Unbiased Hubble constant estimation from binary neutron star mergers

Gravitational-wave (GW) observations of binary neutron star (BNS) mergers can be used to measure luminosity distances and hence, when coupled with estimates for the mergers’ host redshifts, infer the Hubble constant $H_0$. These observations are, however, affected by GW measurement noise, uncertainties in host redshifts and peculiar velocities, and are potentially biased by selection effects and the misspecification of the cosmological model or the BNS population. The estimation of $H_0$ from samples of BNS mergers with optical counterparts is tested here by using a phenomenological model for the GW strains that captures both the data-driven event selection and the distance-inclination degeneracy, while being simple enough to facilitate large numbers of simulations. A rigorous Bayesian approach to analyzing the data from such simulated BNS merger samples is shown to yield results that are unbiased, have the appropriate uncertainties, and are robust to model misspecification. Applying such methods to a sample of $N\simeq 50$ BNS merger events, as $\mathrm{LIGO}+\mathrm{Virgo}$ could produce in the next $\sim 5\mathrm{years}$, should yield robust and accurate Hubble constant estimates that are precise to a level of $\lesssim 2\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, sufficient to reliably resolve the current tension between local and cosmological measurements of $H_0$.

Figure 1

The expected range of $H_0$ estimates from samples of $N=10$, $N=30$, and $N=100$ BNS merger events, as a function of the maximum distance $D_{*}$ to which sources can be detected. The offset from zero comes from the potential bias due to selection effects, which would dominate over sample variance for $N\gtrsim 10$ and $D_{*}\gtrsim 70\mathrm{Mpc}$.

Figure 2

Constraints on the distance $D$ and inclination $\iota$ for three representative BNS mergers from the GW data only, using the simplified model described in Sec. 4. The purple and mauve areas indicate the 68% and 95% highest posterior density credible regions and the dashed lines show the input parameters.

Figure 3

Constraints on the Hubble constant $H_0$ and the cosine of the system inclination $\cos (\iota )$ from a BNS merger designed to mimic GW 170817. These distributions are hence directly comparable to Figs. 1 and 2 of Ref. [45]. The purple and mauve areas indicate the 68% and 95% highest posterior density credible regions.

Figure 4

Distributions of the measured SNR $\widehat{\rho }$ and distance (left) and inclination (right) for a simulated sample of 100 selected BNS merger events (purple) with $\rho \ge {\rho }_{*}=12$. The much larger number of mergers that were not selected are also shown in orange.

Figure 5

Comparison of the fiducial best-fit distance $\widehat{D}$ and the true distance $D$ for both selected (purple) and rejected (orange) sources. The overabundance of sources close to the maximum survey distance with $\widehat{D}<D$ would potentially give a biased estimate of $H_0$, as described in Sec. 3.

Figure 6

Left: $H_0$ posteriors for 25 independent 100-BNS samples, generated assuming a linear Hubble relation. The posteriors are colored dark purple to light orange by their MAP $H_0$ values. Center/right: Distribution of MAP $H_0$ values and posterior standard deviations for the full set of 1000 100-BNS samples.

Figure 7

As in Fig. 6, but generated assuming a quadratic Hubble relation. From left to right: Example $H_0$ posteriors (colored from low to high MAP value) and distributions of MAP $H_0$ values and posterior standard deviations for 1000 100-BNS samples.

Figure 8

Constraints on $H_0$ and $q_0$ for a single 100-event sample, inferred using our fiducial $\mathcal{M}$ prior (purple solid), a prior shifted upwards by one sigma (pink dashed), and a prior with 4 times the variance (orange dot-dashed). The inference is robust to reasonable variations in the chirp mass prior.

Figure 9

Network diagram for the hierarchical model describing the BNS population and data. Quantities in single circles are model parameters, while quantities in double circles are measured values. These quantities are linked by the probability distributions in orange rectangles: the top row are parameter priors, the middle row describes the population model, and the bottom row defines the (object-level) likelihood. The arrows linking the quantities and distributions define the forward/generative model that could be used to simulate samples and measurements. The quantities inside the red rectangular plate are specific to a single BNS merger event, and the detected events are indexed by $i\in \{1,2,\dots N\}$.