Real-world CMB lensing quadratic estimator power spectrum response Julien Carron1 1Universit e de Gen eve D epartement de Physique Th eorique et CAP 24 Quai Ansermet CH-1211 Gen eve 4 Switzerland

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Real-world CMB lensing quadratic estimator power spectrum response

Julien Carron1, ∗

1Universit´

e de Gen`

eve, D´

epartement de Physique Th´

eorique et CAP, 24 Quai Ansermet, CH-1211 Gen`

eve 4, Switzerland

I describe a method to estimate response matrices of Cosmic Microwave Background (CMB) lensing power

spectra estimators to the true sky power under realistic conditions. Applicable to all lensing reconstruction

pipelines based on quadratic estimators (QE), it uses a small number of Gaussian CMB Monte-Carlos and

specially designed QE’s in order to obtain sufﬁciently accurate matrices with little computational effort. This

method may be used to improve the modelling of CMB lensing band-powers by incorporating at least some of

the non-idealities encountered in CMB lensing reconstruction. These non-idealities always include masking,

and often inhomogeneous ﬁltering, either in the harmonic domain or pixel space. I obtain these matrices for

Planck latest lensing reconstructions, and then show that the residual couplings induced by masking explain

very well the residual multiplicative bias seen on the Planck simulations, removing the need for an empirical

correction.

I. INTRODUCTION

The effect of weak gravitational lensing on the Cosmic Mi-

crowave Background (CMB) has now been measured to a cou-

ple of percent precision, providing a clean probe of the late-

time Universe [1]. The relevance of the CMB lensing power

spectrum as a cosmological probe is expected to increase fur-

ther in upcoming years, a major part of its science case be-

ing its ability to tightly constrain the neutrino mass scale in

combination with other cosmological data sets [2–4]. Cur-

rent measurements from the Planck satellite[5,6] and ground-

based telescopes[7–9] use quadratic estimators (QE [10,11])

to extract the signal from CMB maps1. While more efﬁcient

methods to extract the spectrum are known [13–15], they will

be most useful only when the effective instrumental noise

level will be small enough to resolve the lensing-induced po-

larization B-mode over large fractions of the sky, for exam-

ple with CMB-S42[3] or potentially a high-resolution next-

generation space mission such as PICO[16].

CMB observations are always only usable on some fraction

of the sky. Estimators designed to work on an idealized, full-

sky conﬁguration must account for this in some way, since the

presence of the mask will otherwise introduce biases. This

also affects their covariance matrices, and eventually will in-

troduce some level of coupling between different multipoles

of the CMB power spectra extracted from the data. For the

standard CMB spectra, masking effects are relatively straight-

forward to model and are routinely taken into account[17–19,

e.g.]. However, CMB lensing power spectra from quadratic

estimators are four-point functions of the data. This renders

analytical understanding of the impact of masking, or other

non-idealities, considerably more difﬁcult. In practice, correc-

tions obtained from simulations are applied to measured band-

powers, as described later on. These corrections are small

enough, and the lensing spectrum is smooth enough, that this

∗julien.carron@unige.ch

1A recent exception being [12] that uses Bayesian techniques to put an

approximately 10% constraint on the ΛCDM lensing power spectrum am-

plitude Aφ

2https://cmb-s4.org

way of proceeding is believed to be robust enough for the fore-

seeable future. Nevertheless, a more detailed understanding is

certainly desirable. Unfortunately, barely anything quantita-

tive is known on the responses and couplings of CMB lensing

estimators away from idealized conditions. In this short pa-

per I give a method to obtain a good estimate of the measured

lensing spectrum response to the true sky spectrum, when ex-

tracted with quadratic estimators, at a manageable numerical

cost. This makes use of a small number (I will use here 2

for our main results) of noise-free Gaussian CMB simulations

to which quadratic estimators designed for this purpose are

applied. The method is general enough that it can be used

on any quadratic estimator based lensing spectrum extraction

pipeline.

I then use this method to obtain the response matrices of the

Planck lensing reconstructions [5,6] for various quadratic es-

timators, and show that they provide a very good match to the

empirical Monte-Carlo correction observed on the complex

Planck NPIPE [20] simulation suite, that were applied to the

published band-powers and likelihoods. Our rather simple-

minded method is based on Eqs. (3.4) and (3.3), which we

motivate in Sec. III after reviewing the relevant elements of

CMB lensing quadratic estimators in Sec. II. An appendix col-

lects additional details on the four-point contractions of the

CMB relevant to lensing reconstruction.

II. QUADRATIC ESTIMATOR POWER SPECTRA

I follow somewhat closely in this section the notations of

the Planck 2015 lensing paper [21, Appendix A], which pro-

vides a fairly exhaustive presentation of standard quadratic

estimator theory, and to which I refer for complete expres-

sions and more details. The results obtained here extend

straight-forwardly to the generalized minimum variance es-

timator of Ref. [22], or to the κ-ﬁltered versions of Ref. [23].

Bias-hardened [24] or ‘shear-only’ estimators [25–27] have

weights modiﬁed to be more robust to foregrounds, but re-

main quadratic and can also be treated in the exact same way.

Lensing introduces statistical anisotropies in the CMB two-

point statistics. To linear order, one may write for two CMB

ﬁelds Xand Zthe change in covariance due to the lensing

arXiv:2210.05449v3 [astro-ph.CO] 31 Jan 2023

potential φas

∆hX`1m1Z`2m2i=X

(−1)M`1`2L

m1m2−M

× WXZ

`1`2LφLM

(2.1)

with known covariance response functions WXZ given

in [28]. A quadratic estimator ¯x[X, Z]uses a set of ﬁducial

weights Wxto build

¯xLM =(−1)M

`1m1,`2m2

`1`2L

m1m2−M

×Wx

`1`2L¯

X`1m1¯

Z`2m2,

(2.2)

where ¯

X, ¯

Zare inverse-variance weighted X, Z maps, in or-

der to optimize signal to noise by down-weighting noisy pix-

els or harmonic modes (the speciﬁc implementation of this

step can vary). By design, under idealized conditions and full-

sky coverage, the estimator (2.2) will then respond diagonally

to the lensing potential to linear order according to

h¯xLM i=Rxφ

LφLM (2.3)

(φLM is held ﬁxed in the average). Here Rxφ

Lsums the QE

weights against the responses WXZ and the inverse-variance

ﬁlters mapping X, Z to ¯

X, ¯

Z. An unbiased estimator is then

simply obtained inverting the response,

φx

LM ≡¯xLM

Rxφ

(2.4)

Cross-correlating ˆ

φxto another estimator built from

¯y[C, D]probes the CMB trispectrum. Deﬁning

Cˆ

φˆ

L,xy ≡1

2L+ 1

M=−L

φx

LM ˆ

φ∗y

LM ,(2.5)

and neglecting small complications from the large-scale struc-

ture and post-Born bispectra [29–31], as well as any contri-

bution from extra-galactic foregrounds, one has under these

idealized conditions [32]

DCˆ

φˆ

L,xyE=Cφφ

L+N(0)

L,xy +N(1)

L,xy (2.6)

where

• the ﬁrst term Cφφ

Lis sourced by the primary trispec-

trum contractions, the ones that naturally emerge from

Eq. (2.3) involving the CMB sky covariance responses

WXZ and WCD on each of the QE legs, and each of

two legs brings one φ.

•N(0)

Lis the disconnected 4-point function, proportional

to two powers of the data CMB spectra inclusive of the

the lensing contribution and instrumental or other noise.

•N(1)

Lcaptures the secondary trispectrum contractions,

the ones that involve WXC · WZD and WXD · WZC ,

the CMB contracting across the two QE legs. These are

generally smaller than the primary term except at the

highest L0s, but still relevant [33].

On the masked sky, the structure of Eq. (2.6) remains the

same. However the true estimator response of Eq. (2.3) be-

come non-diagonal and position dependent, and is never ex-

actly known. In the lack of a better prescription, one often

sticks to the same QE deﬁnition3and normalization, Eqs (2.2)

and (2.4). This is a useful approximation which is certainly

correct away from the mask boundaries since lensing recon-

struction is very local. One crudely accounts for the missing

sky area with the rescaling

Cˆ

φˆ

L,xy →1

fsky

Cˆ

φˆ

L,xy (2.7)

where fsky is the unmasked sky fraction (or the mean fourth

power of the mask if a non-boolean mask is used, as proposed

by Ref. [34]).

All of the terms in Eq. (2.6) respond to the sky lensing spec-

trum. However, N(0)

Lcan be removed accurately by using

QE’s built from a mix of data and simulations [24,35,36].

Unlike an analytical N(0), this bias estimate built from QE’s

properly contains all masked-induced couplings, and its accu-

racy is not degraded by the approximate normalization since

it is acting on the unnormalized estimators ¯xand ¯y. Within

the ﬁducial cosmological model, the same holds for N(1)

L[36],

which can also be evaluated by QE’s on pairs of simulations

tuned to capture the secondary contractions. The construction

of these accurate bias estimates is reviewed in the appendix.

For these reasons, when averaged over simulations with

consistent cosmology, the subtraction of the lensing biases

is correct using specially designed biases estimates (‘MC-

N(0)’ and ‘MC- ˆ

N(1)’), but the remaining primary term will

be somewhat off. One can use this to provide a deﬁnition of

a spectrum response matrix that captures the primary trispec-

trum contractions only:

DCˆ

φˆ

L,xy −MC- ˆ

N(0)

L−MC- ˆ

N(1)

LEMC ≡X

Lsky

rxy

LLsky Cφφ

Lsky

(2.8)

Within ΛCDM (and probably for most other models) the lens-

ing spectrum is smooth and largely featureless. This results in

a multiplicative bias affecting the reconstructed spectrum at

multipole L. The factor fsky in Eq 2.7 has been included into

this deﬁnition of the matrix, so that we expect the bias to be

smaller then fsky.

3One also subtracts to ˆ

φxthe average of the QE observed on simulations

to remove contributions from anisotropies unrelated to lensing (the ‘mean-

ﬁeld ’). I stick to Eq. (2.4) to deﬁne ˆ

φxin this paper.

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Real-worldCMBlensingquadraticestimatorpowerspectrumresponseJulienCarron1,1Universit´edeGeneve,D´epartementdePhysiqueTh´eoriqueetCAP,24QuaiAnsermet,CH-1211Geneve4,SwitzerlandIdescribeamethodtoestimateresponsematricesofCosmicMicrowaveBackground(CMB)lensingpowerspectraestimatorstothetrueskypowerunde...

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Real-world CMB lensing quadratic estimator power spectrum response Julien Carron1 1Universit e de Gen eve D epartement de Physique Th eorique et CAP 24 Quai Ansermet CH-1211 Gen eve 4 Switzerland

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