Towards a generic test of the strong field dynamics of general relativity using compact binary coalescence

Coalescences of binary neutron stars and/or black holes are amongst the most likely gravitational-wave signals to be observed in ground-based interferometric detectors. Apart from the astrophysical importance of their detection, they will also provide us with our very first empirical access to the genuine strong-field dynamics of general relativity (GR). We present a new framework based on Bayesian model selection aimed at detecting deviations from GR, subject to the constraints of the Advanced Virgo and LIGO detectors. The method tests the consistency of coefficients appearing in the waveform with the predictions made by GR, without relying on any specific alternative theory of gravity. The framework is suitable for low signal-to-noise ratio events through the construction of multiple subtests, most of which involve only a limited number of coefficients. It also naturally allows for the combination of information from multiple sources to increase one’s confidence in GR or a violation thereof. We expect it to be capable of finding a wide range of possible deviations from GR, including ones which, in principle, cannot be accommodated by the model waveforms, on the condition that the induced change in phase at frequencies where the detectors are the most sensitive is comparable to the effect of a few percent change in one or more of the low-order post-Newtonian phase coefficients. In principle, the framework can be used with any GR waveform approximant, with arbitrary parametrized deformations, to serve as model waveforms. In order to illustrate the workings of the method, we perform a range of numerical experiments in which simulated gravitational waves modeled in the restricted post-Newtonian, stationary phase approximation are added to Gaussian and stationary noise that follows the expected Advanced LIGO/Virgo noise curves.

Figure 1

The high-power, zero-detuning noise curve for Advanced LIGO, and the BNS-optimized Advanced Virgo noise curve.

Figure 2

The fitting factors for a range of $\mathcal{M}$, once the other parameters are maximized over. Here a deviation of ${\psi }_3$ from 2.5% to 17.5% is used for a $(1.4,1.4)M_{\odot }$ system. The vertical dashed line represents the true value of $\mathcal{M}$, while the maximum is offset to compensate for the modification in the phase.

Figure 3

The fitting factors for a range of $\eta$, once the other parameters are maximized over. Here a deviation of ${\psi }_3$ from 2.5% to 17.5% is used for a $(1.4,1.4)M_{\odot }$ system. The vertical dashed line represents the true value of $\eta$, while the maximum is offset to compensate for the modification in the phase.

Figure 4

The log odds ratios for individual sources. The blue crosses represent signals with standard GR waveforms, the red circles are signals with a constant 10% relative offset in ${\psi }_3$. A separation between the two is visible for SNR $\gtrsim 10$ and becomes more pronounced as the SNR increases.

Figure 5

Top panel: The normalized distribution $P(\ln O_{\mathrm{GR}}^{\mathrm{modGR}})$ of log odds ratios for individual sources, where the injections are either GR or have $\delta {\chi }_3=0.1$. Bottom panel: The normalized distribution $P(\ln {\mathcal{O}}_{\mathrm{GR}}^{\mathrm{modGR}})$ of logs of the combined odds ratios for GR injections and injections with $\delta {\chi }_3=0.1$, for catalogs of 15 sources each. The effectiveness of the catalog approach to testing for deviations from GR comes from the combination of multiple sources, each source contributing to the overall result in proportion to its own Bayes factors.

Figure 6

Top panel. Curves and left vertical axis: For a given SNR, the cumulative number of times that the Bayes factor against noise for a particular component hypothesis is the largest for injections with that SNR or below, for $\delta {\chi }_3=0.1$. All 1471 simulated sources were used. As expected, $B_{\mathrm{noise}}^3$ dominates, and Bayes factors for hypotheses that let ${\psi }_3$ be non-GR tend to outperform those that do not. The GR model outperforms the one with the largest number of free parameters. Histogram and right vertical axis: The number of sources per SNR bin. Bottom panel: The same as above, but restricting to sources with SNR $<12$. A similar behavior as for the full set of 1471 sources is observed. Note that, already at SNR close to threshold, the GR hypothesis is more likely to be disfavored.

Figure 7

The log odds ratios for individual sources. The blue crosses are for signals with standard GR waveforms, and the red circles are for signals with a constant 2.5% relative offset in ${\psi }_3$. This time, there is little separation between the two, although there are outliers for SNR $\gtrsim 15$.

Figure 8

Top panel: The normalized distribution $P(\ln O_{\mathrm{GR}}^{\mathrm{modGR}})$ of log odds ratios for individual sources, where the injections are either GR or have $\delta {\chi }_3=0.025$. Bottom panel: The normalized distribution $P(\ln {\mathcal{O}}_{\mathrm{GR}}^{\mathrm{modGR}})$ of logs of the combined odds ratios for GR injections and injections with $\delta {\chi }_3=0.025$, for catalogs of 15 sources each. For individual sources, the two distributions essentially lie on top of each other. However, when sources are combined into catalogs, it is possible for an outlier to boost the odds ratio of the entire catalog.

Figure 9

The buildup of cumulative Bayes factors against GR for individual hypotheses, and the odds ratio, for a typical catalog with $\delta {\chi }_3=0.025$. Note that on the basis of the Bayes factor against GR of the most inclusive hypothesis $H_{123}$ alone, one can conclude that the GR model is in fact the favored one. Even the log Bayes factor for $H_{23}$ ends up being negative. Additionally, the hypothesis with the largest Bayes factor is not $H_3$ but $H_1$. This illustrates that it is necessary to include as many hypotheses as possible in the analysis.

Figure 10

The efficiency of detecting a GR violation for sources with $\delta {\chi }_3=0.025$, as a function of catalog size for false alarm probabilities $\beta \in \{0.05,0.01\}$. The median and the 68% confidence interval from 5000 random catalog orderings are shown as the central curve and the error bars, respectively. The efficiency increases as a function of catalog size, once again underscoring the benefit of combining all available data.

Figure 11

Top panel: The change in FAPs in going from two to three testing parameters, but keeping the odds ratio thresholds fixed. Bottom panel: The change in efficiencies when keeping the false alarm probabilities fixed. The plots shown are for the case where the signals have $\delta {\chi }_3=0.025$. We see that increasing the number of testing parameters has only a moderate effect on the FAPs, while for fixed FAPs, the efficiencies do not change appreciably. Note, however, that when the evidence for a deviation from GR is marginal, the use of as many hypotheses as possible can be pivotal in finding the violation (see Figs. 9, 16).

Figure 12

Top panel: The difference of false alarm probabilities, ${}^{(3)}\beta -{}^{(\mathrm{target})}\beta$, for fixed log odds ratio thresholds and signals having $\delta {\chi }_3=0.025$, between our three-parameter test and a “targeted search” which only looks for a deviation in ${\psi }_3$, i.e., only tests the hypothesis $H_3$ against GR. We see that the difference is minor. Bottom panel: The difference in efficiencies (which is more important), ${}^{(3)}\zeta -{}^{(\mathrm{target})}\zeta$, for fixed false alarm probabilities. Especially for a large number of sources per catalog, our three-parameter test is actually more efficient than the “targeted search,” at least for this particular example. This is because the Bayes factor against GR for $H_3$ will not be the largest in every catalog, but our method naturally compensates for that. Of course, we do not expect our method to outperform a targeted search in the case of a more complicated deviation from GR.

Figure 13

The log odds ratios for individual sources for injections with an anomalous frequency dependence in the phase. The blue crosses represent GR injections, while the red circles are for signals that have a contribution to the phase with a frequency dependence in between that of the 1PN and 1.5PN terms (1.25PN). As with the $\delta {\chi }_3=0.1$ example, a separation between the two is visible for SNR $\gtrsim 10$ and becomes more pronounced as the SNR increases.

Figure 14

Top panel: The normalized distribution $P(\ln O_{\mathrm{GR}}^{\mathrm{modGR}})$ of log odds ratios for individual sources, where the injections are either GR or have the anomalous frequency dependence. Bottom panel: The normalized distribution $P(\ln {\mathcal{O}}_{\mathrm{GR}}^{\mathrm{modGR}})$ of logs of the combined odds ratios for GR injections and injections for catalogs of 15 sources each.

Figure 15

Top panel. Curves and left vertical axis: The cumulative number of times that the Bayes factor against noise for a particular component hypothesis is the largest, with increasing SNR, for individual sources with $\delta {\chi }_A=-2.2$. All 854 simulated sources were used. Histogram and right vertical axis: The number of sources per SNR bin. Bottom panel: The same as above, but now for sources with SNR $<12$. Close to threshold, no single hypothesis is the dominant one.

Figure 16

A few examples of how cumulative Bayes factors against GR for individual hypotheses, and the odds ratio, grow as sources with increasing SNR are being added within three different catalogs of 15 sources in total. Top and middle panels: Catalogs where sources have only modest SNRs ($<20$). Note the large differences in contributions from different hypotheses, and in the ordering of Bayes factors, between these two catalogs. Bottom panel: A catalog with a high SNR source; note, however, that the source with the highest SNR does not cause a particularly large “boost,” and the cumulative log odds ratio ends up being considerably lower than in the top and middle examples.

Figure 17

Posterior PDFs for a single GR injection with network SNR of 23.0. Top panel: $\delta {\chi }_1$ measured with a waveform that has $\{\overrightarrow{\theta },\delta {\chi }_1\}$ as free parameters. Middle panel: $\delta {\chi }_2$ measured with a waveform that has $\{\overrightarrow{\theta },\delta {\chi }_2\}$ free. Bottom panel: $\delta {\chi }_3$ measured with a waveform that has $\{\overrightarrow{\theta },\delta {\chi }_3\}$ free. In each case the distribution is tightly centered on zero, with standard deviations of 0.014, 0.015, and 0.019, respectively.

Figure 18

Posterior PDFs for a single injection with $\delta {\chi }_3=0.1$ and network SNR of 23.2. Top panel: $\delta {\chi }_1$ measured with a waveform that has $\{\overrightarrow{\theta },\delta {\chi }_1\}$ free. Middle panel: $\delta {\chi }_2$ measured with a waveform that has $\{\overrightarrow{\theta },\delta {\chi }_2\}$ free. Bottom panel: $\delta {\chi }_3$ measured with a waveform that has $\{\overrightarrow{\theta },\delta {\chi }_3\}$ free. As expected, the bottom PDF is sharply peaked at the correct value of $\delta {\chi }_3=0.1$, with a standard deviation of 0.012. For the top and middle PDFs, the one test parameter that is used differs from the parameter in the signal that has the shift. The parameters in the waveform will rearrange themselves such as to best accommodate the properties of the signal. Both $\delta {\chi }_1$ and $\delta {\chi }_2$ end up being sharply peaked, but not at the correct value of zero.

Figure 19

The posterior PDF for the injection with $\delta {\chi }_3=0.1$ as in the previous figure, but recovered with waveforms where $\{\overrightarrow{\theta },\delta {\chi }_1,\delta {\chi }_2,\delta {\chi }_3\}$ are all free. The peak is near the correct value of $\delta {\chi }_3$ (with a median of 0.083), but this time the spread is considerably larger (with a standard deviation of 0.055), as we are trying to measure more parameters at the same time.

Figure 20

The 68% and 95% confidence contours of the two-dimensional PDF for $\{\delta {\chi }_2,\delta {\chi }_3\}$, still for an injection with $\delta {\chi }_3=0.1$, and with a waveform that has $\{\overrightarrow{\theta },\delta {\chi }_1,\delta {\chi }_2,\delta {\chi }_3\}$ as free parameters. The maximum of the PDF is given by the black dot, and the injection values are represented by the red cross.

Figure 21

The posterior PDFs for $\delta {\chi }_1$ (top panel), $\delta {\chi }_2$ (middle panel), and $\delta {\chi }_3$ (bottom panel) for a single injection with $\delta {\chi }_3=0.025$ and network SNR of 20.6, recovered with waveforms where, respectively, $\{\overrightarrow{\theta },\delta {\chi }_1\}$, $\{\overrightarrow{\theta },\delta {\chi }_2\}$, and $\{\overrightarrow{\theta },\delta {\chi }_3\}$ are free. Again, $\delta {\chi }_3$ is peaked at close to the correct value, with a median of 0.027 and a standard deviation of 0.0092, but both $\delta {\chi }_1$ and $\delta {\chi }_2$ are strongly peaked at incorrect values.

Figure 22

The posterior PDFs for $\delta {\chi }_1$ (top panel), $\delta {\chi }_2$ (middle panel), and $\delta {\chi }_3$ (bottom panel) for a single injection with $\delta {\chi }_A=-2.2$ and network SNR of 22.4, recovered with waveforms where, respectively, $\{\overrightarrow{\theta },\delta {\chi }_1\}$, $\{\overrightarrow{\theta },\delta {\chi }_2\}$, and $\{\overrightarrow{\theta },\delta {\chi }_3\}$ are free. The distribution of $\delta {\chi }_3$ has its median at 0.11 and a standard deviation of 0.017. Note the remarkable resemblance with Fig. 18, where the signal had $\delta {\chi }_3=0.1$. Also for $\delta {\chi }_1$ and $\delta {\chi }_2$, the distributions are very similar to the ones for a signal with $\delta {\chi }_3=0.1$.

Figure 23

The 68% and 95% confidence contours of the two-dimensional PDF for $\{\delta {\chi }_2,\delta {\chi }_3\}$, for an injection with $\delta {\chi }_A=-2.2$, and with a waveform that has $\{\overrightarrow{\theta },\delta {\chi }_1,\delta {\chi }_2,\delta {\chi }_3\}$ as free parameters. The maximum of the PDF is given by the black dot. Here the distribution is quite different from the case with $\delta {\chi }_3=0.1$ (Fig. 20).