Parametrized tests of post-Newtonian theory using Advanced LIGO and Einstein Telescope

General relativity has very specific predictions for the gravitational waveforms from inspiralling compact binaries obtained using the post-Newtonian (PN) approximation. We investigate the extent to which the measurement of the PN coefficients, possible with the second generation gravitational-wave detectors such as the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) and the third generation gravitational-wave detectors such as the Einstein Telescope (ET), could be used to test post-Newtonian theory and to put bounds on a subclass of parametrized-post-Einstein theories which differ from general relativity in a parametrized sense. We demonstrate this possibility by employing the best inspiralling waveform model for nonspinning compact binaries which is 3.5PN accurate in phase and 3PN in amplitude. Within the class of theories considered, Advanced LIGO can test the theory at 1.5PN and thus the leading tail term. Future observations of stellar mass black hole binaries by ET can test the consistency between the various PN coefficients in the gravitational-wave phasing over the mass range of $11–44M_{\odot }$. The choice of the lower frequency cutoff is important for testing post-Newtonian theory using the ET. The bias in the test arising from the assumption of nonspinning binaries is indicated.

Figure 1

Amplitude spectrum of Advanced LIGO and ET.

Figure 2

Plots showing the regions in the $m_1\mathrm{\text{−}}{m}_2$ plane that correspond to $1\mathrm{\text{−}}\sigma$ uncertainties in ${\psi }_0$, ${\psi }_2$, and ${\psi }_{5l}$ (left panel) and those in ${\psi }_0$, ${\psi }_2$, and ${\psi }_{5l\mathrm{mod}}$ (right panel) for a $(2,20)M_{\odot }$ BBH at a luminosity distance of $D_L=300\mathrm{Mpc}$ observed by ET. The low-frequency cutoff is 1 Hz and RWF has been used. For the curves in the right panel we have assumed that the correct theory of gravity is a hypothetical non-GR theory in which the phasing coefficient ${\psi }_{5l}$ and all higher PN coefficients differ from the GR values by 1%.

Figure 3

Plots showing the regions in the $m_1\mathrm{\text{−}}{m}_2$ plane that correspond to $1\mathrm{\text{−}}\sigma$ uncertainties in Newtonian, 1PN, and 2.5PN coefficients in the PN series for a $(10,100)M_{\odot }$ BBH at a luminosity distance of $D_L=3\mathrm{Gpc}$ observed by ET. The low-frequency cutoff is 1 Hz and RWF has been used. The left panel corresponds to GR as the correct theory of gravity while the middle and right panels correspond to hypothetical non-GR theories of gravity which have phasing coefficients ${\psi }_{5l}$ (2.5PN) and higher differing from the GR values by 1% and 10%, respectively.

Figure 4

Plots showing the variation of relative errors $\Delta {\psi }_T/{\psi }_T$ in the test parameters ${\psi }_T={\psi }_3$, ${\psi }_{5l}$ as a function of total mass of binaries in the range $11–110M_{\odot }$ (with component masses having mass ratio of 0.1) located at 300 Mpc observed by Advanced LIGO, using both the RWF and the FWF as a waveform model with the source orientations chosen arbitrarily to be $\theta =\varphi =\pi /6$, $\psi =\pi /4$, and $\iota =\pi /3$. The noise curve corresponds to the one shown in Fig. 1 for the Advanced LIGO case and its analytical fit is given by Eq. (2.1). It is evident from the plot in the left panel that the fractional accuracies with which ${\psi }_3$ can be measured are better than 6% for the entire mass range under consideration when FWF is used and thus can be used to test the theory of gravity. ${\psi }_{5l}$ (right panel) can be measured with fractional accuracies better than 23% for the entire mass range when FWF is used but being a poorly determined parameter it can provide a much less stringent test of the theory of gravity.

Figure 5

Plots showing the variation of relative errors $\Delta {\psi }_T/{\psi }_T$ in the test parameters ${\psi }_T={\psi }_3$, ${\psi }_4$, ${\psi }_{5l}$, ${\psi }_6$, ${\psi }_{6l}$, and ${\psi }_7$ as a function of total mass $M$ for stellar mass black hole binaries (with component masses having mass ratio 0.1) at a luminosity distance of $D_L=300\mathrm{Mpc}$ observed by ET, using both RWF (left panels) and FWF (right panels) as waveform models. The choice of the source orientations is the same as quoted in Fig. 4. The noise curve corresponds to the recent ET-B sensitivity curve. Top panels correspond to the lower frequency cutoff of 1 Hz. By using FWF as the waveform model all ${\psi }_k$’s except ${\psi }_4$ can be tested with fractional accuracy better than 2% in the mass range $11–44M_{\odot }$. Bottom panels correspond to the lower frequency cutoff of 10 Hz. Using FWF, all ${\psi }_k$’s except ${\psi }_4$ can be tested with fractional accuracy better than 7% in the mass range $11–44M_{\odot }$.

Figure 6

Plots showing the regions in the $m_1\mathrm{\text{−}}{m}_2$ plane that correspond to $1\mathrm{\text{−}}\sigma$ uncertainties in ${\psi }_0$, ${\psi }_2$ and various test parameters, which happen to be one of the six test parameters ${\psi }_T={\psi }_3$, ${\psi }_4,{\psi }_{5l}$, ${\psi }_6$, ${\psi }_{6l}$, and ${\psi }_7$ at one time, for a (2, 20) $M_{\odot }$ BBH at a luminosity distance of $D_L=300\mathrm{Mpc}$ observed by ET. In all six plots shown above ${\psi }_0$ and ${\psi }_2$ are chosen as the fundamental parameters (from which we can measure the masses of the two black holes). Each parameter corresponds to a given region in the $m_1\mathrm{\text{−}}{m}_2$ plane and if GR is the correct theory of gravity then all three parameters, ${\psi }_0$, ${\psi }_2$, and ${\psi }_T$ should have a nonempty intersection in the $m_1\mathrm{\text{−}}{m}_2$ plane. A smaller region leads to a stronger test. Notice that all panels have the same scaling except the top middle panel in which the $Y$ axis has been scaled by a factor 10.

Figure 7

Same as Fig. 5 but for intermediate mass black hole binaries (with component masses having mass ratio 0.1) at a luminosity distance of $D_L=3\mathrm{Gpc}$. With lower frequency cutoff of 1 Hz, using FWF as the waveform model, ${\psi }_3$ and ${\psi }_{5l}$ can be tested with fractional accuracy better than 10% for the mass range $55–400M_{\odot }$. On the other hand, with a lower frequency cutoff of 10 Hz, using the FWF, ${\psi }_3$ and ${\psi }_{5l}$ can be tested with fractional accuracy better than 10% for the mass range $90–220M_{\odot }$.

Figure 8

Same as in Fig. 6 but for intermediate black hole binaries in the mass range $(20,200)M_{\odot }$ at a luminosity distance of $D_L=3\mathrm{Gpc}$ observed by ET. Notice that all bottom panels and the top middle panel have the same scaling, whereas the $Y$ axes of the top left panel and the top right panel have been scaled by a factor of 5. Note that there appears just one boundary for ${\psi }_4$ in the plot shown in the top middle panel since the other bound does not exist for the range of values on the $X$ axis.

Figure 9

A comparison of relative errors in the measurement of various PN parameters for binaries with masses in the range $55–220M_{\odot }$ at a luminosity distance of 3 Gpc for two cases: The first is the same as before when ${\psi }_0$ and ${\psi }_2$ are basic parameters and all PN parameters up to 3.5PN (full phasing) except the test parameter are parametrized by ${\psi }_0$ and ${\psi }_2$. The others are similarly constructed but the phasing is truncated at the PN order corresponding to the test parameter. The low-frequency cutoff is 10 Hz and the RWF has been used. The test parameter with truncated phasing is denoted by ${\psi }_{i\mathrm{t}}$ while with full 3.5PN phasing it is denoted by ${\psi }_{i\mathrm{f}}$.

Figure 10

Histogram for the relative error in the estimation of the parameter ${\psi }_3$ using 100 different realizations of angular parameters for a $(10,100)M_{\odot }$ binary located at the luminosity distance of 3 Gpc. The low-frequency cutoff is 1 Hz and RWF has been used.

Figure 11

This plot shows the variation of systematic bias due to spin $F(\beta )$ with the spin parameter $0\le \beta \le 8.5$, where $F(\beta )$ is given by $F(\beta )=4\beta (16\pi -4\beta {)}^{-1}$.