Parametrized tests of the strong-field dynamics of general relativity using gravitational wave signals from coalescing binary black holes: Fast likelihood calculations and sensitivity of the method

Thanks to the recent discoveries of gravitational wave signals from binary black hole mergers by Advanced Laser Interferometer Gravitational Wave Observatory and Advanced Virgo, the genuinely strong-field dynamics of spacetime can now be probed, allowing for stringent tests of general relativity (GR). One set of tests consists of allowing for parametrized deformations away from GR in the template waveform models and then constraining the size of the deviations, as was done for the detected signals in previous work. In this paper, we construct reduced-order quadratures so as to speed up likelihood calculations for parameter estimation on future events. Next, we explicitly demonstrate the robustness of the parametrized tests by showing that they will correctly indicate consistency with GR if the theory is valid. We also check to what extent deviations from GR can be constrained as information from an increasing number of detections is combined. Finally, we evaluate the sensitivity of the method to possible violations of GR.

Figure 1

The three regimes of the IMRPhenomPv2 model. (Top) The modulus of the waveform as a function of frequency for a signal similar to GW150914. (Bottom) Fourier transform to the time domain (top) and instantaneous frequency as a function of time (bottom).

Figure 2

The effect of varying testing parameters on the phase as a function of frequency for $t_c={\phi }_c=0$. In the top of each panel, we plot the GR phase (black) as well as the way the phase varies with a testing parameter (colors); the bottom shows the difference. In the left column, we show results for an event with parameters like that of GW150914 and similarly in the right column for GW151226.

Figure 3

The differences at $f=150\mathrm{Hz}$ between the GR phase and the phase for given values of testing parameters, for inspiral (top), the intermediate regime (middle), and the merger-ringdown regime (bottom). For definiteness, we again set $t_c={\phi }_c=0$. The solid lines are for zero spins, and the dashed lines for aligned spins with $|{\mathbf{S}}_{1,2}|=0.9$.

Figure 4

Schematic illustration of how the original basis for IMRPhenomPv2 from [58] (triangles) is extended in the additional parameter dimension $\delta {\widehat{p}}_i$ (in this example $\delta {\widehat{\phi }}_3$) to form a new training set. The plot only shows a three-dimensional slice of the full parameter space. Note that points with $-0.02\le \delta {\widehat{\phi }}_3\le 0.02$ are not shown to aid visualization.

Figure 5

Distributions of interpolation errors at different validation steps for some of the testing parameters and chirp mass bins. The left column is for the linear bases, the right column for the quadratic ones. (First row) $\delta {\widehat{\phi }}_3$ for chirp mass bin A in Table 1; (second row) $\delta {\widehat{\beta }}_2$ for bin C; (third row) $\delta {\widehat{\alpha }}_2$ for bin C. In some cases no bad points are found, so that the basis does not need to be enlarged and no further validation steps are needed.

Figure 6

A comparison of parameter estimation results on a simulated signal with parameters ${\mathcal{M}}_c=8.9M_{\odot }$, $q=1.99$, $D_{\mathrm{L}}=200\mathrm{Mpc}$, and $\delta {\widehat{p}}_i=0$, in synthetic, stationary, Gaussian noise, analyzed with and without the ROQ. Results are shown for the cases $\delta {\widehat{\phi }}_3$ (top row), $\delta {\widehat{\beta }}_2$ (middle row), and $\delta {\widehat{\alpha }}_2$ (bottom row). In each case, we show the posterior density function for the testing parameter itself (left column) and for chirp mass (right column). The values of the parameters in the signal are indicated by the vertical dashed lines. Results with and without ROQ agree to within sampling uncertainties [72].

Figure 7

Fraction of simulated signals in stationary Gaussian noise for which the value of zero for $\delta {\widehat{p}}_i$ is within a given confidence level. Shown are $p\text{−}p$ plots for $\delta {\widehat{\phi }}_3$ (left), $\delta {\widehat{\beta }}_2$ (middle), and $\delta {\widehat{\alpha }}_2$ (right). The dark and light gray bands indicate the $1\text{−}\sigma$ and $2\text{−}\sigma$ departures from the diagonal that can be expected on theoretical grounds. The results are consistent with a general absence of bias in the measurements.

Figure 8

Ninety percent credible intervals for the PN testing parameters obtained by performing the parametrized tests on a numerical relativity injection in 21 different stretches of realistic detector data. Note how offsets tend to alternate from one PN testing parameter to the next; this is due to partial correlation between them and the alternating signs of the PN parameters themselves.

Figure 9

Histograms of GR quantiles for the PN testing parameters corresponding to the same simulations as for Fig. 8. Though based on analyses of only 21 stretches of data, the results are consistent with the quantiles being uniformly distributed on the interval [0,1].

Figure 10

Sharper constraints on deviation from GR can be obtained by combining posterior density functions for the $\delta {\widehat{p}}_i$ from all available detections. This is illustrated for $\delta {\widehat{\phi }}_3$ (left), $\delta {\widehat{\beta }}_2$ (middle), and $\delta {\widehat{\alpha }}_2$ (right). The black curve shows the median of the joint distribution, and the darker and lighter shadings show the 68% and 95% confidence intervals, respectively.

Figure 11

(Top) Posterior densities for testing parameters for an injection with $\delta {\widehat{\phi }}_3=+0.4$ (orange) and $\delta {\widehat{\phi }}_3=-0.4$ (blue). (Bottom) posteriors for an injection with $\delta {\widehat{\phi }}_4=+3.3$ (orange) and $\delta {\widehat{\phi }}_4=-3.3$ (blue). Note how all the PN testing parameters indicate a deviation from GR, not just the ones that deviate from zero in the signal.

Figure 12

(Top) Posterior densities for testing parameters for an injection with $\delta {\widehat{\beta }}_2=+0.7$ (orange) and $\delta {\widehat{\beta }}_2=-0.7$ (blue). (Bottom) Posteriors for an injection with $\delta {\widehat{\beta }}_3=+0.8$ (orange) and $\delta {\widehat{\beta }}_3=-0.8$ (blue). In each case, the GR violation is also picked up by the other $\delta {\widehat{\beta }}_i$.

Figure 13

(Top) Posterior densities for testing parameters for an injection with $\delta {\widehat{\alpha }}_2=+1.3$ (orange) and $\delta {\widehat{\alpha }}_2=-1.3$ (blue). (Bottom) Posteriors for an injection with $\delta {\widehat{\alpha }}_4=+1.6$ (orange) and $\delta {\widehat{\alpha }}_4=-1.6$ (blue). Here too, in each case, the other $\delta {\widehat{\alpha }}_i$ also pick up the GR violation.

Figure 14

Results for injections where all of the signal’s testing parameters starting from 1.5 PN have a fractional shift $|\delta {\widehat{p}}_i|=0.4$, in one case, all with positive sign (orange), in another case, with a sign that alternates from one parameter to the next (blue); see the main text for details. In both cases, the offsets of the posterior densities follow the way successive PN coefficients are correlated. Also note how in both cases all of the $\delta {\widehat{p}}_i$ clearly indicate a GR violation in the signal. In fact, from 1.5 PN order onwards, the measured violations in PN parameters is larger than the injected deviation at a given order: individual testing parameters will try to accommodate the collective change in the signal resulting from the shifts in all the parameters together.