Joint analysis of CMB temperature and lensing-reconstruction power spectra

JOINT ANALYSIS OF CMB TEMPERATURE AND

LENSING-RECONSTRUCTION POWER SPECTRA

Marcel Schmittfull!!

Berkeley Center for Cosmological Physics (BCCP)!!

arXiv:1308.0286 (PRD 88 063012)!!

Collaborators!Anthony Challinor (IoA/DAMTP Cambridge)

Duncan Hanson (McGill)!Antony Lewis (Sussex)!

!Pasadena 1 April 2014

http://arxiv.org/ct?url=http%3A%2F%2Fdx.doi.org%2F10%252E1103%2FPhysRevD%252E88%252E063012&v=e6b1c96b

CMB LENSING EFFECTS OF LENSING ON THE CMB

(a) Smoothing CMB 2-point function

3

ISW and lensing-SZ, we calculate

TΘ(li,−li, lj ,−lj) =

2 (li · lj)2

!

"

Cφsli

#2

CΘlj+"

Cφslj

#2

CΘli

$

−

!

[li · (li + lj)]2"

Cφsli

#2

+ [lj · (li + lj)]2"

Cφslj

#2$

CΘ|li+lj |

−

!

[li · (li − lj)]2"

Cφsli

#2

+ [lj · (li − lj)]2"

Cφslj

#2$

CΘ|li−lj |

+ 2 [li · (lj − li)] [lj · (lj − li)]CφsliCφs

ljCΘ

|lj−li|

− 2 [li · (li + lj)] [lj · (li + lj)]CφsliCφs

ljCΘ

|li+lj |

(7)

where the s is a place holder denoting either the ISWor SZ contribution.

B. SZ Trispectrum

In addition to the lensing contributions to the trispec-trum above, we consider contributions from the inverseCompton scattering of the CMB photons. The SZ con-tribution to the trispectrum is given by [17, 25]:

TΘij = g4ν

% zmax

0

dzdV

dz

% Mmax

Mmin

dMdn(M, z)

dM

× |yi(M, z)|2 |yj(M, z)|2 , (8)

where gν is the spectral function of the SZ effect,V (z) is the comoving volume of the universe integratedto a redshift of zmax = 4, M is the virial mass suchthat [log10(Mmin), log10(Mmax)] = [11, 16], dn/dM is the

FIG. 1: The impact of varying the lensing scaling parameteron the lensed CMB temperature power spectrum, for AL =[0,2,5,10].

mass function of dark matter halos as rendered by [18]utilizing the linear transfer function of [19], and y is thedimensionless two-dimensional Fourier transform of theprojected Compton y-parameter, given via the Limberapproximation [20] by:

yl =4πrsl2s

% ∞

0

dxx2y3D(x)sin(lx/ls)

lx/ls, (9)

where the scaled radius x = r/rs and ls = dA/rs suchthat dA is the angular diameter distance and rs is thescale radius of the three-dimensional radial profile y3Dof the Compton y-parameter. This profile is a functionof the gas density and temperature profiles as modeledin [21]. Hence, we incorporate the contributions obtainedfrom the SZ effect along with those from lensing, lensing-ISW, and lensing-SZ effects to the covariance matrix inEqn. 3.

C. The Weak Lensing Scaling Parameter AL

To first order in φ, the weak lensing of the CMBanisotropy trispectrum can be expressed as the con-volution of the power spectrum of the unlensed tem-perature Cl and that of the weak lensing potentialClφφ [15, 22, 23]. The magnitude of the lensing poten-tial power spectrum can be parameterized by the scalingparameter AL, defined as

Cφφl → ALC

φφl . (10)

Thus, AL is a measure of the degree to which the ex-pected amount of lensing appears in the CMB, such thata theory with AL = 0 is devoid of lensing, while AL = 1renders a theory with the canonical amount of lensing.Any inconsistency with unity represents an unexpectedamount of lensing that needs to be explained with newphysics, such as dark energy or modified gravity [15, 24].The impact of this scaling parameter on the lensed CMBtemperature power spectrum can be seen in Fig. 1. Qual-itatively, AL smoothes out the peaks in the power spec-trum and can therefore also be viewed as a smoothingparameter in addition to its scaling property. Given thatAL primarily affects the temperature power spectrum onsmall angular scales, we also explore the possibility thatit deviates from unity as secondary non-Gaussianities areaccounted for in the analysis.

Fig. from Smidt et al. 0909.3515

Acoustic peaks/troughs are smoothed out by lensing!(since lensed = convolution of unlensed CTT and Cφφ)

C T T

CMB LENSING EFFECTS OF LENSING ON THE CMB

(a) Smoothing CMB 2-point function

3

ISW and lensing-SZ, we calculate

TΘ(li,−li, lj ,−lj) =

2 (li · lj)2

!

"

Cφsli

#2

CΘlj+"

Cφslj

#2

CΘli

$

−

!

[li · (li + lj)]2"

Cφsli

#2

+ [lj · (li + lj)]2"

Cφslj

#2$

CΘ|li+lj |

−

!

[li · (li − lj)]2"

Cφsli

#2

+ [lj · (li − lj)]2"

Cφslj

#2$

CΘ|li−lj |

+ 2 [li · (lj − li)] [lj · (lj − li)]CφsliCφs

ljCΘ

|lj−li|

− 2 [li · (li + lj)] [lj · (li + lj)]CφsliCφs

ljCΘ

|li+lj |

(7)

where the s is a place holder denoting either the ISWor SZ contribution.

B. SZ Trispectrum

In addition to the lensing contributions to the trispec-trum above, we consider contributions from the inverseCompton scattering of the CMB photons. The SZ con-tribution to the trispectrum is given by [17, 25]:

TΘij = g4ν

% zmax

0

dzdV

dz

% Mmax

Mmin

dMdn(M, z)

dM

× |yi(M, z)|2 |yj(M, z)|2 , (8)

where gν is the spectral function of the SZ effect,V (z) is the comoving volume of the universe integratedto a redshift of zmax = 4, M is the virial mass suchthat [log10(Mmin), log10(Mmax)] = [11, 16], dn/dM is the

FIG. 1: The impact of varying the lensing scaling parameteron the lensed CMB temperature power spectrum, for AL =[0,2,5,10].

mass function of dark matter halos as rendered by [18]utilizing the linear transfer function of [19], and y is thedimensionless two-dimensional Fourier transform of theprojected Compton y-parameter, given via the Limberapproximation [20] by:

yl =4πrsl2s

% ∞

0

dxx2y3D(x)sin(lx/ls)

lx/ls, (9)

where the scaled radius x = r/rs and ls = dA/rs suchthat dA is the angular diameter distance and rs is thescale radius of the three-dimensional radial profile y3Dof the Compton y-parameter. This profile is a functionof the gas density and temperature profiles as modeledin [21]. Hence, we incorporate the contributions obtainedfrom the SZ effect along with those from lensing, lensing-ISW, and lensing-SZ effects to the covariance matrix inEqn. 3.

C. The Weak Lensing Scaling Parameter AL

To first order in φ, the weak lensing of the CMBanisotropy trispectrum can be expressed as the con-volution of the power spectrum of the unlensed tem-perature Cl and that of the weak lensing potentialClφφ [15, 22, 23]. The magnitude of the lensing poten-tial power spectrum can be parameterized by the scalingparameter AL, defined as

Cφφl → ALC

φφl . (10)

Thus, AL is a measure of the degree to which the ex-pected amount of lensing appears in the CMB, such thata theory with AL = 0 is devoid of lensing, while AL = 1renders a theory with the canonical amount of lensing.Any inconsistency with unity represents an unexpectedamount of lensing that needs to be explained with newphysics, such as dark energy or modified gravity [15, 24].The impact of this scaling parameter on the lensed CMBtemperature power spectrum can be seen in Fig. 1. Qual-itatively, AL smoothes out the peaks in the power spec-trum and can therefore also be viewed as a smoothingparameter in addition to its scaling property. Given thatAL primarily affects the temperature power spectrum onsmall angular scales, we also explore the possibility thatit deviates from unity as secondary non-Gaussianities areaccounted for in the analysis.

Fig. from Smidt et al. 0909.3515

Acoustic peaks/troughs are smoothed out by lensing!(since lensed = convolution of unlensed CTT and Cφφ)

C T T

ESA and Planck Collaboration

For fixed realisation of lenses, the lensed temperature fluctuations are anisotropic!

➟ Mode coupling (off-diagonal covariance)!

!➟ Reconstruct lenses from lensed!

!

➟ Get lensing power from trispectrum*

(b) Non-zero CMB 4-point function

T

hT (l+ L)T ⇤(l)iCMB / �(L)

�rec(L) /Z

lT (l)T ⇤(l� L)⇥ weight

T

* All quadrilaterals whose diagonal has length L

C �rec�rec

L /Z

l,l0T (l)T ⇤(l� L)T (�l0)T ⇤(L� l0)

CMB LENSING MOTIVATION

Probe late time DM to break degeneracies of primary CMB, get bias, de-lens BB!

Both (a) and (b) detected at many-𝝈 level (ACT, SPT, Planck; soon ACTPol, SPTpol)

(a) Smoothed automatically included by using lensed power spectrumC T T

(b) Reconstruction can be added, e.g. for Planck:!

Reduction of errors on ΩK and ΩΛ by factor ~2(evidence for flatness and DE from CMB alone)"

Constraint on τ without WMAP polarization!

Neutrino masses: curious preference for large mν!

Consistency with z~1100 CMB physics seen by Planck

�rec

Planck Collaboration: Gravitational lensing by large-scale structures with Planck

0.2 0.4 0.6 0.8 1.0�m

0.30

0.45

0.60

0.75

��

40

45

50

55

60

65

70

75

H0

�0.12 �0.08 �0.04 0.00 0.04�K

40

48

56

64

72

80

H0

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

��

Fig. 15. Two views of the geometric degeneracy in curved ⇤CDM models which is partially broken by lensing. Left: the degeneracyin the⌦m-⌦⇤ plane, with samples from Planck+WP+highL colour coded by the value of H0. The contours delimit the 68% and 95%confidence regions, showing the further improvement from including the lensing likelihood. Right: the degeneracy in the ⌦K-H0plane, with samples colour coded by ⌦⇤. Spatially-flat models lie along the grey dashed lines.

constraint. We see that the CMB alone now constrains the ge-ometry to be flat at the percent level. Previous constraints oncurvature via CMB lensing have been reported by SPT in com-bination with the WMAP-7 data:⌦K = �0.003+0.014

�0.018 (68%; Storyet al. 2012). This constraint is consistent, though almost a factorof two weaker, than that from Planck. Tighter constraints on cur-vature result from combining the Planck data with other astro-physical data, such as baryon acoustic oscillations, as discussedin Planck Collaboration XVI (2013).

Lensing e↵ects provide evidence for dark energy from theCMB alone, independent of other astrophysical data (Sherwinet al. 2011). In curved⇤CDM models, we find marginalised con-straints on ⌦⇤ of

⌦⇤ = 0.57+0.073�0.055 (68%; Planck+WP+highL)

⌦⇤ = 0.67+0.027�0.023 (68%; Planck+lensing+WP+highL).

Again, lensing reconstruction improves the errors by more thana factor of two over those from the temperature power spectrumalone.

6.1.4. Neutrino masses

The unique e↵ect in the unlensed temperature power spectrumof massive neutrinos that are still relativistic at recombinationis small. With the angular scale of the acoustic peaks fixedfrom measurements of the temperature power spectrum, neutrinomasses increase the expansion rate at z > 1 and so suppress clus-tering on scales larger than the horizon size at the non-relativistictransition (Kaplinghat et al. 2003). This e↵ect reduces C��L forL > 10 (see Fig. 12) and gives less smoothing of the acousticpeaks in CTT

` . As discussed in Planck Collaboration XVI (2013),the constraint on

Pm⌫ from the Planck temperature power spec-

trum (and WMAP low-` polarization) is driven by the smoothinge↵ect of lensing:

Pm⌫ < 0.66 eV (95%; Planck+WP+highL).

Curiously, this constraint is weakened by additionally includingthe lensing likelihood to

Xm⌫ < 0.85 eV, (95%; Planck+WP+highL),

reflecting mild tensions between the measured lensing and tem-perature power spectra, with the former preferring larger neu-

trino masses than the latter. Possible origins of this tension areexplored further in Planck Collaboration XVI (2013) and arethought to involve both the C��L measurements and features inthe measured CTT

` on large scales (` < 40) and small scales` > 2000 that are not fit well by the ⇤CDM+foreground model.As regards C��L , Fisher estimates show that the bandpowers inthe range 130 < L < 309 carry most of the statistical weightin determining the marginal error on

Pm⌫, and Fig. 12 reveals

a preference for highP

m⌫ from this part of the spectrum. (Wehave checked that removing the first bandpower from the lensinglikelihood, which is the least stable to data cuts and the detailsof foreground cleaning as discussed in Sect. 7, has little impacton our neutrino mass constraints.) We also note that a similartrend for lower lensing power than the ⇤CDM expectation onintermediate scales is seen in the ACT and SPT measurements(Fig. 11). Adding the high-L information to the likelihood weak-ens the constraint further, pushing the 95% limit to 1.07 eV. Thisis consistent with our small-scale measurement having a signifi-cantly lower amplitude. At this stage it is unclear what to makeof this mild tension between neutrino mass constraints from the4-point function and those from the 2-point, and we cautionover-interpreting the results. We expect to be able to say moreon this issue with the further data, including polarization, thatwill be made available in future Planck data releases.

6.2. Correlation with the ISW Effect

As CMB photons travel to us from the last scattering surface,the gravitational potentials that they traverse may undergo a non-negligible amount of evolution. This produces a net redshift orblueshift of the photons concerned, as they fall into and thenescape from the evolving potentials. The overall result is a con-tribution to the CMB temperature anisotropy known as the late-time integrated Sachs-Wolfe (ISW) e↵ect, or the Rees-Sciama(R-S) e↵ect depending on whether the evolution of the poten-tials concerned is in the linear (ISW) or non-linear (R-S) regimeof structure formation (Sachs & Wolfe 1967; Rees & Sciama1968). In the epoch of dark energy domination, which occurs af-ter z ⇠ 0.5 for the concordance ⇤CDM cosmology, large-scalepotentials tend to decay over time as space expands, resulting

19

Planck Paper XVII

(a) only

(a) + (b)

How do CMB experiments deal with lensing?

Why do we care?

CMB LENSING MOTIVATION

Probe late time DM to break degeneracies of primary CMB, get bias, de-lens BB!

Both (a) and (b) detected at many-𝝈 level (ACT, SPT, Planck; soon ACTPol, SPTpol)

(a) Smoothed automatically included by using lensed power spectrumC T T

(b) Reconstruction can be added, e.g. for Planck:!

Reduction of errors on ΩK and ΩΛ by factor ~2(evidence for flatness and DE from CMB alone)"

Constraint on τ without WMAP polarization!

Neutrino masses: curious preference for large mν!

Consistency with z~1100 CMB physics seen by Planck

�rec

Planck Collaboration: Gravitational lensing by large-scale structures with Planck

0.2 0.4 0.6 0.8 1.0�m

0.30

0.45

0.60

0.75

��

40

45

50

55

60

65

70

75

H0

�0.12 �0.08 �0.04 0.00 0.04�K

40

48

56

64

72

80

H0

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

��

Fig. 15. Two views of the geometric degeneracy in curved ⇤CDM models which is partially broken by lensing. Left: the degeneracyin the⌦m-⌦⇤ plane, with samples from Planck+WP+highL colour coded by the value of H0. The contours delimit the 68% and 95%confidence regions, showing the further improvement from including the lensing likelihood. Right: the degeneracy in the ⌦K-H0plane, with samples colour coded by ⌦⇤. Spatially-flat models lie along the grey dashed lines.

constraint. We see that the CMB alone now constrains the ge-ometry to be flat at the percent level. Previous constraints oncurvature via CMB lensing have been reported by SPT in com-bination with the WMAP-7 data:⌦K = �0.003+0.014

�0.018 (68%; Storyet al. 2012). This constraint is consistent, though almost a factorof two weaker, than that from Planck. Tighter constraints on cur-vature result from combining the Planck data with other astro-physical data, such as baryon acoustic oscillations, as discussedin Planck Collaboration XVI (2013).

Lensing e↵ects provide evidence for dark energy from theCMB alone, independent of other astrophysical data (Sherwinet al. 2011). In curved⇤CDM models, we find marginalised con-straints on ⌦⇤ of

⌦⇤ = 0.57+0.073�0.055 (68%; Planck+WP+highL)

⌦⇤ = 0.67+0.027�0.023 (68%; Planck+lensing+WP+highL).

Again, lensing reconstruction improves the errors by more thana factor of two over those from the temperature power spectrumalone.

6.1.4. Neutrino masses

The unique e↵ect in the unlensed temperature power spectrumof massive neutrinos that are still relativistic at recombinationis small. With the angular scale of the acoustic peaks fixedfrom measurements of the temperature power spectrum, neutrinomasses increase the expansion rate at z > 1 and so suppress clus-tering on scales larger than the horizon size at the non-relativistictransition (Kaplinghat et al. 2003). This e↵ect reduces C��L forL > 10 (see Fig. 12) and gives less smoothing of the acousticpeaks in CTT

` . As discussed in Planck Collaboration XVI (2013),the constraint on

Pm⌫ from the Planck temperature power spec-

trum (and WMAP low-` polarization) is driven by the smoothinge↵ect of lensing:

Pm⌫ < 0.66 eV (95%; Planck+WP+highL).

Curiously, this constraint is weakened by additionally includingthe lensing likelihood to

Xm⌫ < 0.85 eV, (95%; Planck+WP+highL),

reflecting mild tensions between the measured lensing and tem-perature power spectra, with the former preferring larger neu-

trino masses than the latter. Possible origins of this tension areexplored further in Planck Collaboration XVI (2013) and arethought to involve both the C��L measurements and features inthe measured CTT

` on large scales (` < 40) and small scales` > 2000 that are not fit well by the ⇤CDM+foreground model.As regards C��L , Fisher estimates show that the bandpowers inthe range 130 < L < 309 carry most of the statistical weightin determining the marginal error on

Pm⌫, and Fig. 12 reveals

a preference for highP

m⌫ from this part of the spectrum. (Wehave checked that removing the first bandpower from the lensinglikelihood, which is the least stable to data cuts and the detailsof foreground cleaning as discussed in Sect. 7, has little impacton our neutrino mass constraints.) We also note that a similartrend for lower lensing power than the ⇤CDM expectation onintermediate scales is seen in the ACT and SPT measurements(Fig. 11). Adding the high-L information to the likelihood weak-ens the constraint further, pushing the 95% limit to 1.07 eV. Thisis consistent with our small-scale measurement having a signifi-cantly lower amplitude. At this stage it is unclear what to makeof this mild tension between neutrino mass constraints from the4-point function and those from the 2-point, and we cautionover-interpreting the results. We expect to be able to say moreon this issue with the further data, including polarization, thatwill be made available in future Planck data releases.

6.2. Correlation with the ISW Effect

As CMB photons travel to us from the last scattering surface,the gravitational potentials that they traverse may undergo a non-negligible amount of evolution. This produces a net redshift orblueshift of the photons concerned, as they fall into and thenescape from the evolving potentials. The overall result is a con-tribution to the CMB temperature anisotropy known as the late-time integrated Sachs-Wolfe (ISW) e↵ect, or the Rees-Sciama(R-S) e↵ect depending on whether the evolution of the poten-tials concerned is in the linear (ISW) or non-linear (R-S) regimeof structure formation (Sachs & Wolfe 1967; Rees & Sciama1968). In the epoch of dark energy domination, which occurs af-ter z ⇠ 0.5 for the concordance ⇤CDM cosmology, large-scalepotentials tend to decay over time as space expands, resulting

19

Planck Paper XVII

(a) only

(a) + (b)

How do CMB experiments deal with lensing?

Why do we care?

Requires joint likelihood for and !

➟ Non-trivial because derived from same CMB map!

➟ Need

C T T C �rec�rec

cov(

ˆC �rec�rec , ˆC T T)

CMB LENSING RECONSTRUCTION LIKELIHOOD INGREDIENTS

For likelihood based on reconstruction power spectrum , need to know:!

Expectation value!

!!

C �rec�rec ⇠ T 4

101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L

circles±∆C φφL |n d i ag

crosses±∆C φφL |d i ag

hC �rec�rec

L i = N (0)L + C��

L +N (1)L

disconn. 4-point

connected 4-point

MS, Challinor, Hanson, Lewis 2013

Kesden et al. 0302536; Hanson et al. 1008.4403


Expectation value!

!Auto-covariance!➟ Dominant contributions from disconnected 8-point of ,

can be diagonalised with realisation-dependent N(0) subtraction




cov(

ˆC �rec�rec

L , ˆC �rec�rec

L0 )

THanson et al. 1008.4403; MS, Challinor, Hanson, Lewis 1308.0286

hC �rec�rec

L i = N (0)L + C��

L +N (1)L

disconn. 4-point

connected 4-point


Expectation value!

!Auto-covariance!➟ Dominant contributions from disconnected 8-point of ,

can be diagonalised with realisation-dependent N(0) subtraction




cov(

ˆC �rec�rec

L , ˆC �rec�rec

L0 )

THanson et al. 1008.4403; MS, Challinor, Hanson, Lewis 1308.0286

hC �rec�rec

L i = N (0)L + C��

L +N (1)L

disconn. 4-point

connected 4-point

Cross-covariance!➟ 6-point:

cov(

ˆC �rec�rec

L , ˆC T TL0 )

MS, Challinor, Hanson, Lewis 1308.0286

l1 l4

l2 l3L

weights

cov(

ˆC ��L , ˆC T T

L0 ) /X

l1,l2,l3,l4,M,M 0

(�1)

M+M 0✓

l1 l2 Lm1 m2 �M

◆✓l3 l4 Lm3 m4 M

◆gl1l2(L)gl3l4(L)

⇥hh ˜Tl1

˜Tl2˜Tl3

˜Tl4˜TL0M 0 ˜TL0,�M 0i � h ˜Tl1

˜Tl2˜Tl3

˜Tl4ih ˜TL0M 0 ˜TL0,�M 0ii.

(i) connected 6-point (ii) disconnected (iii) connected 4-point (neglect here)

2.3 CMB lensing

Planck collaboration: CMB power spectra & likelihood

2 10 500

1000

2000

3000

4000

5000

6000

D �[µ

K2 ]

90� 18�

500 1000 1500 2000 2500

Multipole moment, �

1� 0.2� 0.1� 0.07�Angular scale

Figure 37. The 2013 Planck CMB temperature angular power spectrum. The error bars include cosmic variance, whose magnitudeis indicated by the green shaded area around the best fit model. The low-� values are plotted at 2, 3, 4, 5, 6, 7, 8, 9.5, 11.5, 13.5, 16,19, 22.5, 27, 34.5, and 44.5.

Table 8. Constraints on the basic six-parameter �CDM model using Planck data. The top section contains constraints on the sixprimary parameters included directly in the estimation process, and the bottom section contains constraints on derived parameters.

Planck Planck+WP

Parameter Best fit 68% limits Best fit 68% limits

�bh2 . . . . . . . . . 0.022068 0.02207 ± 0.00033 0.022032 0.02205 ± 0.00028

�ch2 . . . . . . . . . 0.12029 0.1196 ± 0.0031 0.12038 0.1199 ± 0.0027100�MC . . . . . . . 1.04122 1.04132 ± 0.00068 1.04119 1.04131 ± 0.00063

� . . . . . . . . . . . . 0.0925 0.097 ± 0.038 0.0925 0.089+0.012�0.014

ns . . . . . . . . . . . 0.9624 0.9616 ± 0.0094 0.9619 0.9603 ± 0.0073

ln(1010As) . . . . . 3.098 3.103 ± 0.072 3.0980 3.089+0.024�0.027

�� . . . . . . . . . . 0.6825 0.686 ± 0.020 0.6817 0.685+0.018�0.016

�m . . . . . . . . . . 0.3175 0.314 ± 0.020 0.3183 0.315+0.016�0.018

�8 . . . . . . . . . . . 0.8344 0.834 ± 0.027 0.8347 0.829 ± 0.012

zre . . . . . . . . . . . 11.35 11.4+4.0�2.8 11.37 11.1 ± 1.1

H0 . . . . . . . . . . 67.11 67.4 ± 1.4 67.04 67.3 ± 1.2

109As . . . . . . . . 2.215 2.23 ± 0.16 2.215 2.196+0.051�0.060

�mh2 . . . . . . . . . 0.14300 0.1423 ± 0.0029 0.14305 0.1426 ± 0.0025Age/Gyr . . . . . . 13.819 13.813 ± 0.058 13.8242 13.817 ± 0.048z� . . . . . . . . . . . 1090.43 1090.37 ± 0.65 1090.48 1090.43 ± 0.54100�� . . . . . . . . 1.04139 1.04148 ± 0.00066 1.04136 1.04147 ± 0.00062zeq . . . . . . . . . . . 3402 3386 ± 69 3403 3391 ± 60

33

D l=

l(l+

1)C

TT

l/(2

⇡)

[µK

2]

Multipole moment, l

Angular scale

Figure 2.2: CMB temperature power spectrum measured by Planck (red points),including cosmic variance error (shaded green) and best-fit ⇤CDM model predic-tion (solid green). The plot is from [77].

2.3 CMB lensing

We have so far neglected the gravitational deflection of CMB photons by inter-

vening large-scale structures on their way from the last-scattering surface to the

observer, because the RMS deflection of CMB photons is only ⇠ 2.5 arcmin, which

is a small e↵ect. However, these deflections are coherent over scales of several

degrees and the e↵ect of lensing on the CMB power spectrum is large enough that

it must be included for high-resolution CMB experiments like Planck to obtain

accurate cosmological parameter estimates. Indeed, lensing of the CMB can be

exploited as a precious probe of the inhomogeneous distribution of dark matter

along the line of sight which is otherwise hard to observe. Since most of the lens-

ing e↵ect is caused by dark matter structures around redshift z ⇠ 2 CMB lensing

can be used to break degeneracies that a↵ect the primary CMB to improve con-

straints on spatial curvature, neutrino masses, dark energy and modified gravity

(see e.g. [78, 79, 80, 81, 82, 83, 84, 85], and Section 2.3.3 below). For example,

recent lensing reconstructions provide evidence for dark energy from the CMB

38

Unlensed CMB

101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



N (0)L

2.3 CMB lensing


2 10 500

1000

2000

3000

4000

5000

6000

D �[µ

K2 ]

90� 18�

500 1000 1500 2000 2500


1� 0.2� 0.1� 0.07�Angular scale



Planck Planck+WP


�bh2 . . . . . . . . . 0.022068 0.02207 ± 0.00033 0.022032 0.02205 ± 0.00028

�ch2 . . . . . . . . . 0.12029 0.1196 ± 0.0031 0.12038 0.1199 ± 0.0027100�MC . . . . . . . 1.04122 1.04132 ± 0.00068 1.04119 1.04131 ± 0.00063

� . . . . . . . . . . . . 0.0925 0.097 ± 0.038 0.0925 0.089+0.012�0.014

ns . . . . . . . . . . . 0.9624 0.9616 ± 0.0094 0.9619 0.9603 ± 0.0073

ln(1010As) . . . . . 3.098 3.103 ± 0.072 3.0980 3.089+0.024�0.027

�� . . . . . . . . . . 0.6825 0.686 ± 0.020 0.6817 0.685+0.018�0.016

�m . . . . . . . . . . 0.3175 0.314 ± 0.020 0.3183 0.315+0.016�0.018

�8 . . . . . . . . . . . 0.8344 0.834 ± 0.027 0.8347 0.829 ± 0.012

zre . . . . . . . . . . . 11.35 11.4+4.0�2.8 11.37 11.1 ± 1.1

H0 . . . . . . . . . . 67.11 67.4 ± 1.4 67.04 67.3 ± 1.2

109As . . . . . . . . 2.215 2.23 ± 0.16 2.215 2.196+0.051�0.060

�mh2 . . . . . . . . . 0.14300 0.1423 ± 0.0029 0.14305 0.1426 ± 0.0025Age/Gyr . . . . . . 13.819 13.813 ± 0.058 13.8242 13.817 ± 0.048z� . . . . . . . . . . . 1090.43 1090.37 ± 0.65 1090.48 1090.43 ± 0.54100�� . . . . . . . . 1.04139 1.04148 ± 0.00066 1.04136 1.04147 ± 0.00062zeq . . . . . . . . . . . 3402 3386 ± 69 3403 3391 ± 60

33

D l=

l(l+

1)C

TT

l/(2

⇡)

[µK

2]

Multipole moment, l

Angular scale


2.3 CMB lensing















38

Lenses

Lensed CMB

101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



C��L

C �rec�rec

L C T TL / T 2 / T 4

(i)

2.3 CMB lensing


2 10 500

1000

2000

3000

4000

5000

6000

D �[µ

K2 ]

90� 18�

500 1000 1500 2000 2500


1� 0.2� 0.1� 0.07�Angular scale



Planck Planck+WP


�bh2 . . . . . . . . . 0.022068 0.02207 ± 0.00033 0.022032 0.02205 ± 0.00028

�ch2 . . . . . . . . . 0.12029 0.1196 ± 0.0031 0.12038 0.1199 ± 0.0027100�MC . . . . . . . 1.04122 1.04132 ± 0.00068 1.04119 1.04131 ± 0.00063

� . . . . . . . . . . . . 0.0925 0.097 ± 0.038 0.0925 0.089+0.012�0.014

ns . . . . . . . . . . . 0.9624 0.9616 ± 0.0094 0.9619 0.9603 ± 0.0073

ln(1010As) . . . . . 3.098 3.103 ± 0.072 3.0980 3.089+0.024�0.027

�� . . . . . . . . . . 0.6825 0.686 ± 0.020 0.6817 0.685+0.018�0.016

�m . . . . . . . . . . 0.3175 0.314 ± 0.020 0.3183 0.315+0.016�0.018

�8 . . . . . . . . . . . 0.8344 0.834 ± 0.027 0.8347 0.829 ± 0.012

zre . . . . . . . . . . . 11.35 11.4+4.0�2.8 11.37 11.1 ± 1.1

H0 . . . . . . . . . . 67.11 67.4 ± 1.4 67.04 67.3 ± 1.2

109As . . . . . . . . 2.215 2.23 ± 0.16 2.215 2.196+0.051�0.060

�mh2 . . . . . . . . . 0.14300 0.1423 ± 0.0029 0.14305 0.1426 ± 0.0025Age/Gyr . . . . . . 13.819 13.813 ± 0.058 13.8242 13.817 ± 0.048z� . . . . . . . . . . . 1090.43 1090.37 ± 0.65 1090.48 1090.43 ± 0.54100�� . . . . . . . . 1.04139 1.04148 ± 0.00066 1.04136 1.04147 ± 0.00062zeq . . . . . . . . . . . 3402 3386 ± 69 3403 3391 ± 60

33

D l=

l(l+

1)C

TT

l/(2

⇡)

[µK

2]

Multipole moment, l

Angular scale


2.3 CMB lensing















38

Unlensed CMB

101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



N (0)L

2.3 CMB lensing


2 10 500

1000

2000

3000

4000

5000

6000

D �[µ

K2 ]

90� 18�

500 1000 1500 2000 2500


1� 0.2� 0.1� 0.07�Angular scale



Planck Planck+WP


�bh2 . . . . . . . . . 0.022068 0.02207 ± 0.00033 0.022032 0.02205 ± 0.00028

�ch2 . . . . . . . . . 0.12029 0.1196 ± 0.0031 0.12038 0.1199 ± 0.0027100�MC . . . . . . . 1.04122 1.04132 ± 0.00068 1.04119 1.04131 ± 0.00063

� . . . . . . . . . . . . 0.0925 0.097 ± 0.038 0.0925 0.089+0.012�0.014

ns . . . . . . . . . . . 0.9624 0.9616 ± 0.0094 0.9619 0.9603 ± 0.0073

ln(1010As) . . . . . 3.098 3.103 ± 0.072 3.0980 3.089+0.024�0.027

�� . . . . . . . . . . 0.6825 0.686 ± 0.020 0.6817 0.685+0.018�0.016

�m . . . . . . . . . . 0.3175 0.314 ± 0.020 0.3183 0.315+0.016�0.018

�8 . . . . . . . . . . . 0.8344 0.834 ± 0.027 0.8347 0.829 ± 0.012

zre . . . . . . . . . . . 11.35 11.4+4.0�2.8 11.37 11.1 ± 1.1

H0 . . . . . . . . . . 67.11 67.4 ± 1.4 67.04 67.3 ± 1.2

109As . . . . . . . . 2.215 2.23 ± 0.16 2.215 2.196+0.051�0.060

�mh2 . . . . . . . . . 0.14300 0.1423 ± 0.0029 0.14305 0.1426 ± 0.0025Age/Gyr . . . . . . 13.819 13.813 ± 0.058 13.8242 13.817 ± 0.048z� . . . . . . . . . . . 1090.43 1090.37 ± 0.65 1090.48 1090.43 ± 0.54100�� . . . . . . . . 1.04139 1.04148 ± 0.00066 1.04136 1.04147 ± 0.00062zeq . . . . . . . . . . . 3402 3386 ± 69 3403 3391 ± 60

33

D l=

l(l+

1)C

TT

l/(2

⇡)

[µK

2]

Multipole moment, l

Angular scale


2.3 CMB lensing















38

Lenses

Lensed CMB

101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



C��L

Cosmic variance

C �rec�rec

L C T TL / T 2 / T 4

(i)

LIKELIHOOD INGREDIENTS TEMPERATURE-LENSING POWER-COVARIANCE

(i) If lensing field fluctuates high, CMB power is smoother and reconstruction is high!

➟ Derives from connected CMB 6-point at !

!

Correlation of unbinned power spectra is up to ~0.04% (at low Lφ!):


cov(

ˆCˆ�rec

ˆ�rec

L�, ˆC

˜T ˜TLT ,expt)

O(�4)

conn.6pt. =2

2L� + 1

⇣C��

L�

⌘2 @C

˜T ˜TLT

@C��L�

15

(a) Theoretical matter cosmic variance contribution (D9) (b) Simulations

FIG. 5. (a) Theoretical matter cosmic variance contribution (D9) to the correlation of the unbinned power spectra of recon-structed lensing potential and lensed temperature. The covariance (D9) is converted to a correlation using the same conversion

factor as in (38). (b) Measured correlation ˆcorrel(Cˆ�in

ˆ�in

L�, C

˜T ˜TLT

� CTTLT

) of the input lensing potential power and the di↵erence

of noise-free lensed and unlensed temperature powers in 1000 simulations.

FIG. 6. Left : The approximate contribution to the covariance between [L(L + 1)]2Cˆ�ˆ�L /(2⇡) and L0(L0 + 1)C

˜T ˜TL0 /(2⇡) from

cosmic variance of the lenses, derived from Eq. (D9). Right : The rank-one approximation to the matrix on the left fromretaining only the largest singular value.TODOO: discuss this in main text or appendix

CT� correlations in all calculations we try to eliminate correlations between the lensing potential and the unlensedtemperature in the simulations by considering

ˆcov(C ��L , C T T

L0 )|CV(�) = ˆcov(C �in

�in

L , C T TL0 � CTT

L0 ), (52)

where C �in

�in is the empirical power of the input lensing potential and CTT is the empirical power of the unlensed

temperature. Subtracting the unlensed from the lensed empirical power spectrum also reduces the noise of thecovariance estimate because it eliminates the scatter due to cosmic variance of the unlensed temperature. We estimatethe covariance in simulations similarly to (48). As shown in Fig. 5b these measurements agree with the theoreticalexpectation from (D9).

SimulationTheory

Minima at acoustic peaks of temperature power which are decreased by larger lensing power; maxima at acoustic troughs

O(�4)

2.3 CMB lensing


2 10 500

1000

2000

3000

4000

5000

6000

D �[µ

K2 ]

90� 18�

500 1000 1500 2000 2500


1� 0.2� 0.1� 0.07�Angular scale



Planck Planck+WP


�bh2 . . . . . . . . . 0.022068 0.02207 ± 0.00033 0.022032 0.02205 ± 0.00028

�ch2 . . . . . . . . . 0.12029 0.1196 ± 0.0031 0.12038 0.1199 ± 0.0027100�MC . . . . . . . 1.04122 1.04132 ± 0.00068 1.04119 1.04131 ± 0.00063

� . . . . . . . . . . . . 0.0925 0.097 ± 0.038 0.0925 0.089+0.012�0.014

ns . . . . . . . . . . . 0.9624 0.9616 ± 0.0094 0.9619 0.9603 ± 0.0073

ln(1010As) . . . . . 3.098 3.103 ± 0.072 3.0980 3.089+0.024�0.027

�� . . . . . . . . . . 0.6825 0.686 ± 0.020 0.6817 0.685+0.018�0.016

�m . . . . . . . . . . 0.3175 0.314 ± 0.020 0.3183 0.315+0.016�0.018

�8 . . . . . . . . . . . 0.8344 0.834 ± 0.027 0.8347 0.829 ± 0.012

zre . . . . . . . . . . . 11.35 11.4+4.0�2.8 11.37 11.1 ± 1.1

H0 . . . . . . . . . . 67.11 67.4 ± 1.4 67.04 67.3 ± 1.2

109As . . . . . . . . 2.215 2.23 ± 0.16 2.215 2.196+0.051�0.060

�mh2 . . . . . . . . . 0.14300 0.1423 ± 0.0029 0.14305 0.1426 ± 0.0025Age/Gyr . . . . . . 13.819 13.813 ± 0.058 13.8242 13.817 ± 0.048z� . . . . . . . . . . . 1090.43 1090.37 ± 0.65 1090.48 1090.43 ± 0.54100�� . . . . . . . . 1.04139 1.04148 ± 0.00066 1.04136 1.04147 ± 0.00062zeq . . . . . . . . . . . 3402 3386 ± 69 3403 3391 ± 60

33

D l=

l(l+

1)C

TT

l/(2

⇡)

[µK

2]

Multipole moment, l

Angular scale


2.3 CMB lensing















38

101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



Lenses

Unlensed CMB

101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



N (0)L

2.3 CMB lensing


2 10 500

1000

2000

3000

4000

5000

6000

D �[µ

K2 ]

90� 18�

500 1000 1500 2000 2500


1� 0.2� 0.1� 0.07�Angular scale



Planck Planck+WP


�bh2 . . . . . . . . . 0.022068 0.02207 ± 0.00033 0.022032 0.02205 ± 0.00028

�ch2 . . . . . . . . . 0.12029 0.1196 ± 0.0031 0.12038 0.1199 ± 0.0027100�MC . . . . . . . 1.04122 1.04132 ± 0.00068 1.04119 1.04131 ± 0.00063

� . . . . . . . . . . . . 0.0925 0.097 ± 0.038 0.0925 0.089+0.012�0.014

ns . . . . . . . . . . . 0.9624 0.9616 ± 0.0094 0.9619 0.9603 ± 0.0073

ln(1010As) . . . . . 3.098 3.103 ± 0.072 3.0980 3.089+0.024�0.027

�� . . . . . . . . . . 0.6825 0.686 ± 0.020 0.6817 0.685+0.018�0.016

�m . . . . . . . . . . 0.3175 0.314 ± 0.020 0.3183 0.315+0.016�0.018

�8 . . . . . . . . . . . 0.8344 0.834 ± 0.027 0.8347 0.829 ± 0.012

zre . . . . . . . . . . . 11.35 11.4+4.0�2.8 11.37 11.1 ± 1.1

H0 . . . . . . . . . . 67.11 67.4 ± 1.4 67.04 67.3 ± 1.2

109As . . . . . . . . 2.215 2.23 ± 0.16 2.215 2.196+0.051�0.060

�mh2 . . . . . . . . . 0.14300 0.1423 ± 0.0029 0.14305 0.1426 ± 0.0025Age/Gyr . . . . . . 13.819 13.813 ± 0.058 13.8242 13.817 ± 0.048z� . . . . . . . . . . . 1090.43 1090.37 ± 0.65 1090.48 1090.43 ± 0.54100�� . . . . . . . . 1.04139 1.04148 ± 0.00066 1.04136 1.04147 ± 0.00062zeq . . . . . . . . . . . 3402 3386 ± 69 3403 3391 ± 60

33

D l=

l(l+

1)C

TT

l/(2

⇡)

[µK

2]

Multipole moment, l

Angular scale


2.3 CMB lensing















38

Lensed CMB

101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



C��L

C �rec�rec

L C T TL / T 2 / T 4

(ii)

2.3 CMB lensing


2 10 500

1000

2000

3000

4000

5000

6000

D �[µ

K2 ]

90� 18�

500 1000 1500 2000 2500


1� 0.2� 0.1� 0.07�Angular scale



Planck Planck+WP


�bh2 . . . . . . . . . 0.022068 0.02207 ± 0.00033 0.022032 0.02205 ± 0.00028

�ch2 . . . . . . . . . 0.12029 0.1196 ± 0.0031 0.12038 0.1199 ± 0.0027100�MC . . . . . . . 1.04122 1.04132 ± 0.00068 1.04119 1.04131 ± 0.00063

� . . . . . . . . . . . . 0.0925 0.097 ± 0.038 0.0925 0.089+0.012�0.014

ns . . . . . . . . . . . 0.9624 0.9616 ± 0.0094 0.9619 0.9603 ± 0.0073

ln(1010As) . . . . . 3.098 3.103 ± 0.072 3.0980 3.089+0.024�0.027

�� . . . . . . . . . . 0.6825 0.686 ± 0.020 0.6817 0.685+0.018�0.016

�m . . . . . . . . . . 0.3175 0.314 ± 0.020 0.3183 0.315+0.016�0.018

�8 . . . . . . . . . . . 0.8344 0.834 ± 0.027 0.8347 0.829 ± 0.012

zre . . . . . . . . . . . 11.35 11.4+4.0�2.8 11.37 11.1 ± 1.1

H0 . . . . . . . . . . 67.11 67.4 ± 1.4 67.04 67.3 ± 1.2

109As . . . . . . . . 2.215 2.23 ± 0.16 2.215 2.196+0.051�0.060

�mh2 . . . . . . . . . 0.14300 0.1423 ± 0.0029 0.14305 0.1426 ± 0.0025Age/Gyr . . . . . . 13.819 13.813 ± 0.058 13.8242 13.817 ± 0.048z� . . . . . . . . . . . 1090.43 1090.37 ± 0.65 1090.48 1090.43 ± 0.54100�� . . . . . . . . 1.04139 1.04148 ± 0.00066 1.04136 1.04147 ± 0.00062zeq . . . . . . . . . . . 3402 3386 ± 69 3403 3391 ± 60

33

D l=

l(l+

1)C

TT

l/(2

⇡)

[µK

2]

Multipole moment, l

Angular scale


2.3 CMB lensing















38

101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



Lenses

Unlensed CMB

101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



N (0)L

2.3 CMB lensing


2 10 500

1000

2000

3000

4000

5000

6000

D �[µ

K2 ]

90� 18�

500 1000 1500 2000 2500


1� 0.2� 0.1� 0.07�Angular scale



Planck Planck+WP


�bh2 . . . . . . . . . 0.022068 0.02207 ± 0.00033 0.022032 0.02205 ± 0.00028

�ch2 . . . . . . . . . 0.12029 0.1196 ± 0.0031 0.12038 0.1199 ± 0.0027100�MC . . . . . . . 1.04122 1.04132 ± 0.00068 1.04119 1.04131 ± 0.00063

� . . . . . . . . . . . . 0.0925 0.097 ± 0.038 0.0925 0.089+0.012�0.014

ns . . . . . . . . . . . 0.9624 0.9616 ± 0.0094 0.9619 0.9603 ± 0.0073

ln(1010As) . . . . . 3.098 3.103 ± 0.072 3.0980 3.089+0.024�0.027

�� . . . . . . . . . . 0.6825 0.686 ± 0.020 0.6817 0.685+0.018�0.016

�m . . . . . . . . . . 0.3175 0.314 ± 0.020 0.3183 0.315+0.016�0.018

�8 . . . . . . . . . . . 0.8344 0.834 ± 0.027 0.8347 0.829 ± 0.012

zre . . . . . . . . . . . 11.35 11.4+4.0�2.8 11.37 11.1 ± 1.1

H0 . . . . . . . . . . 67.11 67.4 ± 1.4 67.04 67.3 ± 1.2

109As . . . . . . . . 2.215 2.23 ± 0.16 2.215 2.196+0.051�0.060

�mh2 . . . . . . . . . 0.14300 0.1423 ± 0.0029 0.14305 0.1426 ± 0.0025Age/Gyr . . . . . . 13.819 13.813 ± 0.058 13.8242 13.817 ± 0.048z� . . . . . . . . . . . 1090.43 1090.37 ± 0.65 1090.48 1090.43 ± 0.54100�� . . . . . . . . 1.04139 1.04148 ± 0.00066 1.04136 1.04147 ± 0.00062zeq . . . . . . . . . . . 3402 3386 ± 69 3403 3391 ± 60

33

D l=

l(l+

1)C

TT

l/(2

⇡)

[µK

2]

Multipole moment, l

Angular scale


2.3 CMB lensing















38

Lensed CMB

101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



101

102

10−9

10−8

10−7

10−6

L (log scale)

[L(L

+1)]

2C

L/(2

π)

1000 1500 2000 2500

L (l inear scale)

C φφL − N ( 0 )

L

C φφL − 2N ( 0 )

L + N ( 0 )L

N ( 0 )L

C φφL

C φφL + N ( 1 )

L



C��L

Cosmic variance

C �rec�rec

L C T TL / T 2 / T 4

(ii)


9

(a) Theoretical noise contribution (34) (b) Simulations (c) Simulations with empirical N(0) subtraction

FIG. 4. (a) Theoretical noise contribution (34) to the correlation of unbinned power spectra of lensed temperature and

reconstructed lensing potential correl(Cˆ�ˆ�L�

, C˜T ˜TLT

) as defined in (36). The acoustic peaks of the temperature power spectrum

are visible in the vertical direction. (b) Estimate of the correlation of unbinned power spectra from 1000 simulations. (c) Same

as (b) after subtracting the empirical N (0) bias from Cˆ�ˆ� as described in Section VC.

TODO: maybe replace L and L0 by L�

and LT

below because use this in plots now The unbinned power correlationshown in Fig. 4a is mostly constrained to a cone-like region in the L vs L0 plane, with the maximum correlation of⇠ 0.5% located at the first acoustic peak L0 ⇠ 200 and lensing reconstruction multipoles L ⇠ 1600�1900. While we donot expect this peak to be relevant for Planck because basically all the statistical power comes from the reconstructionat L . 1000, we will discuss its impact on lensing amplitude estimates more rigorously later. To understand the basicstructure of the correlation in Fig. 4a we compute approximations of (34) for the bottom and left regions in Fig. 4a,i.e. in the limits L0 ⌧ L and L ⌧ L0.

At low temperature and high reconstruction multipoles, L0 ⌧ L, the 3j-symbols in (34) restrict the summationfrom l1 = L� L0 to l1 = L+ L0. If we Taylor expand (34) in l1 around L and approximate Ln � Ln�1, we get (see[7] for a similar calculation)

correl(C ��

L

, C T T

L

0,expt)

L

0⌧L

disconn. ⇡[L(L+ 1)]2A

L

8⇡

p(2L0 + 1)(2L+ 1)

L0(L0 + 1)C T T

L

0

L(L+ 1)C T T

L,expt

. (37)

As shown in Fig. 1 the first term slightly increases with L. The last term peaks at the first acoustic peak L0 ⇠ 200 andat the reconstruction multipole L ⇠ 1600 � 1900 where the observed temperature power C T T

L,expt is minimal (for thePlanck-like noise and beam considered here). TODO: 4 In this region (37) gives ⇠ 0.004 � 0.005, which agrees withFig. 4a. Eq. (37) also implies that lower noise in the temperature power spectrum would move the peak position tohigher reconstruction multipoles L. The cone structure in Fig. 4a based at (L,L0) ⇠ (1600� 1900, 200) is due to thefact that the summand that we expanded around L actually depends on the summation multipole l1. The maximumvalue of the summand at l1 ⇠ 1600 � 1900 can be picked up by the sum if L � L0 . (1600 � 1900) . L + L0, whichimposes a cone-like constraint in the L vs. L0 plane. A similar argument can be applied for the cone patterns in theL . 200 region.

For high temperature and low reconstruction multipoles, L ⌧ L0, we can Taylor expand (34) in l1 around L0 to get

correl(C ��

L

, C T T

L

0,expt)

L⌧L

0

disconn. ⇡[L(L+ 1)]2A2

L

2⇡(AL

+ C��

L

)

C T T

L

0

C T T

L

0,expt

!21

2

p(2L+ 1)(2L0 + 1)

8<

:1 +d lnC T T

L

0

d lnL0 +3

8

d lnC T T

L

0

d lnL0

!29=

; ,

(38)

where we neglected N (1)L

and used AL

= N (0)L

. The logarithmic derivative in the curly brackets peaks betweenacoustic peaks and troughs at L0 ⇠ 350, 625, 925, 1225, 1550, 1850 . . . , which agrees with the peak positions of the fullcorrelation shown in Fig. 4a.5 Its values vary between ⇠ �11 and ⇠ 1, i.e. the curly bracket in (38) is at most ⇠ 36

4 TODO: maybe mention that o↵-diagonal � power auto correlation from Duncan also peaks at around (l, l) = (1800� 1900, 1800� 1900)(not visible in Duncan’s plot because it’s cut o↵ at 2000). can probably understand this with similar approximations as done here. thismight be useful because Duncan’s numbers disagreed with earlier evaluation by Cooray et al. However, Duncan has checked againstsims, so it’s clear he is right and Cooray is wrong. maybe put this in footnote? or include plot of theoretical and measured phi autopower correlation? not really needed b/c it’s in 1008 paper already.

5 Peaks at higher L0 are suppressed by the noise in the temperature power spectrum.

SimulationTheory

Peak at Lφ ~ 1800 = minimum of beam-deconvolved noisy temperature power C ˜T ˜Texpt

(ii) If unlensed CMB fluctuates high, CMB power and Gaussian rec. noise N(0) are high!

➟ Derives from disconnected CMB 6-point !

!

Correlation of unbinned power spectra is up to ~0.5% (at very high Lφ):

cov(

ˆCˆ�rec

ˆ�rec

L�, ˆC

˜T ˜TLT ,expt)

O(�0)

disc. =

@(2 ˆN (0)

L�)

@ ˆC ˜T ˜TLT ,expt

2

2LT + 1

⇣C

˜T ˜TLT ,expt

⌘2



Decorrelating power spectra!➟ Can remove noise contribution (ii) with realisation-dependent bias correction

Cˆ�ˆ�L � 2N (0)

L = Cˆ�ˆ�L �

X

L0

@(2N (0)

L )

@C ˜T ˜TL0,expt

C˜T ˜TL0,expt

11

2. Magnitude and structure of the correlation matrix

In Fig. 4a we plot the power correlation resulting from the power covariance (36) (denoting L� = L and LT = L0for convenience),

correl(C ��L�

, C T TLT ,expt) =

cov(C ��L�

, C T TLT ,expt)q

varG(C��L�

+ N(0)L�

+ N(1)L�

) varG(C T TLT ,expt)

. (38)

This is the correlation if the power spectra are not binned. If the covariance is broad-band (i.e. roughly constantover the bin area) the correlation of (su�ciently finely) binned power spectra will increase roughly proportionally tothe square root of the product of the two bin widths.5 TODO: check footnote properly, actually not sure what’s theprecise condition here. The denominator in (38) contains the beam-deconvolved noisy temperature power spectrum(19) so that high temperature multipoles are suppressed.

(a) Theoretical noise contribution (36) (b) Simulations (c) Simulations with empirical N(0) subtraction

FIG. 4. (a) Theoretical noise contribution (36) to the correlation of unbinned power spectra of lensed temperature and

reconstructed lensing potential correl(Cˆ�ˆ�L�

, C˜T ˜TLT

) as defined in (38). The acoustic peaks of the temperature power spectrum

are visible in the vertical direction. (b) Estimate of the correlation of unbinned power spectra from 1000 simulations. (c) Same

as (b) after subtracting the empirical N (0) bias (18) from Cˆ�ˆ�.

The unbinned power correlation shown in Fig. 4a is mostly constrained to a cone-like region in the L� vs LT plane,with the maximum correlation of ⇠ 0.5% located at the first acoustic peak LT ⇠ 200 and lensing reconstructionmultipoles L� ⇠ 1600 � 1900. While we do not expect this peak to be relevant for Planck because basically all thestatistical power comes from the reconstruction at L� . 1000, we will discuss its impact on lensing amplitude estimatesmore rigorously later. To understand the basic structure of the correlation in Fig. 4a we compute approximations of(36) for the bottom and left regions in Fig. 4a, i.e. in the limits LT ⌧ L� and L� ⌧ LT .

At low temperature and high reconstruction multipoles, LT ⌧ L�, the 3j-symbols in (36) restrict the summationfrom l1 = L� � LT to l1 = L� + LT . If we Taylor expand (36) in l1 around L� and approximate Ln

� � Ln�1� , we get

(see [29] for a similar calculation)

correl(C ��L�

, C T TLT ,expt)

LT⌧L�

disconn. ⇡[L�(L� + 1)]2AL�

8⇡

q(2LT + 1)(2L� + 1)

LT (LT + 1)C T TLT

L�(L� + 1)C T TL�,expt

. (39)

As shown in Fig. 1 the first term slightly increases with L�. The last term peaks at the first acoustic peak LT ⇠ 200

and at the reconstruction multipole L� ⇠ 1600�1900 where the observed temperature power C T TL�,expt

is minimal (for

the Planck-like noise and beam considered here). TODO: 6 In this region (39) gives ⇠ 0.4� 0.5%, which agrees with

5 This is because a broad-band covariance is una↵ected by binning while the power variances decrease proportionally to the bin widths(if the power variance and covariance are slowly varying inside a bin). To see this assume a constant covariance covL�LT

= c. Then the

binned covariance isP

L�,LTc/(�L��LT ) = c where the sum goes over the bin area. In contrast, the power variance is due to the auto-

covariance, which is dominated by its diagonal and not broad-band. Assuming that the diagonal part is constant, covL1

L2

= �L1

L2

d,yields for the binned variance

PL

1

,L2

�L1

L2

d/(�L)2 = d/(�L).6 TODO: maybe mention that o↵-diagonal � power auto correlation from Duncan also peaks at around (l, l) = (1800� 1900, 1800� 1900)(not visible in Duncan’s plot because it’s cut o↵ at 2000). can probably understand this with similar approximations as done here. thismight be useful because Duncan’s numbers disagreed with earlier evaluation by Cooray et al. However, Duncan has checked againstsims, so it’s clear he is right and Cooray is wrong. maybe put this in footnote? or include plot of theoretical and measured phi autopower correlation? not really needed b/c it’s in 1008 paper already.

TheorySimulations w/o

realisation-dep. .N (0)Simulations with

realisation-dep. .N (0)

➟ This also mitigates the off-diagonal reconstruction power auto-covariance (for the same reason); can be understood from optimal trispectrum estimation


Obtain realisation-dependent bias mitigation also from optimal trispectrum estimator using Edgeworth-expansion of lensed temperature around Gaussian:!

!

!

!

!

!

!

OPTIMAL TRISPECTRUM ESTIMATION

N (0)

realisation-indep." N(0)

P (T ) =

⇢1 +

1

24hTiTjTkTlic

TiTj TkTl �

�C�1

ij TkTl + 5 perms�+�C�1

ij C�1kl + 2 perms

��

⇥e�TiC�1ij Tj/2

pdet(2⇡C)

realisation-dep.2N (0)

reconstruction power

C �rec�rec +-

Ti = lensed temperature Cij = <TiTj>Ti = C�1

ij Tj

lensing trispectrum

Impact of temperature-lensing power covariance on lensing amplitude A:!

Rescale lensing power spectrum!

Estimate from reconstruction power spectrum,!

and from smoothing of temperature power spectrum, !

➟ is linearly related to

IMPACT OF . LENSING AMPLITUDE

cov(

ˆA, ˆA0)

cov(

ˆC �rec�rec

L , ˆC T TL0 )

cov(

ˆC �rec�rec

L , ˆC T TL0 )


C��L ! AC��

L

A[C ��]

A0[C˜T ˜Texpt

]

102

103

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

lφmax (for fixed lTmax = 2002)

corr

el(A

,A

′)

O(φ0) fully disconnected (noise)O(φ2) connected 4-pt. type AO(φ4) connected 4-pt. type AO(φ2) connected 4-pt. type B primaryO(φ4) conn. 6-pt. (matter cosmic variance)Full theorySimulations

correl(

ˆ A,ˆ A0 )

(i)

(ii)


cov(

ˆC �rec�rec

L , ˆC T TL0 )


102

103

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

lφmax (for fixed lTmax = 2002)

corr

el(A

,A

′)

O(φ2) connected 4-pt. type B primaryO(φ4) conn. 6-pt. (matter cosmic variance)Full theorySimulations

correl(

ˆ A,ˆ A0 )

(i)

With realisation-dependent .N (0)


cov(

ˆC �rec�rec

L , ˆC T TL0 )



cov(

ˆC �rec�rec

L , ˆC T TL0 )

➟ Neglecting underestimates error of joint amplitude estimate bycov(

ˆC �rec�rec

L , ˆC T TL0 )

➟ Temperature-lensing power-covariance is

negligible for combined amplitude estimate(and for cosmological parameters)


��Ajoint

⇠ correl(

ˆA, ˆA0)/2 ⇠ 3.5% (with realisation-dependent ,

otherwise ~ 5%)N (0)


cov(

ˆC �rec�rec

L , ˆC T TL0 )


Physical reasons for smallness of amplitude correlation!

(i) Lens cosmic variance"

cov(A,A’) is limited by the small number of Cφφ modes affecting acoustic region of! (including more lensing modes in A[Cφφ] reduces cov(A,A’) because there are increasingly more lensing modes in A[Cφφ] whose fluctuations don’t enter A’[ ]).!

correl(A,A’) due to cosmic variance of these modes is diluted by CMB cosmic variance and instrumental noise (because A and A’ are not limited by cosmic variance of the lenses)

C T T

C T T


cov(

ˆC �rec�rec

L , ˆC T TL0 )


Physical reasons for smallness of amplitude correlation!

(i) Lens cosmic variance"

cov(A,A’) is limited by the small number of Cφφ modes affecting acoustic region of! (including more lensing modes in A[Cφφ] reduces cov(A,A’) because there are increasingly more lensing modes in A[Cφφ] whose fluctuations don’t enter A’[ ]).!

correl(A,A’) due to cosmic variance of these modes is diluted by CMB cosmic variance and instrumental noise (because A and A’ are not limited by cosmic variance of the lenses)

C T T

C T T

26

400 600 800 1000 1200 1400 1600 1800 2000 22000

1

2

3

4

x 10−3

l

l�max = 10/ s T T

l

/ s ��l

/ s ��l,approx

(a) l�max

= 10

400 600 800 1000 1200 1400 1600 1800 2000 22000

1

2

3

4

x 10−3

l

l�max = 500/ s T T

l

/ s ��l

(b) l�max

= 500

FIG. 12. Contribution of di↵erent CMB multipoles l to the (S/N)2 of the lensing amplitude for estimators based on the CMBpower spectrum [blue; Eq. (A1)] and on the lensing reconstruction [red; Eq. (A2)] for l�

max

= 10 (left) and l�max

= 500 (right).The approximation in Eq. (A3) for large-scale lenses is shown in black dashed in the left plot. Since we are not interested inthe total S/N but only in its distribution over di↵erent CMB scales, all curves are normalised such their integral over l is unity.

sums over L and l1 in Eq. (A2) by restricting ourselves to very large-scale lenses, l�max ⌧ l, and using the large scaleapproximation for N (0) derived in [26] [see their Eq. (19)], to find

s��l ⇡ s��

l,approx = f(l�min, l�max)1

varG(C T Tl,expt)

2

4

C T Tl

d ln(l2C T Tl )

d ln l

!2

+1

2

C T T

ld ln C T T

l

d ln l

!23

5 , [l�max . O(10)]. (A3)

Here, the prefactor f depends on the minimum and maximum reconstruction multipole but not on the CMB multipolel. The terms in square brackets have the form of the quadrature sum of the information in convergence and shear.Convergence changes locally the angular scale of the CMB anisotropies and so would contribute nothing to the (S/N)2

for a scale-invariant spectrum, l2Cl = const., while shear contributes nothing for a white-noise spectrum, Cl = const.15

Thus, for large-scale lenses, the (S/N)2 for the reconstruction-based amplitude gets most contributions from CMBscales where the gradient of the CMB power spectrum is maximal, i.e. between acoustic peaks and troughs (see redand black curves in Fig. 12a).

In reality, temperature multipoles that are not precisely at peaks or troughs and not precisely in between them willa↵ect both amplitude estimates, which implies a small amplitude correlation. Intermediate- and small-scale lenses canmix CMB modes over multipole ranges comparable to the acoustic peak separation so that they are a↵ected by widerranges of CMB multipoles than argued above (see red curve in Fig. 12b), which implies a somewhat larger amplitudecorrelation. However, since the CMB scales that are most important for the reconstruction still have negligible impacton the amplitude estimated from the temperature power, we expect the correlation of the amplitudes to stay rathersmall.

2. Cosmic variance of the lenses

We now consider the contribution of cosmic variance of the lenses to the covariance of the lensing amplitudeestimates given in Eq. (56). It is instructive to consider a toy-model where the reconstruction “noise” power is

proportional to C��, i.e. N (0)l + N (1)

l = �C��l . Taking the limit � ! 0 is equivalent to being able to observe �

directly with no measurement error, while � ! 1 corresponds to there being no information in the reconstruction.

With N (0)l + N (1)

l = �C��l , the weighting of the reconstruction power spectrum in A is the same as for an ideal

reconstruction (i.e. one with no N (0) and N (1) noise). Provided we then determine A from all those � modes that

15 The relation between large-scale lenses and the induced local convergence and shear is discussed in detail by [45], who also findagreement between the (S/N)2 of a combined convergence and shear estimate with the large-scale limit of the (S/N)2 of the trispectrumreconstruction. This correspondence has also been used to approximate the squeezed limit of the ISW-lensing bispectrum [30].

(ii) CMB cosmic variance!

Roughly disjoint (independently fluctuating) scales in the CMB contribute to amplitude determination from peak-smearing and to lens reconstruction!

S/N

per

CM

B m

ode

Impact of on phys. params.!

Joint data vector:!

Joint covariance!

!

!

Fisher matrix: !

➟ Fisher errors increase by at most 0.7% if is included

IMPACT OF . PHYSICAL PARAMETERS

cov(

ˆC �rec�rec

L , ˆC T TL0 )

p = (⌦bh2,⌦ch

2, h, ⌧, As, ns,⌦⌫h2,⌦K)

C = (C˜T ˜Texpt

, Cˆ�ˆ� � 2N (0) +N (0))

Fij =

X

LL0

@CL

@pi(cov

�1

joint

)LL0@CL0

@pj

cov(

ˆC �rec�rec

L , ˆC T TL0 )

cov(

ˆC �rec�rec

L , ˆC T TL0 )

covLL0, joint ⌘ cov(

ˆCL, ˆCL0) =

�LL0

varG(C˜T ˜TL,expt) cov(

ˆC˜T ˜TL , ˆC

ˆ�recˆ�rec

L0 )

cov(

ˆCˆ�rec

ˆ�rec

L , ˆC˜T ˜TL0 ) �LL0

varG(h ˆCˆ�ˆ�L i)

!

➟ Temperature-lensing power-covariance negligible for physical parameter errors


So far assumed likelihood based on reconstruction power instead of map!➟ Well established for temperature, but unclear for (non-Gaussian) reconstruction!

➟ Compare two lensing-likelihood models: !

1. Gaussian in :!

!

!

2. Gaussian in (with parameter-independent covariance):!

!!

!➟ Estimate lensing amplitude (and tilt) from both likelihoods,

compare scatter of best-fit parameter vs. likelihood width

CMB LENSING RECONSTRUCTION LIKELIHOOD FORM

�recC �rec�rec

�rec

C �rec�rec


�2 lnL2(ˆC ��|A) /

X

l,l0

hˆC ��l � (AC��

l +Nl)

i(cov

�1��)ll0

hˆC ��l0 � (AC��

l0 +Nl0)

i

�2 lnL1(�|A) /X

l

(2l + 1)

C ��

l

AC��l +Nl

+ ln��AC��

l +Nl|!

So far assumed likelihood based on reconstruction power instead of map!➟ Well established for temperature, but unclear for (non-Gaussian) reconstruction!

➟ Compare two lensing-likelihood models: !

1. Gaussian in :!

!

!

2. Gaussian in (with parameter-independent covariance):!

!!

!➟ Estimate lensing amplitude (and tilt) from both likelihoods,

compare scatter of best-fit parameter vs. likelihood width

CMB LENSING RECONSTRUCTION LIKELIHOOD FORM

�recC �rec�rec

�rec

C �rec�rec


�2 lnL2(ˆC ��|A) /

X

l,l0

hˆC ��l � (AC��

l +Nl)

i(cov

�1��)ll0

hˆC ��l0 � (AC��

l0 +Nl0)

i

�2 lnL1(�|A) /X

l

(2l + 1)

C ��

l

AC��l +Nl

+ ln��AC��

l +Nl|!

➟ 2. performs better than 1. (see paper)

Understand non-Gaussianity of φrec and correlation with temperature analytically!

!

!

!

!

Found methods to treat/mitigate both!

This has significantly simplified joint analysis of and φrec for Planck:!

➟ Likelihoods can be modeled separately,!

➟ Non-Gaussianity of φrec modeled by likelihood that’s Gaussian in

CONCLUSIONS

cov(

ˆCˆ�rec

ˆ�rec

L�, ˆC

˜T ˜TLT ,expt) =

@(2 ˆN (0)

L�)

@ ˆC ˜T ˜TLT ,expt

2

2LT + 1

⇣C

˜T ˜TLT ,expt

⌘2

"1 + 2

C��L�

AL�

#+

2

2L� + 1

⇣C��

L�

⌘2 @C

˜T ˜TLT

@C��L�

connected 4-point

noise contribution (disconnected 6-point)

matter cosmic variance (connected 6-point)

C �rec�rec


C T T

lnL(C T T ,�rec) = lnL(C T T ) + lnL2(�rec)

no cross-term!

Thanks

Joint analysis of CMB temperature and lensing-reconstruction power spectra

Science

Transcript of Joint analysis of CMB temperature and lensing-reconstruction power spectra