Possibility of determining cosmological parameters from measurements of gravitational waves emitted by coalescing, compact binaries

We explore the feasibility of using LIGO and/or VIRGO gravitational-wave measurements of coalescing, neutron-star-neutron-star (NS-NS) binaries and black-hole-neutron-star (BH-NS) binaries at cosmological distances to determine the cosmological parameters of our Universe. From the observed gravitational waveforms one can infer, as direct observables, the luminosity distance $D$ of the source and the binary's two "redshifted masses," $M_1^{\prime }\equiv M_1\mathrm{}(1+z)\mathrm{}$ and $M_2^{\prime }\equiv M_2\mathrm{}(1+z)\mathrm{}$, where $M_i$ are the actual masses and $z\equiv \frac{\Delta \lambda }{\lambda }$ is the binary's cosmological redshift. Assuming that the NS mass spectrum is sharply peaked about $1.4M_{\odot }$, as binary pulsar and x-ray source observations suggest, the redshift can be estimated as $z=\frac{M_{\mathrm{NS}}^{\prime }}{1.4M_{\odot }}-1\mathrm{}$. The actual distance-redshift relation $D\left(z\right)\mathrm{}$ for our Universe is strongly dependent on its cosmological parameters [the Hubble constant $H_0$, or $h_0\equiv \frac{H_0}{100}$ km ${\mathrm{s}}^{-1\mathrm{}}$M${\mathrm{pc}}^{-1\mathrm{}}$, the mean mass density ${\rho }_m$, or density parameter ${\Omega }_0\equiv \left(\frac{8\pi }{3H_0^2}\right){\rho }_m$, and the cosmological constant $\Lambda$, or ${\lambda }_0\equiv \frac{\Lambda }{\left(3\mathrm{}{H}_0^2\right)}$], so by a statistical study of (necessarily noisy) measurements of $D$ and $z$ for a large number of binaries, one can deduce the cosmological parameters. The various noise sources that will plague such a cosmological study are discussed and estimated, and the accuracies of the inferred parameters are determined as functions of the detectors' noise characteristics, the number of binaries observed, and the neutron-star mass spectrum. The dominant source of error is the detectors' intrinsic noise, though stochastic gravitational lensing of the waves by intervening matter might significantly influence the inferred cosmological constant ${\lambda }_0$, when the detectors reach "advanced" stages of development. The estimated errors of parameters inferred from BH-NS measurements can be described by the following rough analytic fits: $\frac{\Delta h_0}{h_0}\simeq 0.02\left(\frac{N}{h_0}\right){\left(\tau \mathcal{R}\right)}^{-\frac{1}{2}}$ (for $\frac{N}{h_0}\lesssim 2$), where $N$ is the detector's noise level ($\frac{\mathrm{strain}}{\sqrt{\mathrm{Hz}}}$) in units of the "advanced LIGO" noise level, $\mathcal{R}$ is the event rate in units of the best-estimate value, 100 ${\mathrm{yr}}^{-1\mathrm{}}$ G${\mathrm{pc}}^{-3\mathrm{}}$, and $\tau$ is the observation time in years. In a "high density" universe (${\Omega }_0=1\mathrm{}$, ${\lambda }_0=0\mathrm{}$), $\Delta {\Omega }_0\simeq 0.3{\left(\frac{N}{h_0}\right)}^2{\left(\tau \mathcal{R}\right)}^{-\frac{1}{2}}$, $\Delta {\lambda }_0\simeq 0.4{\left(\frac{N}{h_0}\right)}^{1.5}{\left(\tau \mathcal{R}\right)}^{-\frac{1}{2}}$, for $\frac{N}{h_0}\lesssim 1$. In a "low density" universe (${\Omega }_0=0.2\mathrm{}$, ${\lambda }_0=0\mathrm{}$), $\Delta {\Omega }_0\simeq 0.5{\left(\frac{N}{h_0}\right)}^3{\left(\tau \mathcal{R}\right)}^{-\frac{1}{2}}$, $\Delta {\lambda }_0\simeq 0.7{\left(\frac{N}{h_0}\right)}^{2.5}{\left(\tau \mathcal{R}\right)}^{-\frac{1}{2}}$, also for $\frac{N}{h_0}\lesssim 1$. These formulas indicate that, if event rates are those currently estimated (∼3 per year out to 200 Mpc), then when the planned LIGO and/or VIRGO detectors get to be about as sensitive as the so-called "advanced detector level" (presumably in the early 2000s), interesting cosmological measurements can begin.