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

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