Gravitational-wave echoes from numerical-relativity waveforms via spacetime construction near merging compact objects

We propose a new approach toward reconstructing the late-time near-horizon geometry of merging binary black holes, and toward computing gravitational-wave echoes from exotic compact objects. A binary black-hole merger spacetime can be divided by a timelike hypersurface into a black-hole perturbation (BHP) region (in which the spacetime geometry can be approximated by homogeneous linear perturbations of the final Kerr black hole) and a nonlinear region. At late times, the boundary between the two regions is an infalling shell. The BHP region contains late-time gravitational waves emitted toward the future horizon, as well as those emitted toward future null infinity. In this region, by imposing no-ingoing-wave conditions at past null infinity and matching outgoing waves at future null infinity with waveforms computed from numerical relativity, we can obtain waves that travel toward the future horizon. In particular, the Newman-Penrose ${\psi }_0$ associated with the ingoing wave on the horizon is related to tidal deformations measured by fiducial observers floating above the horizon. We further determine the boundary of the BHP region on the future horizon by imposing that ${\psi }_0$ inside the BHP region can be faithfully represented by quasinormal modes. Using a physically motivated method to impose boundary conditions near the horizon and applying the so-called Boltzmann reflectivity, we compute the quasinormal modes of nonrotating exotic compact objects, as well as gravitational-wave echoes. We also investigate the detectability of these echoes in current and future detectors and prospects for parameter estimation.

Figure 1

Spacetime of a BBH merger event. The hybrid method divides the spacetime into an inner PN region (III) and an outer BHP region ($\mathrm{I}+\text{II}$). The two regions communicate via boundary conditions at the world tube ${\mathrm{\Sigma }}_{\text{Shell}}$ (the blue curve), which was assumed to track the motion of the BH. The dynamical horizon (the red curve) lies inside the future horizon, and it eventually settles down to the isolated horizon. The common horizon forms at the time slice ${\mathrm{\Sigma }}_{\mathrm{init}}$ (the horizontal dashed line). The time slice ${\mathrm{\Sigma }}_{\mathrm{init}}$ is not unique and is determined by gauge conditions. The CLA focuses exclusively on region I, where the system is treated as a Cauchy problem—initial data need to be provided on ${\mathrm{\Sigma }}_{\mathrm{init}}$, whereas the hybrid method gives attention to regions both I and II and handles the system as a boundary value problem.

Figure 2

Coefficients $C_{lm\omega }^{\mathrm{in}}$ and $D_{lm\omega }^{\mathrm{in}/\mathrm{out}}$ predicted by the Teukolsky equation, assuming a Schwarzschild BH. The vertical dashed line stands for the real part of the fundamental QNM $(0.374-0.0890i)$. Data were obtained from the Black Hole Perturbation Toolkit [88].

Figure 3

Diagram summarizing the relations between BHP quantities on the horizon, $Z_{lm\omega }^{\mathrm{in}}$ and $Y_{lm\omega }^{\mathrm{in}}$, and those at infinity, $Z_{lm\omega }^{\infty }$ and $Y_{lm\omega }^{\infty }$.

Figure 4

Spacetime diagram illustrating the BHP region $\mathrm{I}+\mathrm{II}$ and its linear extrapolation into region III. Outside the matching shell, curvature perturbations are linear combinations of the up-mode solutions to the homogeneous Teukolsky equation. At infinity ${\mathcal{I}}^{+}$, the values of $Z_{lm\omega }^{\infty }$ and $Y_{lm\omega }^{\infty }$ are chosen to be consistent with the predictions of CCE. The past horizon exists in the strong gravity region III, where $Z_{lm\omega }^{\mathrm{H}\mathrm{out}}$ and $Y_{lm\omega }^{\mathrm{H}\mathrm{out}}$ represent the image wave that gives rise to waves in the region $\mathrm{I}+\text{II}$. They serve the same role as the initial wave packet within the inside prescription [10, 64]. The future horizon lies partially outside the matching shell: only the outside portion $(v>v_{\mathrm{\Sigma }}^{(\mathrm{H})})$ of $Z_{lm}^{\mathrm{H}\mathrm{in}}$ and $Y_{lm}^{\mathrm{H}\mathrm{in}}$ corresponds to the actual wave that falls down the horizon. One natural way to self-consistently determine the location of ${\mathrm{\Sigma }}_{\text{shell}}$ is to evaluate the starting time after which $Y_{lm}^{\mathrm{H}\mathrm{in}}(v)$ can be decomposed as a sum of QNM overtones. More details can be found in Sec. 4b.

Figure 5

Spherical modes $Y_{22}^{\infty }$ and $Z_{22}^{\infty }$ of SXS:BBH:0207 in the time domain (upper panel) and in the frequency domain (lower panel). The vertical lines in the lower panel represent QNM frequencies of a Schwarzschild BH, labeled with the overtone index $n$. The absolute value of $Z_{22}^{\infty }$ is amplified by a factor of 300 for simplicity.

Figure 6

Validity of the TS identity at infinity [Eq. (9a)] using SXS:BBH:0207. The predicted form $\frac{4{\omega }^4}{C^{*}}{Y}_{22}^{\infty }$ (in red) is compared to the actual $Z_{22}^{\infty }$ (in black) in the time domain (the left and center panels) and the frequency domain (the right panel). The comparison for SXS:BBH:1936 is in Fig. 20.

Figure 7

Real part of $Y_{22}^{\mathrm{H}\mathrm{in}}$ [Eqs. (6)] and $Y_{22}^{\infty }$ in the time domain using SXS:BBH:0207. The temporal coordinate for $Y_{22}^{\mathrm{H}\mathrm{in}}$ is $v$, while is $u$ for $Y_{22}^{\infty }$. Both coordinates are in the unit of final mass.

Figure 8

Mismatch as a function of start time (in the unit of remnant mass) for different models [Eqs. (17)]. Each model includes up to $n_{\max }$ overtones. The left panels correspond to the strain $h_{22}^{\infty }$ at infinity, the middle panels correspond to $Y_{22}^{\infty }$, and the right panels correspond to $Y_{22}^{\mathrm{H}\mathrm{in}}$ [see Eqs. (6) and (15b)]. (a) refers to SXS:BBH:0207, whereas (b) refers to SXS:BBH:1936. All waveforms are aligned such that $t=0$ occurs at the peak of $\sqrt{{\sum }_{lm}|h_{lm}(t){|}^2}$.

Figure 9

Real and imaginary parts of QNMs for an irrotational ECO as functions of $\gamma$. They are the solutions to Eq. (31). The Boltzmann reflectivity is used, assuming $T_{\mathrm{QH}}=T_H$. Each mode is labeled with the overtone index $n$. The imaginary part of the QNMs is negative, meaning that the mode is stable.

Figure 10

Transfer function $\mathcal{K}$ of the ECO using $(\gamma ={10}^{-15},T_{\mathrm{QH}}=T_H)$ (blue curve) and $(\gamma ={10}^{-1},T_{\mathrm{QH}}=5T_H)$ (black curve). The QNM resonances are visible in the former case, where the locations of the first three resonances are labeled with the dashed vertical lines, based on the estimation in Eq. (33). By comparison, the red curve corresponds to the absolute value of the filtered horizon wave $Y^{\mathrm{H}\mathrm{Filter}}$ for SXS:BBH:0207 assuming that $v_{\mathrm{\Sigma }}^{\mathrm{H}}=-13$ and $\mathrm{\Delta }v=2/\kappa$ [see Eq. (36)]. Its value is decreased by a factor of 4000 for simplicity.

Figure 11

Echo emitted by SXS:BBH:0207 following the main GW. Here we set $v_{\mathrm{\Sigma }}^{(\mathrm{H})}=-13,\mathrm{\Delta }v=2/\kappa =8,\gamma ={10}^{-15}$, and $T_{\mathrm{QH}}=T_H$.

Figure 12

The echoes emitted by SXS:BBH:0207, with a variety of $T_{\mathrm{QH}}$ and $\gamma$. The width of filter $\mathrm{\Delta }v$ is equal to $2/\kappa$. The total echoes (orange curves) are compared to the first echoes (blue curves). In the upper-left panel, the values of $T_{\mathrm{QH}}$ and $\gamma$ are small enough that the spacing between echoes is greater than the echo duration; hence, the individual pulses are well separated, whereas in the other three panels different pulses overlap and interfere with each other.

Figure 13

The influence of the filter parameter $\mathrm{\Delta }v$ on echo waveforms. Each curve corresponds to the real part of the first echo (with different $\mathrm{\Delta }v$), using SXS:BBH:0207 and the Boltzmann reflectivity ($\gamma ={10}^{-15}$ and $T_{\mathrm{QH}}=T_H$). The filter is applied at the future horizon with $v_{\mathrm{\Sigma }}^{(\mathrm{H})}=-13$.

Figure 14

Comparison between the hybrid approach and the inside prescription using SXS:BBH:0207. We choose a Boltzmann reflectivity with $\gamma ={10}^{-15}$ and $T_{\mathrm{QH}}=T_H$. The upper panel shows the first echo, whereas the bottom panel displays the second echo. The filter is applied at null infinity (red curve labeled “Inside”) and at the future horizon (black curve labeled “Hybrid”). The width of both filters $\mathrm{\Delta }v$ is $2/\kappa$.

Figure 15

Sky-averaged echo SNR across the $T_{\mathrm{QH}}-\gamma$ space using SXS:BBH:0207 (upper panels) and SXS:BBH:1936 (lower panels), as well as aLIGO (left column) and CE (right column). The binary system is 100 Mpc from the detector, with a total mass of $60M_{\odot }$. We set $\mathrm{\Delta }v$ to $2/\kappa$, and the values of $v_{\mathrm{\Sigma }}^{(\mathrm{H})}$ are listed in Table 2.

Figure 16

Sky-averaged echo SNR as a function of filter parameters $v_{\mathrm{\Sigma }}^{(\mathrm{H})}$ and $\mathrm{\Delta }v$ [see Eq. (37)] using CE. The binary system is SXS:BBH:0207 and has the same total mass and distance as Fig. 15. We use the Boltzmann reflectivity with $\gamma ={10}^{-15}$ and $T_{\mathrm{QH}}=T_H$. The vertical dash-dotted line represents the value of $v_{\mathrm{\Sigma }}^{(\mathrm{H})}$ given in Table 2.

Figure 17

Fractional error of $T_{\mathrm{QH}}$ (solid curves) and $\gamma$ (dashed curves) as functions of $T_{\mathrm{QH}}$ using aLIGO (black curves) and CE (red curves). The binary system is SXS:BBH:0207, which has a total mass of $60M_{\odot }$ and is located 100 Mpc from the detector. The two filter parameters $v_{\mathrm{\Sigma }}^{(\mathrm{H})}$ and $\mathrm{\Delta }v$ are still set to $-13$ and $2/\kappa$, respectively. We vary the value of $T_{\mathrm{QH}}$ from 0.4 to 10 while fixing the value of $\gamma$ at ${10}^{-15}$.

Figure 18

Absolute value (left two panels) and phase (right two panels) of the prograde mode ${\mathcal{A}}_n$ and the retrograde mode ${\mathcal{B}}_n$ assuming that (a) SXS:BBH:0207 and (b) SXS:BBH:1936. We fit Eqs. (17) to the data of $h_{22}^{\infty }$ (blue), $Y_{22}^{\infty }$ (black), and $Y_{22}^{\mathrm{H}\mathrm{in}}$ (red) obtained from CCE.

Figure 19

The $(u,v)$ grid cell in a characteristic evolution scheme of the RW equation.

Figure 20

Same as Fig. 6 using SXS:BBH:1936.

Figure 21

Same as Fig. 12 using SXS:BBH:1936.