Spectral Distortions of the Cosmic Microwave Background Dissertation
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Spectral Distortions of the Cosmic Microwave Background Dissertation der Fakult¨at fur¨ Physik der Ludwig-Maximilians-Universit¨at Munc¨ hen angefertigt von Jens Chluba aus Tralee (Irland) Munc¨ hen, den 31. M¨arz 2005 1. Gutachter: Prof. Dr. Rashid Sunyaev, MPA Garching 2. Gutachter: Prof. Dr. Viatcheslav Mukhanov, LMU Munc¨ hen Tag der mundlic¨ hen Prufung:¨ 19. Juli 2005 Die Sterne Ich sehe oft um Mitternacht, wenn ich mein Werk getan und niemand mehr im Hause wacht, die Stern am Himmel an. Sie gehn da, hin und her, zerstreut als L¨ammer auf der Flur; in Rudeln auch, und aufgereiht wie Perlen an der Schnur; und funkeln alle weit und breit, und funkeln rein und sch¨on; ich seh die große Herrlichkeit, und kann nicht satt mich sehn... Dann saget unterm Himmelszelt mein Herz mir in der Brust; Es gibt was Bessers in der Welt als all ihr Schmerz und Lust. Matthias Claudius Contents Abstract ix 1 Introduction 1 1.1 General introduction on CMB . 1 1.2 Spectral distortions of the CMB . 5 1.2.1 The SZ effect . 5 1.2.2 Spectral distortion due to energy release in the early Universe . 8 1.3 In this Thesis . 10 2 SZ clusters of galaxies: influence of the motion of the Solar System 13 2.1 General transformation laws . 14 2.2 Transformation of the cluster signal . 16 2.3 Multi-frequency observations of clusters . 18 2.3.1 Dipolar asymmetry in the number of observed clusters . 19 2.3.2 Estimates for the dipolar asymmetry in the cluster number counts . 19 2.3.3 Source count contribution from non-SZ populations . 20 2.4 Summary and discussion . 22 3 Spectral distortion of the CMB and the superposition of blackbodies 23 3.1 Basic ingredients . 25 3.1.1 Compton y-distortion . 25 3.1.2 Relation between temperature and intensity . 26 3.2 Small spectral distortions due to the superposition of blackbodies . 27 3.2.1 Sum of blackbodies . 28 3.2.2 Superposition of Planck spectra . 33 3.3 Superposition of two Planck spectra . 35 3.3.1 Sum of two Planck spectra . 35 3.3.2 Difference of two Planck spectra . 37 3.4 Spectral distortions due to the CMB dipole . 37 3.4.1 Whole sky beam spectral distortion . 38 3.4.2 Beam spectral distortion due to the CMB dipole . 39 3.4.3 Distortion with respect to any Tref . 40 3.5 Spectral distortions due to higher multipoles . 43 3.6 Spectral distortions induced in differential measurements . 47 3.7 Cross Calibration of frequency channels . 50 3.7.1 Calibration using clusters of galaxies . 50 3.7.2 Cross calibration using the superposition of blackbodies . 51 3.8 Other sources of spectral distortions . 54 3.9 Summary . 54 ii CONTENTS 4 The double Compton process in mildly relativistic thermal plasmas 57 4.1 The current understanding of the double Compton process . 57 4.2 The kinetic equation for DC scattering . 59 4.2.1 General definitions . 59 4.2.2 Standard approach . 61 4.2.3 Kernel approach . 61 4.3 The DC emission kernel (ν ν ν ) for thermal electrons . 64 P 0 ! 2 j 1 4.3.1 The DC kernel for cold electrons . 64 4.3.2 The DC kernel for thermal electrons . 66 4.3.3 Mean photon energy and dispersion of the DC kernel . 66 4.4 DC emission for monochromatic photons and thermal electrons . 67 4.4.1 Cold electrons and soft initial photons . 68 4.4.2 Cold electrons and arbitrary initial photons . 70 4.4.3 Thermal electrons and low energy initial photons . 74 4.4.4 Thermal electrons and arbitrary initial photons . 77 4.4.5 The DC infrared divergence and the role of the low frequency cutoff . 81 4.4.6 Electron heating and cooling due to DC emission . 83 4.5 Analytical treatment of the full kinetic equation for DC scattering . 83 4.5.1 Analytic approximation for the effective DC Gaunt factor . 84 4.5.2 Derivation of the effective DC Gaunt factor in the soft photon limit . 86 4.5.3 Beyond the limit θ 1 . 88 e 4.5.4 Results for different incoming photon spectra . 88 4.5.5 Beyond the soft photon limit . 90 4.5.6 Discussion of the results for Planck, Bose-Einstein and Wien spectra . 90 4.6 Summary and outlook . 94 5 Thermalization of CMB spectral distortions 97 5.1 General formulation of the thermalization problem . 97 5.1.1 The Boltzmann equation in the expanding Universe . 98 5.1.2 Evolution of the number and energy density . 98 5.2 Evolution of the photons in the expanding Universe . 99 5.2.1 Compton scattering . 99 5.2.2 Double Compton scattering and Bremsstrahlung . 100 5.2.3 Expansion term . 101 5.3 Evolution of the electrons and baryons in the Universe . 101 5.3.1 Evolution of the electron temperature . 101 5.3.2 Interactions with the photons . 103 5.4 Towards a numerical solution of the problem . 104 5.4.1 Representation of the photon spectrum . 104 5.4.2 Compton scattering relativistic corrections . 105 5.4.3 Double Compton and Bremsstrahlung . 105 5.4.4 Electron temperature . 105 5.5 Thermalization of small spectral distortions . 105 5.5.1 Time evolution of small chemical potential distortions . 106 5.5.2 Solving the time evolution in the limit of small chemical potential . 109 5.6 Energy injection by different physical mechanisms . 111 5.6.1 Single energy injection at zh . 112 5.6.2 Energy injection from annihilating relict particles . 113 5.6.3 Energy injection by unstable relict particles . 113 5.7 Summary and discussion . 115 Conclusions 117 CONTENTS iii A Relativistic Maxwell-Boltzmann distribution 119 A.1 Number and phase space density . 119 A.2 Low temperature expansion of the relativistic Maxwell-Boltzmann distribution 119 A.3 Energy density and pressure . 120 A.4 Heat capacity . 120 A.5 Solving integrals over the relativistic Maxwell-Boltzmann distribution . 121 B Relations for the photon phase space distribution 123 B.1 Pressure, energy and number density . 123 B.2 Effective temperature . 124 i B.3 Useful relations between n(x; µ(x)) and its derivatives @xn . 124 C Collection of analytic approximations for the Compton scattering kernel 125 C.1 Compton kernel for cold electrons . 125 C.1.1 Normalization, mean energy and dispersion of the Compton kernel for cold electrons . 125 C.2 Compton kernel for thermal electrons . 126 C.2.1 Normalization, mean energy and dispersion of the Compton kernel for thermal electrons . 126 D Double Compton scattering 127 D.1 Squared matrix element for double Compton scattering . 127 D.2 Numerical solution of the Boltzmann integrals . 127 D.3 Results for the integrals I 's up to (θ4) . 130 k O e D.4 Integration over the frequency of the incoming photons . 130 E Summary of useful relations in the cosmological context 133 E.1 Electron, hydrogen and helium number densities . 133 E.2 Photon energy and number density . 134 E.3 Time scales . 134 Bibliography I Acronyms VII Acknowlegdements IX iv CONTENTS List of Figures 1.1 Uniform CMB spectrum as measured by Cobe/Firas . 2 1.2 CMB sky as observed by Wmap . 3 1.3 Wmap T T power spectrum and T E cross-power spectrum . 4 1.4 Change of the CMB brightness due to the thermal and kinetic SZ effect in the non-relativistic limit . 6 1.5 Change of the CMB brightness due to the thermal SZ effect for hot clusters . 7 1.6 Limits on the lifetimes of decaying relict particles . 9 2.1 Frequency dependence of the SZ brightness due to the non-relativistic th- and k-SZ and the corresponding corrections induced by the motion of the Solar System 15 2.2 Motion-induced change of the CMB spectral brightness ∆I and flux ∆F for a cluster of galaxies at rest with respect to the CMB . 18 2.3 log N-log F for clusters of galaxies . 20 2.4 Motion-induced dipolar asymmetry in number counts . 21 2.5 Required sensitivities for a 3σ and 5σ-detection of the motion-induced dipolar asymmetry in number counts . 21 3.1 x^y as a function of ∆ for different values of . 27 3.2 Frequency dependence of g =g ∆T =T δi . 28 i 1 ≡ i RJ 3.3 x^c as a function of ∆ for different values of . 30 3.4 Dependence of the inferred y-parameter on the chosen reference temperature Tref for two blackbodies with temperatures T1 = 2:72162 K and T2 = 2:72838 K 36 ◦ 3.5 Angular distribution of yr for θr = 20 . 40 3.6 Dependence of the y-parameter on the beam radius θr for a circular beam . 41 3.7 Probability density p(y) for different aperture radii . 44 3.8 Cumulative probability p(y y ) for different aperture radii . 45 ≥ 0 3.9 Number of regions on the sky with y y for different aperture radii θ . 46 ≥ 0 r 3.10 Illustration of a differential observing strategy . 48 3.11 Dependence of the y-parameter on the.