Anisotropies in the Astrophysical Gravitational-Wave Background: The Impact of Black Hole Distributions

We use population inference to explore the impact that uncertainties in the distribution of binary black holes (BBH) have on the astrophysical gravitational-wave background (AGWB). Our results show that the AGWB monopole is sensitive to the nature of the BBH population (particularly the local merger rate), while the anisotropic $C_{\ell }$ spectrum is only modified to within a few percent, at a level which is insignificant compared to other sources of uncertainty (such as cosmic variance). This is very promising news for future observational studies of the AGWB, as it shows that (i) the monopole can be used as a new probe of the population of compact objects throughout cosmic history, complementary to direct observations by LIGO and Virgo and (ii) we are able to make surprisingly robust predictions for the $C_{\ell }$ spectrum, even with only very approximate knowledge of the black hole population. As a result, the AGWB anisotropies have enormous potential as a new probe of the large-scale structure of the Universe, and of late-Universe cosmology in general.

Figure 1

Left panel: AGWB anisotropy parameter ${\mathcal{A}}_{\mathrm{gw}}$ plotted against the BH mass power law exponent ${\alpha }_m$ for $\mathcal{O}(10^4)$ possible BBH distributions. The points are colored according to the upper mass cutoff $m_{\max }$, which ranges between $30M_{\odot }$ and $200M_{\odot }$. Right panel: Total AGWB energy density $4\pi {\overline{\mathrm{\Omega }}}_{\mathrm{gw}}$ plotted against the local BBH rate $R_{\mathrm{BBH}}^{(\text{local})}$ for $\mathcal{O}(10^4)$ possible BBH distributions. The points are coloured according to the anisotropy parameter ${\mathcal{A}}_{\mathrm{gw}}$.

Figure 2

Various predictions for the $C_{\ell }$ spectrum of the AGWB overdensity field ${\delta }_{\mathrm{gw}}$, plotted as $\ell (\ell +1)C_{\ell }/(2\pi )$, which is approximately the contribution to the overdensity field variance per logarithmic bin in $\ell$. The blue curve is calculated using the Millennium catalog as in Ref. [22], with the dark blue region indicating the 99% confidence interval when accounting for the BBH population uncertainties (based on $\sim 3000$ random samples of the population hyperparameters), while the pale blue region indicates the $1\sigma$ error due to population uncertainty, cosmic variance, and finite-number Poisson uncertainty in the catalog. The red curve is calculated using Eq. (9), with ${\mathcal{A}}_{\mathrm{gw}}=1.110\times {10}^{-4}$ corresponding to the maximum-likelihood BBH population parameters, and with the galaxy-galaxy two-point correlation power law index and clustering length set to $\gamma =1.67\pm 0.03$ and $d_1=(5.05\pm 0.26)h^{-1}\mathrm{Mpc}$, respectively, to match the Millennium catalog. The pale red region indicates the $1\sigma$ error due to uncertainty in ${\mathcal{A}}_{\mathrm{gw}}$, $\gamma$, and $d_1$, as well as cosmic variance. The green curve is calculated using cmbquick [40] and halofit [41, 42] as in Ref. [23], using the maximum-likelihood BBH population, with the pale green region showing the $1\sigma$ error due to cosmic variance.