Gravitational waves from extreme-mass-ratio systems in astrophysical environments Vitor Cardoso1 2Kyriakos Destounis3 4 5Francisco

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Gravitational waves from extreme-mass-ratio systems

in astrophysical environments

Vitor Cardoso,1, 2 Kyriakos Destounis,3, 4, 5 Francisco

Duque,2Rodrigo Panosso Macedo,6and Andrea Maselli7, 8

1Niels Bohr International Academy, Niels Bohr Institute,

Blegdamsvej 17, 2100 Copenhagen, Denmark

2CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico – IST,

Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal

3Dipartimento di Fisica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy

4INFN, Sezione di Roma, Piazzale Aldo Moro 2, 00185, Roma, Italy

5Theoretical Astrophysics, IAAT, University of T¨ubingen, 72076 T¨ubingen, Germany

6STAG Research Centre, University of Southampton,

University Road SO17 1BJ, Southampton, UK

7Gran Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy

8INFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy

We establish a generic, fully-relativistic formalism to study gravitational-wave emission by

extreme-mass-ratio systems in spherically-symmetric, non-vacuum black-hole spacetimes. The po-

tential applications to astrophysical setups range from black holes accreting baryonic matter to those

within axionic clouds and dark matter environments, allowing to assess the impact of the galactic

potential, of accretion, gravitational drag and halo feedback on the generation and propagation of

gravitational waves. We apply our methods to a black hole within a halo of matter. We ﬁnd ﬂuid

modes imparted to the gravitational-wave signal (a clear evidence of the black-hole fundamental

mode instability) and the tantalizing possibility to infer galactic properties from gravitational-wave

measurements by sensitive, low-frequency detectors.

Introduction. The birth of gravitational-wave

(GW) astronomy ushered in a new era in gravita-

tional physics and high-energy astrophysical phe-

nomena [1,2]. GWs carry unique information

about compact objects, most notably black hole

(BH) systems, and grant us access to exquisite

tests of the gravitational interaction in the strong

ﬁeld, highly dynamical regime [3–9].

They also bear precious information about the

environment where compact binaries live [10–14].

This knowledge is important per se, and may in-

form us on how compact binaries are formed [15] or

how BHs grow and evolve over cosmic times [16].

In addition, GWs are sensitive to accretion disk

properties [17] and even on fundamental aspects,

such as the existence of dark matter spikes in galac-

tic centers [18–21]; on possibly new fundamen-

tal degrees of freedom that can condense around

spinning BHs [22,23]; and ﬁnally on the nature

and existence of BHs, as well as whether they are

well described by the Kerr family, a quest which

demands environmental eﬀects to be disentangled

from purely gravitational ones.

The above questions require a precise modeling

of compact binaries in a fully-relativistic setting.

Unfortunately, the state-of-the-art adopts at least

one of the following approximations: a slow-motion

quadrupole formula to estimate GW emission and

the dynamics [24–27], Newtonian dynamical fric-

tion, or considers vacuum backgrounds. Recent

attempts to reﬁne the analysis by including some

relativistic eﬀects indicate that these can have a

signiﬁcant impact on the conclusions one makes

regarding detectability and parameter estimation

[21,28,29].

Here – based on classical works on pertur-

bation theory [30–37] – we develop a generic,

fully-relativistic formalism to handle environmen-

tal eﬀects in extreme-mass-ratio inspirals (EMRIs)

in spherically-symmetric, but otherwise generic,

backgrounds. These are inherently relativistic sys-

tems, expected to populate galactic centers and be

observable with the upcoming space-based LISA

mission [38–41], and for which Newtonian approx-

imations are ill-suited. Our framework is able to

treat GW generation and propagation, but also in-

cludes matter perturbations and therefore is able

to capture other environmental eﬀects, such as dy-

namical friction [28,29], accretion and halo feed-

back, and will be important to understand mode

excitation or depletion of accretion disks, and even

viscous heating in these systems. We use geometric

units G=c= 1 everywhere.

Setup. We wish to study a static, spherically-

symmetric spacetime describing a BH immersed in

some environment, like an accretion disk or a dark

matter halo, with line element,

ds2=g(0)

µν dxµdxν=−a(r)dt2+dr2

b(r)+r2dΩ2,(1)

where dΩ2is the line element of the 2-sphere, and

characterized by a (anisotropic) stress tensor [42]

Tenv(0)

µν =ρuµuν+prkµkν+ptΠµν ,(2)

arXiv:2210.01133v2 [gr-qc] 21 Nov 2022

where ρis the total energy density of the ﬂuid,

prand ptare its radial and tangential pressure

respectively, uµthe 4-velocity of the ﬂuid, kµa

unit spacelike vector orthogonal to uµ, such that

kµkµ= 1 and uµkµ= 0, and Πµν =gµν +uµuν−

kµkνis a projection operator orthogonal to uµand

kµ(environmental quantities are hereafter denoted

with a superscript “env”). The functions a(r) and

b(r) are to be determined by the physics; to pre-

vent clustering throughout the text we drop the

(t, r) dependence from all functions, unless nec-

essary. We leave them general for most of the

main body, but specialize to the physics of a super-

massive BH surrounded by a halo of matter when

necessary. The corresponding solution, which we

will term galactic BHs (GBHs), was recently de-

rived [43] and is characterized by the BH mass

MBH, halo mass Mand its spatial scale a0(see

also [44,45] for generalizations and applications).

We now envision a secondary object of mass mp

(a star, asteroid or stellar-mass BH for example)

orbiting the above primary BH and causing pertur-

bations to the geometry and matter stress tensor,

gµν =g(0)

µν +g(1)

µν , T env

µν =Tenv(0)

µν +Tenv(1)

µν ,(3)

where a superscript “(1)” denotes perturbations.

The spherical symmetry of the background al-

lows for a separation of variables in the ﬁrst-order

quantities, expanding into tensor spherical har-

monics, classiﬁed as axial and polar, according

to their properties under parity [46–48]. In the

Regge-Wheeler gauge [35,36,46–48], these are

deﬁned by radial functions h`m

0, h`m

1(axial) and

K`m, H`m

0, H`m

1, H`m

2(polar), and a set of angular

basis functions [35,36,49].

The perturbations induced by the orbiting ob-

ject on the environment are known once its pres-

sure, density and velocity ﬂuctuations are com-

puted. These can also be expanded in harmonics.

For example, a scalar quantity X=pt, pr, ρ will

have a perturbation X(1) expanded as

X(1) =

∞

`=2

m=−`

δX`m(t, r)Y`m(θ, φ),(4)

with Y`m(θ, φ) being the standard spherical har-

monics on the two-sphere. A similar procedure is

applied to any vector quantity.

Finally, a barotropic equation of state provides

a further relation between pressure, density varia-

tions and the medium’s speed of sound via

δp`m

t,r (t, r) = c2

st,r (r)δρ`m(t, r).(5)

Here, csr(r) and cst(r) are, respectively, the radial

and transverse sound speeds. The explicit per-

turbed equations are shown in the Supplemental

Material (see also Ref. [33] if a=b).

With the above procedure, perturbations to the

environmental stress-tensor are completely charac-

terized. The source of these perturbations is mod-

eled as a pointlike object with stress tensor

Tµν

p=mpZuµ

puν

δ(4) xµ−xµ

p(τ)

√−gdτ , (6)

where mpis the mass of the secondary, τits proper

time, xµ

p(τ) its world-line and uµ

p=dxµ

p/dτ its

4-velocity. This stress-energy tensor can also be

decomposed in terms of the angular basis [48,49],

thereby separating the equations of motion. We

will always assume that the pointlike secondary is

on a geodesic of the background spacetime (1), and

use this to simplify the equations of motion.

Evolution equations. The perturbations are de-

scribed by wave equations with a principal part ex-

pressed in terms of the operator Lv=v2∂2/∂r2

∗−

∂2/∂t2,with vthe ﬁeld’s characteristic speed

of propagation. Speciﬁcally, axial perturbations

propagate with the speed of light v= 1 and are

simply described in terms of a master variable

χ=h`m

1√ab/r, governed by the equation

L1χ−Vaxχ=Sax ,(7)

Vax =a

r2`(`+ 1) −6m(r)

r+m0(r),(8)

with m(r) = r(1 −b(r)) /2, the tortoise coordi-

nate is deﬁned by dr∗/dr =√ab, and the source

term depends on the motion of the point particle

(explicit expressions for circular motion are shown

in the Supplemental Material). The polar sector

can be re-expressed as a system of 3 “wavelike”

equations for ~

φ= (S, K, δρ)

φ=ˆ

φ,r∗+ˆ

φ+~

S1,(9)

with S=a/r (H0−K), and ˆ

φ=

L1φ1,L1φ2,Lcsrφ3, i.e., φ1, φ2have character-

istic velocity v= 1, and φ3has v=csr.

We also study perturbations in the frequency do-

main by Fourier-transforming the evolution equa-

tions. Instead of a second-order system for the po-

lar sector, we worked instead with the ﬁrst-order

system

dr =ˆ

α~

ψ+~

S2,(10)

with ~

ψ= (H1, H0, K, W, δρ), and Wa ﬂuid ve-

locity quantity. The matrices ˆ

A,ˆ

B,ˆ

α, as well as

source vectors ~

Siare shown in the Supplemental

Material. Particle contributions enter as a source

term ~

S1,~

S2for the metric variables.

We solve the above problem with two indepen-

dent codes, based on diﬀerent approaches, one

BCs

No BCs

0 500 1000 1500 2000

10-5

10-4

0.001

0.010

0.100

(t-tarrival )/MBH

δρ

FIG. 1. Evolution of δρ in a Schwarzschild background

with csr= 0.9, cst= 0 with diﬀerent boundary condi-

tions imposed. tarrival corresponds to the time of ar-

rival of the ﬁrst direct signal. When δρ is left free at the

horizon, an oscillatory tail sets in at late times, consis-

tent with that of a scalar ﬁeld of mass µeﬀ csr. Instead,

when Dirichlet conditions are imposed at some cutoﬀ

radius rcut (here rcut = 3MBH), we ﬁnd a universal

power-law decay independent of rcut and csr.

in the time and the other in the frequency do-

main. Both use a smoothed distribution to ap-

proximate the point particle, √2πσδ(r−rp) =

exp −(r−rp)2/(2σ2)where the width σis var-

ied to assess numerical convergence. In the axial

sector, the time domain code follows Ref. [50,51]

which places the outer boundary condition at fu-

ture null inﬁnity by using the same hyperboloidal

layers employed there. In the polar sector, the

equations are solved in the usual radial tortoise

cooordinate with physical boundaries placed suﬃ-

ciently far, so that the physical quantities are ex-

tracted within the wave equation’s causality do-

main and in a near vacuum region. For example,

if we evolve the system for t= 103MBH and extract

at rext

∗= 500MBH, then the outer boundary should

be placed further than rout

∗= 103MBH to prevent

any signal from being reﬂected back and aﬀect the

ﬁeld values at the extraction radius. Unless stated

otherwise, we use rext

∗= max{102a0,103MBH}

as extraction radius in the time domain code for

the polar sector. The frequency domain code fol-

lows the framework from Refs. [52] in both sec-

tors, with outer physical boundaries placed at

rext = max 103/Ωp,2a0, with Ωpthe orbital an-

gular frequency. For the gravitational perturba-

tions, we impose usual outgoing boundary condi-

tions there and vanishing Dirichlet boundary con-

ditions for the matter variables. The results from

all codes agree within the numerical error when

varying these parameters. Once the metric vari-

ables are computed, ﬂuxes in GWs can be calcu-

lated. Our two codes are made freely available to

the community [53,54].

Boundary conditions and sound speed. Envi-

ronments cause the presence of density waves that

couple to gravity. To understand their asymptotic

behavior, it’s suﬃcient to examine a vacuum BH

background of mass MBH, to which the ﬁeld equa-

tions reduce very far or very close to the hori-

zon. For constant sound speeds, with the ansatz

δρ =rα(r−2MBH)βΨ, we ﬁnd that Ψ is governed

by the wave equation LcsrΨ−VΨ = 0 for

α=1

4−5 + 1+4c2

sr, β =−3

4−1

4c2

,(11)

with V = O(r−2) at inﬁnity and V = 1−c2

8c2

srMBH 2

at the horizon. The explicit form of V and wave

equation for Ψ are identical to that obtained in

Ref. [34] for isotropic ﬂuids, with a suitable change

of wavefunction H, once we identify csr=cst.

Thus, close to the horizon density ﬂuctuations

propagate as an eﬀectively massive scalar of mass

µeﬀ =1−c2

8c2

srMBH . A rigorous analysis of the wave

equation above is required to understand all the

details of the density waves around BHs; however,

based on knowledge of massive ﬁelds around BHs

[55–57], we expect an intermediate-time power-law

tail of the form Ψ ∼t−5/6sin (µeﬀ csr), caused

by back-scattering in the near-horizon region and

probably giving way to another power-law behav-

ior dictated by the asymptotic region far from

the BH [57]. Our numerical results in Fig. 1–

for initial conditions δρ = 0 , ∂tδρ = exp(−(r∗−

100MBH)2/2), extracted at r∗= 1000MBH – sup-

port this claim. We ﬁnd excellent agreement with

an oscillatory term sin (µeﬀ csr) and decay t−5/6.

We ﬁnd a similar behavior for other values of csr.

Conﬁgurations with a matter proﬁle that van-

ishes at the horizon and spatial inﬁnity, have sound

speeds expected to vanish asymptotically. For

sound speed proﬁles that vanish as a power-law at

the boundaries, we ﬁnd that regular density ﬂuctu-

ations δρ must satisfy Dirichlet conditions. We im-

plement this restriction keeping csrconstant every-

where, but imposing Dirichlet conditions on ﬂuid

variables at some cutoﬀ radius rcut close to the

BH. It is now possible to prove that the late time

asymptotics is governed not by the near-horizon

but by the large-rasymptotic behavior and that

the ﬁeld should decrease as t−3,independently of

the multipole `[55]. This is seen clearly in our sim-

ulations in Fig. 1. The direct signal is followed by

a universal power-law tail δρ ∼t−3, independently

of cutoﬀ radius rcut and sound speed csr.

Environment and spectral stability. From

now on, we always work with vanishing sound

speeds at the boundaries. It is clear from the

above that there are two characteristic speeds in

the problem, the radial sound speed csrand the

light speed. Accordingly, and because the polar

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Gravitationalwavesfromextreme-mass-ratiosystemsinastrophysicalenvironmentsVitorCardoso,1,2KyriakosDestounis,3,4,5FranciscoDuque,2RodrigoPanossoMacedo,6andAndreaMaselli7,81NielsBohrInternationalAcademy,NielsBohrInstitute,Blegdamsvej17,2100Copenhagen,Denmark2CENTRA,DepartamentodeFsica,InstitutoSuper...

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Gravitational waves from extreme-mass-ratio systems in astrophysical environments Vitor Cardoso1 2Kyriakos Destounis3 4 5Francisco

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