The tidal deformability of a self-gravitating object leaves an imprint on the
gravitational-wave signal of an inspiral which is paramount to measure the
internal structure of the binary components. We unveil here a surprisingly
unnoticed effect: in the extreme-mass ratio limit the tidal Love number of the
central object (i.e. the quadrupole moment induced by the tidal field of its
companion) affects the gravitational waveform at the leading order in the mass
ratio. This effect acts as a magnifying glass for the tidal deformability of
supermassive objects but was so far neglected, probably because the tidal Love
numbers of a black hole (the most natural candidate for a compact supermassive
object) are identically zero. We argue that extreme-mass ratio inspirals
detectable by the future LISA mission might place constraints on the tidal Love
numbers of the central object which are roughly 8 orders of magnitude more
stringent than current ones on neutron stars, potentially probing all models of
black hole mimickers proposed so far.Comment: Essay selected for an Honorable Mention in the Gravity Research
  Foundation Essay Competition 2019. v2: two references added, version to
  appear in IJMP

Maselli, Andrea

Pani, Paolo

English

arXiv

The tidal deformability of a self-gravitating object leaves an imprint on the gravitational-wave signal of an inspiral which is paramount to measure the internal structure of the binary components. We unveil here a surprisingly unnoticed effect: in the extreme mass-ratio limit the tidal Love number of the central object (i.e. the quadrupole moment induced by the tidal field of its companion) affects the gravitational waveform at the leading order in the mass ratio. This effect acts as a magnifying glass for the tidal deformability of supermassive objects but was so far neglected, probably because the tidal Love numbers of a black hole (the most natural candidate for a compact supermassive object) are identically zero. We argue that extreme mass-ratio inspirals detectable by the future laser interferometric space antenna (LISA) mission might place constraints on the tidal Love numbers of the central object which are roughly eight orders of magnitude more stringent than current ones on neutron stars, potentially probing all models of black hole mimickers proposed so far

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November 18, 2019 15:41 WSPC/S0218-2718 142-IJMPD 1944001International Journal of Modern Physics DVol. 28, No. 14 (2019) 1944001 (5 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218271819440012Love in extrema ratio*Paolo Pani† and Andrea Maselli‡Dipartimento di Fisica,“Sapienza” Università di Roma and Sezione INFN Roma1,Piazzale Aldo Moro 5, Roma 00185, Italy†paolo.pani@uniroma1.it‡andrea.maselli@roma1.infn.itReceived 12 May 2019Accepted 14 May 2019Published 19 June 2019The tidal deformability of a self-gravitating object leaves an imprint on the gravitational-wave signal of an inspiral which is paramount to measure the internal structure of thebinary components. We unveil here a surprisingly unnoticed effect: in the extreme mass-ratio limit the tidal Love number of the central object (i.e. the quadrupole momentinduced by the tidal field of its companion) affects the gravitational waveform atthe leading order in the mass ratio. This effect acts as a magnifying glass for thetidal deformability of supermassive objects but was so far neglected, probably becausethe tidal Love numbers of a black hole (the most natural candidate for a compactsupermassive object) are identically zero. We argue that extreme mass-ratio inspiralsdetectable by the future laser interferometric space antenna (LISA) mission might placeconstraints on the tidal Love numbers of the central object which are roughly eight ordersof magnitude more stringent than current ones on neutron stars, potentially probing allmodels of black hole mimickers proposed so far.“Love all, trust a few, do wrong to none.”William Shakespeare, All’s Well That Ends WellKeywords: Black holes; gravitational waves; extreme mass ratio inspirals.Gravitational-wave (GW) measurements of the tidal deformability of neutronstars1 — through the so-called tidal Love numbers (TLNs)2 — provide one of themost accurate tools to date to probe the microphysics of the neutron-star interior.3,4It has been recently realized that tidal effects in coalescing binaries can also be used∗This essay received an Honorable Mention in the 2019 Essay Competition of the Gravity ResearchFoundation.1944001-1Int. J. Mod. Phys. D 2019.28. Downloaded from www.worldscientific.comby 78.13.19.172 on 05/05/20. Re-use and distribution is strictly not permitted, except for Open Access articles.November 18, 2019 15:41 WSPC/S0218-2718 142-IJMPD 1944001P. Pani and A. Masellito distinguish black holes (BHs) from other ultracompact objects.5–7 A remarkableresult in classical General Relativity is that — owing to the one-way nature ofthe event horizon — the TLNs of a BH are identically zero,8–13 whereas those ofan ultracompact horizonless object are small but finite.6,14,15 Besides posing anintriguing problem of “naturalness” in Einstein’s theory,15 this precise cancelationprovides also an opportunity to test the BH paradigm: measuring a nonvanishingTLN with measurements errors small enough to exclude the null case would providea smoking gun for the existence of new species of ultracompact massive objects.The impact of the tidal deformability on the GW signal from a binary coa-lescence has been so far studied mostly in the case of comparable masses. Thischoice is well motivated for neutron star binaries, but it might be too restrictivein the context of tests of the nature of dark compact objects, especially becausefuture GW observations are expected to unveil binaries with mass ratios departingsignificantly from unity. With this motivation in mind, here we explore the follow-ing question: how much does the tidal deformability affect an extreme mass-ratioinspiral (EMRI )16 when the massive central object has nonvanishing TLN?Let us consider a nonspinning compact binary, with masses mi (i = 1, 2), totalmass m = m1 + m2, and mass ratio q = m1/m2 ≥ 1. At leading post-Newtonianorder, the correction to the instantaneous GW phase due to the tidal deformabilityof the binary components reads1,2 (we use G = c = 1 units)φtidal(f) = −1178(1 + q)2qΛm5v5, (1)where v = (πmf)1/3 is the orbital velocity, f is the GW frequency,Λ =126((1 +12q)λ1 + (1 + 12q)λ2)(2)is the weighted tidal deformability, whereas λi = 23m5i ki and ki are the tidaldeformability and the (dimensionless) TLN of the ith object, respectively. Thesecan be defined in terms of the quadrupole moment Q(i)ab of the ith object inducedby the tidal field G(j)ab produced by its companion, namelyQ(i)ab = λiG(j)ab ∼ λimjr3, i = j, (3)where r ∼ mv−2 is the orbital distance. For a typical neutron star, ki ≈ 1000and Λ/m5i ≈ 600, the exact values depend on the equation-of-state. As a rule ofthumb, the more compact an object, the smaller its TLN, so much so that a BHhas kBH = 0.It is enlightening to expand Eq. (1) in the extreme mass-ratio limit. Let us firstconsider the standard case in which the central object is a BH, so that k1 = 0. Insuch case the first nonvanishing contribution isφtidal(f) ∼ −92k2v5 1q3+ · · · , q  1 (k1 = 0), (4)1944001-2Int. J. Mod. Phys. D 2019.28. Downloaded from www.worldscientific.comby 78.13.19.172 on 05/05/20. Re-use and distribution is strictly not permitted, except for Open Access articles.November 18, 2019 15:41 WSPC/S0218-2718 142-IJMPD 1944001Love in extrema ratiowhich is proportional to the TLN of the small companion, and is suppressed by aq−3 factor. Therefore, for a typical EMRI with q ≈ 106, the above term is negligiblysmall.On the other hand, when k1 = 0, the tidal phase grows linearly in q,φtidal(f) ∼ −38k1v5q + · · · , q  1, (5)and is proportional to the TLN of the central object. It is remarkable that in thiscase the tidal phase enters at the leading order in the mass ratio just like theordinary radiation-reaction term,φN (f) ∼ 3128v−5q + · · · , q  1, (6)although the latter dominates at large binary separation, owing to the differentscaling with the orbital velocity. The tidal phase contribution in Eq. (5) has thesame scaling with q as the spin-induced quadrupolar deformations17 and, as long ask1  1/q, it is even larger that the first-order correction due to the conservative partof the self-force (i.e. the self-interaction of a test-particle with its own gravitationalfield18), the latter being suppressed by a factor 1/q relative to Eq. (6).The above intriguing result case be explained as follows. The GW phase can beobtained by solving ford2φ(f)df2=2πĖdEdf, (7)where E is the binding energy of the binary and Ė is the energy flux emitted inGWs. The TLNs enter both in conservative piece, E(f), and in the dissipative piece,Ė. To the leading order in post-Newtonian theory19E(f) = − m12(1 + q)v2(1 − εc 6q(k1q3 + k2)(1 + q)5v10), (8)Ė(f) = −325q2(1 + q)4v10(1 + εd4(q4(3 + q)k1 + (1 + 3q)k2)(1 + q)5v10), (9)where εc and εd are just book-keeping parameters for the correction to the conser-vative and dissipative term, respectively. Plugging this into Eq. (7) and solving atthe leading order in the corrections, one findsφ(f) = φN (f)(1 − 16εdk1v10), q  1, (10)whereas the correction coming from the conservative term is subleading. It isstraightforward to check that the above equation yields Eq. (5). Thus, the enhance-ment of the tidal effect in the waveform is due to the contribution of the TLNsto the energy flux. When k1 = 0, this term is not suppressed by any power of1/q relative to the leading-order term, as evident from Eq. (9). In turn, this resultcan be obtained straightforwardly by using Eq. (3) and the quadrupole formula,Ė ∼ ∂3t Qtotij ∂3t Qtotij , where Qtotij is the total quadrupole of the binary.1944001-3Int. J. Mod. Phys. D 2019.28. Downloaded from www.worldscientific.comby 78.13.19.172 on 05/05/20. Re-use and distribution is strictly not permitted, except for Open Access articles.November 18, 2019 15:41 WSPC/S0218-2718 142-IJMPD 1944001P. Pani and A. MaselliLet us now discuss the phenomenological implications of this enhancement. Weconsider an EMRI up to the innermost stable circular orbit (ISCO) of the cen-tral object. The total GW phase accumulated between fmin and fmax ∼ fISCO =16π√6m1 fmin due to the tidal deformability is simplyφtottidal = −k196√6q ≈ −0.004k1q. (11)If for instance q = 107, by requiring a detectability threshold φtottidal > 1 rad, we findthat the effect might be measurable even for TLNs as small as k1 ≈ 2 × 10−5.This bound is quite impressive at least for two reasons. First of all, it sug-gests that the future LISA mission,20 which is expected to detect few to thousandsEMRIs per year,16,21 could set constraints on the TLN of the central object whichare approximately eight orders of magnitude smaller than LIGO’s current measure-ments on the tidal deformability of a neutron star.3,4 Furthermore, in the mostextreme models of horizonless compact objects6 — some of which predicting quan-tum corrections at the horizon scale — the TLNs are of the order k1 ≈ 10−3,well above the bound estimated in Eq. (11). Finally, it is easy to show that therelative measurement errors on k1 scale as q−1/2 in the high signal-to-noise ratiolimit. Thus, EMRIs detectable by LISA might provide the ultimate tests for exoticcompact objects (see Ref. 22 for a review).Besides the aforementioned bounds on BH mimickers, the enhancement of thetidal phase in an EMRI might be relevant if primordial BHs with masses m2 ≈10−4 M exist in nature. These objects might form an EMRI around a neutronstar and pass through LIGO’s band in less than a year before plunge. In suchcase the tidal phase would provide an unparalleled way to constrain the neutron-star equation-of-state through a measurement of the TLN at the level given byEq. (11). Unfortunately, the event rates for neutron-star capture of primordial BHsseem extremely small in this mass range.23Our derivation is based on a low-velocity expansion of the field equations, andthe post-Newtonian series converges poorly in the large mass-ratio limit, at least inits dissipative sector.24 Therefore, Eq. (10) cannot be used for a rigorous parameterestimation, but it should nonetheless provide the correct order of magnitude of theeffect of the tidal deformability of the central object. We conclude that it wouldbe very important to incorporate the tidal deformability terms consistently in anextreme mass-ratio expansion of Einstein’s field equation for binary systems beyondpost-Newtonian theory.All in all, Love might be at play even in extreme encounters.AcknowledgmentsWe are grateful to Tiziano Abdelsalhin, Leor Barack, Enrico Barausse, andScott Hughes for interesting comments on the draft. PP acknowledges finan-cial support provided under the European Union’s H2020 ERC, Starting Grant1944001-4Int. J. Mod. Phys. D 2019.28. Downloaded from www.worldscientific.comby 78.13.19.172 on 05/05/20. Re-use and distribution is strictly not permitted, except for Open Access articles.November 18, 2019 15:41 WSPC/S0218-2718 142-IJMPD 1944001Love in extrema ratioAgreement No. DarkGRA–757480. The authors would like to acknowledge net-working support by the COST Action CA16104 and support from the AmaldiResearch Center funded by the MIUR program “Dipartimento di Eccellenza” (CUP:B81I18001170001).References1. E. E. Flanagan and T. Hinderer, Phys. Rev. D 77 (2008) 021502, arXiv:0709.1915[astro-ph].2. E. Poisson and C. Will, Gravity : Newtonian, Post-Newtonian, Relativistic (CambridgeUniversity Press, Cambridge, UK, 2014).3. LIGO Scientific and Virgo Collabs. (B. P. Abbott et al.), Phys. Rev. Lett. 121 (2018)161101, arXiv:1805.11581 [gr-qc].4. S. De, D. Finstad, J. M. Lattimer, D. A. Brown, E. Berger and C. M. Biwer, Phys.Rev. Lett. 121 (2018) 091102, arXiv:1804.08583 [astro-ph.HE].5. M. Wade, J. D. E. Creighton, E. Ochsner and A. B. Nielsen, Phys. Rev. D 88 (2013)083002, arXiv:1306.3901 [gr-qc].6. V. Cardoso, E. Franzin, A. Maselli, P. Pani and G. Raposo, arXiv:1701.01116 [gr-qc].7. N. Sennett, T. Hinderer, J. Steinhoff, A. Buonanno and S. Ossokine, Phys. Rev. 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Love in Extrema Ratio

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