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," ${M}_{1}^{\ensuremath{'}}\ensuremath{\equiv}{M}_{1}(1+z)$ and ${M}_{2}^{\ensuremath{'}}\ensuremath{\equiv}{M}_{2}(1+z)$, where ${M}_{i}$ are the actual masses and $z\ensuremath{\equiv}\frac{\ensuremath{\Delta}\ensuremath{\lambda}}{\ensuremath{\lambda}}$ is the binary's cosmological redshift. Assuming that the NS mass spectrum is sharply peaked about $1.4{M}_{\ensuremath{\bigodot}}$, as binary pulsar and x-ray source observations suggest, the redshift can be estimated as $z=\frac{{M}_{\mathrm{NS}}^{\ensuremath{'}}}{1.4{M}_{\ensuremath{\bigodot}}}\ensuremath{-}1$. The actual distance-redshift relation $D(z)$ for our Universe is strongly dependent on its cosmological parameters [the Hubble constant ${H}_{0}$, or ${h}_{0}\ensuremath{\equiv}\frac{{H}_{0}}{100}$ km ${\mathrm{s}}^{\ensuremath{-}1}$M${\mathrm{pc}}^{\ensuremath{-}1}$, the mean mass density ${\ensuremath{\rho}}_{m}$, or density parameter ${\ensuremath{\Omega}}_{0}\ensuremath{\equiv}(\frac{8\ensuremath{\pi}}{3{H}_{0}^{2}}){\ensuremath{\rho}}_{m}$, and the cosmological constant $\ensuremath{\Lambda}$, or ${\ensuremath{\lambda}}_{0}\ensuremath{\equiv}\frac{\ensuremath{\Lambda}}{(3{H}_{0}^{2})}$], 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 ${\ensuremath{\lambda}}_{0}$, 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: $\frac{\ensuremath{\Delta}{h}_{0}}{{h}_{0}}\ensuremath{\simeq}0.02(\frac{N}{{h}_{0}}){(\ensuremath{\tau}\mathcal{R})}^{\ensuremath{-}\frac{1}{2}}$ (for $\frac{N}{{h}_{0}}\ensuremath{\lesssim}2$), 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 ${\mathrm{yr}}^{\ensuremath{-}1}$ G${\mathrm{pc}}^{\ensuremath{-}3}$, and $\ensuremath{\tau}$ is the observation time in years. In a "high density" universe (${\ensuremath{\Omega}}_{0}=1$, ${\ensuremath{\lambda}}_{0}=0$), $\ensuremath{\Delta}{\ensuremath{\Omega}}_{0}\ensuremath{\simeq}0.3{(\frac{N}{{h}_{0}})}^{2}{(\ensuremath{\tau}\mathcal{R})}^{\ensuremath{-}\frac{1}{2}}$, $\ensuremath{\Delta}{\ensuremath{\lambda}}_{0}\ensuremath{\simeq}0.4{(\frac{N}{{h}_{0}})}^{1.5}{(\ensuremath{\tau}\mathcal{R})}^{\ensuremath{-}\frac{1}{2}}$, for $\frac{N}{{h}_{0}}\ensuremath{\lesssim}1$. In a "low density" universe (${\ensuremath{\Omega}}_{0}=0.2$, ${\ensuremath{\lambda}}_{0}=0$), $\ensuremath{\Delta}{\ensuremath{\Omega}}_{0}\ensuremath{\simeq}0.5{(\frac{N}{{h}_{0}})}^{3}{(\ensuremath{\tau}\mathcal{R})}^{\ensuremath{-}\frac{1}{2}}$, $\ensuremath{\Delta}{\ensuremath{\lambda}}_{0}\ensuremath{\simeq}0.7{(\frac{N}{{h}_{0}})}^{2.5}{(\ensuremath{\tau}\mathcal{R})}^{\ensuremath{-}\frac{1}{2}}$, also for $\frac{N}{{h}_{0}}\ensuremath{\lesssim}1$. These formulas indicate that, if event rates are those currently estimated (\ensuremath{\sim}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.