Cosmological inference using gravitational wave standard sirens: A mock data analysis

The observation of binary neutron star merger GW170817, along with its optical counterpart, provided the first constraint on the Hubble constant $H_0$ using gravitational wave standard sirens. When no counterpart is identified, a galaxy catalog can be used to provide the necessary redshift information. However, the true host might not be contained in a catalog which is not complete out to the limit of gravitational-wave detectability. These electromagnetic and gravitational-wave selection effects must be accounted for. We describe and implement a method to estimate $H_0$ using both the counterpart and the galaxy catalog standard siren methods. We perform a series of mock data analyses using binary neutron star mergers to confirm our ability to recover an unbiased estimate of $H_0$. Our simulations used a simplified universe with no redshift uncertainties or galaxy clustering, but with different magnitude-limited catalogs and assumed host galaxy properties, to test our treatment of both selection effects. We explore how the incompleteness of catalogs affects the final measurement of $H_0$, as well as the effect of weighting each galaxy’s likelihood of being a host by its luminosity. In our most realistic simulation, where the simulated catalog is about three times denser than the density of galaxies in the local universe, we find that a 4.4% measurement precision can be reached using galaxy catalogs with 50% completeness and $\sim 250$ binary neutron star detections with sensitivity similar to that of Advanced LIGO’s second observing run.

Figure 1

Galaxy catalog completeness fractions for MDA2 and MDA3. Left panel: Galaxy number completeness fraction defined in Eq. (12) as a function of luminosity distance for the three MDA2 sub-catalogs. The lines in green, orange and blue correspond to the catalogs with $m_{\mathrm{th}}=19.5$, 18, and 16 respectively; these correspond to completeness fractions of 75%, 50% and 25% out to a fiducial reference distance of 115 Mpc (shown as a vertical grey line). Right panel: The galaxy luminosity completeness fraction defined in Eq. (15) as a function of luminosity distance for the MDA3 catalog, with $m_{\mathrm{th}}=14$. At the reference distance of 115 Mpc (vertical grey line), this is corresponds to a completeness fraction of $\sim 50\%$.

Figure 2

Individual and combined results for MDA0 (known host galaxy or direct counterpart case). The solid thick purple line shows the combined posterior probability density on $H_0$, while the dashed line shows the combined posterior when GW selection effects are neglected. Individual likelihoods (normalized and then scaled by an arbitrary value), for each of the 249 events, are shown as thin lines with shades corresponding to their optimal SNR. The simulated value of $H_0$ is shown as a vertical dashed line.

Figure 3

Comparison of the galaxy catalog method with the known host galaxy case. Joint posterior probability density on $H_0$ using all 249 events for MDA0 (known host galaxy) and MDA1 (complete galaxy catalog) are shown respectively in purple and red. For this set of simulations, uncertainty with the galaxy catalog is only about 1.63 times larger than with known host galaxies.

Figure 4

Individual and combined results for MDA1 (complete galaxy catalog). The thick red line shows the combined posterior probability density on $H_0$. Individual likelihoods (normalized and then scaled by an arbitrary value), for each of the 249 events, are shown as thin lines with shades corresponding to their optimal SNR. The simulated value of $H_0$ is shown as a vertical dashed line. Many of the individual likelihoods do not have sharp features, however the final result converges to the simulated value with redshift information present in the galaxy catalogs. This demonstrates the applicability of our methodology.

Figure 5

Comparison of results with varying galaxy catalog completeness. In MDA2, the simulated apparent magnitude threshold is varied to obtain galaxy catalogs of 100%, 75%, 50%, and 25% completeness. The corresponding posterior probability densities on $H_0$ using all 249 events are shown in red, green, yellow, and blue respectively.

Figure 6

Individual and combined results for MDA2 with a 25% complete galaxy catalog. The thick blue line shows the combined posterior probability density on $H_0$. Individual likelihoods (normalized and then scaled by an arbitrary value), for each of the 249 events, are shown as thin lines with shades corresponding to their optimal SNR. The simulated value of $H_0$ is shown as a vertical dashed line. Compared to MDA0 (Fig. 2) and MDA1 (Fig. 4), fewer individual likelihoods are peaked here. Although the final $H_0$ estimate is less precise, the results converge to the simulated value, demonstrating the applicability of our methodology to threshold-limited galaxy catalogs of about 25% completeness.

Figure 7

Comparison of results with and without luminosity weighting. In MDA3, by construction, the probability of any galaxy hosting a GW event is proportional to its luminosity. The pink curve shows the posterior probability density on $H_0$ for the case where we take this into account in our analysis as a weighting by the galaxy’s luminosity. The blue curve shows the posterior density for the case where we ignore this extra information, and treat every galaxy as equally likely to be hosts. Luminosity weighting improves the precision in the results by a factor of 1.2 for this set of simulations.

Figure 8

Fractional uncertainty in $H_0$ as a function of the number $N$ of the events for the combined $H_0$ posteriors. The fractional uncertainty in $H_0$ is defined as the half-width of the 68.3% highest probability density interval divided by $70\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, and is shown as the plotted dots for all cases. The error bars contain 68% of the scatter arising from different realizations of the events. (left) In purple, red, green, yellow and blue we show the associated host galaxy case (MDA0), complete galaxy catalog (MDA1) case, and the 75%, 50% and 25% completeness cases; we find a fractional $H_0$ uncertainty of 1.13%, 1.84%, 2.21%, 2.48% and 3.20% respectively for the combined $H_0$ posterior from 249 events. (right) convergence for MDA3 (event probability proportional to galaxy luminosity), analyzed with luminosity-weighted likelihood (pink) or equally-weighted likelihood (light blue). We find fractional $H_0$ uncertainties of 4.48% and 5.31% respectively. MDA0 (purple) is included for reference. We plot the expected $1/\sqrt{N}$ scaling behavior for large values of N for all cases with the dashed lines. This scaling behavior is met by all MDAs as the number of events reaches 249, but for the less informative, lower completeness MDAs the trend is slower to emerge. This is even more evident in MDA3, where the density of galaxies is 100 times greater, producing more potential hosts for each event. This is mitigated somewhat by the effect of luminosity-weighting the potential hosts (pink).

Figure 9

A network diagram showing how the main parameters of the methodology interlink. Circular nodes denote ordinary parameters. Hexagonal nodes denote assumed knowns. Rectangular nodes denote binary flags. The arrows indicate the dependence of each parameter on the parameters which feed into them. The parameters grouped in the “galaxies” cluster are those which can be evaluated for every galaxy in the universe.

Figure 10

Probability of detection, $p(D_{\mathrm{GW}}|z,H_0)$, as a function of $z$ for different values of $H_0$. We assume a 2-detector network at O2-like sensitivity, for a population of binary neutron stars.