Gravitational-wave signatures of exotic compact objects and of quantum corrections at the horizon scale

Gravitational waves from binary coalescences provide one of the cleanest signatures of the nature of compact objects. It has been recently argued that the postmerger ringdown waveform of exotic ultracompact objects is initially identical to that of a black hole, and that putative corrections at the horizon scale will appear as secondary pulses after the main burst of radiation. Here we extend this analysis in three important directions: (i) we show that this result applies to a large class of exotic compact objects with a photon sphere for generic orbits in the test-particle limit; (ii) we investigate the late-time ringdown in more detail, showing that it is universally characterized by a modulated and distorted train of “echoes”of the modes of vibration associated with the photon sphere; (iii) we study for the first time equal-mass, head-on collisions of two ultracompact boson stars and compare their gravitational-wave signal to that produced by a pair of black holes. If the initial objects are compact enough as to mimic a binary black-hole collision up to the merger, the final object exceeds the maximum mass for boson stars and collapses to a black hole. This suggests that—in some configurations—the coalescence of compact boson stars might be almost indistinguishable from that of black holes. On the other hand, generic configurations display peculiar signatures that can be searched for in gravitational-wave data as smoking guns of exotic compact objects.

Figure 1

Qualitative features of the effective potential felt by perturbations of a Schwarzschild BH compared to the case of wormholes [12] and of starlike ECOs with a regular center [22]. The precise location of the center of the star is model dependent and was chosen for visual clarity. The maximum and minimum of the potential corresponds approximately to the location of the unstable and stable PS, and the correspondence is exact in the eikonal limit of large angular number $l$. In the wormhole case, modes can be trapped between the PSs in the two “universes.” In the starlike case, modes are trapped between the PS and the centrifugal barrier near the center of the star [28, 29, 30]. In all cases the potential is of finite height, and the modes leak away, with higher-frequency modes leaking on shorter timescales.

Figure 2

Left: A dipolar ($l=1$, $m=0$) scalar wave packet scattered off a Schwarzschild BH and off different ECOs with $\ell ={10}^{-6}M$ ($r_0=2.000001M$). The right panel shows the late-time behavior of the waveform. The result for a wormhole, a gravastar, and a simple empty shell of matter are qualitatively similar and display a series of “echoes” which are modulated in amplitude and distorted in frequency. For this compactness, the delay time in Eq. (6) reads $\mathrm{\Delta }t\approx 110M$ for wormholes, $\mathrm{\Delta }t\approx 82M$ for gravastars, and $\mathrm{\Delta }t\approx 55M$ for empty shells, respectively.

Figure 3

Left panel: The waveform for the radial infall of a particle with specific energy $E=1.5$ into a wormhole with $\ell ={10}^{-6}M$, compared to the BH case. The BH ringdown, caused by oscillations of the outer PS as the particle crosses through, are also present in the wormhole waveform. A part of this pulse travels inwards and is absorbed by the event horizon (for BHs) or then bounces off the inner (centrifugal or PS) barrier for ECOs, giving rise to echoes of the initial pulse. This is a low-pass cavity which cleans the pulse of high-frequency components. At late times, only a lower frequency, long-lived signal is present, well described by the QNMs of the ECO. Right panel: the same for a scattering trajectory, with pericenter $r_{\min }=4.3M$, off a wormhole with $\ell ={10}^{-6}M$. The main pulse is generated now through the bremsstrahlung radiation emitted as the particle approaches the pericenter. The remaining main features are as before. We show only the real part of the waveform, the imaginary part displays the same qualitative behavior.

Figure 4

GW signal produced by a test particle falling radially into a wormhole with $E=1.01$. We consider the same setup as in Ref. [12] but for a wormhole without a PS. Without outer (and inner) PS, the ringdown signal is, clearly, different from that of a BH. Because there is no longer a good resonating cavity, echoes do not appear to be excited.

Figure 5

Mass-radius relation for a solitonic BS with ${\sigma }_0=0.05m_P$. In the inset we show the compactness as a function of the central scalar field, ${\sigma }_c\equiv |\mathrm{\Phi }(r=0)|$. The red and the green markers correspond, respectively, to the light BSs with $M/R\approx 0.118$ and to the medium-mass one with $M/R\approx 0.184$ whose numerical evolutions are discussed in the main text. The blue marker indicates a stable BS with nearly maximum mass and $M/R\approx 1/3$. The horizontal line in the right panel denotes the compactness of the Schwarzschild PS, $M/R=1/3$.

Figure 6

GW (i.e., represented by the $l=m=2$ mode of the Newman-Penrose ${\mathrm{\Psi }}_4$ scalar), as a function of time, emitted during the head-on collision of two solitonic BSs. Left panel (low mass): The final object is not massive enough to collapse to a BH, so the final fate of the system will depend on the BS configuration; a perturbed BS (bs-bs), annihilation of the stars (bs-abs) or two individual stars after multiple inelastic collisions (bs-bsop). Right panel (medium mass): The final object promptly collapses to a BH, although previously—for some of the configurations, i.e., the bs-bs and the bs-abs—there is a signature on the GWs produced by the scalar field interaction.

Figure 7

Maximum value of the scalar field norm, as a function of time, for the head-on collision of two low-mass solitonic BSs. The bs-bs collision forms, after a relatively long transient, a single perturbed BS. The bs-abs configuration has opposite Noether charges that annihilate soon after they merge, destroying the stars and dispersing/radiating their scalar fields. The scalar field interaction in the bs-bsop is repulsive and larger than the gravitational attraction, so the system undergoes several inelastic collisions—which compresses the star and leads to small bumps on the scalar field norm that can be observed at $t\approx \{60,160,240\}$—before relaxing to a binary at rest with the surfaces barely touching.