Physically consistent gravitational waveform for capturing beyond general relativity effects in the compact object merger phase

The merger phase of compact binary coalescences is the strongest gravity regime that can be observed. To test the validity of general relativity (GR) in strong gravitational fields, we propose a gravitational waveform parametrized for deviations from GR in the dynamical and nonlinear regime of gravity. Our fundamental idea is that perturbative modifications to a GR waveform can capture possible deviations in the merger phase that are difficult to model in a specific theory of gravity. One of notable points is that our waveform is physically consistent in the sense that the additional radiative losses of energy and angular momentum associated with beyond-GR modifications are included. Our prescription to ensure physical consistency in the whole coalescence process is expected to be applicable to any deviation from the standard model of compact binary coalescence, such as the extended models of gravity or the environmental effects of compact objects, as long as perturbative modifications are considered. Based on the Fisher analysis and the compatibility with Einstein-dilaton Gauss-Bonnet waveforms, we show that our parametrization is a physically consistent minimal one that captures the deviations in the nonlinear regime.

Figure 1

Ratios of eigenvalues of ${\mathit{F}}_{\widehat{\beta }\widehat{\alpha }}$ with $0.1\le \eta \le 0.25$ fixing ${\chi }_{\mathrm{eff}}=0$. ${\nu }_i$ are the eigenvalues of ${\mathit{F}}_{\widehat{\beta }\widehat{\alpha }}$, ordered from largest to smallest, $i=0$, 1, 2, 3. The left and right figures show the fraction ${\nu }_i/{\nu }_0$ for $\eta$ and ${\chi }_{\mathrm{eff}}$, respectively. These results show that ${\nu }_1/{\nu }_0=\mathcal{O}({10}^{-3})$.

Figure 2

Modified waveforms in the time domain associated with ${\widehat{P}}_{\mathrm{PCA}}(|{\widehat{P}}_{\mathrm{PCA}}|\le 0.1)$.

Figure 3

Modified amplitude with ${\widehat{\gamma }}_1=0.5$ (navy solid line) and GR amplitude (navy dotted line). ${\widehat{\gamma }}_1$ describes the amplification at $f=f_{\mathrm{a}2}$ (orange) and the linear interpolation at $f=f_{\mathrm{int}}$ (red). The former and latter deviations correspond to Eqs. (3.16) and (3.18), respectively.

Figure 4

Physically consistent modified waveforms in the time domain associated with ${\widehat{\gamma }}_1$ ($|{\widehat{\gamma }}_1|\le 0.5$) that describes an amplification in the nonlinear regime. The left figure shows the IMR waveforms and the right shows the ringdown parts. Due to the backreaction inclusion associated with ${\widehat{\gamma }}_1$, the ringdown and damping frequencies are changed. These changes are studied in Fig. 5.

Figure 5

Fractional deviations of the ringdown frequency (left) and the damping frequency (right) associated with ${\widehat{\gamma }}_1$. ${\overline{f}}_{\mathrm{RD}}-f_{\mathrm{RD}}$ and ${\overline{f}}_{\mathrm{damp}}-f_{\mathrm{damp}}$ are monotonically decreasing and increasing functions of ${\widehat{\gamma }}_1$, respectively.

Figure 6

$\mathcal{MM}$ associated with ${\widehat{P}}_{\mathrm{PCA}}$ and ${\widehat{\gamma }}_1$ for various initial configurations. The top two figures show $\mathcal{MM}$ for ${\widehat{P}}_{\mathrm{PCA}}$, and the left and right show the case I and case II, respectively. The bottom two show $\mathcal{MM}$ for ${\widehat{\gamma }}_1$, and the left and right show the case I and case II, respectively. For ${\widehat{P}}_{\mathrm{PCA}}$, $\mathcal{MM}$ is the same for positive and negative values. For ${\widehat{\gamma }}_1$, on the other hand, the values of $\mathcal{MM}$ are not exactly symmetrical for positive and negative values. For a given positive ${\widehat{\gamma }}_1$, $\mathcal{MM}$ for a negative ${\widehat{\gamma }}_1$ with the same absolute value is 1 to 2 times smaller.

Figure 7

Marginalized $1\sigma$ error estimates for $\mathit{\theta }$ in the O4 observation. The blue and the red show the $1\sigma$ regime for case A and case B, respectively.

Figure 8

Marginalized $1\sigma$ error estimates for $\mathit{\theta }$ in the O5 observation. The blue and the red show the $1\sigma$ regime for case A and case B, respectively.

Figure 9

Best-fit parametrized waveforms to EdGB waveforms. The top left panel shows EdGB waveforms for $\sqrt{{\alpha }_{\mathrm{GB}}}/GM=0.030$, 0.035, 0.040 and GR waveform (dotted). From the top right to the bottom right, the best-fit modified waveforms (light blue lines) are shown for $\sqrt{{\alpha }_{\mathrm{GB}}}/GM=0.030$, 0.035, 0.040 (blue lines).

Figure 10

Coefficients of ${\overrightarrow{e}}_{\mathrm{PCA}}$ with $0.1\le \eta \le 0.25$ and ${\chi }_{\mathrm{eff}}=0$. Each line shows ${\widehat{p}}_{{\widehat{\beta }}_2}$ (blue), ${\widehat{p}}_{{\widehat{\beta }}_3}$ (orange), ${\widehat{p}}_{{\widehat{\alpha }}_2}$ (green), and ${\widehat{p}}_{{\widehat{\alpha }}_3}$ (red).