Table Of ContentTesting strong-field gravity with tidal Love numbers
Vitor Cardoso,1,2 Edgardo Franzin,3 Andrea Maselli,4 Paolo Pani,5,1 Guilherme Raposo1
1 CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico – IST,
Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal
2 Perimeter Institute for Theoretical Physics, 31 Caroline Street North Waterloo, Ontario N2L 2Y5, Canada
3 Dipartimento di Fisica, Universita` di Cagliari & Sezione INFN Cagliari,
Cittadella Universitaria, 09042 Monserrato, Italy
4 Theoretical Astrophysics, Eberhard Karls University of Tuebingen, Tuebingen 72076, Germany and
5 Dipartimento di Fisica, “Sapienza” Universita` di Roma & Sezione INFN Roma1, Piazzale Aldo Moro 5, 00185, Roma, Italy
The tidal Love numbers (TLNs) encode the deformability of a self-gravitating object immersed
in a tidal environment and depend significantly both on the object’s internal structure and on the
dynamicsofthegravitationalfield. Anintriguingresultinclassicalgeneralrelativityisthevanishing
of the TLNs of black holes. We extend this result in three ways, aiming at testing the nature of
7 compact objects: (i) we compute the TLNs of exotic compact objects, including different families
1 ofbosonstars,gravastars,wormholes,andothertoymodelsforquantumcorrectionsatthehorizon
0 scale. Intheblack-holelimit,wefindauniversallogarithmicdependenceoftheTLNsonthelocation
2 ofthesurface;(ii)wecomputetheTLNsofblackholesbeyondvacuumgeneralrelativity,including
Einstein-Maxwell,Brans-DickeandChern-Simonsgravity;(iii)Weassesstheabilityofpresentand
r
a futuregravitational-wavedetectorstomeasuretheTLNsoftheseobjects,includingthefirstanalysis
M of TLNs with LISA. Both LIGO, ET and LISA can impose interesting constraints on boson stars,
while LISA is able to probe even extremely compact objects. We argue that the TLNs provide a
1 smoking gun of new physics at the horizon scale, and that future gravitational-wave measurements
1 oftheTLNsinabinaryinspiralprovideanovelwaytotestblackholesandgeneralrelativityinthe
strong-field regime.
]
c
q
CONTENTS C. Tidal perturbations of boson stars 20
-
r 1. Background solutions 20
g
2. Perturbations and TLNs 21
[ I. Introduction 1
A. The naturalness problem 2
3 D. Tidal perturbations of BH-like ECOs 22
B. Quantifying the existence of horizons 2
v 1. Polar-type TLNs of BH-like ECOs 22
6 C. Executive summary 2 2. Axial-type TLNs of BH-like ECOs 23
1
1 II. Setup: Tidal Love numbers of static objects 3
References 24
1
0
III. Tidal perturbations of exotic compact objects 5
.
1 A. Boson stars 5
I. INTRODUCTION
0 B. Models of microscopic corrections at the
7
horizon scale 6
1 Tidal interactions play a fundamental role in astro-
: physics across a broad range of scales, from stellar ob-
v IV. Tidal perturbations of BHs beyond vacuum GR 9
jects like ordinary stars and neutron stars (NSs) to large
i
X A. Scalar-tensor theories 9 celestial systems such as galaxies. Several astrophysical
B. Einstein-Maxwell 10
r structures (e.g., binaries and tidal tails [1, 2]) are conse-
a C. Chern-Simons gravity 11 quences of tidal interactions. Tidal effects can be partic-
ularlystrongandimportantintheregimethatcharacter-
V. Detectability 13 izes compact objects, giving rise to extreme phenomena
A. Model-independent tests with GWs 14 such as tidal disruptions.
B. Detectability of ECOs 14 Thedeformabilityofaself-gravitatingobjectimmersed
C. Detectability of BSs 15 in an external tidal field is measured in terms of its tidal
D. Testing GR 16 Lovenumbers(TLNs)[3,4]. Theseleaveadetectableim-
printinthegravitational-wave(GW)signalemittedbya
VI. Discussion and extensions 17 neutron-starbinaryinthelatestagesofitsorbitalevolu-
tion [5–7]. So far, a relativistic extension [6, 8, 9] of the
Acknowledgments 18 Newtoniantheoryoftidaldeformabilityhasbeenmostly
motivatedbytheprospectofmeasuringtheTLNsofNSs
A. TLNs of neutron stars 18 through GW detections and, in turn, understanding the
behaviorofmatteratsupranucleardensities[10–16]. The
B. Determination of TLNs 18 scope of this paper is to show that tidal effects can also
2
beusedtoexploremorefundamentalquestionsrelatedto naturallyinmodelsofminichargeddarkmatteranddark
thenatureofcompactobjectsandthebehaviorofgravity photons [48]. In several scalar-tensor theories, BHs are
inthestrong-fieldregime(forapreviousrelatedstudyin uniquelydescribedbytheKerrsolution,asinGR[49,50].
the context of aLIGO binaries, cf. Ref. [17]).1 However, thesetheoriesintroduceascalardegreeoffree-
Anintriguingresultinclassicalgeneralrelativity(GR) dom(non-minimally)coupledtogravityandtheresponse
is the fact that the TLNs of a black hole (BH) are pre- ofBHstoexternalperturbationsisgenericallyricher[51].
cisely zero. This property has been originally demon- Intheorieswithseveralorwithcomplexbosons,hairyBH
strated for small tidal deformations of a Schwarzschild solutions might exist that can be seen as BSs with a BH
BH[8,9,19]andhasbeenrecentlyextendedtoarbitrar- at the center [52, 53]. These solutions could be the end-
ilystrongtidalfields[20]andtothespinningcase[21–23], pointofthesuperradiantinstabilityoftheKerrgeometry,
atleastintheaxisymmetriccasetoquadraticorderinthe and may even describe metastable states when a single
spin [22] and generically to linear order in the spin [23]. realfieldispresent[54,55]. Finally,inquadratictheories
ofgravitytheEinstein-Hilbertactionisconsideredasthe
first term of a possibly infinite expansion containing all
A. The naturalness problem curvature invariants, as predicted by some scenarios re-
lated to string theory and to loop quantum gravity [46].
The precise cancellation of the TLNs of BHs within To leading order in the curvature corrections, stationary
Einstein’s theory poses a problem of “naturalness” in BHsinthesetheoriesbelongtoonlytwofamilies[56,57],
classical GR [24–26], one that can be argued to be as usually dubbed the Einstein-dilaton-Gauss-Bonnet solu-
puzzling as the strong CP and the hierarchy problem in tion [58–60] and the Chern-Simons solution [61, 62].
particle physics, or as the cosmological constant prob-
lem. The resolution of this issue in BH physics could
lead to – testable, since they would be encoded in GW B. Quantifying the existence of horizons
data –smoking-gun effects of new physics.
This question can be solved in at least two (related) Theobservationaldeterminationofthetidalproperties
ways, which we explore here. If new physics sets in, of compact objects has also a bearing on another funda-
for example through unexpectedly large quantum back- mental question: do event horizons exist, and how can
reaction or changes in the equation of state, BHs might we quantify their existence in GW data?
simply not be formed, avoiding this and other prob- It was recently shown that ultracompact horizonless
lems (such as the information loss puzzle [27]) alto- geometriesareexpectedtomimicverywellthelaststages
gether. Instead, other objects might be the end prod- of coalescence of two BHs, when they merge to form a
uct of gravitational collapse. These “exotic compact ob- single distorted BH, ringing down to its final Kerr ge-
jects” (ECOs) include boson stars (BSs) [28–32], gravas- ometry [63, 64]. In this scenario, horizonless geometries
tars [33, 34], wormholes [35], and various toy models de- would show up as echoes in the gravitational waveforms
scribing quantum corrections at the horizon scale, like at very late times. The exclusion of echoes up to some
superspinars[36],fuzzballs[37],“2-2holes”[38]andoth- instanttafterthemergerrulesoutstructureinthespace-
ers [39–41]. ECOs might be formed from the collapse of time down to a region r/r −1 ∼ exp(−t/r ), with r
+ + +
exotic fields or by quantum effects at the horizon scale, being the Schwarzschild radius of the spacetime. Thus,
and represent the prototypical example of exotic GW more sensitive detectors will probe regions closer and
sources[42–44]whichmightbesearchedforwithground- closer to the horizon.
or space-based detectors. Theabovepicturereferstothefinal,post-mergerstate.
Alternatively, GR might not be a good description of The understanding of the initial state can start by infer-
the geometry close to horizons. BHs other than Kerr ringfromtheinspiralsignaltheimprintsofthestructure
ariseintheoriesbeyondGRwhicharemotivatedbyboth of the inspiralling objects. Putative structures will show
theoreticalargumentsandbyalternativesolutionstothe upinthewayeachoftheseobjectsreactstothegravita-
dark matter and the dark energy problems (for recent tional field created by the other, in other words, by their
reviews on strong-field tests of gravity in the context of TLNs. As we will show, the TLNs of all ECOs vanish in
GW astronomy, see Refs. [45, 46]). Arguably, the sim- theBHlimit, logarithmically. Thus, observationalupper
plest BHs arise in Einstein-Maxwell theory and are de- bounds on the TLNs can be converted into constraints
scribed by the Reissner-Nordstr¨om solution. Although on the compactness of the inspiralling objects.
astrophysical BHs are expected to be electrically neu- Throughout this work, we use G = c = 1 units and
tral [47], Reissner-Nordstr¨om BHs can be studied as a denote the Planck length by (cid:96) ≈1.6×10−33cm.
P
proxy of BHs beyond vacuum GR and could also emerge
C. Executive summary
1 A related, independent work dealing with tidal effects for bo-
son stars, conducted simultaneously to ours, is due to appear For the busy reader, in this section we summarize our
soon[18]. mainresults;possibleextensionsarediscussedinSec.VI.
3
We focus on spherically symmetric, static background BS models, where the two inspiralling objects are as-
geometries,andcomputetheTLNsundertheassumption sumed to be equal. In moderately optimistic scenarios,
that the only surviving tide at large distances is gravita- a GW detection of a compact-binary coalescence with
tional. In this setting, the TLNs can be divided into two LIGOcanplaceanupperboundontheTLNsofthetwo
classes according to their parity: an electric- or polar- objects at the level of kE ∼ 10, whereas the future Ein-
2
type, and a magnetic or axial-type, and each of these stein Telescope (ET) [69] can potentially improve this
sectors can in turn be expanded into a set of multipoles constraintbyalmostafactorofahundred. Interestingly,
labeled by an integer l. the future space interferometer LISA [70] has the abil-
Our main results are summarized in Table I and in ity to set much tighter constraints [cf. also Fig. 7] and
Fig.1,andarediscussedindetailintherestofthepaper. to rule out several candidates of supermassive ECOs. In
TableIliststhelowestquadrupolar(l=2)andoctupo- essence, both Earth and space-based detectors are able
lar (l = 3) polar and axial TLNs for various models of to discriminate even the most compact BSs, by impos-
ECOs in GR, and for some static BHs in other gravity ing stringent bounds on their TLNs. By contrast, as we
theories. Table I also compares the TLNs of these ob- show in Sec. V, only LISA is able to probe the regime
jects with the corresponding ones for a typical NS (cf. of very compact ECOs, describing geometries which are
also Table III in Appendix A). microscopic corrections at the horizon scale, for which
One of our main results is that the TLNs of several the compactness C =0.48 or higher.
ECOs display a logarithmic dependence in the BH limit,
i.e. when the compactness of the object approaches that
of a BH,
II. SETUP: TIDAL LOVE NUMBERS OF
STATIC OBJECTS
C :=M/r →1/2, (1)
0
whereM andr arethemassandtheradiusoftheobject.
0 Let us consider a compact object immersed in a tidal
As shown in Table I, this property holds for wormholes,
environment [4]. Following Ref. [8], we define the sym-
thin-shell gravastars, and for a simple toy model of a metricandtrace-freepolarandaxial2tidalmultipolemo-
static object with a perfectly reflecting surface [39, 41]. ments of order l as E ≡ [(l−2)!]−1(cid:104)C (cid:105)
It is natural to conjecture that this logarithmic behavior and B ≡[2(l+1a)(1l...−al2)!]−1(cid:104)(cid:15) Cbc 0a10a(cid:105)2,;aw3.h..earle
is model independent and will hold for any ECO whose a1...al 3 a1bc a20;a3...al
C istheWeyltensor,asemicolondenotesacovariant
abcd
exterior spacetime is arbitrarily close to that of a BH
derivative, (cid:15) is the permutation symbol, the angular
abc
in the r → 2M limit. This mild dependence implies
0 brackets denote symmetrization of the indices a and all
i
that even the TLNs of an object with r −2M ≈(cid:96) are
0 P traces are removed. The polar (respectively, axial) mo-
notextremelysmall,contrarilytowhatonecouldexpect.
ments E (respectively, B ) can be decomposed
Indeed,weestimatethatthedimensionlessTLNsdefined a1...al a1...al
inabasisofeven(respectively,odd)paritysphericalhar-
in Eq. (3) below are monics. We denote by Elm and Blm the amplitudes of
thepolarandaxialcomponentsoftheexternaltidalfield
kE,B ≈O(10−3), kE,B ≈O(10−4), (2)
2 3 with harmonic indices (l,m), where m is the azimuthal
for an ECO with r −2M ≈ (cid:96) and in the entire mass number (|m| ≤ l). The structure of the external tidal
0 P
range M ∈ [1,100]M . Note that, with the exception field is entirely encoded in the coefficients Elm and Blm
(cid:12)
of polar TLNs of boson stars, all TLNs of ultracompact (cf. Ref. [8] for details).
exotic objects listed in Table I have the opposite sign As a result of the external perturbation, the mass and
relative to the neutron-star case (cf. the discussion in current multipole moments3 (M and S , respectively)
l l
Refs. [66, 67] applied to a particular model). Negative of the compact object will be deformed. In linear per-
values of TLNs were also found previously for ultracom- turbation theory, these deformations are proportional to
pact anisotropic NSs [68]. the applied tidal field. In the non-rotating case, mass
Furthermore, we show that the TLNs of a charged BH (current) multipoles have even (odd) parity, and there-
in Einstein-Maxwell theory and of an uncharged static foretheyonlydependonpolar(axial)componentsofthe
BH in Brans-Dicke theory vanish, as in GR, whereas the tidal field.4 Hence, we can define the (polar and axial)
TLNsofaBHinChern-Simonsgravityarenon-zero,even
though the static BH solution to this theory is described
by the Schwarzschild metric. The results for Einstein-
Maxwell and Chern-Simons gravity were obtained with
2 It is slightly more common to use the distinction elec-
the assumption that there are no electromagnetic and tric/magnetic components rather than polar/axial. Since we
scalar tidal fields. As expected, the TLNs are propor- shalldiscussalsoelectromagneticfields,weprefertousethefor-
tional to the coupling constant of the theory so that any merdistinction.
3 We adopt the Geroch-Hansen definition of multipole mo-
constraint on them can be potentially converted into a
ments[71,72],equivalent[73]totheonebyThorne[74]inasymp-
test of gravity.
toticallymass-centeredCartesiancoordinates.
The accuracy with which GW detectors can estimate 4 This symmetry is broken if the compact object is spinning due
the TLNs of compact objects is shown in Fig. 1 for three to spin-tidal couplings. In such case, there exists a series of
4
������ �� ����
△ ������� □ □ □
□
× ������� □ 10
1000 □
□ □ ��������� □ 10
500 □□ □ △
%] □□□□□ △ %] 5△ □□□□□□□□□□□ %] 1 □
Λ|[ 100 △△△ Λ|[ Λ|[ × □ □ □
|σ/Λ 50△△△△△△△△△ |σ/Λ0.51× △ △ △ △ △ △ △ △ △ △ △ △ |σ/Λ0.10 △ △
× ×× × △ △
105 ×××××××××× 0.1 × × × × × × × × × × × × 0.01 × × ×
10 20 30 40 50 10 20 30 40 50 1 5 10 50 100
M[M ] M[M ] M[104×M ]
⊙ ⊙ ⊙
FIG. 1. Relative percentage errors on the average tidal deformability Λ for BS-BS binaries observed by AdLIGO (left panel),
ET (middle panel), and LISA (right panel), as a function of the BS mass and for different BS models considered in this work
(for each model, we considered the most compact configuration in the stable branch; see main text for details). For terrestrial
interferometers we assume a prototype binary at d = 100Mpc, while for LISA the source is located at d = 500Mpc. The
horizontaldashedlineidentifiestheupperboundσ /Λ=1. Roughlyspeaking,ameasurementoftheTLNsforsystemswhich
Λ
lie below the threshold line would be incompatible with zero and, therefore, the corresponding BSs can be distinguished from
BHs. Here Λ is given by Eq. (72), the two inspiralling objects have the same mass, and σ /Λ∼σ /kE.
Λ kE 2
2
TABLE I. Tidal Love numbers (TLNs) of some exotic compact objects (ECOs) and BHs in Einstein-Maxwell theory and modified
theories of gravity; details are given in the main text. As a comparison, we provide the order of magnitude of the TLNs for static NSs
withcompactnessC≈0.2(theprecisenumberdependsontheneutron-starequationofstate;seeTableIIIformoreprecisefits). ForBSs,
thetableprovidesthelowestvalueofthecorrespondingTLNsamongdifferentmodels(cf.Sec.IIIA)andvaluesofthecompactness. In
thepolarcase, thelowestTLNscorrespondtosolitonicBSswithcompactnessC ≈0.18orC ≈0.20(whentheradiusisdefinedasthat
containing99%or90%ofthetotalmass,respectively). Intheaxialcase,thelowestTLNscorrespondtoamassiveBSwithC ≈0.16or
C ≈0.2(againforthetwodefinitionsoftheradius,respectively)andinthelimitoflargequarticcoupling. ForotherECOs,weprovide
expressionsforverycompactconfigurationswherethesurfacer0 sitsatr0∼2M andisparametrizedbyξ:=r0/(2M)−1;thefullresults
areavailableonline[65]. IntheChern-Simonscase,theaxiall=3TLNisaffectedbysomeambiguityandisdenotedbyaquestionmark
[seeSec.IVCformoredetails]. NotethattheTLNsforEinstein-MaxwellandChern-Simonsgravitywereobtainedundertheassumption
ofvanishingelectromagneticandscalartides.
Tidal Love numbers
kE kE kB kB
2 3 2 3
NSs 210 1300 11 70
Boson star 41.4 402.8 −13.6 −211.8
Wormhole 4 8 16 16
ECOs 5(8+3logξ) 105(7+2logξ) 5(31+12logξ) 7(209+60logξ)
Perfect mirror 8 8 32 32
5(7+3logξ) 35(10+3logξ) 5(25+12logξ) 7(197+60logξ)
Gravastar 16 16 32 32
5(23−6log2+9logξ) 35(31−6log2+9logξ) 5(43−12log2+18logξ) 7(307−60log2+90logξ)
Einstein-Maxwell 0 0 0 0
Scalar-tensor 0 0 0 0
BHs
Chern-Simons 0 0 1.1α2CS 11.1α2CS?
M4 M4
TLNs as [6, 8]
(cid:114)
1l(l−1) 4π M
kE ≡− l,
l 2 M2l+1 2l+1E
l0 (3)
(cid:114)
3 l(l−1) 4π S
kB ≡− l ,
l 2(l+1)M2l+1 2l+1B
l0
selection rules that allow to define a wider class of “rotational”
TLNs [22, 23, 75, 76]. In this paper, we neglect spin effects to
leadingorder. where M is the mass of the object, whereas E (respec-
l0
5
tively, B ) is the amplitude of the axisymmetric5 com- III. TIDAL PERTURBATIONS OF EXOTIC
l0
ponent of the polar (respectively, axial) tidal field. The COMPACT OBJECTS
factor M2l+1 was introduced to make the above quan-
tities dimensionless. It is customary to normalize the Inthissectionwedescribesomerepresentativemodels
TLNs by powers of the object’s radius R rather than of ECOs and discuss their TLNs. Technical details are
by powers of its mass M. Here we adopted the latter given in the appendices.
non-standardchoice,sincetheradiusofsomeECOs(e.g.
BSs)isnotawelldefinedquantity. Thus,ourdefinitionis
related to those used by Hinderer, Binnington and Pois- A. Boson stars
son (HBP) [6, 8] through
(cid:18)R(cid:19)2l+1 Model Potential Maximummass
kE,B = kE,B . (4) V(|Φ|2) Mmax/M(cid:12)
lours M lHBP Minimal µ2|Φ|2 8(cid:16)10−11eV(cid:17)
Massive µ2|Φ|2+ α|Φ|4 5√α(cid:126)(cid:16)m0.S1GeV(cid:17)2
Modified theories of gravity and ECOs typi- Solitonic µ2|Φ|2(cid:20)1−42|σΦ02|2(cid:21)2 5(cid:104)10σ−012(cid:105)2(cid:16)m5S0m0GSeV(cid:17)
cally require the presence of extra fields which are
(non)minimally coupled to the metric tensor. Here we TABLE II. Scalar potential and maximum mass for the BS
shallconsidersomerepresentativeexampleofbothscalar models considered in this work. In our units, the scalar field
andvectorfields. Afulltreatmentofthisproblemwould Φ is dimensionless and the potential V has dimensions of an
require allowance for an extra degree of freedom, the ex- inverse length squared. The bare mass of the scalar field is
m := µ(cid:126). For minimal BSs, the scaling of the maximum
ternal scalar and electromagnetic (EM) applied fields. It S
massisexact. FormassiveBSsandsolitonicBSs,thescaling
is generically expected that, in astrophysical situations,
of the maximum mass is approximate and holds only when
the ratio of a putative external (scalar or vector) field
α(cid:29)µ2 and when σ (cid:28)1, respectively.
to the ordinary gravitational tidal field should be small. 0
Wewillthereforefocusonlyonsituationswheretheonly
BSsarecomplex6 bosonicconfigurationsheldtogether
surviving field at large distances is gravitational.
by gravity. In the simplest model they are solutions to
We expand the metric, the scalar field, and the
the Einstein-Klein-Gordon theory,
Maxwell field in spherical harmonics as presented in Ap-
pendix B. Since the background is spherically symmet- S =(cid:90) d4x√−g(cid:20) R −gab∂ Φ∗∂ Φ−V (cid:0)|Φ|2(cid:1)(cid:21) . (5)
ric, perturbationswithdifferentparityanddifferenthar- 16π a b
monic index l decouple. In the following we discuss the
BSs have been extensively studied in the past and have
polar and axial sector separately; due to the spherical
been proposed as BH mimickers and dark matter candi-
symmetry of the background, the azimuthal number m
dates, see e.g. Refs. [28, 29, 31, 78, 79].
is degenerate and we drop it.
BSs are typically classified according to the scalar po-
Finally, in order to extract the tidal field and the in-
tential in the above action; here we investigate three of
duced multipole moments from the solution, we have
the most common models: minimal BSs [80, 81], mas-
adopted two (related) techniques. The first one re-
sive BSs [82] and solitonic BSs [83]. The corresponding
lies on an expansion of the metric at large distances
scalarpotentialforthesemodelsandthemaximummass
[cf. Eqs. (B9) and (B10)] in terms of the multipole mo-
for non-spinning solutions are listed in Table II. A more
ments. The second technique relies on the evaluation of
comprehensivelistofBSmodelscanbefoundinRef.[29].
the Riemann tensor in Schwarzschild coordinates, whose
Depending on the model, compact BSs with masses
tidal correction is related to the total tidal field in the
comparable to those of ordinary stars or BHs require a
local asymptotic rest frame [19]. These two procedures certain range of the scalar mass m := µ(cid:126). For mini-
S
agreewitheachotherand–atleastinthecaseofECOs–
mal BSs, the maximum mass in Table II is comparable
the computation of the TLNs is equivalent to the case of
totheChandrasekharlimitforNSsonlyforanultralight
NSs [6, 8]. On the other hand, computing the TLNs of field with m (cid:46) 10−11eV. For massive BSs, the max-
S
BHs in extensions of GR presents some subtleties which
imum mass is of the same order of the Chandrasekhar
are discussed in Sec. IV.
6 If the scalar field is real, action (5) admits compact, self-
gravitating,oscillatingsolutionsknownasoscillatons[77]. These
solutions are metastable, but their decay time scale can largely
5 We consider only non-spinning objects, hence the spacetime is exceedtheageoftheuniverseandtheirpropertiesareverysim-
spherically symmetric and, without loss of generality, we can ilartothoseofBSs. WeexpectthattheTLNsofBSscomputed
definetheTLNsintheaxisymmetric(m=0)case. Clearly,this herearesimilartothoseofanoscillatonstar,althoughadetailed
propertydoesnotholdwhentheobjectisspinning[21,23,75]. computationisleftforfuturework.
6
limit if m ∼ 0.1GeV and the quartic coupling is large, features: (i) the exterior spacetime is described by the
S
α(cid:126) ∼ 1 [82]. Finally, solitonic BSs may reach massive Schwarzschild metric; (ii) the interior is either vacuum
(M (cid:38) M ) or supermassive (M (cid:38) 106M ) configura- or de Sitter and the tidal perturbation equations can be
(cid:12) (cid:12)
tions even for heavy bosons with m ∼ 500GeV if the solved for in closed form; (iii) simple junction or bound-
S
coupling parameter in their potential is σ (cid:46) 10−12 or ary conditions at the radius r of the object can be im-
0 0
σ (cid:46) 10−15, respectively [83]. For massive and solitonic posed to connect the perturbations in the interior with
0
BSs, the scaling of the maximum mass in Table II is ap- those in the exterior. As a result of these properties, the
proximateandvalidonlywhenα(cid:29)µ2andwhenσ (cid:28)1, TLNsofthesemodelscanbecomputedinclosedanalyt-
0
respectively. In our numerical analysis, we have consid- ical form. As we show, the qualitative features are the
ered α=104µ2 and σ =0.05, whereas the mass term µ same and – especially in the BH limit – do not depend
0
can be rescaled away (cf., e.g., discussion in Ref. [43]). strongly on the details of the models. Below, we present
Even though BSs have a wide range of compactness, explicitformulasfortheBHlimit,expressionsforgeneric
which depends basically on their total mass (cf. Fig. 9 compactness are provided online [65]. The details of the
inAppendixC),interactionsbetweenBSstypicallyleads computation are given in Appendix D.
toanetweightgain,clusteringoldBSsclosetothemass
peak[31],whichalsocoincideswiththepeakofcompact-
ness. 1. Wormholes
Thedetailsofthenumericalproceduretocomputethe
TLNs of a BS are presented in Appendix C. Figure 2
The simplest models of wormhole solutions consist in
shows the TLNs of the BS models presented above as a
taking two copies of the ordinary Schwarzschild solution
function of the total mass M, the latter being normal-
and remove from them the four-dimensional regions de-
izedbythetotalmassM ofthecorrespondingmodel.
max scribedbyr ≤r [35]. Withthisprocedure,weobtain
1,2 0
We only show static configurations in the stable branch,
two manifolds whose geodesics terminate at the timelike
i.e. withamasssmallerthanM (cf.discussioninAp-
max hypersurfaces
pendixC).ForminimalBSsandforl=2polarcase,our
results agree with those recently obtained in Ref. [84].
∂Ω ≡{r =r |r >2M} . (6)
1,2 1,2 0 0
In addition, we also present the results for l = 2 and
l = 3, for both axial and polar TLNs, and for the three The two copies are now glued together by identifying
BS models previously discussed. thesetwoboundaries,∂Ω =∂Ω ,suchthattheresulting
1 2
The behavior of the TLNs of BSs is in qualitative spacetime is geodesically complete and comprises of two
agreement with that of NSs. For a given BS model with distinct regions connected by a wormhole with a throat
a given mass, the magnitude of the polar TLN is larger at r =r . Since the wormhole spacetime is composed by
0
than that of an axial TLN with the same l. Further- two Schwarzschild metrics, the stress-energy tensor van-
more, in the Newtonian regime (M →0) the TLNs scale ishes everywhere except on the throat of the wormhole.
as klE ∼ C−(2l+1) and klB ∼ −C−2l. This scaling is in Thepatchingatthethroatrequiresathin-shellofmatter
agreement with the neutron-star case (cf. Ref. [8] and with surface density and surface pressure
Table III), whereas the sign of the axial TLNs is oppo-
site. Finally, all TLNs are monotonic functions of the 1 (cid:114) 2M 1 1−M/r
compactness, so that more compact configurations have σ =− 1− , p= (cid:112) 0 , (7)
smallertidaldeformability. Thephenomenologicalimpli- 2πr0 r0 4πr0 1−2M/r0
cations of these results are discussed in Sec. V.
whichimplythattheweakandthedominantenergycon-
ditions are violated, whereas the null and the strong en-
ergy conditions are satisfied when r < 3M [63]. To
0
B. Models of microscopic corrections at the
coverthetwopatchesofthespacetime, weusetheradial
horizon scale
tortoise coordinate r , which is defined by
∗
Several phenomenological models of quantum BHs in- dr (cid:18) 2M(cid:19)
=± 1− , (8)
troduce a Planck-scale modification near the horizon.
dr r
∗
In this section, we consider three toy models for micro-
scopic corrections at the horizon scale, namely a worm- where the upper and lower sign refer to the two sides
hole [35], a Schwarzschild geometry with a perfectly re- of the wormhole. Without loss of generality, we can
flective surface near the horizon [39, 41], and a thin- assume that the tortoise coordinate at the throat is
shell gravastar [33].7 These models have some common zero, r (r )=0, so that one side corresponds to r > 0
∗ 0 ∗
whereas the other side corresponds to r <0.
∗
In Fig. 3, we show the polar and axial TLNs with
l=2,3 as functions of ξ := r /(2M)−1. Interestingly,
0
7 Manyoftheseobjectsareunstableorrequireexoticmatterdis- inthiscasetheTLNshavetheoppositesigntothoseofa
tributions. Wewillnotbeconcernedwiththeseissueshere. NS. Furthermore, they vanish in the BH limit, i.e. when
7
FIG.2. Polar(toppanels)andaxial(bottompanels)TLNsforminimal,massiveandsolitonicBSs. Leftandrightpanelsrefers
tol=2andl=3,respectively. FormassiveandsolitonicBSswehaveconsideredα=104µ2 andσ =0.05,respectively. With
0
these values, the maximum mass scales approximately as shown in Table II. Numerical data are available online [65]. These
plots include only stars in the stable branch.
r →2M orξ →0. ThebehavioroftheTLNsintheBH forawormholeintheentiremassrangeM ∈[1,100]M .
0 (cid:12)
limit reads
4
kE ∼ , (9)
2 5(8+3logξ)
8
kE ∼ , (10)
3 105(7+2logξ)
16
kB ∼ , (11)
2 5(31+12logξ)
16
kB ∼ , (12) 2. Perfectly-reflective mirror
3 7(209+60logξ)
(cid:16) (cid:17)
where we have omitted subleading terms of O ξ .
(logξ)2
Ontheotherhand,intheNewtonianlimitwegetkE,B ∼ Thermodynamical arguments suggest that any hori-
l
C−(2l+1). Interestingly, while the scaling for polar TLNs zonless microscopic model of BH should act as a mirror,
at least for long wavelength perturbations [39, 41]. Mo-
agrees with that of NSs (cf. Table III), that for the axial
tivated by this scenario, we consider a Schwarzschild ge-
TLNs is different.
ometrywithaperfectmirroratr =r >2M andimpose
ThelogarithmicdependenceoftheTLNsisveryinter- 0
DirichletboundaryconditionsontheRegge-Wheelerand
esting, because it implies that the deviations from zero
Zerilli functions, for the axial and polar sector, respec-
(i.e., from the BH case) are relatively large even when
tively. Thus, our strategy is to consider the station-
the throat is located just a Planck length away from the
would-be horizon r −2M ∼ (cid:96) ≈ 1.6×10−33cm. In ary limit of generically dynamical perturbations (in the
0 P
Fourier space, where ω is the frequency of the perturba-
this case, the above results yield
tion) of a Schwarzschild geometry.
kE ≈−3×10−3, kB ≈−6×10−3,
2 2
(13)
kE ≈−4×10−4, kE ≈−9×10−4 , The final result, in the ξ → 0 limit, reads (cf. Ap-
3 3
8
120
0.050 200 0.050
100 0.010
0.010
0.005
150 0.005
80
0.001
5.×10-4 0.001
60 10-35 10-25 10-15 10-5 10-35 10-25 10-15 10-5
100
40
50
20
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
FIG. 3. The l=2 and l=3, axial- and polar-type TLNs for FIG. 4. TLNs for a toy model of Schwarzschild metric with
a stiff wormhole constructed by patching two Schwarzschild a perfectly reflective surface at r =r0 >2M. The TLNs are
spacetimes at the throat radius r=r0 >2M. The TLNs are all negative and vanish in the BH limit, r0 → 2M. Close to
negativeandallvanishintheBHlimit,r →2M. Thelatter theBHlimit,thepolar-andaxial-typeLovenumbersforthe
0
is better displayed in the inset. same multipolar order are almost identical, as shown in the
inset.
pendix D for details)
Among thin-shell gravastars, we consider the simplest
case where the background metric is Eq. (B2) with,
8
kE ∼ ≈−6×10−3, (14)
2 5(7+3logξ) (cid:40)
1− 2M r >r
kE ∼ 8 ≈−9×10−4, (15) eΓ =e−Λg = 1−2Cr r2 r <r0 . (18)
3 35(10+3logξ) r2 0
0
32
kB ∼ ≈−6×10−3, (16) In this model, the thin shell is described by a fluid with
2 5(25+12logξ)
zero energy density and negative pressure [66]. By ex-
32 tending the formalism developed in Ref. [67] (cf. also
kB ∼ ≈−9×10−4, (17)
3 7(197+60logξ) Refs. [66, 85]), it is easy to compute the TLNs of this
solution. In the BH limit, the computation derived in
wherethelaststepisevaluatedatr0−2M ∼(cid:96)P ≈1.6× Appendix D yields8
10−33cm and it is roughly valid in the entire mass range
16
M ∈[1,100]M(cid:12)duetothemildlogarithmicdependence. kE ∼ ≈−4×10−3, (19)
We note that also for this model all TLNs are negative 2 5(23−6log2+9logξ)
(i.e., they have the opposite sign relative to the neutron- 16
kE ∼ ≈−6×10−4, (20)
star case) and that kE,B ∼ C−(2l+1) in the Newtonian 3 35(31−6log2+9logξ)
l
limit. The TLNs kE,B for this model as functions of the 32
l kB ∼ ≈−4×10−3, (21)
compactness are shown in Fig. 4. 2 5(43−12log2+18logξ)
32
kB ∼ ≈−6×10−4.(22)
3 7(307−60log2+90logξ)
3. Thin-shell gravastars
As already noted in Ref. [66], the Newtonian regime of a
gravastar is peculiar due to the de Sitter interior; conse-
For completeness, here we briefly consider the case of
quently, the TLNs scale as kE,B ∼ −C−2l. In this case,
another ECO, namely gravastars [33]. The interior of l
the scaling of the polar TLNs is different from that of
these objects is described by a patch of de Sitter space,
which is smoothly connected to the Schwarzschild exte-
rior through an intermediate region filled with a perfect
fluid. A particularly simple model is the so-called thin-
8 This result corrects the computation performed in Ref. [66],
shellgravastar[34],inwhichthethicknessoftheinterme-
whichisflawedduetothefactthatitdoesnotimposethecorrect
diate region shrinks to zero. Remarkably, these models boundaryconditionsacrosstheshell. Forastiffequationofstate,
are simple enough that the TLNs can be computed ana- the correct boundary conditions read [[K]]=0=[[dK/dr∗]] as
lytically [66, 67]. derivedinAppendixDandinRef.[67].
9
an ordinary NS and of the other models of microscopic rule of BHs in GR is discussed in Sec. V.
corrections at the horizon scale, whereas the scaling of
the axial TLNs is the same as that for ordinary NSs and
BSs. IV. TIDAL PERTURBATIONS OF BHS
Interestingly, also in the gravastar case, the axial and BEYOND VACUUM GR
polar TLNs have a logarithmic behavior in the BH limit
andtheyarenegative,asdiscussedinRefs.[66,67]forthe In this section, we discuss the TLNs of BHs in other
polarcaseonly. ThebehaviorofkE,B asfunctionsofthe theories of gravity. Technical details are given in Ap-
l
compactness is shown in Fig. 5. The quadrupolar polar- pendix B.
type TLNs for more generic thin-shell gravastar models
are presented in Ref. [67].
A. Scalar-tensor theories
We start with scalar-tensor theories, which generically
40 give rise to stationary BH solutions which are identi-
0.010
0.005 cal to those of GR [46, 49, 50]. Therefore, the back-
ground solution which we deal with is still described by
30 the Schwarzschild geometry. In the Jordan frame, ne-
0.001
5.×10-4 glecting the matter Lagrangian, the simplest example of
10-35 10-25 10-15 10-5
scalar-tensor theory is described by the Brans-Dicke ac-
20
tion (cf., e.g., Ref. [46])
1 (cid:90) √ (cid:16) ω (cid:17)
10 S = d4x −g ΦR− BD∂ Φ∂µΦ , (23)
16π Φ µ
where ω is a dimensionless coupling constant and Φ
0 BD
is a scalar field characteristic of the theory. Action (23)
0.0 0.2 0.4 0.6 0.8 1.0
yields the equations of motion,
(cid:18) (cid:19)
ω 1 1
FIG.5. TLNsforathin-shellgravastarwithzeroenergyden- Gµν = ΦB2D ∂µΦ∂νΦ− 2gµν∂λΦ∂λΦ + Φ∇µ∇νΦ,
sityasafunctionofthecompactness. Moregenericgravastar
models are presented in Ref. [67]. The TLNs are all neg- (24)
ative and vanish in the BH limit, r0 → 2M. Similar to the (cid:3)Φ=0. (25)
perfectly-reflectivemirrorcase,thepolar-andaxial-typeLove
numbers for the same multipolar order coincide in the BH As mentioned above, the background solution is
limit, as shown in the inset. Schwarzschild with a vanishing scalar field.
Following the procedure described in Sec. II, we con-
sider metric perturbations given by Eqs. (B3) and (B4)
for the polar and axial sector, respectively, and a scalar
4. On the universal BH limit field perturbation given by Eqs. (B6) and (B7). Since
the scalar perturbations are even-parity, axial gravita-
It is remarkable that the models described above dis- tionalperturbationsdonotcoupletothem,implyingthat
play a very similar behavior in the BH limit, when the this sector is governed by equations identical to those of
radius r →2M, cf. Table I. Indeed, although all TLNs vacuum GR. Therefore, all axial-type TLNs of a non-
0
vanish in this limit, they have a mild logarithmic de- rotating BH in Brans-Dicke gravity are zero, klB =0.
pendence. On the light of our results, it is natural to On the other hand, in the polar sector, scalar per-
conjecture that this logarithmic dependence is a generic turbations can be obtained from Eq. (25) by using the
featureofultracompactexoticobjects,andwillholdtrue decomposition in Eqs. (B6) and (B7),
foranyECOwhoseexteriorspacetimeisarbitrarilyclose
2(r−M)δφ(cid:48)−l(l+1)δφ
to that of a BH in the r0 →2M limit. δφ(cid:48)(cid:48)+ =0, (26)
Due to this mild dependence, the TLNs are not ex- r(r−2M)
tremely small, as one would have naively expected if the
The solution which is regular at the horizon is
scalingwithξ werepolynomial. Indeed,inthePlanckian
case(r0−2M ≈(cid:96)P)theorderofmagnitudeoftheTLNs (cid:16) r (cid:17)
is the same for all models and it is given by Eq. (2). In δφ=ClPl M −1 , (27)
particular,theTLNsofPlanckianECOsareonlyfiveor-
ders of magnitude smaller than those a typical NS. The where P is a Legendre polynomial, C is an integration
l l
detectability of these deviations from the “zero-Love” constant, and we have expanded the scalar field as in
10
Eq. (B6) with Φ(0) = 0. By comparing the above ex- Because the background is electrically charged, gravita-
pression with the scalar-field expansion in Eq. (B13), we tional and EM perturbations are coupled to each other.
conclude that C ∝ES and that the induced scalar mul- To compute the tidal deformations, we expand the met-
l l
tipoles Φ are zero. Therefore, Eq. (27) represents an ric as in Eqs. (B3) and (B4) and the Maxwell field as in
l
externalscalartidalfieldandthe“scalarTLN”areiden- Eqs.(B5)and(B8). Asbefore,weconsiderthepolarand
tically zero. the axial sectors separately.
Althoughwewishtofocusongravitationaltidalfields,
it is instructive to investigate the role of a scalar tide in
scalar-tensortheory. BysubstitutingEq.(27)inEq.(24) 1. Polar TLNs
weobtainaninhomogeneousdifferentialequationforH ,
0
which one of the polar perturbations of the metric [cf. The polar functions of the metric are coupled to the
Eq. (B3)]. For l=2, we can identify C ≡−2M2ES and EM function u through the field equations. In the
2 3 2 1
we get Lorenz gauge, we find the following coupled equations,
2(r−M) 2(cid:0)2M2−6Mr+3r2(cid:1) (2) 4Q (1)
H0(cid:48)(cid:48) + r(r−2M)H0(cid:48) − r2(r−2M)2 H0 D1 H0+ r3−2Mr2+Q2rD1 u1 =0, (35)
4M2(cid:0)2M2−6Mr+3r2(cid:1) D(2)u + QD(1)H =0, (36)
= ES. (28) 2 1 r 2 0
3r2(r−2M)2 2
where we defined the operators,
The above equation can be solved analytically. The so-
lution which is regular at the horizon reads D(2) = d2 − 2(M −r) d + 1 (cid:2)Q2r(4M −(η−2)r)
1 dr2 r2f dr r6f2
H =−r2E +2MrE − 2M2ES. (29) −r2(cid:0)4M2−2ηMr+ηr2(cid:1)−2Q4(cid:3) ,
0 2 2 3 2
d (cid:0)Q2−r2(cid:1)
The induced quadrupolar moment is zero, and therefore D(1) = + ,
kE = 0, just as in the GR case. It is straightforward 1 dr r(r(r−2M)+Q2)
2
to show that this result generalizes to higher multipoles, d2 4Q2−ηr2 d
(2)
D = + ,
kE = 0. In conclusion, although in Brans-Dicke theory 2 dr2 r4f dr
l
the BH metric perturbations depend on scalar tides, all d 2(cid:0)Mr−Q2(cid:1)
TLNs of a static BH vanish, as in the case of GR. D(1) = + ,
2 dr r3f
with η := l(l+1). This system allows for a closed-form
B. Einstein-Maxwell solution. Forsimplicity,weimposetheabsenceofelectric
tidal fields, which requires that the function u does not
1
We consider Reissner-Nordstr¨om BHs, which are the containr3-termsatlargedistance[cf.Eq.(B11)]. Inthis
unique static solution to Einstein-Maxwell theory, al- case, the regular solution at the horizon for l=2 reads
though our results are valid for any U(1) field minimally Hl=2 =−E r2f, (37)
0 2
coupled to gravity, as in the case of dark photons or
E r2Qf
the hidden U(1) dark-matter sector [48]. The Einstein- ul=2 =− 2 . (38)
1 2
Maxwell field equations read
Due to the gravito-EM coupling, an external tidal field
G =8πT , (30) induces a Maxwell perturbation which is proportional to
µν µν
∇ Fµν =0, (31) the BH charge Q. A simple comparison between the
µ
above results and the expansions in Eqs. (B9) and (B11)
where Fµν =Aν,µ−Aµ,ν is the Maxwell tensor and shows that the multipole moments are all vanishing.
(cid:18) (cid:19) Although the full solutions for l > 2 are cumbersome,
1 1
T = gµγF F − F Fµνg , (32) it can be shown that for any l>2 the large-distance ex-
αβ 4π αµ βγ 4 µν αβ
pansion of the solutions for H and u which are regular
0 1
is the stress-energy tensor of the EM field. The back- atthehorizonistruncatedatthe1/r termforanyl,and
ground spacetime is the well-known Reissner-Nordstr¨om it is an exact solution of the coupled system. Therefore,
metric, whose line element reads as in Eq. (B2) with the above result directly extends to any l, and we ob-
tainthattheTLNsofachargedBHarezerointhepolar
2M Q2 sector, kE =0, like in the Schwarzschild case.
eΓ =e−Λg =1− + ≡f(r), (33) l
r r2
where M and Q denote the mass and the charge of the
2. Axial TLNs
BH, respectively. The background Maxwell 4-potential
reads
ThecalculationsforgravitationalaxialTLNsandmag-
(0)
A =(−Q/r,0,0,0). (34) neticTLNsaresimpler. Inthiscaseweconsidertheaxial
µ
