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Galaxy clusters leave a specific CMB lensing signature, which can be used in order to estimate their masses. A typical cluster produces CMB deflections of the order of a few arcmin around its location in the sky. On arcmin scales, the unlensed CMB is roughly a gradient, and therefore Eq. (1.46) is accurate. If the cluster lensing potential is roughly circularly symmetric, then, locally, the CMB as lensed by a cluster is a dipole centred at the cluster location and aligned with the direction of the unlensed gradient, oriented such that the amplitude of the unlensed gradient is decreased (Seljak and Zaldarriaga,

2000). The magnitude of this effect is proportional to both the unlensed gradient and the deflection field, and so depends on the cluster mass. The r.m.s. temperature gradient is about 10 µK arcmin−1, and so for typical deflections of about 1 arcmin, cluster lensing produces variations of about 10 µK. For polarisation, gradients are of about 1 µK arcmin−1, and therefore a lensing signal of about 1 µK is generated (Lewis and Challinor, 2006).

Current CMB experiments are not sensitive enough to enable the measurement of individual cluster masses to high signal-to-noise, as opposed to other techniques, such as galaxy weak lensing. However, ensemble-averaged cluster masses can be obtained to relatively high significance, which can prove very useful in the context of counts analyses, as they do not depend on the clusters’ dynamical state (see, e.g., Planck 2015 results XXIV 2016, and Chapters 2 and 3). CMB lensing cluster mass determination has other interesting intrinsic virtues. The signal-to-noise does not decrease significantly with redshift (e.g., Melin and Bartlett 2015), and it does not rely on there being a high density of background galaxies, an important limitation of galaxy weak lensing mass estimation at high redshift. This makes CMB lensing mass estimation especially useful in this regime. Moreover, CMB lensing masses and galaxy weak-lensing masses are affected by different systematic effects, which makes CMB lensing valuable at low redshift too. In particular, it does not suffer from galaxy misidentification and from photometric redshift errors. However, it is sensitive to contamination from residual tSZ signals (a cluster’s tSZ signal can be up to an order of magnitude larger than the lensing signal, although it can be in principle subtracted out; see, e.g., Melin and Bartlett 2015; Patil et al. 2020), and, most importantly, from the kSZ signal (e.g.,

Melin and Bartlett 2015; Raghunathan et al. 2017, 2019a). Other potential sources of bias include miscentering of the cluster profile used to model the lensing signal, and deviations of this profile from the true cluster profile (see, e.g., Raghunathan et al. 2017, and Chapters 2, 3, and 4).

Several approaches have been proposed in order to estimate cluster masses from CMB lensing. One possibility is to reconstruct the lensing convergence in a non- parametric way, using the quadratic estimators or any other technique, and then to fit a (mass-dependent) cluster convergence model, as proposed, e.g., in Hu et al. (2007) and Melin and Bartlett(2015). This was the approach followed inPlanck 2015 results XXIV(2016), in which the method presented inMelin and Bartlett (2015), using a T T quadratic estimator and matched-filtering with a convergence model, was applied to estimate the average mass of 433 tSZ-detected Planck clusters, obtaining a 5 σ detection of cluster CMB lensing. Other works in which the cluster convergence is estimated non-parametrically include Baxter et al. (2017), in which the CMB lensing signal of 3 697 clusters optically-selected from the Dark Energy Survey (DES) is detected to 8.1 σ significance from SPT temperature data, and Geach and Peacock (2017), in which the masses of 26 111 optically-selected SDSS clusters, estimated from Planck convergence maps, are used in order to calibrate their mass-richness relation at the 10 %

level. We also follow an approach in which the cluster convergence is reconstructed non-parametrically in Chapter 2.

Other approaches do not include an intermediate CMB lensing reconstruction step, but model the cluster CMB lensing signal directly. These include maximum-likelihood methods, such as those proposed in Lewis and King (2006), Yoo and Zaldarriaga

(2008), and Yoo et al. (2010), and the method applied in Baxter et al. (2015), in which the CMB lensing signal of 513 tSZ-detected SPT clusters is detected to 3.1 σ significance using SPT temperature data. Parametric Bayesian methods have also been proposed (e.g., Raghunathan et al. 2017). Although quadratic-estimator-based methods are nearly optimal for experiments such as Planck, maximum-likelihood and Bayesian methods will outperform them in future experiments such as CMB-S4 (see, e.g., Raghunathan et al. 2017).

Methods in which the unlensed CMB gradient is measured and ‘factored out’ of the high-pass-filtered observed temperature field in order to measure directly the cluster deflection field have also been proposed (Seljak and Zaldarriaga 2000; Vale et al. 2004;

Horowitz et al. 2019), and are also expected to outperform quadratic-estimator-based methods in the future (see, e.g., Horowitz et al. 2019). They rely on the fact that the unlensed CMB gradient has little contribution from scales smaller than l ∼ 2000, and that cluster lensing barely modifies this gradient on scales l < 2000, so it can be in principle measured directly from observations and ‘fitted out’. A method along these lines was applied in Raghunathan et al. (2019c) to SPTpol polarisation maps, detecting the CMB lensing signal of 18 000 DES clusters to 4.8 σ, the first detection of polarisation-only-based CMB lensing by clusters.

CMB lensing by halos smaller than galaxy clusters has also been detected, with

Madhavacheril et al. (2015) reporting a 3.2 σ detection of the CMB lensing signal of 12 000 optically-selected CMASS galaxies from the SDSS-III/BOSS survey with ACTPol temperature data, the first detection of CMB lensing by compact objects. There have also been recent first detections of CMB lensing by filaments (He et al.,

2018) and by voids (Cai et al., 2017;Raghunathan et al., 2020).