Probing the nonlinear structure of general relativity with black hole binaries

Observations of the inspiral of massive binary black holes (BBH) in the Laser Interferometer Space Antenna (LISA) and stellar mass binary black holes in the European Gravitational Wave Observatory (EGO) offer an unique opportunity to test the nonlinear structure of general relativity. For a binary composed of two nonspinning black holes, the nonlinear general relativistic effects depend only on the masses of the constituents. In a recent paper, we explored the possibility of a test to determine all the post-Newtonian coefficients in the gravitational wave phasing. However, mutual covariances dilute the effectiveness of such a test. In this paper, we propose a more powerful test in which the various post-Newtonian coefficients in the gravitational wave phasing are systematically measured by treating three of them as independent parameters and demanding their mutual consistency. LISA (EGO) will observe BBH inspirals with a signal-to-noise ratio of more than 1000 (100) and thereby test the self-consistency of each of the nine post-Newtonian coefficients that have so-far been computed, by measuring the lower order coefficients to a relative accuracy of $\sim {10}^{-5}$ (respectively, $\sim {10}^{-4}$) and the higher order coefficients to a relative accuracy in the range ${10}^{-4}-0.1$ (respectively, ${10}^{-3}-1$).

Figure 1

The signal-to-noise ratio for stellar mass binary black holes (BBH) in Advanced LIGO and EGO and supermassive BBH in LISA for equal mass binaries at a distance of 200 Mpc (for EGO and Advanced LIGO) and $z=1$ (LISA). In the case of LISA we assume that the signal is integrated for a year (last year before coalescence) and in the case of EGO we assume that the signal is integrated over a bandwidth from 10 Hz until the binary reaches its innermost circular orbit. The masses of supermassive BBH in the case of LISA have been scaled down by a factor of ${10}^4$.

Figure 2

Plot showing the relative errors $\Delta {\psi }_T/{\psi }_T$, in the test parameters ${\psi }_T={\psi }_3$, ${\psi }_4$, ${\psi }_{5l}$, ${\psi }_6$, ${\psi }_{6l}$, ${\psi }_7$ as a function of the total mass $M$ of a supermassive BBH at a redshift of $z=1$ observed by LISA (right panel) and of a stellar mass compact binary at a distance of $D_L=200\mathrm{Mpc}$ observed by EGO (left panel). The rest of the details as in Fig. 1.

Figure 3

Plot showing the regions in the $m_1–m_2$ plane that correspond to 1-$\sigma$ uncertainties in the test parameters ${\psi }_T={\psi }_3$, ${\psi }_4$, ${\psi }_{5l}$, ${\psi }_6$, ${\psi }_{6l}$, ${\psi }_7$ for a $({10}^6,{10}^6)M_{\odot }$ supermassive black hole binary at a redshift of $z=1$ as observed for a year by LISA. (Note that the $1\mathrm{\text{−}}\sigma$ uncertainty in ${\psi }_3$ is smaller than the thickness of the line.)