Parameter estimation of inspiralling compact binaries using 3.5 post-Newtonian gravitational wave phasing: The nonspinning case

We revisit the problem of parameter estimation of gravitational-wave chirp signals from inspiralling nonspinning compact binaries in the light of the recent extension of the post-Newtonian (PN) phasing formula to order $(v/c{)}^7$ beyond the leading Newtonian order. We study in detail the implications of higher post-Newtonian orders from 1PN up to 3.5PN in steps of 0.5PN ($\sim v/c$), and examine their convergence. In both initial and advanced detectors the estimation of the chirp mass ($\mathcal{M}$) and symmetric mass ratio ($\eta$) improve at higher PN orders but oscillate with every half-a-PN order. In initial LIGO, for a $10M_{\odot }–10M_{\odot }$ binary at a signal-to-noise ratio (SNR) of 10, the improvement in the estimation of $\mathcal{M}$ ($\eta$) at 3.5PN relative to 2PN is $\sim 19\%$ (52%). We compare parameter estimation in different detectors and assess their relative performance in two different ways: at a fixed SNR, with the aim of understanding how the bandwidth improves parameter estimation, and for a fixed source, to gauge the importance of sensitivity. Errors in parameter estimation at a fixed SNR are smaller for VIRGO than for both initial and advanced LIGO. This is because of the larger bandwidth over which it observes the signals. However, for sources at a fixed distance it is advanced LIGO that achieves the lowest errors owing to its greater sensitivity. Finally, we compute the amplitude corrections due to the “frequency-sweep” in the Fourier domain representation of the waveform within the stationary phase approximation and discuss its implication on parameter estimation. We find that the amplitude corrections change the errors in $\mathcal{M}$ and $\eta$ by less than 10% for initial LIGO at a signal-to-noise ratio of 10. Our analysis makes explicit the significance of higher PN order modeling of the inspiralling compact binary on parameter estimation.

Figure 1

Amplitude spectrum (top panel) of initial LIGO, VIRGO and advanced LIGO together with the luminosity distance (bottom panel) at which RMS-oriented binaries would produce a SNR of 5.

Figure 2

Comparison of errors in the estimation of $t_c$, $\mathcal{M}$ and $\eta$ for sources with a fixed SNR of 10 (left panels) with those for systems at a fixed distance of 300 Mpc (right panels).

Figure 3

Left-hand panel is the plot of the derivative $d{N}_{\mathrm{useful}}/d(\ln f)$ against the frequency (in arbitrary normalization) for the three detectors. Similarly, right panel gives the number of useful cycles as a function of the total mass of the binary for the three detectors.