Lensing anomalies from the epoch of reionisation

TL;DR

This paper investigates reionisation blurring, a non-linear correction to the CMB similar to lensing, analyzing its impact and correlation with lensing potential, and its potential to cause observable anomalies.

Contribution

It provides the first comprehensive calculation of reionisation blurring's effect on the CMB temperature power spectrum and its correlation with lensing potential.

Findings

Reionisation blurring's impact is about two orders of magnitude smaller than lensing.

Reionisation blurring is strongly correlated with lensing potential.

It can potentially induce observable lensing anomalies.

Abstract

Reionisation blurring is a non-linear correction to the cosmic microwave background that acts similar to weak gravitational lensing and that can be computed from linear perturbations through a blurring potential. Its impact on the cosmic microwave background is roughly two order of magnitude smaller than that of lensing, in isolation. But the blurring potential is strongly correlated with the lensing potential thereby generating a potentially observable cross-correlation. We compute for the first time the inclusive impact of reionisation blurring on the temperature angular power spectrum and discuss how much it could induce lensing anomalies

Tables

Table 1. Table 1 : Free parameters modelling uncertainties on perturbed ionization spectra, with respective top-hat prior edges. More details can be found in the appendix  A .

Name	prior range	description
$a_1$	$[-1,1]$	correlation sign between ${\delta }_b$ and ${\delta }_{x_e}$
$a_2$	$[0,20]$	overall ${\tilde{x}}_e$-dependent amplitude of ${\delta }_{x_e}$
$R_1$	$[-500,500]$	linear bubble size (in Mpc)
$R_2$	$[-500,500]$	quadratic bubble size (in Mpc)
$R_3$	$[-500,500]$	cubic bubble size (in Mpc)
${\alpha }_0$	$[-2,100]$	$k$-dependent weight
${\alpha }_1$	$[0,100]$	${\tilde{x}}_e$-dependent correction
${\gamma }_0$	$[-10,55]$	power-law typical value
${\gamma }_1$	$[2,10]$	${\tilde{x}}_e$-dependent correction
${\gamma }_2$	$[1,10]$	logarithmic correction

Table 2. Table 2 : Best fit values for the parameters entering into the functional dependency of the ionising power spectra of equations ( 2.16 ) and ( 2.17 ) with respect to the ionised fraction x 𝑥 x . The functions A ​ ( x ) 𝐴 𝑥 A(x) encode their amplitude, R ​ ( x ) 𝑅 𝑥 R(x) the typical size of ionised structures while the other functions represent different power-law exponents.

Function	Parameter	Value	Parameter	Value	Parameter	Value
$A_i\left(x\right)$	$a_{i_1}$	$0.09$	$a_{i_2}$	$9.64$
$R_i\left(x\right)$	$R_{i_1}$	$17.81\mathrm{Mpc}$	$R_{i_2}$	$-58.03\mathrm{Mpc}$	$R_{i_3}$	$55.29\mathrm{Mpc}$
${\alpha }_i\left(x\right)$	${\alpha }_{i_0}$	$-2.93$	${\alpha }_{i_1}$	$10.04$
${\gamma }_i\left(x\right)$	${\gamma }_{i_0}$	$16.13$	${\gamma }_{i_1}$	$2.20$	${\gamma }_{i_2}$	$1.23$
$A_{\times }\left(x\right)$	$a_{{\times }_1}$	$0.306$	$a_{{\times }_2}$	$4.727$
$R_{\times }\left(x\right)$	$R_{{\times }_0}$	$0.59\mathrm{Mpc}$	$R_{{\times }_1}$	$26.24$	$R_{{\times }_2}$	$0.66$
${\alpha }_{\times }\left(x\right)$	${\alpha }_{{\times }_0}$	$-0.10$	${\alpha }_{{\times }_2}$	$10.73$

Equations

$$\mathrm{d}{s}^2=a^2\left\{(1+2A)\mathrm{d}{\eta }^2+2B_i\mathrm{d}\eta \mathrm{d}{x}^i-\left[(1+2D){\delta }_{ij}+2E_{ij}\right]\mathrm{d}{x}^i\mathrm{d}{x}^j\right\},$$

$$T_{\mu \nu }=(\rho +P)u_{\mu }{u}_{\nu }-P{g}_{\mu \nu }+{\Sigma }_{\mu \nu },$$

$$\Delta (\eta ,\mathbf{x},\mathbf{n})\equiv \frac{\int \mathrm{d}q{q}^{3}f(\eta ,x,q\mathbf{n})}{\int \mathrm{d}q{q}^3{f}^{(0)}(q)},$$

$$A(\mathbf{x})=\int \frac{{\mathrm{d}}^3\mathbf{k}}{(2\pi {)}^3}{e}^{i\mathbf{k}\cdot \mathbf{x}}A(\mathbf{k}),$$

$$A({\mathbf{k}}_1)\cdot B({\mathbf{k}}_2)\equiv \int \frac{{\mathrm{d}}^3{\mathbf{k}}_1}{(2\pi {)}^3}\int \frac{{\mathrm{d}}^3{\mathbf{k}}_2}{(2\pi {)}^3}(2\pi {)}^3{\delta }^3(\mathbf{k}-{\mathbf{k}}_1-{\mathbf{k}}_2)A({\mathbf{k}}_1)B({\mathbf{k}}_2).$$

$${\dot{\Delta }}^{(2)}+i\mathbf{n}\cdot \mathbf{k}{\Delta }^{(2)}=-|\dot{\kappa }|{\Delta }^{(2)}-|\dot{\kappa }|\left(A^{(1)}+{\delta }_{\mathrm{b}}^{(1)}+{\delta }_{x_{\mathrm{e}}}^{(1)}-\mathbf{n}\cdot {\mathbf{v}}_{\mathrm{b}}^{(1)}\right)({\mathbf{k}}_1)\cdot {\Delta }^{(1)}({\mathbf{k}}_2),$$

$${\kappa }_{\mathrm{blur}}^{(1)}(\eta ,{\mathbf{k}}_1,\mathbf{n})$$

$$\times \left[\mathbf{n}\cdot {\mathbf{v}}_{\mathrm{b}}^{(1)}({\eta }^{\prime },{\mathbf{k}}_1)-A^{(1)}({\eta }^{\prime },{\mathbf{k}}_1)-{\delta }_{\mathrm{b}}^{(1)}({\eta }^{\prime },{\mathbf{k}}_1)-{\delta }_{x_{\mathrm{e}}}^{(1)}({\eta }^{\prime },{\mathbf{k}}_1)\right],$$

$${\Delta }^{(2)}(\eta ,k)=-{\kappa }_{\mathrm{blur}}^{(1)}({\mathbf{k}}_1)\cdot {\Delta }^{(1)}({\mathbf{k}}_2).$$

$${\Delta }^{(2)}(\eta ,\mathbf{x},\mathbf{n})=-{\kappa }_{\mathrm{blur}}^{(1)}(\eta ,\mathbf{x},\mathbf{n}){\Delta }^{(1)}(\eta ,\mathbf{x},\mathbf{n}).$$

$$\Delta (\eta ,\mathbf{x},\mathbf{n})$$

$$\simeq e^{-\kappa (\eta )}\left[{\Delta }^{(1)}(\eta ,\mathbf{x},\mathbf{n})-{\kappa }_{\mathrm{blur}}^{(1)}(\eta ,\mathbf{x},\mathbf{n}){\Delta }^{(1)}(\eta ,\mathbf{x},\mathbf{n})\right].$$

$${\Delta }_{\mathrm{lens}}^{(2)}(\eta ,\mathbf{x},\mathbf{n})={\nabla }^a{\Psi }_{\mathrm{lens}}^{(1)}(\eta ,\mathbf{x},\mathbf{n}){\nabla }_a{\Delta }^{(1)}(\eta ,\mathbf{x},\mathbf{n}),$$

$${\Delta }_{\mathrm{blur},I}^{(2)}(l)=-\int \frac{{\mathrm{d}}^2{l}^{\prime }}{(2\pi {)}^2}{\kappa }_{\mathrm{blur}}^{(1)}(l-l^{\prime }){\Delta }_I^{(1)}(l^{\prime }),$$

$${\mathcal{C}}_{\mathrm{blur},I}(l)=\int \frac{{\mathrm{d}}^2{l}^{\prime }}{(2\pi {)}^2}{\mathcal{C}}^{\kappa }(l-l^{\prime }){\mathcal{C}}_I(l^{\prime })-{\mathcal{C}}_I(l)\int \frac{{\mathrm{d}}^2{l}^{\prime }}{(2\pi {)}^2}{\mathcal{C}}^{\kappa }(l^{\prime }),$$

$${\mathcal{C}}_{\mathrm{lens},I}(l)=\int \frac{{\mathrm{d}}^2{l}^{\prime }}{(2\pi {)}^2}{\left[{\mathbf{l}}^{\prime }\cdot (\mathbf{l}-{\mathbf{l}}^{\prime })\right]}^2{\mathcal{C}}^{\Psi }(l-l^{\prime }){\mathcal{C}}_I(l^{\prime })-{\mathcal{C}}_I(l)\int \frac{{\mathrm{d}}^2{l}^{\prime }}{(2\pi {)}^2}(\mathbf{l}\cdot {\mathbf{l}}^{\prime }{)}^2{\mathcal{C}}^{\Psi }(l^{\prime }).$$

$${\mathcal{C}}_{\mathrm{blur}-\mathrm{lens},I}(l)=\int \frac{{\mathrm{d}}^2{l}^{\prime }}{(2\pi {)}^2}\left[{\mathbf{l}}^{\prime }\cdot (\mathbf{l}-{\mathbf{l}}^{\prime })\right]{\mathcal{C}}^{\kappa \Psi }(l-l^{\prime }){\mathcal{C}}_I(l^{\prime })-{\mathcal{C}}_I(l)\int \frac{{\mathrm{d}}^2{l}^{\prime }}{(2\pi {)}^2}(\mathbf{l}\cdot {\mathbf{l}}^{\prime }){\mathcal{C}}^{\kappa \Psi }(l^{\prime }).$$

$${\tilde{x}}_{\mathrm{e}}^2{P}_{{\delta }_{x_{\mathrm{e}}}{\delta }_{x_{\mathrm{e}}}}(\eta ,k)$$

$${\tilde{x}}_{\mathrm{e}}{P}_{{\delta }_{\mathrm{b}}{\delta }_{x_{\mathrm{e}}}}(\eta ,k)$$

$${\delta }_{x_{\mathrm{e}}}(\eta ,k)=A({\tilde{x}}_{\mathrm{e}})\left(1-{\tilde{x}}_{\mathrm{e}}\right){\left\{1+\alpha ({\tilde{x}}_{\mathrm{e}})kR({\tilde{x}}_{\mathrm{e}})+{\left[kR{\tilde{x}}_{\mathrm{e}})\right]}^2\right\}}^{-\gamma ({\tilde{x}}_{\mathrm{e}})/4}{\delta }_{\mathrm{b}}(\eta ,k).$$

$$A_{\mathrm{i}}^2(x)$$

$$R_{\mathrm{i}}(x)$$

$${\alpha }_{\mathrm{i}}(x)$$

$${\gamma }_{\mathrm{i}}(x)$$

$${\delta }_{x_{\mathrm{e}}}(\eta ,k)=A({\tilde{x}}_{\mathrm{e}})\left(1-{\tilde{x}}_{\mathrm{e}}\right){\left\{1+\alpha ({\tilde{x}}_{\mathrm{e}})k{R}_{\mathrm{i}}({\tilde{x}}_{\mathrm{e}})+{\left[kR({\tilde{x}}_{\mathrm{e}})\right]}^2\right\}}^{-\gamma ({\tilde{x}}_{\mathrm{e}})/4}{\delta }_{\mathrm{b}}(\eta ,k).$$

$$A(x)$$

$$R(x)$$

$$\alpha (x)$$

$$\gamma (x)$$

Full text

1 Introduction

The cosmic microwave background has shaped our understanding of the Universe over the past decades and remains of central importance for precision cosmology. Parameter estimation relies crucially on the Planck data [1] and thus the accurate analysis of secondary effects that may contaminate the cosmic microwave background (CMB) is an important task. Secondary CMB anisotropies are not only foregrounds for cosmological data analysis, they also carry additional and unique information on the late time universe and the non-linear growth of cosmic structures. The Planck data are accurate enough to be sensitive to various of these secondary anisotropies. The non-Gaussianities contained in the CMB have been measured for the first time in Ref. [2], they are primarily induced by weak gravitational lensing which describes the bending of light by the gravitational potential of the matter along the line-of-sight [3, 4]. Planck is also sensitive to the Cosmic Infrared Background (CIB) and thousands of galaxy clusters have been detected through localised Sunyaev-Zel’dovich (SZ) signals in the CMB maps [5, 6]. Some cosmological parameters can also be extracted from SZ cluster counts and lensing measurements, alone, and this allows for an intricate testing of the $\Lambda$CDM model over the whole cosmological history. Although the lensing data are in agreement with CMB primary anisotropies on a wide range of length scales, a three-sigma deficit in “curl power” has been consistently reported within the multipole range $\ell\in[264,901]$ [7, 8].

The reionisation of the universe by the first stars is another source of unavoidable secondary CMB anisotropies which is usually accounted for in data analysis through an unique homogeneous parameter, the reionisation optical depth. It describes the suppression of perturbations and, although strongly correlated with the primordial power spectrum amplitude, its value can be accurately inferred from the $E$-mode polarisation induced by the free electrons created during the Epoch of Reionisation (EoR) [9]. By definition, reionisation is a highly non-linear process in which the physics of very non-linear structures (stars) backreacts onto the largest length scales of the universe. One may therefore question the accuracy of modelling reionisation effects on the CMB by only the homogeneous optical depth [10], while in reality reionisation is always a patchy process. In Ref. [11], we have shown that anisotropies in the reionisation optical depth induces various new secondary distortions on the CMB anisotropies. They are induced by inhomogeneities in the baryon density, in the ionisation fraction, in gravitational redshift, while some of these effects can be identified as the diffuse analogues of blurring and kinetic SZ effects [12, 13, 14, 15]. Because inhomogeneities in the ionisation fraction trace cosmic structures, one may wonder if it could not be the source of the systematic lensing deviations currently measured in the CMB data. Among all these effects, the diffuse blurring at reionisation could be the main suspect. It is a loss term, i.e., it describes an incoming bundle of light rays which is scattered out of the observer line-of-sight due to, from some parts, collisions with ionised gas created by the ionising radiation of the first stars. As we discuss below, it is expected to be correlated with lensing and this correlation has never been estimated before. Let us stress that the impact of a ionised clouds has been well studied in the literature [16, 17, 18], focusing on the so-called “screening” contribution related to the ionisation fraction and density of the absorbing cloud. This is the main driver of the blurring effect, however, in this analysis, we consider two additional terms that have not been studied before; a relativistic and a kinetic correction that naturally appear from our inclusive second-order Boltzmann analysis. These terms are particularly relevant on intermediate scales, around $l=100$ [11].

Mathematically lensing can be described as a modulation of the primary CMB by the deflections that the gravitational potentials of the forming overdensities induce. Lensing is implemented in the most popular Boltzmann codes [19, 20], employing the decomposition of the non-linear effect into a convolution of two independent linear terms, the primary CMB and the lensing potential [4]. Let us notice that lensing is not the only modulation of the microwave background along the line-of-sight. Gravitational redshift modulations describe the impact of an inhomogeneous expansion, and time-delay effects change the distance to the last-scattering surface. But both are significantly smaller than lensing and cannot explain the observed missing power [21, 22, 23].

While the physical origin of lensing and reionisation blurring are different, the way both effects impact the CMB is similar. Lensing describes how the primary CMB is distorted by the late Universe gravitational potentials and blurring is the distortion accompanying ionised clouds. As for the lensing, we can implement the blurring effect by defining a blurring potential, that is sourced at late times and correlates with the matter overdensities [11]. Beyond linear perturbations, the probability to be scattered depends on the distribution of matter along the path of the CMB photons such that the overall effect is more than just dimming the amplitude, the angular power spectrum is distorted, or blurred. In Ref. [11], we have shown that this blurring contribution is the dominant second-order effect along the line-of-sight after lensing. It provides corrections that are typically two orders of magnitude smaller, a priori not large enough to be visible by Planck. But lensing and blurring may be correlated, generating a cross-term that could potentially cause much larger corrections (see Ref. [24] for an analysis based on the screening terms only).

In this paper, we derive and compute the non-linear blurring-lensing correlation for the intensity $I$, and we estimate whether it can lead to significant corrections to the lensing signal. The paper is organised as follows. In section 2 we summarise our notations, review the blurring effect, and describe our method to compute the cross-correlation with lensing. In section 3 we discuss our numerical results and perform a statistical analysis using the latest public Planck likelihoods and data, which currently are the 2015 ones [25]. Finally, we discuss our results and conclude.

2 Reionisation blurring

2.1 Notation and conventions

Our convention follows the one presented in Refs. [26, 27, 11] that we briefly recap below. The metric is assumed to be of the form

[TABLE]

where $A$ is the lapse perturbation, ${\boldsymbol{B}}\equiv\{B^{i}\}$ the shift vector, $D$ stands for the spatial trace perturbation and $E_{ij}$ is the symmetric spatial stress tensor. We use second-order perturbation theory in which the stress tensor, the Einstein equation and the Boltzmann equation are all expanded as $X=X^{(1)}+X^{(2)}+\cdots$. Simplifications are made using $B_{i}^{(1)}=E^{(1)}_{ij}=0$, corresponding to the so-called Poisson gauge at linear order. We will further assume that all non-scalar modes are of second or higher order.

The stress-energy tensor is decomposed as

[TABLE]

where $\rho$ is the energy density, $P$ the pressure density, $\Sigma_{\mu\nu}$ the anisotropic stress tensor and $u_{\mu}$ the rest-frame 4-velocity.

Cold species, such as massive particles, are uniquely described by their energy density $\rho$ and their 3-velocity ${\boldsymbol{v}}$. The latter is defined as the spatial part of $u$ at linear order, $v^{(1)}_{i}\equiv au^{(1)}_{i}$. On the contrary, the complex phase-space of relativistic species requires to specify the full distribution function $f(\eta,x,q{\boldsymbol{n}})$ where the comoving momentum ${\boldsymbol{q}}=a{\boldsymbol{p}}$ has been expressed as a direction ${\boldsymbol{n}}$ and a magnitude $q$. We define the integrated distribution

[TABLE]

that can be understood as a temperature perturbation and which contains all the information required to evaluate the stress-energy tensor of a relativistic specie. We use the Fourier conventions

[TABLE]

Finally, we use a shortcut notation for convolutions which appear as multiplications of different wavenumbers, namely

[TABLE]

2.2 Blurring, lensing and their correlation

The differential equation for non-linear blurring of the intensity perturbation to second order in perturbation theory, $\Delta^{(2)}$, has been derived in Ref. [11] and reads, in Fourier space,

[TABLE]

where $\delta_{\mathrm{b}}=\delta\rho_{\mathrm{b}}/\bar{\rho}_{\mathrm{b}}$ is the relative perturbation to the baryon density, and $\delta_{x_{\mathrm{e}}}$ that to the ionisation fraction. The left-hand side describes the free propagation of photons in the direction ${\boldsymbol{n}}$ while the right-hand side represents collisions scattering photons out of the observer line-of-sight. The first term is “purely” second-order. It features an isotropic collision rate $|\dot{\kappa}|$, and it leads to an overall suppression similar in all points to the linear theory. The following terms describe an anisotropic modulation of the reaction rate. They enhance the scattering probability in overdense regions, $\delta_{\mathrm{b}}^{(1)}>0$, and in ionised patches, $\delta^{(1)}_{x_{\mathrm{e}}}>0$. These are commonly referred to as “screening” while the term in $\delta_{\mathrm{b}}^{(1)}$ is the diffuse analog of the so-called blurring SZ effect [14, 11]. There are further a kinematic enhancement based on the motion of the ionised gas with respect to the CMB rest frame, ${\boldsymbol{n}}\cdot{\boldsymbol{v}}^{(1)}_{\mathrm{b}}$ (blurring Doppler term), plus a relativistic correction $A^{(1)}$ due to shift between the cosmological time $\eta$ and the time coordinate of the inertial frame comoving with the baryons (blurring GR term).

As explained in Ref. [11], we may integrate the probability for collisions along the line-of-sight. This allows us to define the blurring potential:

[TABLE]

with the help of which equation (2.6) can be integrated as

[TABLE]

In real space, we then find a simple equation separating the second-order terms into a product of linear perturbations at the present time

[TABLE]

This expression shows that the blurring potential mathematically appears as an inhomogeneous correction to the usual (homogeneous) optical depth $\kappa$. As such, at second order, it can be conveniently included in the line of sight integrations by exponentiation

[TABLE]

where the first term gives back the linear result and the second term equation (2.9). Defining the blurring potential as above implicitly requires that we neglect any reionisation blurring on the late-time Integrated Sachs-Wolf (ISW) perturbations. This additional effect may lead to small corrections on the largest angular scales, but the same approximation is routinely applied in the standard lensing computation and is necessary to be able to separate lensing into a product of two linear terms without an intrinsic second-order contribution. Our equation indeed shows a remarkable similarity to lensing, where the angular power spectrum can also be written as a modulation of the linear result [4]:

[TABLE]

with $\nabla_{a}$ the angular derivative .

Next we employ the flat sky limit, accurate on scales $l>10$. In this limit, the curvature of the celestial sphere is neglected, simplifying the angular multipole decomposition into a Fourier analysis. For the blurring, the flat sky limits gives

[TABLE]

where ${\boldsymbol{l}}$ is a 2-vector parametrising the flat sky, that would be replaced by a discrete set of multipole coefficients $(l,m)$ on the sphere.

A reasonable hypothesis is to assume that the correlation between $\kappa_{\mathrm{blur}}$ and $\Delta^{(1)}$ is suppressed since the linear perturbations arise from the time of recombination in the early Universe while the blurring potential is sourced only in the late Universe. In the flat sky limit, the angular power spectra for the intensity is then given by

[TABLE]

where the last term is obtained by expanding equation (2.10) up to second order and with the present day angular temperature power spectrum $16\delta(l-l^{\prime})\mathcal{C}_{I}(l)=\langle\Delta(l)\Delta(l^{\prime})^{*}\rangle$ and the present day spectra of the blurring and lensing potentials, $\delta(l-l^{\prime})\mathcal{C}^{\Psi}(l)=\langle\Psi(l)\Psi^{*}(l^{\prime})\rangle$, $\delta(l-l^{\prime})\mathcal{C}^{\kappa}(l)=\langle\kappa^{\phantom{*}}_{\mathrm{blur}}(l)\kappa^{*}_{\mathrm{blur}}(l^{\prime})\rangle$ and $\delta(l-l^{\prime})\mathcal{C}^{\kappa\Psi}(l)=\langle\kappa_{\mathrm{blur}}(l)\Psi^{*}(l^{\prime})\rangle$. This allows us to compare this expression with the one used to compute the lensing, namely [4]

[TABLE]

The only differences are exchanging the blurring potential for the lensing one and the appearance of additional factors ${\boldsymbol{l}}$ due to the angular derivatives.

The correlation between lensing and blurring can be derived in a similar manner and one gets

[TABLE]

Notice that the second term vanishes due to the antisymmetry of the integrand in ${\boldsymbol{l}}^{\prime}$.

The blurring potential can be computed using a linear Boltzmann code following the implementation of the lensing potential. In the flat sky limit the angular power spectrum can then be constructed via the above integrals. We have implemented the blurring computation using our equations in the Boltzmann code class [28, 20], in the Newtonian gauge.

2.3 Epoch of Reionisation

While most of the sources for the blurring potential can be directly extracted from the linear perturbations computed in class, the perturbation of the ionisation fraction $\delta_{x_{\mathrm{e}}}$ requires a different framework. We have used a similar approach as in Refs. [29, 30, 11], namely, using analytical expressions fitted against numerical simulation results [31, 32, 33, 34, 35, 36, 37]. More precisely, the functional form of the relevant power spectra is the same as in Ref. [29] and given by

[TABLE]

where $\tilde{x}_{\mathrm{e}}(\eta)\equiv x_{\mathrm{e}}(\eta)/x_{\mathrm{e}}(\eta_{0})$ is the background normalised ionised fraction, and $x_{\mathrm{e}}(\eta_{0})$ is the total ionised fraction once the Universe is completely reionised. The various functions of $\tilde{x}_{\mathrm{e}}$ entering equations (2.16) and (2.17) encode various physical evolving properties. For instance the length scale $R(x_{\mathrm{e}})$ keeps track of the typical ionised bubble size while $P_{\delta_{\mathrm{b}}}(\eta,k)$ is the baryon power spectrum whose evolution is solved in class. The fitting form, with respect to $\tilde{x}_{\mathrm{e}}$, of all the functions appearing in equations (2.17) are detailed in Appendix A.

In figure 1, we have represented the typical evolution of these spectra for a given reionisation model with an optical depth $\tau_{\mathrm{reio}}=0.0952$ and a reionisation redshift $z_{\mathrm{reio}}\simeq 12$. This figure shows that, up to different relative amplitudes, the spectrum of the ionisation fraction as well as its correlation with the baryons, and the power spectrum of baryons all evolve in parallel up to some relatively small scales $k\gtrsim 0.2$ (from the CMB point of view). For this reason, we have adopted a much simplified model and, in class, we choose directly $\delta_{x_{\mathrm{e}}}$ to be of a functional form given by the square root of its power spectrum, namely

[TABLE]

The unknown functions in this equation, $A(\tilde{x}_{\mathrm{e}})$, $\alpha(\tilde{x}_{\mathrm{e}})$, $R(\tilde{x}_{\mathrm{e}})$ and $\gamma(\tilde{x}_{\mathrm{e}})$ should be viewed as nuisance. They have been chosen of the same functional form as their $A_{\mathrm{i}}(\tilde{x}_{\mathrm{e}})$, $\alpha_{\mathrm{i}}(\tilde{x}_{\mathrm{e}})$, $R_{\mathrm{i}}(\tilde{x}_{\mathrm{e}})$ and $\gamma_{\mathrm{i}}(\tilde{x}_{\mathrm{e}})$ counterparts but instead of being fitted to reionisation simulation data, we let all their parameters free to vary over some conservative prior range encompassing simulation results. In total, we end up with ten new free parameters. More details on the modelling are given in appendix A. Let us stress again that our goal here is not to have the most accurate reionisation sources model but rather to marginalise over the nuisances they are expected to induce, see section 4.

3 Lensing anomalies

By computing the level of correlation between blurring and lensing, we are in the position to calculate the total impact of blurring on the angular power spectrum. Including lensing, but omitting blurring in theoretical predictions, is therefore expected to artificially induce deviations between the measured power spectrum and the predictions, thereby potentially triggering an artificial “lensing anomaly”. In this section, we calculate the amplitude of such an effect onto the power spectrum.

3.1 Lensing and blurring potentials

The correlation between the blurring and lensing potential comes from the fact that both effects are induced by the late Universe matter overdensities. These overdensities source the gravitational potentials, which are responsible for lensing, and are the seeds of reionisation.

In order to solve equation (2.15), one needs the correlator $\mathcal{C}^{\kappa\Psi}$. Both potentials, $\kappa_{\mathrm{blur}}$ and $\phi$, can be computed from the linear perturbations, and thus by using class. As discussed before, the only caveat in the calculation of blurring is the use of a analytical fit for the perturbed ionisation fraction, see equations (2.18). Taking the best fit values mentioned in the appendix A, our results for the spectrum of the two potentials and their cross-correlation are shown in figure 2. By comparing the three curves, we see that blurring and lensing are correlated on all but the largest scales. This is expected because, on very large scales, reionisation is no longer tracing overdensities, but becomes dominated by the expansion of the ionised bubbles instead. Note that these scales are particularly relevant for lensing.

3.2 Lensing-blurring corrections to the angular power spectrum

Having computed the auto- and cross-correlation spectra of the the blurring and lensing potentials, we employ equations (2.13) to (2.15) to obtain the final impact of lensing and blurring on the temperature angular power spectrum, shown in the lower panel of figure 2, where the effect of blurring and of its correlation to lensing have been multiplied by a factor of $100$ for better visibility. We notice that the correlation deceptively ends up being of similar magnitude as the blurring effect. Looking at the potential spectra in the upper panel of figure 2, one could have expected the correlation effect to be much larger than the blurring one. This ends up not being the case due to the absence of correlation between the lensing and blurring potentials on the largest scales, which are responsible for a significant part of the lensing signal.

The correlation also shows oscillations comparable to the lensing signal. In principle, when fitting the observational data (which are affected by all these effects) by a theoretical model in which blurring is neglected, one would therefore expect to get a small mismatch, that could be misinterpreted as an excess, or a deficit, of lensing. This could be the case especially if the oscillations in the lensing and in the lensing-blurring correlation signal had the same phase. However, the lower panel of figure 2 shows that the lensing-blurring oscillations are shifted in phase, roughly by $\pi/2$. It is thus less clear whether the lensing-blurring effect can be confused with a lensing effect of different magnitude, or if this shift could actually be preferred by the data. Furthermore the amplitude of these corrections is small, typically around $1\%$.

4 CMB data analysis with lensing-blurring

Because reionisation blurring is a non-speculative contribution that is present in the CMB, we now address the question whether neglecting it may bias the extracted cosmological parameters, or could be misinterpreted as some kind of anomaly in the data.

For this purpose, we fitted the Planck 2015 TT+lowP data [25] to different models and sets of free parameters, using our modified version of class111http://class-code.net v2.6.3 [20] in combination with the parameter extraction code MontePython222https://github.com/brinckmann/montepython_public v3.1.0 [38, 39]. As discussed in section 2.3, the details of the reionisation which affect the blurring are encoded in ten nuisance parameters that we incorporate in the data analysis. This allows us to check if CMB observations prefer some inhomogeneous reionisation evolution. Note that we do not include the Planck CMB lensing likelihood in our analysis. This likelihood accounts for the extraction of the lensing signal from the trispectrum of observed temperature and polarization anisotropies. In principle, blurring effects also contribute to this trispectrum, but the standard lensing extraction process neglects this contribution. It is beyond the scope of this paper to compute whether the blurring effect has a significant impact on the lensing extraction pipeline or not, and whether the CMB lensing estimator should be corrected accordingly. Thus, we conservatively ignore lensing extraction, and focus only on the temperature (and low-l polarization) power spectra measured by Planck. See Refs. [40, 41] for searches on the trispectrum that would be induced by the screening terms only.

4.1 Bias on cosmological parameters

To check the impact of neglecting blurring on the measurement of standard cosmological parameters, we first fitted the data with the exact same baseline $\Lambda$CDM model as in [25] (with flat priors on the six standard parameters $\{\omega_{\mathrm{b}},\omega_{\mathrm{cdm}},\theta_{\mathrm{s}},\ln(10^{10}A_{\mathrm{s}}),n_{\mathrm{s}}$, $\tau_{\mathrm{reio}}\}$ and standard priors on the Planck nuisance parameters), switching off the blurring effect as in all previous analyses. We checked the level of agreement between runs performed with MontePython+class in the synchronous gauge, in the newtonian gauge (that we will always use when switching on blurring), and with CosmoMC+camb [42, 19] in the synchronous gauge. For the latter we use the results publicly available on the Planck Legacy Archive333https://pla.esac.esa.int, quoted in subsection 2.1 of the PDF parameter table document. All three cases are based on 8 Markov chains that have reached a Gelmann-Rubin convergence criterium $R-1\leq 0.01$ for all parameters. We find that between the two runs in the synchronous gauge, error bars always agree with each other within $5\%$, and mean values up to $\pm 0.09\sigma$. Between the class runs in the two different gauges, error bars agree within $4\%$, and mean values up to $\pm 0.09\sigma$ again. We conclude that in general, with standard precision settings and convergence criteria, results can be trusted up to approximately $\pm 0.1\sigma$ for the mean and $5\%$ for the error. This finite precision comes from: the level of convergence of the MCMC chains; numerical errors when deriving confidence limits from binned chains; and the precision of camb and class when they are used with default precision444With very high precision settings, but much longer computing times, the codes agree in the synchronous gauge up to a very high level that would not contribute to the present errors..

We then turned on the blurring and lensing-blurring effects, while adding the ten new free parameters encoding reionisation sources, (see appendix A), modelling our ignorance on reionisation details with conservative flat priors ranges shown in table 1. These prior ranges have been taken by multiplying (and dividing) the expected value by typically two orders of magnitude for amplitude parameters, and by one order of magnitude for power law exponents.

We ran again 8 Markov chains until all parameters reached a Gelmann-Rubin convergence criterium $R-1\leq 0.018$. Going even below would be extremely CPU-consuming due to presence of several reionisation parameters poorly constrained by the data. Compared to the previous run in the Newtonian gauge, the largest variation of the the error is for $(n_{\mathrm{s}},\tau_{\mathrm{reio}})$, with respectively a $(10\%,6\%)$ increase. All other errors vary by $5\%$ at most. The mean value of $\tau_{\mathrm{reio}}$ is shifted by 0.12$\sigma$, while all other shifts are below $\pm 0.1\sigma$. We conclude that within the accuracy of Planck and of a standard parameter extraction pipeline, blurring has a negligible impact on the measurement of cosmological parameters, excepted for a very small widening of the error on $\Lambda$CDM parameters by $\sim 10\%$ at most. In two-dimensional posteriors, we find a very small correlation between $\{n_{\mathrm{s}},\tau_{\mathrm{reio}},\ln(10^{10}A_{\mathrm{s}})\}$ and the reionisation parameter $a_{2}$ when $a_{2}$ exceeds $\sim 10$. This parameter has a very specific meaning: it controls the response of the ionisation-to-baryon bias $\delta_{x_{\mathrm{e}}}/\delta_{\mathrm{b}}$ to the average ionisation fraction $\tilde{x}_{\mathrm{e}}$ in the large wavelength limit. However, with our parametrisation and prior ranges, $a_{2}$ plays a simple role: it is the main parameter controlling the overall order of magnitude of the ionisation fluctuations $\delta_{x_{\mathrm{e}}}$. For relatively large value, $a_{2}\geq 10$, and thus for quite inhomogeneous reionisation scenarios, the posterior distribution for $n_{\mathrm{s}}$ gets a bit wider. The current fit to simulation being at $a_{2}=9.64$, this suggests that lensing-blurring may actually slightly affect the accuracy in the determination of the $\Lambda$CDM parameters.

We also checked that in most cases, the posterior of Planck nuisance parameters does not change significantly when blurring effects are switched on. Thus the absence of sensitivity of the data to blurring is really due to the smallness of the effect, rather that to marginalization over foregrounds and systematics. The only exception, clearly seen in two-dimensional contours, is a degeneracy between $a_{2}$ and the nuisance parameters modelling the amplitude of point-source foregrounds in difference frequency bands: $A^{\mathrm{PS}}_{100-100}$, $A^{\mathrm{PS}}_{143-143}$, $A^{\mathrm{PS}}_{143-217}$, $A^{\mathrm{PS}}_{217-217}$ [43]. For large values $a_{2}\geq 10$, these amplitudes can be reduced. This is particularly true at the frequency of $143\,\mathrm{GHz}$: we find that with $a_{2}\simeq 13$, the parameters $A^{\mathrm{PS}}_{143-143}$ and $A^{\mathrm{PS}}_{143-217}$ are compatible with zero. This means that a large blurring effect mimics the impact of point source contamination on the temperature spectrum. Blurring effects can thus be relevant at least in one case: when one tries to infer the point source temperature spectrum from CMB observations. But again, if one assumes the fiducial reionisation model for which $a_{2}=9.64$, the impact of such a correlation remains small.

Even though our Bayesian analysis of current data is only very weakly sensitive to blurring, there are some interesting parameter combinations in which the blurring effect is enhanced. This happens for example when the blurring potential is specifically tuned such that it is more strongly correlated with lensing. But because these are so finely tuned models they are irrelevant in the result of a Bayesian analysis.

4.2 Reionisation parameters

We investigated whether the blurring signal is sufficiently large to constrain the parameters of perturbed reionisation. We expect that when reionisation is assumed to be strongly inhomogeneous, the blurring effects will become too large to be compatible with the data. Indeed we do find a bound on one parameter: $a_{2}<16$ ($68\%$ confidence limit), assuming a flat prior on $a_{2}>0$). Compared to the fiducial value expected from reionisation simulations, $a_{2}=9.64$, such a limit confirms the weak sensitivity of the data to the amount of inhomogeneities. Let us notice however that both values are not orders of magnitude apart such that competitive bounds could be in the reach of the future CMB experiments.

The other reionisation parameters are, however, totally unconstrained in the ranges described in Table 1. This is not surprising, considering that we are only probing these parameters through small modulations of the CMB sky, and that we have introduced ten of them.

4.3 Investigating a possible connection with the $A_{\mathrm{L}}$ anomaly

The community identified a peculiarity in the Planck data. The plain $\Lambda$CDM model does provide a very good fit to the data, with a very satisfactory reduced $\chi^{2}$, $p$-value and level of residuals. In $\Lambda$CDM, the amplitude of the CMB lensing effect in the temperature power spectrum is fixed, because the six model parameters define the amplitude of the matter power spectrum in the recent universe, and thus the variance of the CMB lensing potential and of the associated lensing deflection field.

However, as an academic exercise, one can multiply the amplitude of the lensing potential spectrum used to compute the lensing effect in the temperature power spectrum by a factor $A_{\mathrm{L}}$. Values $A_{\mathrm{L}}\neq 1$ are in principle unphysical. It is however worth noticing that when $A_{\mathrm{L}}$ is treated as a free parameter, the data prefer higher than unity values, $A_{\mathrm{L}}=1.22\pm 0.10$ ($68\%$CL, Planck 2015 TT+lowP) [25] or $A_{\mathrm{L}}=1.243\pm 0.096$ ($68\%$CL, Planck 2018 TT+lowE) [1]. The level of tension with respect to $A_{\mathrm{L}}=1$ does not decrease when adding high-$\ell$ polarization data. All this means that assuming an artificially large level of smoothing of the temperature spectrum (and to a much lesser extent, of the $E$-polarisation spectrum) by lensing allows to absorb a non-negligible part of the residuals of the $\Lambda$CDM best fit. Note that the parameter $A_{\mathrm{L}}$ should not be confused with another parameter $A_{\mathrm{L}}^{\phi\phi}$ discussed in [25]. That parameter is a test of the compatibility between the temperature spectrum and the extracted CMB lensing spectrum. That test is independent of the previous one, and returns no anomalous behaviour.

It is not obvious that there is something to learn from this $A_{\mathrm{L}}$ test. A priori, we could transform each one of the few hundreds of coefficients appearing in the equations defining linear cosmological perturbation theory into a free parameter. We would then expect that $\sim 1.3\%$ of them have a best-fit value $2.5\sigma$ away from their theoretical value. $A_{\mathrm{L}}$ could just be within these $1.3\%$. On the other hand, it is always worth considering whether there could be a convincing physical explanation mimicking this effect and allowing to increase the likelihood of the best fit model.

The residuals of the best-fit $\Lambda$CDM model to the Planck data show some vaguely oscillatory features roughly in phase with lensing effects the range $1100<\ell<2000$ (see Fig. 24 in Ref. [1]). Since the blurring-lensing correlation effect also has oscillatory patterns, it is worth checking whether the observed “anomaly” could actually arise from neglecting this effect. Our strategy is to run simultaneously with blurring effects turned on and a free parameter $A_{\mathrm{L}}$. If the blurring effects could fit some of the residuals giving rise to the “$A_{\mathrm{L}}$ anomaly”, the posterior for $A_{\mathrm{L}}$ would become more compatible with $A_{\mathrm{L}}=1$ and the “anomaly” would be explained.

Our run with $17$ model parameters ($6$ $\Lambda$CDM parameters, $10$ perturbed reionisation parameters and $A_{\mathrm{L}}$) plus Planck nuisance parameters gives some marginalized bounds $A_{\mathrm{L}}=1.23^{+0.09}_{-0.11}$ ($68\%$CL, Planck 2015 TT+lowP), to be compared with $A_{\mathrm{L}}=1.22\pm 0.10$ in absence of blurring. Once again, we find that the impact of blurring is negligible. We also checked the absence of correlation in the two-dimensional contour plots between $A_{\mathrm{L}}$ and the parameters describing perturbed reionisation. The reason is the $\pi/2$ phase shift between the oscillations in the lensing-lensing and blurring-lensing effects. This shift, that makes it impossible to reproduce a lensing-like signal in the data from blurring, appears to be a generic feature, present even when the parameters of perturbed reionisation are varied.

5 Conclusions

We have computed, for the first time, the inclusive impact of the reionisation blurring and its correlation with weak gravitational lensing. The blurring of the cosmic microwave background is induced from ionised gas at the time of reionisation, plus other velocity- and GR-induced effects that we have considered, and is a prediction in the standard cosmological model. In principle it offers a probe into the epoch of reionisation allowing to study not only the optical depth of reionisation, but also the dynamics of perturbations during reionisation.

However, we find that the signal is small and that a reasonable parameter space for reionisation cannot be constrained with current data. Since Planck is already cosmic variance limited in the lower and medium temperature multipoles, this is unlikely to change in the near future.

We have studied in particular the correlation between lensing and blurring that in principle could have been observable and explain the measured lensing anomaly. However we find that the correlation between the lensing and blurring potential is weak on the large scales that are particularly important in the lensing computation. As a result the overall correlation is small. In addition we find a generic phase shift between lensing and the lensing-blurring correlation that makes it impossible to mistake a blurring signal for enhanced lensing. Our analysis shows that blurring cannot explain the lensing anomaly and that $A_{\mathrm{L}}$ remains unaffected within the statistical errors.

Since reionisation blurring is a non-speculative effect, it should be included in the data analysis to obtain unbiased parameters. However, we find that the cosmological parameters are not affected much by the blurring signal. In the case of a larger than expected reionisation parameter $a_{2}>10$, the most affected parameter is $n_{\mathrm{s}}$, whose bounds get broadened by $10\%$. In that case, blurring effects can be partially degenerate with foregrounds from point sources. However, for $a_{2}<10$, the blurring impact remains negligible. In the future, reionisation models could be better inferred from a combination of observations and N-body/reionisation simulations, such that all reionisation parameters including $a_{2}$ could be fixed to their expected value.

Overall we conclude that for the current experiments blurring does not play an important role in the CMB data analysis based on the angular power spectra. However it is much bigger than any other non-linear correction along the line-of-sight, beyond lensing, causing corrections of a few percent compared to lensing. This is at the limit of the PLANCK data sensitivity. With a more precise experiment, reionisation blurring may be observable and this could open a new window into the epoch of reionisation.

It should be noted that our analysis does not include several other secondary effects. While we consider photons scattered out of the line-of-sight, we are not including photons that are emitted into our line-of-sight. This contribution was discussed in our previous paper, see Ref. [11], and is structurally very different from the blurring signal. It dominates the small scales and acts similar to the Ostriker-Vishniak effect [44, 45, 46]. Both non-linear effects originating from reionisation are of roughly comparable order of magnitude, but only blurring is correlated to lensing, providing a boost in amplitude.

Finally, a word of caution about our analysis is in order. We have been focused on the impact of blurring onto the power spectra only. There is still the possibility that blurring impacts various $N$-point functions of the temperature and polarisation anisotropies, in particular any blurring imprints onto the trispectrum could, in principle, bias the lensing extraction pipeline [41]. We leave however such a study for a future work.

Appendix A Reionisation parametrisation

A.1 Ionising power spectra

The power spectra for the ionized fraction and its correlation to baryons are given in equations (2.16) and (2.17). The functions entering these equations have been fitted against the ionised background fraction $\tilde{x}_{\mathrm{e}}$, from the tabulated points obtained in the reionisation history model considered in Ref. [29] (see figure 2 and table 3), which is based on the numerical simulations of Ref. [31]. The shape of these functions is motivated by their good fit to the tabulated data, and, their robustness with respect to extrapolation [30, 11]. For instance, the amplitude functions $A_{\mathrm{i}}$ (and $A_{\times}$) are modelled as exponential functions to ensure their positivity whereas a simple Taylor expansion has been chosen for the typical radius $R_{\mathrm{i}}$ (and $R_{\times}$) of ionised structures. They read

[TABLE]

The fiducial values reproducing the results of Refs. [29, 31] are given in Table 2 and the resulting ionising power spectra have been plotted in figure 1.

A.2 Simpler model for CMB marginalisation

For the CMB data analysis in presence of blurring and blurring-lensing, because we are only interested in marginalising over the ionising sources, as explained in section 4, we have chosen a much simplified approach in which we postulate that the ionised fraction itself behaves as:

[TABLE]

This form is simply the same as the square root of the power spectrum $\tilde{x}_{\mathrm{e}}^{2}P_{\delta_{x_{\mathrm{e}}}\delta_{x_{\mathrm{e}}}}$, with all the parametrising functions $A$, $R$, $\alpha$ and $\gamma$ assuming the same functional dependency with respect to $\tilde{x}_{\mathrm{e}}$ as in equation (A.1), namely

[TABLE]

However, we do no longer fix their parameters to the realistic values given in Table 2 but we rather vary them over a wide prior range given in Table 1. Strictly speaking, equation (A.2) cannot be straightforwardly used for the computation of the cross power spectrum with baryons, $P_{\delta_{\mathrm{b}}\delta_{x_{\mathrm{e}}}}$. However, as can be checked in figure 1, for the best fit parameters, both spectra are of very similar shape on scales larger than typically $k=0.2\,\mathrm{Mpc}^{-1}$, which are the ones relevant for our CMB analysis. Therefore, for the purpose of CMB marginalisation over nuisances, equation (A.2) is sufficient. Notice however that the sign of $a_{1}$ in equation (A.3) does not play a role for $P_{\delta_{x_{\mathrm{e}}}\delta_{x_{\mathrm{e}}}}$ but does encode the correlation sign between $\delta_{\mathrm{b}}$ and $\delta_{x_{\mathrm{e}}}$.

Acknowledgments

Simulations were performed with computing resources granted by JARA-HPC from RWTH Aachen University under project jara0184. The work of C. R. is supported by the “Fonds de la Recherche Scientifique - FNRS” under Grant $\mathrm{N^{\circ}T}.0198.19$.