10. Anisotropies

THE ANISOTROPY SPECTRUM TODAY 245

^ 40

^ 30

c\2 20

10 o +

0

"I I I I I I I I I I I I I I I I r

I I I I I I I I

10 15 20 1

Figure 8.12. Large-scale anisotropies. Hatched boxes show measurements by COBE satellite (Bennett et al, 1996). Curves show the spectra for standard CDM and ACDM (both with n = 1). The tilted model is identical to standard CDM, except n = 0.5. The late time integrated Sachs-Wolfe effect enhances anisotropy on the largest scales in ACDM. Note that here, and in subsequent Ci figures, the root mean square anisotropy is plotted, proportional to C/ . Ci is dimensionless so the units of /iK come from multiplying by the present background temperature, T = 2.73K.

The monopole at recombination (©o + ^)(/c,7y*) free-streams to us today, cre-ating anisotropies on angular scales / ~ fcr/o- This is what we expected back in Figure 8.4, showed to be true in Eq. (8.56), and can now see directly in the top panel of Figure 8.13. There are two interesting features of the quantitative aspect of the free-streaming process. First, note that the "zeroes" in the monopole spec-trum, here at 400, 650, and 970, are smoothed out because many modes contribute to anisotropy on a given angular scale. If only the k — 400/7yo modes contributed to the anisotropy at / = 400, then C400 would really be zero. But many nonzero modes, with wavenumber greater than 400/r]o, contribute. These change the zero to a trough in the Ci spectrum.

The second feature of free-streaming worth noticing is that our initial estimate tof the peak positions is not exactly right. Inhomogeneity on scale k does not show up as anisotropy precisely on angular scale / = kr]0' Rather, There is a notica-ble shift in the top panel, suggesting that a given /c-mode contributes to slightly smaller / than we anticipated. This shift arises from the spherical Bessel function in Eq. (8.56). As indicated in Figure 8.14, the peak in the Bessel function comes not when / = krjo, but rather at slightly smaller values of /. In addition, our initial

Dodelson 2003

Dodelson 2003

Wayne Hu’s CMB tutorial

254 ANISOTROPIES

+

100

3 80 h

60 h

T—I— r—I—I—I—I—I—r T ^ T n„h«-o„hV2

200 400 600 800 1000 1

Figure 8.19. Changes in the anisotropy spectrum as baryon density, matter density, and cos-mological constant vary. Same base model as Figure 8.18.

Each of these parameters induces a small shift in the locations of the peaks and troughs in the spectrum. To understand these shifts, it is important to recall that since inhomogenities on scales k show up at / == kijo in a flat universe, the peaks in a flat universe will show up at Ip c^ /cp7/o — n7rrjo/rs{rj^) (Eq. (8.25), but also see the discussion on page 247 that argues that the actual value of Ip is ~ 25% lower). Figure 8.20 shows this ratio as a function of matter and baryon density. It is more sensitive to the matter density, so the peak spacing increases as the matter density goes down. But there is also a little sensitivity to the baryon density. With the densities fixed, introducing a cosmological constant does not change the sound horizon, but it does slightly affect rjo, so the peaks shift in that cases as well. Baryon density. In addition to the lateral shift in the spectrum due to the change in the sound horizon, changes in the baryon density affect the heights of the peaks as well. We have already touched on the ways in which the anisotropy spectrum depends on the baryon density. The foremost, clearly visible in Figure 8.19, is that odd peaks (first and third in the figure) are higher than the even peaks when the baryon density is large. This is a direct ramification of the lower frequency of oscillations due to the massive baryons. This change is virtually unique, making the baryon density one of the easiest parameters to extract from the CMB. Observations as of 2001 (e.g., Pryke et aL, 2001) pin down Q^/i = 0.022ib0.04, and this constraint will undoubtedly get tighter with data from the Map and Planck satellites. The second change due to U^h^ is that an increased baryon density reduces the diffusion length. Therefore, a larger baryon density means damping moves to smaller angular scales, so the anisotropy spectrum on scales / > 1000 is larger in a high-r^^/i^ model.

Dodelson 2003

Wayne Hu’s CMB tutorial

TCMB≈2.725KΔT/T~10-4

Cosmic Microwave Background

Planck collab. 2013

Planck Collaboration: The Planck mission

2 10 500

1000

2000

3000

4000

5000

6000

D�[µK

2 ]90� 18�

500 1000 1500 2000 2500

Multipole moment, �

1� 0.2� 0.1� 0.07�Angular scale

Fig. 19. The temperature angular power spectrum of the primary CMB from Planck, showing a precise measurement of seven acoustic peaks, thatare well fit by a simple six-parameter⇤CDM theoretical model (the model plotted is the one labelled [Planck+WP+highL] in Planck CollaborationXVI (2013)). The shaded area around the best-fit curve represents cosmic variance, including the sky cut used. The error bars on individual pointsalso include cosmic variance. The horizontal axis is logarithmic up to ` = 50, and linear beyond. The vertical scale is `(`+ 1)Cl/2⇡. The measuredspectrum shown here is exactly the same as the one shown in Fig. 1 of Planck Collaboration XVI (2013), but it has been rebinned to show betterthe low-` region.

2 10 500

1000

2000

3000

4000

5000

6000

D�[µK

2 ]

90� 18�

500 1000 1500 2000

Multipole moment, �

1� 0.2� 0.1�Angular scale

Fig. 20. The temperature angular power spectrum of the CMB, esti-mated from the SMICA Planck map. The model plotted is the one la-belled [Planck+WP+highL] in Planck Collaboration XVI (2013). Theshaded area around the best-fit curve represents cosmic variance, in-cluding the sky cut used. The error bars on individual points do not in-clude cosmic variance. The horizontal axis is logarithmic up to ` = 50,and linear beyond. The vertical scale is `(` + 1)Cl/2⇡. The binningscheme is the same as in Fig. 19.

8.1.1. Main catalogue

The Planck Catalogue of Compact Sources (PCCS, PlanckCollaboration XXVIII (2013)) is a list of compact sources de-

tected by Planck over the entire sky, and which therefore con-tains both Galactic and extragalactic objects. No polarization in-formation is provided for the sources at this time. The PCCSdi↵ers from the ERCSC in its extraction philosophy: more e↵orthas been made on the completeness of the catalogue, without re-ducing notably the reliability of the detected sources, whereasthe ERCSC was built in the spirit of releasing a reliable catalogsuitable for quick follow-up (in particular with the short-livedHerschel telescope). The greater amount of data, di↵erent selec-tion process and the improvements in the calibration and map-making processing (references) help the PCCS to improve theperformance (in depth and numbers) with respect to the previ-ous ERCSC.

The sources were extracted from the 2013 Planck frequencymaps (Sect. 6), which include data acquired over more than twosky coverages. This implies that the flux densities of most ofthe sources are an average of three or more di↵erent observa-tions over a period of 15.5 months. The Mexican Hat Waveletalgorithm (Lopez-Caniego et al. 2006) has been selected as thebaseline method for the production of the PCCS. However, oneadditional methods, MTXF (Gonzalez-Nuevo et al. 2006) wasimplemented in order to support the validation and characteriza-tion of the PCCS.

The source selection for the PCCS is made on the basis ofSignal-to-Noise Ratio (SNR). However, the properties of thebackground in the Planck maps vary substantially depending onfrequency and part of the sky. Up to 217 GHz, the CMB is the

27

Planck XVI 2013

Planck Collaboration: Cosmological parameters

Planck+WP Planck+WP+highL Planck+lensing+WP+highL Planck+WP+highL+BAO

Parameter Best fit 68% limits Best fit 68% limits Best fit 68% limits Best fit 68% limits

⌦bh2 . . . . . . . . . . 0.022032 0.02205 ± 0.00028 0.022069 0.02207 ± 0.00027 0.022199 0.02218 ± 0.00026 0.022161 0.02214 ± 0.00024

⌦ch2 . . . . . . . . . . 0.12038 0.1199 ± 0.0027 0.12025 0.1198 ± 0.0026 0.11847 0.1186 ± 0.0022 0.11889 0.1187 ± 0.0017

100✓MC . . . . . . . . 1.04119 1.04131 ± 0.00063 1.04130 1.04132 ± 0.00063 1.04146 1.04144 ± 0.00061 1.04148 1.04147 ± 0.00056

⌧ . . . . . . . . . . . . 0.0925 0.089+0.012�0.014 0.0927 0.091+0.013

�0.014 0.0943 0.090+0.013�0.014 0.0952 0.092 ± 0.013

ns . . . . . . . . . . . 0.9619 0.9603 ± 0.0073 0.9582 0.9585 ± 0.0070 0.9624 0.9614 ± 0.0063 0.9611 0.9608 ± 0.0054

ln(1010As) . . . . . . . 3.0980 3.089+0.024�0.027 3.0959 3.090 ± 0.025 3.0947 3.087 ± 0.024 3.0973 3.091 ± 0.025

APS100 . . . . . . . . . . 152 171 ± 60 209 212 ± 50 204 213 ± 50 204 212 ± 50

APS143 . . . . . . . . . . 63.3 54 ± 10 72.6 73 ± 8 72.2 72 ± 8 71.8 72.4 ± 8.0

APS217 . . . . . . . . . . 117.0 107+20

�10 59.5 59 ± 10 60.2 58 ± 10 59.4 59 ± 10

ACIB143 . . . . . . . . . . 0.0 < 10.7 3.57 3.24 ± 0.83 3.25 3.24 ± 0.83 3.30 3.25 ± 0.83

ACIB217 . . . . . . . . . . 27.2 29+6

�9 53.9 49.6 ± 5.0 52.3 50.0 ± 4.9 53.0 49.7 ± 5.0

AtSZ143 . . . . . . . . . . 6.80 . . . 5.17 2.54+1.1

�1.9 4.64 2.51+1.2�1.8 4.86 2.54+1.2

�1.8

rPS143⇥217 . . . . . . . . 0.916 > 0.850 0.825 0.823+0.069

�0.077 0.814 0.825 ± 0.071 0.824 0.823 ± 0.070

rCIB143⇥217 . . . . . . . . 0.406 0.42 ± 0.22 1.0000 > 0.930 1.0000 > 0.928 1.0000 > 0.930

�CIB . . . . . . . . . . 0.601 0.53+0.13�0.12 0.674 0.638 ± 0.081 0.656 0.643 ± 0.080 0.667 0.639 ± 0.081

⇠tSZ⇥CIB . . . . . . . . 0.03 . . . 0.000 < 0.409 0.000 < 0.389 0.000 < 0.410

AkSZ . . . . . . . . . . 0.9 . . . 0.89 5.34+2.8�1.9 1.14 4.74+2.6

�2.1 1.58 5.34+2.8�2.0

⌦⇤ . . . . . . . . . . . 0.6817 0.685+0.018�0.016 0.6830 0.685+0.017

�0.016 0.6939 0.693 ± 0.013 0.6914 0.692 ± 0.010

�8 . . . . . . . . . . . 0.8347 0.829 ± 0.012 0.8322 0.828 ± 0.012 0.8271 0.8233 ± 0.0097 0.8288 0.826 ± 0.012

zre . . . . . . . . . . . 11.37 11.1 ± 1.1 11.38 11.1 ± 1.1 11.42 11.1 ± 1.1 11.52 11.3 ± 1.1

H0 . . . . . . . . . . . 67.04 67.3 ± 1.2 67.15 67.3 ± 1.2 67.94 67.9 ± 1.0 67.77 67.80 ± 0.77

Age/Gyr . . . . . . . 13.8242 13.817 ± 0.048 13.8170 13.813 ± 0.047 13.7914 13.794 ± 0.044 13.7965 13.798 ± 0.037

100✓⇤ . . . . . . . . . 1.04136 1.04147 ± 0.00062 1.04146 1.04148 ± 0.00062 1.04161 1.04159 ± 0.00060 1.04163 1.04162 ± 0.00056

rdrag . . . . . . . . . . 147.36 147.49 ± 0.59 147.35 147.47 ± 0.59 147.68 147.67 ± 0.50 147.611 147.68 ± 0.45

Table 5. Best-fit values and 68% confidence limits for the base ⇤CDM model. Beam and calibration parameters, and addi-tional nuisance parameters for “highL” data sets are not listed for brevity but may be found in the Explanatory Supplement(Planck Collaboration ES 2013).

strongly degenerate with the Poisson point source ampli-tude at 100 GHz. This degeneracy is broken when the high-resolution CMB data are added to Planck.

The last two points are demonstrated clearly in Fig. 7, whichshows the residuals of the Planck spectra with respect to thebest-fit cosmology for the Planck+WP analysis compared to thePlanck+WP+highL fits. The addition of high-resolution CMBdata also strongly constrains the net contribution from the kSZand tSZ⇥CIB components (dotted lines), though these compo-nents are degenerate with each other (and tend to cancel).

Although the foreground parameters for the Planck+WP fitscan di↵er substantially from those for Planck+WP+highL, thetotal foreground spectra are rather insensitive to the addition ofthe high-resolution CMB data. For example, for the 217 ⇥ 217spectrum, the di↵erences in the total foreground solution are lessthan 10 µK2 at ` = 2500. The net residuals after subtracting boththe foregrounds and CMB spectrum (shown in the lower panelsof each sub-plot in Fig. 7) are similarly insensitive to the addi-tion of the high-resolution CMB data. The foreground model issu�ciently complex that it has a high “absorptive capacity” toany smoothly-varying frequency-dependent di↵erences betweenspectra (including beam errors).

Table 6. Goodness-of-fit tests for the Planck spectra. The ��2 =�2 � N` is the di↵erence from the mean assuming the model iscorrect, and the last column expresses ��2 in units of the disper-sion

p2N`.

Spectrum `min `max �2 �2/N` ��2/p

2N`

100 ⇥ 100 50 1200 1158 1.01 0.14143 ⇥ 143 50 2000 1883 0.97 �1.09217 ⇥ 217 500 2500 2079 1.04 1.23143 ⇥ 217 500 2500 1930 0.96 �1.13

All 50 2500 2564 1.05 1.62

To quantify the consistency of the model fits shown in Fig. 7for Planck we compute the �2 statistic

�2 =X

``0(Cdata` �CCMB

` �Cfg` )M�1

``0 (Cdata`0 �CCMB

`0 �Cfg`0 ), (33)

for each of the spectra, where the sums extend over the mul-tipole ranges `min and `max used in the likelihood, M``0 isthe covariance matrix for the spectrum Cdata

` (including cor-rections for beam eigenmodes and calibrations), CCMB

` is thebest-fit primordial CMB spectrum and Cfg

` is the best-fit fore-

22

Planck Collaboration: Cosmological parameters

Planck+WP Planck+WP+highL Planck+lensing+WP+highL Planck+WP+highL+BAO

Parameter Best fit 68% limits Best fit 68% limits Best fit 68% limits Best fit 68% limits

⌦bh2 . . . . . . . . . . 0.022032 0.02205 ± 0.00028 0.022069 0.02207 ± 0.00027 0.022199 0.02218 ± 0.00026 0.022161 0.02214 ± 0.00024

⌦ch2 . . . . . . . . . . 0.12038 0.1199 ± 0.0027 0.12025 0.1198 ± 0.0026 0.11847 0.1186 ± 0.0022 0.11889 0.1187 ± 0.0017

100✓MC . . . . . . . . 1.04119 1.04131 ± 0.00063 1.04130 1.04132 ± 0.00063 1.04146 1.04144 ± 0.00061 1.04148 1.04147 ± 0.00056

⌧ . . . . . . . . . . . . 0.0925 0.089+0.012�0.014 0.0927 0.091+0.013

�0.014 0.0943 0.090+0.013�0.014 0.0952 0.092 ± 0.013

ns . . . . . . . . . . . 0.9619 0.9603 ± 0.0073 0.9582 0.9585 ± 0.0070 0.9624 0.9614 ± 0.0063 0.9611 0.9608 ± 0.0054

ln(1010As) . . . . . . . 3.0980 3.089+0.024�0.027 3.0959 3.090 ± 0.025 3.0947 3.087 ± 0.024 3.0973 3.091 ± 0.025

APS100 . . . . . . . . . . 152 171 ± 60 209 212 ± 50 204 213 ± 50 204 212 ± 50

APS143 . . . . . . . . . . 63.3 54 ± 10 72.6 73 ± 8 72.2 72 ± 8 71.8 72.4 ± 8.0

APS217 . . . . . . . . . . 117.0 107+20

�10 59.5 59 ± 10 60.2 58 ± 10 59.4 59 ± 10

ACIB143 . . . . . . . . . . 0.0 < 10.7 3.57 3.24 ± 0.83 3.25 3.24 ± 0.83 3.30 3.25 ± 0.83

ACIB217 . . . . . . . . . . 27.2 29+6

�9 53.9 49.6 ± 5.0 52.3 50.0 ± 4.9 53.0 49.7 ± 5.0

AtSZ143 . . . . . . . . . . 6.80 . . . 5.17 2.54+1.1

�1.9 4.64 2.51+1.2�1.8 4.86 2.54+1.2

�1.8

rPS143⇥217 . . . . . . . . 0.916 > 0.850 0.825 0.823+0.069

�0.077 0.814 0.825 ± 0.071 0.824 0.823 ± 0.070

rCIB143⇥217 . . . . . . . . 0.406 0.42 ± 0.22 1.0000 > 0.930 1.0000 > 0.928 1.0000 > 0.930

�CIB . . . . . . . . . . 0.601 0.53+0.13�0.12 0.674 0.638 ± 0.081 0.656 0.643 ± 0.080 0.667 0.639 ± 0.081

⇠tSZ⇥CIB . . . . . . . . 0.03 . . . 0.000 < 0.409 0.000 < 0.389 0.000 < 0.410

AkSZ . . . . . . . . . . 0.9 . . . 0.89 5.34+2.8�1.9 1.14 4.74+2.6

�2.1 1.58 5.34+2.8�2.0

⌦⇤ . . . . . . . . . . . 0.6817 0.685+0.018�0.016 0.6830 0.685+0.017

�0.016 0.6939 0.693 ± 0.013 0.6914 0.692 ± 0.010

�8 . . . . . . . . . . . 0.8347 0.829 ± 0.012 0.8322 0.828 ± 0.012 0.8271 0.8233 ± 0.0097 0.8288 0.826 ± 0.012

zre . . . . . . . . . . . 11.37 11.1 ± 1.1 11.38 11.1 ± 1.1 11.42 11.1 ± 1.1 11.52 11.3 ± 1.1

H0 . . . . . . . . . . . 67.04 67.3 ± 1.2 67.15 67.3 ± 1.2 67.94 67.9 ± 1.0 67.77 67.80 ± 0.77

Age/Gyr . . . . . . . 13.8242 13.817 ± 0.048 13.8170 13.813 ± 0.047 13.7914 13.794 ± 0.044 13.7965 13.798 ± 0.037

100✓⇤ . . . . . . . . . 1.04136 1.04147 ± 0.00062 1.04146 1.04148 ± 0.00062 1.04161 1.04159 ± 0.00060 1.04163 1.04162 ± 0.00056

rdrag . . . . . . . . . . 147.36 147.49 ± 0.59 147.35 147.47 ± 0.59 147.68 147.67 ± 0.50 147.611 147.68 ± 0.45

Table 5. Best-fit values and 68% confidence limits for the base ⇤CDM model. Beam and calibration parameters, and addi-tional nuisance parameters for “highL” data sets are not listed for brevity but may be found in the Explanatory Supplement(Planck Collaboration ES 2013).

strongly degenerate with the Poisson point source ampli-tude at 100 GHz. This degeneracy is broken when the high-resolution CMB data are added to Planck.

The last two points are demonstrated clearly in Fig. 7, whichshows the residuals of the Planck spectra with respect to thebest-fit cosmology for the Planck+WP analysis compared to thePlanck+WP+highL fits. The addition of high-resolution CMBdata also strongly constrains the net contribution from the kSZand tSZ⇥CIB components (dotted lines), though these compo-nents are degenerate with each other (and tend to cancel).

Although the foreground parameters for the Planck+WP fitscan di↵er substantially from those for Planck+WP+highL, thetotal foreground spectra are rather insensitive to the addition ofthe high-resolution CMB data. For example, for the 217 ⇥ 217spectrum, the di↵erences in the total foreground solution are lessthan 10 µK2 at ` = 2500. The net residuals after subtracting boththe foregrounds and CMB spectrum (shown in the lower panelsof each sub-plot in Fig. 7) are similarly insensitive to the addi-tion of the high-resolution CMB data. The foreground model issu�ciently complex that it has a high “absorptive capacity” toany smoothly-varying frequency-dependent di↵erences betweenspectra (including beam errors).

Table 6. Goodness-of-fit tests for the Planck spectra. The ��2 =�2 � N` is the di↵erence from the mean assuming the model iscorrect, and the last column expresses ��2 in units of the disper-sion

p2N`.

Spectrum `min `max �2 �2/N` ��2/p

2N`

100 ⇥ 100 50 1200 1158 1.01 0.14143 ⇥ 143 50 2000 1883 0.97 �1.09217 ⇥ 217 500 2500 2079 1.04 1.23143 ⇥ 217 500 2500 1930 0.96 �1.13

All 50 2500 2564 1.05 1.62

To quantify the consistency of the model fits shown in Fig. 7for Planck we compute the �2 statistic

�2 =X

``0(Cdata` �CCMB

` �Cfg` )M�1

``0 (Cdata`0 �CCMB

`0 �Cfg`0 ), (33)

for each of the spectra, where the sums extend over the mul-tipole ranges `min and `max used in the likelihood, M``0 isthe covariance matrix for the spectrum Cdata

` (including cor-rections for beam eigenmodes and calibrations), CCMB

` is thebest-fit primordial CMB spectrum and Cfg

` is the best-fit fore-

22

Planck XVI 2013