# Are gravitational wave ringdown echoes always equal-interval?

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## Abstract

Gravitational wave (GW) ringdown waveforms may contain “echoes” that encode new physics in the strong gravity regime. It is commonly assumed that the new physics gives rise to the GW echoes whose intervals are constant. We point out that this assumption is not always applicable. In particular, if the post-merger object is initially a wormhole, which slowly pinches off and eventually collapses into a black hole, the late-time ringdown waveform exhibit a series of echoes whose intervals are increasing with time. We also assess how this affects the ability of Advanced LIGO/Virgo to detect these new signals.

## 1 Introduction

Recently, the LIGO Scientific and Virgo Collaborations, using ground based laser interferometers, have detected gravitational wave (GW) signals of binary black hole (BH) [1] and binary neutron stars [2] coalescences, which opened a new window to probe gravity physics, particularly in the strong field regime, and the origin of universe.

Inflation is the current paradigm of the early universe. The domain-wall bubbles (or relevant objects) can spontaneously nucleate in de Sitter space and be stretched by the inflation to astrophysical scales [3], see also [4]. In Refs. [5, 6], it has been argued that under certain conditions the interior of a large bubble will develop into a baby universe, which is connected to the exterior region through a wormhole (WH), see also [7]. The throat of the WH is dynamic, which will pinch off shortly after the WH enters into the cosmological horizon, or see [8]. The resulting BHs might be candidates for seeding the supermassive objects at the center of galaxies [9]. Thus, it is possible that the primordial WHs, created and enlarged in the inflationary phase, might be slowly pinching till today, and merge with another compact object (a neutron star or BH). In any case, one could speculate a scenario where a WH may appear as an intermediate state in the coalescences of some compact objects (BH/BH, WH/BH, WH/WH mergers, etc.).

In this paper, we will show that if the post-merger object is a WH, which is slowly pinching off (and eventually will collapse into a black hole), the late-time ringdown waveform will exhibit a series of interval-increasing echoes. It is commonly assumed, after Cardoso et.al.’s seminal work Refs. [10, 11], that the intervals between the neighboring GW echoes are constant, which has been widely used in searching for the signals of echoes in GW data [12, 13, 14]. However, this assumption could bring bias that causes systematic errors in the parameter estimation of signals, e.g. [15], as we find that the GW echoes may not be equal-interval. Our result suggests a more general pool of templates for the echo searches might be desirable.

## 2 Setup and ringdown echoes

*i*th echo \(\delta t_i\) is approximately

*i*th echoes.

*L*between the barriers will become \({\tilde{L}} =L+\Delta L\). In the approximation \(\Delta L\ll L\), we could regard the moving of barriers as the perturbation for the QNFs \(\omega _{L}\), which will give rise to the shifts of \(\omega _{L}\) to \(\omega _{{\tilde{L}}}\). Thus in the frequency domain, we can write Eq. (2) as

*L*, the effect of \(\Delta L\) (\(\ll L\)) is absorbed into \(\omega _{L}^2\). Equation (7) is the Regge–Wheeler equation for the static WH, and its QNFs have been calculated in Ref. [21],

## 3 Effect of the interval shift

We calculate the SNR in a fixed segment \(T=N_{echo}\Delta t\sim 3N_{echo}\times 10^{-2}\)s. We plot the SNR with respect to \(\delta t/\Delta t\) in left panel of Fig. 3, where \(\Delta t=\Delta t_{echo}^{1,0}\) is set. We take \(N_{echo}=20\) (so \(T\simeq 3N_{echo}\times 10^{-2}\)s\(=0.6\)s), and see that if all echoes are equal-interval (\(\delta t=0\)), i.e.\({\tilde{N}}_{echo}=N_{echo}\) for (17), we have the SNR \(\rho \simeq 9.4\), but if \(\delta t/\Delta t\simeq 0.1\), the SNR will reduce to \(\rho \simeq 9.3\), since we only have \({\tilde{N}}_{echo}\simeq 12\) in this segment. Thus the larger is \(\delta t/\Delta t\), the less is the number of echoes in fixed segment, so the lower is the SNR. In right panel of Fig. 3, we also show how the different values of \(\delta t/\Delta t\) alter the SNR of the signals with the echo width \(\sigma \).

The shift of echo interval is encoded in \(\delta t\). The LIGO/Virgo collaborations modelled the ringdown waveform without the echoes as \(\Psi ^{BH}(t)\), see (17), and found the SNR \(\rho \sim 8\) [24]. Generally, the inclusion of echoes will enhance the SNR, e.g. [15]. Our result indicates that the shift of echo interval could significantly affect the parameter estimation of echo signals, when one searched for the corresponding signals in GW data.

## 4 Discussion

Even though after the merger a BH/BH binary (or BH/WH binary) eventually develop into a BH, an exotic intermediate state might exist. We show that if such a state is a WH, which is slowly pinching off (and eventually will collapse into a BH), the ringdown waveform will exhibit a series of echoes, as pointed out in [10]. However, we have found that the usual assumption that the GW echoes are equal-interval is not always applicable. In particular, in our scenario the intervals between the neighboring echoes will increase with time. We have argued the significant effect of the shift of echo interval on the search for the signals of echoes in GW data released by LIGO/Virgo.

The viability of WH depends on special models, which is still a developing subject, e.g. [25, 26, 27, 28]. Some of the issues might be better understood by performing numerical simulations of binary mergers with WHs. The physics of GW echoes has recently been extensively studied, see also [29, 30, 31]. While the post-merger object we considered is a WH, our result may also be applicable for other exotic compact objects (e.g. [32, 33, 34]), as well as the BHs with the correction of modified/quantum gravity [35, 36], with the shift of their reflector surface towards the Schwarzschild radius. However, if initial state is not a BH, the inspiral stage could in principle be used to discriminate against a two-BH initial state, since the quadrupole moment, tidal love numbers or absorption of the initial state is different from that of a BH, see e.g. [37, 38].

## Notes

### Acknowledgements

We would like to thank Raul Carballo-Rubio and Leo C. Stein for valuable comments, and Zhoujian Cao, Bin Hu for discussions. YSP is supported by NSFC, Nos. 11575188, 11690021, and also supported by the Strategic Priority Research Program of CAS, No. XDB23010100. YTW is supported in part by the sixty-second batch of China Postdoctoral Fund. SYZ acknowledges support from the starting grant of USTC and the 1000 Young Talent Program of China. JZ is supported by the National Science and Engineering Research Council through a Discovery grant.

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