Detectability of the Cross-Correlation between CMB Lensing and Stochastic GW Background from Compact Object Mergers

Abstract

The anisotropies of the Stochastic Gravitational-Wave Background (SGWB), produced by merging compact binaries, constitute a possible new probe of the Large-Scale Structure (LSS). However, the significant shot noise contribution caused by the discreteness of the GW sources and the poor angular resolution of the instruments hampers the detection of the intrinsic anisotropies induced by the LSS. In this work, we investigate the potential of cross-correlating forthcoming high precision measurements of the SGWB energy density and the Cosmic Microwave Background (CMB) lensing convergence to mitigate the effect of shot noise. Combining a detailed model of stellar and galactic astrophysics with a novel framework to distribute the GW emitters in the sky, we compute the auto- and cross-correlation power spectra for the two cosmic fields, evaluate the shot noise contribution and predict the signal-to-noise ratio. The results of our analysis show that the SGWB energy density correlates significantly with the CMB lensing convergence and that the cross-correlation between these two cosmic fields reduces the impact of instrumental and shot noise. Unfortunately, the S/N is not high enough to detect the intrinsic SGWB anisotropies. Nevertheless, a network composed of both present and future generation GW interferometers, operating for at least 10 yrs, should be able to measure the shot noise contribution.

1. Introduction

The direct detection of gravitational waves (GW) [1] achieved by the network of ground-based interferometers constituted by the Advanced Laser Interferometer Gravitatio nal-Wave Observatory (LIGO) [2], Advanced Virgo [3] and the Kamioka Gravitational Wave Detector (KAGRA) [4] has opened a new observational window on the Universe. The past five years have witnessed a real revolution in astronomy, with the detection of more than 50 compact binary coalescences during the first three LIGO/Virgo/KAGRA observing runs [5,6,7], the last of which is still ongoing. Each of these detections is associated with a single loud event, but we expect that all the GW signals too faint or too numerous to be individually resolved combine together to create a stochastic gravitational-wave background (SGWB). The measurement and characterisation of the SGWB are one of the main observational targets for current and future generation GW detectors. Existing data from the LIGO/Virgo/KAGRA network have already placed upper bounds on both the isotropic and anisotropic components of the SGWB [8,9].

Many physical mechanisms, both astrophysical and cosmological, can produce different backgrounds with distinct properties (see [10] for a review). Among them, the SGWB given by the incoherent superposition of GW events produced by merging stellar remnant compact binaries has raised significant interest among the scientific community [11,12,13,14,15,16,17,18,19,20]. This astrophysical SGWB is one of the dominant contributions in the frequency range explored by ground-based detectors (Hz-kHz) and is likely to be the first one to be detected. Moreover, the merging compact binaries that create this SGWB are the outcomes of stellar evolution and mainly reside in galaxies. Consequently, the SGWB anisotropies reflect the distribution of galaxies in the Universe and could constitute a new tracer of the Large-Scale Structure (LSS). For these reasons, the anisotropic SGWB has been extensively studied during the last few years, with relevant effort given to theoretical modeling [21,22,23,24,25,26,27,28,29,30,31,32], observational searches [33,34,35,36] and data analysis techniques [37,38,39,40,41,42,43].

The two main obstacles to the detection of the SGWB anisotropies in the Hz-kHz band are the poor angular resolution of GW detectors to a diffuse SGWB mapping and the presence of a large shot noise contribution. The former issue is related to the noise properties of the detector and how they are mapped onto the sky, but it also depends on the network configuration and the scan strategy [44]. Instead, the shot noise arises because the SGWB is composed of discrete transient events occurring at a low rate. Therefore, the predicted SGWB energy density is affected by a high level of uncertainty. Several recent studies have addressed the issue of shot noise, showing that its expected amplitude is orders of magnitude higher than the correlation induced by the LSS [45,46,47]. Cross-correlation with other tracers of the LSS has already been proposed as a possible solution to reduce the impact of shot noise. In particular, cross-correlations with galaxy number counts [22,27,45,48,49,50], weak lensing [22,27] and Cosmic Microwave Background (CMB) temperature fluctuations [51,52] have been investigated.

Polarisation CMB measurements from ongoing [53,54,55,56,57] and planned [58,59,60,61,62,63] CMB observations are targeting the cosmological background of GWs generated in the Early Universe. Moreover, deflections of CMB photons through weak gravitational lensing caused by forming cosmological structures are explicitly targeted by arcminute scale CMB probes, in both total intensity and polarisation, and this aspect is actually central for the present work.

In this paper, for the first time to our knowledge, we cross-correlate the astrophysical SGWB produced by merging compact binaries in galaxies with the CMB lensing convergence 1.The purpose of this study is to verify if the cross-correlation with another cosmic field tracing the same underlying dark matter distribution can bring out the intrinsic SGWB anisotropies given the presence not only of the shot noise but also of the more severe instrumental one. For this reason, we opted for the CMB lensing convergence, which is an integrated tracer of the LSS, just as the SGWB, and will be constrained by future CMB experiments with exquisite precision. Indeed, given the huge uncertainties still affecting GW measurements, it is crucial that the other cosmic field is tightly constrained. Starting from a detailed model of stellar astrophysics [66,67] and galaxy evolution [68,69], we compute the anisotropies of the SGWB for three types of sources—binary black holes (BH-BH), binary neutron stars (NS-NS) and black hole-neutron star binaries (BH-NS)—and for different present and future ground-based detectors: LIGO/Virgo/KAGRA, Einstein Telescope (ET) [70] and Cosmic Explorer (CE) [71]. Exploiting the novel framework to distribute the GW emitters in the sky and simulate a full-sky map of the SGWB developed in [30], we estimate the shot noise contribution. Finally, we investigate the potential of cross-correlation with forthcoming high precision measurements of CMB lensing by the Simons Observatory (SO) [60], a powerful future probe observing arcminute scale CMB anisotropies with high sensitivity that will allow us to characterise the lensing signal with unprecedented precision and to exploit it to enhance the sensitivity on cosmological GWs [72].

The paper is structured as follows: in Section 2 we go through the theoretical background, in Section 3 we present and discuss our results and in Section 4 we draw our conclusions. In Appendix A and Appendix B we review the characterisation of the SGWB produced by merging compact binaries and the derivation of the shot noise contribution, both described in [30] and adopted also for this study.

Throughout this work we assume the standard flat $\mathsf{\Lambda }$CDM cosmology with parameter values from the Planck 2018 legacy release [73], with Hubble rate today corresponding to $H_0=67.4$ km s${}^{-1}$ Mpc${}^{-1}$, Cold Dark Matter (CDM) and baryon abundances with respect to the critical density corresponding to ${\mathrm{\Omega }}_{\mathrm{CDM}}{h}^2=0.120$ and ${\mathrm{\Omega }}_b{h}^2=0.022$, respectively, reionisation optical depth $\tau =0.054$, amplitude and spectral index of primordial scalar perturbations corresponding to ln$\left({10}^{10}{A}_S\right)=3.045$ and $n_S=0.965$, respectively.

2. Methods

In this Section, we review the description of CMB lensing, closely following [74,75] and references therein. Subsequently, we characterise the SGWB as a tracer of the LSS and we derive an expression of its kernel. Finally, we outline the theoretical background for the cross-correlation of two cosmic fields.

2.1. CMB Lensing

The LSS deflects CMB photons during their travel from the last-scattering surface to the observer, leaving small imprints on the observed CMB anisotropies. In particular, the effect of gravitational lensing on CMB photons can be described as a remapping of the unlensed temperature anisotropies $\mathbf{\Theta }\left({\widehat{\mathit{e}}}_{\mathbf{o}}\right)$ by a two-dimensional vector field in the sky, namely the deflection field $\mathbf{d}\left({\widehat{\mathit{e}}}_{\mathit{o}}\right)$:

$$\begin{array}{cc}\hfill \tilde{\mathbf{\Theta }}\left({\widehat{\mathit{e}}}_{\mathbf{o}}\right)& =\mathbf{\Theta }({\widehat{\mathit{e}}}_{\mathbf{o}}+\mathit{d}\left({\widehat{\mathit{e}}}_{\mathbf{o}}\right))\hfill \\ \hfill & =\mathbf{\Theta }({\widehat{\mathit{e}}}_{\mathbf{o}}+\nabla \varphi \left({\widehat{\mathit{e}}}_{\mathbf{o}}\right))\hfill \\ \hfill & =\mathbf{\Theta }\left({\widehat{\mathit{e}}}_{\mathbf{o}}\right)+{\nabla }^i\varphi \left({\widehat{\mathit{e}}}_{\mathbf{o}}\right){\nabla }_i\mathbf{\Theta }\left({\widehat{\mathit{e}}}_{\mathbf{o}}\right)+\mathcal{O}\left({\varphi }^2\right),\hfill \end{array}$$

(1)

where $\tilde{\mathbf{\Theta }}\left({\widehat{\mathit{e}}}_{\mathbf{o}}\right)$ is the lensed temperature fluctuation and $\varphi \left({\widehat{\mathit{e}}}_{\mathbf{o}}\right)$ is the CMB lensing potential:

$$\varphi \left({\widehat{\mathit{e}}}_{\mathbf{o}}\right)=-2{\int }_0^{z_{\star }}{\displaystyle \frac{cdz}{H\left(z\right)}}{\displaystyle \frac{{\chi }_{\star }-\chi \left(z\right)}{{\chi }_{\star }\chi \left(z\right)}}\mathrm{\Psi }(\chi \left(z\right){\widehat{\mathit{e}}}_{\mathbf{o}},z).$$

(2)

In the previous equation, $\chi \left(z\right)$ is the comoving distance to redshift z, ${\chi }_{\star }$ is the comoving distance to the last-scattering surface at $z_{\star }=1090$, $H\left(z\right)$ is the Hubble parameter at redshift z, c is the speed of light and $\mathrm{\Psi }(\chi \left(z\right){\widehat{\mathit{e}}}_{\mathbf{o}},z)$ is the three-dimensional gravitational potential at the point on the photon path given by $\chi \left(z\right){\widehat{\mathit{e}}}_{\mathbf{o}}$. The deflection field is defined as $\mathit{d}\left({\widehat{\mathit{e}}}_{\mathbf{o}}\right)=\nabla \varphi \left({\widehat{\mathit{e}}}_{\mathbf{o}}\right)$, where ∇ is the the two-dimensional gradient on the sphere. Since the lensing potential is an integrated measure of the gravitational potential, it is convenient to describe the CMB lensing by means of the lensing convergence, which is proportional to the two dimensional Laplacian of the lensing potential and can be written as a weighted integral over redshift of the projected dark matter density contrast $\delta$:

$$\kappa \left({\widehat{\mathit{e}}}_{\mathbf{o}}\right)=-\frac{1}{2}{\nabla }^2\varphi \left({\widehat{\mathit{e}}}_{\mathbf{o}}\right)={\int }_0^{z_{*}}{W}^{\kappa }\left(z\right)\delta (\chi \left(z\right){\widehat{\mathit{e}}}_{\mathbf{o}},z).$$

(3)

The weight inside the integral is the lensing kernel $W^{\kappa }$, which describes the lensing efficiency of the matter distribution and is given by:

$$W^{\kappa }\left(z\right)={\displaystyle \frac{3{\mathrm{\Omega }}_m}{2c}}{\displaystyle \frac{H_0^2}{H\left(z\right)}}(1+z)\chi \left(z\right){\displaystyle \frac{{\chi }_{*}-\chi \left(z\right)}{{\chi }_{\star }}},$$

(4)

where ${\mathrm{\Omega }}_m$ and $H_0$ are the present-day value of the matter density and the Hubble parameter, respectively.

2.2. Stochastic Gravitational Wave Background

The stochastic gravitational wave background is usually described in terms of the dimensionless energy density parameter:

$${\mathrm{\Omega }}_{\mathrm{gw}}(f_{\mathrm{o}},{\widehat{\mathit{e}}}_{\mathbf{o}})={\displaystyle \frac{1}{{\rho }_c}}{\displaystyle \frac{d^3{\rho }_{\mathrm{gw}}(f_{\mathrm{o}},{\widehat{\mathit{e}}}_{\mathbf{o}})}{dln{f}_{\mathrm{o}}{d}^2{\omega }_{\mathrm{o}}}}={\displaystyle \frac{8\pi G{f}_{\mathrm{o}}}{3H_0^2{c}^2}}{\displaystyle \frac{d^3{\rho }_{\mathrm{gw}}(f_{\mathrm{o}},{\widehat{\mathit{e}}}_{\mathbf{o}})}{d{f}_{\mathrm{o}}{d}^2{\omega }_{\mathrm{o}}}},$$

(5)

where ${\rho }_c=3H_0^2{c}^2/8\pi G$ is the critical density and ${\rho }_{\mathrm{gw}}$ is the SGWB energy density at the observed frequency $f_{\mathrm{o}}$, arriving from a solid angle ${\omega }_{\mathbf{o}}$ centred on the observed direction ${\widehat{\mathit{e}}}_{\mathbf{o}}$. The energy density parameter can be split into an isotropic term ${\overline{\mathrm{\Omega }}}_{\mathrm{gw}}\left(f_{\mathrm{o}}\right)$ and a directional dependent term $\delta {\mathrm{\Omega }}_{\mathrm{gw}}(f_{\mathrm{o}},{\widehat{\mathit{e}}}_{\mathbf{o}})$:

$${\mathrm{\Omega }}_{\mathrm{gw}}(f_{\mathrm{o}},{\widehat{\mathit{e}}}_{\mathbf{o}})={\displaystyle \frac{{\overline{\mathrm{\Omega }}}_{\mathrm{gw}}\left(f_{\mathrm{o}}\right)}{4\pi }}+\delta {\mathrm{\Omega }}_{\mathrm{gw}}(f_{\mathrm{o}},{\widehat{\mathit{e}}}_{\mathbf{o}}).$$

(6)

In this work we consider the astrophysical SGWB given by the incoherent superposition of GW signals produced during the merger of compact binaries inside galaxies. Assuming that the SGWB, as well as the galaxies that host the merging binaries, traces the peaks of the dark matter distribution, the energy density contrast ${\delta }_{\mathrm{gw}}=\delta {\mathrm{\Omega }}_{\mathrm{gw}}/{\overline{\mathrm{\Omega }}}_{\mathrm{gw}}$ can be expressed as a line-of-sight integral of the dark matter density contrast:

$${\delta }_{\mathrm{gw}}(f_{\mathrm{o}},{\widehat{\mathit{e}}}_{\mathbf{o}})={\int }_0^{z_{\star }}{W}^{\mathrm{\Omega }}(f_{\mathrm{o}},z)\delta (\chi \left(z\right){\widehat{\mathit{e}}}_{\mathbf{o}},z),$$

(7)

where the SGWB kernel $W^{\mathrm{\Omega }}(f_{\mathrm{o}},z)$ is the sum of two terms:

$$W^{\mathrm{\Omega }}(f_{\mathrm{o}},z)={\displaystyle \frac{b_{\mathrm{\Omega }}(f_{\mathrm{o}},z){\displaystyle \frac{d\mathrm{\Omega }}{dz}}(f_{\mathrm{o}},z)}{(\int d{z}^{\prime }\frac{d\mathrm{\Omega }}{d{z}^{\prime }})}}+\mu (f_{\mathrm{o}},z).$$

(8)

The first term is the product of the linear bias $b_{\mathrm{\Omega }}$, which quantifies the mismatch between the distribution of the SGWB and the total matter density, and the SGWB redshift distribution $d\mathrm{\Omega }/dz$. The second term takes into account the effect of lensing magnification on the observed SGWB energy density and it is given by:

$$\mu (f_{\mathrm{o}},z)={\displaystyle \frac{3{\mathrm{\Omega }}_{\mathrm{m}}}{2c}}{\displaystyle \frac{H_0^2}{H\left(z\right)}}(1+z)\chi \left(z\right){\int }_z^{z_{\star }}d{z}^{\prime }(1-{\displaystyle \frac{\chi \left(z\right)}{\chi \left(z^{\prime }\right)}})(s_{\mathrm{\Omega }}(f_{\mathrm{o}},z)-1){\displaystyle \frac{{\displaystyle \frac{d\mathrm{\Omega }}{dz}}(f_{\mathrm{o}},z)}{(\int d{z}^{\prime }\frac{d\mathrm{\Omega }}{d{z}^{\prime }})}},$$

(9)

where $s_{\mathrm{\Omega }}$ is the magnification bias.

The expressions of the redshift distribution $d\mathrm{\Omega }/dz$, the bias $b_{\mathrm{\Omega }}$ and the magnification bias $s_{\mathrm{\Omega }}$ are derived and discussed in [30]: we review them in Appendix A for the interested reader. All these quantities can be evaluated once a specific model of stellar astrophysics and galaxy evolution has been chosen: in this work, we adopt the astrophysical prescriptions presented in [68,69]. There, the authors combine the results of the STARTRACK population synthesis simulations2, specifically the ‘reference B’ model in [66,67], with different observationally derived prescriptions for the host galaxies. For our analysis we use the merger rates computed using the empirical Star Formation Rate Function (SFRF) as galaxy statistics and the Fundamental Metallicity Relation (FMR) to assign metallicity to galaxies (see Figure 8 of [69]). In order to reduce the impact of the uncertainties in the astrophysical modeling, we decide to re-scale all the merger rates per unit comoving volume to match the local values measured by LIGO/Virgo/KAGRA [76,77]: $23.9\begin{array}{c}+14.9\\ -8.6\end{array}{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$ for BH-BH, $320\begin{array}{c}+490\\ -240\end{array}{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$ for NS-NS and $45\begin{array}{c}+75\\ -33\end{array}{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$ for BH-NS.

2.3. Cross-Correlation Technique

Because the kernels of both our cosmic fields are broad functions of redshift, the angular cross-correlation can be computed using the Limber approximation [78] in the following way:

$$C_{\ell }^{\kappa \mathrm{\Omega }}={\int }_0^{z_{\star }}{\displaystyle \frac{dz}{c}}{\displaystyle \frac{H\left(z\right)}{{\chi }^2\left(z\right)}}{W}^{\kappa }\left(z\right)W^{\mathrm{\Omega }}\left(z\right)P(k=\frac{l}{\chi \left(z\right)},z),$$

(10)

where $P(k,z)$ is the matter power spectrum, which we computed using the CLASS3 public code [79,80]. The nonlinear evolution of the matter power spectrum was taken into account using the HALOFIT prescription [81]. Assuming that both the SGWB and the CMB lensing behave as Gaussian random fields, the variance of $C_{\ell }^{\kappa \mathrm{\Omega }}$ is given by:

$$(\mathrm{\Delta }{C}_{\ell }^{\kappa \mathrm{\Omega }}{)}^2={\displaystyle \frac{1}{(2\ell +1)f_{\mathrm{sky}}}}[(C_{\ell }^{\kappa \mathrm{\Omega }}{)}^2+(C_{\ell }^{\kappa \kappa }+N_{\ell }^{\kappa \kappa }\left)\right(C_{\ell }^{\mathrm{\Omega }\mathrm{\Omega }}+{\mathbf{S}}_{\mathbf{\ell }}^{\mathbf{\Omega }\mathbf{\Omega }}+N_{\ell }^{\mathrm{\Omega }\mathrm{\Omega }}\left)\right],$$

(11)

where $f_{\mathrm{sky}}$ is the sky fraction covered by both the SGWB and the CMB lensing surveys, $N_{\ell }^{\kappa \kappa }$ is the lensing noise, $S_{\ell }^{\mathrm{\Omega }\mathrm{\Omega }}$ and $N_{\ell }^{\mathrm{\Omega }\mathrm{\Omega }}$ are the SGWB the shot and instrumental noise respectively. For our analysis, we employ the Simons Observatory lensing noise curves [60] and we adopt the fiducial Large Aperture Telescope value $f_{\mathrm{sky}}^{\kappa }=0.4$ for the sky fraction. Since GW experiments cover the entire sky (that is, $f_{\mathrm{sky}}^{\mathrm{\Omega }}=1$), we use the limiting value $f_{\mathrm{sky}}=f_{\mathrm{sky}}^{\kappa }$ for the cross-correlation. As for the SGWB, we compute the instrumental noise curves for different detector network configurations using the software schNell4 [44], while we evaluate the SGWB shot noise contribution through the novel map-making technique developed in [30] and summarised in Appendix B. The signal-to-noise ratio (S/N) at multipole ℓ is then given by:

$$({\displaystyle \frac{S}{N}}{)}_{\ell }^2={\displaystyle \frac{(C_{\ell }^{\kappa \mathrm{\Omega }}{)}^2}{(\mathrm{\Delta }{C}_{\ell }^{\kappa \mathrm{\Omega }}{)}^2}}={\displaystyle \frac{(2\ell +1)f_{\mathrm{sky}}(C_{\ell }^{\kappa \mathrm{\Omega }}{)}^2}{(C_{\ell }^{\kappa \mathrm{\Omega }}{)}^2+(C_{\ell }^{\kappa \kappa }+N_{\ell }^{\kappa \kappa }\left)\right(C_{\ell }^{\mathrm{\Omega }\mathrm{\Omega }}+S_{\ell }^{\mathrm{\Omega }\mathrm{\Omega }}+N_{\ell }^{\mathrm{\Omega }\mathrm{\Omega }})}}$$

(12)

and the cumulative S/N for multipoles up to ${\ell }_{\max }$ is

$$\left({\displaystyle \frac{S}{N}}\right)(\ell <{\ell }_{\max })=\sqrt{\sum _{\ell ={\ell }_{\min }}^{{\ell }_{\max }}({\displaystyle \frac{S}{N}}{)}_{\ell }^2}.$$

(13)

Similarly, the auto-correlation power spectra can be evaluated as:

$$C_{\ell }^{XX}={\int }_0^{z_{\star }}{\displaystyle \frac{dz}{c}}{\displaystyle \frac{H\left(z\right)}{{\chi }^2\left(z\right)}}\left[W^X\left(z\right){]}^{2}P\right(k=\frac{l}{\chi \left(z\right)},z),$$

(14)

where X is either $\kappa$ or $\mathrm{\Omega }$, and the associated S/N is given by:

$$({\displaystyle \frac{S}{N}}{)}_{\ell }^2={\displaystyle \frac{(2\ell +1)}{2}}{f}_{\mathrm{sky}}^X{\displaystyle \frac{(C_{\ell }^{XX}{)}^2}{(C_{\ell }^{XX}+S_{\ell }^{XX}+N_{\ell }^{XX}{)}^2}}.$$

(15)

where $S_{\ell }^{XX}$, $N_{\ell }^{XX}$ and $f_{\mathrm{sky}}^X$ are the shot noise, the instrumental noise and the sky fraction covered by the considered survey for the tracer X.

3. Results

We study the cross-correlation between the astrophysical SGWB and the CMB lensing convergence in two different scenarios. First, we examine the case where all GW events are taken into account when computing the SGWB energy density. Second, we consider the residual SGWB as potentially measured by a specific detector, that is, we only take into account the contribution of the unresolved events. The events that can be resolved individually are filtered out by means of a signal-to-noise threshold, which we fix at the reference value of $\overline{\rho }=8$ (see Appendix A for more details). We include in our analysis the present and future ground-based GW detectors LIGO/Virgo/KAGRA, ET and CE. Indeed, to detect a diffuse GW background, it is necessary to cross-correlate the outputs of several interferometers handled as a network. In

Table 1. Location coordinates and orientation angles for the detectors considered in this work. For the future 3G detectors ET and CE, we adopted arbitrary locations and orientations, assuming that the detectors will be built somewhere in one of their proposed sites: Sardinia (Italy) and Utah (USA), respectively. See [44] for further details on the definition of the coordinates and the orientation angles.

Detector	Latitude (deg)	Longitude (deg)	Orientation (deg)
LIGO Hanford	46.6	−119.4	171.8
LIGO Livingston	30.7	−90.8	243.0
Virgo	43.6	10.5	116.5
KAGRA	36.3	137.2	225.0
ET	40.1	9.0	90.0
CE	40.8	−113.8	90.0

, we report the detector coordinates and orientation angles that we plug in the schNell to compute the noise curves for specific network configurations. In particular, for the anisotropies of the residual SGWB detected by LIGO/Virgo/KAGRA, we consider a network composed of the first four interferometers in Table 1 and the associated noise curve. Instead, to investigate the detectability of the anisotropies of the residual SGWB measured by ET or CE, we consider a network composed of LIGO/Virgo/KAGRA and ET or CE, respectively. In any case, we consider an integration time of $T=1$ yr. Finally, we also consider an extended network given by all the detectors taken into account in this work. We use its noise curves for $T=1$ yr and $T=10$ yrs to study the detectability of the total background, given by all resolved and unresolved events. We work at the reference frequency of 65 Hz, which falls into the middle of the sensitivity bands of all the considered detectors.

In Figure 1, we compare the CMB lensing kernel with the SGWB kernel for different sources (BH-BH, BH-NS and NS-NS) and detectors, computed using Equations (4) and (8), respectively. The lensing kernel is a broad function of redshift that peaks at $z\simeq 2$ and decreases slowly at higher redshifts, making the CMB lensing convergence a powerful probe of LSS up to the last-scattering surface. In the upper-left panel, we consider the SGWB obtained integrating all the GW events, resolved and unresolved. The kernel is quite broad and peaks at $z\lesssim$ 1 for all types of sources. The exact position of the peak and the shape of the kernel are the outcomes of the complex interplay among the behaviour of three factors: the merger rate, the flux carried by each GW event and the SGWB bias. On the one hand, the shape of the redshift distribution $d\mathrm{\Omega }/dz$ reflects the fact that the merger rate peaks with the cosmic star formation rate at $z\simeq 2$ and rapidly declines at higher redshifts. On the other hand, nearer events largely contribute to the SGWB energy density because their flux is less diluted. Finally, the SGWB bias is an increasing function of redshift (see upper panels of Figure A2) so that its effect is to give more weight to more distant objects. The BH-BH kernel is slightly broader than BH-NS and NS-NS ones because BH-BH binaries have a greater chirp mass and produce more energetic GW signals, contributing significantly up to higher redshift. The upper-right panel shows the kernels of the residual SGWB, which we obtain filtering out the events individually resolved by LIGO/Virgo/KAGRA. Typically, this operation removes many of the nearby events, especially the energetic BH-BH mergers, conferring a proportional higher weight to more distant ones. For this reason, the kernels are broader and peak at slightly higher redshifts than in the previous case, where we integrated all GW events to obtain the SGWB.

As we can see in the lower panels of Figure 1, these effects are even more evident for 3G detectors such as ET and CE, which will be able to resolve BH-BH mergers individually up to $z\simeq 10$. As a result, the kernels are substantially broader than previous cases and peak at higher redshifts. It is also evident that these SGWB configurations present a more extended overlap with the CMB lensing kernel. As we will see in the following, this has a considerable impact on the cross-correlation results. The non-monotonic behaviour that characterises the SGWB kernels at low redshifts for ET and CE is due to the lensing term in Equation (8). As pointed out in Appendix A and extensively discussed in [30], the effect of lensing is to reduce the SGWB energy density, boosting some events above the detection threshold and making them directly resolvable by the instrument so that they do not contribute to the SGWB. This process is more effective for 3G detectors and produces a dip at $z\simeq 0.2-0.3$.

In Figure 2, we show the auto- and cross-correlation power spectra for all the sources and detectors. Although we perform all the calculations in the Limber approximation, the SGWB auto-correlation power spectra are in good agreement5 with the results obtained in [30] with the public code CLASS [79,80]. At large angular scales, all the spectra behave as a power-law whose slope depends on the source type and the detector. For ET and CE, the large-scale behaviour is significantly affected by the magnification bias and the power-law is broken. As expected, the $\mathrm{\Omega }\times \kappa$ cross-correlation power spectrum always lies between the two auto-correlation power spectra. The fact that the cross-correlation is strong is quite remarkable and constitutes one of the most important findings of this paper since it proves that the large-scale distribution of late-time objects such as merging compact binaries is well correlated with the linear structures probed by CMB lensing.

The subtle SGWB anisotropies produced by the LSS, described by the power spectra in Figure 2, coexist with the much larger ones caused by the spatial and temporal discreteness of the GW sources that make up the SGWB. Indeed, the shot noise power spectrum outreaches the SGWB auto-correlation power spectrum by several orders of magnitude (see e.g., Figure A3 and related comments in Appendix B). Moreover, the instrumental noise is an even more significant killing factor for measuring intrinsic SGWB anisotropies. For all the considered detector networks, the noise curve $N_{\ell }$ is orders of magnitude larger than the shot noise power spectrum $S_{\ell }$. For these reasons, as we can see in Figure 3, the S/N of the auto-correlation power spectra are broadly smaller than unity for all types of sources detector configurations.

The cumulative S/N of the $\mathrm{\Omega }\times \kappa$ cross-correlation, displayed in Figure 4, shows that cross-correlating with CMB lensing convergence is an effective way to mitigate the impact of instrumental and shot noise: indeed, it is around three orders of magnitude larger than the one of auto-correlation. The cross-correlation with another tracer of the same underlying dark matter distribution actually enhances the SGWB anisotropies induced by the LSS. Unfortunately, the instrumental noise is too high and this substantial improvement is not sufficient to guarantee a direct detection of the cross-correlation signal.

As a final step, we try to increase as much as possible the S/N using a network with the five instruments considered in this work to detect the total SGWB given by all resolved and unresolved events. Combining the outputs of more detectors is an effective way to reduce the noise curve $N_{\ell }$, whereas considering the total background enhances the signal because the amplitude of the monopole is higher than the one of the residual background. Moreover, we also explore the benefits of measuring the GW signal for a longer integration time, $T=10$ yrs. In Figure 5 we show the results of this analysis. As we can see in the left panel, where we show the noise curves for $T=1$ yr and $T=10$ yrs together with the shot noise and auto-correlation power spectra, the improved sensitivity of the extended network allows us to observe the flat shot noise power spectrum. Unfortunately, the amplitude of the intrinsic anisotropies is well below the noise level, as we can see in the right panels: despite a two-orders-of-magnitude enhancement, the cumulative S/N of both auto- and cross-correlation is still too low to allow a direct detection even for $T=10$ yrs.

4. Conclusions

The anisotropies of the SGWB produced by merging compact binaries in galaxies contain a wealth of information about the physical properties of GW emitters, their distribution in redshift and their position in the sky. However, the shot noise caused by the spatial and temporal discreteness of the GW emitters covers the signal. Moreover, the poor angular resolution of GW instruments constitutes a significant obstacle to the detection of SGWB anisotropies.

Other works have already shown that cross-correlating the SGWB with other cosmological probes mitigates the impact of shot noise. In this paper, we studied the potential of cross-correlating the SGWB with CMB lensing. We produced forecasts for the cross-correlation of forthcoming high precision measurements of the SGWB energy density and the CMB lensing convergence. Specifically, we considered LIGO/Virgo/KAGRA at design sensitivity, ET and CE for the gravitational waves and the SO for the CMB.

Starting from a detailed model of stellar and galactic astrophysics, based on simulations and observationally driven prescriptions, we characterised the SGWB energy density as a tracer of LSS, derived its kernel and compared it with the CMB lensing one. We computed the auto- and cross-correlation power spectra in the Limber approximation for three types of SGWB sources: BH-BH, BH-NS and NS-NS merging binaries. The cross-correlation power spectrum always lies between the auto-correlation power spectra, showing a good correlation between the two cosmic fields for each source and detector. This result is not trivial since it implies that the distribution of merging compact binaries traced by the SGWB anisotropies is well correlated with the linear structures probed by CMB lensing. Finally, we computed the S/N for both the auto- and the cross-correlation power spectra. To this purpose, we used the shot noise estimation naturally provided by a novel framework to distribute the GW emitters in the sky and computed the instrumental noise curves with the public package schNell.

We found that the auto-correlation cumulative S/N is extremely low, since the instrumental noise outreaches both the signal and the shot noise by several orders of magnitude, even when considering third-generation GW detectors. The cross-correlation with CMB lensing effectively enhances the S/N of at least three orders of magnitude, but this improvement is not sufficient to ensure a direct detection of the intrinsic SGWB anisotropies induced by the LSS. On the other hand, we found that by combining all the instruments considered in this work as a single network operating for ten years, it will be possible at least to constrain the shot noise contribution.

In conclusion, we have shown that cross-correlating the astrophysical SGWB with CMB lensing improves the S/N by orders of magnitude. However, we found that the instrumental noise constitutes a major limiting factor for the S/N of both auto- and cross-correlation. Nevertheless, this study represents a novel aspect, directly impacting the characterisation of the SGWB anisotropies produced by the LSS.