Modeling Compact Binary Merger Waveforms Beyond General Relativity

For completeness, we include the case here where we do not include a post-inspiral correction at all, and instead truncate the ppE waveform at a particular frequency $f_{\mathrm{IM}}$.

Although the IMRPhenomD model is constructed to be $C^{1}$ in its phase, we include the $C^{0}$ correction here to be able to make comparisons against a naive post-inspiral correction that is not physically motivated.

As seen in Fig. 1, such a correction may not necessarily result in an obviously pathological waveform.

The $C^{1}$ correction is the first post-inspiral correction that is physically motivated.

As seen in Fig. 2, the additional post-inspiral term induces a shift in the merger time and phase that leaves the two waveforms identical in the merger after this shift, which is linear in frequency, has been subtracted off. As the ppE-$\beta$ correction is only applied in the inspiral regime, this scheme is a good candidate for extending the correction into the merger/ringdown regime with minimal assumptions.

We also include the $C^{\infty}$ correction for the sake of completeness:

For some values of $b$ and $\beta$ the extension of the inspiral correction into the merger regime leads to significant distortion of the waveform. We compute an upper bound on the value of $\beta$ that is compatible with this type of correction. This motivates the need for a novel extension of the ppE correction into the intermediate and merger regimes.

These shifts are given by:

This corresponds to the displacement of the peak GW amplitude in time and can be arranged such that at $t=t_{c}$, the GW reaches its peak amplitude in both the GR case as well as the ppE corrected case. The $\beta$-ppE correction used is the $C^{1}$ correction $\Delta\phi^{C1}_{\mathrm{ppE}}$ (see Eq. 10).

In Fig. 3, we show the effects of promoting the second of the $\beta$s, which affects the $u^{3}$ term, into an additional free parameter $\epsilon$ that controls the merger time.

The IMRPhenomD model uses a GR phase approximant that is C1 in frequency, so both the $\Delta\phi^{\mathrm{C0}}_{\mathrm{ppE}}$ and $\Delta\phi^{\mathrm{C1}}_{\mathrm{ppE}}$ post-inspiral phase corrections can be absorbed into a redefinition of the IMRPhenomD coefficients. This leaves the phase evolution $\Delta\phi^{\prime}_{\mathrm{ppE}}(f)$ entirely expressible in terms of the IMRPhenomD parameters. This means that an IMRPhenomD waveform corresponding to a GR signal can be found such that the two frequency waveforms match exactly in the inspiral. In the C0 case, this corresponds to a change in the coalescence phase of the original GR waveform. In the C1 case, this corresponds to a change in both the coalescence phase and coalescence time of the original GR waveform. In Fig. 4 we show how this $C^{1}$ agreement between the IMRPhenomD model and the ppE correction scheme is reflected in the fact that the derivatives of the Fourier phases are identical in the merger/ringdown regime, barring a constant offset corresponding to the time shift incurred by the ppE modification.

Note that the time shift in the coalescence time incurred by extending the ppE-$\beta$ correction into the merger regime in a C1 fashion is not a prediction of the original beyond-GR theory in the ppE-$\beta$ framework. The ppE-$\beta$ correction is calculated using an assumption of a quasicircular inspiral. One should not expect that the dynamics of a merger in which this assumption breaks down can be approximated in this way. This motivates us to extend the ppE-$\beta$ correction by generalizing the time shift such that it is controlled by a new parameter $\epsilon$ (cf. Eq.(13)):

This new parameter $\epsilon$ then parameterizes the change in the merger time with respect to a GR merger with the same intrinsic parameters. Note that the time shift $t_{\beta}$ in the ppE-$\beta$ framework is determined by the C1 requirement imposed on the waveform across the frequency regimes. In the ppE-$\beta\epsilon$ framework, the C1 requirement is still imposed; the requirement is satisfied by modifying the waveform in the intermediate regime through changes in the parameters $\beta_{0}$, $\beta_{1}$, $\beta_{2}$, and $\beta_{3}$, which are now functions of $\epsilon$. The phase change in the inspiral-merger regime is then given by:

The form of $\Delta\phi_{\mathrm{Int}}(f,\beta,\epsilon)$ is determined by the coefficients $\beta_{0}$, $\beta_{1}$, $\beta_{2}$, and $\beta_{3}$ that solve the following matrix equation:

While Eq. (15) gives the modification of the waveform, this is not the modification that is applied when performing parameter estimation. The presence of the factor that is linear in frequency in the ringdown regime causes the coalescence time to incur a shift after Fourier transforming to the time domain; this linear factor is subtracted off in order for the coalescence time $t_{c}$ to remain unchanged for different choices of $\beta$ and $\epsilon$. Note that when a correction of non-zero $\beta$ is applied to the inspiral regime in the $\beta\epsilon$ correction scheme, a corresponding correction of $\epsilon=\beta$ must also be applied to the rest of the waveform such that the resultant waveform is equivalent to a ppE-$\beta$ corrected waveform satisfying the C1 continuity condition. For example, in Fig. 6, a choice of $\epsilon=\beta=-2$ would lead to the dotted orange curve lying on top of of the blue curve. A choice of $\epsilon\neq\beta$ leads to the merger portion of the waveform being displaced. As this leaves the rest of the waveform intact, it demonstrates how $\epsilon$ is a nearly independent parameter from $\beta$, which only affects the inspiral portion of the waveform, and thus can be measured separately. In Fig. 5 the GW strain amplitudes are plotted demonstrating that the monotonicity of the amplitude is preserved after the $\epsilon$ modification of the merger-ringdown portion of the waveform, indicating that the time-shifting property of the $\epsilon$ modification is a physically plausible one.

The parameter estimation is done via a Markov Chain Monte Carlo (MCMC) algorithm within PyCBC Inference [27], in which trial waveforms are sampled from a parameter space and then compared against the simulated data. Using the likelihood, subsequent waveforms are sampled in a probabilistic fashion, resulting in a chain of samples. If done correctly, the resulting distribution of samples approximates the posterior distribution. One detail of MCMC is that the sampling process resembles a random walk, and the samples that make up the initial steps of the random walk may not be representative of the posterior distribution, because of being initialized in, or stepping into, a region that should be exponentially less populated. These samples are identified and eliminated in a procedure known as burn-in. There are multiple procedures for estimating when the chain has burned in; we use the nacl and max_posterior methods defined in PyCBC.

In our parameter estimation runs we use an extension of MCMC that makes use of what is known as parallel tempering. Parallel tempering initializes multiple chains that explore the parameter space according to the “temperature,” which modifies the effect the likelihood has on the next iteration of the chain. This allows the chains to more efficiently explore the parameter space than could be done using only a single temperature.

The particular MCMC settings that have been used are given in table 2.

In this section we discuss the effect the choice of prior has on the recovery of the parameters of a GR injection. At present, no measurable non-zero $\beta$-deviation from GR has been detected in GW data [28]. However, we can do better than to start with an arbitrarily wide uniform prior on $\beta$. From our match approximant, we can identify values of $\beta$ for which the match would drop as low as 95%; we do not need to consider values of $\beta$ that would cause a higher mismatch as such deviations from GR would have been identified already [4]. For a fixed mismatch, we can also examine what effect the mass has on the value of $\beta$ needed to produce such a mismatch. At higher total masses, the lower frequency portions of the waveform, which are identically GR in the ppE-$\beta$ framework, dominate the SNR and a higher value for $\beta$ is needed for mismatches arising from the modified inspiral to be significant. This effect become more pronounced as one changes the value of the ppE parameter $b$ in the exponent of the phase change term. In this paper we are examining posteriors in $\beta$ and $\epsilon$ for only a single value of $b$, so even though the masses are not held fixed, a fixed prior over $\beta$ and $\epsilon$ is sufficient for our purposes.

The particular prior choices that have been used are given in table 3.

In Table 4, we list the different injections over which we have done parameter estimation runs. These values are the same as those used in the parameter estimation example in the PyCBC documentation. The masses probed are those most most likely to be found by detectors with noise curves similar to LIGO [6]. In this paper we only consider equal-mass injections, but the parameter estimation does not assume this.

In order to have a baseline for how well we can expect the beyond-GR parameters to be recovered, we must first examine the degree to which the ordinary GR parameters can be recovered using the unmodified IMRPhenomD approximant. In Fig. 7 we show the posterior recovered for two chosen parameters, the coalescence time $t_{c}$ and the chirp mass $\mathcal{M}$. In Fig. 8, we show the posterior distribution recovered for the chirp mass for each GR injection. Using the widths of the initial posteriors as a baseline, we repeat the parameter estimation on the GR injection using the ppE-$\beta$ approximant. If using the $\beta$-modified ppE approximant causes the spreads to widen too broadly, then parameter estimation may not be too useful anymore.

We now examine the results of parameter estimation runs where the injection is a GR waveform, but the template used to recover parameters is of a ppE nature. In this case, the parameter estimation will attempt to recover values of $\beta$ and $\epsilon$ in addition to the standard GR parameters. Since the underlying signal is GR in nature, we expect the recovered values of $\beta$ and $\epsilon$ to be consistent with zero. As we only vary the injections over the mass of the system, there are only five injections created. In Fig. 9, we show the posteriors recovered for four chosen parameters, the coalescence time $t_{c}$ and the chirp mass $\mathcal{M}$ as before, and now the ppE parameters $\beta$ and $\epsilon$. We note that the peaks of the posteriors for $t_{c}$ and $\mathcal{M}$ are offset slightly from the posteriors recovered in Fig. 7. This offset reflects the change in the parameters recovered by the introduction of the additional parameters $\beta$ and $\epsilon$. Despite this offset, the injected value still lies within the 90% contour lines. In Fig. 10 we show the corresponding plot to Fig. 8, which demonstrates how the posteriors for the chirp mass change after the introduction of $\beta$ and $\epsilon$ to the approximant. The injected value is still contained within each posterior across the masses probed, but the posteriors have also widened by a factor of roughly four compared to the GR approximant case.

In this section, we examine the results of parameter estimation where the injection is a ppE waveform, while the template is of a GR nature. In this case, there are no $\beta$ or $\epsilon$ parameters in the template, and therefore there are no $\beta$ nor $\epsilon$ posteriors recovered. However, because the injections are of a ppE nature, they are created with varying $\beta$ and $\epsilon$ values. In this way, we can simulate how the presence of beyond-GR effects might instead be determined to be a GR signal, albeit one with slightly different parameters. In Fig. 11 we can see an example of this; the posteriors recovered correspond to that of a more massive system.

Finally, we examine the results of parameter estimation where both the injections as well as the templates are of a ppE nature. In this case, each injection is created with differing values of $\beta$ and $\epsilon$, and the templates will attempt to recover the standard GR parameters as well as $\beta$ and $\epsilon$ for each of these cases. In Fig. 12 we see how both the $\beta$ and $\epsilon$ posteriors contain the injected value, indicating that both are simultaneously recoverable. The joint posterior distribution indicates that the correlation between $\beta$ and $\epsilon$ is less than or comparable to the correlation between either of these to the remaining parameters.

In the previous sections, we have examined whether the introduction of the $\beta$ and $\epsilon$ parameters spoils the recovery of GR parameters. To do this, we compared the posteriors for coalescence time $t_{c}$ and chirp mass $\mathcal{M}$ before and after this introduction, and found that the recovery is not spoiled.

In Figs. 8 and 10 we show this result is robust for the mass range probed by LIGO. where the injection is generated using IMRPhenomD and the template is the $\beta\epsilon$-corrected IMRPhenomD, in the limit of zero noise.

Assuming the SNR is high enough to recover an accurate estimate of $\beta$, we can ask whether it is possible to recover the value of $\epsilon$ as well. One advantage of the $\epsilon$ extension to the ppE framework is that it is not necessary for $\beta$ to be recoverable for $\epsilon$ to be recovered as well. Depending on the total mass of the binary, a different part of the GW is in the LIGO band. Smaller mass binaries inspiral at higher frequencies, leaving more cycles in GW data. The longer inspirals aid in the recovery of $\beta$. On the other hand, if the GW is from a larger mass system, the power that is in the LIGO band is concentrated in the intermediate/merger-ringdown regime. As the $\epsilon$ parameter controls the deviation from GR in this region, it may still be recoverable in cases where $\beta$ is not. With multiple runs one can quantify how the width of the posterior behaves in the $\beta\epsilon$ parameter space. When we plot the posterior distributions from different injections on top of one another, we see how sensitive the shape and location of the posteriors are on the beyond-GR parameters. To make clear how the parameters $\beta$ and $\epsilon$ control the waveform in different frequency regimes, we repeat the parameter estimation runs on injections of different total masses. As expected, when the total mass is smaller, is it the higher-frequency inspiral portion that lies within the LIGO band, and so the distinguishability of different values of $\beta$ is higher for lower system masses. Likewise, for higher mass systems, the lower-frequency merger falls within the LIGO band, and so the distinguishability of the different values of $\epsilon$ are higher here.

That $\beta$ and $\epsilon$ are independently recoverable is demonstrated best by posterior distributions like those shown in Fig. 12. Even as the late-inspiral parameter $\epsilon$ is changed, the resulting posterior distribution for $\beta$ appears to still contain the injected value. Demonstrating that this is the case is left to future work. While this relationship changes as the total mass of the system changes, depending on which parts of the inspiral fall into LIGO’s sensitivity curve, it can be seen that both parameters are sufficiently distinguishable when both the inspiral and merger fall within LIGO’s sensitivity curve.