# The isospectrality of asymptotic quasinormal modes of large Gauss-Bonnet
$d$-dimensional black holes

Filipe Mouraa and João Rodriguesb

a Departamento de Matemática, Escola de Tecnologias e Arquitetura,
ISCTE - Instituto Universitário de Lisboa
and Instituto de Telecomunicações,
Av. das Forças Armadas, 1649-026 Lisboa, Portugal

b Centro de Análise Matemática, Geometria e Sistemas Dinâmicos,
Departamento de Matemática,
Instituto Superior Técnico,
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

We compute the quasinormal frequencies of $d$-dimensional large spherically
symmetric black holes with Gauss-Bonnet corrections in the highly damped
regime. We solve perturbatively the master differential equation and we
compute the monodromies of the master perturbation variable (analytically
continued to the complex plane) in different contours, in order to obtain the
quasinormal mode spectra. We consider tensorial, vectorial and scalar
gravitational perturbations, obtaining the same frequencies for the three
cases like in Einstein gravity. We also separately perform the same
calculation for test scalar fields.

##  1 Introduction

Gravitational wave detectors can directly measure quasinormal ringing
frequencies, which carry unique information about parameters of the black hole
in the ringdown phase resulting from a binary black hole coalescence. Indeed,
the spectrum of black hole quasinormal modes (QNMs) does not depend on what
drives the perturbations: it is completely determined by intrinsic physical
quantities of the black hole such as mass, charge or spin, and eventually
(beyond Einstein gravity) some other parameters of the theory. This feature
turns QNMs into preferential probes for testing theories of gravity beyond
Einstein, since the ringing frequencies represent a universal part of the
gravitational wave signals. With the advent of gravitational wave astronomy,
therefore, interest in the study of black hole QNMs has raised, as their
measurement can provide a test to modified gravity theories. Some recent works
computing QNMs of black holes with higher derivative corrections coming from
different theories are [1, 2, 3, 4].

Black hole quasinormal modes are associated to perturbations that can be
related either to the black hole metric (gravitational perturbations) or to
external fields. They are solutions of a master differential equation (whose
form in general depends on the action/black hole solution one is considering)
with a potential (depending on the kind of perturbation/external field). Most
of the times, QNMs have to be computed numerically. Nonetheless, different
analytical methods have been developed in order to compute QNMs in some
limiting cases. Two of such cases are the eikonal limit and the asymptotic
(highly damped) limit.

Concretely, the master equation and the corresponding higher-derivative
corrected potential have been obtained for tensorial perturbations of black
holes with leading $\alpha^{\prime}$ corrections in string theory [5, 6].
Using this result, we have computed the corresponding quasinormal modes (and
also for test scalar fields) in the eikonal limit [7] and in the asymptotic
highly damped limit [8].

The higher-derivative corrected potential has also been obtained for
tensorial, vectorial and scalar perturbations of $d$-dimensional black holes
with Gauss-Bonnet corrections. The corresponding quasinormal modes in the
eikonal limit have been computed in [3]. In this article we compute these
quasinormal modes in the asymptotic highly damped limit, like we did in [8] in
string theory.

Quasinormal modes associated to different kinds of gravitational perturbations
have different spectra in general, but they share the same spectrum in
Einstein gravity in $d$ dimensions in some limiting cases. One of such cases
is the eikonal limit. Such isospectrality is observed in that limit for
spherically symmetric $d$ dimensional black holes in Einstein gravity, but not
any longer in the presence of perturbative Gauss-Bonnet corrections. As shown
in [3], the leading terms are equal, but the first order corrections to the
real parts of quasinormal frequencies are distinct for the different kinds of
gravitational perturbations; the imaginary parts of the same quasinormal
frequencies are still equal to first order in the Gauss-Bonnet parameter, but
in the same article they have been shown to be distinct to second order in
such parameter.

Another limiting case in which there is an isospectrality of quasinormal modes
for different kinds of perturbations in Einstein gravity in $d$ dimensions is
the asymptotic (highly damped) limit. This is the limit we will address in
this work, for all different kinds of perturbations of spherically symmetric
black holes, in the presence of perturbative Gauss-Bonnet corrections, hoping
to clarify if the isospectrality of quasinormal modes is preserved or not with
such corrections. We will also consider quasinormal modes of massless scalar
test field in the background of these black holes. This way, for tensorial
perturbations and test scalar fields, we will be able to compare the results
for the two theories: string theory and Einstein-Gauss-Bonnet theory.

The article is organized as follows. In section 2 we review a $d$-dimensional
spherically symmetric black hole solution in Einstein-Gauss-Bonnet gravity and
its large black hole limit. In section 3 we review the gravitational
perturbation theory of these black holes, writing down the master equation and
the respective potentials corresponding to these perturbations and also to
test scalar fields. Given this information, in section 4 we compute the
quasinormal spectrum in the highly damped limit corresponding to these
perturbations and fields using the monodromy method. We conclude by discussing
our results.

##  2 Einstein-Gauss-Bonnet black holes

###  2.1 The Einstein-Gauss-Bonnet action

In $d$–dimensional Gauss-Bonnet gravity one considers the following action
with higher derivative corrections:

$\frac{1}{16\pi
G_{d}}\int_{\mathcal{M}}\sqrt{-g}\left(\mathcal{R}+\alpha(\mathcal{R}_{mnpq}\mathcal{R}^{mnpq}-4\mathcal{R}_{mn}\mathcal{R}^{mn}+\mathcal{R}^{2})\right)\mbox{d}^{d}x,$
(1)

where $\mathcal{M}$ is our $d$-dimensional space time manifold, $G_{d}$ is the
generalized gravitational constant and $\alpha$ is a positive coupling
constant.

In the context of superstring theories, this action (also always coupled to a
dilaton $\phi$ and eventually to other string fields) represents an
$\alpha^{\prime}$ correction to Einstein gravity. The whole set of corrections
is expressed in a perturbative expansion in the inverse string tension
$\alpha^{\prime}$, of which the higher order Gauss-Bonnet term in (1) is the
leading correction term. From this point of view, (1) (or its equivalent in
string theory) is an effective action. The parameter $\alpha$ in (1) is to be
identified (up to a numerical constant) with the inverse string tension
$\alpha^{\prime}$. Besides $\alpha$ (or $\alpha^{\prime}$), in string theory
there is also a dilaton term multiplying the Gauss-Bonnet correction, since
$e^{\phi}$ represents the string coupling.

In Gauss-Bonnet gravity one takes a different point of view, considering (1)
as a complete (and not effective) action and computing exact solutions to its
field equations (and not just solutions that are perturbative in $\alpha$; in
particular, not assuming that $\alpha$ is necessarily a small parameter). This
is one of the two very important conceptual differences between Gauss-Bonnet
gravity and superstring gravity, the other one being the presence/absence of
the dilaton.

###  2.2 A spherically symmetric black hole solution

A general static spherically symmetric metric in $d$ dimensions can always be
written depending only on a metric function $f(r)$:

$ds^{2}=-f(r)\ dt^{2}+f^{-1}(r)\ dr^{2}+r^{2}d\Omega^{2}_{d-2}.$ (2)

$d\,\Omega^{2}_{d-2}=\sum_{i=2}^{d-1}\prod_{j=2}^{i-1}\sin^{2}\theta_{j}d\,\theta_{i}^{2}$
is the canonical metric tensor field of the unit ($d$-2)-sphere.

In references [9, 10, 11, 12], black hole solutions of the field equations
from (1) are found and discussed. We are interested in a spherically symmetric
solution of the form (2), with

$f(r)=1+\frac{r^{2}}{\alpha(d-3)(d-4)}(1-q(r))\hskip 10.0pt,\hskip
10.0ptq(r)=\sqrt{1+\frac{4\alpha(d-3)(d-4)\mu}{(d-2)r^{d-1}}}.$ (3)

The parameter $\mu$ is related to the black hole mass through

$M=\frac{\Omega_{d-2}}{8\pi
G_{d}}\mu,\,\Omega_{d-2}=\frac{2\pi^{\frac{d-1}{2}}}{\Gamma\left(\frac{d-1}{2}\right)}.$

We introduce the parameter $R_{0}$ which we define by

$\mu=\frac{(d-2)}{2}R_{0}^{d-3}\hskip 10.0pt\Leftrightarrow\hskip
10.0ptR_{0}=\left(\frac{2\mu}{d-2}\right)^{\frac{1}{d-3}}.$ (4)

$R_{0}$ would correspond to the horizon radius of the Tangherlini solution, in
the absence of Gauss-Bonnet corrections (i.e. setting $\alpha=0$). For the
solution we are considering, this parameter has no physical meaning; we
introduce it for convenience, and because it is simply related to $\mu$. The
true horizon radius $R_{H}$ of the black hole (3) is related to $R_{0}$
through

$R_{0}^{d-3}=R_{H}^{d-3}\left(1+\alpha\frac{(d-3)(d-4)}{2R_{H}^{2}}\right).$
(5)

Gravitational perturbations for this black hole solution have been studied in
[13, 14]. The stability of this solution under such perturbations has also
been studied in [15].

###  2.3 The perturbative large black hole limit

In this article, we take the limit of a small coupling constant $\alpha$. In
this limit one is allowed to take a perturbative expansion in $\alpha$. This
procedure is similar to the one in string theory, where one also takes a
perturbative expansion in $\alpha^{\prime}$. But the situation is different,
as we are not considering any dilaton. Because of that, even in the
perturbative small $\alpha$ limit Einstein-Gauss-Bonnet theory is different
from string theory.

More concretely, concerning the black hole solution (3) in which we are
interested, defining the dimensionless parameter

$\lambda^{\prime}=\frac{\alpha}{R_{0}^{2}},$ (6)

this limit corresponds to the condition $\lambda^{\prime}\ll 1$ and, from (5),
it is equivalent to having $\alpha\ll R_{H}^{2}$ i.e. the limit of large black
holes. We take the perturbative expansion in $f$ and consider only terms to
first order in $\lambda^{\prime}$ as

$f(r)=1-\left(\frac{R_{0}}{r}\right)^{d-3}+\lambda^{\prime}\frac{(d-3)(d-4)}{2}\left(\frac{R_{0}}{r}\right)^{2d-4}.$
(7)

In order to simplify the notation we define the functions

$f_{0}(r)=1-\left(\frac{R_{0}}{r}\right)^{d-3}\hskip 10.0pt,\hskip
10.0pt\delta
f(r)=\frac{(d-3)(d-4)}{2}\left(\frac{R_{0}}{r}\right)^{2d-4}\frac{1}{1-\left(\frac{R_{0}}{r}\right)^{d-3}},$
(8)

and rewrite (7) as

$f(r)=f_{0}(r)(1+\lambda^{\prime}\delta f(r)).$ (9)

We can also consider a perturbative expansion in $\lambda^{\prime}$ in
obtaining the physical quantities related to the black hole (3) in the limit
(9). The horizon radius $R_{H}$ can be obtained from taking such expansion in
(5), with the result

$R_{H}=R_{0}\left(1-\frac{d-4}{2}\lambda^{\prime}\right).$ (10)

The temperature of a spherically symmetric black hole of the form (2), like
(3), is given by $T_{H}=\frac{f^{\prime}(R_{H})}{4\pi}.$ In the large black
hole perturbative limit we are considering, to first order in
$\lambda^{\prime}$, this temperature reads

$T_{H}=\frac{d-3}{4\pi
R_{0}}\left(1-\lambda^{\prime}\frac{(d-4)(d-2)}{2}\right).$ (11)

We see that, even in this perturbative limit, the black hole solution (3) is
different and has distinct properties from the $d-$dimensional stringy
solutions with quadratic curvature corrections obtained in [16, 17].

##  3 Gravitational perturbations in the perturbative limit

General tensors of rank at least 2 on the $(d-2)$-sphere $\mathbb{S}^{d-2}$
can be uniquely decomposed in their tensorial, vectorial and scalar
components. That is the case of general perturbations $h_{\mu\nu}=\delta
g_{\mu\nu}$ of a $d-$dimensional spherically symmetric metric like (2). For
such metric we have then scalar, vectorial and (for $d>4$) tensorial
gravitational perturbations.

Each type of perturbation is described in terms of master variables
$\Psi_{a}(r,t)$ (the subscript $a$ indicates the kind of perturbation). In
Einstein gravity, each master variable obeys a second order differential
equation (“master equation”):

$\frac{\partial^{2}\Psi_{a}}{\partial
x^{2}}-\frac{\partial^{2}\Psi_{a}}{\partial
t^{2}}=V_{a}\left[f(r)\right]\Phi_{a}.$ (12)

This “master equation” is given in terms of the tortoise coordinate $x$ for
the metric (2) defined by

$dx=\frac{dr}{f(r)},$ (13)

and of a potential $V_{a}\left[f(r)\right]$ that depends on the kind of
perturbation one considers [18]:
$V_{\textsf{S}},V_{\textsf{V}},V_{\textsf{T}}$. We assume the time dependence
of the master variables to be of the form

$\Psi(x,t)=e^{i\omega t}\psi(x),$ (14)

such that $\frac{\partial\Psi}{\partial t}=i\omega\Psi$ (for simplicity we
drop the subscript $a$). In this way the master equation (12) may be written
in Schrödinger form, for a generic potential $V,$ as

$\frac{d^{2}\psi}{d\,x^{2}}+\omega^{2}\psi=V\left[f(r)\right]\psi.$ (15)

In the presence of higher derivative corrections in the lagrangian, one still
has spherically symmetric black holes of the form (2) and the same type of
gravitational perturbations. The master equation obeyed by each perturbation
variable may change, though: generically, it may become a higher order
differential equation. But for the specific case of an action with corrections
quadratic in the Riemann tensor it has been shown that, perturbing the field
equations, one also obtains for each perturbation variable a second order
master equation like (15).

Concretely, the master equation and the corresponding higher-derivative
corrected potential have been obtained for tensorial perturbations of black
holes with leading $\alpha^{\prime}$ corrections in string theory [5, 6]. The
higher-derivative corrected potential has also been obtained for tensorial
[13], vectorial and scalar [14] perturbations of $d$-dimensional black holes
with Gauss-Bonnet corrections. These are the cases we will address in this
work. Therefore we will now review the master equations and potentials
associated with each of these cases, corresponding to perturbing the solution
(3) of the field equations from (1). But first we consider the simpler case of
a massless scalar test field.

###  3.1 Massless scalar test field

The field equation of a minimally coupled massless scalar test field in the
background of a spherically symmetric $d$-dimensional black hole is of the
form of the Schrödinger like master equation (15), with the minimal effective
potential given by

$V_{\textsf{M}}(r)=f(r)\left(\frac{\ell(\ell+d-3)}{r^{2}}+\frac{(d-2)(d-4)f(r)}{4r^{2}}+\frac{(d-2)f^{\prime}(r)}{2r}\right).$
(16)

$\ell\in\mathbb{N}$ is the multipole number, and $\ell(\ell+d-3)$ are the
eigenvalues of the laplacian on the $(d-2)$-sphere $\mathbb{S}^{d-2}$.

Since the test field is minimally coupled, $V_{\textsf{M}}(r)$ does not have
explicit $\lambda^{\prime}$ corrections. Once we replace $f(r)$ by its
$\lambda^{\prime}$-dependent expression (7), we can expand this effective
potential considering terms up to first order in $\lambda^{\prime}$, with the
result

$\displaystyle V_{\textsf{M}}(r)$ $\displaystyle=$ $\displaystyle
V^{0}_{\textsf{M}}(r)+\lambda^{\prime}V^{1}_{\textsf{M}}(r),$ (17)
$\displaystyle V^{0}_{\textsf{M}}(r)$ $\displaystyle=$ $\displaystyle
V_{\textsf{M}}\left[f_{0}(r)\right]=f_{0}(r)\left(\frac{(d-4)(d-2)f_{0}(r)}{4r^{2}}+\frac{(d-2)f_{0}^{\prime}(r)}{2r}+\frac{\ell(\ell+d-3)}{r^{2}}\right),$
(18) $\displaystyle V^{1}_{\textsf{M}}(r)$ $\displaystyle=$
$\displaystyle-\frac{(d-4)(d-3)}{4}\frac{R_{0}^{2d-4}}{r^{3d-2}}\left(\left(d^{2}-2\ell^{2}+6\ell-2d(\ell+1)\right)r^{d}-(2d-3)(d-2)R_{0}^{d-3}r^{3}\right).$
(19)

###  3.2 Scalar type gravitational perturbations

Scalar type gravitational perturbations of (3) are governed by the master
equation (15), with the effective potential taking the form

$V_{\textsf{S}}(r)=\frac{f(r)U(r)}{64r^{2}(d-3)^{2}A(r)^{2}q(r)^{8}\left(4c\,q(r)+(d-1)R(r)\left(q(r)^{2}-1\right)\right)^{2}}$
(20)

where $q(r)$ was defined in (3) and

$\displaystyle A(r)$ $\displaystyle=$
$\displaystyle\frac{1}{q(r)^{2}}\left(\frac{1}{2}+\frac{1}{d-3}\right)+\left(\frac{1}{2}-\frac{1}{d-3}\right),$
(21) $\displaystyle R(r)$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{\lambda^{\prime}R_{0}^{2}(d-3)(d-4)}\hskip 10.0pt,$
(22) $\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{\ell(\ell+d-3)}{d-2}-1.$ (23)

Finally, the function $U$ takes a rather lengthy form (see appendix A).

Expanding the effective potential above to first order in $\lambda^{\prime}$,
we obtain the potential corresponding to scalar perturbations of the solution
(7) we are looking for as

$V_{\textsf{S}}(r)=V^{0}_{\textsf{S}}(r)+\lambda^{\prime}V^{1}_{\textsf{S}}(r),$
(24)

where $V^{0}_{\textsf{S}}$ is the uncorrected effective potential present in
the master equation associated with scalar type gravitational perturbations of
the $d$-dimensional Tangherlini black hole, taking the form [18] of the
corresponding minimal effective potential given by (18):

$V^{0}_{\textsf{S}}(r)=V^{0}_{\textsf{M}}(r).$ (25)

The expression for $V^{1}_{\textsf{S}}$ is rather lengthy (see appendix A).

###  3.3 Vector type gravitational perturbations

Vector type gravitational perturbations of (3) are governed by the master
equation (15), with the effective potential taking the form

$V_{\textsf{V}}(r)=f(r)\left[\frac{(d-2)c}{r^{2}}A(r)+K(r)\left(\frac{d^{2}K}{dr^{2}}(r)+\frac{df}{dr}(r)\frac{dK}{dr}(r)\right)\right]$
(26)

where

$K(r)=\frac{1}{\sqrt{r^{d-2}A(r)q(r)}},$ (27)

$A(r)$ is given by (21) and $q(r)$ was defined in (3). Since we are only
considering terms up to first order in $\lambda^{\prime}$, we expand the
effective potential above as

$V_{\textsf{V}}(r)=V^{0}_{\textsf{V}}(r)+\lambda^{\prime}V^{1}_{\textsf{V}}(r)$
(28)

where $V^{0}_{\textsf{V}}$ is the uncorrected effective potential present in
the master equation associated with vector type gravitational perturbations of
the $d$-dimensional Tangherlini black hole, taking the form [18]

$V^{0}_{\textsf{V}}(r)=f_{0}(r)\frac{4(\ell-1)(d+\ell-2)+(d-2)\left(d-3(d-2)\left(\frac{R_{0}}{r}\right)^{d-3}\right)}{4r^{2}}$
(29)

and

$\begin{split}V^{1}_{\textsf{V}}(r)=-\frac{R_{0}^{d-1}}{r^{3d+1}}\frac{(d-4)}{2}\left[-R_{0}^{d-3}\left((3d-5)\ell^{2}+(d-3)(3d-5)\ell+\frac{3}{2}(d-12)(d-3)d-36\right)r^{d+3}\right.\\\
+2(d-1)\left.\left(d(\ell-3)+\ell(\ell-3)+4\right)r^{2d}+\left(d(d(4d-39)+91)-62\right)\frac{R_{0}^{2d-6}}{2}r^{6}\right]\end{split}$
(30)

###  3.4 Tensor type gravitational perturbations

Tensor type gravitational perturbations of (3) are governed by the master
equation (15), with the effective potential taking the form

$V_{\textsf{T}}(r)=f(r)\left[\frac{\ell(\ell+d-3)}{r^{2}}\left(3-\frac{B(r)}{A(r)}\right)+K(r)\left(\frac{d^{2}K}{dr^{2}}(r)+\frac{df}{dr}(r)\frac{dK}{dr}(r)\right)\right]$
(31)

where

$B(r)=A(r)^{2}\left(1+\frac{1}{d-4}\right)+\left(1-\frac{1}{d-4}\right).$ (32)

Since we are only considering terms up to first order in $\lambda^{\prime}$,
we expand the effective potential above as

$V_{\textsf{T}}(r)=V^{0}_{\textsf{T}}(r)+\lambda^{\prime}V^{1}_{\textsf{T}}(r).$
(33)

We obtain for $V^{0}_{\textsf{T}}(r)$ the same $\lambda^{\prime}=0$ potential
as for minimally coupled test fields, given by (18), and for scalar
perturbations, given by (25):

$V^{0}_{\textsf{T}}(r)=V^{0}_{\textsf{S}}(r)=V^{0}_{\textsf{M}}(r).$ (34)

For $V^{1}_{\textsf{T}}(r)$ we obtain

$\begin{split}V^{1}_{\textsf{T}}(r)=\frac{R_{0}^{d-1}}{2r^{3d+1}}\left(R_{0}^{d-3}\left(((d-11)d+16)\ell^{2}+d(d-7)^{2}\ell-\frac{d}{2}((d-7)(d-6)d-20)\right.-48\ell+8\right)r^{d+3}\\\
\left.+4(d-1)(d(\ell-1)+(\ell-3)\ell+4)r^{2d}+(d-4)((d-5)d(2d-7)-22)\frac{R_{0}^{2d-6}}{2}r^{6}\right).\end{split}$
(35)

##  4 Quasinormal modes, the asymptotic limit and the monodromy method

Quasinormal modes associated to gravitational perturbations of asymptotically
flat spherically symmetric black holes are solutions to the corresponding
master equation (15) subject to the boundary conditions

$\displaystyle\psi\propto e^{-i\omega x}\hskip 3.0pt,\hskip
3.0ptx\to+\infty\hskip 3.0pt\left(r\to+\infty\right);$ (36)
$\displaystyle\psi\propto e^{i\omega x}\hskip 3.0pt,\hskip
3.0ptx\to-\infty\hskip 3.0pt\left(r\to R_{H}\right).$ (37)

These boundary conditions state that waves can escape into infinity or inside
the black hole. They reflect the fact that black holes are dissipative
systems; hence, quasinormal frequencies are complex. The time dependence of
the form (14) requires the imaginary part of the quasinormal frequencies to be
positive. Thus, imposing boundary conditions (36) and (37) is equivalent to
the existence of two terms of the form $e^{\pm i\omega x}.$ For $x\to+\infty$
one of such terms is exponentially growing, while the other is exponentially
vanishing. The same is true for $x\to-\infty$, but with the two terms
switching their behavior. This poses an operational problem in handling and
distinguishing these two terms.

But one can allow $r$ (and $x$) to take complex values and consequently assume
an analytic continuation of functions of $r$ to the complex plane. In this
case, near the event horizon we can distinguish the two exponential terms by
computing the respective monodromies around it: indeed, the boundary condition
(37) can be set as a monodromy condition. Furthermore, one can take the
contour of a Stokes line defined by $\mathop{\rm Im}\left(\omega x\right)=0$
in the complex $r$ plane. Through Stokes lines we have $|e^{\pm i\omega
x}|=1$: the asymptotic behavior of $e^{\pm i\omega x}$ is always oscillatory
and there will be no problems with exponentially growing versus exponentially
vanishing terms. Thus if one considers the Stokes lines, imposing the boundary
condition (36) in the complex $r$ plane no longer poses a challenge to an
approximate analytical method.

In order to compute analytical expressions for quasinormal frequencies, we
resort to the monodromy method [19] (for a recent application to quasinormal
modes of regular black holes see [20]). In order to apply this method, as we
mentioned, we need to consider the coordinate $r$ as complex-valued, and the
master equations (15) as differential equations defined on the complex plane.
The general idea of the method is to pick two closed homotopic contours on the
complex $r$-plane. Both these contours enclose only the physical horizon
$r=R_{H}$: none of them encloses the origin of the complex $r$-plane nor any
other complex root of the metric function $f(r)$ (“fictitious horizons”). One
of these contours, the “big contour”, seeks to encode information of the
boundary condition (36) on the monodromy of $\psi$ associated with a full loop
around it. The other contour, the “small contour”, seeks to encode information
of the boundary condition (37) on the monodromy of $\psi$ associated with a
full loop around it. Since both contours are homotopic, the monodromy theorem
asserts that the respective monodromies must be the same. Thus, equating them
yields an analytic solution to the values of the quasinormal frequencies
$\omega$.

We restrict our analysis in this article to the highly damped regime of
quasinormal modes defined by the condition

$\mathop{\rm Im}\left(\omega\right)\gg\mathop{\rm Re}\left(\omega\right).$
(38)

This condition is equivalent to $\omega$ being approximately imaginary. The
definition of a Stokes line comes thus as

$\mathop{\rm Im}\left(\omega x\right)=0\Rightarrow\mathop{\rm
Re}\left(x\right)=0.$ (39)

###  4.1 The big contour

####  4.1.1 Change of variable and perturbation theory

The tortoise coordinate (13) for the metric (3) we are working with is given,
to first order in $\lambda^{\prime}$, in terms of the Gauss hypergeometric
function ${}_{2}F_{1}$, up to an integration constant $C_{X}$, by

$\displaystyle x$ $\displaystyle=$
${}_{2}F_{1}\left(1,-\frac{1}{d-3};\frac{d-4}{d-3};\left(\frac{R_{0}}{r}\right)^{d-3}\right)r$
(40) $\displaystyle-$
${}_{2}F_{1}\left(2,\frac{2d-5}{d-3};1+\frac{2d-5}{d-3};\left(\frac{R_{0}}{r}\right)^{d-3}\right)\frac{(d-3)(d-4)}{2(2d-5)}\left(\frac{R_{0}}{r}\right)^{2d-6}r\lambda^{\prime}+C_{X}.$

We see that, because of the $\lambda^{\prime}$ correction, there is a
singularity in the coordinate $x$ at $r=0$. In order to apply the monodromy
method, we need to consider the Stokes lines given by the condition
$\mathop{\rm Re}(x)=0$; because of such singularity, close to the origin those
lines are very difficult to handle. Since in this article we are taking a
perturbative approach, we extend it to the tortoise coordinate and we rather
consider the tortoise coordinate $z$ of the Tangherlini solution, given by

$dz=\frac{dr}{f_{0}(r)}$ (41)

with $f_{0}(r)$ defined in (8). This definition of $z$ corresponds to the
$\lambda^{\prime}=0$ limit of the perturbative expansion (9); after
integration, the explicit result corresponds to the same limit in (40).

Taking the second order derivative in (15) with respect to $z$, we can rewrite
the master equations as

$\left(\frac{dz}{dx}\right)^{2}\frac{d^{2}\psi}{dz^{2}}+\frac{d}{dr}\left(\frac{dz}{dx}\right)\frac{dr}{dx}\frac{d\psi}{dz}+\left(\omega^{2}-V_{a}\right)\psi=0.$
(42)

with $a=\textsf{S},\textsf{V},\textsf{T},\textsf{M}$. Also from (9) and the
definitions (13), (41) we can write, to first order in $\lambda^{\prime}$,

$\left(\frac{dz}{dx}\right)^{2}=1+2\lambda^{\prime}\delta f(r)$ (43)
$\frac{d}{dr}\left(\frac{dz}{dx}\right)\frac{dr}{dx}=\lambda^{\prime}\frac{d(\delta
f)}{dr}(r).$ (44)

Expanding the solution of (42) as

$\psi(r)=\psi_{0}(r)+\lambda^{\prime}\psi_{1}(r),$ (45)

replacing the expansion above in (42) and solving the equation perturbatively
in powers of $\lambda^{\prime}$, we obtain two distinct linear ordinary
differential equations. The first one, of zeroth order in $\lambda^{\prime}$,
is

$\frac{d^{2}\psi_{0}}{dz^{2}}+\left(\omega^{2}-V_{a}^{0}\right)\psi_{0}=0.$
(46)

The second one, of first order in $\lambda^{\prime}$, is

$\frac{d^{2}\psi_{1}}{dz^{2}}+\left(\omega^{2}-V_{a}^{0}\right)\psi_{1}=\xi_{a}$
(47)

where

$\xi_{a}=\xi_{1}\frac{d^{2}\psi_{0}}{dz^{2}}+\xi_{2}\frac{d\psi_{0}}{dz}+(\xi_{a})_{3}\psi_{0}.$
(48)

In the expression above, we defined functions

$\xi_{1}(r)=-2\delta f(r)$ (49) $\xi_{2}(r)=-\frac{d(\delta f)}{dr}(r)f(r)$
(50) $(\xi_{a})_{3}(r)=V_{a}^{1}(r).$ (51)

The subscript $a$ reflects the dependence of $\xi_{a}$ on the kind of
perturbation/potential that one is considering, a dependence which is explicit
through $(\xi_{a})_{3}$ but, from (46), also implicit in $\psi_{0}$.

####  4.1.2 Behavior close to the origin

In the sequence, namely for the differential equations (46), (47) and the
functions $(\xi_{a})_{3}$ in (51), we will need the asymptotic behavior near
the origin of the complex $r$-plane of $\xi_{1},\xi_{2}$ and each of the
effective potentials. We will also need to rewrite all these asymptotic
expansions with respect to the tortoise coordinate $z$ of $d$-dimensional
Tangherlini black hole. When defining $z$ in (41), we chose the integration
constant such that

$z(r)=r+\frac{1}{2}\sum_{n=0}^{d-4}\frac{1}{k_{n}}\log\left(1-\frac{r}{R_{n}}\right)$
(52)

where

$k_{n}=\frac{1}{2}f_{0}^{\prime}(R_{n})\hskip 10.0pt;\hskip
10.0ptR_{n}=R_{0}e^{2\pi i\frac{n}{d-3}}$ (53)

for $0\leq n\leq d-4$. Near the origin of the complex $r$-plane we have

$z(r)\sim-\frac{r^{d-2}}{(d-2)R_{0}^{d-3}}.$ (54)

The asymptotic behaviors for the effective potentials can be obtained after
some algebraic manipulations.

For the potentials (18) and (19) for minimally coupled scalar fields, we get

$\displaystyle V^{0}_{\textsf{M}}(r)$ $\displaystyle\sim$
$\displaystyle-\frac{(d-2)^{2}R_{0}^{2d-6}}{4r^{2d-4}},$ (55) $\displaystyle
V^{1}_{\textsf{M}}(r)$ $\displaystyle\sim$
$\displaystyle\frac{(d-4)(d-3)(2d-3)(d-2)}{4}\frac{R_{0}^{3d-7}}{r^{3d-5}}.$
(56)

For the potentials (25) and (110) for scalar perturbations,

$\displaystyle V^{0}_{\textsf{S}}(r)$ $\displaystyle\sim$
$\displaystyle-\frac{(d-2)^{2}R_{0}^{2d-6}}{4r^{2d-4}},$ (57) $\displaystyle
V^{1}_{\textsf{S}}(r)$ $\displaystyle\sim$
$\displaystyle\frac{(d-4)((d-5)d(2d-7)-22)}{4}\frac{R_{0}^{3d-7}}{r^{3d-5}}.$
(58)

For the potentials (29) and (30) for vectorial perturbations,

$\displaystyle V^{0}_{\textsf{V}}(r)$ $\displaystyle\sim$
$\displaystyle\frac{3(d-2)^{2}R_{0}^{2d-6}}{4r^{2d-4}},$ (59) $\displaystyle
V^{1}_{\textsf{V}}(r)$ $\displaystyle\sim$
$\displaystyle-\frac{(d-4)(d(d(4d-39)+91)-62)}{4}\frac{R_{0}^{3d-7}}{r^{3d-5}}.$
(60)

For the potentials (34) and (35) for tensorial perturbations,

$\displaystyle V^{0}_{\textsf{T}}(r)$ $\displaystyle\sim$
$\displaystyle-\frac{(d-2)^{2}R_{0}^{2d-6}}{4r^{2d-4}},$ (61) $\displaystyle
V^{1}_{\textsf{T}}(r)$ $\displaystyle\sim$
$\displaystyle\frac{(d-4)((d-5)d(2d-7)-22)}{4}\frac{R_{0}^{3d-7}}{r^{3d-5}}.$
(62)

We knew that $V^{0}_{\textsf{M}}=V^{0}_{\textsf{T}}=V^{0}_{\textsf{S}}$; it is
not a surprise that their asymptotic behaviors are also the same. From (58)
and (62) we immediately notice that asymptotically we also have
$V^{1}_{\textsf{T}}=V^{1}_{\textsf{S}}.$

Finally, replacing (54) in the asymptotic expressions for the potentials above
yields

$V^{0}_{\textsf{M}}(z)=V^{0}_{\textsf{S}}(z)=V^{0}_{\textsf{T}}(z)\sim-\frac{1}{4z^{2}}.$
(63)

For minimally coupled fields we also have

$V^{1}_{\textsf{M}}(z)\sim\frac{(d-4)(d-3)(2d-3)(d-2)}{4}\frac{R_{0}^{\frac{d-1}{d-2}}}{(-(d-2)z)^{\frac{3d-5}{d-2}}},$
(64)

while, for scalar and tensorial perturbations,

$V^{1}_{\textsf{S}}(z)\sim
V^{1}_{\textsf{T}}(z)\sim\frac{(d-4)((d-5)d(2d-7)-22)}{4}\frac{R_{0}^{\frac{d-1}{d-2}}}{(-(d-2)z)^{\frac{3d-5}{d-2}}}.$
(65)

For vectorial perturbations the corresponding results are

$\displaystyle V^{0}_{\textsf{V}}(z)$ $\displaystyle\sim$
$\displaystyle\frac{3}{4z^{2}},$ (66) $\displaystyle V^{1}_{\textsf{V}}(z)$
$\displaystyle\sim$
$\displaystyle-\frac{(d-4)(d(d(4d-39)+91)-62)}{4}\frac{R_{0}^{\frac{d-1}{d-2}}}{(-(d-2)z)^{\frac{3d-5}{d-2}}}.$
(67)

Near the origin of the complex $r$-plane, from (8), (49), (50) we can write
the asymptotic expansions for $\xi_{1},\xi_{2}$:

$\displaystyle\xi_{1}(r)$ $\displaystyle\sim$
$\displaystyle(d-4)(d-3)\frac{R_{0}^{d-1}}{r^{d-1}},$ (68)
$\displaystyle\xi_{2}(r)$ $\displaystyle\sim$
$\displaystyle\frac{(d-4)(d-3)(d-1)}{2}\frac{R_{0}^{2d-4}}{r^{2d-3}}.$ (69)

Rewriting these expressions with respect to the tortoise coordinate $z$, given
by (54), yields

$\xi_{1}(z)\sim(d-4)(d-3)\frac{R_{0}^{\frac{d-1}{d-2}}}{(-(d-2)z)^{\frac{d-1}{d-2}}}$
(70)

and

$\xi_{2}(z)\sim\frac{(d-4)(d-3)(d-1)}{2}\frac{R_{0}^{\frac{d-1}{d-2}}}{(-(d-2)z)^{\frac{2d-3}{d-2}}}.$
(71)

Finally, from (51), the asymptotic expansions for $(\xi_{a})_{3}$ for
$a=\textsf{M, T, V, S}$ can be obtained from the corresponding expansions of
$V_{a}^{1}$ given by (64), (65) and (67).

####  4.1.3 Calculation of the monodromy

Now, we compute the monodromy of $\psi$, associated with a full clockwise loop
around the big contour, depicted in figure (1).

Figure 1: Schematic depiction of the big contour, as the blue dashed line. The
Stokes lines are depicted as red curves. Naturally, not all Stokes lines are
depicted. Furthermore, we marked by $D$ and $U$ the regions where the boundary
condition (36) may be imposed.

We want to compute the monodromy of $\psi$, associated with a full clockwise
loop around the big contour, starting and ending at $D$. In order to attain
reliable information on how $\psi$ changes around the big contour, we resort
to WKB theory. Near the origin of the complex $r$-plane, the WKB approximation
fails as $r=0$ is a singular point of $V_{a}^{0}$ for every
$a=\textsf{S},\textsf{V},\textsf{T},\textsf{M}$. Thus, we need to analytically
solve the differential equations (46) and (47) in this region.

From the asymptotic limits (63), (66), equation (46) can be written as

$\frac{d^{2}\psi_{0}}{dz^{2}}+\left(\omega^{2}-\frac{j^{2}-1}{4z^{2}}\right)\psi_{0}=0,$
(72)

with $j=0$ for the potentials in (63), and $j=2$ for the potential in (66).
Following the procedure of [19, 21] we considered the general solution, for
arbitrary $j$, of the above differential equation, and at the end taking the
adequate limit $j\rightarrow 0$ or $j\rightarrow 2$. Such solution is given by

$\psi_{0}(z)=A_{+}\sqrt{2\pi}\sqrt{\omega z}J_{\frac{j}{2}}(\omega
z)+A_{-}\sqrt{2\pi}\sqrt{\omega z}J_{-\frac{j}{2}}(\omega z),$ (73)

where $J_{\pm\frac{j}{2}}(\omega z)$ are Bessel functions of the first kind
and $A_{+},A_{-}$ arbitrary constants.

From here we can proceed in an identical fashion to the analogous computation
performed in [8]. The final monodromies are given in terms of quantities that
depend (in a complicated way) on the type of perturbation through the
perturbation potentials $V_{a}$ and the value of $j$. We have

$\left(\mathcal{M}_{a}\right)_{1}=\left(\frac{A_{+}e^{5i\alpha_{+}}+A_{-}e^{5i\alpha_{-}}}{A_{+}e^{i\alpha_{+}}+A_{-}e^{i\alpha_{-}}}\right)\left(1+\lambda^{\prime}\delta\left(\mathcal{M}_{a}\right)_{1}\right)e^{-i\omega\Delta_{z}}$
(74)

with

$\displaystyle\Delta_{z}\ =-\frac{2\pi iR_{0}}{d-3},$ (75)
$\displaystyle\alpha_{\pm}=\frac{\pi}{4}\left(1\pm j\right),$ (76)
$\displaystyle
A_{\pm}(j)=\mp\frac{ie^{-i\alpha_{\mp}}}{2}\csc\left(\frac{j\pi}{2}\right).$
(77)

Taking both the limit $j\to 2$ (for vectorial perturbations) and $j\to 0$ (for
the other cases), we get in both cases the same result for the uncorrected
part of all monodromies [21]:

$\frac{A_{+}e^{i5\alpha_{+}}+A_{-}e^{i5\alpha_{-}}}{A_{+}e^{i\alpha_{+}}+A_{-}e^{i\alpha_{-}}}=-3.$
(78)

The correction term to the monodromies in (74) is given by

$\delta\left(\mathcal{M}_{a}\right)_{1}=\frac{\Lambda_{F}^{+}e^{i5\alpha_{+}}+\Lambda_{F}^{-}e^{i5\alpha_{-}}}{A_{+}e^{5i\alpha_{+}}+A_{-}e^{5i\alpha_{-}}}-\frac{\Lambda_{I}^{+}e^{i\alpha_{+}}+\Lambda_{I}^{-}e^{i\alpha_{-}}}{A_{+}e^{i\alpha_{+}}+A_{-}e^{i\alpha_{-}}}$
(79)

with

$\displaystyle\Lambda_{I}^{\pm}(d,j,\omega,R_{0})=\left(R_{0}\omega\right)^{\frac{d-1}{d-2}}\sum_{k=1}^{3}\Theta_{k}^{\pm}(d,j,1,1),$
(80) $\displaystyle
B_{\pm}(d,j,\omega,R_{0})=A_{\pm}\left(\Lambda_{I}^{+}e^{-i\alpha_{+}}+\Lambda_{I}^{-}e^{-i\alpha_{-}}\right),$
(81)
$\displaystyle\Lambda^{\pm}_{F}(d,j,\omega,R_{0})=\left(R_{0}\omega\right)^{\frac{d-1}{d-2}}\sum_{k=1}^{3}\Xi_{k}^{\pm}(d,j,1,1)+B_{\pm}.$
(82)

Explicit expressions for the functions
$\Theta_{k}^{\pm},\Xi_{k}^{\pm},\,k=1,2,3$ can be found in [8].

The sums in $k$ in the definitions of $\Lambda_{I}^{\pm},\Lambda^{\pm}_{F}$
have their origin in the analogous sum in the definition of $\xi_{a}$: each of
the functions $\Theta_{k}^{\pm},\Xi_{k}^{\pm},\,k=1,2,3$ in (80) and (82)
corresponds to the term including $\xi_{k}$ in (48), with $\xi_{k}$ given in
(49), (50), (51). Each of these functions $\Theta_{k}^{\pm},\Xi_{k}^{\pm}$ has
a different form, because of the different coefficients multiplying each
$\xi_{k}$ in (48), but always involving sums of products/quotients of Gamma
functions, which we designate by $\mathcal{H}(m,n,k)$, given by

$\mathcal{H}(m,n,k):=\frac{\Gamma\left(\frac{1}{2}-\frac{k}{2}\right)\Gamma\left(-\frac{k}{2}\right)\Gamma\left(\frac{k}{2}+\frac{m}{2}+\frac{n}{2}+\frac{1}{2}\right)}{2\sqrt{\pi}\Gamma\left(-\frac{k}{2}+\frac{m}{2}-\frac{n}{2}+\frac{1}{2}\right)\Gamma\left(-\frac{k}{2}+\frac{n}{2}-\frac{m}{2}+\frac{1}{2}\right)\Gamma\left(-\frac{k}{2}+\frac{m}{2}+\frac{n}{2}+\frac{1}{2}\right)}.$
(83)

The complete expression for $\delta\left(\mathcal{M}_{a}\right)_{1}$ in (79)
is a sum of 96 of these terms, all with different coefficients. Specifically
there are terms depending on

$\mathcal{H}\left(\pm\frac{j}{2},\pm\frac{j}{2},-\frac{1}{d-2}\right),\mathcal{H}\left(\pm\frac{j}{2},\pm\frac{j}{2},-\frac{d-1}{d-2}\right),\mathcal{H}\left(\pm\frac{j}{2},\pm\frac{j}{2},-\frac{2d-3}{d-2}\right).$
(84)

We will not reproduce these terms here, since they can all be found in [8].
The calculation of all these expressions for a generic value of $j$ is
evidently a formidably complicated task.

For the case $j=0$ (one of the two cases we will need), from (84) the
functions $\mathcal{H}(m,n,k)$ that arise are always of the form
$\mathcal{H}\left(0,0,k\right)$. Using the known properties of the Gamma
functions

$\displaystyle\Gamma(x+1)=x\Gamma(x),$ (85)
$\displaystyle\Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin(\pi x)},$ (86)

one can easily show that these functions are given by

$\mathcal{H}\left(0,0,k\right)=\frac{\sqrt{\pi}\Gamma\left(-\frac{k}{2}\right)}{2\Gamma\left(\frac{1-k}{2}\right)^{3}\sin\left(\frac{1-k}{2}\pi\right)}.$
(87)

One can then evaluate $\mathcal{H}\left(0,0,k\right)$ for the values of $k$
referred in (84).

The dependence of the monodromy $\left(\mathcal{M}_{a}\right)_{1}$ on the type
of perturbation is expressed in the value of $j$, as we mentioned (or
equivalently on the potential $V^{0}_{a}$), but also on $(\xi_{a})_{3}$ (or
equivalently on the potential $V^{1}_{a}$). The result for
$\delta\left(\mathcal{M}_{a}\right)_{1}$ in (79) is always of the form

$\delta\left(\mathcal{M}_{a}\right)_{1}=\left(\frac{R_{0}\omega}{d-2}\right)^{\frac{d-1}{d-2}}e^{-\frac{2i\pi}{d-2}}\Pi_{a},$
(88)

with a factor $\Pi_{a}$ depending on the type of perturbation.

For test scalar fields, with $j=0$ and $(\xi_{\textsf{M}})_{3}$ given by (64),
we obtained

$\Pi_{\textsf{M}}=\frac{8}{3}\frac{\pi^{2}}{2^{\frac{1}{d-2}}}\frac{(d-4)(d-3)(d-2)}{d-1}\frac{\Gamma\left(\frac{1}{d-2}\right)}{\left[\Gamma\left(\frac{d-1}{2d-4}\right)\right]^{4}}\sin\left(\frac{\pi}{2(d-2)}\right).$
(89)

For tensorial perturbations, we also take $j=0$ but considering
$(\xi_{\textsf{T}})_{3}$ given by (65). For scalar perturbations we should
consider $j=0$ and $(\xi_{\textsf{S}})_{3}$ but, as we know from (65),
$(\xi_{\textsf{S}})_{3}=(\xi_{\textsf{T}})_{3}$. This way, the result for
$\Pi_{\textsf{S}}$ will be the same as the one for $\Pi_{\textsf{T}}$.

For vectorial perturbations we must take $j=2$. From (84) this means the
functions $\mathcal{H}(m,n,k)$ that arise in the calculation of
$\left(\mathcal{M}_{\textsf{V}}\right)_{1}$ are always of the form
$\mathcal{H}\left(\pm 1,\pm 1,k\right)$. From the definition (83) and using
(86), one can show that

$\mathcal{H}\left(1,1,k\right)=\mathcal{H}\left(-1,-1,k\right)=-\mathcal{H}\left(-1,1,k\right)=-\mathcal{H}\left(1,-1,k\right)=\frac{1+k}{1-k}\mathcal{H}\left(0,0,k\right)$
(90)

with $\mathcal{H}\left(0,0,k\right)$ given by (87). One can this way evaluate
$\mathcal{H}\left(\pm 1,\pm 1,k\right)$ for the values of $k$ referred in
(84). We can then proceed computing $\Pi_{\textsf{V}}$, taking $j=2$ also in
the adequate coefficients and the value (67) for $(\xi_{\textsf{V}})_{3}$.
Rather surprisingly, because of the changes motivated by taking $j=2$ in (90)
and in the adequate coefficients, $\Pi_{\textsf{V}}$ turned out to be
identical to $\Pi_{\textsf{T}}$ and $\Pi_{\textsf{S}}$, despite
$(\xi_{\textsf{V}})_{3}\neq(\xi_{\textsf{T}})_{3},\,(\xi_{\textsf{S}})_{3}$:

$\Pi_{\textsf{T}}=\Pi_{\textsf{V}}=\Pi_{\textsf{S}}=\frac{2\sqrt{\pi}}{3}\frac{(d(d-5)+2)(d-4)}{d-1}\frac{\Gamma\left(\frac{1}{2(d-2)}\right)\Gamma\left(\frac{d-3}{2(d-2)}\right)}{\Gamma\left(\frac{d-1}{2(d-2)}\right)^{2}}\sin\left(\frac{\pi}{d-2}\right).$
(91)

To summarize, the monodromy of $\psi$ associated with a full clockwise loop
around the big contour is given by

$\left(\mathcal{M}_{a}\right)_{1}=-3\left(1+\lambda^{\prime}\left(\frac{R_{0}\omega}{d-2}\right)^{\frac{d-1}{d-2}}e^{-\frac{2i\pi}{d-2}}\Pi_{a}\right)e^{-i\omega\Delta_{z}},$
(92)

with $\Delta_{z}$ given by (75) and $\Pi_{a}$ given by (89), for test scalar
fields, and by (91), for all kinds of gravitational perturbations.

###  4.2 Small contour

Now we compute the monodromy of $\psi$, associated with a full clockwise loop
around the small contour, depicted in figure (2).

From (10) we write, up to first order in $\lambda^{\prime}$, the asymptotic
expansion near the real horizon $R_{H}$

$\frac{1}{f(r)}\sim\frac{1}{(d-3)\left(1-\frac{R_{H}}{r}\right)}\left[1+\lambda^{\prime}\frac{(d-4)(d-1)}{2}\right].$
(93)

Figure 2: Schematic depiction of the small and big contours as the orange and
blue dashed lines respectively. The orange contour is to be interpreted as
arbitrarily close to $R_{H}$. The Stokes lines are depicted by red curves.
Naturally, not all Stokes lines are depicted.

From the definition (13), integrating the expression above yields the
asymptotic expansion of the tortoise coordinate $x$ near $R_{H}$:

$x(r)\sim\frac{R_{H}}{d-3}\log\left(r-R_{H}\right)\left[1+\lambda^{\prime}\frac{(d-4)(d-1)}{2}\right].$
(94)

The monodromy of $x(r)$ in (94), associated with a full clockwise loop around
the real horizon $R_{H}$, is given by

$\Delta_{x}=-\frac{2\pi
iR_{H}}{d-3}\left[1+\lambda^{\prime}\frac{(d-4)(d-1)}{2}\right]=-\frac{2\pi
iR_{0}}{d-3}\left[1+\lambda^{\prime}\frac{(d-4)(d-2)}{2}\right].$ (95)

One easily checks that

$\lim_{r\to R_{H}}V_{a}(r)=0$ (96)

for all $a=\textsf{S},\textsf{V},\textsf{T},\textsf{M}$. Thus, near the event
horizon $R_{H}$, we can write the master equations as

$\frac{d^{2}\psi}{dx^{2}}+\omega^{2}\psi=0$ (97)

whose solution, imposing the boundary condition (37) in the event horizon, is

$\psi(x)=\mathcal{C}_{+}e^{i\omega x}.$ (98)

for some $\mathcal{C}_{-}\in\mathbb{C}$. Computing the monodromy of the
expression above, associated with a full clockwise loop around the event
horizon $R_{H}$, yields

$(\mathcal{M}_{a})_{2}=e^{i\omega\Delta_{x}}.$ (99)

###  4.3 Equating monodromies: the final result

Now, we want to relate the monodromies $(\mathcal{M}_{a})_{1}$ and
$(\mathcal{M}_{a})_{2}$.

We notice the big contour is homotopic to the small one. This is so, because
one can continuously deform the big contour into the small one. Thus, using
the monodromy theorem, we know the monodromies of $\psi$, associated with the
full clockwise loops around the big and the small contours, are equal. Hence,
the equation

$(\mathcal{M}_{a})_{1}=(\mathcal{M}_{a})_{2}$ (100)

must hold. Using equations (92) and (99), we can rewrite the equation above as

$-3(1+\lambda^{\prime}\delta(\mathcal{M}_{a})_{1})e^{-i\omega(\Delta_{x}+\Delta_{z})}=1.$
(101)

Taking the logarithm on both sides of the equation above yields

$\ln\left(3\right)+(2k+1)\pi
i-i\omega(\Delta_{x}+\Delta_{z})+\log(1+\lambda^{\prime}\delta(\mathcal{M}_{a})_{1})=0$
(102)

for $k\in\mathbb{Z}$. Taylor expanding the last logarithm in the equation
above, up to first order in $\lambda^{\prime}$, yields

$\ln(3)+(2k+1)\pi
i-i\omega(\Delta_{x}+\Delta_{z})+\lambda^{\prime}\delta(\mathcal{M}_{a})_{1}=0.$
(103)

Considering the expressions for $\Delta_{z},\Delta_{x}$ (75) and (95), we can
write

$\Delta_{x}+\Delta_{z}=\frac{-4\pi
iR_{0}}{d-3}\left(1+\lambda^{\prime}\frac{(d-4)(d-2)}{4}\right).$ (104)

The equation above and (88) allow us to rewrite (103) as

$\ln(3)+(2k+1)\pi i=\omega\frac{4\pi
R_{0}}{d-3}\left(1+\lambda^{\prime}\frac{(d-4)(d-2)}{4}\right)-\lambda^{\prime}\left(\frac{R_{0}\omega}{d-2}\right)^{\frac{d-1}{d-2}}e^{-\frac{2i\pi}{d-2}}\Pi_{a}.$
(105)

The results should be written in terms of physical variables, instead of the
parameter $R_{0}$. It is customary to write the QNM frequencies with respect
to the Hawking temperature $T_{H}$ of the black hole given by (11). We can
solve (11) for $R_{0}$ as a function of $T_{H}$ and replace it in (105) and in
the definition (6) of $\lambda^{\prime}$, obtaining
$\lambda^{\prime}=\alpha\left(\frac{4\pi}{d-3}\right)^{2}T_{H}^{2}$. Solving
for $\frac{\omega}{T_{H}}$ one gets

$\frac{\omega}{T_{H}}=\left[\ln(3)+(2k+1)\pi
i\right]\left[1+\alpha\left(\frac{4\pi}{d-3}\right)^{2}T_{H}^{2}\left(\frac{(d-4)(d-2)}{4}+\left[\frac{d-3}{4\pi(d-2)}\right]^{\frac{d-1}{d-2}}\left[\frac{\omega}{T_{H}}\right]^{\frac{1}{d-2}}\Pi_{a}\right)\right].$
(106)

This is a transcendental equation for $\frac{\omega}{T_{H}}$. Recalling that
we have been working in the highly damped limit (38), we can take
$\left.\frac{\omega}{T_{H}}\right|_{\alpha=0}\approx(2k+1)\pi i$ in the
$\alpha$ correction in (106), obtaining this way the final expression for the
$\alpha$-corrected asymptotic quasinormal spectra:

$\frac{\omega}{T_{H}}=\left[\ln(3)+(2k+1)\pi
i\right]\left[1+\alpha\left(\frac{4\pi}{d-3}\right)^{2}T_{H}^{2}\left(\frac{(d-4)(d-2)}{4}+\left[\frac{d-3}{d-2}\right]^{\frac{d-1}{d-2}}\left[\frac{2k+1}{4}\right]^{\frac{1}{d-2}}\frac{\Pi_{a}}{4\pi}\,\mathrm{e}^{-\frac{3\pi
i}{2(d-2)}}\right)\right],$ (107)

with $\Pi_{a}$ given by (89), for test scalar fields, and by (91), for all
kinds of gravitational perturbations. We also take $k\in\mathbb{N}$ in order
to have $\mathop{\rm Im}\left(\omega\right)>0$ which, from our conventions in
(14), is a necessary condition for the stability of the black hole. 111Some
authors use the opposite convention.

This expression has the same form as the ones previously derived in [8] for
the black hole solution with higher derivatives from string theory obtained in
[16], the only difference coming from the distinct values of the Hawking
temperature, giving rise to a different value of the $k-$independent
correction term in (107). This term will be actually negligible in the large
$k$ limit, which is the asymptotic limit.

Because of the $\mathrm{e}^{-\frac{3\pi i}{2(d-2)}}$ term the $\alpha$
correction is complex, which means it will affect both the real and the
imaginary parts of $\omega$. The real part of the asymptotic limit of
$\frac{\omega}{T_{H}}$ is then no longer equal to the universal value
$\ln(3)$: because of the $\alpha$-correction, it now also depends on the
spacetime dimension $d$ and on the mode number $k$. We have evaluated
numerically
$\left[\frac{d-3}{d-2}\right]^{\frac{d-1}{d-2}}\frac{\Pi_{a}}{4\pi}$ for the
relevant values of $d$ in the context of superstring theory. This factor grows
monotonically with $d$, varying from approximately 0.19 (corresponding to
$d=5$) to approximately 13.5 (corresponding to $d=10$), for
$\Pi_{\textsf{T}},\,\Pi_{\textsf{V}},\,\Pi_{\textsf{S}}$. The values of
$\Pi_{\textsf{M}}$, on the same range, are slightly larger, varying from
approximately 0.58 (corresponding to $d=5$) to approximately 14.48. Just for
comparison, $\frac{(d-2)(d-4)}{4}$ varies between 0.75 and $12$ on the same
range. In order to compare the orders of magnitude of the two correction terms
in (107), one must also consider the factor
$\left[\frac{2k+1}{4}\right]^{\frac{1}{d-2}}$ multiplying the term with
$\Pi_{a}$. For values of $k$ that are not very large, the two correction terms
in (107) are of comparable orders of magnitude. In a perturbative expansion
the correction terms should be relatively small. The perturbation parameter
$\lambda^{\prime}$ defined in (6) is supposed to be small in magnitude in our
perturbative approach, but for the $k$-dependent correction term to be small
(even if multiplied by $\lambda^{\prime}$), $k$ cannot be too large. As we
argued in [8], consistency with the condition (38) defining the highly damped
limit used in our derivation requires $k$ to be large enough such that the
condition $(2k+1)\pi\gg\ln(3)$ is valid, but $k$ should not be arbitrarily
large.

##  5 Conclusions

In this article, we have studied quasinormal modes of spherically symmetric
$d$ dimensional black holes in Gauss-Bonnet gravity, also considered
perturbatively, equivalently to having large black holes, having taken the
asymptotic limit and computed them analytically, using the monodromy method.
We have obtained the same result for the three different kinds of
perturbations (tensorial, vectorial and scalar), which means that the
isospectrality of quasinormal modes for different kinds of perturbations in
the asymptotic limit is preserved in the presence of perturbative Gauss-Bonnet
corrections.

This isospectrality is a remarkable result, since as we have seen in section
4.1 the expression for the monodromy of the big contour depends explicitly on
the potential corresponding to the perturbation being considered (more
precisely, on its asymptotic expansion close to the origin). For vectorial
perturbations, these asymptotic expansions are different than for the other
kinds, both at $\alpha=0$ and to first order in $\alpha$ (the Gauss-Bonnet
parameter in which we have made a perturbative expansion). This difference
gives rise to different asymptotic master equations (which we also considered
perturbatively in $\alpha$) close to the origin. At order $\alpha=0$, the
asymptotic master equation close to the origin (72) is such that the
corresponding value of $j$ for vectorial perturbations is different than for
the other kinds. This affects the coefficients of many terms in the monodromy,
as we mentioned. To first order in $\alpha$, the asymptotic potential
corresponding to vectorial perturbations is different than for the other
kinds, giving rise to different values of $(\xi_{a})_{3}$ in (51) that
directly affect the monodromy. The net effect of the two changes (of $j$ and
$(\xi_{a})_{3}$) for vectorial perturbations is such that the total value of
the monodromy does not change: it is the same for all kinds of gravitational
perturbations, and so is the spectrum of asymptotic quasinormal modes with
Gauss-Bonnet corrections, to first order in $\alpha$.

We have also considered test scalar fields propagating in the same kind of
black holes. These fields satisfy the same field equation, with the same
potential (and logically the same spectrum of quasinormal modes), as tensorial
gravitational perturbations in Einstein gravity. In the presence of higher
derivative corrections the potentials become distinct and, as we have shown,
so do the spectra of quasinormal modes (although the functional form of the
corrected frequencies is analogous in both cases). The isospectrality of
asymptotic quasinormal modes with Gauss-Bonnet corrections we observed is,
therefore, valid only for gravitational perturbations.

There is a potential conflict between the asymptotic limit of quasinormal
modes and the perturbative limit of large black holes that we have taken. The
imaginary parts of these frequencies grow with the mode number $k$ to
infinity. Although the growth of the correction terms is more moderate than
the one of the uncorrected parts, they also grow arbitrarily with $k$. But the
corrections to the quasinormal mode frequencies, like any perturbative
correction, are supposed to be small. This means that our result is valid for
values of $k$ which are large enough for the asymptotic condition to be
verified, but not arbitrarily large. The maximum allowed value of $k$ depends
on the magnitudes of the Gauss-Bonnet constant (which is expected to be very
small), but also of the dimensional-dependent factors of the corrections. We
have evaluated numerically these factors, both for the gravitational
perturbations and for the test scalar fields. We concluded that, although
growing with the spacetime dimension $d$, the magnitude of these factors is
perfectly compatible with a perturbative expansion for the relevant values of
$d$ in the context of superstring theory.

Our results have the same form as the ones for the black hole solution with
higher derivatives from string theory obtained in [8]. In the large black hole
limit there is no difference between Gauss-Bonnet gravity and string theory
concerning asymptotic quasinormal mode frequencies for spherically symmetric
black holes, at least for the cases whose results are known in string theory
(tensorial perturbations and test fields). In future works we plan to study
the remaining cases in string theory.

The obtained results indicate a tendency of the highly damped quasinormal mode
frequencies for large black holes in Gauss-Bonnet gravity. In this limit, one
cannot distinguish the spectra of QNMs associated to different gravitational
perturbations. It would be interesting to extend this study to black hole
solutions with higher order corrections (namely $\alpha^{\prime 3}$
corrections from string theory), in order to figure out if this isospectrality
is preserved also by such corrections or if it is just a feature of Einstein
and Gauss-Bonnet gravity.

##### Acknowledgements

This work has been supported by Fundação para a Ciência e a Tecnologia under
grants IT (UIDB/50008/2020 and UIDP/50008/2020), CAMGSD/IST-ID
(UIDB/04459/2020 and UIDP/04459/2020) and project CERN/FIS-PAR/0023/2019. João
Rodrigues is supported by Fundação para a Ciência e a Tecnologia through the
doctoral fellowship UI/BD/151499/2021.

##  Appendix A Effective potential functions

The function $U(r)$, present in the effective potential $V_{\textsf{S}}(r)$
associated with scalar type gravitational perturbations, reads

$\displaystyle U(r)$ $\displaystyle=$ $\displaystyle
5(d-1)R(r)^{2}(R(r)+1)-3(d-1)^{5}R(r)q(r)\left(24c(R(r)+1)+(d-1)R(r)^{2}\right)+$
(108) $\displaystyle
2(d-1)^{4}q(r)^{2}\left(168c^{2}(R(r)+1)+24c(d-1)R(r)^{2}-(d-1)R(r)^{2}(7d(R(r)+1)+5R(r)-3)\right)+$
$\displaystyle+2(d-1)^{4}R(r)q(r)^{3}\left(c(84d(R(r)+1)+44R(r)-84)-184\
2c+(d-1)(d+13)R(r)^{2}\right)+$
$\displaystyle+(d-1)^{3}\left(384c^{3}-48c((3d-5)d+2)R(r)^{2}+192c^{2}\left((d-15)R(r)^{2}+d-11\right)+\right.$
$\displaystyle\left.+(d-1)R(r)^{2}(d(7d(R(r)+1)+106R(r)+26)-3(55R(r)+7))\right)q(r)^{4}+$
$\displaystyle+(d-1)^{3}R(r)\left(-64c^{2}(d-38)+(d-1)((7d-90)d+71)R(r)^{2}+\right.$
$\displaystyle\left.+16c\left(13d^{2}(R(r)+1)-2d(81R(r)+73)+255R(r)+303\right)\right)q(r)^{5}+$
$\displaystyle+4(d-1)^{2}\left(96c^{3}(d-7)-8c(d-1)\left(6d^{2}-74d+145\right)R(r)^{2}-\right.$
$\displaystyle-8c^{2}(d(11d(R(r)+1)-34R(r)-58)-175R(r)+9)+d-1R(r)^{2}(-5(23R(r)+79)+$
$\displaystyle+d(d(7d(R(r)+1)-89R(r)-81)+5(41R(r)+57))))q(r)^{6}-$
$\displaystyle-4(d-1)^{2}R(r)\left(8c^{2}(d(72-13d)+43)+(d-1)(d(d(5d-49)+99)-63)+R(r)^{2}+\right.$
$\displaystyle+4c(d(d(17d(R(r)+1)-107R(r)-123)-39R(r)+121)+465R(r)+321))q(r)^{7}+$
$\displaystyle+(d-1)\left(128c^{3}(d-9)(d-5)+32c(d-1)(d(d(8d-55)+9)+246)R(r)^{2}+\right.$
$\displaystyle+64c^{2}(d-5)\left(d^{2}+((d-4)d+49)R(r)-3\right)-$
$\displaystyle\left.-(d-1)R(r)^{2}(d(d(d(45d(R(r)+1)-452R(r)-548)+6(217R(r)+393))-\right.$
$\displaystyle\left.-4(349R(r)+997))+565R(r)+1173)\right)q(r)^{8}+$
$\displaystyle+(d-1)R(r)\left(-64c^{2}(d-5)(d(3d-13)+36)+(d-1)(d(3d(d(9d-92)+294)-1204)+635)R(r)^{2}-\right.$
$\displaystyle-8c(d-5)(d(d((d-79)R(r)+d-47)+191R(r)+127)+31R(r)+63))q(r)^{9}+$
$\displaystyle+2d-5\left(64c^{3}(d-5)(d-3)+8c(d-1)(d((d-43)d+141)-27)R(r)^{2}+\right.$
$\displaystyle+8c^{2}(d-5)(d((d-18)R(r)+d-2)+77R(r)-3)+(d-1)^{2}R(r)^{2}(-33(R(r)-7)+$
$\displaystyle+d(d(9d(R(r)+1)-35R(r)-59)+43R(r)+59)))q(r)^{10}-$
$\displaystyle-2d-5R(r)\left(24c^{2}(d-11)(d-5)(d-3)+(d-1)^{2}(d((7d-39)d+81)-65)R(r)^{2}+\right.$
$\displaystyle+12c(d-7)(d-5)(d-3)(d-1)(R(r)+1))q(r)^{11}+$
$\displaystyle+(d-5)^{2}(d-1)R(r)^{2}q(r)^{12}(16c((d-9)d+26)+(d-1)(d((d-2)R(r)+d-18)-3R(r)+77))+$
$\displaystyle+(d-5)^{2}(d-3)^{2}(d-1)^{2}R(r)^{3}q(r)^{13}.$

$R(r)$ is given by (22), $c$ was defined in (23) and $q(r)$ was defined in
(3).

The expression for $V^{1}_{\textsf{S}}$, the first-order in $\lambda^{\prime}$
correction to $V_{\textsf{S}}(r)$, is given by

$\displaystyle
V^{1}_{\textsf{S}}(r)=-\frac{d-4}{4(d-2)^{2}\left((\ell-1)(d+\ell-2)+(d-1)\left(\frac{d-2}{2}R_{0}^{d-3}\right)r^{3-d}\right)^{3}}R_{0}^{d-1}r^{-6d-1}$
$\displaystyle\left(-8(d-2)^{2}(d-1)(\ell-1)^{3}(d+\ell-2)^{3}\left((d-3)\ell+(d-2)^{2}+\ell^{2}\right)r^{5d}+\right.2(d-2)(\ell-1)^{2}(d+\ell-2)^{2}$
$\displaystyle\left((d-2)^{2}(d(d(4d-71)+201)-152)+\right.$
$\displaystyle(d-3)(d-2)(d(19d-103)+106)\ell+(d(d(29d-215)+486)-338)\ell^{2}+$
$\displaystyle\left.2(5d-7)\ell^{4}+4(d-3)(5d-7)\ell^{3}\right)r^{4d+3}\left(\frac{d-2}{2}R_{0}^{d-3}\right)-$
$\displaystyle
2(d-2)(\ell-1)(d+\ell-2)\left(-(d-2)^{2}(d(d(15d-268)+747)-554)+\right.$
$\displaystyle(d-3)(d-2)(d(d(15d-242)+801)-694)\ell+$
$\displaystyle((d-11)d+22)(d(37d-147)+140)\ell^{2}+4(d-3)(d(11d-75)+94)\ell^{3}+$
(109)
$\displaystyle\left.2(d(11d-75)+94)\ell^{4}\right)r^{3d+6}\left(\frac{d-2}{2}R_{0}^{d-3}\right)^{2}-2(d-1)\left(-(d-2)^{3}(d(d(7d-60)-113)+370)+\right.$
$\displaystyle(d-3)(d-2)^{2}(d(7(d-6)d-255)+698)\ell+$
$\displaystyle(d(d(d(-3(d-48)d-1661)+7112)-12992)+8624)\ell^{2}-$
$\displaystyle\left.4(d-3)^{2}(d(5d-62)+108)\ell^{3}-2(d-3)(d(5d-62)+108)\ell^{4}\right)$
$\displaystyle\left(\frac{d-2}{2}R_{0}^{d-3}\right)^{3}r^{2d+9}-2(d-1)^{2}\left((d-2)^{2}(d(d(d+8)-131)+134)-\right.$
$\displaystyle\left.2(d((69-7d)d-172)+116)\ell^{2}+2(d-3)(d(d(7d-69)+172)-116)\ell\right)$
$\displaystyle\left.\left(\frac{d-2}{2}R_{0}^{d-3}\right)^{4}r^{d+12}+4(d-1)^{3}((d-5)d(2d-7)-22)\left(\frac{d-2}{2}R_{0}^{d-3}\right)^{5}r^{15}\right).$
(110)

## References

  * [1] J. L. Blázquez-Salcedo, C. F. B. Macedo, V. Cardoso, V. Ferrari, L. Gualtieri, F. S. Khoo, J. Kunz and P. Pani, _Perturbed black holes in Einstein-dilaton-Gauss-Bonnet gravity: Stability, ringdown, and gravitational-wave emission_ , Phys. Rev. D94 (2016) 10, 104024 [arXiv:1609.01286 [gr-qc]].
  * [2] P. A. Cano, K. Fransen and T. Hertog, _Ringing of rotating black holes in higher-derivative gravity_ , Phys. Rev. D102 (2020) 4, 044047 [arXiv:2005.03671 [gr-qc]].
  * [3] R. A. Konoplya and Z. Stuchlík, _Are eikonal quasinormal modes linked to the unstable circular null geodesics?_ , Phys. Lett. B771 (2017) 597 [arXiv:1705.05928 [gr-qc]].
  * [4] L. Pierini and L. Gualtieri, _Quasi-normal modes of rotating black holes in Einstein-dilaton Gauss-Bonnet gravity: the first order in rotation_ [arXiv:2103.09870 [gr-qc]].
  * [5] F. Moura and R. Schiappa, Higher-derivative corrected black holes: Perturbative stability and absorption cross-section in heterotic string theory, Class. Quant. Grav. 24 (2007) 361 [hep-th/0605001].
  * [6] F. Moura, _Tensorial perturbations and stability of spherically symmetric $d$-dimensional black holes in string theory_, Phys. Rev. D87 (2013), 044036 [arXiv:1212.2904 [hep-th]].
  * [7] F. Moura and J. Rodrigues, _Eikonal quasinormal modes and shadow of string-corrected $d$-dimensional black holes_, Phys. Lett. B819 (2021), 136407 [arXiv:2103.09302 [hep-th]].
  * [8] F. Moura and J. Rodrigues, _Asymptotic quasinormal modes of string-theoretical $d$-dimensional black holes_, JHEP 08 (2021) 078 [arXiv:2105.02616 [hep-th]].
  * [9] D. G. Boulware and S. Deser, _String Generated Gravity Models_ , Phys. Rev. Lett. 55 (1985), 2656.
  * [10] D. L. Wiltshire, _Black Holes in String Generated Gravity Models_ , Phys. Rev. D38 (1988), 2445.
  * [11] J. T. Wheeler, _Symmetric Solutions to the Gauss-Bonnet Extended Einstein Equations_ , Nucl. Phys. B268 (1986), 737.
  * [12] J. T. Wheeler, _Symmetric Solutions to the Maximally Gauss-Bonnet Extended Einstein Equations_ , Nucl. Phys. B273 (1986), 732.
  * [13] G. Dotti and R. J. Gleiser, _Linear stability of Einstein-Gauss-Bonnet static spacetimes. Part I. Tensor perturbations_ , Phys. Rev. D72 (2005), 044018 [arXiv:gr-qc/0503117 [gr-qc]].
  * [14] R. J. Gleiser and G. Dotti, _Linear stability of Einstein-Gauss-Bonnet static spacetimes. Part II: Vector and scalar perturbations_ , Phys. Rev. D72 (2005), 124002 [arXiv:gr-qc/0510069 [gr-qc]].
  * [15] R. A. Konoplya and A. Zhidenko, _(In)stability of D-dimensional black holes in Gauss-Bonnet theory_ , Phys. Rev. D77 (2008), 104004 [arXiv:0802.0267 [hep-th]].
  * [16] C. G. Callan, R. C. Myers and M. J. Perry, Black Holes in String Theory, Nucl. Phys. B311 (1989) 673.
  * [17] F. Moura, _String-corrected dilatonic black holes in d dimensions_ , Phys. Rev. D83 (2011), 044002 [arXiv:0912.3051 [hep-th]].
  * [18] A. Ishibashi and H. Kodama, A Master Equation for Gravitational Perturbations of Maximally Symmetric Black Holes in Higher Dimensions, Prog. Theor. Phys. 110 (2003) 701, [hep-th/0305147].
  * [19] L. Motl and A. Neitzke, _Asymptotic black hole quasinormal frequencies_ , Adv. Theor. Math. Phys. 7 (2003) 2, 307-330 [arXiv:hep-th/0301173 [hep-th]].
  * [20] C. Lan and Y. F. Wang, _Singularities of regular black holes and the art of monodromy method for asymptotic quasinormal modes_ [arXiv:2205.05935 [gr-qc]].
  * [21] J. Natário and R. Schiappa, _On the classification of asymptotic quasinormal frequencies for d-dimensional black holes and quantum gravity_ , Adv. Theor. Math. Phys. 8 (2004) 6, 1001 [hep-th/0411267].