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--> --> -->In these models, the horizon is not a smooth sphere, but a shell with a finite thickness. Consequently, tail wavelets referred to as "echoes" may appear after emission of the main GWs [6-9]. The waveform of the echo model suggested in [6] may be ruled out by analysis in [10, 11]. However, the tail wavelet may assume a different form compared with Ref. [6]. This problem can be dealt with from an opposite direction, that is, to find what happens if there is absolutely no signal in the data after the main wave [10, 11]. A detailed analysis in Ref. [12] demonstrates that signals appear for several GW events with a p-value of order 1% or occasionally significantly less. Further approaches for the signal reconnaissance of tail wavelets are presented in Ref. [13]. In this context, observation of the tail wavelets in GW events of the binary neutron stars, besides those of binary black holes, is surprising. While explanations are given in terms of the resulting black hole remnant [14], the observed events also imply the possibility that the tail wavelet may be related to specific generic properties of the binary system rather than the near-horizon quantum structure [15]. In fact, it is quite challenging to interpret the appearance of tail wavelets in terms of the properties of the horizon if the latter does not exist in the first place.
Following this train of thought, we proceed by discussing some of the generic properties of compact objects, which might potentially give rise to tail wavelets. Generally, a black hole/neutron star is surrounded by dark matter, whose density is significantly higher than that of the average value in the universe. We show later that dark matter distributed in the vicinity of the compact object can provide a mechanism for the observed tail wavelets. Further quantitative studies of the wavelets may be helpful to extract the distribution of dark matter around the compact objects. Extensive studies were conducted on the distribution of dark matter around celestial bodies. Some pioneering studies [16, 17] explored specific properties of the dark matter distribution in the halos of various mass scales. A remarkable feature of the distribution of the dark matter is its universality. Halos of different scales, namely, from the mass of the celestial body to that of a cluster of galaxies, all exhibit similar distributions [18, 19]. The density profile of supermassive black holes in elliptical galaxies and barred/ordinary spiral galaxies, such as our Milky Way, is studied quantitatively [20, 21]. For the stellar mass black holes, direct observation of the dark matter halo is still absent. The distribution of dark matter around neutron stars, whose space environment is very similar to stellar mass black holes, was studied in detail [22–24]. Regarding the universal distribution of dark matter halos, it is reasonable that binary compact stars/black holes are immersed within them.
Our proposed scenario is described as follows. The passage of the main GWs from the binary system merger pushes up the dark matter to a certain altitude. After the initial waves traverse through, the matter falls back towards the black hole region and subsequently produces the tail wavelets. The relevant physical quantities include the energy density, pressure, energy flux of the GWs. For linear plane GWs with frequency
In a general case, the stress-energy tensor for the GWs can be defined as a second order perturbation of the Einstein tensor. The general definition of GWs in the strong field region leads to some uncertainties, which depend on the specific choice of the background for the GWs in question. For example, the obtained waves exhibit different forms when the metric about a Minkowski spacetime is expanded as compared to about a Schwarzschild spacetime. Nonetheless, the relation between power per area
$ P = S/c, $  | (1) |
$ P = \frac{w}{4\pi r_h^2 c}\sim \frac{3\times 10^{49}}{4\pi (62\times 3\times 10^3)^2\times 3\times 10^8} = 2.3\times 10^{29} {\rm {Pa}}. $  | (2) |
At the region near the horizon, the Newton gravity fails. We only make an assumption to determine which force, i.e., pressure or attraction, dominates for an object around the black hole. Considering an object whose mass is 1 kg and area is 1
Furthermore, we assume that the surrounding matter can only absorb a tiny fraction of GW. A rigorous treatment of the scattering between the graviton and matter demands a full-fledged theory of quantum gravity, which remains unknown. Tree level calculations may also be affected by loop corrections. Nevertheless, some preliminary studies on the scattering of gravitons have been carried out, for example in studies of Refs. [25, 26]. By very generic arguments, similar to those for earlier approaches of quantum physics, one can estimate the lower bound of total cross-section of a graviton scattered by a matter particle
Because a rigorous approach is not yet feasible, in the remainder of the present study, we carry out a phenomenological approach to investigate this interaction between GWs and dark matter.
$ {\rm d}s^2 = -f{\rm d}t^2+f^{-1}{\rm d}r^2+r^2{\rm d}\Omega^2,\; $  | (3) |
The second one is a short-hair black hole, which describes non-homogeneous fluids [29], where
$ f = 1-2M/r+Q_m^{2k}/r^{2k}. $  | (4) |
We estimate the amplitude of the GW around the horizon. The energy density is proportional to the inverse square of the distance from the source
From previous studies on tail wavelets (echoes), we estate that the amplitude of the wavelet is one tenth of the main wave or smaller. Thus, the energy carried by the wavelet is about
Before presenting the waveform of the wavelet, we explore the time scale between the emission of the main GWs and tail wavelets. This can be estimated by simply investigating the equation of motion of a free-falling particle from a given height. For simplicity, we consider motions along radial geodesics. The results are presented in Fig. 1. There, we consider a particle splashing down from an initial altitude of
Figure1. (color online) Estimation of epochs of tail wavelets after the main GWs. Left panel: case without backreaction, i.e., Schwarzschild. Middle panel: Black hole with DM halos, i.e., case of SdS. Right panel: Black hole with short-hair. Location of event horizon is depicted by horizontal dashed purple line.The waveform of the tail wavelet generated by the infalling of the matter is similar to that of the ordinary quasinormal modes [31–33]. Therefore, in our model, the observed tail wavelets are also a manifestation of the black hole quasinormal modes. However, there is a subtlety. Because the infalling process is continuous, the initial disturbance period carries the information not only on the black hole itself, but also on the matter distribution surrounding it. It is therefore expected that the waveform in the initial disturbance period will be characteristically different from the ordinary quasinormal modes. The Navarro-Frenk-White (NFW) profile is the mostly studied distribution of dark matter around galaxies and clusters [18, 19]. As a more realistic study, we consider an NFW distribution of dark matter around the hole. The NFW distribution can be described by,
$ \rho(r) = \frac{\rho_0}{\dfrac{r}{R_s}\left(1+\dfrac{r}{R_s}\right)^2}. $  | (5) |
$ \rho(x) = \frac{\rho_0}{Kx(1+Kx)^2}. $  | (6) |
$\begin{split} M_d = 4\pi\int_{r_h}^{100r_h} {\rm d}r r^2 \rho(r) = 4r_h^3\pi\int_{1}^{100}{\rm d}x x^2 \rho(x) = 4\pi r_h^3 \rho_0\int_{1}^{100}{\rm d}x \frac{x^2}{Kx(1+Kx)^2} = 4\pi r_h^3 \rho_0 \frac{1}{K^3}\left(\frac{1}{1+100K}-\frac{1}{1+K}+\ln \frac{1+100K}{1+K}\right). \end{split}$  | (7) |
$ M_{dh} = \rho_{0h}4\pi\int_{r_h}^{100r_h} {\rm d}r r^2 = \rho_{0h}\frac{4\pi}{3}10^6 r_h^3.$  | (8) |
In this section, we present numerical calculations of the waveform associated with the gravitational perturbation of the metric owing to the infalling of matter.
The radial master equation of the gravitational fluctuation around the black hole reads [35, 36]
$ f(r)\frac{\partial}{\partial r} \left(f(r)\frac{\partial\Phi}{\partial r}\right)-\frac{1}{c^2}\frac{\partial^2\Phi}{\partial t^2}-V(r)\Phi = 0. $  | (9) |
Based on the analysis in the previous sections, we assume the following initial conditions. For convenience, we choose
Case I,
Case II,
The first case denotes dark matter that is initially in a homogeneous distribution. For comparison, we plot the second case, which displays a pulse excitation that is the ordinary QNM of the black hole. Figs. 2 and 3 demonstrate the waveforms of the odd and even fluctuations for all geometries, including the Schwarzschild, the geometry with backreaction of dark matter halos (SdS), and the geometry of a short-hair black hole. According to the present estimations, the magnitude of the tail wavelets is at most ~10% of the main GWs [12, 13]. Thus, we take the initial perturbation to be
Figure2. (color online) Waveform of odd gravitational fluctuations, including cases of a bare Schwarzschild black hole and dressed black holes, i.e., SdS and short-haired black hole.
Figure3. (color online) Waveform of even gravitational fluctuations, including cases of a bare Schwarzschild black hole and dressed black holes, i.e., SdS and short-haired black hole.We expect that forthcoming data with a better resolution for the waveform, especially those from the third generation GW detectors like the Einstein Telescope [37, 38] and Cosmic Explorer [39], may shed light on the feasibility of the present model.
Generally, the astrophysical black hole is a Kerr black hole. We study the Schwarzschild black hole as an approximation of the realistic case. Hence, we present a preliminary investigation of rotating black holes. We plot the corresponding wavelets for the Kerr holes in Fig. 4 and Fig. 5.
Figure4. (color online) Even modes of wavelets for rotating black hole. Left panel depicts wavelets corresponding to shell initial distribution, and right panel depicts wavelets corresponding to homogeneous distribution. The even mode is sensitive to the rotating parameter.
Figure5. (color online) Odd modes of wavelets for rotating black hole. Left panel depicts wavelets corresponding to shell initial distribution, and right panel depicts wavelets corresponding to homogeneous distribution. The odd mode is not sensitive to the rotating parameter.Following this line of thought, in this work, we present a more natural and straightforward scenario for the generation of tail wavelets. In our model, the cause of the phenomenon is attributed to the matter or dark matter surrounding the binary system. In particular, the dark matter distribution around a compact object has been studied at different scales for many years. We carefully investigate the pressure of the GWs and demonstrate that they can perturb and even push away the matter distributed around a compact star. After the main wave traverses through, the matter falls back towards the black hole horizon, which in turn excites the tail wavelets. Based on our analysis, we assume four different initial conditions for the related gravitational perturbations and evaluate the corresponding waveforms. Furthermore, we make a preliminary exploration of the wavelet for Kerr black holes.
In this study, we concentrate on the physics after inspirals. We explore the effects of energy emission in the merge process and particularly the splashing down of dark matter to the black hole. This is not closely related to the physics before merging. According to the previous studies of intermediate-mass-ratio inspirals (IMRI) without dark matter [40], and especially an upcoming study [41], there is almost no wavelet in IMRI. However, we show that dark matter in such a system can produce significant tail wavelets in the merger of binary compact objects. The effect of dark matter in the inspiraling stage of the IMRI system was investigated in Ref. [42], which demonstrates that dark matter significantly enhances the merging process. We show that dark matter has a significant effect after the merging process.
We look forward to testing the proposed model against forthcoming data from the GW detectors of the third generation. Further studies concerning specific dark matter distributions are in progress. Furthermore, in the frame of this model, we investigate the distribution of dark matter surrounding the black holes by analyzing the waveform of the tail wavelet from the forthcoming data at better resolution, especially those from the third generation GW detectors.
Our heartfelt thanks to Professor Anzhong Wang, Zhoujian Cao, and Wenbiao Han for helpful discussions.
