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1.4 Galaxy cluster lensing of the cosmic microwave background

1.4.3 CMB lensing reconstruction

The lensing potential (or, equivalently, the deflection field, or the convergence) can be estimated, or ‘reconstructed’, with CMB observations alone due to the specific correlations lensing induces in the CMB fields.

The simplest technique for CMB lensing reconstruction involves CMB lensing quadratic estimators, so called because they are quadratic in the CMB fields (Hu,2001;

Hu and Okamoto,2002). Here, we present the T T quadratic estimator, which takes as input two copies of the observed temperature field, following the discussion of Lewis and Challinor (2006). For simplicity, we will work in the flat-sky approximation, which is valid if the observed fields are small enough such that the curvature of the sky can be neglected. Within this approximation, Fourier transforms can be used instead of

spherical harmonics, greatly simplifying calculations. An analogous estimator can be derived for the spherical sky (see Okamoto and Hu 2003).

Consider the lensed CMB temperature, ˜T (ˆn) = T (ˆn + ∇ψ). It can be expanded

to linear order in the lensing potential,

˜

T (ˆn) = T (ˆn + ∇ˆnψ) ≃ T (ˆn) + ∇ˆnψ(ˆn) · ∇T (ˆn), (1.46)

which in Fourier space reads

˜

T (l) ≃ T (l) −

Z d2l

l

· (l − l)ψ(l − l)T (l). (1.47)

This ‘gradient approximation’ is valid on scales on which deflections are small compared to the wavelength of the unlensed fields, roughly for l < 2000. However, on very small scales, l > 3000, this approximation is also valid, since due to damping the unlensed CMB is very smooth on these scales, essentially just a gradient. Under this approximation, the correlation between two different modes of the lensed temperature field can be written to linear order in ψ as

D ˜ T (l) ˜T(l − L)E T = δ(L)C T T l + 1 h (L − l) · LC|l−L|T T + l · LClT Tiψ(L), (1.48) where ClT T is the unlensed T T power spectrum, and where the angular brackets denote ensemble averaging over the unlensed temperature for fixed lensing potential. Here, the first term on the left-hand side is the isotropic unlensed power, and the second term corresponds to the lensing-induced off-diagonal correlations, which are proportional to the lensing potential, ψ(L). Thus, these off-diagonal correlations can be used in order to construct an estimator of ψ(L), ˆψ(L), by summing over all possible combinations of

the quadratic product ˜T (l) ˜T(l − L) with L ̸= 0,

ˆ ψ(L) = N (L) Z d2l ˆ T (l) ˆT (l − L) g(l, L), (1.49)

where ˆT denotes a measurement (estimate) of the lensed temperature, ˜T, g(l, L) is a weight function chosen to ensure optimality, and N (L) is a normalisation factor. This is the T T quadratic estimator. Although here it is presented as an estimator of the lensing potential, estimators for the deflection field, the shear, and the convergence can be trivially obtained, as these quantities are just derivatives of the lensing potential; e.g., the quadratic estimator of the lensing convergence is ˆκ(L) = L2ψ(L)/2.ˆ

By requiring that ˆψ(L) is unbiased when ensemble averaging over the unlensed temperature, Dψ(L)ˆ E T = ψ(L), N (L) is N (L) = " Z d2l (2π)2 h (L − l) · LC|l−L|T T + l · LClT Tig(l, L) #−1 . (1.50)

By further requiring that the estimator has minimum variance (to linear order in ψ), the weight function can be written as (Hu, 2001)

g(l, L) = (L − l) · LC T T |l−L|+ l · LClT T 2CT ˆˆT l C ˆ T ˆT |l−L| , (1.51)

where ClT ˆˆT is the total power spectrum of the observed temperature, which, to linear order in ψ, can be written as ClT ˆˆT = ˜ClT ˜˜T + NlT T, where ˜ClT ˜˜T is the lensed T T power spectrum, and where NlT T is the beam-deconvolved temperature noise power spectrum. This guarantees that the T T quadratic estimator has minimum variance, although it does not imply that this is the optimal temperature-based lensing estimator.

With this choice of weight function, N (L) is also the estimator variance to linear order in ψ. As is apparent from Eq. (1.50), this variance arises not only from instrumental noise, but also from the unlensed temperature fluctuations themselves, which limit how well the lensing potential can be estimated.

This T T quadratic estimator can be derived from a maximum-likelihood approach in which the likelihood of the observed temperature given the lensing potential is assumed to be Gaussian, arising as the first step in a Newton-Raphson-like approach to maximise the likelihood (seeHirata and Seljak 2003andHanson et al. 2010). Moreover, it can be understood in a more intuitive way in the ‘large-scale lens regime’, in which small-scale (roughly l > 300) temperature modes are used in order to reconstruct large scale (roughly l < 300) lensing modes (see Schaan and Ferraro 2019and Fabbian et al. 2019). In this regime, the T T quadratic estimator can be expressed as a sum of a magnification-only estimator and a shear-only estimator. Intuitively, the magnification- only estimator reconstructs the lensing potential by comparing the local size of the hot and cold spots in the CMB anisotropies with their average size as expected from the unlensed T T power spectrum. For example, if in a region the anisotropies are larger than average, a positive convergence (i.e., an overdensity) is predicted. Similarly, the shear-only estimator looks for local deviations from statistical isotropy in order to reconstruct the potential. Thus, the quadratic estimator can be understood as using both information from the local magnification and shearing of the CMB anisotropies. It should be noted that both the magnification and shear-only parts are unbiased on all

scales, although their interpretation as convergence and shear-only estimators is only clear in the large-scale lens regime. They can also be used independently from each other. In fact, it is argued in Schaan and Ferraro(2019) that the shear-only estimator is essentially insensitive to uncleaned foregrounds, such as the difficult-to-remove kSZ signal, which makes it a very interesting tool, especially for future experiments, which will probe high multipoles (l > 2000), where the kSZ signal is expected to be relevant.

Similar quadratic estimators can be derived for all possible combinations of the CMB temperature and the polarisation fields, T T , T E, T B, EE, and EB (seeHu and Okamoto 2002). In addition, they can all be jointly combined into a minimum-variance quadratic estimator. Their associated reconstruction noise per mode (N (L) in Eq. 1.50 for the T T estimator, with analogous expressions for the other estimators) for the Planck 2015 lensing analysis are shown, as a function of lensing multipole, L, in Figure 1.10. It can be seen how, for Planck, the T T estimator dominates the total signal-to-noise, with a significant contribution from polarisation only on large scales. The situation, however, will be different in future experiments, in which polarisation will make a significant part, if not the most important part, of the total signal-to-noise (see, e.g., Abazajian et al. 2016and Simons Observatory Collaboration 2019). Furthermore, polarisation also provides different sensitivity to foreground contamination, being in general more robust against biases from uncleaned foregrounds. For instance, whereas the kSZ effect is a difficult-to-remove source of bias in the temperature at high l, the polarised total SZ signal is very small, about 10–100 nK (Carlstrom et al.,2011), so polarisation estimators are essentially unaffected by it.

Due to their simplicity and their computational efficiency, quadratic estimators have been widely used in lensing analyses. Moreover, for the resolution and noise levels of experiments like Planck, they are very close to optimal. In Chapter 2, for instance, we use the T T quadratic estimator, which is shown in its computationally efficient form in Eq. (2.3), in order to estimate the lensing convergence around the sky locations of the Planck galaxy clusters. However, for future experiments, the quadratic estimators are suboptimal. More optimal approaches, in which the full lensed CMB likelihood (or posterior) is maximised, or explored, have been developed (see Anderes et al. 2015,

Carron and Lewis 2017, and Millea et al. 2019).

CMB lensing reconstruction allows one to obtain the lensing convergence power spectrum, Cκκ

l , from which constraints on parameters such as Ωm and σ8 can be derived

(e.g., Planck 2018 results VIII 2018). In addition, reconstructed lensing maps can also be used for delensing (e.g., Carron et al. 2017), and for cross-correlation with other tracers of the cosmic density field, such as galaxies (e.g., Bianchini et al. 2015),

Fig. 1.10 CMB lensing quadratic estimator reconstruction noise as a function of lensing multipole, L, for the different quadratic estimators and their minimum-variance combination for the Planck 2015 lensing analysis. In addition, the lensing potential power spectrum is shown in black. Figure credit: Planck 2015 results XV (2016).

quasars (e.g., Hirata et al. 2008), the tSZ signal across the sky (e.g., Hill and Spergel 2014), the X-ray signal across the sky (e.g.,Hurier et al. 2019), and the cosmic infrared background (e.g.,Planck 2013 results XVIII 2014).