Physics of the Cosmic Microwave BackgroundBasic deﬁnitions [note: c = 1]CMB = electromagnetic radiation permeating the Universe.It can be described by its speciﬁc intensityfrequency

Yacine Ali-Haïmoud New York University

ICTP school on cosmology, January 2021

Physics of the Cosmic Microwave Background

Lecture I: setting the stage

Basic definitions [note: c = 1]

CMB = electromagnetic radiation permeating the Universe.

It can be described by its specific intensity

frequency direction of propagation

AAACJnicbVBNSwMxEM36bf2qevQSLIKCtLui6EUQvehNwarQXUs2nbbBJLsks2Jd+mu8+Fe8eFBEvPlTTGsPan3DwOO9GZJ5cSqFRd//8EZGx8YnJqemCzOzc/MLxcWlC5tkhkOVJzIxVzGzIIWGKgqUcJUaYCqWcBnfHPX8y1swViT6HDspRIq1tGgKztBJ9eL+yXqos00athnmurtBw6GqhQh3aFQOplWxleP7ijUVrrrXW1G9WPLLfh90mAQDUiIDnNaLL2Ej4ZkCjVwya2uBn2KUM4OCS+gWwsxCyvgNa0HNUc0U2Cjvn9mla05p0GZiXGukffXnRs6UtR0Vu0nFsG3/ej3xP6+WYXMvyoVOMwTNvx9qZpJiQnuZ0YYwwFF2HGHcCPdXytvMMI4u2YILIfh78jC52CoHO2X/bLt0cDiIY4qskFWyTgKySw7IMTklVcLJA3kiL+TVe/SevTfv/Xt0xBvsLJNf8D6/AJKyotU=

I(⌫, n̂) [erg/s/Hz/sr/cm2]

…or equivalently by its phase-space density = (2/h3)f [number of photons per d3x d3p]

…or equivalently by its photon occupation number f [number of photons per quantum state]

AAACHXicbVDLSgMxFM3UV62vqks3wSJUkDJTK7oRim50V8E+oFNLJs10QpPMkGSEMvRH3Pgrblwo4sKN+Dem7Sxs64HA4ZxzubnHixhV2rZ/rMzS8srqWnY9t7G5tb2T391rqDCWmNRxyELZ8pAijApS11Qz0ookQdxjpOkNrsd+85FIRUNxr4cR6XDUF9SnGGkjdfOV26Ir4hPoBkgnYnQML2EZBt3ElRzWRhAa8+EU+rOhbr5gl+wJ4CJxUlIAKWrd/JfbC3HMidCYIaXajh3pToKkppiRUc6NFYkQHqA+aRsqECeqk0yuG8Ejo/SgH0rzhIYT9e9EgrhSQ+6ZJEc6UPPeWPzPa8fav+gkVESxJgJPF/kxgzqE46pgj0qCNRsagrCk5q8QB0girE2hOVOCM3/yImmUS85Zyb6rFKpXaR1ZcAAOQRE44BxUwQ2ogTrA4Am8gDfwbj1br9aH9TmNZqx0Zh/MwPr+BS8Fn4U=

I(⌫, n̂) = 2hP⌫3f(⌫, n̂)

T (⌫, n) = T0AAAB+HicbZDLSsNAFIZP6q3WS6Mu3QwWoYKURATdCEU3Liv0Bm0Ik+mkHTqZhJmJUEOfxI0LRdz6KO58G6dtFtr6w8DHf87hnPmDhDOlHefbKqytb2xuFbdLO7t7+2X74LCt4lQS2iIxj2U3wIpyJmhLM81pN5EURwGnnWB8N6t3HqlULBZNPUmoF+GhYCEjWBvLt8vNal+k50icoRvU9B3frjg1Zy60Cm4OFcjV8O2v/iAmaUSFJhwr1XOdRHsZlpoRTqelfqpogskYD2nPoMARVV42P3yKTo0zQGEszRMazd3fExmOlJpEgemMsB6p5drM/K/WS3V47WVMJKmmgiwWhSlHOkazFNCASUo0nxjARDJzKyIjLDHRJquSCcFd/vIqtC9qruGHy0r9No+jCMdwAlVw4QrqcA8NaAGBFJ7hFd6sJ+vFerc+Fq0FK585gj+yPn8A2GmRPw==AAAB+HicbZDLSsNAFIZP6q3WS6Mu3QwWoYKURATdCEU3Liv0Bm0Ik+mkHTqZhJmJUEOfxI0LRdz6KO58G6dtFtr6w8DHf87hnPmDhDOlHefbKqytb2xuFbdLO7t7+2X74LCt4lQS2iIxj2U3wIpyJmhLM81pN5EURwGnnWB8N6t3HqlULBZNPUmoF+GhYCEjWBvLt8vNal+k50icoRvU9B3frjg1Zy60Cm4OFcjV8O2v/iAmaUSFJhwr1XOdRHsZlpoRTqelfqpogskYD2nPoMARVV42P3yKTo0zQGEszRMazd3fExmOlJpEgemMsB6p5drM/K/WS3V47WVMJKmmgiwWhSlHOkazFNCASUo0nxjARDJzKyIjLDHRJquSCcFd/vIqtC9qruGHy0r9No+jCMdwAlVw4QrqcA8NaAGBFJ7hFd6sJ+vFerc+Fq0FK585gj+yPn8A2GmRPw==AAAB+HicbZDLSsNAFIZP6q3WS6Mu3QwWoYKURATdCEU3Liv0Bm0Ik+mkHTqZhJmJUEOfxI0LRdz6KO58G6dtFtr6w8DHf87hnPmDhDOlHefbKqytb2xuFbdLO7t7+2X74LCt4lQS2iIxj2U3wIpyJmhLM81pN5EURwGnnWB8N6t3HqlULBZNPUmoF+GhYCEjWBvLt8vNal+k50icoRvU9B3frjg1Zy60Cm4OFcjV8O2v/iAmaUSFJhwr1XOdRHsZlpoRTqelfqpogskYD2nPoMARVV42P3yKTo0zQGEszRMazd3fExmOlJpEgemMsB6p5drM/K/WS3V47WVMJKmmgiwWhSlHOkazFNCASUo0nxjARDJzKyIjLDHRJquSCcFd/vIqtC9qruGHy0r9No+jCMdwAlVw4QrqcA8NaAGBFJ7hFd6sJ+vFerc+Fq0FK585gj+yPn8A2GmRPw==AAAB+HicbZDLSsNAFIZP6q3WS6Mu3QwWoYKURATdCEU3Liv0Bm0Ik+mkHTqZhJmJUEOfxI0LRdz6KO58G6dtFtr6w8DHf87hnPmDhDOlHefbKqytb2xuFbdLO7t7+2X74LCt4lQS2iIxj2U3wIpyJmhLM81pN5EURwGnnWB8N6t3HqlULBZNPUmoF+GhYCEjWBvLt8vNal+k50icoRvU9B3frjg1Zy60Cm4OFcjV8O2v/iAmaUSFJhwr1XOdRHsZlpoRTqelfqpogskYD2nPoMARVV42P3yKTo0zQGEszRMazd3fExmOlJpEgemMsB6p5drM/K/WS3V47WVMJKmmgiwWhSlHOkazFNCASUo0nxjARDJzKyIjLDHRJquSCcFd/vIqtC9qruGHy0r9No+jCMdwAlVw4QrqcA8NaAGBFJ7hFd6sJ+vFerc+Fq0FK585gj+yPn8A2GmRPw==

+�T (⌫, n)AAAB/HicdZDLSsNAFIZP6q3WW7VLN4NFqCglFUWXRTcuK/QGTSiTyaQdOpmEmYlQQn0VNy4UceuDuPNtnLQRVPSHgZ/vnMM583sxZ0rb9odVWFpeWV0rrpc2Nre2d8q7e10VJZLQDol4JPseVpQzQTuaaU77saQ49DjteZPrrN67o1KxSLT1NKZuiEeCBYxgbdCwXDlGjk+5xqhdc0RygsQRGpardv3czoTsuv1lctLISRVytYbld8ePSBJSoQnHSg0adqzdFEvNCKezkpMoGmMywSM6MFbgkCo3nR8/Q4eG+CiIpHlCozn9PpHiUKlp6JnOEOux+l3L4F+1QaKDSzdlIk40FWSxKEg40hHKkkA+k5RoPjUGE8nMrYiMscREm7xKJoSvn6L/Tfe03jD+9qzavMrjKMI+HEANGnABTbiBFnSAwBQe4AmerXvr0XqxXhetBSufqcAPWW+f8gKTAg==AAAB/HicdZDLSsNAFIZP6q3WW7VLN4NFqCglFUWXRTcuK/QGTSiTyaQdOpmEmYlQQn0VNy4UceuDuPNtnLQRVPSHgZ/vnMM583sxZ0rb9odVWFpeWV0rrpc2Nre2d8q7e10VJZLQDol4JPseVpQzQTuaaU77saQ49DjteZPrrN67o1KxSLT1NKZuiEeCBYxgbdCwXDlGjk+5xqhdc0RygsQRGpardv3czoTsuv1lctLISRVytYbld8ePSBJSoQnHSg0adqzdFEvNCKezkpMoGmMywSM6MFbgkCo3nR8/Q4eG+CiIpHlCozn9PpHiUKlp6JnOEOux+l3L4F+1QaKDSzdlIk40FWSxKEg40hHKkkA+k5RoPjUGE8nMrYiMscREm7xKJoSvn6L/Tfe03jD+9qzavMrjKMI+HEANGnABTbiBFnSAwBQe4AmerXvr0XqxXhetBSufqcAPWW+f8gKTAg==AAAB/HicdZDLSsNAFIZP6q3WW7VLN4NFqCglFUWXRTcuK/QGTSiTyaQdOpmEmYlQQn0VNy4UceuDuPNtnLQRVPSHgZ/vnMM583sxZ0rb9odVWFpeWV0rrpc2Nre2d8q7e10VJZLQDol4JPseVpQzQTuaaU77saQ49DjteZPrrN67o1KxSLT1NKZuiEeCBYxgbdCwXDlGjk+5xqhdc0RygsQRGpardv3czoTsuv1lctLISRVytYbld8ePSBJSoQnHSg0adqzdFEvNCKezkpMoGmMywSM6MFbgkCo3nR8/Q4eG+CiIpHlCozn9PpHiUKlp6JnOEOux+l3L4F+1QaKDSzdlIk40FWSxKEg40hHKkkA+k5RoPjUGE8nMrYiMscREm7xKJoSvn6L/Tfe03jD+9qzavMrjKMI+HEANGnABTbiBFnSAwBQe4AmerXvr0XqxXhetBSufqcAPWW+f8gKTAg==AAAB/HicdZDLSsNAFIZP6q3WW7VLN4NFqCglFUWXRTcuK/QGTSiTyaQdOpmEmYlQQn0VNy4UceuDuPNtnLQRVPSHgZ/vnMM583sxZ0rb9odVWFpeWV0rrpc2Nre2d8q7e10VJZLQDol4JPseVpQzQTuaaU77saQ49DjteZPrrN67o1KxSLT1NKZuiEeCBYxgbdCwXDlGjk+5xqhdc0RygsQRGpardv3czoTsuv1lctLISRVytYbld8ePSBJSoQnHSg0adqzdFEvNCKezkpMoGmMywSM6MFbgkCo3nR8/Q4eG+CiIpHlCozn9PpHiUKlp6JnOEOux+l3L4F+1QaKDSzdlIk40FWSxKEg40hHKkkA+k5RoPjUGE8nMrYiMscREm7xKJoSvn6L/Tfe03jD+9qzavMrjKMI+HEANGnABTbiBFnSAwBQe4AmerXvr0XqxXhetBSufqcAPWW+f8gKTAg==

spectral-spatial distortions (e.g. SZ)

+�T (⌫)AAAB+HicdVDLSsNAFJ34rPXRqEs3g0WoCCWJoa27oi5cVugLmlAm00k7dDIJMxOhln6JGxeKuPVT3Pk3TtoKKnrgwuGce7n3niBhVCrL+jBWVtfWNzZzW/ntnd29grl/0JZxKjBp4ZjFohsgSRjlpKWoYqSbCIKigJFOML7K/M4dEZLGvKkmCfEjNOQ0pBgpLfXNwhn0rglTCDZLHk9P+2bRKl/UKo5bgVbZsqq2Y2fEqbrnLrS1kqEIlmj0zXdvEOM0IlxhhqTs2Vai/CkSimJGZnkvlSRBeIyGpKcpRxGR/nR++AyeaGUAw1jo4grO1e8TUxRJOYkC3RkhNZK/vUz8y+ulKqz5U8qTVBGOF4vClEEVwywFOKCCYMUmmiAsqL4V4hESCCudVV6H8PUp/J+0nbKt+a1brF8u48iBI3AMSsAGVVAHN6ABWgCDFDyAJ/Bs3BuPxovxumhdMZYzh+AHjLdPHluSGA==AAAB+HicdVDLSsNAFJ34rPXRqEs3g0WoCCWJoa27oi5cVugLmlAm00k7dDIJMxOhln6JGxeKuPVT3Pk3TtoKKnrgwuGce7n3niBhVCrL+jBWVtfWNzZzW/ntnd29grl/0JZxKjBp4ZjFohsgSRjlpKWoYqSbCIKigJFOML7K/M4dEZLGvKkmCfEjNOQ0pBgpLfXNwhn0rglTCDZLHk9P+2bRKl/UKo5bgVbZsqq2Y2fEqbrnLrS1kqEIlmj0zXdvEOM0IlxhhqTs2Vai/CkSimJGZnkvlSRBeIyGpKcpRxGR/nR++AyeaGUAw1jo4grO1e8TUxRJOYkC3RkhNZK/vUz8y+ulKqz5U8qTVBGOF4vClEEVwywFOKCCYMUmmiAsqL4V4hESCCudVV6H8PUp/J+0nbKt+a1brF8u48iBI3AMSsAGVVAHN6ABWgCDFDyAJ/Bs3BuPxovxumhdMZYzh+AHjLdPHluSGA==AAAB+HicdVDLSsNAFJ34rPXRqEs3g0WoCCWJoa27oi5cVugLmlAm00k7dDIJMxOhln6JGxeKuPVT3Pk3TtoKKnrgwuGce7n3niBhVCrL+jBWVtfWNzZzW/ntnd29grl/0JZxKjBp4ZjFohsgSRjlpKWoYqSbCIKigJFOML7K/M4dEZLGvKkmCfEjNOQ0pBgpLfXNwhn0rglTCDZLHk9P+2bRKl/UKo5bgVbZsqq2Y2fEqbrnLrS1kqEIlmj0zXdvEOM0IlxhhqTs2Vai/CkSimJGZnkvlSRBeIyGpKcpRxGR/nR++AyeaGUAw1jo4grO1e8TUxRJOYkC3RkhNZK/vUz8y+ulKqz5U8qTVBGOF4vClEEVwywFOKCCYMUmmiAsqL4V4hESCCudVV6H8PUp/J+0nbKt+a1brF8u48iBI3AMSsAGVVAHN6ABWgCDFDyAJ/Bs3BuPxovxumhdMZYzh+AHjLdPHluSGA==AAAB+HicdVDLSsNAFJ34rPXRqEs3g0WoCCWJoa27oi5cVugLmlAm00k7dDIJMxOhln6JGxeKuPVT3Pk3TtoKKnrgwuGce7n3niBhVCrL+jBWVtfWNzZzW/ntnd29grl/0JZxKjBp4ZjFohsgSRjlpKWoYqSbCIKigJFOML7K/M4dEZLGvKkmCfEjNOQ0pBgpLfXNwhn0rglTCDZLHk9P+2bRKl/UKo5bgVbZsqq2Y2fEqbrnLrS1kqEIlmj0zXdvEOM0IlxhhqTs2Vai/CkSimJGZnkvlSRBeIyGpKcpRxGR/nR++AyeaGUAw1jo4grO1e8TUxRJOYkC3RkhNZK/vUz8y+ulKqz5U8qTVBGOF4vClEEVwywFOKCCYMUmmiAsqL4V4hESCCudVV6H8PUp/J+0nbKt+a1brF8u48iBI3AMSsAGVVAHN6ABWgCDFDyAJ/Bs3BuPxovxumhdMZYzh+AHjLdPHluSGA==

spectral distortions

+�T (n)AAAB9HicdZDLSgMxFIYzXmu9VV26CRahIpSkiG13RV24rNAbtEPJpJk2NJMZk0yhDH0ONy4UcevDuPNtzLQVVPRA4OP/z+Gc/F4kuDYIfTgrq2vrG5uZrez2zu7efu7gsKXDWFHWpKEIVccjmgkuWdNwI1gnUowEnmBtb3yd+u0JU5qHsmGmEXMDMpTc55QYK7nnsHfDhCGwUZBn/VweFRFCGGOYAi5fIgvVaqWEKxCnlq08WFa9n3vvDUIaB0waKojWXYwi4yZEGU4Fm2V7sWYRoWMyZF2LkgRMu8n86Bk8tcoA+qGyTxo4V79PJCTQehp4tjMgZqR/e6n4l9eNjV9xEy6j2DBJF4v8WEATwjQBOOCKUSOmFghV3N4K6YgoQo3NKWtD+Pop/B9apSK2fHeRr10t48iAY3ACCgCDMqiBW1AHTUDBPXgAT+DZmTiPzovzumhdcZYzR+BHOW+fBGCQ9g==AAAB9HicdZDLSgMxFIYzXmu9VV26CRahIpSkiG13RV24rNAbtEPJpJk2NJMZk0yhDH0ONy4UcevDuPNtzLQVVPRA4OP/z+Gc/F4kuDYIfTgrq2vrG5uZrez2zu7efu7gsKXDWFHWpKEIVccjmgkuWdNwI1gnUowEnmBtb3yd+u0JU5qHsmGmEXMDMpTc55QYK7nnsHfDhCGwUZBn/VweFRFCGGOYAi5fIgvVaqWEKxCnlq08WFa9n3vvDUIaB0waKojWXYwi4yZEGU4Fm2V7sWYRoWMyZF2LkgRMu8n86Bk8tcoA+qGyTxo4V79PJCTQehp4tjMgZqR/e6n4l9eNjV9xEy6j2DBJF4v8WEATwjQBOOCKUSOmFghV3N4K6YgoQo3NKWtD+Pop/B9apSK2fHeRr10t48iAY3ACCgCDMqiBW1AHTUDBPXgAT+DZmTiPzovzumhdcZYzR+BHOW+fBGCQ9g==AAAB9HicdZDLSgMxFIYzXmu9VV26CRahIpSkiG13RV24rNAbtEPJpJk2NJMZk0yhDH0ONy4UcevDuPNtzLQVVPRA4OP/z+Gc/F4kuDYIfTgrq2vrG5uZrez2zu7efu7gsKXDWFHWpKEIVccjmgkuWdNwI1gnUowEnmBtb3yd+u0JU5qHsmGmEXMDMpTc55QYK7nnsHfDhCGwUZBn/VweFRFCGGOYAi5fIgvVaqWEKxCnlq08WFa9n3vvDUIaB0waKojWXYwi4yZEGU4Fm2V7sWYRoWMyZF2LkgRMu8n86Bk8tcoA+qGyTxo4V79PJCTQehp4tjMgZqR/e6n4l9eNjV9xEy6j2DBJF4v8WEATwjQBOOCKUSOmFghV3N4K6YgoQo3NKWtD+Pop/B9apSK2fHeRr10t48iAY3ACCgCDMqiBW1AHTUDBPXgAT+DZmTiPzovzumhdcZYzR+BHOW+fBGCQ9g==AAAB9HicdZDLSgMxFIYzXmu9VV26CRahIpSkiG13RV24rNAbtEPJpJk2NJMZk0yhDH0ONy4UcevDuPNtzLQVVPRA4OP/z+Gc/F4kuDYIfTgrq2vrG5uZrez2zu7efu7gsKXDWFHWpKEIVccjmgkuWdNwI1gnUowEnmBtb3yd+u0JU5qHsmGmEXMDMpTc55QYK7nnsHfDhCGwUZBn/VweFRFCGGOYAi5fIgvVaqWEKxCnlq08WFa9n3vvDUIaB0waKojWXYwi4yZEGU4Fm2V7sWYRoWMyZF2LkgRMu8n86Bk8tcoA+qGyTxo4V79PJCTQehp4tjMgZqR/e6n4l9eNjV9xEy6j2DBJF4v8WEATwjQBOOCKUSOmFghV3N4K6YgoQo3NKWtD+Pop/B9apSK2fHeRr10t48iAY3ACCgCDMqiBW1AHTUDBPXgAT+DZmTiPzovzumhdcZYzR+BHOW+fBGCQ9g==

anisotropies

Basic definitions [note: kB = 1, Eγ = hP ν]

For a perfect blackbody at temperature T:

For generic radiation (non-blackbody), can always define T:AAACMHicbVDLSgMxFM34rPVVdekmWISKUmdE0Y1QFNFlBdsKnVLupJk2NMkMSUYow3ySGz9FNwqKuPUrTGsFXwcCh3PO5eaeIOZMG9d9ciYmp6ZnZnNz+fmFxaXlwspqXUeJIrRGIh6p6wA05UzSmmGG0+tYURABp42gfzr0GzdUaRbJKzOIaUtAV7KQETBWahfOr0pnbb8LQsAO9ntgUplt4WPshwpI+mVlqc9lydsN/wtvY28raxeKbtkdAf8l3pgU0RjVduHe70QkEVQawkHrpufGppWCMoxwmuX9RNMYSB+6tGmpBEF1Kx0dnOFNq3RwGCn7pMEj9ftECkLrgQhsUoDp6d/eUPzPayYmPGqlTMaJoZJ8LgoTjk2Eh+3hDlOUGD6wBIhi9q+Y9MBWZWzHeVuC9/vkv6S+V/YOyu7lfrFyMq4jh9bRBiohDx2iCrpAVVRDBN2iB/SMXpw759F5dd4+oxPOeGYN/YDz/gFlq6dY

T (E� , n̂) =E�

ln(1/f(E� , n̂) + 1)

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

f =1

exp(hP⌫/T )� 1=

1

exp(E�/T )� 1

1965: Discovery of the CMB

Penzias & Wilson detect “excess radiation” at ν = 4 GHz Corresponds to T ≈ 3 K radiation interpreted as the CMB by

Dicke, Peebles, Roll & Wilkinson

Additional measurements are required to confirm the interpretation

Discovery of the CMB

Vor.UME 16,NvMsza 10 PHYSICAL RKVIEW LKTTKRS 7 M&RGB 1966

Table I. Potential energy of 0' for Yale potential.Matrix elementFirst Secondorder order Weight

State nl +I {MeV) (MeV) factor

Contributionto the potential

energy(Mev)

00'S, 00S 003si 00Si 10Sp QQ'S, 00Sp QQSp 00Sp 10I' 013J'p 01Pi 013P2 01Dp 02'D, 02'D, 02'D, 02

00 -2.0201 -2.0202 -2.0210 -2.0200 -0.6600 -8.0301 -8.0302 -8.0310 -8.0300 -7.2400 4.8600 -1.6300 2.7000 -0.8400 -0.5000 1.0800 -2.0100 0.07

-6,98—6 ~ 74—6.53-6.57-6.97

~ ~ ~

3 -27.09 —78.8

15/2 —64.1-12.9

2 -11.43 -24.19 72 03

15/2 -60.2—12.0

2 —10.86 +24.26 -9.818 48.630 -25.215/2 -3.8

1.625 -5,0

0.2Total = -337.8

about 6.5 MeV per particle after correctingfor the Coulomb energy and the center-of-mass

motion. Results of Hartree- Fock calculationsusing the harmonic-oscillator basis will be re-ported shortly as well as further calculationaldetails. We want to emphasize at this stagethat in the framework of our theory we havealready obtained reasonable values for the bind-ing energy, spin-orbit splittings, and the P-shell effective interaction.We are grateful to Professor F. Villars for

stimulating discussions and comments duringthis work and to M. Tomaselli for preliminarycalculations of the second-order terms,

*This work is supported in part through funds pro-vided by the Atomic Energy Commission under Con-tract No. AT(30-1}-2098.F. Villars, in Proceedings of the Enrico Fermi

International School of Physics, Course XXIII, 1961(Academic Press, Inc. , New York, 1963); J. Da Provi-

. dencia and C. M. Shakin, Ann. Phys. (N.Y.) 30, 96(1964).2S. A. Moszkowski and B. L. Scott, Ann. Phys. (N.Y.}11, 66 (1960); M. H. Hull, Jr. , and C. M. Shakin, Phys.Letters 19, 506 (1965).3T. T. S. Kuo and O. E. Brown, Phys. Letters 18, 54(1965).A. D. MacKellar, thesis, Texas A. 5 M. University,

January 1966 (unpublished).

COSMIC BACKGROUND RADIATION AT 3.2 cm —SUPPORT FOR COSMIC BLACK-BODY RADIATION*P. G. Rollt and David T. Wilkinson

Palmer Physical Laboratory, Princeton University, Princeton, New Jersey(Received 27 January 1966}

Dicke et al. ' have suggested that the universemay be filled with black-body radiation whichoriginated at a time when the matter and radi-ation were in a hot, highly contracted, state—the primordial fireball. As the universe ex-panded, the cosmological red shift would havecooled the cosmic black-body radiation to theextent that one should now look for it in themicrowave band. Concurrent with this sugges-tion, Penzias and Wilson' reported the discov-ery of an excess background radiation at a, wave-length of 7.35 cm. The measurement of thespectrum of this new microwave backgroundprovides a severe test of the cosmic black-body-radiation hypothesis. This Letter reports ameasurement of the microwave background ata wavelength of 3.2 cm; the flux found is that

which would be emitted by a black body at 3.0+0.5 K. A more complete description of theexperiment will appear elsewhere.Figure 1 shows a schematic diagram of the

instrument. It is a Dicke-type radiometer'in which the receiver input is periodically switchedbetween a horn antenna and a reference source(cold load). The output of the receiver at theswitching frequency is synchronously detect-ed and recorded. The record is a measureof the difference between the temperature ofthe reference source and the apparent temper-ature of the radiation collected by the antenna.The horn antenna is shielded to exclude radi-ation from the ground and has a main lobe half-angle (10 dB down) of 10'. The cold-load ter-mination is immersed in liquid helium to es-

405

VOLUME 16, NUMBER 10 PHYSI t" AL REVIEW LETTERS 7 MARcH 1966

ic errors stem from inaccurate knowledge ofthe wall radiation in the horn antenna and coldload. The error quoted for THI, (see Table I)is the estimated limit of error as a result ofuncertainties in the wall heating experimentsmentioned above. The estimated limits of er-ror in TCL are based upon bench measurementsof the cold-load wall losses at room temper-ature and at 77'K, and upon wall heating exper-iments similar to those employed to measureTHL

The result of this experiment is shown inFig. 2 along with the results of Penzias andWilson. ' lt is seen that the measurements todate of the microwave background are consis-tent with a cosmic black-body radiation tem-perature of 3'K. Also, the brightness of themicrowave background is about 100 times great-er than that expected by extrapolating long-wave-length measurements' of the galactic and ex-tragalactic background. On the basis of themeasurements at 7.35- and 3.2-cm wavelength,the spectral index (brightness = constx &+) ofthe microwave background is found to be -2.4- e - -1.4. This should be compared with e= -2.0 for black-body radiation in this wavelengthrange, and with 0.5 ~ e ~ 1.0 for most nonther-mal radio sources. '~' Thus the results of mea-surements of the microwave background at wave-lengths of 7.35 and 3.2 cm lend support to thecosmic black-body radiation hypothesis and,at the very least, indicate a new source of cos-mic microwaves.

The proposed cosmic black-body radiationis expected to be isotropic, and so TBG wasmeasured with the antenna pointing in variousdirections along the +40' celestial parallel (seecolumn 3 in Table I). The results indicate thatTBG is the same in these directions to within+10'%%uo. Isotropy measurements are continuingat Princeton.

We would like to acknowledge many valuablediscussions with R. H. Dicke and P. J. E. Pee-bles concerning the experimental and theoret-ical aspects of this work.

lo '4-(flV)QJ e

CLI-O~lO l6K

OtA

0 E lO-isXxtOO p0 g)Z~0

lO-20

P R I NC E TON

(3.5(3.i

llo' IO IO IWAVELENGTH (c m )

!0 '

FIG. 2. Measurements to date of the microwavebackground radiation. The galactic radio background isextrapolated with a spectral index of n =0.5. Thisfigure due to P. J. E. Peebles.

*This work was supported in part by the NationalScience Foundation and the Office of Naval Research ofthe U. S. Navy.

)Presently on leave to the Commission on CollegePhysics, Physics and Astronomy Building, Universityof Michigan, Ann Arbor, Michigan.

~R. H. Dicke, P. J. E. Peebles, P. G. Roll, and D. T.Wilkinson, Astrophys. J. 142, 414 (1965); P. J. E.Peebles, Phys. Rev. Letters 16, 410 (1966).

2A. A, Penzias and R. W. Wilson, Astrophys. J. 142,419 (1965). The result reported in this paper is (3.5+1)'K. A more recent measurement with a modifiedhorn feed gave (3.1+ 1)'K (private communication).

3R. H. Dicke, Rev. Sci. Instr. 17, 268 (1946),4R. H. Dicke, Robert Beringer, Robert L. Kyhl, and

A. B.Vane, Phys. Rev. 70, 340 (1946).5A. J. Turtle, J. F. Pugh, S. Kenderdine, and I. I. K.

Pauliny-Toth, Monthly Notices Roy. Astron. Soc. 124,297 (1962).

6R. G. Conway, K. I. Kellerman, and R. T. I ong,Monthly Notices Roy. Astron. Soc. 125, 313 (1963).

7W. A. Dent and F. T. Haddock, Quasi-Stellar Sourcesand Gravitational Collapse, edited by I. Robinson, A.Schild, and K. L. Schucking (University of Chicago Press,Chicago, 1965), p. 381.

407

COBE measures the CMB spectrum, consistent with blackbody spectrum (i.e. no detected spectral distortions)

(best limit: spectral distortions < 0.01%, Mather et al. 1999)

19 90

ApJ.

. .35

4L. .

37M

No. 2, 1990 COBE MEASUREMENT OF BACKGROUND RADIATION L39

Fig. 2.—Preliminary spectrum of the cosmic microwave background from the FIRAS instrument at the north Galactic pole, compared to a blackbody. Boxes are measured points and show size of assumed 1% error band. The units for the vertical axis are 10“4 ergs s -1 cm-2 sr~1 cm.

The error band in Figure 2 is a conservative estimate of the systematic errors in our current calibration algorithm, taken to be 1% of the peak intensity of the spectrum. Since the data show a good null both when the FIRAS is looking at the external calibrator and at the sky, one can determine from the interferograms alone that the spectrum of the sky is close to a blackbody, regardless of the details of the data reduction and calibration.

IV. DISCUSSION The CMBR temperature reported here lies between the

average of direct ground-based measurements, 2.655 ± 0.036 K (see Smoot et al 1988 for a tabulation), and the precise measurement of 2.783 ± 0.025 K (1 o) at 0.8 cm"1 made from a balloon by Johnson and Wilkinson (1987). At the CN tran- sition frequency, the temperature measured by FIRAS is 2.735 ± 0.06 K, compared to 2.70 ± 0.04 K from Meyer and Jura (1985), 2.796( +0.014, -0.039) K from Crane et al. (1989), and 2.77 ± 0.4 K from Kaiser and Wright (1990). The FIRAS data are not consistent with the departures from a blackbody spectrum reported by Matsumoto et al. (1988).

Using the conservative 1% error bands, these new data set a 3 a upper limit on the Comptonization y parameter of 0.001 and on the chemical potential g of 0.009. This value of g is based on a fit to a pure Bose-Einstein spectrum with g inde- pendent of frequency. The hot smooth intergalactic medium (IGM) suggested to explain the cosmic X-ray background by

Fig. 3.—Composite plot of recent measurements of the temperature of the sky (temperature of the cosmic background vs. wavelength). A = Sironi et al. (1987), B = Levin et al. (1987), C = Sironi and Bonelli (1986), D = De Amici et al. (1988), E = Mandolesi et al. (1986), F = Kogut et al. (1988), G = Johnson and Wilkinson (1987), H = Smoot et al. (1985), I = Smoot et al. (1987), J = Crâne et al. (1989), K = Meyer et al. (1989), Palazzi et al. (1990), L = Matsumoto et al. (1988).

Field and Perrenod (1977), Guilbert and Fabian (1986), and recalculated by Taylor and Wright (1989) can be ruled out, since the predicted X-ray background scales as y2. The new limits on y would limit the X-ray background to only 1/36 of the observed value, even at a heating redshift as small as zc = 2. Many other sources of distortions of the CMBR spectrum (Bond, Carr, and Hogan 1986) are also severely constrained.

A more accurate determination of the spectrum will be made after further sky observations, calibrations, and refinement of the calibration algorithm. The ultimate accuracy of any measured spectrum distortions should be limited only by the optical design and stability of the external calibrator and by the models of radiation from interstellar dust.

It is a pleasure to acknowledge the vital contributions of all those at GSFC who devoted their efforts to making this chal- lenging mission not only possible but enjoyable as well. Special thanks are due to Paul Richards and Patrick Thaddeus for their early encouragement to the lead author, to Robert Maichle and Michael Roberto for leading the engineering effort on the FIRAS instrument, and to Shirley Read, Robert Kümmerer, and Leonard Olson for their leadership in software development for the FIRAS.

REFERENCES Bond, J. R., Carr, B. J., and Hogan, C. J. 1986, Ap. J., 306,428. Crane, P., Hegyi, D. J., Kutner, M. L., and Mandolesi, N. 1989, Ap. J., 346,136. De Amici, G., Smoot, G., Aymon, J., Bersanelli, M., Kogut, A., Levin, S., and

Witebsky, C. 1988, Ap. J., 329,556. Field, G. B., and Perrenod, S. C. 1977, Ap. J., 215,717. Guilbert, P. W., and Fabian, A. C. 1986, M.N.R.A.S., 220,439. Gulkis, S., Lubin, P. M., Meyer, S. S., and Silverberg, R. F. 1990, Sei. Am., Vol.

262, No. l,p. 132. Gush, H. P. Í981, Phys. Rev. Letters, 47,745. Hemmati, H., Mather, J. C, and Eichhorn, W. L. 1985, Appl. Optics, 24,4489. Johnson, D. G., and Wilkinson, D. T. 1987, Ap. J. (Letters), 313, LI. Kaiser, M. E., and Wright, E. L. 1990, Ap. J. (Letters), submitted. Kogut, A., et al. 1988, Ap. J., 325,1. Levin, S., Witebsky, C, Bensadoun, M., Bersanelli, M., De Amici, G., Kogut,

A., and Smoot, G. 1987, Ap. J., 334,14.

Mandolesi, N., Calzolari, P., Cortiglioni, S., and Morigi, G. 1986, Ap. J., 310, 561.

Mather, J. C. 1982, Opt. Eng., 21,769. . 1984, Appl. Opt., 23,3181. Mather, J. C, Toral, M., and Hemmati, H. 1986, Appl. Opt., 25,2826. Matsumoto, T., Hayakawa, S., Matsuo, H., Murakami, H., Sato, S., Lange,

A. E., and Richards, P. L. 1988, Ap. J., 329,567. Meyer, D. M., and Jura, M. 1985, Ap. J., 297,119. Meyer, D. M., Roth, K. C, and Hawkins, 1.1989, Ap. J. (Letters), 343, LI. Palazzi, E., Mandolesi, N., Crane, P., Kutner, M. L., Blades, J. C, and Hegyi,

D. J. 1990, Ap. J., in press. Peebles, P. J. E. 1971, Physical Cosmology (Princeton: Princeton University

Press). Serlemitsos, A. 1988, in Cryogenic Optical Systems and Instruments, ed. R. K.

Melugin (Proc. SPIE, Vol. 973), p. 314.

© American Astronomical Society • Provided by the NASA Astrophysics Data System

Mather et al. 1990 (COBE-FIRAS)

VOLUME 65, NUMBER 5 PHYSICAL REVIEW LETTERS 30 JuLY 1990

xIO "W/cm'str. cm '1.2—1.0 sky brightness

(a)

0.8 ction at T=2.736K.

0.60.40.2

—0.002» I I I I I » I « I I I5. 10.cm-' 15. I I I I I I I I I I I20. 25. 30.

I I I I I I II[

at(b)

JW CII i 2.8

S

SS. work

2.6

SI IIII I I 1 I IIIII I I I I I IIII I I I 2.4

O1 1. 10.cm-&

FIG. 3. (a) The intensity of the sky as a function of wavenumber as measured by COBRA. It is derived from ananalysis of all the interferograms recorded while the observa-tion door was open. The smooth curve is a Planck functionwhich fits the data well. The curve oscillating along the abscis-sa axis is the diff'erence between the experimental and Planckcurves. (b) The equivalent blackbody temperature as a func-tion of frequency. Results of COBRA inferred from the datain (a) are shown as circles. The scatter at the band ends wascaused by microphonic pickup. Other recent data have beenplotted on the same graph: s, Smoot (Ref. 18); JW, Johnsonand Wilkinson (Ref. 19); c, Crane et al. (Ref. 13); mrh,Meyer, Roth, and Hawkins (Ref. 14); mat, Matsumoto et al.(Ref. 16); m, Mandolesi et al. (Ref. 20). The dotted lines areupper and lower limits found by FIRAS on COBE (Ref. 17).Note added: Kaiser and Wright (Ref. 15) measure T=2.74+ 0.05 from CN at k =2.64 mm.

diffuse galactic radiation, "being free of strong sources.During flight the Sun and Moon were both beneath theEarth's horizon, which itself was always more than 90from the telescope axis. Laboratory measurements ofthe telescopes beam pattern limit the response at this an-gle to be less than 10

A composite graph showing a selection of some rele-vant data during the flight is shown in Fig. 2. Averageinterferograms for each of the three conditions of cali-brator temperature have been produced for each detec-tor, from which it is possible to make an absolute cali-bration of the sensitivity, and evaluate the spectrum of

sky emission. The average of the two detector results isshown in Fig. 3(a) as an intensity spectrum, and in Fig.3(b) as an equivalent blackbody temperature as a func-tion of frequency.

In Fig. 3(a), in addition to the measured data, aPlanck function has been drawn corresponding to a tem-perature of 2.736 K, as well as the deviation between thedata and the calculated curve. The fit is quite good. Thequality of the fit may be judged from Fig. 3(b). In thefrequency region 3 ~ v ~ 16 cm ', where the signal-to-noise ratio is high, the rms scatter of the temperature is9.5 mK, about —,' % of the mean. The standard deviationof the mean temperature equals 2 mK. Outside of thisband the confidence of the temperature measurement isnot good, in part because the power available to abso-lutely calibrate the instrument is very small, and in partbecause the power is insensitive to T. Nevertheless, onecan say with considerable certainty that for 20 ~ v ~ 30cm ' the temperature is less than 3 K, a new upper limitin this region. Further analysis is underway to establishprecise confidence levels for this limit.

Various factors contribute to uncertainty in the mea-sured value for the mean temperature: (1) internal cali-brator nonuniformity, contributing an estimated 5 mK;(2) uncertainties in offset corrections, 3 mK (the un-corrected offsets are of the order of 30 mK); (3) possiblenoncancellation of microphonics which developed duringliftoff, 2 mK (the microphonic disturbance is common tothe two detectors whereas optical signals are exactly outof phase, so microphonics cancel in taking the differencebetween interferograms measured by the two detectors);(4) statistical fluctuations in the measured spectrum, 2mK; (5) uncertainties due to heating of the sky telescopeby temporary coupling to the warm door during liftoff, 5mK (this perturbation is shown to be at least this smallby comparing interferograms measured at various horntemperatures). The linear sum of these uncertaintiesequals 17 mK. It is expected that further analysis mayreduce the magnitude of uncertainties (I), (3), and (5).

The above conservative estimate of the uncertainty ofthe temperature of the CBR is the smallest of any mea-surement of this quantity yet reported and provides areference against which other measurements can be com-pared in search for spectral distortions. Referring to Fig.3(b), this measurement falls slightly below previoussingle-frequency (CN) determinations in the same fre-quency range, ' ' and particularly below the broadbandrocket measurements' which differ from our results byabout 4~, 12o., and 17o. in their three bands. The latterwere interpreted as due to large infrared distortions inthe CBR; these distortions are inconsistent with our re-sults. In due course, the systematic errors of the COBEexperiment' are expected to be understood and the cor-responding error bars reduced. At that time it will befruitful to make a detailed comparison between the re-sults of these two precise experiments. The fact that themean temperatures agree to within 1 mK, much less

539

Gush, Halpern & Wishnow 1990

(COBRA)

1990: The CMB frequency spectrum

COBE-FIRAS + other data (Fixsen 2009):T0 = 2.7255 ± 0.0006 K

Aside on CMB spectral distortions• Suppose photons start with a blackbody spectrum.

• Suppose some process injects energy into the photons (either fresh photons, or heats up photons).

• If energy injection is at t ≲ 2 months, it gets fully thermalized

Free-free (Bremsstrahlung) e� + p $ e� + p+ �Double-Compton scattering e� + � $ e� + �0 + �00

Efficiently change photon number and energy; recover perfect blackbody at different T

• Absent injection of additional energy, phase-space density is conserved, photons retain a blackbody spectrum Tγ ∝ 1/a

Aside on CMB spectral distortions

• If energy injection is at 2 months ≲ t ≲ 300 yrs, photons can no longer be efficiently created or destroyed, but Thomson scattering efficiently changes photon energy (not number)

e� + � $ e� + �

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

f�(E�) =1

exp (E�/T�+µ)� 1

➡ Photons acquire a Bose-Einstein distribution with non-zero chemical potential µ

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

µ ⇠ 1⇢�

Z 300 yr

2 modt ⇢̇inj

Aside on CMB spectral distortions

• If energy injection is after 300 yrs, it cannot be thermalized.

➡ Photon spectrum gets distorted from blackbody. Specific shape of distortion depends on energy injection process.

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

f�(E�) = fBB� (E�) [1 +�(E�)]

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

� ⇠ 1⇢�

Z t0

300 yrdt ⇢̇inj

➡ Upper limits on CMB spectral distortions imply stringent constraints on exotic energy injection at t ≳ 2 months

For now we will assume that CMB has a perfect blackbody spectrum and focus on CMB anisotropies

COBE-DMR1992

CMB temperature anisotropies

PSfragreplacements

-160 160 µK0.41 µK

CMB polarization Planck map

CMB temperature anisotropies

CMB photons gravitationally lensed by structure between and now => probes structure formation (+ neutrino masses).

Planck map of the lensing potentialCMB lensing

Ongoing and future measurements: SPT, ACT, Simons Observatory, CMB Stage IV

CMB lensing

-

(μK)2

(null)(null)(null)(null)

(μK)2Temperature

Polarization

baryons

dark matter


Planck collaboration 2018

Planck Collaboration: Cosmological parameters

Fig. 5. Constraints on parameters of the base-⇤CDM model from the separate Planck EE, T E, and TT high-` spectra combinedwith low-` polarization (lowE), and, in the case of EE also with BAO (described in Sect. 5.1), compared to the joint result usingPlanck TT,TE,EE+lowE. Parameters on the bottom axis are our sampled MCMC parameters with flat priors, and parameters on theleft axis are derived parameters (with H0 in km s�1Mpc�1). Contours contain 68 % and 95 % of the probability.

Table 1. Base-⇤CDM cosmological parameters from Planck TT,TE,EE+lowE+lensing. Results for the parameter best fits,marginalized means and 68 % errors from our default analysis using the Plik likelihood are given in the first two numericalcolumns. The CamSpec likelihood results give some idea of the remaining modelling uncertainty in the high-` polarization, thoughparts of the small shifts are due to slightly di↵erent sky areas in polarization. The “Combined” column give the average of thePlik and CamSpec results, assuming equal weight. The combined errors are from the equal-weighted probabilities, hence includingsome uncertainty from the systematic di↵erence between them; however, the di↵erences between the high-` likelihoods are so smallthat they have little e↵ect on the 1� errors. The errors do not include modelling uncertainties in the lensing and low-` likelihoodsor other modelling errors (such as temperature foregrounds) common to both high-` likelihoods. A total systematic uncertainty ofaround 0.5� may be more realistic, and values should not be overinterpreted beyond this level. The best-fit values give a represen-tative model that is an excellent fit to the baseline likelihood, though models nearby in the parameter space may have very similarlikelihoods. The first six parameters here are the ones on which we impose flat priors and use as sampling parameters; the remainingparameters are derived from the first six. Note that ⌦m includes the contribution from one neutrino with a mass of 0.06 eV. Thequantity ✓MC is an approximation to the acoustic scale angle, while ✓⇤ is the full numerical result.

Parameter Plik best fit Plik [1] CamSpec [2] ([2] � [1])/�1 Combined

⌦bh2 . . . . . . . . . . . . . 0.022383 0.02237 ± 0.00015 0.02229 ± 0.00015 �0.5 0.02233 ± 0.00015⌦ch2 . . . . . . . . . . . . . 0.12011 0.1200 ± 0.0012 0.1197 ± 0.0012 �0.3 0.1198 ± 0.0012100✓MC . . . . . . . . . . . 1.040909 1.04092 ± 0.00031 1.04087 ± 0.00031 �0.2 1.04089 ± 0.00031⌧ . . . . . . . . . . . . . . . . 0.0543 0.0544 ± 0.0073 0.0536+0.0069

�0.0077 �0.1 0.0540 ± 0.0074ln(1010As) . . . . . . . . . 3.0448 3.044 ± 0.014 3.041 ± 0.015 �0.3 3.043 ± 0.014ns . . . . . . . . . . . . . . . 0.96605 0.9649 ± 0.0042 0.9656 ± 0.0042 +0.2 0.9652 ± 0.0042

⌦mh2 . . . . . . . . . . . . 0.14314 0.1430 ± 0.0011 0.1426 ± 0.0011 �0.3 0.1428 ± 0.0011H0 [ km s�1Mpc�1] . . . 67.32 67.36 ± 0.54 67.39 ± 0.54 +0.1 67.37 ± 0.54⌦m . . . . . . . . . . . . . . 0.3158 0.3153 ± 0.0073 0.3142 ± 0.0074 �0.2 0.3147 ± 0.0074Age [Gyr] . . . . . . . . . 13.7971 13.797 ± 0.023 13.805 ± 0.023 +0.4 13.801 ± 0.024�8 . . . . . . . . . . . . . . . 0.8120 0.8111 ± 0.0060 0.8091 ± 0.0060 �0.3 0.8101 ± 0.0061S 8 ⌘ �8(⌦m/0.3)0.5 . . 0.8331 0.832 ± 0.013 0.828 ± 0.013 �0.3 0.830 ± 0.013zre . . . . . . . . . . . . . . . 7.68 7.67 ± 0.73 7.61 ± 0.75 �0.1 7.64 ± 0.74100✓⇤ . . . . . . . . . . . . 1.041085 1.04110 ± 0.00031 1.04106 ± 0.00031 �0.1 1.04108 ± 0.00031rdrag [Mpc] . . . . . . . . . 147.049 147.09 ± 0.26 147.26 ± 0.28 +0.6 147.18 ± 0.29

14





⌦bh2 . . . . . . . . . . . . . 0.022383 0.02237 ± 0.00015 0.02229 ± 0.00015 �0.5 0.02233 ± 0.00015⌦ch2 . . . . . . . . . . . . . 0.12011 0.1200 ± 0.0012 0.1197 ± 0.0012 �0.3 0.1198 ± 0.0012100✓MC . . . . . . . . . . . 1.040909 1.04092 ± 0.00031 1.04087 ± 0.00031 �0.2 1.04089 ± 0.00031⌧ . . . . . . . . . . . . . . . . 0.0543 0.0544 ± 0.0073 0.0536+0.0069

�0.0077 �0.1 0.0540 ± 0.0074ln(1010As) . . . . . . . . . 3.0448 3.044 ± 0.014 3.041 ± 0.015 �0.3 3.043 ± 0.014ns . . . . . . . . . . . . . . . 0.96605 0.9649 ± 0.0042 0.9656 ± 0.0042 +0.2 0.9652 ± 0.0042


14

The CMB: a pillar of high-precision cosmology

Protagonists and stage

first ~4e5 yrs

Thomson scatteringphotons (CMB)

neutrinos (collisionless, “hot”)

“baryons”

e-

p+

He++

cold dark matter(pressureless, collisionless)

? ??

gravity

The stage: FLRW spacetimeFriedmann-Lemaître-Robertson-Walker (FLRW) metric for a homogeneous and isotropic (and spatially flat) Universe:

ds2 = �dt2 + a2(t)d~x2 a: scale factor (a=1 today) x : comoving coordinate

η: conformal time= a2(⌘)

⇥�d⌘2 + d~x2

⇤

In the absence of interactions, particle momenta (as observed by comoving observers) “redshift” as

AAACDXicbVDLSgMxFM3UV62vUZduglWoCHVGFF0W3bisYB/QGYZMmrahmUlMMoU69Afc+CtuXCji1r07/8a0nYW2HggczrmXm3NCwajSjvNt5RYWl5ZX8quFtfWNzS17e6eueCIxqWHOuGyGSBFGY1LTVDPSFJKgKGSkEfavx35jQKSiPL7TQ0H8CHVj2qEYaSMF9oEIUk9GkIdqBD0hudAcuicIeuQ+oQNYco8fjgK76JSdCeA8cTNSBBmqgf3ltTlOIhJrzJBSLdcR2k+R1BQzMip4iSIC4T7qkpahMYqI8tNJmhE8NEobdrg0L9Zwov7eSFGk1DAKzWSEdE/NemPxP6+V6M6ln9JYJJrEeHqokzBoEo+rgW0qCdZsaAjCkpq/QtxDEmFtCiyYEtzZyPOkflp2z8vO7VmxcpXVkQd7YB+UgAsuQAXcgCqoAQwewTN4BW/Wk/VivVsf09Gcle3sgj+wPn8AVMqaaw==

pobs / 1/a ⌘ (1 + z)

Cosmological redshift: AAAB8HicbVBNSwMxEJ2tX7V+VT16CRZBEOpuUfQiFL14rGA/pF1KNs22oUl2SbJCXforvHhQxKs/x5v/xrTdg7Y+GHi8N8PMvCDmTBvX/XZyS8srq2v59cLG5tb2TnF3r6GjRBFaJxGPVCvAmnImad0ww2krVhSLgNNmMLyZ+M1HqjSL5L0ZxdQXuC9ZyAg2VnrwTp7QFfJOcbdYcsvuFGiReBkpQYZat/jV6UUkEVQawrHWbc+NjZ9iZRjhdFzoJJrGmAxxn7YtlVhQ7afTg8foyCo9FEbKljRoqv6eSLHQeiQC2ymwGeh5byL+57UTE176KZNxYqgks0VhwpGJ0OR71GOKEsNHlmCimL0VkQFWmBibUcGG4M2/vEgalbJ3XnbvzkrV6yyOPBzAIRyDBxdQhVuoQR0ICHiGV3hzlPPivDsfs9ack83swx84nz+HpY7s

1 + z = 1/a

The stage: FLRW spacetime

Hubble expansion rate: AAAB/nicbVBNS8NAEN3Ur1q/ouLJy2IRPJVEFL0IRS89VrAf0JSy2WzapZtN2J0IJQT8K148KOLV3+HNf+O2zUFbHww83pthZp6fCK7Bcb6t0srq2vpGebOytb2zu2fvH7R1nCrKWjQWser6RDPBJWsBB8G6iWIk8gXr+OO7qd95ZErzWD7AJGH9iAwlDzklYKSBfdS4wV6oCM0C7AmJSZ4FkA/sqlNzZsDLxC1IFRVoDuwvL4hpGjEJVBCte66TQD8jCjgVLK94qWYJoWMyZD1DJYmY7mez83N8apQAh7EyJQHP1N8TGYm0nkS+6YwIjPSiNxX/83ophNf9jMskBSbpfFGYCgwxnmaBA64YBTExhFDFza2YjogJA0xiFROCu/jyMmmf19zLmnN/Ua3fFnGU0TE6QWfIRVeojhqoiVqIogw9o1f0Zj1ZL9a79TFvLVnFzCH6A+vzB2LhlSE=

H =d ln a

dt

Hubble “constant” H0 = H(a = 1) = H(today) 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

H0 = (67.4± 0.5)km/s/Mpc [Planck 2018]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

H0 = (73.5± 1.4)km/s/Mpc [local measurements]

The famous “Hubble tension”.

Friedman’s equation (in a spatially flat Universe): 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

H2 =

8⇡G

3⇢tot =

8⇡G

3

�⇢� + ⇢⌫ + ⇢c + ⇢b + ⇢⇤

�

The stage: FLRW spacetime

•Mean number densities:

•For massive neutrinos: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

⇢⌫ / a�4 while T⌫ � m⌫ ,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

⇢⌫ / a�3 once T⌫ ⌧ m⌫ ,

AAACBHicbVC7TsMwFHV4lvIKMHaxqJBYqBIegrGChQkViT6kJlSO67RWHTuyHaQqysDCr7AwgBArH8HG3+C0GaDlSJaOzrlX1+cEMaNKO863tbC4tLyyWlorr29sbm3bO7stJRKJSRMLJmQnQIowyklTU81IJ5YERQEj7WB0lfvtByIVFfxOj2PiR2jAaUgx0kbq2RVPGDvfTm8y6MVSxFpAdJ8enWQ9u+rUnAngPHELUgUFGj37y+sLnESEa8yQUl3XibWfIqkpZiQre4kiMcIjNCBdQzmKiPLTSYgMHhilD0MhzeMaTtTfGymKlBpHgZmMkB6qWS8X//O6iQ4v/JTyONGE4+mhMGHQ5MwbgX0qCdZsbAjCkpq/QjxEEmFteiubEtzZyPOkdVxzz2rO7Wm1flnUUQIVsA8OgQvOQR1cgwZoAgwewTN4BW/Wk/VivVsf09EFq9jZA39gff4ArxKYGg==

N / a�3

•For radiation: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

⇢� = N�hp�i / a�4

• Non-relativistic species (cold dark matter, baryons):

AAACH3icbVDNS8MwHE3n15xfVY9egkPw4mj9vghDL55kgvuAtZY0y7awtClJKoxS/xIv/itePCgi3vbfmG4FdfMHCY/3fo/kPT9iVCrLGhmFufmFxaXicmlldW19w9zcakgeC0zqmDMuWj6ShNGQ1BVVjLQiQVDgM9L0B1eZ3nwgQlIe3qlhRNwA9ULapRgpTXnmqcO1nLkTR/R56rXgBfzhbjLiMdCXEwkeKQ7RfXJwlHpm2apY44GzwM5BGeRT88wvp8NxHJBQYYakbNtWpNwECUUxI2nJiSWJEB6gHmlrGKKASDcZ50vhnmY6sMuFPqGCY/a3I0GBlMPA15sBUn05rWXkf1o7Vt1zN6FhFCsS4slD3ZhBnTMrC3aoIFixoQYIC6r/CnEfCYSVrrSkS7CnI8+CxmHFPqlYt8fl6mVeRxHsgF2wD2xwBqrgGtRAHWDwBF7AG3g3no1X48P4nKwWjNyzDf6MMfoGdlijMg==

⇢X = NX mX / a�3

The stage: FLRW spacetimeDimensionless density parameters:

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

⌦X =8⇡G ⇢X,0

3H20

AAAB/HicbZDLSsNAFIZP6q3WW7RLN4NFcFUSUXQjFN24s4K9QBPCZDpph84kYWYilFBfxY0LRdz6IO58G6dtFtr6w8DHf87hnPnDlDOlHefbKq2srq1vlDcrW9s7u3v2/kFbJZkktEUSnshuiBXlLKYtzTSn3VRSLEJOO+HoZlrvPFKpWBI/6HFKfYEHMYsYwdpYgV31VCaCLvLuBB1gA1fIDeyaU3dmQsvgFlCDQs3A/vL6CckEjTXhWKme66Taz7HUjHA6qXiZoikmIzygPYMxFlT5+ez4CTo2Th9FiTQv1mjm/p7IsVBqLELTKbAeqsXa1Pyv1st0dOnnLE4zTWMyXxRlHOkETZNAfSYp0XxsABPJzK2IDLHERJu8KiYEd/HLy9A+rbvndef+rNa4LuIowyEcwQm4cAENuIUmtIDAGJ7hFd6sJ+vFerc+5q0lq5ipwh9Znz/l3ZOfX

X

⌦X = 1

Equivalently:

AAACCHicbVC7SgNBFJ31GeNr1dLCwSBYJbOiaCMEbdIIEcwDkhBmJ5NkyMzuMnNXDEvS2fgrNhaK2PoJdv6Nk0ehiQcuHM65l3vv8SMpDBDy7SwsLi2vrKbW0usbm1vb7s5u2YSxZrzEQhnqqk8NlyLgJRAgeTXSnCpf8orfux75lXuujQiDO+hHvKFoJxBtwShYqekeFJoEX2KPkGEXD3Ed+ANolfRUzuRuIjZouhmSJWPgeeJNSQZNUWy6X/VWyGLFA2CSGlPzSASNhGoQTPJBuh4bHlHWox1eszSgiptGMn5kgI+s0sLtUNsKAI/V3xMJVcb0lW87FYWumfVG4n9eLYb2RSMRQRQDD9hkUTuWGEI8SgW3hOYMZN8SyrSwt2LWpZoysNmlbQje7MvzpHyS9c6y5PY0k7+axpFC++gQHSMPnaM8KqAiKiGGHtEzekVvzpPz4rw7H5PWBWc6s4f+wPn8ASR3mC8=

H0 = 100 h km/s/MpcAAACBXicbVC7TsMwFHXKq5RXgBEGiwqJqUoqEIwVLGwUiT6kJkSOe9NadR7YTqUq6sLCr7AwgBAr/8DG3+C2GaDlSFc6Pude+d7jJ5xJZVnfRmFpeWV1rbhe2tjc2t4xd/eaMk4FhQaNeSzaPpHAWQQNxRSHdiKAhD6Hlj+4mvitIQjJ4uhOjRJwQ9KLWMAoUVryzEMnDqFHvDZ24CFlQ+zc5O/+fdUzy1bFmgIvEjsnZZSj7plfTjemaQiRopxI2bGtRLkZEYpRDuOSk0pICB2QHnQ0jUgI0s2mV4zxsVa6OIiFrkjhqfp7IiOhlKPQ150hUX05703E/7xOqoILN2NRkiqI6OyjIOVYxXgSCe4yAVTxkSaECqZ3xbRPBKFKB1fSIdjzJy+SZrVin1Ws29Ny7TKPo4gO0BE6QTY6RzV0jeqogSh6RM/oFb0ZT8aL8W58zFoLRj6zj/7A+PwBN7WXvQ==

!X ⌘ ⌦Xh2

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

H2(a) = H20

⌦⇤ + (⌦c + ⌦b)a

�3 + ⌦�a�4 + ⌦⌫

⇢⌫(a)

⇢⌫,0

�

Perturbed FLRW metric: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

ds2 = a2(⌘)⇥�(1+2 )d⌘2 + (1�2�)d~x2

⇤+ a2(⌘)hGWij dx

idxj

Cold dark matter

Approximated as a collisionless and pressureless ideal fluid entirely described by its density and velocity fields

AAACAXicbVDLSgMxFM3UV62vqhvBTbAIFaTMiKLLohuXFewDOkPJpLdtaCYzJJliGcaNv+LGhSJu/Qt3/o1pOwttPXDh5Jx7yb3HjzhT2ra/rdzS8srqWn69sLG5tb1T3N1rqDCWFOo05KFs+UQBZwLqmmkOrUgCCXwOTX94M/GbI5CKheJejyPwAtIXrMco0UbqFA/cEdBklHZo2QVNTvH0/ZCedIolu2JPgReJk5ESylDrFL/cbkjjAISmnCjVduxIewmRmlEOacGNFUSEDkkf2oYKEoDykukFKT42Shf3QmlKaDxVf08kJFBqHPimMyB6oOa9ifif145178pLmIhiDYLOPurFHOsQT+LAXSaBaj42hFDJzK6YDogkVJvQCiYEZ/7kRdI4qzgXFfvuvFS9zuLIo0N0hMrIQZeoim5RDdURRY/oGb2iN+vJerHerY9Za87KZvbRH1ifPxcQlqY=

~vc(⌘, ~x)AAACOHicbVBNSwMxEM36bf2qevQSLEKLUnZF0YsgevGmglWhW0o2ndrQ7GZJZotlqf/Kiz/Dm3jxoIhXf4HZuge1Pgg83ryZybwglsKg6z45Y+MTk1PTM7OFufmFxaXi8sqlUYnmUONKKn0dMANSRFBDgRKuYw0sDCRcBd3jrH7VA22Eii6wH0MjZDeRaAvO0ErN4qmvO6rJyz4g26J+D3h6O6jQA0pp2aOb1G+BRDZiqNz5yo7NtqbZhEHuqDSLJbfqDkFHiZeTEslx1iw++i3FkxAi5JIZU/fcGBsp0yi4hEHBTwzEjHfZDdQtjVgIppEODx/QDau0aFtp+yKkQ/VnR8pCY/phYJ0hw475W8vE/2r1BNv7jVREcYIQ8e9F7URSVDRLkbaEBo6ybwnjWti/Ut5hmnG0WRdsCN7fk0fJ5XbV26265zulw6M8jhmyRtZJmXhkjxySE3JGaoSTe/JMXsmb8+C8OO/Ox7d1zMl7VskvOJ9fRf2q/A==

⇢c(⌘, ~x) = (1 + �c(⌘, ~x)) ⇢c(⌘)

Fluid equations: relativistic generalization of 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

@⇢c@t

+ ~r · (⇢c~vc) = 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

@~vc@t

+ (~vc · ~r)~vc = �~r�Newt

continuity (mass conservation)

momentum equation

See Fabian Schmidt’s lectures for the rich physics of CDM gravitational collapse and structure formation.

Massive neutrinos

• Three neutrino (+ antineutrino) flavors

• Three mass eigenstates

AAACAXicbZDLSgMxFIYz9VbrbdSN4CZYBBdSZkTRZdGNywr2Ap1hyKSZNjTJDLkIpdSNr+LGhSJufQt3vo2ZdhbaeiDk4//PITl/nDGqtOd9O6Wl5ZXVtfJ6ZWNza3vH3d1rqdRITJo4ZansxEgRRgVpaqoZ6WSSIB4z0o6HN7nffiBS0VTc61FGQo76giYUI22lyD0IhInIKcyvgJsCNDKRW/Vq3rTgIvgFVEFRjcj9CnopNpwIjRlSqut7mQ7HSGqKGZlUAqNIhvAQ9UnXokCcqHA83WACj63Sg0kq7REaTtXfE2PElRrx2HZypAdq3svF/7yu0clVOKYiM5oIPHsoMQzqFOZxwB6VBGs2soCwpPavEA+QRFjb0Co2BH9+5UVondX8i5p3d16tXxdxlMEhOAInwAeXoA5uQQM0AQaP4Bm8gjfnyXlx3p2PWWvJKWb2wZ9yPn8Am46WWg==⌫e, ⌫µ, ⌫⌧AAACGnicbZBLSwMxEMez9VXrq+rRS7AIHqTsVkWPRS8eK9gHdEvJptM2NJtd8xBK6efw4lfx4kERb+LFb2N2uwdtHUjmx39mSOYfxJwp7brfTm5peWV1Lb9e2Njc2t4p7u41VGQkhTqNeCRbAVHAmYC6ZppDK5ZAwoBDMxhdJ/XmA0jFInGnxzF0QjIQrM8o0VbqFj1fmK53gpNUmaVTe8N9ijBT/NBkoInpFktu2U0DL4KXQQllUesWP/1eRE0IQlNOlGp7bqw7EyI1oxymBd8oiAkdkQG0LQoSgupM0tWm+MgqPdyPpD1C41T9PTEhoVLjMLCdIdFDNV9LxP9qbaP7l50JE7HRIOjsob7hWEc48Qn3mASq+dgCoZLZv2I6JJJQbd0sWBO8+ZUXoVEpe+dl9/asVL3K7MijA3SIjpGHLlAV3aAaqiOKHtEzekVvzpPz4rw7H7PWnJPN7KM/4Xz9AFpCnzw=

⌫1, ⌫2, ⌫3 6= ⌫e, ⌫µ, ⌫⌧

• Individual neutrino masses are still unknown, but sum of neutrino masses constrained to Σ mν < 0.12 eV (Planck 2018)

• Neutrinos decouple from the plasma at T ~ MeV, while ultra-relativistic. They have a (perturbed) relativistic Fermi-Dirac occupation number

AAACGnicbVDLSgMxFM3UV62vUZdugkVoEeqMKLoRim5cVugLOqVk0kwbmsmEJCOWYb7Djb/ixoUi7sSNf2Om7UJbDwQO59zDzT2+YFRpx/m2ckvLK6tr+fXCxubW9o69u9dUUSwxaeCIRbLtI0UY5aShqWakLSRBoc9Iyx/dZH7rnkhFI17XY0G6IRpwGlCMtJF6tutFxs7SSZD2PB6XRBleQS+QCCdumnjkQZTEST2zyvAYumnPLjoVZwK4SNwZKYIZaj370+tHOA4J15ghpTquI3Q3QVJTzEha8GJFBMIjNCAdQzkKieomk9NSeGSUPgwiaR7XcKL+TiQoVGoc+mYyRHqo5r1M/M/rxDq47CaUi1gTjqeLgphBHcGsJ9inkmDNxoYgLKn5K8RDZFrRps2CKcGdP3mRNE8r7nnFuTsrVq9ndeTBATgEJeCCC1AFt6AGGgCDR/AMXsGb9WS9WO/Wx3Q0Z80y++APrK8feACf0w==

f⌫(p) =1

exp(p/T⌫) + 1AAACBnicbVDLSgNBEJz1GeNr1aMIg0HwtOyKEo9BLx4j5AXZEHonk2TIzOwyMyuGJV68+CtePCji1W/w5t84eRw0saChqOqmuytKONPG97+dpeWV1bX13EZ+c2t7Z9fd26/pOFWEVknMY9WIQFPOJK0aZjhtJIqCiDitR4PrsV+/o0qzWFbMMKEtAT3JuoyAsVLbPaq0Q5niEJJExffY94oBfsBW7IEQ0HYLvudPgBdJMCMFNEO57X6FnZikgkpDOGjdDPzEtDJQhhFOR/kw1TQBMoAebVoqQVDdyiZvjPCJVTq4Gytb0uCJ+nsiA6H1UES2U4Dp63lvLP7nNVPTvWxlTCapoZJMF3VTjk2Mx5ngDlOUGD60BIhi9lZM+qCAGJtc3oYQzL+8SGpnXnDh+bfnhdLVLI4cOkTH6BQFqIhK6AaVURUR9Iie0St6c56cF+fd+Zi2LjmzmQP0B87nDwWYl48=

T⌫ ⇡ 0.71 T�

• Epochs relevant to observable CMB properties correspond to T

•“baryons” in cosmology= ions, electrons and neutral atoms.

•At T

Baryons

• Baryons are an ideal fluid, fully described by their density, velocity and temperature (same temperature for H, He, e-)

AAACOHicbVBNSwMxEM36bf2qevQSLEKLUnZF0YsgevGmglWhW0o2ndrQ7GZJZotlqf/Kiz/Dm3jxoIhXf4HZuge1Pgg83ryZybwglsKg6z45Y+MTk1PTM7OFufmFxaXi8sqlUYnmUONKKn0dMANSRFBDgRKuYw0sDCRcBd3jrH7VA22Eii6wH0MjZDeRaAvO0ErN4qmvO6oZlH1AtkX9HvD0dlChB5TSskc3qd8CiWzEULnzlR2bbU2zCYPcUWkWS27VHYKOEi8nJZLjrFl89FuKJyFEyCUzpu65MTZSplFwCYOCnxiIGe+yG6hbGrEQTCMdHj6gG1Zp0bbS9kVIh+rPjpSFxvTDwDpDhh3zt5aJ/9XqCbb3G6mI4gQh4t+L2omkqGiWIm0JDRxl3xLGtbB/pbzDNONosy7YELy/J4+Sy+2qt1t1z3dKh0d5HDNkjayTMvHIHjkkJ+SM1Agn9+SZvJI358F5cd6dj2/rmJP3rJJfcD6/AED6qvk=

⇢b(⌘, ~x) = (1 + �b(⌘, ~x)) ⇢b(⌘)AAACAXicbVDLSgMxFM3UV62vqhvBTbAIFaTMiKLLohuXFewDOkPJpLdtaCYzJJliGcaNv+LGhSJu/Qt3/o1pOwttPXDh5Jx7yb3HjzhT2ra/rdzS8srqWn69sLG5tb1T3N1rqDCWFOo05KFs+UQBZwLqmmkOrUgCCXwOTX94M/GbI5CKheJejyPwAtIXrMco0UbqFA/cEdBklHb8sguanOLp+yE96RRLdsWeAi8SJyMllKHWKX653ZDGAQhNOVGq7diR9hIiNaMc0oIbK4gIHZI+tA0VJADlJdMLUnxslC7uhdKU0Hiq/p5ISKDUOPBNZ0D0QM17E/E/rx3r3pWXMBHFGgSdfdSLOdYhnsSBu0wC1XxsCKGSmV0xHRBJqDahFUwIzvzJi6RxVnEuKvbdeal6ncWRR4foCJWRgy5RFd2iGqojih7RM3pFb9aT9WK9Wx+z1pyVzeyjP7A+fwAVfZal

~vb(⌘, ~x)AAAB+3icbVDLSsNAFJ3UV62vWJduBotQQUoiii6LblxW6AuaECbTm3bo5MHMpLSE/oobF4q49Ufc+TdO2yy09cCFwzn3cu89fsKZVJb1bRQ2Nre2d4q7pb39g8Mj87jclnEqKLRozGPR9YkEziJoKaY4dBMBJPQ5dPzRw9zvjEFIFkdNNU3ADckgYgGjRGnJM8tNz686oMgldsZAs8nswjMrVs1aAK8TOycVlKPhmV9OP6ZpCJGinEjZs61EuRkRilEOs5KTSkgIHZEB9DSNSAjSzRa3z/C5Vvo4iIWuSOGF+nsiI6GU09DXnSFRQ7nqzcX/vF6qgjs3Y1GSKojoclGQcqxiPA8C95kAqvhUE0IF07diOiSCUKXjKukQ7NWX10n7qmbf1Kyn60r9Po+jiE7RGaoiG92iOnpEDdRCFE3QM3pFb8bMeDHejY9la8HIZ07QHxifP+bEk7U=

Tb(⌘, ~x)

continuity (mass conservation)

momentum equation

• Fluid equations: relativistic generalization of 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

@⇢b@t

+ ~r · (⇢b~vb) = 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

d~vb@t

= �~r�Newt + �Thomson (~v� � ~vb)

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

dTb@t

= �̃Thomson (T� � Tb) heat equation

× × × × ×Time

(years)10,000 100,000 1 million

He++➞He+ He+➞HeoH+➞Ho

Computed with HyRec [Ali-Haïmoud & Hirata 2011]io

nize

d fr

actio

n x e

Recombination H and He start fully ionized, and eventually “recombine” with electrons, to form neutral atoms. Recombination is a crucial piece of CMB physics, we’ll study it in lecture 2.

Photons (i.e. the CMB)

• Described by photon occupation number fγ

•Not an ideal fluid and not collisionless in general — in fact, they are either one or the other

We will spend lectures 3-4 deriving this equation and discussing it solutions in some limiting regimes.

Described by the collisional Boltzmann equationAAACH3icbVDLSsNAFJ34tr6qLt0MFsFVScTXRhDduFSwtdCEcDOd1NGZJMzcCCXmT9z4K25cKCLu/BunD0GtBwYO55zL3HuiTAqDrvvpTExOTc/Mzs1XFhaXlleqq2tNk+aa8QZLZapbERguRcIbKFDyVqY5qEjyq+j2tO9f3XFtRJpcYi/jgYJuImLBAK0UVvf9WAMrOjQO/S4oBaXlWPonolvcl2Hha0VRw01Jj+hp+zsUhNWaW3cHoOPEG5EaGeE8rH74nZTliifIJBjT9twMgwI0CiZ5WfFzwzNgt9DlbUsTUNwExeC+km5ZxW6YavsSpAP150QBypieimxSAV6bv15f/M9r5xgfBoVIshx5woYfxbmkmNJ+WbQjNGcoe5YA08LuStk12MLQVlqxJXh/Tx4nzZ26t1d3L3ZrxyejOubIBtkk28QjB+SYnJFz0iCMPJAn8kJenUfn2Xlz3ofRCWc0s05+wfn8AnOZozQ=

df�dt

��traj

= C[f� ]Thomson collision operator

Task at hand: solve linear coupled differential equations

Thomson scattering

neutrinos

e-

p+

He++

cold dark matter

? ??

gravityAAACBHicbVC7TsMwFHXKq5RXgLGLRYXEgKoEgWCsYICxSPQhNSFyXKc12E5kO0hVlIGFX2FhACFWPoKNv8FtM0DLke7V0Tn3yr4nTBhV2nG+rdLC4tLySnm1sra+sbllb++0VZxKTFo4ZrHshkgRRgVpaaoZ6SaSIB4y0gnvL8Z+54FIRWNxo0cJ8TkaCBpRjLSRArvqJUN6CL1EmT4MMnqX32ae5PCykwd2zak7E8B54hakBgo0A/vL68c45URozJBSPddJtJ8hqSlmJK94qSIJwvdoQHqGCsSJ8rPJETncN0ofRrE0JTScqL83MsSVGvHQTHKkh2rWG4v/eb1UR2d+RkWSaiLw9KEoZVDHcJwI7FNJsGYjQxCW1PwV4iGSCGuTW8WE4M6ePE/aR3X3pO5cH9ca50UcZVAFe+AAuOAUNMAVaIIWwOARPINX8GY9WS/Wu/UxHS1Zxc4u+APr8wetpJd7

�, , hGWijAAAB/HicdVDLSsNAFJ34rPUV7dLNYBFcSEhKfSyLblxWsA9oSphMb9qhkwczk0II9VfcuFDErR/izr9x0geo6IHLPZxzL3Pn+AlnUtn2p7Gyura+sVnaKm/v7O7tmweHbRmngkKLxjwWXZ9I4CyClmKKQzcRQEKfQ8cf3xR+ZwJCsji6V1kC/ZAMIxYwSpSWPLPiDoAr4tEz7E6A5pOpRz2zalsXdgFsW/Ulqc2JY826XUULND3zwx3ENA0hUpQTKXuOnah+ToRilMO07KYSEkLHZAg9TSMSguzns+On+EQrAxzEQlek8Ez9vpGTUMos9PVkSNRI/vYK8S+vl6rgqp+zKEkVRHT+UJByrGJcJIEHTABVPNOEUMH0rZiOiCBU6bzKOoTlT/H/pF2znHPLvqtXG9eLOEroCB2jU+SgS9RAt6iJWoiiDD2iZ/RiPBhPxqvxNh9dMRY7FfQDxvsXjyCUtw==

�c,~vc

baryonsphotons (CMB)

AAACB3icdVDLSgMxFM34rPVVdSlIsAgVZEjfLotuXFawD+iUkkkzbWgmMySZYhm6c+OvuHGhiFt/wZ1/Y9qOoKIHLhzOuZd773FDzpRG6MNaWl5ZXVtPbaQ3t7Z3djN7+00VRJLQBgl4INsuVpQzQRuaaU7boaTYdzltuaPLmd8aU6lYIG70JKRdHw8E8xjB2ki9zJHXc0SU02fQGVMS304TEk5n+mkvk0U2qlYrxQJEdgWVymVkSLlUrBYqMG+jObIgQb2XeXf6AYl8KjThWKlOHoW6G2OpGeF0mnYiRUNMRnhAO4YK7FPVjed/TOGJUfrQC6QpoeFc/T4RY1+pie+aTh/rofrtzcS/vE6kvfNuzEQYaSrIYpEXcagDOAsF9pmkRPOJIZhIZm6FZIglJtpElzYhfH0K/yfNgp0v2+i6lK1dJHGkwCE4BjmQB1VQA1egDhqAgDvwAJ7As3VvPVov1uuidclKZg7AD1hvn5+QmS4=

f⌫(t, ~x, ~p⌫)

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

f�(t, ~x, ~p�)AAACBnicdZDLSgMxFIYzXmu9VV2KECyCCymJeGl3RTcuK/QGbRky6WkbzFxIMsUydOXGV3HjQhG3PoM738ZMW0FFDyR8/P85JOf3Iim0IeTDmZtfWFxazqxkV9fWNzZzW9t1HcaKQ42HMlRNj2mQIoCaEUZCM1LAfE9Cw7u5TP3GEJQWYVA1owg6PusHoic4M1Zyc3vtLkjDXO8It4fAk+E4xWp63brg5vKkQAihlOIU6PkZsVAqFY9pEdPUspVHs6q4ufd2N+SxD4HhkmndoiQynYQpI7iEcbYda4gYv2F9aFkMmA+6k0zWGOMDq3RxL1T2BAZP1O8TCfO1Hvme7fSZGejfXir+5bVi0yt2EhFEsYGATx/qxRKbEKeZ4K5QwI0cWWBcCftXzAdMMW5sclkbwtem+H+oHxfoaYFcn+TLF7M4MmgX7aNDRNE5KqMrVEE1xNEdekBP6Nm5dx6dF+d12jrnzGZ20I9y3j4B+HSYLA==

�b,~vb, Tb, xe

Initial conditions

• Observations are consistent with adiabatic initial conditions: all species have equal relative fluctuations in their number density:

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

�NbNb

=�NcNc

=�N⌫N⌫

=�N�N�

• non-relativistic species (CDM and baryons): 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

�N

N=

�⇢

⇢⌘ �

• relativistic species (neutrinos and photons): AAACDnicbVBLSwMxGMzWV62vVY9egqXgQcquVvRY9OJJKvQF3bVk02wbmt2EJCuUpb/Ai3/FiwdFvHr25r8xbfdQWycEhpnvI5kJBKNKO86PlVtZXVvfyG8WtrZ3dvfs/YOm4onEpIE547IdIEUYjUlDU81IW0iCooCRVjC8mfitRyIV5XFdjwTxI9SPaUgx0kbq2qU76AnJheaw/nB+Cr3ZkQM+p1e6dtEpO1PAZeJmpAgy1Lr2t9fjOIlIrDFDSnVcR2g/RVJTzMi44CWKCISHqE86hsYoIspPp3HGsGSUHgy5NDfWcKrOb6QoUmoUBWYyQnqgFr2J+J/XSXR45ac0FokmMZ49FCYMmpCTbmCPSoI1GxmCsKTmrxAPkERYmwYLpgR3MfIyaZ6V3Yuyc18pVq+zOvLgCByDE+CCS1AFt6AGGgCDJ/AC3sC79Wy9Wh/W52w0Z2U7h+APrK9f5PCaHw==

N / T 3, ⇢ / T 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

) �NN

=3

4

�⇢

⇢=

3

4�

• On very large scales (>> Hubble radius), peculiar velocities initially vanish

Initial conditions

• Metric perturbations can be decomposed into ``scalar” [φ, ψ], “vector” and “tensor” modes (see Valerie Domcke’s lecture 4)

• “vector” modes decay and are neglected

• we will assume scalar modes (and will briefly touch on tensor modes [i.e. gravitational waves] in lectures 4/5).

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

�N

N

��i= ⇣, �i = i = �

2

3⇣

primordial curvature perturbation

• Initial conditions are only described statistically

Initial conditions• Initial conditions are observed to be consistent with Gaussian

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

h⇣(~k)⇣⇤(~k0)i = (2⇡)3�D(~k0 � ~k)P⇣(k) primordial curvature power spectrumcomoving Fourier wavenumber

• Variance of primordial curvature perturbations: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

h[⇣(~x)]2i =Z

d3k

(2⇡)3P⇣(k) =

Zd ln k

k3

2⇡2P⇣(k)

• Slow-roll inflation predicts quasi scale-invariant power spectrum:(see Valerie Domcke’s lectures)

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

k3

2⇡2P⇣(k) = As(k/k⇤)

ns�1, ns ⇡ 1

temperature

time

last scattering epoch

~400,000 yr

~3000 K

e-

p+ H

Qualitative description of what’s next

time

modern galaxies

proto- galaxies

first stars diffuse hot gas

H

last scattering epoch

time

time

Big BangBi

g Ba

ng

Big Ba

ng

last scattering

“surface”

provide pressure provide containment(impede free streaming)

What are we seeing exactly?

e-

pHe

“baryons”photons

together: ideal fluid with large pressure

before last scattering:

Small initial overdensities generate sound waves in photon-baryon fluid, propagating for ~400,000 years

Superposition of many incoherent sound waves, oscillating for ~400,000 years

±300 μK

0 0.2

0.4

0.6

0.8

1.0

T-+300 μK-300 μK 0 μK

Last ``snapshot” is imprinted on the last scattering surface

PSfragreplacements

-160 160 µK0.41 µK

Planck collaboration

Temperature

Polarization

Mathew Madhavacheril, Perimeter InstituteCourtesy of Mathew

Madhavacheril

CMB Angular power spectrum = variance of temperature fluctuations as a function of angular scale

--

-

-

Temperature x Polarization Polarization

--

-

-

Temperature (error bars x 5)(μK)2

(μK)2


(null)(null)(null)(null)(null)(null)(null)(null)

Planck 2018

Best-fit model(6 parameters)

Temperature x Polarization (μK)2 Polarization (μK)2

Ionization fraction xe

Time since Big Bang (years)(null)(null)(null)(null)

(null)(null)(null)(null) (null)(null)(null)(null)

Temperature (μK)2

He++➞He+

He+➞Heo H+➞Ho

Effect of non-standard xe on CMB power spectra

-

(μK)2


(μK)2Temperature

Polarization

baryons

dark matter


Planck collaboration 2018





⌦bh2 . . . . . . . . . . . . . 0.022383 0.02237 ± 0.00015 0.02229 ± 0.00015 �0.5 0.02233 ± 0.00015⌦ch2 . . . . . . . . . . . . . 0.12011 0.1200 ± 0.0012 0.1197 ± 0.0012 �0.3 0.1198 ± 0.0012100✓MC . . . . . . . . . . . 1.040909 1.04092 ± 0.00031 1.04087 ± 0.00031 �0.2 1.04089 ± 0.00031⌧ . . . . . . . . . . . . . . . . 0.0543 0.0544 ± 0.0073 0.0536+0.0069

�0.0077 �0.1 0.0540 ± 0.0074ln(1010As) . . . . . . . . . 3.0448 3.044 ± 0.014 3.041 ± 0.015 �0.3 3.043 ± 0.014ns . . . . . . . . . . . . . . . 0.96605 0.9649 ± 0.0042 0.9656 ± 0.0042 +0.2 0.9652 ± 0.0042


14





⌦bh2 . . . . . . . . . . . . . 0.022383 0.02237 ± 0.00015 0.02229 ± 0.00015 �0.5 0.02233 ± 0.00015⌦ch2 . . . . . . . . . . . . . 0.12011 0.1200 ± 0.0012 0.1197 ± 0.0012 �0.3 0.1198 ± 0.0012100✓MC . . . . . . . . . . . 1.040909 1.04092 ± 0.00031 1.04087 ± 0.00031 �0.2 1.04089 ± 0.00031⌧ . . . . . . . . . . . . . . . . 0.0543 0.0544 ± 0.0073 0.0536+0.0069

�0.0077 �0.1 0.0540 ± 0.0074ln(1010As) . . . . . . . . . 3.0448 3.044 ± 0.014 3.041 ± 0.015 �0.3 3.043 ± 0.014ns . . . . . . . . . . . . . . . 0.96605 0.9649 ± 0.0042 0.9656 ± 0.0042 +0.2 0.9652 ± 0.0042


14

Plan for the next lectures

• Lecture 2: cosmological recombination

• Lecture 3: derivation of the photon Boltzmann equation

• Lecture 4: solutions of the photon Boltzmann equation

• Lecture 5: introduction to CMB polarization and lensing, and/or cosmological parameter estimation

Documents

Physics of the Cosmic Microwave BackgroundBasic deﬁnitions [note: c = 1]CMB = electromagnetic radiation permeating the Universe.It can be described by its speciﬁc intensityfrequency