Measuring the Hubble constant with black sirens

We investigate a recently proposed method for measuring the Hubble constant from gravitational wave detections of binary black hole coalescences without electromagnetic counterparts. In the absence of a direct redshift measurement, the missing information on the left-hand side of the Hubble-Lemaître law is provided by the statistical knowledge on the redshift distribution of sources. We assume that source distribution in redshift depends on unknown hyperparameters, modeling our ignorance of the astrophysical binary black hole distribution. With tens of thousands of these “black sirens”—a realistic figure for the third generation detectors Einstein Telescope and Cosmic Explorer—an observational constraint on the value of the Hubble parameter at percent level can be obtained. This method has the advantage of not relying on electromagnetic counterparts, which accompany a very small fraction of gravitational wave detections, nor on often unavailable or incomplete galaxy catalogs.

Figure 1

Second and third generation spectral noise densities. The 2G noise curve has been obtained by a fit to the LIGO Hanford O3 data around the event GW190814 based on the “Zero Detuning High Power” spectral noise density implemented in LALSuite [48]. For the 3G curve we adopted the noise spectral density “D” from Hall and Evans [49].

Figure 2

Distribution of the comoving distance $d_C$ for the events satisfying $\mathrm{SNR}>8$, see Eq. (17), averaged over masses and orientations for 2G and 3G detectors. The curve shows the fit of Eq. (21) to the tail of the distribution. Note the different scale for both axis in the two figures.

Figure 3

Comoving distance-redshift relationship for standard $\mathrm{\Lambda }\mathrm{CDM}$ model.

Figure 4

Toy redshift prior of Eq. (28) for $H_0^{(\text{fid})}=70\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, $z_f=0.5$, and $d_L^{(\text{cut})}=700\mathrm{Mpc}$ (top) or $d_L^{(\text{cut})}=4.0\mathrm{Gpc}$ (bottom). The vertical lines mark the mean redshifts.

Figure 5

Forecasted constraints on $H_0$ relative to the toy model of Eq. (29) for a second generation (blue line) and third generation (orange line) detector.

Figure 6

Distribution of detections and merger rate (assuming $\tau =5\mathrm{Gyr}$). The 2G curve is normalized to unity, the 3G curve and merger rate have normalization consistent with the 2G curve.

Figure 7

Normalized distribution of simulated detections for a 3G detector.

Figure 8

Marginalized constraints on $H_0$, $\tau$, and ${\mathrm{\Omega }}_m$ for 10,000 and 20,000 simulated injections for the 3G case with 5% relative errors in the measurement of $d_L$. The fiducial values of the parameters are marked with red lines. The prior on ${\mathrm{\Omega }}_m$ is displayed with a green dashed line.

Figure 9

$1\sigma$ covariance regions from averaged Fisher matrix.

Figure 10

The star formation rate of Eqs. (14) and (a1).

Figure 11

Comparison of normalized detected merger rate assuming the DM star formation rate (14), the RE one (a1), or the one of Eq. (14) with $C=4.5$.

Figure 12

Statistical inference for injections generated according to the star formation rate of Eq. (a1) but analyzed using the star formation rate of Eq. (14), for 5% relative errors in the measurement of $d_L$, with 20,000 and 10,000 injections. Here, the model includes the nuisance parameter $C$, which absorbs the effect of a different star formation rate between injection and recovery, with the result of keeping $H_0$ unbiased.