Parametrized quasi-normal mode framework for non-Schwarzschild metrics Nicola Franchini1 2 3 4and Sebastian H. Völkel1 2 5 1SISSA Via Bonomea 265 34136 Trieste Italy and INFN Sezione di Trieste

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Parametrized quasi-normal mode framework for non-Schwarzschild metrics

Nicola Franchini1, 2, 3, 4 and Sebastian H. Völkel1, 2, 5

1SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste

2IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy

3Université Paris Cité, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France

4CNRS-UCB International Research Laboratory, Centre Pierre Binétruy, IRL2007, CPB-IN2P3, Berkeley, US

5Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-14476 Potsdam, Germany

In this work we comment in more detail on what happens to the parametrized framework ﬁrst

presented by Cardoso et al. when there are departures from the Schwarzschild background metric, as

well as possible deviations in the “dynamics”. We treat possible deviations in the background metric

with additional coeﬃcients with respect to the original works. The advantages of this reformulation

are clear when applied to a parameter estimation problem, since the coeﬃcients are always real,

and many of them do not depend on the overtone number and angular momentum of the frequency,

thus eventually reducing the total amount of parameters to be inferred.

I. INTRODUCTION

Ongoing and future eﬀorts of gravitational wave de-

tectors provide unprecedented access to test general rel-

ativity (GR) in the strong ﬁeld regime [1–4]. Among the

several exciting possibilities, one is to perform black hole

spectroscopy in order to verify the validity of the require-

ments of the no-hair theorem and also general relativity

itself [5–7]. Our current understanding of binary black

hole mergers predicts that the ﬁnal stage of such events

should be described by a superposition of linear pertur-

bations of the background, the so-called quasi-normal

modes [8–10], see also Refs. [11–14] for reviews.

The spectrum of quasi-normal modes is sensitive to

the background perturbed metric, but also to the dy-

namical behaviour of the theory in which the perturba-

tions are excited. The purpose of this work is to pro-

vide a framework which allows one to study quasi-normal

modes in spherical symmetry for metrics which are dif-

ferent from the Schwarzschild metric when the modiﬁ-

cations are small. Our framework follows a prescription

similar to the one presented by Cardoso et al. [15] and

McManus et al. [16]. In these works, a modiﬁed (sys-

tem) of wave equations has been introduced and the cor-

responding quasi-normal mode spectrum has been com-

puted up to quadratic order for a set of small deviation

parameters. We explicitly show how possible deviations

in the background metric could aﬀect the quasi-normal

frequencies, in a diﬀerent fashion compared to [15,16].

While the general idea is similar, our approach allows for

a better description of non-Schwarzschild backgrounds,

purely real deviation parameters in the framework, as

well as a diﬀerentiation between the parameters that de-

pend on the angular momentum of the frequencies and

those which do not.

This work is structured as follows. In Sec. II we intro-

duce the metric dependent parametrized framework. We

apply it to two diﬀerent cases in Sec. III. Conclusions are

discussed in Sec. IV. Throughout this work we use units

in which G=c= 1.

II. THEORETICAL FRAMEWORK

Let us start by qualitatively reviewing how gravita-

tional perturbations in gravity theories can be studied.

In general, one assumes a linear perturbation of the met-

ric as

gab ≃gab +hab ,(1)

for which the background metric gab and the perturba-

tion metric hab satisfy the 0th order and 1st order equa-

tions of motion, respectively. They are obtained from the

linear expansion of the full system of equations

Gab ≃Gab +Hab = 0 ,(2)

where Gab are the equations of motion of a given grav-

ity theory. Let us assume that the background metric

is given by some non-GR metric. We remark that we

do not know a priori which equations of motion this

metric is satisfying. However, as long as it is pertur-

batively close, with respect to some parameter δ, to

the Schwarzschild/Kerr metric, it must satisfy Einstein’s

equations of GR up to some error linear in δ,

GGR

ab [gcd] = 0 + O(δ).(3)

The presence of the O(δ)term can be interpreted as the

fact that the equations of motion of a new theory whose

static solution is approximated by the metric gab must

have diﬀerent dynamics, not captured with the GR equa-

tions by a linear factor.

We can conjecture that the ﬁrst order equations share

the same property. For GR, one would have

HGR

ab hcd;gSch

cd = 0 ,(4)

but when gSch

cd is substituted by gcd, the equations become

HGR

ab [hcd;gcd] = 0 + O(δ).(5)

One must bear in mind that this notation generically

includes the 0th and 1st order metric as well as their

derivatives.

arXiv:2210.14020v2 [gr-qc] 2 Oct 2023

Let us now take into account a spherically symmetric

background metric deﬁned as follows

gabdxadxb=−f(r)A(r)dt2+B2(r)

f(r)A(r)dr2+r2dΩ ,(6)

with f(r)=1−rH/r,rHbeing the location of the

event horizon. Then, we can expand the functions A

and Bwith a set of real coeﬃcients a(k)

M, b(k)

Min a post-

Newtonian fashion [17]

A(r) = 1 + δ

k=1

a(k)

MrH

rk

,(7)

B(r) = 1 + δ

k=1

b(k)

MrH

rk

.(8)

We make the series terminate at some ﬁnite number K

starting from k= 1 to ensure asymptotic ﬂatness.

From the prescription suggested by passing from

Eq. (4) to Eq. (5) one could guess the following form

for the master perturbation equation

fZ d

drfZ dΦ

dr+hω2W−fZV ℓiΦ=0,(9)

where one has enough generality to deﬁne the per-

turbation variable Φto choose Z=A/B such that

the tortoise coordinate r∗arising from this equation is

dr∗fA/B = dr. Finally, we expect small deviations from

GR in the other parameters as well: W= 1 + δw(r),

Vℓ=VGR

ℓ(r) + δV ℓ(r). Note that in this deﬁnition we

made explicit that radial functions modifying the poten-

tial depend on the angular momentum ℓof the quasi-

normal frequencies.

In the following we try to provide a heuristic explana-

tion for why we expect this form for the master equation.

The ﬁrst step is to write down the master equation for a

perturbation in GR

FGR

drFGR

dΦ

dr+hω2−VGR iΦ=0,(10)

where FGR ≡pgSch

tt grr

Sch =f(r)is the “natural” tortoise

function for the Schwarzschild solution, and

VGR =gSch

r2+G(r)FGR

dFGR

dr,(11)

where Λ = ℓ(ℓ+ 1),G(r) = 1 denotes scalar perturba-

tions, G(r) = −3for axial perturbations and

G(r) = −31+3F2

GR (r)−Λ2

1−3FGR (r)+Λ2(12)

for polar perturbations. We intentionally wrote the

eikonal term gSch

tt Λ/r2to make its correspondence with

geodesic motion evident.

If we assume that the background metric is the metric

of Eq. (6), we can substitute all the gSch

ab terms with gab

everywhere in equation (10). For example, one should

transform FGR into fA/B. Moreover, to take into ac-

count that the theory can be diﬀerent, one should add a

correction δZ to the tortoise coordinate and one δV to

the potential. These two corrections take into account

the fact that the derivation of the perturbation equation

might be done in a theory which is not GR [cf. Eq. (5)].

The resulting equation is

f(Z+δZ)d

drf(Z+δZ)dΦ

dr

+ω2−fZ VGR

ℓ+δV ℓΦ=0.

(13)

One can always re-deﬁne the perturbation function Φand

re-scale the equation to remove the δZ term from the

derivative term. This would ensure that the second order

derivative is acted upon the tortoise coordinate of the

metric. However, by doing so, one would introduce some

terms multiplying the frequency, obtaining Eq. (9). The

term multiplying the frequency can be mapped into the

modiﬁcation of the tortoise coordinate δw(r) = −2δZ(r),

and we expand it as

δw(r) =

k=1

w(k)rH

rk

.(14)

It is worth noting that this term appears because the

background metric (and thus the tortoise coordinate) is

diﬀerent from the Schwarzschild metric.

On the potential side, from the construction, we can

split the contribution to the potential coming from the

metric and from the “dynamics”. We have that

VGR

ℓ+δV ℓ=Vmod

ℓ+δV D

ℓ,(15)

where we made the following identiﬁcations:

Vmod

ℓ=Bℓ(ℓ+ 1)

r2+G(r)

drfA

B,(16)

δV D

ℓ=δ

k=0

β(k)

D,ℓ rH

rk

.(17)

This last term, linear in δ, expresses in a 1/r basis our

ignorance on the correct equations of motion for the dy-

namics of the theories under scrutiny.

Collectively we label all the real coeﬃcients, from met-

ric and from dynamics as

β(i)=β(0)

D,ℓ, z(1), b(1)

M, w(1), β(1)

D,ℓ, z(2), b(2)

M, w(2), β(2)

D,ℓ, . . . ,

(18)

where we identiﬁed z(k)=a(k)

M−b(k)

Mfrom the deﬁnition

of Z. In the rest of the paper we will refer to this set of

parameters as mixed parametrization or metric-potential

parametrization to distinguish it from the potential-only

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Parametrizedquasi-normalmodeframeworkfornon-SchwarzschildmetricsNicolaFranchini1,2,3,4andSebastianH.Völkel1,2,51SISSA,ViaBonomea265,34136Trieste,ItalyandINFNSezionediTrieste2IFPU-InstituteforFundamentalPhysicsoftheUniverse,ViaBeirut2,34014Trieste,Italy3UniversitéParisCité,CNRS,AstroparticuleetCosmol...

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Parametrized quasi-normal mode framework for non-Schwarzschild metrics Nicola Franchini1 2 3 4and Sebastian H. Völkel1 2 5 1SISSA Via Bonomea 265 34136 Trieste Italy and INFN Sezione di Trieste

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