Distinguishing black holes from horizonless objects through the excitation of resonances during inspiral

How well is the vacuum Kerr geometry a good description of the dark, compact objects in our universe? Precision measurements of accreting matter in the deep infrared and gravitational-wave measurements of coalescing objects are finally providing answers to this question. Here, we study the possibility of resonant excitation of the modes of the central object – taken to be very compact but horizonless – during an extreme-mass-ratio inspiral. We show that for very compact objects resonances are indeed excited. However, the impact of such excitation on the phase of the gravitational-wave signal is negligible, since resonances are crossed very quickly during inspiral.

How well is the vacuum Kerr geometry a good description of the dark, compact objects in our universe? Precision measurements of accreting matter in the deep infrared and gravitational-wave measurements of coalescing objects are finally providing answers to this question. Here, we study the possibility of resonant excitation of the modes of the central object -taken to be very compact but horizonless -during an extreme-mass-ratio inspiral. We show that for very compact objects resonances are indeed excited. However, the impact of such excitation on the phase of the gravitational-wave signal is negligible, since resonances are crossed very quickly during inspiral.

I. INTRODUCTION
A remarkable feature of classical General Relativity is that vacuum spacetime can be curled to the extreme point of producing horizons, the boundaries of causally disconnected regions of spacetime that cloak singularities from far away observers. Such extraordinary property requires strong observational evidence for black holes (BHs), a quest that should be placed alongside tests of the equivalence principle. In addition, dark compact objects are predicted to arise, at a phenomenological level, either when quantum effects are included or when beyond-the-standard model of particle physics is considered [1].
Here, we study instead the possibility that the proper modes of oscillation of compact objects are excited and play a role in the inspiralling process. Previous studies focused on a special class of solutions -boson starswhich have a well defined underlying theory and are of interest from a particle-physics point of view. Resonant excitation of modes was found to be possible [19]. However, such self-gravitating solutions are never as compact as to be able to mimic the ringdown stage of BHs [1]. Therefore, here we turn to a (artificial) model describing the physics of objects whose surface sits deep down the gravitational potential 1 . The compact object is assumed 1 Ultracompact objects -so-called gravastars -were investigated in Ref. [20]; it was shown that resonances can be excited during inspiral, but a proper detectability analysis was not performed.
to be spherically symmetric. The exterior is vacuum and therefore described by the Schwarzschild geometry, down to the (hard) surface at We consider both a toy model where a scalar particle orbits the compact object and a more realistic extrememass-ratio inspiral, driven by GW emission.

II. RESULTS
We used matched asymptotic expansions to solve the relevant wave equations describing linearized scalar (s = 0) and gravitational (s = 2, odd parity only) fields, excited by a pointlike charge γ and mass m 0 , respectively. The pointlike object orbits the central mass M on a circular geodesic of radius r p ≫ M . The technical details are relegated to appendices. The system emits scalar and gravitational waves. We are mostly interested in the possibility of excitation of the internal degrees of freedom of the massive object, and on the possibility that such excitations show up in the detected signal. We will assume, for simplicity, Dirichlet boundary conditions on the relevant master variables at the surface of the compact object. Gravitational fluctuations will most likely not interact significantly with any putative surface, but would cross unimpeded towards the center of the star, where they would be reflected. Thus, these artificial conditions are expected to mimic the physics we want to study.

A. Resonant frequencies
The angular dependence of the fields is separated using spin-s spherical harmonics labeled by an angular number ℓ and an azimuthal number m. Define ̟ = 2M ω. In the small ǫ regime, we find resonant or quasinormal frequencies ω = ω R + iω I at (n = 1, 2, ...) These results agree with previous studies [1,21]. To excite such quasinormal modes, the orbital frequency Ω of the particle needs to be tuned, ω R = mΩ. Thus, resonances occur when the radius of the circular orbit is r p = M 1/3 /|ω R /m| 2/3 (see (A7)) with r 0 ω R ≃ nπ/ log ǫ. For the orbit to be stable, the radius should be r > 6M , which implies that we focus on

B. Fluxes on and off resonance
In the absence of resonance with the central object, a particle on a circular orbit of radius r p gives rise to a flux whose dominant component iṡ These fluxes correspond to the dominant dipolar and quadrupolar modes of the field for s = 0, 2 respectively and agree with known expressions in the literature (most notably Einstein's quadrupole formula). The structure of the central object is irrelevant in this regime (and thus a central BH would give rise to identical fluxes) [22].
However, when the orbital frequency approaches a resonance frequency, the flux has a sharp peak equal tȯ These resonances have a radial width in orbital frequency of δΩ ∼ ω I , see (A51). Notice that for most parameters of interest and for the dominant modesĖ at resonance is indeed larger than off-resonance.

C. Impact of resonances on EMRIs
The pace at which inspiral proceeds is determinedwithin a quasi-adiabatic approach -by energy conservation. An increased flux at resonance implies that the inspiral towards an exotic horizonless objects proceeds faster, when compared to BH binaries. In turn, this effect might lead to an observable dephasing in gravitational waveforms. On the other hand, these are very narrow resonances and thus the accumulated energy release may be small enough that the effect is negligible.
To estimate the impact on the GW phase, one can compute the number of cycles accumulated during the resonant stage, where f f − f i ∼ ω I /2π is the resonance width and the variation of the frequency can be computed from the orbital parameters in a quasi-adiabatic fashion, Here is the gravitational binding energy of the small point particle. The number of cycles should be compared to those in a BH vacuum spacetime, N BH , obtained by using the flux (4) in the previous expressions (we ignore fluxes through the horizon, since these are subdominant [22]). We find, for l = m = 2 and for the dominant fundamental n = 1 mode N res ∼ 2.4 × 10 −11 10 −6 q 10 | log ǫ| N res N BH ∼ 9.5 × 10 −11 10 | log ǫ| where q ≡ m 0 /M is the mass ratio. Thus, the small object passes through resonances without any noticeable effect on the GW output. Higher modes are necessarily suppressed even further, since Eq. (2) forces ǫ to decrease exponentially with n. It is, in principle, possible that the number of cycles spent in resonance is small, yet the signal is observable. However, the time δt that it takes to cross the resonance can be estimated using δt ∼ ω I /Ω orb , witḣ Ω orb = dΩ/dr dr/dE orbĖ . We find This corresponds to a high-frequency "glitch", inaccessible by current or planned GW detectors.

D. Discussion
In conclusion, the inspiral of a small pointlike particle around an ultracompact object can excite the characteristic modes of the central object, which carry important information on the nature of the latter. However, our results indicate that such excitation does not have a significant impact of the phase of the GW, and leads to only a very high-frequency glitch. Thus, resonant excitation of modes during inspiral is not a promising mechanism to help constraining the nature of dark, ultracompact objects. Our results are based on a simple-minded model for the supermassive object, it would certainly be desirable to extend the analysis to other self-gravitating objects whose surface lies extremely close to the Schwarzschild radius.
We start with a very simple toy problem, that of a massless scalar field Φ around a compact horizonless ob-ject of mass M in a spacetime background of metric g µν . The scalar field will be excited by introducing a pointlike particle of mass m 0 coupled to it and orbiting around the central object. The full dynamics is described by the action (A1) Here R[g] denotes the Ricci scalar of the metric, γ > 0 is a coupling constant, and T is the trace of the stress tensor of the particle. We consider the point particle to be a small perturbation. Thus, the background spacetime is fixed and taken to be described by Schwarzschild exterior geometry, with coordinates {t, r, θ, φ}. All that remains is to solve the scalar field equation of motion coupled to the point-like particle: For this matter we expand the fields in Fourier modes of frequency ω and in spherical harmonics Y ℓm as: For static backgrounds, one finds the equation (A5) with f = 1 − 2M/r, and r * denoting the tortoise coordinate. The object has a surface at r = r 0 , or in tortoise coordinates r * = r 0 * , where we shall impose reflective conditions.

The source
If τ denotes the proper time of the point particle along the world line z µ (τ ) = (T (τ ), R(τ ), ϑ(τ ), ϕ(τ )), the corresponding stress-energy tensor is given by where the definition of the Dirac delta is taken as δ (4) (x) √ −gd 4 x = 1. We shall consider a stable circular geodesic taking place in the equatorial plane.
The particle will have an orbit of radius r = r p > 6M and an orbital frequency given by Kepler's law, Then We can now solve the multipolar moments T ℓm (ω). Equating (A4) with (A8), multiplying both sides by e iω ′ t Y * ℓ ′ m ′ and integrating on the sphere and in time, we get 2

The formal solution
Define two independent solutions of the homogeneous ODE (A5) as The former one is considered to have reflective boundary conditions on the central object surface r 0 * , while the latter describes purely outgoing waves at spatial infinity. The two of them are found to be linearly independent by computing their Wronskian, which gives 2iωA in . The Green's function reads Then it is easy to show that, at large distances, the inhomogeneous solution is

Energy flux
The energy flux emitted to infinity by the scalar field is determined bẏ where T rt is the relevant component of the scalar stressenergy tensor, Taking into account the expansion in spherical harmonics (A3) and the asymptotic behavior Z ℓm ∼ Z ∞ ℓm e iωr δ(mΩ− ω) at large radial distances, we find The normalization condition for the spherical harmonics reduce this toĖ

Matched asymptotic expansions
We now want to have an analytical understanding of the solutions of the homogeneous equation at small frequencies. The homogeneous equation can be written in 2 We take the normalization of the spherical harmonics as two equivalent forms where primes stand for radial derivatives.

a. The near-region solution
We follow the procedure in Refs. [21,23]. Consider first a "near-region" where r − r 0 ≪ 1/ω (we assume r 0 ∼ 2M to good approximation). Then, the second equation above can be written as where now primes are derivatives with respect to x ≡ f , and we introduced the dimensionaless frequency ̟ ≡ ωr H . Notice that at the object surface r = r 0 , Defining now Z/r = x i̟ (1 − x) ℓ+ǫ+1 F , and neglecting O(̟ 2 ) and O(ǫ) terms in the coefficient of F , one finds the standard hypergeometric equation, Given that c is not an integer, around x = 0 two linearly independent solutions are 2 F 1 (a, b, c, x) and . The general solution in the nearregion is then For BHs one imposes boundary conditions corresponding to purely ingoing waves at the horizon, and that implies B = 0. For "exotic compact objects" (ECOs) with Dirichlet BCs at the surface To understand the far-region behavior of the above solution we use the transformation properties of hypergeometric functions, Therefore, the large r behavior is

The far-region solution
In the far-region (i.e. when r ≫ r 0 ), the wave equation reduces to with solutions which for small r reduces to From the behavior of the Bessel functions at r → ∞, the solution (A27) is asymptotic to and by demanding now the behaviour (A11) we get Now we want to express α and β in terms of the boundary parameters A, B. In order to do this we proceed to match the small r behaviour of the far-region solution to the large r behaviour of the near-region solution.
First of all, we use the result On the other hand, we can also expand the Gamma function to find (A31) Then, in the limit in which ̟ ≪ 1, we get Finally, we can write (A34)

The QNMs of ECOs
For ECOs we have B = −Ax −2i̟ 0 . The energy flux is determined by (A17) with (A13). We expect resonances then at the poles of (A34). These are the quasi-normal mode (QNM) frequencies of these objects [1,21].  We compute the QNM frequencies as the roots of A in (̟) = 0. By setting (A34) equal to zero we can write We proceed to solve the equation iteratively, by solving e i2̟i+1 log x0 R(̟ i ) = 1 for ̟ i+1 , with the initial input ̟ 0 = 0. In the first iteration one obtains for all n ∈ Z. The second iteration gives (since x 0 < 1 the logarithm is negative) As argued in the main text x 0 ≪ e −6 √ 6nπ/m , so | log x 0 | ≫ nπ, and we can approximate F s=0 (ℓ) can be further simplified and then the next-toleading order result for the QNMs is (A43) The presence of a non-vanishing imaginary part with the correct sign accounts for the exponential decay of the mode in time (stability). This analytical prediction works reasonably well, as checked with the numerical implementation (see Table I).

Resonance widths
Given the Kepler relation (A7), the energy flux (A46) is a function of the radial distance only. To find the width of the ECO resonances we look for a value r of the radial distance that fulfillsĖ(r) = 1 2Ė (r QNM ), where r QNM de-notes the value that gives the resonant frequencies ̟ QNM written in (A41), i.e. ̟ 2 QNM m 2 = 4M 3 /r 3 QNM . We assume that the frequency band will be small, so that we can expand the function m̟(r) = 4M 3 /r 3 in Taylor series at r QNM as: (A48) The scalar flux (A46) can be written now aṡ E s=0 (r) = a ℓm ̟ 2ℓ/3+4 (r) cos 2 (̟(r) log x 0 ) b ℓ ̟ 4ℓ+4 (r) cos 2 (̟(r) log x 0 ) + c ℓ sin 2 (̟(r) log x 0 ) , with b ℓ ≡ Γ(−ℓ + 1/2) 2 Γ(ℓ + 1) 10 > 0 and c ℓ ≡ 4ℓ 2 πΓ(2ℓ) 2 Γ(2ℓ + 2) 4 > 0 (a ℓm will not be needed). The flux peaks at r QNM , and givesĖ s=0 (r QNM ) = a ℓm b ℓ ̟ −10/3ℓ QNM . By expanding the energy flux up to second order we get 4 3 It is proportional to Im̟ log x 0 , and according to (A43) it is tiny. 4 That the coefficient in front of r − r QNM is not identically zero is just a residue due to the approximations. In an exact approach Demanding nowĖ s=0 (r) = 1 2Ė s=0 (r QNM ) one gets a this term would not appear. It does not play any role in what follows.
quadratic equation for (r − r QNM )/r QNM ≡ y, which leads to two solutions, r + and r − , corresponding to the outer and inner radius of the width, respectively. The solution, in the approximation in which ̟ QNM ≪ 1 (recall (A43)), takes the simple form: From Eq. (A40) it can be readily checked that b ℓ c ℓ = F s=0 (ℓ)(−1) ℓ+1 . Thus we finally arrive at Im̟ QNM Re̟ QNM (A50) where we used the form (A42). The width in frequency around the value ̟ QNM can be inferred from (A48) and (A50). It reads: Appendix B: Gravitational case

The setup
We consider gravitational radiation within the framework of metric perturbations around the Schwarzschild geometry. These perturbations are excited by the inspiral of a small mass around the Schwarzschild BH. We will study these perturbations using the Newman-Penrose formalism combined with the Regge-Wheeler analysis, following Ref. [25].
The fundamental perturbation field is the Weyl scalar Ψ 4 , which is well adapted to analyze outgoing GWs. We expand it as where −2 Y ℓm denote the spherical harmonics of spinweight s = −2 [26]. The sums are restricted to ℓ ≥ 2 and −ℓ ≤ m ≤ ℓ. With this ansatz for the Weyl scalar, the linearised Newman-Penrose equations lead to with and source term T ωℓm given in detail in (2.6) of [25]. We employ the Green's function method to obtain a solution of the previous equation. To build the Green's function we need two linearly independent solutions of the homogeneous equation. We take R H ωℓm and R ∞ ωℓm . The latter one will describe purely outgoing waves escaping to infinity. The former one shall denote purely ingoing waves in the BH case (corresponding to a perfect absorber), and purely reflected waves in the ECO case (corresponding to a perfect mirror). For the considered ODE the Wronskian of these two solutions must be a constant, and so it can be evaluated at any value of r. According the their definitions, these solutions should have the following asymptotic behaviours and the Wronskian yields, Here, primes denote differentiation with respect to r. Thus, both solutions are linearly independent as long as B in ωℓm = 0 and ω = 0. The Green's function is Following the standard theory, we can write the inhomogeneous solution as where in the last step we considered the limit r → ∞.

Energy flux
To calculate the energy flux it is helpful to introduce an auxiliary quantity Z ℓm by R ωℓm (r → ∞) = m 0 Z ℓm δ(ω − mΩ)r 3 e iωr * . Then the flux formula readṡ The auxiliary function can be calculated from the stressenergy tensor of a point particle orbiting around a Schwarzschild BH and the result is [25] with coefficients From the identity s Y ℓ−m π 2 , 0 = s Y ℓm π 2 , 0 (−1) s+ℓ it is easy to see that Z ℓ−m = (−1) ℓZ ℓm . Consequently, the final expression for the energy flux iṡ with ω = mΩ.

Reducing the problem to solving the Regge-Wheeler equation
Chandrasekhar [27] showed that if X ωℓm (r) is a solution to the Regge-Wheeler equation with effective potential is a solution to the homogeneous equation (B2). So rather than working with the ODE derived from the Newman-Penrose formalism, it is more convenient to solve the Regge-Wheeler equation first and then apply the Chandrasekar transformation to obtain the Weyl component R H ωℓm (r) that governs the energy flux (B13). The relevant solution X H ωℓm (r) for our problem has the following asymptotic conditions, inherited from (B4): As in the scalar case, for BHs we shall fix B = 0, while for ECOs we have B = −Ax −2iω 0 .

Some simplifications
Although equation (B9) seems complicated at first, it can be further simplified. Recall that we consider the problem of having the particle far away from the BH, so that r p ≫ M . Using the Kepler Law M ω/m = M Ω = (M/r p ) 3/2 this automatically implies that ̟ = 2M ω ≪ 1. Inspection of (B10)-(B12) shows that This means that Z ℓm in (B9) is dominated by the contribution involving 0 b ℓm , unless 0 Y ℓm π 2 , 0 vanishes, which according to (A36) happens when ℓ + m is an odd number. When this happens, Z ℓm is dominated by the term involving −1 b ℓm , but this is suppressed in our approximation. For a given ℓ therefore, the energy flux (B13) will be dominated by modes for which ℓ + m is even. The leading order expression for Z ℓm is then All that remains now is to calculate A in (ω) and X H ωℓm (r). From (B19) and (B16), we get B in (ω) and R H ωℓm , respectively. Then finally we will use (B24) to calculate (B13). For the purpose of calculating A in (ω) and X H ωℓm (r) we shall follow closely the strategy done in the scalar case.

Matched asymptotic expansions
The homogeneous Regge-Wheeler equation can be written in two equivalent forms where primes stand for radial derivatives, and s = 2. The scalar case is recovered by taking s = 0.

a. Near-region solution
We analyze first the regime in which ω ≪ 1 r−2M . Comparing (B26) with (A19) it is clear that we can recycle the results done for the scalar case, as long as we keep track of the new factor s 2 . Equation (A20) reads now where here primes are derivatives with respect to x ≡ f and ̟ ≡ ωr H . Making the substitution X/r = x i̟ (1 − x) ℓ+ǫ+1 F we find, after neglecting O(̟ 2 ) and O(ǫ) terms, with a, b, c the parameters (A22). Now we want to reabsorbe the s 2 term into these parameters. Define a ′ and b ′ by ab − s 2 = a ′ b ′ and a + b = a ′ + b ′ . Solving this system leads to The general solution in the near-region is then To study the far-region behavior of the above solution use again the transformation properties of hypergeometric functions (A24). Then, we find the large r behavior (x ∼ 1)
Putting these expressions into (B33) we find