On gravitational-wave echoes from neutron-star binary coalescences Paolo Pani∗ and Valeria Ferrariy Dipartimento di Fisica, \Sapienza" Universit`adi Roma & Sezione INFN Roma1, Piazzale Aldo Moro 5, 00185, Roma, Italy A tentative detection of gravitational-wave echoes in the post-merger signal of GW170817 has been recently claimed at 4:2σ significance level. It has been speculated that the signal might provide evidence for near-horizon quantum structures in the remnant exotic object. We point out that if the remnant object is an ultracompact neutron star, echoes are expected for objects with radius only slightly smaller than that of an ordinary neutron star. The reported echoes at ≈ 72 Hz are compatible with a toy model of incompressible star with mass approximately M 2 (2; 3)M and radius close to the Buchdahl limit, R ≈ 9GM=(4c2). If confirmed, low-frequency gravitational- wave echoes would be in tension with all current neutron-star models and would have dramatic implications for nuclear physics and gravity. Introduction. It has been recently suggested that figurations which are so compact [26, 27] (see Fig. 1). gravitational-wave (GW) echoes [1, 2] in the post-merger Therefore, should echoes be confirmed in GW170817, GW signal from a binary coalescence might be a generic they might provide evidence for a very exotic state of feature of quantum corrections at the horizon scale, and matter formed after the merger, but they are not neces- might provide a smoking-gun signature of exotic compact sarily associated with near-horizon quantum structures. objects (see [3, 4] for a review). In the last two years, ten- tative evidence for echoes in the combined LIGO/Virgo 3.0 binary black-hole (BH) events have been reported [5, 6] with controversial results [7{10]. This has also motivated several studies on the modeling of the echo waveform [11{ 2.5 photon-sphere 17]. Very recently, a tentative detection of echoes in the black hole post-merger signal of neutron-star (NS) binary coales- 2.0 Buchdahl's limit cence GW170817 [18] has been claimed at 4:2σ signifi- GW echoes cance level [19]. It has been speculated that these GW ⊙ M echoes could be related to the quantum properties of an / 1.5 exotic \BH remnant". M If GW echoes are confirmed in the event GW170817, it is of utmost importance to understand whether they are 1.0 related to new physics at the horizon scale [3, 4] or if sim- ilar repeated signals might arise also in other scenarios, 0.5 for example as low-frequency quasiperiodic oscillations in the post-merger environment [19], or as quasinormal modes [20] of a remnant (possibly exotic) star. 0.0 In this letter we point out that GW echoes are not a 6 8 10 12 14 16 prerogative of quantum corrections at the horizon scale. R/km Similar signals have been long known to arise in ultra- compact stars [21, 22] featuring a photon-sphere [3, 4, 23], FIG. 1. Mass-radius diagram of nonspinning NSs for sev- the latter being able to effectively trap radiation within eral representative equations of state [26, 27] (data taken the stellar interior [21, 24] (for a mathematical discus- from [28]). GW echoes require the star to feature a photon- arXiv:1804.01444v3 [gr-qc] 12 Jun 2018 sion on photon surfaces in general relativity, see [25]). sphere (red shaded area) and low-frequency echoes require We show that echoes at the reported frequency f ≈ configurations located far deep into this region. Echoes in echo the post-merger signal would imply that the remnant is more 72 Hz [19] would arise naturally if the remnant object is compact than a NS with an ordinary equation of state. an ultracompact star only slightly more compact than an ordinary NS, and are thus not necessarily related to Echoes from ultracompact stars. As an illustrative Planckian corrections at the horizon scale. In the static model, we consider an incompressible, constant-density case, the existence of a photon-sphere requires R < 3M, NS in general relativity, described by the Schwarzschild where M and R are the stellar mass and radius, respec- solution [29, 30]. In the static case, the metric reads tively (henceforth, we use G = c = 1 units). Ordinary 2 ds2 = −e2Φdt2 + dr + r2dΩ2, with equations of state do not support self-gravitating con- 1−2M(r)=r ! 4π p1 − 2Mr2=R3 − p1 − 2M=R M = ρr3 ;P = ρ ; 3 3p1 − 2M=R − p1 − 2Mr2=R3 ∗ [email protected] 3p 1p y [email protected] eΦ = 1 − 2M=R − 1 − 2Mr2=R3 ; (1) 2 2 2 when r < R. Here P and ρ = 3M=(4πR3) are the pres- of ordinary NSs is expected to be small compared to the sure and the density of the fluid. When r > R the p 3 2:7M mass-shedding limit, ΩK = M=R ≈ 22 kHz metric reduces to the Schwarzschild vacuum solution, M (for a star with R ≈ 9=4M), one cannot exclude the pos- M(r) = M, e2Φ = 1 − 2M=r, P = ρ = 0. The min- sibility of highly-spinning merger remnants, or the fact imum radius of the object is R ≡ 9=4M [31], so that B that exotic stars can spin faster than ordinary NSs. It is the external spacetime can feature a photon-sphere when therefore relevant to include spin effects in our model. R < R < 3M. B This can be done perturbatively (i.e., for Ω Ω ) The characteristic echo time scale is related to the light K through the Hartle-Thorne formalism [35, 36]. As in the crossing time from the center of the star to the photon- Kerr case [5, 17], one could estimate the echo frequency sphere, through the crossing time of principal null geodesics, i.e. Z 3M 1 those geodesics that more directly fall into the object, τecho = dr ; (2) having angular-momentum-to-energy ratio equal to the p 2Φ 0 e (1 − 2M=r) angular velocity of the central object [37]. and the echo frequency is well approximated by the - 5.× 10 3 roundtrip frequency, fecho ∼ π/τecho [1{4]. Models of quantum corrections at the horizon scales typically assume that R ∼ 2M + lp, where lp M is the Planck length. In this case, the echo delay 2.× 10 -3 time reduces to the BH \scrambling" [32, 33] time, τecho ∼ Mj log(lp=M)j, irrespectively of the object in- terior [2]. This is the case studied so far for worm- M 1.× 10 -3 holes [1], gravastars [2], Kerr-like objects with a reflective echo f surface [2, 5, 34], and other quantum-dressed BH-like ob- jects [6] (for a review, see Refs. [3, 4]). 5.× 10 -4 For an ultracompact star the situation is dramatically ϵ=4 10-5 different, because redshift at the surface is moderate and -6 most of the crossing time accumulates in the interior of ϵ=4 10 -6 the object. By defining = R=R − 1, it is straightfor- ϵ=2 10 B 2.× 10 -4 ward to obtain 0.0 0.2 0.4 0.6 0.8 1.0 τ 27( + 1)2 p p Ω/Ω echo = p cot−1 + cot−1 3 K M 16 FIG. 2. Dimensionless echo frequency f M as a function 9 − 3 9 + 1 27π echo + + 2 log ∼ −1=2 ; (3) of the angular velocity Ω of the object (normalized by the 4 4 16 p 3 mass-shedding limit, ΩK = M=R ) for different values of = R=R − 1 and to quadratic order in Ω=Ω . The echo where the first and second lines come from the internal B K frequency monotonically decreases with the spin. Shaded (0 < r < R) and external (R < r < 3M) contributions +0:5 areas refer to a reference measurement of M = 2:5−0:5M , to the integral (2), respectively. The last approximation +0:05 Ω = 0:3 ΩK , and f ≈ 72 Hz. is valid when ! 0 and depends only on the internal −0:05 echo contribution. The echo delay time diverges in the Buch- Figure 2 shows the echo frequency as a function of the dahl's limit, but not logarithmically as for near-horizon angular velocity Ω of the star for several values of , as quantum corrections. computed by solving the Tolman-Oppenheimer-Volkoff Therefore, for a nonspinning constant-density star, the equations numerically to second order in the spin and echo frequency reads computing the principal null geodesics on the equator of 1=2 this metric. The echo frequency decreases monotonically π 2:7M fecho ∼ ≈ 46 Hz : (4) with the spin, because spinning configurations can be τ M 10−6 echo more compact than their nonspinning counterpart. Thus, For ∼ O(10−6 − 10−5), the above frequency is compa- although low-frequency echoes can be explained also by rable to what reported in Ref. [19] for GW170817. In nonspinning models [see Eq. (4)], including the angular momentum would permit to match the same echo fre- this case, R − 2M ≈ M km, i.e. the difference be- 2:7M quency with a less compact configuration. tween the stellar radius and the Schwarzschild radius is Ultracompact exotic objects are unstable against the macroscopic, not Planckian. When 1, Eq. (4) agrees ergoregion instability [38{40] when they spin suffi- with the time-domain results of Ref. [21] and with the ciently fast [41{45]. However, the instability time scale fundamental quasinormal mode frequency of this object. τergoregion M (see, e.g., Fig. 4.19 in Ref. [40]) and Discussion. The above estimates are valid for nonrotat- is comparable to (and typically longer than) τecho [34].