Cosmology using advanced gravitational-wave detectors alone

We investigate a novel approach to measuring the Hubble constant using gravitational-wave (GW) signals from compact binaries by exploiting the narrowness of the distribution of masses of the underlying neutron-star population. Gravitational-wave observations with a network of detectors will permit a direct, independent measurement of the distance to the source systems. If the redshift of the source is known, these inspiraling double-neutron-star binary systems can be used as standard sirens to extract cosmological information. Unfortunately, the redshift and the system chirp mass are degenerate in GW observations. Thus, most previous work has assumed that the source redshift is obtained from electromagnetic counterparts. However, we investigate a novel method of using these systems as standard sirens with GW observations alone. In this paper, we explore what we can learn about the background cosmology and the mass distribution of neutron stars from the set of neutron-star (NS) mergers detected by such a network. We use a Bayesian formalism to analyze catalogs of NS-NS inspiral detections. We find that it is possible to constrain the Hubble constant, $H_0$, and the parameters of the NS mass function using gravitational-wave data alone, without relying on electromagnetic counterparts. Under reasonable assumptions, we will be able to determine $H_0$ to $\pm 10\%$ using $\sim 100$ observations, provided the Gaussian half-width of the underlying double NS mass distribution is less than $0.04M_{\odot }$. The expected precision depends linearly on the intrinsic width of the NS mass function, but has only a weak dependence on $H_0$ near the default parameter values. Finally, we consider what happens if, for some fraction of our data catalog, we have an electromagnetically measured redshift. The detection, and cataloging, of these compact-object mergers will allow precision astronomy, and provide a determination of $H_0$ which is independent of the local distance scale.

Figure 1

Recovered 1D posterior distributions for $H_0$ (left), ${\sigma }_{\mathrm{NS}}$ ( center) and ${\mu }_{\mathrm{NS}}$ (right), computed for one realization. The black lines represent best-fit Gaussian distributions to $H_0$, $\ln ({\sigma }_{\mathrm{NS}})$ and ${\mu }_{\mathrm{NS}}$, which were obtained via a least-squares fitting procedure. The vertical lines indicate the values of these parameters used to generate the data set.

Figure 2

Recovered 2D posterior distribution in $H_0$ and ${\mu }_{\mathrm{NS}}$ space, showing a correlation between these recovered parameters. The model parameter values used to generate the data are the reference values. There appears to be negligible correlation between ${\sigma }_{\mathrm{NS}}$ with $H_0$ or ${\mu }_{\mathrm{NS}}$.

Figure 3

Distribution of the Gaussian-fit standard deviations (top) and errors (bottom) of the recovered posteriors over 100 realizations, for $H_0$ (left), $\ln ({\sigma }_{\mathrm{NS}})$ ( center) and ${\mu }_{\mathrm{NS}}$ (right). More details are given in the text.

Figure 4

A comparison of the best-fit $\sigma$ distributions over 100 realizations, between the case of no errors present in the data catalog, and the case of errors applied to system properties in the catalog. We have attempted to compensate for the errors in the data. (i), (iii) and (v) show the best-fit $\sigma$ distributions when no errors are applied to the system properties in the data catalog. (ii), (iv) and (vi) show the best-fit $\sigma$ distributions when the received data has errors.

Figure 5

The “measurement accuracy” of each parameter (represented by the standard deviation of the Gaussian fit to the posterior) plotted against the number of observed events, $N_o$. The intrinsic parameters are kept fixed while the local merger-rate density, ${\dot{n}}_0$, is scaled up and down. The number of observed events scales linearly with the observation time and the local merger-rate density, such that the same result is achieved for twice the local merger-rate density if the observation time is halved. We see that a $1/\sqrt{N_o}$ relation is favored. The points and solid lines correspond to the case when we ignore errors, where the curves have gradients $108\pm 2\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, $0.737\pm 0.001$ and $0.131\pm 0.002M_{\odot }$ respectively. The dashed lines are best-fit curves for the same analysis, but with measurement errors included and accounted for, for which the gradients are $136\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, 0.917 and $0.152M_{\odot }$, respectively.

Figure 6

The variation of measurement accuracy with instrumental distance reach is shown. Each point represents the weighted mean of the $H_0$ measurement accuracy from 10 realizations at a particular $r_{0,\mathrm{net}}$, where the error bars show the maximum and minimum values of $\sigma$ out of the 10 values. All other parameters are at their reference values, and the total number of detections is scaled to 100. The reference distance reach is $r_{0,\mathrm{net}}\sim 176\mathrm{Mpc}$. The curve is a ($1/r_{0,\mathrm{net}}$) fit to the data, with gradient $10.8\pm 0.2\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$.

Figure 7

The variation of the weighted mean (over 10 realizations) of the Gaussian-fit standard deviations with the parameters of the underlying NS mass distribution. All other parameters are fixed at their reference values. The variation of the expected number of detections with ${\sigma }_{\mathrm{NS}}$ is less than one event, while for ${\mu }_{\mathrm{NS}}$ it is more significant. Thus all the posterior fit $\sigma$ values are scaled to the average number of detections for a given ${\mu }_{\mathrm{NS}}$, e.g., for ${\mu }_{\mathrm{NS}}=1.33M_{\odot }$ this average event number is $\sim 95$, while for $1.39M_{\odot }$ it is $\sim 110$, in 1 yr.

Figure 8

Plots of the measurement precision versus the beaming fraction of SGRBs, $f$. Approximately $f{N}_o$ events in the catalog were denoted as SGRBs, while the remaining $\sim (1-f)N_o$ events in the catalog were assumed to be GW-only events. A single data catalog was used repeatedly, with larger and larger fractions of it assumed to be observable as SGRBs. The data was generated with the reference parameters. The fitted curves are of the form $a\exp (-b\sqrt{x})+c$, where $(a,b,c)=(9.04,3.64,1.59)$ and (0.00932, 4.46, 0.00439) for the $H_0$ and ${\mu }_{\mathrm{NS}}$ precisions, respectively. The corresponding plot for ${\sigma }_{\mathrm{NS}}$ shows no obvious trend.