT 2 2 C� = dk k Θ� (k), (4.10) �

whereΘ �(k) is the multipole moment of the temperatureﬁeld of �k mode, 1 1 dµ � Θ�(k)= � P�(µ)Θ( k,ˆγ,η0), (4.11) ( i) 1 2 − �− whereµ= kˆ ˆγ. · Using the plane-wave pattern (see Fig. 4.5)

Θ(�k,ˆγ,η ) [Θ +Φ]( �k,ˆγ,η ) 0 � 0 � iˆγ�k(η0 η�) = [Θ +Φ](k,η ) e · − , (4.12) 0 � ×

iˆγ�k(η0 η�) where [Θ0 +Φ](k,η �) is the oscillation amplitude [Eq. (4.8)] ande · − is the spatial pattern, we have

1 1 dµ iµk(η0 η�) − Θ�(k) � P�(µ)[Θ0 +Φ](k,η �)e , � ( i) 1 2 − �− = ( 1)�[Θ +Φ](k,η )j [k(η η )], (4.13) − 0 � � 0 − � wherej [k(η η )] is a spherical Bessel function, which peaks at� k(η η ). Therefore, � 0 − � � 0 − � we expect the TT power at mode� is mainly sourced by [Θ +Φ](k,η ) by modek 0 � � 1Strictly speaking, it is more appropriate to write [Θ+Φ](�x=0,ˆγ,η=η ) [Θ +Φ](�x=ˆγ(η 0 � 0 0 − η�),ˆγ,η=η �), but the local potential todayΦ(�x=0,η=η 0) has no direction dependence, thus has no inﬂuence on the CMB anisotropies.

42 �/(η η ). A similar argument yields that EE and TE are mainly sourced byΠ(k,η ) 0 − � � and [(Θ +Φ) Π](k,η ), respectively [see e.g. 49, 129, 137, 138]. As a result, 0 × �

DTT [Θ +Φ] 2(kr ) cos 2(�θ ), � ∼ 0 s,� ∝ � DEE Π 2(kr ) sin 2(�θ ), � ∼ s,� ∝ � DTE [Θ +Φ] Π (kr ) sin(2�θ ), (4.14) � ∼{ 0 × } s,� ∝ � whereD XX �(� + 1)/(2π)C XX, with XX = TT,TE, EE, andθ is the angular size of � ≡ � � 2 the sound horizon at recombination,θ r /(η η ) = 1.04 10 − [e.g. 2]. Therefore � ≡ s,� 0 − � × TT EE TE D� ,D� ,D� reach theirp-th peak at

0 �p(TT) = 302 p, �0(EE) = 302(p 0.5), p − 0 �p(TE) = 151(p+0.5), (4.15) respectively (Throughout this paper, we refer to both the maxima and the minima in the TE power spectrum as peaks due to the arbitrary sign of E mode, and we also refer to the zero points of the TE power spectrum as troughs). We are only interested in the peaks in the power spectra which correspond to the extrema of sources [Θ +Φ],Π,[Θ +Φ] Π, 0 0 × and so carry phase information of the acoustic oscillation. The troughs in the spectra corresponding to the zero points of the sources also carry phase information, but baryon drag shifts the zero points and introduces extra uncertainty. Therefore the troughs in the spectra, and the reionization bumps in EE and TE power spectra are not investigated in this paper. In Fig. 4.1, we compare the theoretical spectra of theﬁducial cosmology with the baseline model. The baseline model is roughly correct in its prediction for peak spacing and for the relative locations of the peaks in diﬀerent spectra. But peak locations predicted by the baseline model do not coincide with the true locations, and the typical phase shift

δ�p is about one fourth of the oscillation period. In addition, the baseline model also predicts that EE peaks are located halfway between TT peaks and also halfway between TE peaks, which is not exactly true either.

43 ��

Figure 4.1. Comparison of the spectra of theﬁducial cosmology (solid curves) and the peak locations predicted by the baseline model (vertical dashed lines).

Despite its deficiencies, wefind the baseline model to be a useful starting point. In the remainder of the paper we explain the differences between these baseline predictions and the predictions of theΛCDM model when calculated much more precisely.

4.3 Evolution of phase shifts in the photon perturbations

According to the baseline model,Θ 0(Θ1) can be described as a simple harmonic oscillator under two assumptions: tight coupling between photons and baryons, and negligible impact of gravitational driving. In fact, both the decoupling eﬀect and decaying gravitational potentials aﬀect the amplitude and the phase of the acoustic oscillation. Taking these into account, we may formally write the solution as

[Θ +Φ] cos(kr +φ ), (4.16) 0 ∝ s tot whereφ φ +φ withφ ,φ denoting the phase shift induced by decoupling tot ≡ dcp gr dcp gr and gravitational driving, respectively. The latter can be further decomposed asφ gr =

φgr,γ +φ gr,ν, due to the fact that the decay ofΦ+Ψ is caused by photon pressure and

neutrino free-streaming. To distinguish them, we callφ gr,γ the gravitational potential

transient induced phase shift, and callφ gr,ν the neutrino induced phase shift. The reason for this decomposition shall be clear later.

44 ��

��

� ��

��

Figure 4.2. The intervalsΔ(kr s) of neighboring peak-trough of [Θ0 +Φ](k,η) for 1 modek=0.5 Mpc − . Dots are numerical results of peak-trough intervals. Solid line is the analytic result of high-order correction to the tight coupling approximationφ dcp. Dashed line is the result of corrections from both late-time high-order correctionφ dcp and early-time gravitational drivingφ gr,γ sourced by photon perturbations.

In the remainder of this section, we will analytically derive the phase shift induced by each eﬀect and numerically measure these phase shifts.

4.3.1 Decoupling:φ dcp

After a mode enters the horizon (kη� 1), the tight coupling approximation becomes less reliable, and the small decoupling effect induces both diffusion damping and phase shift to the evolution of photon perturbations. The diffusion damping was analytically studied in [e.g. 76, 129, 139] by expanding the correction to tight coupling approximation toO(k/˙τ) 2. For our purpose of exploring the phase shift induced by decoupling, we extend the correction toO(k/˙τ) 3 andfind the analytic expression of the phase shift induced by decouplingφ (see Appendix 4.8.1). Consequently, the intervalsΔ(kr (η )) kr (η ) dcp s p ≡ s p − krs(ηp 1) of thep-th and (p 1)-th extrema inΘ 0(k,η) are no longer equal toπ. Instead, − − Δ(kr ) =π Δ(φ ). The intervalsΔ(kr ) measured from the Boltzmann code Class s − dcp s [108, 140] and obtained from the analytical result in Appendix 4.8.1 are shown in Fig. 4.2. They are in good agreement except at late time when the correction to tight coupling up toO(k/˙τ) 3 is no longer accurate and at early time when the gravitational driving is important.

45 4.3.2 Transient:φ gr,γ Both the amplitude and the phase of the acoustic oscillation are modulated by the gravitational driving at early time when the potentials have not decayed. Following Bashinsky and Seljak [75], we deﬁne the overdensity of photon number with respect to coordinate

volume,d γ 3(Θ0 Ψ) and two potentialsΦ =Φ Ψ, which satisfy the dynamical ≡ − ± ± equation accurate toO(k/˙τ) [Eq. (4.6)],

d�� +d = 3Φ , (4.17) γ γ − +

where the prime denotes diﬀerentiation with respect to krs. Assuming negligible neu-

trinos, bothΦ andd γ can be analytically solved during radiation domination,d γ ± ∝ cos(krs +θ(kr s)), where the phase shiftθ(kr s) decays with time as 2 θ(krs) , (4.18) � krs krs 1 � � � (see Appendix 4.8.2 for accurate expressions). We� have assumed radiation domination in the above derivation, so we expect it is correct only forη�η eq. After the transition to matter domination, the gravitational potentials keep roughly constant, andθ also freezes at

θ(krs) θ(kr s) . (4.19) η>ηeq � η=ηeq � � The above analysis shows that,� for largek,θ(kr �) decays with time to zero, as the � �s potentials decay to zero; while for smallk,θ(kr s) does not decay to zero, as the potentials

do not completely decay. To summarize,θ(kr s) traces the potential transient. This also

explains why we callθ(kr s) the gravitational potential transient induced phase shift. A minor point is thatθ is the phase shift ford = 3(Θ Ψ), while the quantity more γ 0 − relevant to the TT power spectrum is the eﬀective temperature perturbation [Θ0 +Φ].

Hence what we plot in Fig. 4.2 is the peak-trough spacing of [Θ0 +Φ], i.e.,φ gr,γ instead

ofθ. According to Fig. 4.2, the transient induced phase shiftφ gr,γ accounts for most of

the residual phase shifts of theﬁrst few extrema in [Θ 0 +Φ].

4.3.3 Neutrinos:φ gr,ν Another important component during radiation domination is free-streaming neutrinos,

which recast the potential transient and introduce a new phase shift componentφ gr,ν.

46 ��

��

Figure 4.3. Phase shifts of sources induced by 3.046 neutrinos and measured at the LSS.

Under the assumption of radiation domination (neglecting matter and dark energy), the potentialΦ (kr ) completely decays, and so does the transient induced phase shift + s → ∞ φ (kr ). More generally, previous studies [75, 141, 142] show that a nonzero gr,γ s → ∞ phase shiftφ (kr ) requires modes propagating faster than the sound speed of gr s → ∞ photon-baryon plasmac s. Neutrinos freely stream at the light speed,c>c s, so a nonzero phase shiftφ (kr ) is expected. Accurate toO(R ), Bashinsky and Seljak [75] gr,ν s → ∞ ν and Baumann et al. [142] obtained a scale-independent phase shift

φ (kr ) = 0.191R π, (4.20) gr,ν s → ∞ ν

whereR ν =ρ ν/(ρν +ρ γ) is the energy fraction of neutrinos in the radiation. Taking account of matter domination, the above scale-independent result only applies for modes entering the horizon during radiation domination; while for modes entering the horizon during matter domination,φ approaches zero as k 2 (see Appendix 4.8.3 for details). gr,ν ∼ In contrast toφ gr,γ, the neutrino induced phase shiftφ gr,ν does not trace the potential transient, though neutrinos indeed aﬀect the transient.

4.4 Phase shifts in photon perturbations at the LSS

In this section, we numerically measure the phase shift of photon perturbations at the LSS, single out the contribution from each eﬀect, and analytically interpret them.

47 ��

��

Figure 4.4. The phase shifts of [Θ0 +Φ] (left panel) and ofΠ (right panel) induced by diﬀerent physical eﬀects measured at the LSS.

To measure the total phase shiftφ tot of the monopole source [Θ0 +Φ](kr s,�) for the

ﬁducial cosmology, weﬁxη=η � and match its extrema as a function ofk to those of

cos(krs,� +φ tot) by adjustingφ tot. In a similar way, the total phase shiftφ tot[Π] of the

polarization sourceΠ(kr s,�) is also measured by matching with sin(krs,� +φ tot[Π]).

To single out the neutrino induced phase shiftφ gr,ν from the total phase shift, we need toﬁlter out the other two eﬀects. For this purpose, we construct a comparison cosmology

without neutrinos and withz eq,θ �,θ D,ω b ﬁxed to the corresponding values of theﬁducial

cosmology, by adjusting the cold dark matter densityω c, dark energy fractionΩ Λ and the

helium fractionY P [143]. Then we measure the monopole source [Θ0 +Φ](krs,�) at the LSS for both theﬁducial cosmology and the comparison cosmology. The displacement between

the extrema locations of the two isφ gr,ν, which is plotted in Fig. 4.3. As expected,φ gr,ν approaches zero for smallk modes and approaches a constant for largek modes. We ﬁndφ (k ) = 0.067π, which is about 15% lower than the lowest-order analytically gr,ν →∞ derived value 0.078π. Note that theφ gr,ν derived in Appendix 4.8.3 is actually the phase

shift of the monopoleφ gr,ν[Θ0], not the phase shift of the polarizationφ gr,ν[Π]. Both of

them (and alsoφ gr,ν[Θ1]) are wellﬁtted by

1 1 krs,� φ (k,η ) = tan− 0.5 , (4.21) gr,ν � 7.5 π − � �

at least forkr s,� �π.

48 To single out the gravitational potential transient induced phase shiftφ gr,γ, we solve

the Einstein-Boltzmann equations in the strict tight coupling limit (Θ� 2 = 0) for the ≥

comparison cosmology constructed above, and evaluate the monopole atη �. In this way,

we get rid of bothφ dcp andφ gr,ν, therefore the only phase shift left isφ gr,γ (see left panel of Fig. 4.4). Analytic study [Eqs.(4.18,4.19)] shows that 2 φgr,γ(k,η �) , (4.22) � krs(ηeq) krs(ηeq) 1 � � � which is consistent with the numerical result for large� k modes (left panel of Fig. 4.4).

The transient induced phase shiftφ gr,γ[Π] in the polarization source is more subtle to tap, because the polarization sourceΠ (4k/3˙τ)Θ vanishes in the strict tight coupling �− 1 limit. We tentatively extractφ [Π] by matching the extrema ink Θ (kr ) with those gr,γ × 1 s,� of sin(krs,� +φ gr,γ[Π]) (see right panel of Fig. 4.4 for the numerical results).

Withφ gr,ν andφ gr,γ singled out, the residual part is certainly the decoupling induced phase shiftφ given byφ (φ +φ ), which as expected scales as k 3 for small dcp tot − gr,ν gr,γ ∼ k modes (see left panel of Fig.4.4). The decoupling induced phase shiftφ dcp[Π] in the polarization source is also extracted in the same way, which shows more structures than that of the monopole (see right panel of Fig.4.4). According to the analytic study in Appendix 4.8.1,φ [Π] scales as O(k 3) O(k), where the former term is the same to dcp ∼ − φdcp of the monopole and the latter term comes from the fact thatΠ andΘ 0 are out of

phase by slightly less than 90 degrees. The scaling explains the overall shape ofφ dcp[Π]. The ‘anomaly’ of thefirst point is due to the rise in the amplitude ofΠ ask increases. The polarization is sourced by the gradient of the velocityfield,Π kΘ , and so the ∝ 1 factor ofk drives the extrema inΠ to largerk modes. It is straightforward to estimate that thefirst extremum is driven away byδ(kr ) 0.1π, which is exactly the anomaly s � dip we observed, and the effect on other extrema is weak as their changes ink have a smaller dynamic range.

4.5 Projection

With our description of the phase shiftφ tot in sources at the LSS complete, we are ready to study the peak shift in the spectra. In this section, weﬁrst give a more rigorous

49 treatment of the projection process from photon perturbations at the LSS to the power spectra, then point out the corrections to the baseline model, and measure the peak shift induced by each correction.

4.5.1 A Rigorous Treatment of Projection In the baseline model, the pictorial argument of projection yields a qualitative understanding on the peak structure in the spectra. But for a quatitative understanding, we need a rigorous treament of the projection process. Let us start from the well-known line-of-sight solutions to Eq. (4.2) [e.g. 135, 137],

η0 ikµ(η η0) τ(η) Θ(k, µ,η 0) = dη S˜(k, µ,η)e − − , (4.23) �0 where the source

1 S˜(k, µ,η)= Ψ˙ ikµΦ ˙τ Θ + µV P (µ)Π . (4.24) − − 0 b − 2 2 � � From solution Eq. (4.23), we obtain the multipoles

η0 Θ (k) = dηg(η)[Θ (k,η)+Φ(k,η)]j [k(η η)] � 0 � 0 − �0 η0 iV d dηg(η) b j [k(η η)] − k dη � 0 − �0 η0 3 Π d2 + dηg(η) 2 2 j�[k(η0 η)] 0 4 k dη − � η0 τ + dηe − [Φ˙ + Ψ˙ ]j [k(η η)], (4.25) � 0 − �0 τ whereg(η) ˙τe − is the visibility function which narrowly peaks at the LSSη=η , ≡− � Θ0(k,η) is the amplitude we have investigated in Section 4.4 in detail. The above rigorous formula not only verifies the naive expectation of the baseline model, but also takes account of contributions from doppler effect (dipole), polarization, and integrated Sachs- Wolfe (ISW) effect. In addition,j [k(η η)] is nearly zero fork(η η) �. The � 0 − 0 − � asymmetric projection can be understood with the following pictorial argument. As shown in Fig. 4.5, in the direction perpendicular to the wavevector,µ = 0, the peak-trough separation of thek mode matches that ofP (µ) with� =k(η η ). �m m 0 − � Whereas in the direction parallel to the wavevector,µ = 1, the peak-trough separation

50 ��

� ��

��

Figure 4.5. Illustration of the projection from three to two dimensions. The round circle is the LSS, the vertical solid(dashed) lines are the peaks(troughs) of mode �k atη �. The wiggling curve around the LSS is a Legendre polynomialP (µ) with�=k(η η ), � 0 − � whereµ= kˆ ˆγ, is the cosine of the angle subtended by the wavevector �k and the · direction of observation ˆγ. In theµ = 0 direction, the peak-trough separation of � the k mode matches that ofP �m (µ) �m=k(η0 η�) (shown). In theµ = 1 direction, the � | − peak-trough separation of the k mode matches that ofP �m (µ) �m k(η0 η�) (not shown). | � −

subtends a larger angle and so is better matched with aP (µ) with� k(η η ). �m m � 0 − � Therefore modek distributes its power on all modes satisfying� k(η η ). In other � 0 − � words, the power of mode� is contributed by all modes satisfyingk(η η ) � [e.g. 137]. 0 − � � Using the transfer functionΔ (k) deﬁned byΔ (k) Θ (k)/Φ(k, 0) and the primordial � � ≡ � potential power spectrumP(k) deﬁned by

� 3 (3) Φ(�k, 0)Φ (�k�, 0) = (2π) δ (�k �k�)P(k), (4.26) − � � TT C� is written as

TT 2 2 C� = dk k P(k)Δ � (k), (4.27) � up to some constant factor [e.g. 122, 135].

4.5.2 Corrections to the Baseline Model In the baseline model, we simplify the monopole to be purely cosine, [Θ +Φ] cos(kr ), 0 ∝ s simplify the LSS to be inﬁnitely thin, i.e.g(η)=δ(η η ), and simplify the projection − �

51 TT Table 4.1. The shift of thep-th peak in the temperature power spectrumD � deﬁned byδ� p 302p � p(TT), consists ofδ� monopole,δ� non monopole andδ� lensing, where the ≡ − − 2 former can be decomposed asδ� monopole =δ�[φ gr,ν] +δ�[φ gr,γ] +δ�[φ dcp] +δ�[k P(k)] + δ�[j�] +δ�[g(η)]. 2 p-th peak δ�[φgr,ν] δ�[φgr,γ] δ�[φdcp] δ�[k P(k)] δ�[j �] δ�[g(η)] δ�monopole δ�non monopole δ�lensing δ�p − 1st 5 23 4 20 19 7 78 4 0 82 = 302 220 − 2nd 12 25 1 7 29 9 83 16 1 68 = 604 536 − − 3rd 15 26 0 6 43 7 97 6 2 93 = 906 813 − − 4th 16 21 0 4 40 3 84 6 3 81 = 1208 1127 − − 5th 17 20 2 3 45 1 86 3 6 89 = 1510 1421 − − − 6th 17 17 6 3 39 9 73 1 15 87 = 1812 1725 − − −

EE Table 4.2. The shift of thep-th peak in the polarization power spectrumD � deﬁned by δ�p 302(p 0.5) � p(EE), consists of δ�Π and δ�lensing, where the former is ≡ − − 2 decomposed asδ� Π =δ�[φ gr,ν] +δ�[φ gr,γ] +δ�[φ dcp] +δ�[k P(k)] +δ�[j �] +δ�[g(η)]. 2 p-th peak δ�[φgr,ν] δ�[φgr,γ] δ�[φdcp] δ�[k P(k)] δ�[j �] δ�[g(η)] δ�Π δ�lensing δ�p 1st 3 12 30 15 4 15 11 0 11 = 151 140 − − − 2nd 10 16 17 11 19 19 58 0 58 = 453 395 − − 3rd 14 15 23 6 34 21 67 0 67 = 755 688 − − 4th 16 14 31 4 43 20 66 1 67 = 1057 990 − − 5th 17 13 36 3 46 16 59 1 60 = 1359 1299 − − 6th 17 12 39 2 48 11 51 2 53 = 1661 1608 − −

TE Table 4.3. The shift of thep-th peak in the power spectrumD � is deﬁned by δ� 151(p+0.5) � (TE), and notations used here are similar to those in Table 4.1 p ≡ − p . 2 p-th peak δ�[φtot] δ�[k P(k)] δ�[j �] δ�[g(η)] δ�monopole δ�non monopole δ�lensing δ�p − 1st 32 11 21 7 71 4 0 75 = 227 152 − 2nd 27 8 28 8 71 2 0 69 = 378 309 − − 3rd 23 5 28 11 67 6 0 61 = 529 468 − − 4th 32 4 41 10 87 3 1 85 = 680 595 − − 5th 28 4 40 13 85 3 1 83 = 831 748 − − 6th 19 2 41 11 73 5 0 68 = 982 914 − − 7th 15 3 39 9 66 5 0 61 = 1133 1072 − − 8th 18 2 41 8 69 5 0 64 = 1284 1220 − − 9th 16 2 45 6 69 5 1 65 = 1435 1370 − − 10th 11 1 34 8 54 6 0 48 = 1586 1538 − − 11th 11 1 41 4 49 5 0 44 = 1737 1693 − − − 12th 13 0 28 6 45 7 0 38 = 1888 1850 − − −

52 fromk modes to� modes as one-to-one,�=k(η η ). In fact, all above simplifications 0 − � are not exactly correct. The phase shiftφ tot of multipolesΘ �, thefinite width of the LSS and the fact that the projection fromk modes to� modes is not one-to-one, introduce peak shifts to the spectra. In addition, dipoleΘ 1, polarizationΠ and ISW effect also contribute a sub-dominant part to the power spectra. Taking TT as an example, we define the total peak shift relative to the prediction of the baseline model, δ� 302p � (TT), where p ≡ − p TT �p(TT) is the location ofp-th peak in the theoretical temperature power spectrumD � of thefiducial cosmology (Fig. 4.1). In the remainder of this section, we shall investigate each correction contributing to the total peak shiftδ� p individually. Note that a positive

δ�p denotes a shift to smaller�.

4.5.2.1 Phase Shifts in Sources: δ�[φtot]

The phase shiftφ tot in the monopole [Θ0 +Φ] at the LSS induces a peak shift in TT,

δ�[φtot] =φ tot/θ�, which is decomposed into three componentsδ�[φ gr,ν]+δ�[φgr,γ]+δ�[φdcp] (Table 4.1 and Fig. 4.4). Similar analysis is also done for EE (Table 4.2 and Fig. 4.4) and TE (Table 4.3). We choose not to do the decomposition for TE, whose source [Θ +Φ] Π 0 × is not an independent quantity. 4.5.2.2 Primordial Power Spectrum: δ�[k2P(k)]

Eachk mode carries diﬀerent amount of power which is speciﬁed by the primordial power spectrumP(k), a detail not included in the baseline model. In fact, the temperature

TT power spectrumD � is modulated by the primordial power spectrumP(k) as follows, 1 DTT k 2P(k) , (4.28) � � ∼ �

2 1 where we have used the scale-invariant primordial power spectrumk P(k) k − , and the ∼ simpliﬁed correspondence�=k(η η ). 0 − � The modulation of the primordial power can be derived in a more rigorous way. Using Limber approximation [92, 107, 144],

∞ π dxf(x)j (x) f(L) , (4.29) � � 2L �0 �

53 whereL=�+1/2, the transfer function is written as

1 π Θ0(k,η)+Φ(k,η) Δ�(k)= g(η) , (4.30) k 2L Φ(k, 0) � � � � �k(η0 η)=L � − � and the temperature power spectrum is simpliﬁed as �

η0 TT 2 2 D� = dηk P(k)g (η) �0 Θ (k,η)+Φ(k,η) 2 0 , (4.31) × Φ(k, 0) � � � �k=L/(η0 η) � − � where we have used the deﬁnition (4.27), and changed the� integration variable fromk to

η. Taking a step further, using facts that the visibility function is narrowly peaked atη � and [Θ +Φ](k,η )/Φ(k, 0) cos(kr +φ ), we have 0 � ∝ s,� tot cos2(�θ +φ ) DTT[j=δ, g=δ] � tot , (4.32) � ∝ �

TT whereD � [j=δ, g=δ] denotes the approximate power spectrum calculated using the Limber approximation and instantaneous recombination. The modulation by the primordial powerk 2P(k) 1/� drives the TT peaks to smaller ∼ � from the predictions of the baseline model. Analytically, it is straightforward to obtain

1 48 δ�[k2P(k)] , (4.33) � 2θ (pπ φ ) � pπ φ � − tot − tot which is consistent with numerical results (see Table 4.1). Similar analysis is also done for EE and TE.

4.5.2.3 Asymmetric Projection: δ�[j�]

Assuming the monopole [Θ0 +Φ](k,η �) peaks atk p, the baseline model predict a peak in the power spectrum at� =k (η η ). But the projection fromk modes to� modes is not p p 0 − � one-to-one, instead allk modesk �/(η η ) contribute to�, and modesk �/(η η ) � 0 − � � 0 − � contribute no power to� (Fig.4.5 and Eq.(4.13)). As a result, a slightly smaller� (than

�p) receives power from a wider range ofk modes aroundk p. Therefore the asymmetric projection drives TT peaks to smaller� from the baseline model predicted peak locations.

54 ��

��

Figure 4.6. The comparison between the asymmetric visibility functiong(η) and two Gaussian functions both withσ = 20 Mpc, peaking atη � = 281 Mpc and ¯η = 293 Mpc, respectively, where ¯η is the mean decoupling time deﬁned by ¯η= g(η)ηdη/ g(η)dη. � � Thek modes and� modes are connected by the transfer functionΔ �(k) [Eq.(4.25)], which is simpliﬁed as

Θ (k,η ) +Φ(k,η ) Δ (k) j (k(η η )) 0 � � , (4.34) � � � 0 − � × Φ(k, 0) � � under the approximation ofg(η)=δ(η η ). Quantitatively, we compute an approximate − � TT spectrumD � [g=δ] from Eqs.(4.27, 4.34), and numerically measure δ�[j �] from differ- TT TT ences between the peak locations ofD � [j=δ, g=δ] andD � [g=δ] (see Table 4.1). Similar analysis is also done for EE and TE power. According to Table 4.1, 4.2 and 4.3, the asymmetric projection drives the peaks in the spectra to smaller� except thefirst EE peak. The anomaly also comes from the rise in the amplitude ofΠ ask increases. Thefirst peak inΠ is tiny compared to following few peaks. As a result, thefirst EE peak gains more power from largerk modes ofΠ, therefore is driven to larger� (negativeδ�). 4.5.2.4 Visibility Function: δ�[g(η)]

TT Due to theﬁnite width of the visibility functiong(η),D � is powered by the source

g(η)[Θ0(k,η)+Φ(k,η)] from a time interval instead of a single time sliceη �. The visibility

functiong(η) is positively skewed (Fig. 4.6). But [Θ 0 +Φ](k,η) decreases in amplitude over time due to diﬀusion damping, so is negatively skewed. In addition, the monopole

55 ��

��

� ��

TT Figure 4.7. The contribution to the unlensedD � from each component: dominant monopole, subdominant dipole and early ISW, and negligible polarization and late ISW.

[Θ +Φ](k,η) cos(kr (η)+φ ) peaks at smallerk modes at later time (largerr (η)). 0 ∝ s tot s Therefore theﬁrst few TT peaks would be driven to smaller�, where the damping is small and the asymmetry ofg(η) dominates; and other TT peaks would be driven to large�,

where the asymmetry of [Θ0 +Φ](k,η) dominates due to stronger damping. For polarization, the source termΠ increases in amplitude over time, generated by free streaming. Consequently,Π(η) has similar asymmetry to the visibilityg(η), and so EE and TE peaks are driven to even smaller�. Quantitatively, the phase shift induced by the visibility function δ�[g(η)] is obtained

TT TT by the diﬀerences between peak locations of trueD � and ofD � [g=δ]. The numerical results are consistent with our qualitative analysis (Table 4.1, 4.2 and 4.3). 4.5.2.5 ISW, Dipole, Polarization: δ�[non monopole] − The TT and TE power spectra also receive contributions from other components: dipole

Θ1, polarization sourceΠ, and ISW eﬀect Φ˙ + Ψ˙ (see Fig. 4.7 for the decomposition of TT). The early ISW power peaks near the particle horizon scale of decoupling, which is larger than the sound horizon so drives theﬁrst peak to lower�. The late ISW only

operates at very large scale (�� 10), therefore has almost no impact on the peak locations [49].

56 According to the baseline model,Θ 1 is expected to be 90 degrees out of phase withΘ 0, in which caseΘ 1 would have no inﬂuence on the peak locations. Taking the decoupling eﬀect into account,Θ 1 andΘ 0 are found to be out of phase by more than 90 degrees

(see Appendix 4.8.1). As a result,Θ 1 drives the peaks to larger�. The impact ofΠ is negligible. To summarize,δ�[non monopole] of TT and TE are positive at small� modes where − the (early) ISW dominates, and are negative at large� modes whereΘ 1 dominates (see Fig. 4.7, Table 4.1 and 4.3). 4.5.2.6 Lensing: δ�[lensing]

Gravitational lensing tends to smooth the power spectra by redistributing the power among� modes (see the review of weak lensing by [91]). The net eﬀect is that peaks lose power and troughs gain power. If a peak is symmetric, the modes on both sides of the peak would lose the same amount of power, thus the peak amplitude is suppressed with the peak location unaﬀected. If a peak is asymmetric due to more damping at larger� modes, the modes on the right side of the peak would lose more power than modes on the left side, therefore the peak is driven to smaller�. The EE and TE peaks are more symmetric than TT peaks; as a result, the lensing driven phase shifts in EE and TE are smaller (see Table 4.1 and Table 4.2, 4.3 for comparison) .

4.6 Comparison of predicted and measured peak locations

In contrast to theoretical power spectra, it is impossible to directly read the peak locations out of data points in the presence of noise. To measure the peak locations from the data points, afitting procedure is required. Taking the TT power spectrum as an example, the Planck collaboration [2]first removed the damping tail, and thenfit Gaussian functions

TT to the peaks inD � . The peak locations measured in this specific procedure cannot be compared with the literal peak locations in the theoretical power spectrum of the fiducial cosmology. To compare theΛCDM model predictions with the peak location measurements , we apply the samefitting procedure on the theoretical spectrum of the

57 Table 4.4. Locations of the peaks in the power spectra. The peak locations measured from the Planck 2015 data are listed in the 3rd column [Table E.2. in 2], and the peak locations predicated by thefiducial cosmology are listed in the 2nd column (Note that these peak locations are determined by thefitting procedure used on the data, therefore are different from the literal peak locations of theoretical power spectra). p-th peak multipole (model) multipole (data)

TT power spectrum

1st 220.9 220.0 0.5 ± 2nd 538.5 537.5 0.7 ± 3rd 809.5 810.8 0.7 ± 4th 1122.5 1120.9 1.0 ± 5th 1445.7 1444.2 1.1 ± 6th 1774 1776 5 ± 7th 2071 2081 25 ± 8th 2429 2395 24 ±

EE power spectrum

1st 135.3 137 6 ± 2nd 395.4 397.2 0.5 ± 3rd 689.7 690.8 0.6 ± 4th 992.1 992.1 1.3 ± 5th 1299 1296 4 ±

TE power spectrum

1st 149.0 150.0 0.8 ± 2nd 307.8 308.5 0.4 ± 3rd 471.3 471.2 0.4 ± 4th 593.4 595.3 0.7 ± 5th 747.3 746.7 0.6 ± 6th 915.9 916.9 0.5 ± 7th 1073.3 1070.4 1.0 ± 8th 1224.8 1224.0 1.0 ± 9th 1371.9 1371.7 1.2 ± 10th 1542.1 1536.0 2.8 ± 11th 1700.6 1693.0 3.3 ± 12th 1865 1861 4 ±

58 ﬁducial cosmology (Table 4.6). Weﬁnd that peak locations measured from the data points and from the theoretical spectra are in agreement. Most of the relative displacements are within 1σ, and all the relative displacements are less than 3σ.

4.7 Conclusions

The acoustic peak locations of the angular power spectra have been studied in detail. We start from a baseline model, which assumes tight coupling between photons and baryons before instantaneous recombination, and a simpliﬁed one-to-one projection fromk modes of the photon perturbations at the LSS to the� modes of the angular power spectra. Taking the temperature power spectrum as an example, the baseline model predict that

TT 2 2 [Θ0 +Φ](k,η) cos(kr s(η)) andD � cos (krs,�) �=k(η0 η�) = cos (�θ�), which peaks ∝ ∝ | − 0 at� p =pπ/θ � = 302p. The baseline model is roughly correct in its prediction for the peak spacing and for the relative positions of the peaks in different spectra, but is off by a large margin in its absolute predictions of peak locations. For example, thefirst peak

TT inD � is at� 1 = 220, which is shifted by δ�1 = 82 from the baseline model prediction 0 �1 = 302. The shift of the true power spectra locations relative to the baseline model predictions comes both from the phase shiftφ tot in the acoustic oscillations of the photon perturbations [Θ +Φ](k,η) cos(kr (η)+φ ), and the fact that the projection from 0 ∝ s tot photon perturbations at the LSS to the angular power spectra is far more complicated than assumed in the baseline model.

The phase shiftφ tot(k,η �) consists of two componentsφ dcp andφ gr, whereφ dcp is the phase shift induced by decoupling and dominates for largek modes (kη � � 1), andφ gr is the phase shift induced by the gravitational driving and dominates for smallk modes

(kη� � 1). The latter component can be further decomposed asφ gr,γ +φ gr,ν, whereφ gr,γ is the transient induced phase shift andφ gr,ν is the neutrino induced phase shift. A key difference between the two is thatφ gr,γ decays with increasingk, whileφ gr,ν grows with increasingk and approaches a nonzero constant. This difference stems from the time dependence of these effects, and the fact that at higherk there is more time, in units of the natural period of the oscillator 2π/(krs), between horizon-crossing and matter-radiation

59 equality. The projection fromk modes to� modes is not one-to-one as assumed in the baseline model. All perturbation modes satisfyingk �/(η η ) contribute to the angular � 0 − � power spectra at a given�. In addition, the LSS has non-zero width. Both of these differences with the baseline model introduce peak shifts to the power spectra. Other effects including the modulation of the primordial power spectrumP(k), (early) ISW effect, dipole moment of photon perturbation and lensing also contribute subdominant shifts to the peak locations. We also compare each peak location determined from Planck measurements to the location predicted under the assumption of the best-fitΛCDM model, andfind consistency.

4.8 Appendix

4.8.1φ dcp High-order corrections to the tight coupling approximation are only important at late time

after a mode enters the horizon (kη� 1), and the gravitational potentials have decayed. Following Zaldarriaga and Harari [129], we setΦ = 0 andΨ = 0 in this subsection (Refer to Blas et al. [140] for rigorous high-order correction to the tight coupling approximation). Assuming the formal solutions

i ωdη Θi,Θ p,i,Vb e , (4.35) ∼ � the dipole moment of Eq.(4.2) is written as

i 1 2 1 Θ V = iωΘ +k Θ Θ . (4.36) 1 − 3 b ˙τ 1 3 2 − 3 0 � � �� Accurate toO(k/˙τ) 3 on both sides of the above equation, Eqs.(4.2, 4.3) are decomposed as

ik Θ = Θ , (4.37) 0 ω 1 8 k 11 iω Θ = Θ 1 + , 2 − 15 ˙τ 1 6 ˙τ � � 5 3 iω Π= Θ 1 + , 2 2 2 ˙τ � �

60 and Eq.(4.4) is expanded as

i ωR ωR 2 ωR 3 Θ V =Θ i b + b +i b , (4.38) 1 − 3 b 1 − ˙τ ˙τ ˙τ � � � � � �

Rb ˙a where we have dropped a term ˙τ a in the bracket on the right-hand side, which is smaller

Rb than ˙τ ω when the mode is within the horizonkη> 1.

Plugging Eqs.(4.37,4.38) into Eq.(4.36), we obtain, forω=ω 0 +δω 0 +iγ,

2 2 16 γ k csRb + 45 ω0 = kcs, = , (4.39) ω0 − ˙τ�2cs(1 +R b)� δω k2 c2R2 + 88 3 c2R2 + 16 5c2R2 + 16 0 s b 135 − 4 s b 45 s b 45 = 2 . ω0 ˙τ � � 2(1� +R b) � � � 3 Note that˙τ is negative, and soγ is positive. Therefore, accurate toO(k/˙τ) ,Θ 0(k,η) can be described as a damped oscillator with a time-dependent phase shift, i.e.

k2/k2 Θ (k,η) cos(kr +φ )e− D , (4.40) 0 ∝ s dcp

where

φ = δω dη O(k/˙τ) 2, dcp 0 ∼ � k2/k2 = γdη O(k/˙τ). (4.41) D ∼ � Other useful multipoles obtained are

k2/k2 Θ c sin(kr +φ +φ )e− D , 1 ∝ s s dcp 1 4k k2/k2 Π c sin (kr +φ +φ )e − D , (4.42) ∝− 3˙τ s s dcp 2

where

1 γ 1 21γ φ = tan− ,φ = tan − , (4.43) 1 ω 2 − 4ω � 0 � � 0 �

andφ 1(φ2) comes from the fact thatΘ 1(Π) andΘ 0 are not exactly 90 degrees out of phase due to the diﬀusion damping.

61 4.8.2φ gr,γ The solution to the equation of the forced oscillator [Eq.(4.17)] is written as (e.g. Eq. (8.24) of Dodelson [135])

dγ(krs) =d γ(0) cos(krs) krs 3 d(kr� )Φ (kr� ) sin(kr kr � ), (4.44) − s + s s − s �0 which can be simpliﬁed as

dγ(krs) = [d (0) + 3A(kr )] cos(kr ) 3B(kr ) sin(kr ), γ s s − s s 2 2 = [dγ(0) + 3A] + (3B) cos(krs +θ), (4.45) � whered (0) is the initial amplitude denoted asd (0) 3ζ, and γ γ ≡−

krs A(krs) = Φ+(krs� ) sin(krs� )d(kr s� ), �0 krs B(krs) = Φ+(krs� ) cos(krs� )d(kr s� ), (4.46) �0

1 3B θ = sin− . (4.47) 2 2 � (3A+d γ(0)) + (3B) �

To solveA andB,Φ + is required.� In the radiation domination,Φ + is sourced byΦ − (e.g. Eq.(2.50) of Baumann et al. [142])

4 2 Φ+�� + Φ+� +Φ + =S(Φ ) Φ �� + Φ� + 3Φ , (4.48) krs − ≡ − krs − − andΦ is sourced by the radiation stress which is dominated by free-streaming neutrinos − (e.g. Eq.(5.33) of Dodelson [135])

2 2 k Φ = 32πGa ρν 2, (4.49) − − N

where we have dropped the negligible stress of photons.

0 Assuming a cosmology without neutrinos, or accurate toO(R ν), whereR ν is the energy (0) (0) fraction of neutrinos in radiation,R ν ρ ν/(ρν +ρ γ), we haveΦ = 0, andΦ + can be ≡ −

62 analytically solved as

(0) sin(krs) kr s cos(krs) Φ+ (krs) = 4ζ − 3 . (4.50) − (krs)

Consequently,A andB are integrated as

2 sin (krs) krs A(0)(kr ) = 2ζ 1 →∞ 2ζ, s − (kr )2 −−−−→ � s � krs cos(krs) sin(krs) krs (0) →∞ B (krs) = 2ζ − 2 0. (4.51) (krs) −−−−→

PluggingA (0) andB (0) into Eq.(4.47), we obtain the phase shiftθ for [Θ Ψ]. Withd 0 − γ (0) andΦ known, it is straightforward to obtain both [Θ0 +Φ] and its phase shiftφ gr,γ, ± which is indistinguishable fromθ after the mode enters the horizon.

4.8.3φ gr,ν

Following Baumann et al. [142],φ gr,ν can be analytically studied by a perturbation approach, whose earlier version was orginally developed for probing the impact of neutrino free-streaming on tensor modes [e.g. 145–149]. Accurate toO(R ν), the two potential are written as (1) (0) (1) Φ =R νΦ ,Φ + =Φ + +R νΦ+ . (4.52) − − In the radiation domination, Eq.(4.49) can be rewritten as

4 4 Φ = 2 2 Rν 2 = 2 2 RνDν,2, (4.53) − − k rs N − 3k rs whereD is deﬁned byD 3( Ψ), andD is the quadrupole moment,D = 3 . ν ν ≡ N− ν,2 ν,2 N2 0 To determineΦ toO(R ν), we only need to specifyD ν,2 toO(R ν). − The evolution of neutrino perturbation is governed by (e.g. Eq.(4.107) of Dodelson [135]), D˙ + ikµD = 3ikµΦ , (4.54) ν ν − + and the solution is

ikµη Dν(η) =D ν(0)e− η ikµ(η η ) 3ikµ dη� e− − � Φ (η�), (4.55) − + �0

63 whereD ν(0) is determined by the inﬂation inspired initial conditionD ν(0) =D ν,0(0) = 3ζ. Plugging the zeroth-order potentialΦ (0) into the above solution,D (η) and conse- − + ν 0 quentlyD ν,2 ofO(R ν) are obtained. (1) (1) Combining Eq.(4.53) and Eq.(4.48),Φ (krs) andΦ + (krs) are obtained. Plugging − (1) Φ+ (krs) into Eqs.(4.46),A andB can be numerically obtained, accurate toO(R ν) [142],

A(kr ) = 2ζ (1 0.134R ), s → ∞ − ν B(kr ) = 0.600ζR . (4.56) s → ∞ ν

PluggingA andB into Eq.(4.47), the phase shift induced by neutrinos accurate toO(R ν) is obtained B φ = = 0.191R π. (4.57) gr,ν ζ ν

The aboveφ gr,ν is derived under the assumption of radiation domination, so is appropriate only for largek modes. Taking the matter domination epoch into account,φ gr,ν is expected to approach zero as k 2 for smallk modes [142]. ∼

64 Chapter 5

Searching for Signatures of Dark Matter-Dark Radiation Interaction in Observations of Large-scale Structure

This chapter is based on a paper (arXiv:1801.07348) submitted to Phys. Rev. D.

5.1 Introduction

Dark matter is an essential component of the standardΛCDM cosmology, whose existence has been established from many cosmological and astrophysical lines of evidence [see e.g. 150–152, for a brief summary]. On the other hand, increasingly sensitive eﬀorts at direct detection of canonical candidates such as WIMPs and axions have only resulted in upper limits [153, 154]. The lack of direct detection signatures implies that the dark matter is weakly coupled to the standard model, but it does not preclude a large coupling to a hidden sector. The idea of hidden sector dark matter has broadened the experimental search possibilities, while retaining some of the virtues of WIMP models such as concrete thermal and non-thermal production mechanisms [e.g., 155–158] and opening up new cosmological signatures [e.g., 159, 160]. The richer phenomenology expands the possible ways in which dark matter properties may be revealed through observations of the large-scale structure (LSS) of the universe.

65 Precision measurement of the Cosmic Microwave Background (CMB) temperature and polarization, as well as large-scale photometric and spectroscopic galaxy surveys can be used to detect the inﬂuence of non-trivial dark matter properties or to limit them [161].

Indeed, theσ 8 tension inΛCDM cosmology, that LSS surveys yield lowerσ 8 values than that derived from CMB observations, is potentially due to non-trivial dark matter interactions and has also served to renew interest in exploration of broader classes of dark matter models [e.g. 81, 160, 162–177]. In this chapter, we focus on the non-Abelian dark sector scenario proposed by [81], where the dark matter is a Dirac fermion that transforms under a non-Abelian gauge group, and the dark radiation is the associated gaugefield, a massless “dark gluon”. The strong self-interaction of the dark radiation makes it behave like afluid, instead of a free-streaming species. The interaction between dark matter and the dark radiationfluid (dm-drf) acts to suppress the matter power spectrum, improving agreement with lower

σ8 values derived from LSS measurements [4–7, 9, 10, 178–180]. Some previous works [169, 174, 176] show an inference of non-zero dark matter-dark radiationﬂuid (dm-drf) interaction at 3σ signiﬁcance jointly using Planck CMB and LSS measurements, including Planck CMB lensing [7], CFHTLens weak lensing (WL) [4] and Planck Sunyaev-Zeldovich (SZ) cluster counters [5, 6]. The authors of [169] also emphasize the possibility for these

1 models to alleviate theH 0 tension [181, 182] as well. Throughout this chapter, we examine this non-Abelian dark sector model with these LSS measurements one by one. We critically review the analyses done previously and ﬁnd that Planck SZ is essential for the claimed inference of non-zero dm-drf interaction in previous analyses. But the SZ cluster constraint is limited by a large uncertainty in the cluster mass scale determination, which is usually parametrized by a mass bias parameterb. The bias parameterb itself is constrained by several diﬀerent analyses of the gravitational lensing induced by SZ galaxy clusters including two using the distorted shapes of background galaxies (“Weighing the Giants” [184] and the Canadian Cluster Comparison Project [185]) and one using distortions of the CMB [186]. The inferred value

1Recently, Das et al. [183] have proposed to use the history of cosmic reionization to constrain general dark matter-dark radiation interactions.

66 ofσ 8 from the observations of the SZ clusters depends sensitively on the mass estimates of the clusters and therefore on the mass bias parameterb. While previous dm-drf analyses effectively assumed zero uncertainty inb, wefind that including an uncertainty based on any of the above inferences ofb, the claimed inference of non-zero dm-drf interaction turns into an upper limit. Next we introduce a data set not previously used for constraints on the dm-drf interaction, the latest inferences of the matter power spectrum from Lyman-α (Lyα) forest data [187–189] . Compared with the other LSS measurements mentioned above, the Lyα forest 1 power spectrum is sensitive to the matter power spectrum at smaller scalek Mpc − , ∼ a scale that is more sensitive to the strength of the interaction in the dm-drf model. We find that the matter power spectrum derived from the latest Lyα forest data is much steeper than that derived from Planck CMB data, assumingΛCDM. Finally, we examine whether this Lyα-CMB tension can be resolved by the dm-drf interaction, and whether the dm-drf interaction can lead to consistency across all the datasets we consider here. The chapter is organized as follows. In Section 5.2, we briefly introduce the non- Abelian dark sector model and its impacts on the CMB power spectra and the matter power spectrum. In Section 5.3, we show that using SZ data with the mass bias parameter fixed or varying makes a huge difference in the constraints of cosmological parameters. In Section 5.4, we point out the Lyα-CMB tension in theΛCDM cosmology, and that the joint dataset favors a non-zero dm-drf interaction. In Section 5.5, we extend the exploration to more general dm-drf interaction models characterized by interaction rates scaling with temperature in different ways, and we also examine these models against CMB data and LSS measurements. We provide a summary in Section 5.6.

5.2 Canonical DM-DRF Interaction Model

Following Ref. [81], we useΓ for the momentum transfer rate (i.e., time scale for momentum of dark matter particles to change by (1)) due to the dm-drf scattering. This O momentum transfer leads to a drag force on the non-relativistic dark matter particles such that �˙v =aΓ(�v �v ), wherea is the scale factor and throughout this paper dots dm drf − dm

67 denote conformal time derivatives. In terms of physical quanties,Γ is approximately

Γ (T drf /mdm)ndrf σdm drf , (5.1) � − wherem dm is the mass of dark matter particles,σ dm drf is the cross section of dm-drf − scattering, andT drf andn drf are the temperature and the number density of dark radiation, 2 respectively. For the non-Abelian dark sector model proposed by [81],σ dm drf T drf− , − ∝ 2 thus we can writeΓ=Γ 0(T/T0) , whereΓ 0 denotes the velocity change rate today. With this parametrization, the evolution equations of dark matter density and velocity perturbations,δ dm andθ dm, and dark radiation density and velocity perturbations,δ drf andθ drf , are written as [169]

δ˙ = θ + 3φ˙, dm − dm ˙a θ˙ = θ +k 2ψ+aΓ(θ θ ), dm a dm drf − dm 4 (5.2) δ˙ = θ + 4φ˙, drf − 3 drf 2 δdr 2 3ρdm θ˙drf =k +k ψ+ aΓ(θdm θ drf ). 4 4ρdrf − in the Conformal Newtonian gauge, whereρ dm andρ drf are the average densities of dark matter and dark radiation, respectively;ψ andφ are the Newtonian potential and the perturbation to the spatial curvature, respectively.2 In the remainder of this section, we qualitatively explain the impacts of dm-drf interaction on cosmological observables, by comparing the matter power spectra (Figure 5.1) and CMB power spectra (Figure 5.2) of three cosmologies with diﬀerent parameters N ,N ,Γ and same other parameters (see Table 5.1). Similar numerical comparisons { ν drf 0} were also given in previous works [169, 176], and here we focus on connecting the impacts on observables with underlying physics.

5.2.1 LSS

In the standardΛCDM cosmology, the dark matter over-densityδ dm grows logarithmically in the radiation-dominated era, and grows linearly in the matter-dominated era [e.g., 49].

2The perturbation evolution equations in the Boltzmann code CAMB are written in the Synchronous gauge. In our modﬁed version CAMB, we keep a tiny amount of non-interacting dark matter to carry the Synchronous gauge.

68 7 1 Nν Ndrf 10 Γ0(Mpc− ) Model 1 3.546 0.0 0 Model 2 3.046 0.5 0 Model 3 3.046 0.5 2 Table 5.1. Three models used for clarifying the impact of dark radiationﬂuid on cosmological observables, with diﬀerent parameters N ,N ,Γ and sameΛCDM pa- { ν drf 0} rametersω = 0.02253,ω = 0.1122,A = 2.42 10 9, n = 0.967,τ=0.0845,H = b dm s × − s 0 70.4 km/s/Mpc.

A small dm-drf interaction does not remove these growth modes, instead it decreases the corresponding growth rates. We deﬁne the over-density suppression function (k,η) S ≡ [δ ] /[δ ] , and plot (k,η)=[δ ] /[δ ] (Table 5.1) for diﬀerentk dm Γ0>0 dm Γ0=0 S dm Model 3 dm Model 2 modes in Figure 5.1. We see that (k,η) shows a “self-similar” behavior for small-scale S modesk k : approximately, � eq

(k,η)=1 (kη� 1), S (5.3) (k,η) 1 A log(kη) (1 kη kη ), S � − � � eq and after radiation-matter equality, the evolution of (k,η) is similar for all diﬀerent S modes, where the suppression at radiation-matter transition and that today diﬀer by a constant number, (k,η ) (k,η ) B, (5.4) S eq −S 0 � where 1/k η 100 Mpc,η is the conformal time today,A andB are numbers eq ≈ eq ≈ 0 independent of modek and timeη (for the example shown in Figure 5.1,A 0.04 and ≈ B 0.05). The self-similar behavior for modesk k originates from the factΓ/H is ≈ � eq a constant in the radiation-dominated era, and therefore introduces no new timescale or length scale. With the approximations above, we can estimate the power spectrum suppression today as [P(k)] Γ0>0 = (1 A log(k/k ) B) 2 [P(k)] − eq − Γ0=0 (5.5) 1 2B 2A log(k/k ), ≈ − − eq

69 where we have ignored quadratic terms in the second line. This estimate explains the dm-drf interaction induced logarithmic suppression in the matter power spectrumP(k) for modesk k (see Figure 5.1 for the matter power suppression computed from CAMB � eq and the logarithmicﬁt).

5.2.2 CMB The imprint of the dm-drf interaction on the CMB power spectra is much more subtle as shown in Figure 5.2. Comparison of Models 1 and 2 conﬁrms the signatures of free- streaming neutrinos in the CMB spectra: namely power suppression and a (very small) shift in acoustic peak locations [1, 49, 77, 80, 94, 141, 142, 190]. Comparison of Models 2 and 3 shows that the dm-drf interaction very slightly increases the amplitude of modes

�� 500 and decreases that of modes�� 500. For modes�� 500, the increased amplitude can be explained by the near-resonant driving of the baryon-photonﬂuid perturbation amplitude by gravitational potential decay as modes enter the horizon [49, 77, 94]. The resistance to dark matter free fall from the dm-drf interaction contributes to gravitational potential decay, at least on scales large enough that, at the time of horizon crossing, the dark matter contributes a signiﬁcant fraction of the total energy density.

For modes�� 500, instead of an enhancement, we see instead a very small suppression of power. and a small suppression of even-odd peak height difference due to the dm-drf interaction. The extra potential decay arising from the interaction changes the photon overdensity in two ways: amplitude suppression and baryon loading alleviation. At these small scales, we numericallyfind that the extra potential decay leads to a nearly uniform suppression of the photon perturbation amplitude in a low-baryon cosmology. We also find that the baryon loading effect is weaker in Model 3 than in Model 2. The two changes (amplitude suppression and baryon loading alleviaion) add up constructively for the odd

3 extrema (krs, = 3π,5π,7π) and destructively for the even extrema (kr s, = 2π,4π,6π). ∗ ∗ To summarize, the dm-drf interaction has a much smaller impact on the CMB power

3In fact, Planck CMB data is sensitive to the small suppression of the odd-even peak height diﬀerence. Our MCMC results show that the dm-drf model prefers a higherω b than in theΛCDM model.

70 ��

��

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Figure 5.1. Upper Panel: the evolution of dark matter over-densityδ dm suppression for diﬀerentk modes. Middle Panel: the “self-similar” behavior of the over-density 1 suppression, where we displace the suppression of modesk=0.3 Mpc − andk= 1 1.0 Mpc− by 0.05 and 0.096 respectively. Lower Panel: the dm-drf interaction induced matter power spectrum suppression today, where the dashed line is an analyticﬁt in the form of Equation (5.5).

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Figure 5.2. Comparison of TT and EE spectra of the three models listed in Table 5.1, where in the upper panel we plot the TT spectra with damping eﬀect largely removed by multiplying a factorK = exp 2 (�/1267) 1.18 , in the lower panel we � × plot the EE spectra, and in the two inset plots, we normalize the spectra amplitudes � � to allow one to see the impact of the very small shift in peak locations induced by free-streaming species as done in [1]. In these insets the red curve (the model with additional freestreaming neutrinos) is slightly shifted to the left relative to the dashed line and blue line which overlap each other.

72 spectra than on the matter power spectrum. The impact on the matter power spectrum arises from interactions in the radiation-dominated era when the dark radiation has more inertia than the dark matter. The impact on the CMB power spectrum is through the impact on the dark matter evolution. On small scales, where the impact on dark matter is sizeable, the dark matter contribution to the gravitational potential at the time of horizon crossing is very small and thus the net impact on the photon distribution is small.

5.3 Parameter constraints from LSS data

In this section, weﬁrst brieﬂy review previous analyses of the implications of cosmological data for the extension ofΛCDM to include the dm-drf interaction model, identify Planck SZ as the major driver for the previously claimed inference of the non-zero dm-drf interaction, and redo the analysis with a treatment of uncertainties in the SZ-mass observable relationship.

5.3.1 Previous Analyses In previous analyses [e.g. 169, 174, 176], Planck CMB and LSS measurements, including Planck CMB Lensing [7], CFHTLens [4] and Planck SZ [5, 6],

σ (Ω /0.27)0.25 = 0.820 0.029 [CMB Lensing], (5.6a) 8 m ± σ (Ω /0.27)0.46 = 0.774 0.040 [CFHTLens], (5.6b) 8 m ± σ (Ω /0.27)0.30 = 0.782 0.010 [Planck SZ]. (5.6c) 8 m ± were used to constrain the dm-drf model, and the dm-drf interaction was detected at 3σ confidence level. Tofigure out which dataset is essential to the claimed inference of non-zero dm-drf interaction, we show theσ 8 tension of theΛCDM cosmology in Figure 5.3, which clearly shows that the SZ-CMB tension is the strongest. Thisfinding also suggests that Planck SZ is driving the inference of non-zero dm-drf interaction. The cosmological implications of the Planck SZ cluster counts depends on assumptions about the relationship between SZflux and cluster mass [e.g., 6]. This is usually expressed as uncertainty in the hydrostatic mass bias parameterb whereM = (1 b)M ,M is a X − 500 X

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Figure 5.3. Theσ 8 tension in theΛCDM cosmology, where the redfilled contours (1σ and 2σ) are derived from Planck 2015 temperature and polarization, and the unfilled contours corresponding to the three LSS measurements are given at 2σ level, where the SZ contour is the constraintfixing the mass bias parameter as the baseline value 1 b=0.8. − mass proxy derived from observed SZflux with an assumption of hydrostatic equilibrium,

andM 500 is the true cluster halo mass (see [5, 6] for more details). The bias parameter b itself is not well known today, e.g., constraints derived from gravitational shear mass measurements Weighing the Giants (WtG) [184], from Canadian Cluster Comparison Project (CCCP) [185], and from CMB Lensing (Lens) [186, 191] listed as follows show signiﬁcant uncertainties:

1 b=0.688 0.072 [WtG], (5.7a) − ± 1 b=0.780 0.092 [CCCP], (5.7b) − ± 1 b=0.74 0.07 [Lens]. (5.7c) − ±

As shown in [6], theσ 8 constraint derived from SZ cluster counts is sensitive to the prior

used: the WtG prior almost eliminates theσ 8 tension between SZ and Planck CMB, while the CCCP prior remains in noticeable tension. In addition, a reference modelﬁxing the bias parameter as the baseline value, 1 b=0.8, was also investigated in the PlanckSZ − analysis [5], which yields theσ 8 constraint of Equation (5.6c), in tension with that derived from Planck CMB at�3σ conﬁdence level (see also Figure 5.3).

In previous analyses, theσ 8 constraint of Equation (5.6c) was usually used as an approximation to the full Planck SZ data. It is natural to ask whether it is valid toﬁx

74 the bias parameter as the baseline value in constraining the dm-drf interaction model, considering the large uncertainty in the bias parameter, the sensitive dependence of the

σ8 constraint on the bias parameter and the mild tension between the WtG constraint and the baseline value (see [192] for a summary of recent bias parameter inferences). We discuss this next.

5.3.2 Anaysis with SZ data: the impact of the mass bias parameter To highlight the impact of the uncertainty in the SZ cluster counts on the model parameter constraints, we use both CMB and SZ data with the mass bias parameterfixed or varying, and compare the resulting constraints. For CMB data, we use Planck 2015 CMB temperature and polarization data TTTEEE + lowTEB [2] (PlanckTP). For SZ data with varying mass bias parameter, we use Planck 2015 SZ cluster counts data (PlanckSZ) with the CCCP prior, while for SZ data withfixed mass bias parameter, we use the single data point of Equation (5.6c), as done in previous analyses. We use CosmoMC to run MCMC chains, withflat priors onN 0.07 andΓ 0, drf ≥ 0 ≥ and CosmoMC default priors forΛCDM parameters and other nuisance parameters, where

Ndrf = 0.07 andΓ 0 = 0 are the physical lower limits of these parameters in the dm-drf interaction model we are considering [81, 169]. The data disfavor large values for these

7 1 two parameters,N 1 andΓ 10− Mpc− , so there is no need to truncate the prior drf � 0 � ranges for these parameters. Our adopted priors are consistent with those used previously [169]. We use the Raferty and Lewis statisticR 1 0.02 as the convergence criterion, − ≤ and we summarize the MCMC results in Figure 5.4 and Table 5.2. We perform two different MCMC runs jointly using PlanckTP and PlanckSZ. In the first run, wefix the bias parameter as 1 b=0.8 and obtain an inference of non-zero − Γ0 at 3σ significance (red/dot-dashed lines in Figure 5.4), which is similar to previous works. In the second run, we let the bias parameter vary and impose the CCCP prior

(blue/dashed line in Figure 5.4). As a result, we onlyﬁnd an upper limit ofΓ 0 (blue/solid lines in Figure 5.4 ). In the second run, the posterior of the mass bias parameter turns out to converge at 1 b=0.647 0.044, which is about 3σ lower than the baseline value − ±

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Figure 5.4. The results of the two MCMC runs (red vs. blue) for the canonical dm- drf model using joint dataset PlanckTP+PlanckSZ. Upper Left Panel: the comparison of the CCCP prior (blue/dashed) and the resulting posterior (blue/solid) with the baseline value 1 b=0.8 (red/dot-dashed). Lower Left Panel: the posterior contour − of 1 b vs.Γ . Lower Right Panel: the marginalized posteriors ofΓ with the bias − 0 0 parameterﬁxed (red/dot-dashed) or varying (blue/solid).

PlanckTP+PlanckSZ Dataset PlanckTP PlanckTP+Lensing+DES 1 b=0.78 0.092 1 b=0.8 − ± − 1 b 0.647 0.044 − ± 7 1 Γ (10− Mpc− ) <1.28 <1.36 1.61 0.54 <1.43 0 ± Ndrf <0.57 <0.62 <0.64 <0.67 σ 0.817 0.022 0.807 0.019 0.758 0.015 0.800 0.016 8 ± ± ± ± Table 5.2. Constraints on the dm-drf model parameters using datasets PlanckTP, PlanckSZ, Lensing and DES, where the uncertainties are 1σ values, and the upper limits are given at 2σ conﬁdence level.

76 0.8 and 2 times tighter than the CCCP prior imposed, due to the overriding Planck CMB preference for lower 1 b (higherσ ); and the preference of non-zeroΓ is gone due to − 8 0 the tight positive correlation between 1 b andΓ . But the degeneracy of 1 b andΓ − 0 − 0 7 1 breaks down at 10 Γ /Mpc− 1.5 since Planck CMB power spectra disfavor largeΓ 0 ∼ 0 (see Table 5.2). We also checked the approximation of using the single data point of Equation (5.6c) rather than the full SZ likelihood. Equation (5.6c) follows from the full SZ likelihood given theΛCDM model and that 1 b=0.8 with no uncertainty. Wefind the approx- − imation works well. Wefind very similar constraints on the dm-drf interaction model parameters whether we use the full SZ likelihood (and 1 b=0.8) or approixmate it with − Equation (5.6c). Both of them result in an inference of non-zero dm-drf interaction at 3σ significance, with tiny differences in the mean values and the uncertainties, which ∼ do not affect our conclusion. We therefore do not distinguish the two cases in this paper. From Table 5.2, we also see that PlanckSZ with the CCCP prior is not highly constraining; adding it to PlanckTP only slightly increases the upper limits ofΓ 0 andN drf .

It is clear that the other two priors would lead to even less of a non-zeroΓ 0 preference, which can be veriﬁed by the fact that the tension of the 1 b posterior with the CCCP − prior is greater than its tension with the WtG/Lens prior.

5.3.3 Analysis with only CMB Lensing and DES data Since the SZ data (with the bias parameter allowed tofloat) is not highly constraining, we drop it from further consideration as we we examine the dm-drf model with PlanckTP and the following two LSS datasets: (1) Lensing: Planck 2015 lensing data [7]. (2) DES:σ (Ω /0.3)0.5 = 0.789 0.026, which is derived from the Dark Energy Survey 8 m ± (DES)first-year cosmic shear data [9] and is a slightly tighter constraint than that derived from CFHTLens (Eq.5.6b) or from KiDS-450 [8]. Strictly speaking, we should use the DES likelihood with all the relevant nuisance parameters (e.g. the intrinsic alignment of galaxies) varying, instead of using this single data point. But the likelihood code is not publicly available, and as we will see later, wefind no detection of the dm-drf interaction.

77 Therefore we expect no qualitative diﬀerence using the single data point versus using a full likelihood with proper treatment of uncertainties. The MCMC results are summarized in Table 5.2. Again, weﬁnd no detection of the dm-drf interaction using the joint dataset PlanckTP+Lensing+DES, though it is more constraining than another joint dataset PlanckTP+PlanckSZ with the CCCP prior.

5.4 Lyman-α forest data

Lyα forest observations have been used as a cosmological probe for the past two decades [e.g. 193–195]. Lyα absorption is sensitive to the density of neutral gas in a relatively low-density, smooth environment, which is tightly correlated with the underlying dark matter density on large scales. Many of these observational results are based on a direct measurement of the Lyα forest power spectrumP F (k), a statistical property of the transmittedﬂuxﬂuctuations

τ(λ) τ(λ) δ (λ) =e − / e− 1, (5.8) F � � − whereλ is the observed wavelength of Lyα emission, andτ is the optical depth to Lyα absorption. The tight correlation between the neutral gas density and the underlying dark matter density allows a determination of the matter power spectrum from the Lyα forest power spectrumP F (k). For this purpose, hydrodynamical simulations are required to computeP F (k) for a given initial linear matter power spectrumP L(k, zi) at some high redshiftz i, due to the complexities in the non-linear evolution of dark matter and hydrodynamical processes. Compared with CMB data and LSS measurements including DES and Planck SZ, the latest measurements [187–189] of the Lyα forestﬂux power spectrum from the Baryon Oscillation Spectroscopic Survey extends sensitivity to the matter power spectrum to

2 3 2 smaller scales. The constraints on the amplitudeΔ L =k PL(k, z)/2π and the slope n =d lnP (k, z)/d lnk atk=0.009(s/km) H(z)/(1 +z) andz = 3 are explicitly eﬀ L × given in [188], whereH(z) is the Hubble expansion rate. As pointed out in [196], the matter power spectrum derived from the Lyα forest data 1 yields a comparable amplitude but a much steeper slope at scalek Mpc − , compared ∼

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Figure 5.5. The tension between Lyα and CMB inΛCDM cosmology with neu- 2 trino massﬁxed as 0.06 eV (ΛCDM) or varying (νΛCDM) whereΔ L andn eﬀ are the 1 amplitude and the slope of the linear matter power spectra atk h Mpc − and at � z = 3.

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2 Figure 5.6. Comparison of the Lyα constraints on the amplitudeΔ L and the slopen eff of the matter power spectrum with those derived fromΛCDM (bluefilled contours) and the dm-drf interaction model (red unfilled contours) using PlanckTP+Lensing+DES, 7 1 where the color points denote differentΓ 0 values in unit of 10− Mpc− .

with those derived from Planck CMB data, assumingΛCDM. We plot these constraints in Figure 5.5, which clearly shows that inferences of the matter power spectrum from the Lyα forest data are highly inconsistent with the Planck CMB data, assumingΛCDM. The discrepancy has increased from theﬁrst release of Planck data to the second, since

the second yields aﬂatter slopen eﬀ with a reduced uncertainty (likely due to a largern s

and a tighter constraint onω m [2]) Allowing neutrino mass to vary does not do much to

reconcile the discrepancy in the matter power slopen eﬀ . 1 The steeper slopen derived from Lyα data at scalek Mpc − implies a scale- eﬀ ∼

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Figure 5.7. Left Panel: the matter power spectrum suppression from diﬀerent dm-drf β 1 5 7 11 interaction ratesΓ=Γ 0(T/T0) (withΓ 0/Mpc− = 10− , 10− , 10− forβ=1,2, 3, respectively), where the red band denotes the modesσ 8 is sensitive to and the blue band denotes the modes Lyα measurement is sensitive to. Right Panel: the comparison between the Hubble expansion rate and the dm-drf interaction rates.

dependent matter power suppression which aligns well with the dm-drf interaction picture. To examine whether the Lyα data is in agreement with other datasets in the dm-drf interaction model, we plot theΔ 2 n contours derived from PlanckTP+Lensing+DES L − eﬀ in Figure 5.6. We see that the joint dataset favors the dm-drf interaction model (with

7 1 interaction rate 10 Γ0/Mpc− in the range of [0.9,1.6]). Our results serve to highlight the potential importance of these inferences of the matter power spectrum from the Lyα data. If they are substantially free from bias and have adequately captured all signiﬁcant sources of uncertainty, then the discrepancy with the Planck-conditioned predictions ofΛCDM are extremely interesting. Possible solutions to this discrepancy include the dark matter model we are studying here, as well as a negative

running dns/d lnk [189, 196] or possibly a diﬀerent dm-drf interaction.

5.5 Generalized dm-drf models

In this section, we extend our exploration from the canonical dm-drf model with inter-

2 β action rateΓ=Γ 0(T/T0) to generalized models with interaction rateΓ=Γ 0(T/T0) (β=1,2, 3). Wefirst briefly discuss the imprints of the generalized dm-drf interaction rates on the CMB power spectra and the matter power spectrum, then constrain these models using CMB data and LSS measurements. The imprints of the three different models on the matter power spectrum suppres-

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2 Figure 5.8. Comparison of the Lyα constraint on the amplitudeΔ L and the slopen eff of the matter power spectrum with those derived fromΛCDM and dm-drf interaction models using PlanckTP+Lensing+DES, where the bluefilled contours are the results of theΛCDM cosmology, the red unfilled contours are the results of the dm-drf models, and the color bars denoteσ 8 for dm-drf withβ=1, 3, respectively. Two right panels show the sameβ = 3 model but with different ranges on the x-axis.

sion are pretty distinct at small scales, as shown in Figure 5.7. Forβ = 3, dark matter and dark radiation are tightly coupled deep in the radiation-dominated era. Dark matter perturbations oscillate instead of growing, therefore all modes entering the horizon

whenΓ�H are strongly suppressed. In theβ = 1 case, the interaction is negligible at early time. For suﬃciently small modes entering horizon early, the power suppression is dominated by late time when the dm-drf interaction becomes important. For large

modes entering horizon whenρ drf /ρdm becomes vanishingly small, the dark matter overdensity growth is unaﬀected by the interaction. For the intermittent modes, the power

suppression is determined by several factors, including the coupling strength todayΓ 0, the matter-radiation equality where theΓ/H dependence on the scale factor changes, and

the dark matter and dark radiation energy density ratioρ drf /ρdm. Therefore we see a power suppression plateau on the small-scale end, no suppression on the large-scale end, and a smooth transition in between. Different from theβ = 2 case, bothβ = 1 andβ=3 interactions introduces new length scales to the matter power suppression. Following the argument given in Section 5.2.2, it is not hard tofigure out the impacts of the general interaction rates on the temperature and polarization power spectra. For example, we expect that theβ = 3 interaction tends to suppress the amplitudes of large k modes entering the horizon at radiation domination and whenΓ/H is noticeable, while leaving no imprint on the amplitudes of smallk modes entering the horizon whenΓ/H is negligible; on the contrary, theβ = 1 interaction should only affect largek modes. We

81 have modified CAMB to allow for all the three interaction models, and numerical results confirm our qualitative expectations above. Wefind that the imprints of the three different models are too subtle to be distinguished via Planck CMB data, so we do not plot them here. We constrain the two models using the joint dataset PlanckTP+Lensing+DES,finding no detection of interaction for either of the two new cases. Similar to previous section, we also examine whether these two models reconcile the Lyα-CMB tension. As shown in Figure 5.8, theβ = 1 interaction does not change the amplitude and the slope much, and theβ = 3 interaction leads to an overwhelming suppression. We see that neither of the two help to reconcile the Lyα-CMB tension.

5.6 Summary

In this paper, we reinvestigated the non-Abelian dark sector model proposed by [81]. We examined the impact of the dm-drf interaction on the CMB power spectra and the matter power spectrum in detail. We found that the dm-drf interaction affects the amplitudes of CMB power spectra by modifying the gravitational potential decay, but only slightly. We verified the presence of a logarithmic suppression in the matter power spectrum that originates from the self-similar suppression of the matter overdensity. We also constrained the dm-drf model using CMB and LSS measurements in a more systematic way. We found that Planck SZ plays the key role in the previously claimed inference of non- zero dm-drf interaction. However the SZ cluster counts constraint is limited by uncertainty in the cluster mass scale determination, which is usually parametrized as the mass bias parameterb. We confirmed the inference of non-zero dm-drf interaction at 3σ significance using the Planck data and and the SZ datafixing the bias parameter to be constant, 1 b=0.8, as done in previous analyses. But, when we included uncertainties in 1 b, − − the preference of non-zero dm-drf interaction essentially disappeared. We also show that the latest inferences of the matter power spectrum from Lyα forest data are highly inconsistent with the Planck CMB data, assumingΛCDM, and that the joint data sets favor a non-zero dark sector interaction. Thus if these matter power

82 spectrum inferences are free from significant systematic error, and if the reported uncertainties accurately include all sources of uncertainty, these data are more sensitive to the impact of dm-drf interactions and provide us with a significant detection. Even so, there are other possible ways to reconcile the Planck and Lyα forest data such as a non-zero running of the scalar spectral index dns/d lnk [189, 196]. We also explored two different phenomenological dm-drf interaction models characterized by interaction rates scaling with temperature in different power laws, and found neither of these interactions is favored by current CMB and LSS data. We are unsure what to make of these inferences of the matter power spectrum from the Lyα forest data. We hope our work serves to motivate further study of these data. Were a different group to reach similar conclusions independently, even if from the same data, that would bolster our confidence. Another avenue for progress is measurements that can decrease uncertainty in 1 b as the cluster mass function has the statistical − power to make a detection absent that uncertainty, if the interaction strength is at the higher end of the range consistent with Lyα data.

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