Determining the Hubble Constant through Cross-Correlation of Galaxies and Gravitational Waves

Abstract

Gravitational wave (GW) standard sirens have the potential to measure the Hubble constant $H_0$ in the local universe independently of the distance ladder, and thus offer unique new insights into the Hubble tension. A key challenge with standard sirens is detecting their electromagnetic counterparts, and therefore assigning redshifts to the measured distances. One promising way to proceed is to utilize GW `dark sirens' -- events without an identified electromagnetic counterpart -- and cross-correlate their angular distribution with that of galaxies. We present a quantitative study of how precisely the Hubble constant can be measured using tomographic cross-correlation between galaxies and GW sources. Overall, we find that the constraints on $H_0$ will be limited by the quality and quantity of GW data. We find that percent-level constraints on $H_0$ will primarily depend on achieving small distance uncertainties ($σ_{d_L}=0.1\,d_L$), obtaining a large number of GW dark sirens ($\gtrsim$$5{,}000$), and accurate sky localization in the tomographic analysis.

Figures (8)

• Figure 1: Angular power spectra $C_{\ell}$ for galaxies and gravitational-wave events in the first bin with $z\in[0.1,0.15)$, assuming fifteen radial bins in total. The horizontal lines indicate the corresponding shot noise for each population. We show the auto-correlation power spectra of galaxies and GWs, as well as their cross-correlation power spectrum. Note that the shot noise in the GW auto-correlation is particularly large due to the relatively small number density of GW sources (here, we assume 1,000 GW events over a 10,000 deg$^2$ area). The shot noise cross terms are suppressed at high $\ell$ due to large angular localization error of GW sources.
• Figure 2: Forecasted constraints on key cosmological parameters from the cross-correlation between galaxies and GW dark sirens. The plot includes results for both bin 2 and bin 15 cases, and assuming GW event number $N_{\rm GW}=1000$. The red stars denote the true values of the parameters. The shaded regions represent the 1$\sigma$ and 2$\sigma$ (68.3% and 95.4% credible levels in 2D) intervals. Increased number of radial bins breaks the parameter degeneracies, thereby providing more stringent constraints on the Hubble constant $H_0$. Note that the uncertainty on the first bias parameter, $b^{\rm gal}_1$, is comparable in the 15-bin and 2-bin cases because there are fewer objects in that bin in the former case, but the constraints on the Hubble constant nevertheless improve with more tomographic bins as expected.
• Figure 3: Forecasted relative uncertainty on the Hubble constant $H_0$ as a function of the number of tomographic radial bins. The fiducial configuration (red) assumes $z\in[0.1,0.7]$, $10\,\mathrm{deg}^2$ localization uncertainty, $N_{\rm GW}=10^3$, and $\sigma_{d_L}=0.2\,d_L$. Variants shown are: improved luminosity-distance errors $\sigma_{d_L}=0.1\,d_L$ (green); reduced localization errors to $1\,\mathrm{deg}^2$ (brown); a larger GW sample size $N_{\rm GW}=10{,}000$ (black); the combined case with $1\,\mathrm{deg}^2$, $\sigma_{d_L}=0.1\,d_L$, and $N_{\rm GW}=5000$ (orange); and the same combined case but with $N_{\rm GW}=10{,}000$ (orange dashed). Gray dashed lines mark the $10\%$ and $1\%$ levels. This illustrates that percent-level constraints require simultaneously large $N_{\rm GW}$, precise angular localization, and small luminosity-distance errors.
• Figure 4: Forecasted fractional uncertainty on the Hubble constant $H_0$ as a function of the maximum multipole $\ell_{\max}$. We compare scenarios with different localization areas (10 deg$^2$ and 1 deg$^2$) and with broader galaxy redshift distributions ($0.1<z<0.7$, $0.1<z<1.0$, and $0.1<z<1.3$). All cases assume $N_{\rm GW}=1000$ and a distance error of $\sigma_{d_L}=0.2\,d_L$.
• Figure 5: Dependence of forecasted errors in $H_0$ on the galaxy redshift distribution, where the GW events are drawn. We show cases with three different galaxy redshift distributions, assuming throughout a fixed galaxy number density (so that the total number of galaxies varies between the three cases). Annotations indicate the marginalized error in $H_0$ assuming in each case fifteen tomographic bins and $N_{\rm GW}=1000$. The results indicate that low-redshift GW samples are very important for tighter $H_0$ constraints.