The Route to Unveil the Cosmic Genealogy of Supermassive Black Hole Binaries Using Nano-Hertz Gravitational Waves and Galaxy Surveys

TL;DR

This work tackles how to extract the cosmic genealogy of SMBHBs from nano-Hertz gravitational waves by jointly using the SGWB energy density $\Omega_{\rm GW}(f)$, its anisotropies $C^{\rm GWGW}_{\ell}(f)$, and cross-correlations with galaxy surveys $C^{\rm gGW}_{\ell}(f)$. The authors build a parametric SMBHB population model linking host galaxy properties to black hole masses and merger rates, and they forecast constraints with Fisher analysis for SKA-like PTAs and LSST-like galaxy data. They find that $\Omega_{\rm GW}(f)$ alone yields strong degeneracies between mass and redshift-evolution parameters, but incorporating anisotropic signals, especially the cross-correlation with galaxies and the SGWB auto-correlation, dramatically improves constraints on the redshift evolution of the SMBHB population and the frequency distribution of emitters. The results demonstrate a viable route for joint GW-galaxy studies to illuminate SMBH growth across cosmic time, with practical gains anticipated from upcoming facilities like SKA and Rubin LSST, and potential synergy with space-based detectors like LISA.

Abstract

The nano-hertz (nHz) stochastic gravitational wave background (SGWB), generated by unresolved supermassive black hole binaries (SMBHBs), provides a unique probe of their population and its cosmic evolution. In this work, we explore the potential of uncovering the SMBHB population and its redshift dependence by combining the SGWB signal and its anisotropies with galaxy distribution through cross-correlation analyses. Using a Fisher analysis technique, we show that the SGWB power spectrum alone can not provide any information on the evolutionary history of SMBHBs, whereas the inclusion of the angular power spectrum of the SGWB and its cross-correlation with the galaxy distribution substantially improves constraints on the redshift evolution parameters. Assuming pulsar timing array (PTA) configurations achievable in the Square Kilometre Array (SKA) era, we find that the combined use of isotropic and anisotropic SGWB signals, together with galaxy surveys, can provide valuable measurements of the redshift evolution of the SMBH-galaxy connection and the frequency distribution of SMBHBs. These results highlight the potential of joint GW-galaxy studies to address the long-standing open question of SMBH growth and evolution across cosmic time.

Figures (16)

• Figure 1: The SGWB energy density spectrum, $\Omega_{\rm GW}(f)$, as a function of frequency for different SMBHB population models. Various population parameters influence the spectrum differently: $\eta$, $\nu$, and $\xi$ primarily determine the overall amplitude, while $\alpha$ and $\lambda$ control the shape of the spectrum.
• Figure 2: The angular power spectrum of SGWB density, $C_{\ell}^{\rm GWGW}(f)$ at $\ell = 15$, as a function of frequency for different SMBHB population models. The $C_{\ell}^{\rm GWGW}(f)$ signal is expected to be shot noise-dominated, leading to a flat spectrum in spherical harmonic mode $\ell$.
• Figure 3: The cross-correlation power spectrum between SGWB density and the galaxy distribution, $C_{\ell}^{\rm gGW}(f)$, as a function of spherical harmonic mode $\ell$ at a GW frequency of $6 \times 10^{9}$ Hz, for different population models. The shape and structure of the signal are primarily influenced by the population parameters $\nu$, $\alpha$, $\lambda$, and $\xi$ while $\eta$, which controls the overall amplitude of $\Omega_{\rm GW}(f)$, has little to no effect on $C_{\ell}^{\rm gGW}$.
• Figure 4: The cross-correlation power spectrum between SGWB density and the galaxy distribution, $C_{\ell}^{\rm gGW}$ at $\ell = 15$, as a function of GW frequency for different population models. The curves are flat for all cases except when $\lambda \neq 0$. This is because the window function $\phi_{\rm GW}(f,{\rm z})$ is independent of frequency in cases where the frequency distribution does not depend on redshift.
• Figure 5: Mapping between the SGWB power spectrum ($\Omega_{\rm GW}(f)$) features and physical parameter, $\eta$, $\nu$, $\alpha$ and $\lambda$. The $\log_{10}(\Omega_{\rm GW}(f))$ is fitted to linear curve with an intercept $Amp$ and slopes $\beta_1$ and $\beta_2$ with a parameters $f_t$ denoting the transition frequency of the curve from slope $\beta_1$ to $\beta_2$.