Cosmological Tensions: Amplitude of Matter Density Fluctuations

(1)

Cosmological Tensions:

Amplitude of Matter Density Fluctuations

Eiichiro Komatsu

(Max-Planck-Institut für Astrophysik)

“The Extragalactic Distance Scale in the Gaia Era”,

MIAPP, June 12, 2018

(2)

E2E Test of Cosmology

• ^H ⁰ oﬀers an E2E test of the evolution of the cosmological background

• Amplitude of matter density fluctuations oﬀers an E2E test of the evolution of the fluctuations

• Cosmology as an initial-value problem: given the initial

condition given by the CMB, can we reproduce late-time observations?

This talk

!2

(3)

This talk

H

<latexit sha1_base64="OAU+kFZV1vXCswPFH4ylTz6u0NM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeilx4r2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJaPZpqgH9GR5CFn1FjpoTFwB+WKW3UXIOvEy0kFcjQH5a/+MGZphNIwQbXueW5i/Iwqw5nAWamfakwom9AR9iyVNELtZ4tTZ+TCKkMSxsqWNGSh/p7IaKT1NApsZ0TNWK96c/E/r5ea8MbPuExSg5ItF4WpICYm87/JkCtkRkwtoUxxeythY6ooMzadkg3BW315nbSvqp5b9e6vK/XbPI4inME5XIIHNahDA5rQAgYjeIZXeHOE8+K8Ox/L1oKTz5zCHzifP8D9jW8=</latexit><latexit sha1_base64="OAU+kFZV1vXCswPFH4ylTz6u0NM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeilx4r2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJaPZpqgH9GR5CFn1FjpoTFwB+WKW3UXIOvEy0kFcjQH5a/+MGZphNIwQbXueW5i/Iwqw5nAWamfakwom9AR9iyVNELtZ4tTZ+TCKkMSxsqWNGSh/p7IaKT1NApsZ0TNWK96c/E/r5ea8MbPuExSg5ItF4WpICYm87/JkCtkRkwtoUxxeythY6ooMzadkg3BW315nbSvqp5b9e6vK/XbPI4inME5XIIHNahDA5rQAgYjeIZXeHOE8+K8Ox/L1oKTz5zCHzifP8D9jW8=</latexit><latexit sha1_base64="OAU+kFZV1vXCswPFH4ylTz6u0NM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeilx4r2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJaPZpqgH9GR5CFn1FjpoTFwB+WKW3UXIOvEy0kFcjQH5a/+MGZphNIwQbXueW5i/Iwqw5nAWamfakwom9AR9iyVNELtZ4tTZ+TCKkMSxsqWNGSh/p7IaKT1NApsZ0TNWK96c/E/r5ea8MbPuExSg5ItF4WpICYm87/JkCtkRkwtoUxxeythY6ooMzadkg3BW315nbSvqp5b9e6vK/XbPI4inME5XIIHNahDA5rQAgYjeIZXeHOE8+K8Ox/L1oKTz5zCHzifP8D9jW8=</latexit><latexit sha1_base64="OAU+kFZV1vXCswPFH4ylTz6u0NM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeilx4r2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJaPZpqgH9GR5CFn1FjpoTFwB+WKW3UXIOvEy0kFcjQH5a/+MGZphNIwQbXueW5i/Iwqw5nAWamfakwom9AR9iyVNELtZ4tTZ+TCKkMSxsqWNGSh/p7IaKT1NApsZ0TNWK96c/E/r5ea8MbPuExSg5ItF4WpICYm87/JkCtkRkwtoUxxeythY6ooMzadkg3BW315nbSvqp5b9e6vK/XbPI4inME5XIIHNahDA5rQAgYjeIZXeHOE8+K8Ox/L1oKTz5zCHzifP8D9jW8=</latexit>

₀

8

<latexit sha1_base64="U3aI1xkBy+T9nxI8X6j9nKGusgY=">AAAB73icbVDLSgNBEOz1GeMr6tHLYBA8hV0RzDHoxWME84BkCb2TSTJkZnadmRXCkp/w4kERr/6ON//GSbIHTSxoKKq66e6KEsGN9f1vb219Y3Nru7BT3N3bPzgsHR03TZxqyho0FrFuR2iY4Io1LLeCtRPNUEaCtaLx7cxvPTFteKwe7CRhocSh4gNO0Tqp3TV8KLFX7ZXKfsWfg6ySICdlyFHvlb66/ZimkilLBRrTCfzEhhlqy6lg02I3NSxBOsYh6ziqUDITZvN7p+TcKX0yiLUrZclc/T2RoTRmIiPXKdGOzLI3E//zOqkdVMOMqyS1TNHFokEqiI3J7HnS55pRKyaOINXc3UroCDVS6yIquhCC5ZdXSfOyEviV4P6qXLvJ4yjAKZzBBQRwDTW4gzo0gIKAZ3iFN+/Re/HevY9F65qXz5zAH3ifP9Bhj84=</latexit><latexit sha1_base64="U3aI1xkBy+T9nxI8X6j9nKGusgY=">AAAB73icbVDLSgNBEOz1GeMr6tHLYBA8hV0RzDHoxWME84BkCb2TSTJkZnadmRXCkp/w4kERr/6ON//GSbIHTSxoKKq66e6KEsGN9f1vb219Y3Nru7BT3N3bPzgsHR03TZxqyho0FrFuR2iY4Io1LLeCtRPNUEaCtaLx7cxvPTFteKwe7CRhocSh4gNO0Tqp3TV8KLFX7ZXKfsWfg6ySICdlyFHvlb66/ZimkilLBRrTCfzEhhlqy6lg02I3NSxBOsYh6ziqUDITZvN7p+TcKX0yiLUrZclc/T2RoTRmIiPXKdGOzLI3E//zOqkdVMOMqyS1TNHFokEqiI3J7HnS55pRKyaOINXc3UroCDVS6yIquhCC5ZdXSfOyEviV4P6qXLvJ4yjAKZzBBQRwDTW4gzo0gIKAZ3iFN+/Re/HevY9F65qXz5zAH3ifP9Bhj84=</latexit><latexit sha1_base64="U3aI1xkBy+T9nxI8X6j9nKGusgY=">AAAB73icbVDLSgNBEOz1GeMr6tHLYBA8hV0RzDHoxWME84BkCb2TSTJkZnadmRXCkp/w4kERr/6ON//GSbIHTSxoKKq66e6KEsGN9f1vb219Y3Nru7BT3N3bPzgsHR03TZxqyho0FrFuR2iY4Io1LLeCtRPNUEaCtaLx7cxvPTFteKwe7CRhocSh4gNO0Tqp3TV8KLFX7ZXKfsWfg6ySICdlyFHvlb66/ZimkilLBRrTCfzEhhlqy6lg02I3NSxBOsYh6ziqUDITZvN7p+TcKX0yiLUrZclc/T2RoTRmIiPXKdGOzLI3E//zOqkdVMOMqyS1TNHFokEqiI3J7HnS55pRKyaOINXc3UroCDVS6yIquhCC5ZdXSfOyEviV4P6qXLvJ4yjAKZzBBQRwDTW4gzo0gIKAZ3iFN+/Re/HevY9F65qXz5zAH3ifP9Bhj84=</latexit><latexit sha1_base64="U3aI1xkBy+T9nxI8X6j9nKGusgY=">AAAB73icbVDLSgNBEOz1GeMr6tHLYBA8hV0RzDHoxWME84BkCb2TSTJkZnadmRXCkp/w4kERr/6ON//GSbIHTSxoKKq66e6KEsGN9f1vb219Y3Nru7BT3N3bPzgsHR03TZxqyho0FrFuR2iY4Io1LLeCtRPNUEaCtaLx7cxvPTFteKwe7CRhocSh4gNO0Tqp3TV8KLFX7ZXKfsWfg6ySICdlyFHvlb66/ZimkilLBRrTCfzEhhlqy6lg02I3NSxBOsYh6ziqUDITZvN7p+TcKX0yiLUrZclc/T2RoTRmIiPXKdGOzLI3E//zOqkdVMOMqyS1TNHFokEqiI3J7HnS55pRKyaOINXc3UroCDVS6yIquhCC5ZdXSfOyEviV4P6qXLvJ4yjAKZzBBQRwDTW4gzo0gIKAZ3iFN+/Re/HevY9F65qXz5zAH3ifP9Bhj84=</latexit>

Normalisation for the linear power spectrum of matter density fluctuations at present

A

<latexit sha1_base64="RZwV7gjffrlk8GsoWl2PBBYCIzI=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeqF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0cN3X/XLFrbpzkFXi5aQCORr98ldvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s/mpU3JmlQEJY2VLGjJXf09kNNJ6EgW2M6JmpJe9mfif101NeOVnXCapQckWi8JUEBOT2d9kwBUyIyaWUKa4vZWwEVWUGZtOyYbgLb+8SloXVc+teveXlfpNHkcRTuAUzsGDGtThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifPxvujas=</latexit><latexit sha1_base64="RZwV7gjffrlk8GsoWl2PBBYCIzI=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeqF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0cN3X/XLFrbpzkFXi5aQCORr98ldvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s/mpU3JmlQEJY2VLGjJXf09kNNJ6EgW2M6JmpJe9mfif101NeOVnXCapQckWi8JUEBOT2d9kwBUyIyaWUKa4vZWwEVWUGZtOyYbgLb+8SloXVc+teveXlfpNHkcRTuAUzsGDGtThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifPxvujas=</latexit><latexit sha1_base64="RZwV7gjffrlk8GsoWl2PBBYCIzI=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeqF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0cN3X/XLFrbpzkFXi5aQCORr98ldvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s/mpU3JmlQEJY2VLGjJXf09kNNJ6EgW2M6JmpJe9mfif101NeOVnXCapQckWi8JUEBOT2d9kwBUyIyaWUKa4vZWwEVWUGZtOyYbgLb+8SloXVc+teveXlfpNHkcRTuAUzsGDGtThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifPxvujas=</latexit><latexit sha1_base64="RZwV7gjffrlk8GsoWl2PBBYCIzI=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeqF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0cN3X/XLFrbpzkFXi5aQCORr98ldvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s/mpU3JmlQEJY2VLGjJXf09kNNJ6EgW2M6JmpJe9mfif101NeOVnXCapQckWi8JUEBOT2d9kwBUyIyaWUKa4vZWwEVWUGZtOyYbgLb+8SloXVc+teveXlfpNHkcRTuAUzsGDGtThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifPxvujas=</latexit>

_s

8 Primordial amplitude constrained by the CMB

Present-day amplitude constrained by late-time

observations

(4)

matter-density perturbation

we observe at a given z primordial perturbation in curvature we observe by CMB

“transfer function” fixed by the CMB physics

(⌦

<latexit sha1_base64="mfbxcE2qnL9TCsYGkZyDtV5HGrM=">AAACFHicbVDLSgMxFM3UV62vqks3wSJUlDJTBF0W3bjSCvYBnTpk0jttaDIzJBmhDP0IN/6KGxeKuHXhzr8xbWeh1QOBc8+5l5t7/JgzpW37y8otLC4tr+RXC2vrG5tbxe2dpooSSaFBIx7Jtk8UcBZCQzPNoR1LIMLn0PKHFxO/dQ9SsSi81aMYuoL0QxYwSrSRvOJR2b0W0CeewIO76rGLs9LPyisvdaXAEATjQ69Ysiv2FPgvcTJSQhnqXvHT7UU0ERBqyolSHceOdTclUjPKYVxwEwUxoUPSh46hIRGguun0qDE+MEoPB5E0L9R4qv6cSIlQaiR80ymIHqh5byL+53USHZx1UxbGiYaQzhYFCcc6wpOEcI9JoJqPDCFUMvNXTAdEEqpNjgUTgjN/8l/SrFYcu+LcnJRq51kcebSH9lEZOegU1dAlqqMGougBPaEX9Go9Ws/Wm/U+a81Z2cwu+gXr4xtEM5xy</latexit><latexit sha1_base64="mfbxcE2qnL9TCsYGkZyDtV5HGrM=">AAACFHicbVDLSgMxFM3UV62vqks3wSJUlDJTBF0W3bjSCvYBnTpk0jttaDIzJBmhDP0IN/6KGxeKuHXhzr8xbWeh1QOBc8+5l5t7/JgzpW37y8otLC4tr+RXC2vrG5tbxe2dpooSSaFBIx7Jtk8UcBZCQzPNoR1LIMLn0PKHFxO/dQ9SsSi81aMYuoL0QxYwSrSRvOJR2b0W0CeewIO76rGLs9LPyisvdaXAEATjQ69Ysiv2FPgvcTJSQhnqXvHT7UU0ERBqyolSHceOdTclUjPKYVxwEwUxoUPSh46hIRGguun0qDE+MEoPB5E0L9R4qv6cSIlQaiR80ymIHqh5byL+53USHZx1UxbGiYaQzhYFCcc6wpOEcI9JoJqPDCFUMvNXTAdEEqpNjgUTgjN/8l/SrFYcu+LcnJRq51kcebSH9lEZOegU1dAlqqMGougBPaEX9Go9Ws/Wm/U+a81Z2cwu+gXr4xtEM5xy</latexit><latexit sha1_base64="mfbxcE2qnL9TCsYGkZyDtV5HGrM=">AAACFHicbVDLSgMxFM3UV62vqks3wSJUlDJTBF0W3bjSCvYBnTpk0jttaDIzJBmhDP0IN/6KGxeKuHXhzr8xbWeh1QOBc8+5l5t7/JgzpW37y8otLC4tr+RXC2vrG5tbxe2dpooSSaFBIx7Jtk8UcBZCQzPNoR1LIMLn0PKHFxO/dQ9SsSi81aMYuoL0QxYwSrSRvOJR2b0W0CeewIO76rGLs9LPyisvdaXAEATjQ69Ysiv2FPgvcTJSQhnqXvHT7UU0ERBqyolSHceOdTclUjPKYVxwEwUxoUPSh46hIRGguun0qDE+MEoPB5E0L9R4qv6cSIlQaiR80ymIHqh5byL+53USHZx1UxbGiYaQzhYFCcc6wpOEcI9JoJqPDCFUMvNXTAdEEqpNjgUTgjN/8l/SrFYcu+LcnJRq51kcebSH9lEZOegU1dAlqqMGougBPaEX9Go9Ws/Wm/U+a81Z2cwu+gXr4xtEM5xy</latexit><latexit sha1_base64="mfbxcE2qnL9TCsYGkZyDtV5HGrM=">AAACFHicbVDLSgMxFM3UV62vqks3wSJUlDJTBF0W3bjSCvYBnTpk0jttaDIzJBmhDP0IN/6KGxeKuHXhzr8xbWeh1QOBc8+5l5t7/JgzpW37y8otLC4tr+RXC2vrG5tbxe2dpooSSaFBIx7Jtk8UcBZCQzPNoR1LIMLn0PKHFxO/dQ9SsSi81aMYuoL0QxYwSrSRvOJR2b0W0CeewIO76rGLs9LPyisvdaXAEATjQ69Ysiv2FPgvcTJSQhnqXvHT7UU0ERBqyolSHceOdTclUjPKYVxwEwUxoUPSh46hIRGguun0qDE+MEoPB5E0L9R4qv6cSIlQaiR80ymIHqh5byL+53USHZx1UxbGiYaQzhYFCcc6wpOEcI9JoJqPDCFUMvNXTAdEEqpNjgUTgjN/8l/SrFYcu+LcnJRq51kcebSH9lEZOegU1dAlqqMGougBPaEX9Go9Ws/Wm/U+a81Z2cwu+gXr4xtEM5xy</latexit>

_m h ² , ⌦ _b h ² , N _e↵ )

Poisson equation

!4

(5)

matter-density perturbation

we observe at a given z primordial perturbation in curvature we observe by CMB

“transfer function” fixed by the CMB physics

“linear growth factor”, relating the CMB amplitude to the late-time matter fluctuation amplitude

(⌦

_m h ² , ⌦ _b h ² , N _e↵ )

Poisson equation

!5

(6)

matter-density perturbation

we observe at a given z primordial perturbation in curvature we observe by CMB

“transfer function” fixed by the CMB physics

d ² g

d ln(1 + z ) ²

 5

2 + 1

2 (⌦ _k (z ) 3w (z )⌦ _de (z )) dg

d ln(1 + z ) +



2⌦ _k (z ) + 3

2 (1 w(z ))⌦ _de (z ) g (z ) = 0

D (z ) / g (z ) 1 + z

<latexit sha1_base64="P0dv04pZLdi3BgmeGeyvhiRaAzQ=">AAACBXicbVDLSgMxFM3UV62vUZe6CBahIpQZEXRZ1IXLCvYBnaFk0kwbmkmGJCO0w2zc+CtuXCji1n9w59+YtrPQ1gOBwzn3cnNOEDOqtON8W4Wl5ZXVteJ6aWNza3vH3t1rKpFITBpYMCHbAVKEUU4ammpG2rEkKAoYaQXD64nfeiBSUcHv9SgmfoT6nIYUI22krn14UxmfeLEUsRbQCyXCad8oWeqejrOuXXaqzhRwkbg5KYMc9a795fUETiLCNWZIqY7rxNpPkdQUM5KVvESRGOEh6pOOoRxFRPnpNEUGj43Sg6GQ5nENp+rvjRRFSo2iwExGSA/UvDcR//M6iQ4v/ZTyONGE49mhMGHQJJ5UAntUEqzZyBCEJTV/hXiATBXaFFcyJbjzkRdJ86zqOlX37rxcu8rrKIIDcAQqwAUXoAZuQR00AAaP4Bm8gjfryXqx3q2P2WjBynf2wR9Ynz9umpfi</latexit><latexit sha1_base64="P0dv04pZLdi3BgmeGeyvhiRaAzQ=">AAACBXicbVDLSgMxFM3UV62vUZe6CBahIpQZEXRZ1IXLCvYBnaFk0kwbmkmGJCO0w2zc+CtuXCji1n9w59+YtrPQ1gOBwzn3cnNOEDOqtON8W4Wl5ZXVteJ6aWNza3vH3t1rKpFITBpYMCHbAVKEUU4ammpG2rEkKAoYaQXD64nfeiBSUcHv9SgmfoT6nIYUI22krn14UxmfeLEUsRbQCyXCad8oWeqejrOuXXaqzhRwkbg5KYMc9a795fUETiLCNWZIqY7rxNpPkdQUM5KVvESRGOEh6pOOoRxFRPnpNEUGj43Sg6GQ5nENp+rvjRRFSo2iwExGSA/UvDcR//M6iQ4v/ZTyONGE49mhMGHQJJ5UAntUEqzZyBCEJTV/hXiATBXaFFcyJbjzkRdJ86zqOlX37rxcu8rrKIIDcAQqwAUXoAZuQR00AAaP4Bm8gjfryXqx3q2P2WjBynf2wR9Ynz9umpfi</latexit><latexit sha1_base64="P0dv04pZLdi3BgmeGeyvhiRaAzQ=">AAACBXicbVDLSgMxFM3UV62vUZe6CBahIpQZEXRZ1IXLCvYBnaFk0kwbmkmGJCO0w2zc+CtuXCji1n9w59+YtrPQ1gOBwzn3cnNOEDOqtON8W4Wl5ZXVteJ6aWNza3vH3t1rKpFITBpYMCHbAVKEUU4ammpG2rEkKAoYaQXD64nfeiBSUcHv9SgmfoT6nIYUI22krn14UxmfeLEUsRbQCyXCad8oWeqejrOuXXaqzhRwkbg5KYMc9a795fUETiLCNWZIqY7rxNpPkdQUM5KVvESRGOEh6pOOoRxFRPnpNEUGj43Sg6GQ5nENp+rvjRRFSo2iwExGSA/UvDcR//M6iQ4v/ZTyONGE49mhMGHQJJ5UAntUEqzZyBCEJTV/hXiATBXaFFcyJbjzkRdJ86zqOlX37rxcu8rrKIIDcAQqwAUXoAZuQR00AAaP4Bm8gjfryXqx3q2P2WjBynf2wR9Ynz9umpfi</latexit><latexit sha1_base64="P0dv04pZLdi3BgmeGeyvhiRaAzQ=">AAACBXicbVDLSgMxFM3UV62vUZe6CBahIpQZEXRZ1IXLCvYBnaFk0kwbmkmGJCO0w2zc+CtuXCji1n9w59+YtrPQ1gOBwzn3cnNOEDOqtON8W4Wl5ZXVteJ6aWNza3vH3t1rKpFITBpYMCHbAVKEUU4ammpG2rEkKAoYaQXD64nfeiBSUcHv9SgmfoT6nIYUI22krn14UxmfeLEUsRbQCyXCad8oWeqejrOuXXaqzhRwkbg5KYMc9a795fUETiLCNWZIqY7rxNpPkdQUM5KVvESRGOEh6pOOoRxFRPnpNEUGj43Sg6GQ5nENp+rvjRRFSo2iwExGSA/UvDcR//M6iQ4v/ZTyONGE49mhMGHQJJ5UAntUEqzZyBCEJTV/hXiATBXaFFcyJbjzkRdJ86zqOlX37rxcu8rrKIIDcAQqwAUXoAZuQR00AAaP4Bm8gjfryXqx3q2P2WjBynf2wR9Ynz9umpfi</latexit>

Writing

**the growth factor is given by(*)**

() Strictly speaking, this formula is valid when we ignore massive neutrinos, and the contribution* of dark energy fluctuations to the gravitational potential is negligible compared to matter

“linear growth factor”, relating the CMB amplitude to the late-time matter fluctuation amplitude

(⌦

_m h ² , ⌦ _b h ² , N _e↵ )

Poisson equation

!6

(7)

A s

0.05/Mpc

[linear matter power]

!7

(8)

A s

[linear matter power]

A s

where

0.05/Mpc

(9)

WMAP5

!9

(10)

• The growth is normalised to unity at high redshift, g(z) -> 1 for z >> 1

• w>–1: For a given Ω de today, DE

becomes dominant earlier for w>–1, giving earlier/more suppression in

the growth of matter perturbations

0.75 0.8 0.85 0.9 0.95 1

0 1 2 3 4 5 6

Linear growth, g(z)=(1+z)D(z)

Redshift, z

’redshift_g_w1.txt’

’redshift_g_w09.txt’

’redshift_g_w11.txt’

w=–0.9 w=–1.1

⌦ _m = 0.3

⌦ _de = 0.7

10

(11)

WMAP5

BAO and SN help here because they fix H 0 by

the inverse distance ladder, hence Ω m , given Ω m h 2 from CMB

!11

(12)

WMAP5 BAO and SN help here because they fix H 0 by

the inverse distance ladder, hence Ω m , given Ω m h 2 from CMB

!12

(13)

Optical Depth

• Extra scattering by electrons in a low-redshift Universe damps temperature anisotropy

• ^C ^l ^{-> C} ^l ^exp(–2τ) at l >~ 10

• where τ is the optical depth

re-ionisation _!13

(14)

!14

(15)

!15

(16)

• Since the power spectrum is uniformly suppressed by

exp(–2τ) at l>~10, we cannot determine A s independently of τ.

• Namely, what we constrain is the combination:

exp(–2τ)A s

Important consequence of the optical depth

• Breaking this degeneracy requires an independent determination of the optical depth. This requires

polarisation of the CMB, but the accuracy ^{is not}

suﬃcient yet

!16

(17)

!17

(18)

South Pole Telescope Collaboration

T

<latexit sha1_base64="nicjjCiJEDA8X6LtwDfr+lwihmU=">AAACF3icbVBNS8NAEN34bf2KevSyWARFCIkIehGKXjwqtFVoQplst+3SzSbsToQS+i+8+Fe8eFDEq978N27bHPx6MPB4b4aZeXEmhUHf/3RmZufmFxaXlisrq2vrG+7mVtOkuWa8wVKZ6tsYDJdC8QYKlPw20xySWPKbeHAx9m/uuDYiVXUcZjxKoKdEVzBAK7Vdr74f9gGpOjirt4tQJ1Qo1EIZwUalcxgqiCWEWV8ctN2q7/kT0L8kKEmVlLhqux9hJ2V5whUyCca0Aj/DqACNgkk+qoS54RmwAfR4y1IFCTdRMflrRPes0qHdVNtSSCfq94kCEmOGSWw7E8C++e2Nxf+8Vo7d06gQKsuRKzZd1M0lxZSOQ6IdoTlDObQEmBb2Vsr6oIGhjbJiQwh+v/yXNI+8wPeC6+Nq7byMY4nskF2yTwJyQmrkklyRBmHknjySZ/LiPDhPzqvzNm2dccqZbfIDzvsXmY+e8w==</latexit><latexit sha1_base64="nicjjCiJEDA8X6LtwDfr+lwihmU=">AAACF3icbVBNS8NAEN34bf2KevSyWARFCIkIehGKXjwqtFVoQplst+3SzSbsToQS+i+8+Fe8eFDEq978N27bHPx6MPB4b4aZeXEmhUHf/3RmZufmFxaXlisrq2vrG+7mVtOkuWa8wVKZ6tsYDJdC8QYKlPw20xySWPKbeHAx9m/uuDYiVXUcZjxKoKdEVzBAK7Vdr74f9gGpOjirt4tQJ1Qo1EIZwUalcxgqiCWEWV8ctN2q7/kT0L8kKEmVlLhqux9hJ2V5whUyCca0Aj/DqACNgkk+qoS54RmwAfR4y1IFCTdRMflrRPes0qHdVNtSSCfq94kCEmOGSWw7E8C++e2Nxf+8Vo7d06gQKsuRKzZd1M0lxZSOQ6IdoTlDObQEmBb2Vsr6oIGhjbJiQwh+v/yXNI+8wPeC6+Nq7byMY4nskF2yTwJyQmrkklyRBmHknjySZ/LiPDhPzqvzNm2dccqZbfIDzvsXmY+e8w==</latexit><latexit sha1_base64="nicjjCiJEDA8X6LtwDfr+lwihmU=">AAACF3icbVBNS8NAEN34bf2KevSyWARFCIkIehGKXjwqtFVoQplst+3SzSbsToQS+i+8+Fe8eFDEq978N27bHPx6MPB4b4aZeXEmhUHf/3RmZufmFxaXlisrq2vrG+7mVtOkuWa8wVKZ6tsYDJdC8QYKlPw20xySWPKbeHAx9m/uuDYiVXUcZjxKoKdEVzBAK7Vdr74f9gGpOjirt4tQJ1Qo1EIZwUalcxgqiCWEWV8ctN2q7/kT0L8kKEmVlLhqux9hJ2V5whUyCca0Aj/DqACNgkk+qoS54RmwAfR4y1IFCTdRMflrRPes0qHdVNtSSCfq94kCEmOGSWw7E8C++e2Nxf+8Vo7d06gQKsuRKzZd1M0lxZSOQ6IdoTlDObQEmBb2Vsr6oIGhjbJiQwh+v/yXNI+8wPeC6+Nq7byMY4nskF2yTwJyQmrkklyRBmHknjySZ/LiPDhPzqvzNm2dccqZbfIDzvsXmY+e8w==</latexit><latexit sha1_base64="nicjjCiJEDA8X6LtwDfr+lwihmU=">AAACF3icbVBNS8NAEN34bf2KevSyWARFCIkIehGKXjwqtFVoQplst+3SzSbsToQS+i+8+Fe8eFDEq978N27bHPx6MPB4b4aZeXEmhUHf/3RmZufmFxaXlisrq2vrG+7mVtOkuWa8wVKZ6tsYDJdC8QYKlPw20xySWPKbeHAx9m/uuDYiVXUcZjxKoKdEVzBAK7Vdr74f9gGpOjirt4tQJ1Qo1EIZwUalcxgqiCWEWV8ctN2q7/kT0L8kKEmVlLhqux9hJ2V5whUyCca0Aj/DqACNgkk+qoS54RmwAfR4y1IFCTdRMflrRPes0qHdVNtSSCfq94kCEmOGSWw7E8C++e2Nxf+8Vo7d06gQKsuRKzZd1M0lxZSOQ6IdoTlDObQEmBb2Vsr6oIGhjbJiQwh+v/yXNI+8wPeC6+Nq7byMY4nskF2yTwJyQmrkklyRBmHknjySZ/LiPDhPzqvzNm2dccqZbfIDzvsXmY+e8w==</latexit>

(ˆ n) = T _intrinsic (ˆ n + r )

“lens potential”

map of “convergence”

!18

(19)

South Pole Telescope Collaboration

T

<latexit sha1_base64="nicjjCiJEDA8X6LtwDfr+lwihmU=">AAACF3icbVBNS8NAEN34bf2KevSyWARFCIkIehGKXjwqtFVoQplst+3SzSbsToQS+i+8+Fe8eFDEq978N27bHPx6MPB4b4aZeXEmhUHf/3RmZufmFxaXlisrq2vrG+7mVtOkuWa8wVKZ6tsYDJdC8QYKlPw20xySWPKbeHAx9m/uuDYiVXUcZjxKoKdEVzBAK7Vdr74f9gGpOjirt4tQJ1Qo1EIZwUalcxgqiCWEWV8ctN2q7/kT0L8kKEmVlLhqux9hJ2V5whUyCca0Aj/DqACNgkk+qoS54RmwAfR4y1IFCTdRMflrRPes0qHdVNtSSCfq94kCEmOGSWw7E8C++e2Nxf+8Vo7d06gQKsuRKzZd1M0lxZSOQ6IdoTlDObQEmBb2Vsr6oIGhjbJiQwh+v/yXNI+8wPeC6+Nq7byMY4nskF2yTwJyQmrkklyRBmHknjySZ/LiPDhPzqvzNm2dccqZbfIDzvsXmY+e8w==</latexit><latexit sha1_base64="nicjjCiJEDA8X6LtwDfr+lwihmU=">AAACF3icbVBNS8NAEN34bf2KevSyWARFCIkIehGKXjwqtFVoQplst+3SzSbsToQS+i+8+Fe8eFDEq978N27bHPx6MPB4b4aZeXEmhUHf/3RmZufmFxaXlisrq2vrG+7mVtOkuWa8wVKZ6tsYDJdC8QYKlPw20xySWPKbeHAx9m/uuDYiVXUcZjxKoKdEVzBAK7Vdr74f9gGpOjirt4tQJ1Qo1EIZwUalcxgqiCWEWV8ctN2q7/kT0L8kKEmVlLhqux9hJ2V5whUyCca0Aj/DqACNgkk+qoS54RmwAfR4y1IFCTdRMflrRPes0qHdVNtSSCfq94kCEmOGSWw7E8C++e2Nxf+8Vo7d06gQKsuRKzZd1M0lxZSOQ6IdoTlDObQEmBb2Vsr6oIGhjbJiQwh+v/yXNI+8wPeC6+Nq7byMY4nskF2yTwJyQmrkklyRBmHknjySZ/LiPDhPzqvzNm2dccqZbfIDzvsXmY+e8w==</latexit><latexit sha1_base64="nicjjCiJEDA8X6LtwDfr+lwihmU=">AAACF3icbVBNS8NAEN34bf2KevSyWARFCIkIehGKXjwqtFVoQplst+3SzSbsToQS+i+8+Fe8eFDEq978N27bHPx6MPB4b4aZeXEmhUHf/3RmZufmFxaXlisrq2vrG+7mVtOkuWa8wVKZ6tsYDJdC8QYKlPw20xySWPKbeHAx9m/uuDYiVXUcZjxKoKdEVzBAK7Vdr74f9gGpOjirt4tQJ1Qo1EIZwUalcxgqiCWEWV8ctN2q7/kT0L8kKEmVlLhqux9hJ2V5whUyCca0Aj/DqACNgkk+qoS54RmwAfR4y1IFCTdRMflrRPes0qHdVNtSSCfq94kCEmOGSWw7E8C++e2Nxf+8Vo7d06gQKsuRKzZd1M0lxZSOQ6IdoTlDObQEmBb2Vsr6oIGhjbJiQwh+v/yXNI+8wPeC6+Nq7byMY4nskF2yTwJyQmrkklyRBmHknjySZ/LiPDhPzqvzNm2dccqZbfIDzvsXmY+e8w==</latexit><latexit sha1_base64="nicjjCiJEDA8X6LtwDfr+lwihmU=">AAACF3icbVBNS8NAEN34bf2KevSyWARFCIkIehGKXjwqtFVoQplst+3SzSbsToQS+i+8+Fe8eFDEq978N27bHPx6MPB4b4aZeXEmhUHf/3RmZufmFxaXlisrq2vrG+7mVtOkuWa8wVKZ6tsYDJdC8QYKlPw20xySWPKbeHAx9m/uuDYiVXUcZjxKoKdEVzBAK7Vdr74f9gGpOjirt4tQJ1Qo1EIZwUalcxgqiCWEWV8ctN2q7/kT0L8kKEmVlLhqux9hJ2V5whUyCca0Aj/DqACNgkk+qoS54RmwAfR4y1IFCTdRMflrRPes0qHdVNtSSCfq94kCEmOGSWw7E8C++e2Nxf+8Vo7d06gQKsuRKzZd1M0lxZSOQ6IdoTlDObQEmBb2Vsr6oIGhjbJiQwh+v/yXNI+8wPeC6+Nq7byMY4nskF2yTwJyQmrkklyRBmHknjySZ/LiPDhPzqvzNm2dccqZbfIDzvsXmY+e8w==</latexit>

(ˆ n) = T _intrinsic (ˆ n + r )

“lens potential”

!19

(20)

+CMB Lensing Planck

[100 Myr]

Cosmological Parameters Derived from the Power Spectrum

(21)

Reported tensions:

σ 8 seems low compared to σ 8 =0.83?

• ^σ ⁸ from the peculiar velocity of galaxies as measured by the so-called “redshift space distortion” seems low

• ^σ ⁸ from (non-CMB) gravitational lensing of galaxy images by the large-scale structure seems low

• ^σ ⁸ from the number count of clusters of galaxies seems low

• ^σ ⁸ from the gas pressure fluctuations in the Universe seems low

Each tension may not be significant enough,

but they all go in the same direction

(22)

Redshift Space Distortion

Figure from Shi et al., arXiv:1712.04163

f ⌘ d ln D

d ln(1 + z )

<latexit sha1_base64="bjWxwY7eOZYpmkowzFP3SCi//BM=">AAACCnicbVDLSsNAFJ3UV62vqEs3o0WoiCURQZdFXbisYB/QhDKZTNqhk0mcmRRqyNqNv+LGhSJu/QJ3/o3TNgttPXC5h3PuZeYeL2ZUKsv6NgoLi0vLK8XV0tr6xuaWub3TlFEiMGngiEWi7SFJGOWkoahipB0LgkKPkZY3uBr7rSERkkb8To1i4oaox2lAMVJa6pr7gUPuEzqEJ04gEE59h3F4nU16xT5+OMq6ZtmqWhPAeWLnpAxy1Lvml+NHOAkJV5ghKTu2FSs3RUJRzEhWchJJYoQHqEc6mnIUEummk1MyeKgVHwaR0MUVnKi/N1IUSjkKPT0ZItWXs95Y/M/rJCq4cFPK40QRjqcPBQmDKoLjXKBPBcGKjTRBWFD9V4j7SEeidHolHYI9e/I8aZ5Wbatq356Va5d5HEWwBw5ABdjgHNTADaiDBsDgETyDV/BmPBkvxrvxMR0tGPnOLvgD4/MHEAKZ1Q==</latexit><latexit sha1_base64="bjWxwY7eOZYpmkowzFP3SCi//BM=">AAACCnicbVDLSsNAFJ3UV62vqEs3o0WoiCURQZdFXbisYB/QhDKZTNqhk0mcmRRqyNqNv+LGhSJu/QJ3/o3TNgttPXC5h3PuZeYeL2ZUKsv6NgoLi0vLK8XV0tr6xuaWub3TlFEiMGngiEWi7SFJGOWkoahipB0LgkKPkZY3uBr7rSERkkb8To1i4oaox2lAMVJa6pr7gUPuEzqEJ04gEE59h3F4nU16xT5+OMq6ZtmqWhPAeWLnpAxy1Lvml+NHOAkJV5ghKTu2FSs3RUJRzEhWchJJYoQHqEc6mnIUEummk1MyeKgVHwaR0MUVnKi/N1IUSjkKPT0ZItWXs95Y/M/rJCq4cFPK40QRjqcPBQmDKoLjXKBPBcGKjTRBWFD9V4j7SEeidHolHYI9e/I8aZ5Wbatq356Va5d5HEWwBw5ABdjgHNTADaiDBsDgETyDV/BmPBkvxrvxMR0tGPnOLvgD4/MHEAKZ1Q==</latexit><latexit sha1_base64="bjWxwY7eOZYpmkowzFP3SCi//BM=">AAACCnicbVDLSsNAFJ3UV62vqEs3o0WoiCURQZdFXbisYB/QhDKZTNqhk0mcmRRqyNqNv+LGhSJu/QJ3/o3TNgttPXC5h3PuZeYeL2ZUKsv6NgoLi0vLK8XV0tr6xuaWub3TlFEiMGngiEWi7SFJGOWkoahipB0LgkKPkZY3uBr7rSERkkb8To1i4oaox2lAMVJa6pr7gUPuEzqEJ04gEE59h3F4nU16xT5+OMq6ZtmqWhPAeWLnpAxy1Lvml+NHOAkJV5ghKTu2FSs3RUJRzEhWchJJYoQHqEc6mnIUEummk1MyeKgVHwaR0MUVnKi/N1IUSjkKPT0ZItWXs95Y/M/rJCq4cFPK40QRjqcPBQmDKoLjXKBPBcGKjTRBWFD9V4j7SEeidHolHYI9e/I8aZ5Wbatq356Va5d5HEWwBw5ABdjgHNTADaiDBsDgETyDV/BmPBkvxrvxMR0tGPnOLvgD4/MHEAKZ1Q==</latexit><latexit sha1_base64="bjWxwY7eOZYpmkowzFP3SCi//BM=">AAACCnicbVDLSsNAFJ3UV62vqEs3o0WoiCURQZdFXbisYB/QhDKZTNqhk0mcmRRqyNqNv+LGhSJu/QJ3/o3TNgttPXC5h3PuZeYeL2ZUKsv6NgoLi0vLK8XV0tr6xuaWub3TlFEiMGngiEWi7SFJGOWkoahipB0LgkKPkZY3uBr7rSERkkb8To1i4oaox2lAMVJa6pr7gUPuEzqEJ04gEE59h3F4nU16xT5+OMq6ZtmqWhPAeWLnpAxy1Lvml+NHOAkJV5ghKTu2FSs3RUJRzEhWchJJYoQHqEc6mnIUEummk1MyeKgVHwaR0MUVnKi/N1IUSjkKPT0ZItWXs95Y/M/rJCq4cFPK40QRjqcPBQmDKoLjXKBPBcGKjTRBWFD9V4j7SEeidHolHYI9e/I8aZ5Wbatq356Va5d5HEWwBw5ABdjgHNTADaiDBsDgETyDV/BmPBkvxrvxMR0tGPnOLvgD4/MHEAKZ1Q==</latexit>

f

<latexit sha1_base64="VVirdkQNRVfmCZcsHAGPVFpcKK0=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFqJsyI4JdFt24rGAv0A5DJs20oUlmSDJCW/okblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsbm1vbNb2CvuHxweldzjk5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehuXm8/UaVZIh/NOKWBwAPJYkawsVboluLK5LKn2UDgsGYxdMte1VsIrYOfQxlyNUL3q9dPSCaoNIRjrbu+l5pgipVhhNNZsZdpmmIywgPatSixoDqYLg6foQvr9FGcKPukQQv398QUC63HIrKdApuhXq3Nzf9q3czEtWDKZJoZKslyUZxxZBI0TwH1maLE8LEFTBSztyIyxAoTY7Mq2hD81S+vQ+uq6ntV/+G6XL/N4yjAGZxDBXy4gTrcQwOaQCCDZ3iFN2fivDjvzseydcPJZ07hj5zPH2YGkkE=</latexit><latexit sha1_base64="VVirdkQNRVfmCZcsHAGPVFpcKK0=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFqJsyI4JdFt24rGAv0A5DJs20oUlmSDJCW/okblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsbm1vbNb2CvuHxweldzjk5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehuXm8/UaVZIh/NOKWBwAPJYkawsVboluLK5LKn2UDgsGYxdMte1VsIrYOfQxlyNUL3q9dPSCaoNIRjrbu+l5pgipVhhNNZsZdpmmIywgPatSixoDqYLg6foQvr9FGcKPukQQv398QUC63HIrKdApuhXq3Nzf9q3czEtWDKZJoZKslyUZxxZBI0TwH1maLE8LEFTBSztyIyxAoTY7Mq2hD81S+vQ+uq6ntV/+G6XL/N4yjAGZxDBXy4gTrcQwOaQCCDZ3iFN2fivDjvzseydcPJZ07hj5zPH2YGkkE=</latexit><latexit sha1_base64="VVirdkQNRVfmCZcsHAGPVFpcKK0=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFqJsyI4JdFt24rGAv0A5DJs20oUlmSDJCW/okblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsbm1vbNb2CvuHxweldzjk5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehuXm8/UaVZIh/NOKWBwAPJYkawsVboluLK5LKn2UDgsGYxdMte1VsIrYOfQxlyNUL3q9dPSCaoNIRjrbu+l5pgipVhhNNZsZdpmmIywgPatSixoDqYLg6foQvr9FGcKPukQQv398QUC63HIrKdApuhXq3Nzf9q3czEtWDKZJoZKslyUZxxZBI0TwH1maLE8LEFTBSztyIyxAoTY7Mq2hD81S+vQ+uq6ntV/+G6XL/N4yjAGZxDBXy4gTrcQwOaQCCDZ3iFN2fivDjvzseydcPJZ07hj5zPH2YGkkE=</latexit><latexit sha1_base64="VVirdkQNRVfmCZcsHAGPVFpcKK0=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFqJsyI4JdFt24rGAv0A5DJs20oUlmSDJCW/okblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsbm1vbNb2CvuHxweldzjk5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehuXm8/UaVZIh/NOKWBwAPJYkawsVboluLK5LKn2UDgsGYxdMte1VsIrYOfQxlyNUL3q9dPSCaoNIRjrbu+l5pgipVhhNNZsZdpmmIywgPatSixoDqYLg6foQvr9FGcKPukQQv398QUC63HIrKdApuhXq3Nzf9q3czEtWDKZJoZKslyUZxxZBI0TwH1maLE8LEFTBSztyIyxAoTY7Mq2hD81S+vQ+uq6ntV/+G6XL/N4yjAGZxDBXy4gTrcQwOaQCCDZ3iFN2fivDjvzseydcPJZ07hj5zPH2YGkkE=</latexit>

(z ) ₈ (z )

where

!22

(23)

Cosmological Gravitational Lensing (Cosmic Shear)

Dark Energy Survey Collaboration

!23

(24)

Cosmological Gravitational Lensing (Cosmic Shear)

Dark Energy Survey Collaboration

with neutrino mass varied

⌦

<latexit sha1_base64="NTjbw/24frSZUz6FLMeT1u1sq58=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoY2cEEwPJEfY2c8mS3b1zd08IIX/CxkIRW/+Onf/GTXKFJj4YeLw3w8y8KBXcWN//9gorq2vrG8XN0tb2zu5eef+gaZJMM2ywRCS6FVGDgitsWG4FtlKNVEYCH6Lh9dR/eEJteKLu7SjFUNK+4jFn1Dqp1bmV2Kdd2S1X/Ko/A1kmQU4qkKPeLX91egnLJCrLBDWmHfipDcdUW84ETkqdzGBK2ZD2se2oohJNOJ7dOyEnTumRONGulCUz9ffEmEpjRjJynZLagVn0puJ/Xjuz8WU45irNLCo2XxRngtiETJ8nPa6RWTFyhDLN3a2EDaimzLqISi6EYPHlZdI8qwZ+Nbg7r9Su8jiKcATHcAoBXEANbqAODWAg4Ble4c179F68d+9j3lrw8plD+APv8wfjN4/b</latexit><latexit sha1_base64="NTjbw/24frSZUz6FLMeT1u1sq58=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoY2cEEwPJEfY2c8mS3b1zd08IIX/CxkIRW/+Onf/GTXKFJj4YeLw3w8y8KBXcWN//9gorq2vrG8XN0tb2zu5eef+gaZJMM2ywRCS6FVGDgitsWG4FtlKNVEYCH6Lh9dR/eEJteKLu7SjFUNK+4jFn1Dqp1bmV2Kdd2S1X/Ko/A1kmQU4qkKPeLX91egnLJCrLBDWmHfipDcdUW84ETkqdzGBK2ZD2se2oohJNOJ7dOyEnTumRONGulCUz9ffEmEpjRjJynZLagVn0puJ/Xjuz8WU45irNLCo2XxRngtiETJ8nPa6RWTFyhDLN3a2EDaimzLqISi6EYPHlZdI8qwZ+Nbg7r9Su8jiKcATHcAoBXEANbqAODWAg4Ble4c179F68d+9j3lrw8plD+APv8wfjN4/b</latexit><latexit sha1_base64="NTjbw/24frSZUz6FLMeT1u1sq58=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoY2cEEwPJEfY2c8mS3b1zd08IIX/CxkIRW/+Onf/GTXKFJj4YeLw3w8y8KBXcWN//9gorq2vrG8XN0tb2zu5eef+gaZJMM2ywRCS6FVGDgitsWG4FtlKNVEYCH6Lh9dR/eEJteKLu7SjFUNK+4jFn1Dqp1bmV2Kdd2S1X/Ko/A1kmQU4qkKPeLX91egnLJCrLBDWmHfipDcdUW84ETkqdzGBK2ZD2se2oohJNOJ7dOyEnTumRONGulCUz9ffEmEpjRjJynZLagVn0puJ/Xjuz8WU45irNLCo2XxRngtiETJ8nPa6RWTFyhDLN3a2EDaimzLqISi6EYPHlZdI8qwZ+Nbg7r9Su8jiKcATHcAoBXEANbqAODWAg4Ble4c179F68d+9j3lrw8plD+APv8wfjN4/b</latexit><latexit sha1_base64="NTjbw/24frSZUz6FLMeT1u1sq58=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoY2cEEwPJEfY2c8mS3b1zd08IIX/CxkIRW/+Onf/GTXKFJj4YeLw3w8y8KBXcWN//9gorq2vrG8XN0tb2zu5eef+gaZJMM2ywRCS6FVGDgitsWG4FtlKNVEYCH6Lh9dR/eEJteKLu7SjFUNK+4jFn1Dqp1bmV2Kdd2S1X/Ko/A1kmQU4qkKPeLX91egnLJCrLBDWmHfipDcdUW84ETkqdzGBK2ZD2se2oohJNOJ7dOyEnTumRONGulCUz9ffEmEpjRjJynZLagVn0puJ/Xjuz8WU45irNLCo2XxRngtiETJ8nPa6RWTFyhDLN3a2EDaimzLqISi6EYPHlZdI8qwZ+Nbg7r9Su8jiKcATHcAoBXEANbqAODWAg4Ble4c179F68d+9j3lrw8plD+APv8wfjN4/b</latexit>

_m

8 !24

(25)

Where is a galaxy cluster?

Subaru image of RXJ1347-1145 (Medezinski et al. 2010) http://wise-obs.tau.ac.il/~elinor/clusters

25

(26)

Where is a galaxy cluster?

Subaru image of RXJ1347-1145 (Medezinski et al. 2010) http://wise-obs.tau.ac.il/~elinor/clusters

26

(27)

Hubble image of RXJ1347-1145 (Bradac et al. 2008)

Hubble

!27

(28)

Chandra X-ray image of RXJ1347-1145 (Johnson et al. 2012)

Chandra

!28

(29)

Chandra X-ray image of RXJ1347-1145 (Johnson et al. 2012)

ALMA Band-3 Image of the

Sunyaev-Zel’dovich effect at 92 GHz (Kitayama et al. 2016)

ALMA!

5” resolution

!29

1σ=17 μJy/beam

=120 μK CMB

T. Kitayama

(30)

Galaxy Cluster Counts

Optical:

•10 ^2–3 galaxies

•velocity dispersion

•gravitational lensing

X-ray:

•hot gas (10 ^7–8 K)

•spectroscopic T X

•Intensity ~ n e2 L

I _X = Z

dl n ² _e ⇤(T _X )

SZ [microwave]:

•hot gas (10 ^7-8 K)

•electron pressure

•Intensity ~ n e T e L

I _SZ = g _⌫ ^T k _B m _e c ²

Z

dl n _e T _e

(31)

The Biggest Enemy:

Mass Bias

B=M true /M estimated

(People more often use 1–b = 1/B)

!31

(32)

Planck SZ Cluster Count, N(z)

Planck CMB prediction with B=1.25

Planck CMB+SZ best fit with B=1.67

Planck Collaboration XX, arXiv:1303.5080v2

!32

(33)

Galaxy Cluster Counts [SZ]

If the galaxy

cluster mass can be calibrated

accurately

Planck Collaboration 2015

!33

(34)

Galaxy Cluster Counts [SZ]

This plot is for B=M true /M est =1.28

Planck Collaboration 2015

!34

(35)

B=1.45 ±0.15

B=1.28±0.15 B=0.99±0.19

B=M true /M estimated

Planck Collaboration 2015

!35

(36)

B=1.45 ±0.15

B=1.28±0.15 B=0.99±0.19

B=M true /M estimated

Planck Collaboration 2015

(B=1.45) (B=1.28)

(B=0.99)

!36

(37)

Galaxy cluster counts South Pole Telescope Collaboration

X-ray

[from S. Allen’s group]

Sunyaev-Zeldovich by SPT

from SZ+H 0 +BBN prior on baryon density

(38)

!38

(39)

Adding a bunch…

R. A. Burenin, arXiv:1806.03261

!39

(40)

Adding a bunch…

R. A. Burenin, arXiv:1806.03261

No τ constraint

included

!40

(41)

Adding a bunch…

R. A. Burenin, arXiv:1806.03261

!41

(42)

!42

(43)

Full-sky Thermal Pressure Map

North Galactic Pole South Galactic Pole

Planck Collaboration ⁴³

(44)

Simple Interpretation

• Randomly-distributed point sources

= Poisson spectrum = ∑ i (flux i ) ² / 4π

multipole C l [not “l 2 C l ”]

44

(45)

Simple Interpretation

• Extended sources = the power

spectrum reflects intensity profiles

multipole C l [not “l 2 C l ”]

45

(46)

Multipole l(l+1)C l /2 π [ μ K 2 ]

>2x10 ¹⁵ M sun

>10 ¹⁵ M sun

>5x10 ¹⁴ M sun

>5x10 ¹³ M sun

Adding smaller clusters

!46

(47)

Simple Formula

• y l with small l just gives the total thermal pressure, MT ~ M ^5/3

• Heavily weighted by massive clusters

• The mass function, dn/dM, is sensitive to the amplitude of fluctuations, σ 8

C _` =

Z

dz dV dz

Z

dM dn

dM | y _` (M, z ) | ²

2d Fourier transform of pressure

47

(48)

Komatsu & Kitayama (1999)

Degree-scale SZ power spectrum

is less sensitive to astrophysics in cluster cores

1999

!48

(49)

McCarthy et al. (2014)

2014

confirmed by simulations with

varying AGN feedback

!49

(50)

It is very sensitive to the amplitude of fluctuations

Komatsu & Kitayama (1999) Komatsu & Seljak (2002)

1999

!50

(51)

McCarthy et al. (2014)

tension?

Planck13 parameters

2014

!51

(52)

McCarthy et al. (2014)

Planck13 parameters

similar to planck15+CMBlens

2014

!52

(53)

Closer look at the

parameter dependence

Bolliet, Comis, EK, Macias-Perez (2017)

Mass Bias

Hubble σ 8

Ω m w n s

53

(54)

Closer look at the

parameter dependence

Bolliet, Comis, EK, Macias-Perez (2017)

2.6% measurement!

Essentially cosmological

model-independent ₅₄

(55)

Mass Bias in ΛCDM

• Constraining the ΛCDM parameters by the Planck (TT+lowP+lensing) chain, we find

• B = 1.54 ± 0.098 (68%CL; Makiya, Ando & EK, arXiv:1804.05008)

• or, 1–b = 1/B = 0.649 ± 0.041

• Cf: Simulation by Dolag, EK & Sunyaev: B ~ 1.2.

55

(56)

Towards “Tomography”

• Cross-correlating the Compton-Y map with galaxies with known redshifts!

56

(57)

2MASS Redshift Survey

• ~40K galaxies with the median redshift of 0.02

Huchra et al. (2012)

57

(58)

2MASS Redshift Survey

• ~40K galaxies with the median redshift of 0.02

Huchra et al. (2012)

58

(59)

2MRS Auto Power

Dominated by 1-halo term in most of the angular scales => Good for cross-correlation with Compton-Y

Ando, Benoit-Levy & EK (2018)

59

(60)

2MRS Auto Power

Ando, Benoit-Levy & EK (2018)

60

(61)

Cross-power! Makiya, Ando & EK (2018)

61 R. Makiya

(62)

Mass-bias Consistency

We get consistent mass bias from Compton-Y and 2MRS cross. Neat.

[for Planck TT+lowP+lensing]

62 Makiya, Ando & EK (2018)

(63)

Summary

• ^σ ⁸ ^Ω ^mN [where N depends on data sets] seems low compared to the prediction of the ΛCDM model constrained by the Planck CMB data

• No single data set is in a significant tension, but they all go in the same direction

• Cluster counts and the pressure fluctuation data yield consistent results for the mass bias (B) required to

reconcile themselves with the Planck CMB data

• But, the required B seems too high compared to the expectation. We need to understand this!

!63

(64)

SZ Map of RXJ1347–1145

ALMA

on-source integration times 5.6 hours with 7-m array 2.6 hours with 12-m array

Thank you TAC!

(65)

Planck Mass Bias

• The key ingredient of the power spectrum is a profile of thermal pressure of electrons

C _` =

Z

dz dV dz

Z

dM dn

dM | y _` (M, z ) | ²

M ˜ _500c = M _500c,true /B

<latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">AAACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82llZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoIICJdQiBcz3JFS9/sWwXr0DpUUY3OAggqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JJpXXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xxTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit><latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">AAACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82llZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoIICJdQiBcz3JFS9/sWwXr0DpUUY3OAggqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JJpXXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xxTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit><latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">AAACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82llZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoIICJdQiBcz3JFS9/sWwXr0DpUUY3OAggqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JJpXXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xxTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit><latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">AAACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82llZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoIICJdQiBcz3JFS9/sWwXr0DpUUY3OAggqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JJpXXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xxTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit>

65

(66)

Mass Dependence

66 Makiya, Ando & EK (2018)

(67)

Mass Dependence

Cross is sensitive to less massive halos: We can use this to explore the mass bias as a function of mass!

67 Makiya, Ando & EK (2018)

(68)

Planck Mass Bias

• The key ingredient of the power spectrum is a profile of thermal pressure of electrons

C _` =

Z

dz dV dz

Z

dM dn

dM | y _` (M, z ) | ²

M ˜ _500c = M _500c,true /B

<latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">AAACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82llZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoIICJdQiBcz3JFS9/sWwXr0DpUUY3OAggqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JJpXXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xxTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit><latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">AAACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82llZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoIICJdQiBcz3JFS9/sWwXr0DpUUY3OAggqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JJpXXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xxTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit><latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">AAACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82llZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoIICJdQiBcz3JFS9/sWwXr0DpUUY3OAggqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JJpXXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xxTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit><latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">AAACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82llZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoIICJdQiBcz3JFS9/sWwXr0DpUUY3OAggqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JJpXXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xxTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit>

Cosmological Tensions: Amplitude of Matter Density Fluctuations