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
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?
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
For likelihood based on reconstruction power spectrum , need to know:!
Expectation value!
!Auto-covariance!➟ Dominant contributions from disconnected 8-point of ,
can be diagonalised with realisation-dependent N(0) subtraction
CMB LENSING RECONSTRUCTION LIKELIHOOD INGREDIENTS
C �rec�rec ⇠ T 4
Kesden et al. 0302536; Hanson et al. 1008.4403
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
For likelihood based on reconstruction power spectrum , need to know:!
Expectation value!
!Auto-covariance!➟ Dominant contributions from disconnected 8-point of ,
can be diagonalised with realisation-dependent N(0) subtraction
CMB LENSING RECONSTRUCTION LIKELIHOOD INGREDIENTS
C �rec�rec ⇠ T 4
Kesden et al. 0302536; Hanson et al. 1008.4403
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
circles±∆C φφL |n d i ag
crosses±∆C φφL |d i ag
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
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
N (0)L
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
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
circles±∆C φφL |n d i ag
crosses±∆C φφL |d i ag
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
C��L
C �rec�rec
L C T TL / T 2 / T 4
(i)
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
circles±∆C φφL |n d i ag
crosses±∆C φφL |d i ag
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
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
N (0)L
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
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
circles±∆C φφL |n d i ag
crosses±∆C φφL |d i ag
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
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!):
MS, Challinor, Hanson, Lewis 1308.0286
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
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
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
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
circles±∆C φφL |n d i ag
crosses±∆C φφL |d i ag
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
N (0)L
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
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
circles±∆C φφL |n d i ag
crosses±∆C φφL |d i ag
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
C��L
C �rec�rec
L C T TL / T 2 / T 4
(ii)
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
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
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
circles±∆C φφL |n d i ag
crosses±∆C φφL |d i ag
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
N (0)L
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
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
circles±∆C φφL |n d i ag
crosses±∆C φφL |d i ag
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
C��L
Cosmic variance
C �rec�rec
L C T TL / T 2 / T 4
(ii)
LIKELIHOOD INGREDIENTS TEMPERATURE-LENSING POWER-COVARIANCE
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
MS, Challinor, Hanson, Lewis 1308.0286
LIKELIHOOD INGREDIENTS TEMPERATURE-LENSING POWER-COVARIANCE
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
MS, Challinor, Hanson, Lewis 1308.0286
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 )
MS, Challinor, Hanson, Lewis 1308.0286
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)
IMPACT OF . LENSING AMPLITUDE
cov(
ˆC �rec�rec
L , ˆC T TL0 )
MS, Challinor, Hanson, Lewis 1308.0286
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)
IMPACT OF . LENSING AMPLITUDE
cov(
ˆC �rec�rec
L , ˆC T TL0 )
MS, Challinor, Hanson, Lewis 1308.0286
IMPACT OF . LENSING AMPLITUDE
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)
MS, Challinor, Hanson, Lewis 1308.0286
��Ajoint
⇠ correl(
ˆA, ˆA0)/2 ⇠ 3.5% (with realisation-dependent ,
otherwise ~ 5%)N (0)
IMPACT OF . LENSING AMPLITUDE
cov(
ˆC �rec�rec
L , ˆC T TL0 )
MS, Challinor, Hanson, Lewis 1308.0286
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
IMPACT OF . LENSING AMPLITUDE
cov(
ˆC �rec�rec
L , ˆC T TL0 )
MS, Challinor, Hanson, Lewis 1308.0286
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
MS, Challinor, Hanson, Lewis 1308.0286
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
MS, Challinor, Hanson, Lewis 1308.0286
�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
MS, Challinor, Hanson, Lewis 1308.0286
�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
MS, Challinor, Hanson, Lewis 1308.0286
C T T
lnL(C T T ,�rec) = lnL(C T T ) + lnL2(�rec)
no cross-term!
Thanks