TOLERANCIAS EN DIMENS IONES DE ELEMENTOS Las siguientes figuras muestran las tolerancias dimensionales

Given the increasing sensitivity of current and future CMB experiments, there is at present a large interest in finding smaller, secondary signals in CMB observations, and in using them as cosmological probes. Some of these signals are generated as the CMB photons propagate from the last-scattering surface towards us; in this context, the CMB can be thought of as a ‘backlight’. These include the various SZ effects discussed above, the ISW and Rees-Sciama effects (Rees and Sciama 1968; see, e.g., Cai et al. 2010), and CMB lensing.

As the CMB photons free-stream from the last scattering surface towards us, they are deflected as a result of the gravity of the matter distribution through which they travel. This effect is known as CMB lensing (Blanchard and Schneider 1987; Cole and Efstathiou 1989; see Lewis and Challinor 2006for a review). First detected about a decade ago (Smith et al.,2007; Das et al.,2011), CMB lensing has now been detected to 40 σ and has become a powerful cosmological probe (Planck 2018 results VIII,2018).

Let X be an ‘unlensed’ CMB field, i.e., a CMB field as it would have been observed if there was no lensing, where X can be T , Q, or U . Lensing remaps the CMB fields so that the ‘lensed’ field ˜X(ˆn) along the line-of-sight direction ˆn is the unlensed field

at ˆn + α(ˆn), i.e., ˜X(ˆn) = X(ˆn + α).

The total deflection angle, α, can be seen as the addition of all the deflections due to all the matter that a given photon encounters along its way. Consider a deflection at a comoving distance χ from the observer, and let us assume that the CMB comes from a single source plane (the distance to last scattering is about 14 000 Mpc comoving, which is large compared to the width of the last-scattering surface, about 100 Mpc comoving, so this ought to be a good approximation, Lewis and Challinor 2006). The local deflection angle, δβ, can be written as δβ = −2∇⊥Ψδχ (e.g., Lewis and Challinor

Fig. 1.9 CMB lensing geometry. Here, δβ denotes the local deflection angle due to Ψ, which gives rise to an observed deflection angle δθ; χ and χ⋆ denote, respectively, the

comoving distances to the lens and to the source (the last scattering surface). Figure credit: Lewis and Challinor (2006).

2006), where Ψ is the Newtonian gravitational potential (or, in a general relativistic framework, the Weyl potential), δχ denotes a small comoving distance interval, and

∇⊥ denotes the two-dimensional gradient perpendicular to the direction of travel,

along which δχ runs. For simplicity, let us assume that the Universe is spatially flat. Following a simple geometric argument (see Figure 1.9), this local deflection angle translates into an observed deflection angle

δθ = χ⋆− χ χ⋆ δβ = −2χ⋆− χ χ⋆ ∇⊥Ψδχ = −2 χ⋆− χ χ⋆χ ∇ˆnΨδχ, (1.43)

where χ⋆ is the comoving distance to the source, i.e., to last scattering, and where ∇nˆ

denotes the gradient with respect to the observation angular coordinates.

In the limit in which the deflection angles δβ and their angular derivatives are small, known as the ‘weak lensing’ limit, the total deflection angle, α, can be computed by adding all the deflections evaluated along the undeflected line of sight (the Born approximation; see Lewis and Challinor 2006). That is,

α = −2∇nˆ Z χ⋆ 0 χ⋆− χ χ⋆χ Ψ(χˆn; z(χ))dχ ≡ ∇nˆψ, (1.44)

where Ψ is evaluated at the appropriate location, χˆn, and redshift, z(χ). The lensing

the gravitational potential along the line of sight all the way back to last scattering, weighted by the lensing kernel.

An important quantity to consider is the Jacobian of the deflected directions, which contains information about how the sizes and shapes of the anisotropies are distorted. To linear order, it can be written as (Lewis and Challinor, 2006)

Aij ≡ δij + ∇iαj =   1 − κ − γ1 −γ2 −γ2 1 − κ + γ1  . (1.45)

Here, κ is the lensing convergence, which is responsible for the trace of the distortion, i.e., it quantifies the apparent local size change due to lensing. On the other hand,

γ1 and γ2 are the two components of the lensing shear, which are responsible for the

traceless distortion, i.e., they quantify the apparent area-preserving local shape change. Both quantities can be obtained from the lensing potential, and thus contain the same information. In particular, the convergence can be written as κ = −1₂∇2

ˆ

nψ, which is a

projected version of Poisson’s equation. By using the definition of ψ (see Eq. 1.44) in this equation, and then using Poisson’s equation, it can be seen that the lensing convergence is a weighted integral of the matter distribution along the line of sight.

CMB lensing is a small effect, with associated changes of the order of 10 µK in the observed CMB temperature. The deflections are typically small, of about a few arcmin for linear LSS and for galaxy clusters. Compact objects such as black holes can produce much larger deflection angles, but only a small fraction of lines of sight pass near them, so they can be ignored. The weak lensing approximation is thus expected to be a good approximation, at least for current observations (Lewis and Challinor,2006). However, despite being small, the lensing deflections are correlated over larger scales. For linear structure, the main source of CMB lensing, a typical potential has a size of around 300 Mpc comoving (about the peak of the matter power spectrum). If it is half-way back to the last scattering surface, χ ∼ 7 000 Mpc, its observed size is around 2◦ (Lewis and Challinor, 2006). The CMB lensing deflections are thus correlated over degree scales, which means that CMB observables are expected to be affected by lensing on these scales.

Indeed, the T T , EE, and T E power spectra are modified by lensing on degree scales due to lensing by LSS. In particular, the acoustic peaks are smoothed at a several percent level (Lewis and Challinor, 2006). Intuitively, lensing, by distorting the sizes and shapes of the hot and cold spots of the CMB anisotropies, ‘blurs’ the BAO scale, which translates into smoothing of the BAO oscillations in harmonic space. The total variance, however, is conserved, as lensing only remaps the anisotropies: lensing drives

power towards smaller scales, where power is increased. For the T T power spectrum, at l > 2000, where there is little unlensed power, lensing is in fact an order unity effect (Lewis and Challinor, 2006).

In addition, lensing of the E-mode polarisation field generates a B-mode polarisation field with an approximately white noise power spectrum on large scales, even if no primordial B-modes are present (Zaldarriaga and Seljak,1998). These lensing B-modes, whose detection was first reported in Hanson et al. (2013), have now been detected to 18.1 σ (Sayre et al., 2019), and currently constitute the main limitation for the detection of primordial, inflation-sourced B-modes. To look optimally for primordial B- modes, the CMB can be ‘delensed’ by remapping back the particular realisation of the anisotropies that we see with an estimate of the lensing potential, thus circumventing the B-mode sample variance due to lensing (e.g., Seljak and Hirata 2004).

Lensing also generates a non-Gaussian signal in the CMB anisotropies. In effect, even if the unlensed CMB and the lensing potential are assumed to be Gaussian, remapping of a Gaussian field by a Gaussian field results in a non-Gaussian field. This lensing-induced non-Gaussianity can be probed by measuring the three-point, and higher-order, correlation functions that it induces (or, equivalently, the bispectrum, trispectrum, etc. in harmonic space); see, e.g., Lewis and Challinor (2006).

Finally, it should be noted that lensing, as a purely geometric effect, does not alter the CMB frequency spectrum. Thus, multi-frequency observations cannot be used in order to separate the lensing signal from the unlensed CMB anisotropies, as is the case, e.g., of the tSZ effect. However, lensing can be probed at the map level thanks to our very good statistical knowledge of the angular distribution of the unlensed anisotropies, as we will see in the next section.