# Quasinormal modes of Einstein–scalar–Gauss–Bonnet black holes

Prosenjit Paul<EMAIL_ADDRESS>Indian Institute Of Engineering
Science and Technology (IIEST), Shibpur-711103, WB, India

###### Abstract

In this paper, we investigate quasinormal modes of scalar and electromagnetic
fields in the background of Einstein–scalar–Gauss–Bonnet (EsGB) black holes.
Using the scalar and electromagnetic field equations in the vicinity of the
EsGB black hole, we study nature of the effective potentials. The dependence
of real and imaginary parts of the fundamental quasinormal modes on parameter
$p$ (which is related to the Gauss–Bonnet coupling parameter $\alpha$) for
different values of multipole numbers $l$ are studied. We analyzed the effects
of massive scalar fields on the EsGB black hole, which tells us the existence
of quasi–resonances. In the eikonal regime, we find the analytical expression
for the quasinormal frequency and show that the correspondence between the
eikonal quasinormal modes and null geodesics is valid in the EsGB theory for
the test fields. Finally, we study grey-body factors of the electromagnetic
fields for different multipole numbers $l$, which deviates from
Schwarzschild’s black hole.

##  1 Introduction

General Relativity (GR) is the simplest theory of gravitation, which is
consistent with various astrophysical observations, like gravitational waves,
black holes, etc. Nevertheless, GR opens some challenging problems. These are
the existence of singularities at the center of the black holes, a complete
theory of quantum gravity, the problem of dark matter/energy, cosmic
inflation, and others. To solve these problems theorists introduce alternative
approaches to gravity. There are a number of such approaches to gravity
available in the literature, e.g., string–theory–inspired gravities, adding
higher-order terms in the curvature tensor to the Einstein–-Hilbert action of
GR.

One of the well-motivated alternative theories of gravity is the EsGB theory.
In this theory, the scalar field is nonminimally coupled to the Gauss–Bonnet
(GB) term. The EsGB theories also arise in the low energy limit of string
theory [1]. The lowest order correction to the Einstein-Hilbert action is the
GB term, which is quadratic in curvature tensor, but this theory has pure
divergence in $4D$ spacetime. Alternatively, in recent years Glavan and Lin
[2] have removed the divergence in $4D$ GB gravity by replacing the GB
parameter $\alpha$ to $\alpha/D-4$, where $D$ is the number of spacetime
dimensions. However, later it was shown that this naive regulariation scheme
does not lead to the well-defined theory of gravity. Nevertheless, the black
hole solutions [3, 4] obtained as results of such regularazation proved out to
be also solutions in the well-defined theories [5]. On the contrary, the EsGB
theories, we are interested here, are free of such kind of problems, because
in EsGB theories the divergence is absent due to the coupling between the GB
term and scalar fields.

Isolated black holes are very simple objects and can be described by the three
parameters: mass, charge, and spin. However, the actual situation differs from
this. A black hole at the center of galaxies is surrounded by matter
distribution, such as an accretion disk, jets and outflows, stars, etc.
Therefore, a black hole interacts with its surroundings matter distribution.
Even in the absence of any matter distribution around black hole, it will
interact with the vacuum and produce a pair of particles and anti-particle,
which is known as Hawking radiation. Therefore, a dynamical, interacting black
hole can not be described by mass, charge, and spin only, and one needs to
consider the perturbations theory of black holes. There has been a growing
interest in the perturbation theory of black holes [6, 7, 8]. There are a
number of reasons for interest in proper frequencies of such out-of-
equilibrium black holes, called quasinormal modes. First of all, the LIGO and
VIRGO scientific collaborations detect gravitational-wave signals from black
holes [9] and it is consistent with Einstein’s theories of gravity, though,
due to the large uncertainty in the determining of the spin and mass of the
black hole, the large window for alternative theories remains [10, 11]. The
dominating influence on such a signal is expected to come from the quasinormal
modes exhibiting the lowest frequency, referred to as the fundamental mode.

Various black hole solutions in higher-order theories of gravity, its
quasinormal modes, and Hawking radiation are studied extensively. The EsGB
gravity was studied in Refs. [12, 13, 14], where the most important black hole
solutions were found numerically. The analytical black hole solution of EsGB
theories using continued fraction approximation (CFA) and its shadow was
studied in Ref. [15]. The gravitational quasinormal modes of
Einstein–dilaton–Gauss–Bonnet (EdGB) solutions are studied in Refs. [16, 17].
The spontaneous scalarization, black hole sensitivities, and linear stability
of the EsGB black hole are investigated in Refs. [18, 19, 20]. Black hole
solutions in EdGB, Einstein–Wely, and Einstein cubic gravity investigated in
Refs. [21, 22, 23, 24], while its quasinormal modes and Hawking radiation are
studied in Refs. [25, 26, 27, 28, 29]. In addition, quasinormal modes of
Kaluza-Klein-like black holes in the Einstein-Gauss-Bonnet theory were
considered in [30]. Furthermore, quasinormal modes of string-corrected
$d$-dimensional black holes and noncommutative Schwarzschild black holes are
analysed in Refs. [31, 32, 33, 34, 35, 36]. The gravitational perturbations of
numerically obtained EdGB black holes are studied in Ref. [16]. In the eikonal
limit, the quasinormal modes for gravitational perturbations of EsGB black
holes were obtained in Ref. [17]. However, the scalar and electromagnetic
perturbations either of analytically or numerically obtained EsGB black hole
solution are not analyzed, which provides us an opportunity to fill this gap.
The scalar field will be studied not only in the massless limit, but also for
the non-zero massive term. The latter has a number of motivations, because an
effective mass term appears in the wave equation as a result of introduction
of extra dimensions [37, 38], and magnetic fields [39, 40]. After all, massive
long-lived modes and oscillatory tails [41] may contribute into the very long
gravitational waves observed recently via the Time Pulsar Array [42, 43].

The paper is organized as follows. In section 2, we outline the basics of EsGB
theories in four dimensions and discuss the metric functions for black holes
in EsGB gravity. The quasinormal modes for massless/massive scalar fields and
electromagnetic fields are studied in section 3 for different scalar coupling
functions. Furthermore, we derived the analytical formula for quasinormal
frequency in the eikonal regime. In section 4, we discuss the grey–body
factors for electromagnetic fields. Finally, we summarize our results in the
conclusions section.

##  2 Einstein–scalar–Gauss-Bonnet Black Holes

The action for EsGB theories in $4D$ can be written as

$I=\int dx^{4}\sqrt{-g}\biggr{[}\frac{R}{\kappa^{2}}+\alpha
f(\varphi)R_{GB}^{2}-\frac{1}{2}\nabla_{\mu}\varphi\nabla^{\mu}\varphi\biggr{]},$
(1)

where $g$ is determinant of the metric $g_{\rho\sigma}$, we take $\kappa=16\pi
Gc^{-4}=1$, $\alpha$ is the GB coupling constant and $R_{GB}^{2}$ is defined
as

$R_{GB}^{2}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2},$
(2)

and $f(\varphi)$ is the arbitrary smooth function of the scalar field
$\varphi$, which is known as GB coupling functional. The metric of static and
spherically symmetric black holes in EsGB theory can be written in the
following form [15]

$ds^{2}=-g_{tt}dt^{2}+g_{rr}dr^{2}+r^{2}\Bigl{(}d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\Bigl{)},$
(3)

where the analytically approximated metric functions were found in Ref. [15].
The functions $g_{tt}$ and $g_{rr}$ are written up to fourth order for
different GB coupling functional in the Appendix A. Here we consider
quadratic, cubic, quartic, inverse and Logarithmic Gauss-Bonnet coupling
functional.

To parameterize the family of EsGB black holes solution, we will introduce the
dimension–less parameter p as

$p\equiv\frac{96\alpha^{2}f^{\prime}(\varphi)^{2}}{r_{0}^{4}},\\\ 0\leq p<1,$
(4)

where $r_{0}$ is the position of the event horizon, $\varphi_{0}$ is the
scalar fields at the event horizon. In the limit $p=0$, one can obtain
Schwarzschild black hole.

##  3 Quasinormal Modes of Test Fields

In this paper, we consider quasinormal modes of test fields in the background
of EsGB black holes. The equation for scalar, and electromagnetic in the
background of the EsGB black hole is given by

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}\Bigl{(}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\Bigl{)}-m^{2}\Phi$
$\displaystyle=0,$ (5a)
$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}\Bigl{(}\sqrt{-g}g^{\sigma\mu}g^{\rho\nu}F_{\rho\sigma}\Bigl{)}$
$\displaystyle=0,$ (5b)

where $F_{\rho\sigma}=\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho}$ and
$A_{\mu}$ is the vector potential. To do the separation of variables we will
introduce the radial function $R_{\omega l}(r)$ and spherical harmonics
$Y_{l}(\theta,\phi)$ as

$\Phi(t,r,\theta,\phi)=e^{\pm i\omega t}R_{\omega l}(r)Y_{l}(\theta,\phi),$
(6)

where $l$ is the angular number. After the separation of variables equation
(5) can be written in the following Schrodinger-like wave equation [44, 26]

$\frac{d^{2}\Psi}{dr_{*}^{2}}+\Bigl{(}\omega^{2}-V_{i}(r)\Bigl{)}\Psi=0,$ (7)

where $r_{*}$ is the “ _tortoise coordinate_ ”, defined as

$dr_{*}=dr\sqrt{\frac{g_{rr}}{g_{tt}}}.$ (8)

The effective potentials of scalar $(i=s)$ and electromagnetic $(i=e)$ fields
in the background of EsGB black hole are given by

$\displaystyle V_{s}$
$\displaystyle=\frac{g_{rr}g_{tt}^{\prime}-g_{tt}g_{rr}^{\prime}}{2rg_{rr}^{2}}+g_{tt}\frac{l(l+1)}{r^{2}}+g_{tt}m,$
(9a) $\displaystyle V_{e}$ $\displaystyle=g_{tt}\frac{l(l+1)}{r^{2}}.$ (9b)

The dependence of effective potential on various parameters (e.g. $p$ and $l$)
for massless scalar fields and electromagnetic fields are shown in Fig. 1 for
cubic and logarithmic coupling functional. To express the radial coordinates
in units of event horizon radius, we set $r_{0}=1$. The form of effective
potential is positive definite and it diminishes at both the event horizon and
infinity. For massless scalar and electromagnetic fields as we increase the
parameter $p$ height of the potential barrier decreases and the barrier height
is maximum at $p=0$. It can be seen that as we increase the angular number $l$
height of the potential barrier becomes higher. For massive scalar fields, the
height of the potential barrier (Fig. 2) increases with mass and the height of
the potential barrier is minimum for $m=0$. The effective potential for others
coupling functional exhibits similar behaviour.

The master equation (7) can be solved using WKB approximation [45]. The wave
function $\Psi$ satisfies the following boundary condition

$\Psi\sim e^{\pm i\omega r_{*}},$ (10)

at $r_{*}\to\pm\infty$, i.e. there are only incoming waves at the event
horizon and purely outgoing waves at spatial infinity. In the asymptotic
region, using the boundary condition one can solve the master equation using
the WKB approximations up to the required order. Expanding the potential into
the Taylor series and using the master equation, the solution of differential
equation (7) is obtained near the peak of the potential. Matching these two
solutions the quasinormal frequency $\omega$ is derived. Finally, the
frequency can be expressed as $\omega=\text{Re}(\omega)+i\text{Im}(\omega)$,
where $\text{Re}(\omega)$ represent real oscillation frequency and
$\text{Im}(\omega)$ represent damping rate. The quasinormal frequency upto
6-th order WKB method has the following form

$\frac{i(\omega^{2}-V_{0})}{\sqrt{-2V_{0}^{\prime\prime}}}-\sum_{i=2}^{i=6}\Lambda_{i}=n+\frac{1}{2},$
(11)

where $\Lambda_{i}$ are the correction term up to sixth-order [45, 46, 47,
48]. $n=0$, $1$, $2$ is the overtone number. To compute the quasinormal modes
of massless scalar and vector fields we will use sixth-order WKB formula with
Padé approximation [49, 50] $\tilde{m}=5$.

It is worth of mentioning that special attention will be devoted to the $l\gg
1$, eikonal, regime, first of all because there is a correspondence between
the eikonal quasinormal modes and some parameters of the null geodesics
suggested in [51] and further constrained in [52, 53, 54] . The correspondence
is expected to be broken for gravitational perturbations of the Einstein-
dilaton-Gauss-Bonnet theory [26]. Here we will check whether this
correspondence is fulfilled for the test fields in the EsGB theory.

(a) $f(\varphi)=\varphi^{3}$: $l=1$ & $s=0$

(b) $f(\varphi)=\varphi^{3}$: $l=2$ & $s=0$

(c) $f(\varphi)=\varphi^{3}$: $l=1$ & $s=1$

(d) $f(\varphi)=\varphi^{3}$: $l=2$ & $s=1$

(e) $f(\varphi)=\ln(\varphi)$: $l=1$ & $s=0$

(f) $f(\varphi)=\ln(\varphi)$: $l=2$ & $s=0$

(g) $f(\varphi)=\ln(\varphi)$: $l=1$ & $s=1$

(h) $f(\varphi)=\ln(\varphi)$: $l=2$ & $s=1$

Figure 1: Effective potential with $r_{0}=1$, black line denotes $p=0.0$, red
line denotes $p=0.4$ and blue lines denotes $p=0.8$

(a) $f(\varphi)=\varphi^{3}$: $l=15$ & $s=0$

(b) $f(\varphi)=\varphi^{-1}$: $l=15$ & $s=1$

(c) $f(\varphi)=\ln(\varphi)$: $l=15$ & $s=0$

Figure 2: Effective potential for massive scalar fields with $r_{0}=1$,
$p=0.5$, black line denotes $m=0.0$, red line denotes $m=2.0$, blue lines
denotes $m=4.0$ and magenta line denotes $m=6.0$.

###  3.1 Quadratic GB-Coupling Functional: $f(\varphi)=\varphi^{2}$

The quasinormal modes of massless scalar $(s=0)$ and electromagnetic $(s=1)$
fields for $l=1$, $2$ and $l=1$, $2$, $3$ are shown in Figs. 3-7. From Figs.
3-7 one can be seen that the behaviour of real and imaginary parts of the
quasinormal frequency as a function of parameter $p$ is almost linear for
different values of $l$. Therefore, we use in-build Mathematica function
FindFormula, to express real and imaginary parts of the quasinormal frequency
as a function of parameter $p$ for different values of $l$. In formula 12 and
13 the parameter $p$ runs from $p=0.0$ to $p=0.8$. The approximate linear laws
of scalar and electromagnetic fields for different values of $l$ are given by

$\displaystyle\text{Re}(\omega_{s=0,l=1})$ $\displaystyle\approx
0.586146-0.0192317p,$ (12a) $\displaystyle\text{Im}(\omega_{s=0,l=1})$
$\displaystyle\approx 0.0114263p-0.194999,$ (12b)
$\displaystyle\text{Re}(\omega_{s=0,l=2})$ $\displaystyle\approx
0.967948-0.0299163p,$ (12c) $\displaystyle\text{Im}(\omega_{s=0,l=2})$
$\displaystyle\approx 0.0108174p-0.193500.$ (12d)

$\displaystyle\text{Re}(\omega_{s=1,l=1})$ $\displaystyle\approx
0.498029-0.0176951p,$ (13a) $\displaystyle\text{Im}(\omega_{s=1,l=1})$
$\displaystyle\approx 0.00835913p-0.184913,$ (13b)
$\displaystyle\text{Re}(\omega_{s=1,l=2})$ $\displaystyle\approx
0.916163-0.0281436p,$ (13c) $\displaystyle\text{Im}(\omega_{s=1,l=2})$
$\displaystyle\approx 0.00940289p-0.189931,$ (13d)
$\displaystyle\text{Re}(\omega_{s=1,l=3})$ $\displaystyle\approx
1.31512-0.0405207p,$ (13e) $\displaystyle\text{Im}(\omega_{s=1,l=3})$
$\displaystyle\approx 0.0103759p-0.191264.$ (13f)

(a)

(b)

Figure 3: Fundamental quasinormal mode $(n=0)$ for massless scalar fields with
$r_{0}=1$; red line denotes real part of frequency and blue lines denote the
imaginary part of the frequency.

(a)

(b)

Figure 4: Fundamental quasinormal mode $(n=0)$ for massless scalar fields with
$r_{0}=1$; red line denotes real part of frequency and blue lines denote the
imaginary part of the frequency.

(a)

(b)

Figure 5: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields with
$r_{0}=1$; red line denotes real part of frequency and blue lines denote
imaginary part of the frequency.

(a)

(b)

Figure 6: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields with
$r_{0}=1$; red line denotes real part of frequency and blue lines denote
imaginary part of the frequency.

(a)

(b)

Figure 7: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields with
$r_{0}=1$; red line denotes real part of frequency and blue lines denote
imaginary part of the frequency.

The behaviour of real and imaginary parts of the quasinormal modes for massive
scalar fields as a function of mass ($m$) is shown in Fig. 8. The WKB formula
[45, 46, 47, 48] can not describe the quasi resonances accurately [55], but in
the eikonal regime WKB method is exact, providing sufficiently accurate
results at $l\geq n$. Numerous examples of usage of the WKB method (see, for
instance [56, 50, 57]) say that, the sixth-order WKB formula with Padé
approximants is usually the best for the computation of the quasinormal
frequency of massive scalar fields. From Fig. 8(b) we can see that the
imaginary part of the quasinormal frequency approaches zero, which indicates
the existence of arbitrarily long-lived frequencies, called quasi-resonances.

(a)

(b)

Figure 8: The dependence of quasinormal frequency on mass $(m)$ for massive
scalar fields with $r_{0}=1$, $p=0.5$, red line denotes real part of frequency
and blue lines denote imaginary part of the frequency.

####  3.1.1 Analytical Formula For QNMS in the Eikonal Regime

In the domain of high multipole numbers $l$ (eikonal), test fields with
varying spin follows a common law up to the leading order. In this context, we
examine an electromagnetic field with the effective potential 9(b). In the
eikonal regime, one can use the first-order WKB formula given by

$\omega=\sqrt{V_{0}-i\Bigl{(}n+\frac{1}{2}\Bigl{)}\sqrt{-2V_{0}^{\prime\prime}}},$
(14)

where $n=0$, $1$, $2$ is the overtone number, $V_{0}$ is the effective
potential at $r=r_{max}$ and $V_{0}^{\prime\prime}$ is the double derivative
of effective potential with respect to radial coordinate $r$, evaluated at
$r=r_{max}$. Using the Ref. [58] we find that the maximum of effective
potential occurs at $r=r_{max}$ which is given by

$r_{max}=\frac{3r_{0}}{2}+0.0387r_{0}p+\mathcal{O}(p^{2}).$ (15)

Now, substituting $r_{max}$ and equation 9(b) into equation (14) one can
obtain the analytical expression for quasinormal frequency in the eikonal
regime as

$\omega=\frac{(1+2l)(1-0.0258p)-i(1+2n)(1-0.0447p)}{3\sqrt{3}r_{0}}.$ (16)

In the limit $p\to 0$, we obtained the analytical expression of quasinormal
frequency for Schwarzschild black hole.

###  3.2 Cubic GB-Coupling Functional: $f(\varphi)=\varphi^{3}$

The quasinormal modes of massless scalar $(s=0)$ and electromagnetic $(s=1)$
fields for $l=1$, $2$ and $l=1$, $2$, $3$ are shown in Figs. 9-13. From Figs.
9-13 one can be seen that the behaviour of real and imaginary parts of the
quasinormal frequency as a function of parameter $p$ is almost linear for
different values of $l$. In formula 17 and 18 the parameter $p$ runs from
$p=0.0$ to $p=0.6$. After $p=0.6$, the behaviour of real and imaginary parts
of quasinormal frequency is non-linear, especially for $l=2$ (scalar fields,
Fig. 10) and $l=2$, $3$ (electromagnetic fields, Figs. 12 & 13). The
approximated linear laws of scalar and electromagnetic fields for different
values of $l$ are given by

$\displaystyle\text{Re}(\omega_{s=0,l=1})$ $\displaystyle\approx
0.586345-0.0231319p,$ (17a) $\displaystyle\text{Im}(\omega_{s=0,l=1})$
$\displaystyle\approx 0.0236755p-0.258812,$ (17b)
$\displaystyle\text{Re}(\omega_{s=0,l=2})$ $\displaystyle\approx
0.967896-0.0355368p,$ (17c) $\displaystyle\text{Im}(\omega_{s=0,l=2})$
$\displaystyle\approx 0.0114912p-0.193244.$ (17d)

$\displaystyle\text{Re}(\omega_{s=1,l=1})$ $\displaystyle\approx
0.498397-0.0219789p,$ (18a) $\displaystyle\text{Im}(\omega_{s=1,l=1})$
$\displaystyle\approx 0.0134254p-0.202685,$ (18b)
$\displaystyle\text{Re}(\omega_{s=1,l=2})$ $\displaystyle\approx
0.916381-0.0344771p,$ (18c) $\displaystyle\text{Im}(\omega_{s=1,l=2})$
$\displaystyle\approx 0.0147288p-0.239645,$ (18d)
$\displaystyle\text{Re}(\omega_{s=1,l=3})$ $\displaystyle\approx
1.31503-0.0482737p,$ (18e) $\displaystyle\text{Im}(\omega_{s=1,l=3})$
$\displaystyle\approx 0.0174108p-0.233444.$ (18f)

(a)

(b)

Figure 9: Fundamental quasinormal mode $(n=0)$ for massless scalar fields with
$r_{0}=1$, red line denotes the real part of frequency and blue lines denote
the imaginary part of the frequency.

(a)

(b)

Figure 10: Fundamental quasinormal mode $(n=0)$ for massless scalar fields
with $r_{0}=1$, red line denotes the real part of frequency and blue lines
denote the imaginary part of the frequency.

(a)

(b)

Figure 11: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

(a)

(b)

Figure 12: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

(a)

(b)

Figure 13: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

(a)

(b)

Figure 14: The dependence of quasinormal frequency on mass $(m)$ for massive
scalar fields with $r_{0}=1$, $p=0.5$ red line denote the real part of
frequency and blue lines denote the imaginary part of the frequency.

The behaviour of real and imaginary parts of the quasinormal modes for massive
scalar fields as a function of mass ($m$) is shown in Fig. 14. From Fig. 14(a)
we can see that the real part of quasinormal frequency is an increasing
function of mass $m$ and the imaginary part of the quasinormal frequency (Fig.
14b) approaches zero, which indicates the existence of quasi-resonances.

####  3.2.1 Analytical Formula For QNMS in the Eikonal Regime

In the domain of high multipole numbers $l$ (eikonal), test fields with
varying spin follow a common law up to the leading order. In this context, we
examine an electromagnetic field with the effective potential 9(b). In the
eikonal regime, one can use the first-order WKB formula given in equation
(14). With the help of Ref. [58] we find that the maximum of effective
potential occurs at $r=r_{max}$ which is given by

$r_{max}=\frac{3r_{0}}{2}+0.0637r_{0}p+\mathcal{O}(p^{2}).$ (19)

Now, substituting $r_{max}$ and equation 9(b) into equation (14) one can
obtain the analytical expression for quasinormal frequency in the eikonal
regime for cubic GB-coupling functional as

$\omega=\frac{(1+2l)(1-0.0596p)-i(1+2n)(1-0.1071p)}{3\sqrt{3}r_{0}}.$ (20)

In the limit $p\to 0$, we obtained the analytical expression of quasinormal
frequency for Schwarzschild black hole.

###  3.3 Quartic GB-Coupling Functional: $f(\varphi)=\varphi^{4}$

The quasinormal modes of massless scalar $(s=0)$ and electromagnetic $(s=1)$
fields for $l=1$, $2$ and $l=1$, $2$, $3$ are shown in Figs. 15-19. From Figs.
15-19 one can be seen that the behaviour of real and imaginary parts of the
quasinormal frequency as a function of parameter $p$ is almost linear for
different values of $l$. In formula 21 and 22 the approximated analytical
expression for quasinormal frequency is written down for different values of
$l$, where parameter $p$ runs from $p=0.0$ to $p=0.7$. After $p=0.7$, the non-
linear behaviour is visible. The approximated linear laws of scalar and
electromagnetic fields for different values of $l$ are given by

$\displaystyle\text{Re}(\omega_{s=0,l=1})$ $\displaystyle\approx
0.583828-0.0130698p,$ (21a) $\displaystyle\text{Im}(\omega_{s=0,l=1})$
$\displaystyle\approx 0.0230895p-0.272239,$ (21b)
$\displaystyle\text{Re}(\omega_{s=0,l=2})$ $\displaystyle\approx
1.00388-0.0383934p,$ (21c) $\displaystyle\text{Im}(\omega_{s=0,l=2})$
$\displaystyle\approx 0.00397625p-0.192472.$ (21d)

$\displaystyle\text{Re}(\omega_{s=1,l=1})$ $\displaystyle\approx
0.495561-0.0117364p,$ (22a) $\displaystyle\text{Im}(\omega_{s=1,l=1})$
$\displaystyle\approx 0.00632363p-0.184499,$ (22b)
$\displaystyle\text{Re}(\omega_{s=1,l=2})$ $\displaystyle\approx
0.943469-0.0345489p,$ (22c) $\displaystyle\text{Im}(\omega_{s=1,l=2})$
$\displaystyle\approx 0.00445784p-0.189155,$ (22d)
$\displaystyle\text{Re}(\omega_{s=1,l=3})$ $\displaystyle\approx
1.36289-0.0510792p,$ (22e) $\displaystyle\text{Im}(\omega_{s=1,l=3})$
$\displaystyle\approx 0.00418319p-0.190336.$ (22f)

(a)

(b)

Figure 15: Fundamental quasinormal mode $(n=0)$ for massless scalar fields
with $r_{0}=1$, red line denotes the real part of frequency and blue lines
denote the imaginary part of the frequency.

(a)

(b)

Figure 16: Fundamental quasinormal mode $(n=0)$ for massless scalar fields
with $r_{0}=1$, red line denotes the real part of frequency and blue lines
denote the imaginary part of the frequency.

(a)

(b)

Figure 17: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

(a)

(b)

Figure 18: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

(a)

(b)

Figure 19: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

The behaviour of real and imaginary parts of the quasinormal modes for massive
scalar fields as a function of mass ($m$) is shown in Fig. 20 for quartic GB-
coupling functional. The real part of the quasinormal frequency is shown in
Fig. 20(a), which is an increasing function of mass $m$. The imaginary part is
shown in Fig. 20(b) and it approaches zero as mass increases. Therefore, the
behaviour of the imaginary part of quasinormal frequency indicates the
existence of quasi-resonances for quartic GB-coupling functional.

(a)

(b)

Figure 20: The dependence of quasinormal frequency on mass $(m)$ for massive
fields with $r_{0}=1$, $p=0.5$ red line denotes real part of frequency and
blue lines denote the imaginary part of the frequency.

####  3.3.1 Analytical Formula For QNMS in the Eikonal Regime

In the domain of high multipole numbers $l$ (eikonal), test fields with
varying spin follow a common law up to the leading order. In this context, we
examine an electromagnetic field with the effective potential 9(b). In the
eikonal regime, one can use the first-order WKB formula given in equation
(14). With the help of numerical analysis given in Ref. [58] we find that the
maximum of effective potential occurs at $r=r_{max}$ which is given by

$r_{max}=\frac{3r_{0}}{2}+0.0527r_{0}p+\mathcal{O}(p^{2}).$ (23)

Now, substituting $r_{max}$ and equation 9(b) into equation (14) one can
obtain the analytical expression for quasinormal frequency in the eikonal
regime as

$\omega=\frac{(1+2l)(1-0.0570p)-i(1+2n)(1-0.0971p)}{3\sqrt{3}r_{0}}.$ (24)

In the limit $p\to 0$, we obtained the analytical expression of quasinormal
frequency for Schwarzschild black hole.

###  3.4  Inverse GB-Coupling Functional: $f(\varphi)=\varphi{-1}$

The quasinormal modes of massless scalar $(s=0)$ and electromagnetic $(s=1)$
fields for $l=1$, $2$ and $l=1$, $2$, $3$ are shown in Figs. 21-25. From Figs.
21-25 one can see that the behaviour of real and imaginary parts of the
quasinormal frequency as a function of parameter $p$ is almost linear for
different values of $l$. In formula 25 and 26 the parameter $p$ runs from
$p=0.0$ to $p=0.65$. The approximated linear law of scalar and electromagnetic
fields for different values of $l$ are given by

$\displaystyle\text{Re}(\omega_{s=0,l=1})$ $\displaystyle\approx
0.58558-0.0330224p,$ (25a) $\displaystyle\text{Im}(\omega_{s=0,l=1})$
$\displaystyle\approx 0.0183859p-0.194868,$ (25b)
$\displaystyle\text{Re}(\omega_{s=0,l=2})$ $\displaystyle\approx
0.966147-0.0558721p,$ (25c) $\displaystyle\text{Im}(\omega_{s=0,l=2})$
$\displaystyle\approx 0.0231498p-0.215501.$ (25d)

$\displaystyle\text{Re}(\omega_{s=1,l=1})$ $\displaystyle\approx
0.495763-0.0233709p,$ (26a) $\displaystyle\text{Im}(\omega_{s=1,l=1})$
$\displaystyle\approx 0.0139608p-0.184918,$ (26b)
$\displaystyle\text{Re}(\omega_{s=1,l=2})$ $\displaystyle\approx
0.915373-0.0367994p,$ (26c) $\displaystyle\text{Im}(\omega_{s=1,l=2})$
$\displaystyle\approx 0.0122145p-0.188642,$ (26d)
$\displaystyle\text{Re}(\omega_{s=1,l=3})$ $\displaystyle\approx
1.31159-0.0728545p,$ (26e) $\displaystyle\text{Im}(\omega_{s=1,l=3})$
$\displaystyle\approx 0.011762p-0.189603.$ (26f)

(a)

(b)

Figure 21: Fundamental quasinormal mode $(n=0)$ for massless scalar fields
with $r_{0}=1$, red line denotes the real part of frequency and blue lines
denote the imaginary part of the frequency.

(a)

(b)

Figure 22: Fundamental quasinormal mode $(n=0)$ for massless scalar fields
with $r_{0}=1$, red line denotes the real part of frequency and blue lines
denote the imaginary part of the frequency.

(a)

(b)

Figure 23: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

(a)

(b)

Figure 24: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

(a)

(b)

Figure 25: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

(a)

(b)

Figure 26: The dependence of quasinormal frequency on mass $(m)$ for massive
scalar fields with $r_{0}=1$, $p=0.5$, red line denotes real part of frequency
and blue lines denote the imaginary part of the frequency.

The behaviour of real and imaginary parts of the quasinormal modes for massive
scalar fields as a function of mass ($m$) is shown in Fig. 26 for inverse GB-
coupling functional. From Fig. 26(b) we can see that the imaginary part of the
quasinormal frequency slowly approaches zero as mass $m$ increases, which
indicates the existence of arbitrarily long-lived frequencies, called quasi-
resonances.

####  3.4.1 Analytical Formula For QNMS in the Eikonal Regime

In the domain of high multipole numbers $l$ (eikonal), test fields with
varying spin follow a common law up to the leading order. In this context, we
examine an electromagnetic field with the effective potential 9(b). In the
eikonal regime, one can use the first-order WKB formula given in equation
(14). Using the Ref. [58] we find that the maximum of effective potential
occurs at $r=r_{max}$ which is given by

$r_{max}=\frac{3r_{0}}{2}+0.0563r_{0}p+\mathcal{O}(p^{2}).$ (27)

Now, substituting $r_{max}$ and equation 9(b) into equation (14) one can
obtain the analytical expression for quasinormal frequency in the eikonal
regime as

$\omega=\frac{(1+2l)(1-0.0699p)-i(1+2n)(1-0.1201p)}{3\sqrt{3}r_{0}}.$ (28)

In the limit $p\to 0$, we obtained the analytical expression of quasinormal
frequency for Schwarzschild black hole.

###  3.5 Logarithmic GB-Coupling Functional: $f(\varphi)=\ln{(\varphi)}$

The quasinormal modes of massless scalar $(s=0)$ and electromagnetic $(s=1)$
fields for $l=1$, $2$ and $l=1$, $2$, $3$ are shown in Figs. 27-31. From Figs.
27-31 one can see that the behaviour of real and imaginary parts of the
quasinormal frequency as a function of parameter $p$ is almost linear for
different values of $l$. In formulas 29 and 30 the parameter $p$ runs from
$p=0.0$ to $p=0.70$. The approximated linear laws for scalar and
electromagnetic fields for different values of $l$ are given by

$\displaystyle\text{Re}(\omega_{s=0,l=1})$ $\displaystyle\approx
0.587079-0.043569p,$ (29a) $\displaystyle\text{Im}(\omega_{s=0,l=1})$
$\displaystyle\approx 0.0231869p-0.195646,$ (29b)
$\displaystyle\text{Re}(\omega_{s=0,l=2})$ $\displaystyle\approx
0.969141-0.0766068p,$ (29c) $\displaystyle\text{Im}(\omega_{s=0,l=2})$
$\displaystyle\approx 0.0245713p-0.19412.$ (29d)

$\displaystyle\text{Re}(\omega_{s=1,l=1})$ $\displaystyle\approx
0.497885-0.0368047p,$ (30a) $\displaystyle\text{Im}(\omega_{s=1,l=1})$
$\displaystyle\approx 0.0271431p-0.187726,$ (30b)
$\displaystyle\text{Re}(\omega_{s=1,l=2})$ $\displaystyle\approx
0.915132-0.0449956p,$ (30c) $\displaystyle\text{Im}(\omega_{s=1,l=2})$
$\displaystyle\approx 0.0277317p-0.191956,$ (30d)
$\displaystyle\text{Re}(\omega_{s=1,l=3})$ $\displaystyle\approx
1.31625-0.103508p,$ (30e) $\displaystyle\text{Im}(\omega_{s=1,l=3})$
$\displaystyle\approx 0.0252856p-0.192324.$ (30f)

(a)

(b)

Figure 27: Fundamental quasinormal mode $(n=0)$ for massless scalar fields
with $r_{0}=1$, red line denotes the real part of frequency and blue lines
denote the imaginary part of the frequency.

(a)

(b)

Figure 28: Fundamental quasinormal mode $(n=0)$ for massless scalar fields
with $r_{0}=1$, red line denotes the real part of frequency and blue lines
denote the imaginary part of the frequency.

(a)

(b)

Figure 29: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

(a)

(b)

Figure 30: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

(a)

(b)

Figure 31: Fundamental quasinormal mode $(n=0)$ for electromagnetic fields
with $r_{0}=1$, red line denotes real part of frequency and blue lines denote
the imaginary part of the frequency.

The behaviour of real and imaginary parts of the quasinormal modes for massive
scalar fields as a function of mass ($m$) is shown in Fig. 32. From Fig. 32(b)
we can see that the imaginary part of the quasinormal frequency approaches
zero, which indicates the existence of quasi-resonances.

(a)

(b)

Figure 32: The dependence of quasinormal frequency on mass $(m)$ for massive
scalar fields with $r_{0}=1$, $p=0.5$, red line denotes real part of frequency
and blue lines denote the imaginary part of the frequency.

####  3.5.1 Analytical Formula For QNMS in the Eikonal Regime

In the domain of high multipole numbers $l$ (eikonal), test fields with
varying spin follow a common law up to the leading order. In this context, we
examine an electromagnetic field with the effective potential 9(b). In the
eikonal regime, one can use the first-order WKB formula given in equation
(14). Using the Ref. [58] we find that the maximum of effective potential 9(b)
occurs at $r=r_{max}$ which is given by

$r_{max}=\frac{3r_{0}}{2}+0.0544r_{0}p+\mathcal{O}(p^{2}).$ (31)

Now, substituting $r_{max}$ and equation 9(b) into equation (14) one can
obtain the analytical expression for quasinormal frequency in the eikonal
regime for logarithmic GB-coupling functional as

$\omega=\frac{(1+2l)(1-0.0690p)-i(1+2n)(1-0.1181p)}{3\sqrt{3}r_{0}}.$ (32)

In the limit $p\to 0$, we obtained the analytical expression of quasinormal
frequency for Schwarzschild black hole.

As one can see from this and previous analytical relations for the eikonal
regime, the quasinormal modes are fully reproduced by the WKB formula and,
therefore, the correspondence between eikonal qnms and null geodesics
suggested in [51] is indeed valid for the test fields under consideration.

##  4 Grey–Body Factors

The strength of Hawking radiation does not fully reach a distant observer, it
is partially suppressed by the effective potential surrounding the black holes
and reflected to the black hole event horizon. To find the number of particles
reflected due to the effective potential, it is essential to determine the
grey–body factors. This involves solving the classical scattering problem to
estimate the number of particles that undergo reflection.

We will investigate the wave equation (7) under the boundary conditions that
allow the inclusion of incoming waves from infinity,

$\displaystyle\Psi$ $\displaystyle=e^{-i\omega r_{*}}+Re^{i\omega
r_{*}},r_{*}\to\infty,$ (33a) $\displaystyle\Psi$ $\displaystyle=Te^{-i\omega
r_{*}},r_{*}\to-\infty,$ (33b)

where R and T are the reflection and transmission coefficients. This scenario
is equivalent to the scattering of a wave originating from the horizon.

The nature of the effective potential is such that it decreases at both
infinity, therefore we can safely apply the WKB method [45, 46, 47, 48] to
find reflection and transmission coefficients. The reflection and transmission
coefficients satisfy the following relations

$|R|^{2}+|T|^{2}=1.$ (34)

Once the reflection coefficient is computed, we can find the transmission
coefficient for each multipole number as follows

$|A_{l}|^{2}=1-|R_{l}|^{2}.$ (35)

Numerous approaches are available in the literature for the computation of the
transmission and reflection coefficients. For an accurate computation of the
transmission and reflection coefficients, we employed the 6th-order WKB
formula [45, 46, 47, 48]. In accordance with the findings presented in [45,
46, 47, 48], the reflection coefficient can be written as

$R=(1+e^{-2i\pi K})^{-\frac{1}{2}},$ (36)

where $K$ is defined by the following relations

$K-i\frac{(\omega^{2}-V_{0})}{\sqrt{-2V_{0}^{\prime\prime}}}-\sum_{i=2}^{i=6}\Lambda_{i}(K)=0,$
(37)

where $V_{0}$ is the effective potential at $r=r_{max}$,
$V_{0}^{\prime\prime}$ is the double derivative of effective potential with
respect to radial coordinate $r$, evaluated at $r=r_{max}$ and $\Lambda_{i}$
is the higher-order correction to the WKB formula.

###  4.1 Quadratic GB-Coupling Functional

(a)

(b)

(c)

Figure 33: The dependence of grey-body factor on $\omega$, for the
electromagnetic field $(s=1)$, with $r_{0}=1$, black line denotes $p=0.0$,
magenta line denotes $p=0.2$, blue line denotes $p=0.4$, red line denotes
$p=0.6$ and green line denotes $p=0.8$.

The grey–body factors for quadratic GB-coupling functional are shown in Fig.
33 as a function of $\omega$ for electromagnetic fields with different values
of multiple numbers $l$. From Fig. 33 it can be seen that the grey–body factor
is higher for EsGB black holes compared to Schwarzschild black holes. As the
parameter $p$ increases transmission rate of the particles also increases,
which is consistent with the nature of the effective potential, i.e., as the
parameter $p$ increases the height of the potential barrier becomes lower
resulting in a high transmission rate for the particles and vice-versa.

###  4.2 Cubic GB-Coupling Functional

(a)

(b)

(c)

Figure 34: The dependence of grey-body factor on $\omega$, for the
electromagnetic field $(s=1)$, with $r_{0}=1$, black line denotes $p=0.0$,
magenta line denotes $p=0.2$, blue line denotes $p=0.4$, red line denotes
$p=0.6$ and green line denotes $p=0.8$.

The grey–body factors for cubic GB-coupling functional are shown in Fig. 34 as
a function of $\omega$ for electromagnetic fields with different values of
multiple numbers $l$. From Fig. 34 it can be seen that the grey–body factor is
higher for EsGB black holes compared to Schwarzschild black holes. As the
parameter $p$ increases transmission rate of the particles also increases,
which is consistent with the nature of the effective potential, i.e., as the
parameter $p$ increases the height of the potential barrier becomes lower
resulting in a high transmission rate for the particles and vice-versa.

###  4.3 Quartic GB-Coupling Functional

(a)

(b)

(c)

Figure 35: The dependence of grey-body factor on $\omega$, for the
electromagnetic field $(s=1)$, with $r_{0}=1$, black line denotes $p=0.0$,
magenta line denotes $p=0.2$, blue line denotes $p=0.4$, red line denotes
$p=0.6$ and green line denotes $p=0.8$.

The grey–body factors for quartic GB-coupling functional are shown in Fig. 35
as a function of $\omega$ for electromagnetic fields with different values of
multiple numbers $l$. From Fig. 35 it can be seen that the grey–body factor is
higher for EsGB black holes compared to Schwarzschild black holes. As the
parameter $p$ increases transmission rate of the particles also increases,
which is consistent with the nature of the effective potential, i.e., as the
parameter $p$ increases the height of the potential barrier becomes lower
resulting in a high transmission rate for the particles and vice-versa.

###  4.4 Inverse GB-Coupling Functional

(a)

(b)

(c)

Figure 36: The dependence of grey-body factor on $\omega$, for the
electromagnetic field $(s=1)$, with $r_{0}=1$, black line denotes $p=0.0$,
magenta line denotes $p=0.2$, blue line denotes $p=0.4$, red line denotes
$p=0.6$ and green line denotes $p=0.8$.

The grey–body factors for inverse GB-coupling functional are shown in Fig. 36
as a function of $\omega$ for electromagnetic fields with different values of
multiple numbers $l$. From Fig. 36 it can be seen that the grey–body factor is
higher for EsGB black holes compared to Schwarzschild black holes. As the
parameter $p$ increases transmission rate of the particles also increases,
which is consistent with the nature of the effective potential, i.e., as the
parameter $p$ increases the height of the potential barrier becomes lower
resulting in a high transmission rate for the particles and vice-versa.

###  4.5 Logarithmic GB-Coupling Functional

(a)

(b)

(c)

Figure 37: The dependence of grey-body factor on $\omega$, for the
electromagnetic field $(s=1)$, with $r_{0}=1$, black line denotes $p=0.0$,
magenta line denotes $p=0.2$, blue line denotes $p=0.4$, red line denotes
$p=0.6$ and green line denotes $p=0.8$.

The grey–body factors for logarithmic GB-coupling functional are shown in Fig.
37 as a function of $\omega$ for electromagnetic fields with different values
of multiple numbers $l$. From Fig. 37 it can be seen that the grey–body factor
is higher for EsGB black holes compared to Schwarzschild black holes. As the
parameter $p$ increases transmission rate of the particles also increases,
which is consistent with the nature of the effective potential, i.e., as the
parameter $p$ increases the height of the potential barrier becomes lower
resulting in a high transmission rate for the particles and vice-versa.

Here, we will write a brief analysis of how different GB-coupling functions
affect the quasinormal modes of the black hole. From above analysis of the
quasinormal modes of EsGB black hole as a function of parameter $p$, we see a
strong deviation of qnms for both massless scalar and electromagnetic field
compared to Schwarzschild black hole when GB-coupling functions are
logarithmic. The deviation of quasinormal modes is lowest for quadratic GB-
coupling functional. The transmission rate of the particles is highest for
logarithmic GB-coupling functional (i.e., the strongest deviation of
transmission rate occurs compared to Schwarzschild black hole) and lowest for
quadratic GB-coupling functional (i.e., the lowest deviation of transmission
rate occurs compared to Schwarzschild black hole).

##  5 Conclusions

In this study, we address previously unexplored aspects of quasinormal modes
for scalar and electromagnetic fields in the vicinity of analytically obtained
EsGB black holes solutions. Here, we investigate:

  * •

The quasinormal modes of scalar and electromagnetic fields in the background
of EsGB black hole for different GB coupling functional (quadratic, cubic,
quartic, inverse, and Logarithmic). The dependence of the real and imaginary
parts of the quasinormal modes on the parameter $p$ is shown in Figures. For
$p=0$, quasinormal modes of EsGB black hole match with Schwarzschild one. For
non–zero values of $p$ quasinormal frequency of EsGB black hole take smaller
values compared to Schwarzschild black hole.

  * •

We analyzed massive scalar fields in the EsGB background. The height of the
effective potential for massive scalar fields is higher as the mass increases.
For large values of mass, it shows undamped oscillations occur, which commonly
known as quasi–resonances.

  * •

We derived an analytical formula of quasinormal frequency in the eikonal
regime for each GB coupling function. In the limit $p\to 0$, it reduced to the
quasinormal frequency of Schwarzschild black hole. We have shown that the
correspondence between the eikonal quasinormal frequencies and null geodesics
[51] is hold in the EsGB theory for the test fields under consideration.

  * •

Finally, we studied the grey-body factors for electromagnetic fields with
different values of multiple numbers. Our analysis showed that for higher
values of parameter $p$ grey-body factors are larger.

##  6 Acknowledgements

I would like to thank Roman Konoplya for useful discussions and help with the
Mathematica WKB code.

##  Appendix A Appendix: Analytical expressions for the metric functions up to
fourth order

Here, we will write down the analytical expression for the metric functions up
to the fourth order in the CFA method. Therefore, the metric functions are
given by

$g_{tt}(r)\approx\frac{\mathcal{N}^{1}}{\mathcal{D}^{1}}(1-\frac{r_{0}}{r}),$
(A1)
$\sqrt{g_{tt}(r)g_{rr}(r)}\approx\frac{\mathcal{N}^{2}}{\mathcal{D}^{2}},$
(A2)

where the numerators, $\mathcal{N}^{1}$, $\mathcal{N}^{2}$ and denominators,
$\mathcal{D}^{1}$, $\mathcal{D}^{2}$ are separately given for each GB coupling
functional.

###  A.1 Quadratic GB-Coupling Functional: $f(\varphi)=\varphi^{2}$

$g_{tt}(r)\approx\frac{\mathcal{N}^{1}_{eve2}}{\mathcal{D}^{1}_{eve2}}(1-\frac{r_{0}}{r}),$
(A3)
$\sqrt{g_{tt}(r)g_{rr}(r)}\approx\frac{\mathcal{N}^{2}_{eve2}}{\mathcal{D}^{2}_{eve2}},$
(A4)

where

$\displaystyle\mathcal{N}^{1}_{\text{eve}2}$
$\displaystyle=p^{11}(-0.027339r^{3}r_{0}+0.027339r^{2}r_{0}^{2}+0.027339r{r_{0}}^{3}-0.027339{r_{0}}^{4})+p^{10}(-1.65064r^{3}{r_{0}}$
(A5)
$\displaystyle+r^{4}+0.639158r^{2}{r_{0}}^{2}-0.349358r{r_{0}}^{3}+0.360842{r_{0}}^{4})+p^{9}(-13.5993r^{4}+25.9764r^{3}{r_{0}}$
$\displaystyle-12.2194r^{2}{r_{0}}^{2}+1.96871r{r_{0}}^{3}-2.12636{r_{0}}^{4})+p^{8}(66.7223r^{4}-136.15r^{3}{r_{0}}+68.5796r^{2}{r_{0}}^{2}$
$\displaystyle-6.96288r{r_{0}}^{3}+7.81103{r_{0}}^{4})+p^{7}(-142.79r^{4}+317.427r^{3}{r_{0}}-172.292r^{2}{r_{0}}^{2}+15.8938r{r_{0}}^{3}$
$\displaystyle-18.2388{r_{0}}^{4})+p^{6}(101.122r^{4}-301.89r^{3}{r_{0}}+197.051r^{2}{r_{0}}^{2}-22.9918r{r_{0}}^{3}+26.7025{r_{0}}^{4})$
$\displaystyle+p^{5}(120.603r^{4}-78.7254r^{3}{r_{0}}-38.3503r^{2}{r_{0}}^{2}+20.5997r{r_{0}}^{3}-24.0645{r_{0}}^{4})+p^{4}(-295.148r^{4}$
$\displaystyle+437.019r^{3}{r_{0}}-143.957r^{2}{r_{0}}^{2}-10.8823r{r_{0}}^{3}+12.7195{r_{0}}^{4})+p^{3}(236.149r^{4}-388.645r^{3}{r_{0}}$
$\displaystyle+153.456r^{2}{r_{0}}^{2}+3.01323r{r_{0}}^{3}-3.47898{r_{0}}^{4})+p^{2}(-85.9513r^{4}+147.171r^{3}{r_{0}}-61.7593r^{2}{r_{0}}^{2}$
$\displaystyle-0.319982r{r_{0}}^{3}+0.336681{r_{0}}^{4})+p(12.5012r^{4}-21.3996r^{3}{r_{0}}+9.1727r^{2}{r_{0}}^{2}+0.00359074r{r_{0}}^{3}$
$\displaystyle+0.0054195{r_{0}}^{4})-0.608781r^{4}+0.894103r^{3}{r_{0}}-0.347264r^{2}{r_{0}}^{2},$

$\displaystyle\mathcal{D}^{1}_{\text{eve}2}$
$\displaystyle=p^{10}r^{2}(r^{2}-2r{r_{0}}+{r_{0}}^{2})+p^{9}r^{2}(-13.5993r^{2}+27.6187r{r_{0}}-14.0194{r_{0}}^{2})+p^{8}r^{2}(66.7223r^{2}$
(A6)
$\displaystyle-139.556r{r_{0}}+72.8336{r_{0}}^{2})+p^{7}r^{2}(-142.79r^{2}+319.724r{r_{0}}-176.934{r_{0}}^{2})+p^{6}r^{2}(101.122r^{2}$
$\displaystyle-298.896r{r_{0}}+197.772{r_{0}}^{2})+p^{5}r^{2}(120.603r^{2}-85.7535r{r_{0}}-34.8151{r_{0}}^{2})+p^{4}r^{2}(-295.148r^{2}$
$\displaystyle+442.573r{r_{0}}-147.595{r_{0}}^{2})+p^{3}r^{2}(236.149r^{2}-390.653r{r_{0}}+154.893{r_{0}}^{2})+p^{2}r^{2}(-85.9513r^{2}$
$\displaystyle+147.462r{r_{0}}-61.9667{r_{0}}^{2})+pr^{2}(12.5012r^{2}-21.4138r{r_{0}}+9.17934{r_{0}}^{2})+r^{2}(-0.608781r^{2}$
$\displaystyle+0.894103r{r_{0}}-0.347264{r_{0}}^{2}),$

$\displaystyle\mathcal{N}^{2}_{\text{eve}2}$
$\displaystyle=p^{9}(r^{3}-2r^{2}{r_{0}}+r{r_{0}}^{2})+p^{8}(-8.86767r^{3}+17.7041r^{2}{r_{0}}-8.8364r{r_{0}}^{2})+p^{7}(29.1248r^{3}$
(A7)
$\displaystyle-57.7325r^{2}{r_{0}}+28.8414r{r_{0}}^{2}-0.233666{r_{0}}^{3})+p^{6}(-56.6765r^{3}+110.257r^{2}{r_{0}}-54.735r{r_{0}}^{2}$
$\displaystyle+1.1405{r_{0}}^{3})+p^{5}(90.3253r^{3}-171.817r^{2}{r_{0}}+83.6835r{r_{0}}^{2}-2.08071{r_{0}}^{3})+p^{4}(-119.877r^{3}$
$\displaystyle+226.049r^{2}{r_{0}}-108.183r{r_{0}}^{2}+1.65941{r_{0}}^{3})+p^{3}(105.82r^{3}-199.251r^{2}{r_{0}}+94.4276r{r_{0}}^{2}$
$\displaystyle-0.411268{r_{0}}^{3})+p^{2}(-51.741r^{3}+96.6288r^{2}{r_{0}}-45.2948r{r_{0}}^{2}-0.132824{r_{0}}^{3})+p(12.1153r^{3}$
$\displaystyle-21.8254r^{2}{r_{0}}+9.91398r{r_{0}}^{2}+0.058561{r_{0}}^{3})-1.22281r^{3}+1.98757r^{2}{r_{0}}-0.817317r{r_{0}}^{2},$

$\displaystyle\mathcal{D}^{2}_{\text{eve}2}$
$\displaystyle=p^{9}r(1.r^{2}-2.r{r_{0}}+1.{r_{0}}^{2})+p^{8}r(-8.86767r^{2}+17.7041r{r_{0}}-8.8364{r_{0}}^{2})+p^{7}r(29.1248r^{2}$
(A8)
$\displaystyle-57.7325r{r_{0}}+28.6077{r_{0}}^{2})+p^{6}r(-56.6765r^{2}+110.257r{r_{0}}-53.5872{r_{0}}^{2})+p^{5}r(90.3253r^{2}$
$\displaystyle-171.817r{r_{0}}+81.5611{r_{0}}^{2})+p^{4}r(-119.877r^{2}+226.049r{r_{0}}-106.429{r_{0}}^{2})+p^{3}r(105.82r^{2}$
$\displaystyle-199.251r{r_{0}}+93.9092{r_{0}}^{2})+p^{2}r(-51.741r^{2}+96.6288r{r_{0}}-45.3674{r_{0}}^{2})+pr(12.1153r^{2}$
$\displaystyle-21.8254r{r_{0}}+9.95903{r_{0}}^{2})+r(-1.22281r^{2}+1.98757r{r_{0}}-0.817317{r_{0}}^{2}).$

###  A.2 Cubic GB-Coupling Functional: $f(\varphi)=\varphi^{3}$

$g_{tt}(r)\approx\frac{\mathcal{N}^{1}_{odd3}}{\mathcal{D}^{1}_{odd3}}(1-\frac{r_{0}}{r}),$
(A9)
$\sqrt{g_{tt}(r)g_{rr}(r)}\approx\frac{\mathcal{N}^{2}_{odd3}}{\mathcal{D}^{2}_{odd3}},$
(A10)

where

$\displaystyle\mathcal{N}^{1}_{\text{odd}3}$
$\displaystyle=p^{12}(0.00537634r^{3}r_{0}-0.00537634r^{2}r_{0}^{2}-0.00537634rr_{0}^{3}+0.00537634r_{0}^{4})$
(A11)
$\displaystyle+p^{11}(-0.0670504r^{3}r_{0}+0.0683756r^{2}r_{0}^{2}+0.0670504rr_{0}^{3}-0.0683756r_{0}^{4})+p^{10}(1.r^{4}$
$\displaystyle-1.70036r^{3}r_{0}+0.683358r^{2}r_{0}^{2}-0.299635rr_{0}^{3}+0.316642r_{0}^{4})+p^{9}(-6.78182r^{4}+13.1727r^{3}r_{0}$
$\displaystyle-6.31042r^{2}r_{0}^{2}+0.737004rr_{0}^{3}-0.817608r_{0}^{4})+p^{8}(17.7971r^{4}-36.7027r^{3}r_{0}+18.714r^{2}r_{0}^{2}$
$\displaystyle-1.12269rr_{0}^{3}+1.3156r_{0}^{4})+p^{7}(-21.7877r^{4}+48.5071r^{3}r_{0}-26.4795r^{2}r_{0}^{2}+1.08356rr_{0}^{3}$
$\displaystyle-1.34315r_{0}^{4})+p^{6}(9.51766r^{4}-26.9583r^{3}r_{0}+17.3344r^{2}r_{0}^{2}-0.638392rr_{0}^{3}+0.834102r_{0}^{4})$
$\displaystyle+p^{5}(4.59852r^{4}-2.91241r^{3}r_{0}-1.79709r^{2}r_{0}^{2}+0.19656rr_{0}^{3}-0.266067r_{0}^{4})+p^{4}(-5.37404r^{4}$
$\displaystyle+8.71817r^{3}r_{0}-3.1644r^{2}r_{0}^{2}-0.0116997rr_{0}^{3}+0.0128034r_{0}^{4})+p^{3}(0.540319r^{4}-1.28023r^{3}r_{0}$
$\displaystyle+0.652328r^{2}r_{0}^{2}-0.00618425rr_{0}^{3}+0.0102582r_{0}^{4})+p^{2}(0.451911r^{4}-0.727369r^{3}r_{0}+0.284135r^{2}r_{0}^{2}$
$\displaystyle-0.00017735rr_{0}^{3}+0.000404216r_{0}^{4})+p(0.0379465r^{4}-0.0549725r^{3}r_{0}+0.0202381r^{2}r_{0}^{2}-0.0000201357rr_{0}^{3}$
$\displaystyle+0.0000179163r_{0}^{4})+0.000140052r^{4}-0.0000971891r^{3}r_{0},$

$\displaystyle\mathcal{D}^{1}_{\text{odd}3}$
$\displaystyle=p^{10}r^{2}(1.r^{2}-2.rr_{0}+1.r_{0}^{2})+p^{9}r^{2}(-6.78182r^{2}+13.8101rr_{0}-7.02829r_{0}^{2})$
(A12)
$\displaystyle+p^{8}r^{2}(17.7971r^{2}-37.355rr_{0}+19.5579r_{0}^{2})+p^{7}r^{2}(-21.7877r^{2}+48.6892rr_{0}-26.9117r_{0}^{2})$
$\displaystyle+p^{6}r^{2}(9.51766r^{2}-26.7448rr_{0}+17.2901r_{0}^{2})+p^{5}r^{2}(4.59852r^{2}-3.06912rr_{0}-1.67657r_{0}^{2})$
$\displaystyle+p^{4}r^{2}(-5.37404r^{2}+8.71215rr_{0}-3.17516r_{0}^{2})+p^{3}r^{2}(0.540319r^{2}-1.2622rr_{0}+0.6403r_{0}^{2})$
$\displaystyle+p^{2}r^{2}(0.451911r^{2}-0.725183rr_{0}+0.28313r_{0}^{2})+pr^{2}(0.0379465r^{2}-0.0549644rr_{0}+0.0202407r_{0}^{2})$
$\displaystyle+r^{2}(0.000140052r^{2}-0.0000971891rr_{0}),$

$\displaystyle\mathcal{N}^{2}_{\text{odd}3}$
$\displaystyle=p^{9}(1.r^{3}-2.r^{2}r_{0}+1.rr_{0}^{2})+p^{8}(-15.624r^{3}+31.3197r^{2}r_{0}-15.587rr_{0}^{2}-0.108754r_{0}^{3})+p^{7}(75.8827r^{3}$
(A13)
$\displaystyle-153.129r^{2}r_{0}+75.7277rr_{0}^{2}+1.51139r_{0}^{3})+p^{6}(-135.885r^{3}+279.413r^{2}r_{0}-138.956rr_{0}^{2}-4.46757r_{0}^{3})$
$\displaystyle+p^{5}(72.0953r^{3}-161.504r^{2}r_{0}+83.7029rr_{0}^{2}+5.22405r_{0}^{3})+p^{4}(47.4153r^{3}-75.3996r^{2}r_{0}+31.4976rr_{0}^{2}$
$\displaystyle-2.47728r_{0}^{3})+p^{3}(-61.6661r^{3}+111.092r^{2}r_{0}-50.8295rr_{0}^{2}+0.253284r_{0}^{3})+p^{2}(18.923r^{3}-32.9427r^{2}r_{0}$
$\displaystyle+14.5941rr_{0}^{2}+0.0638083r_{0}^{3})+p(-2.11497r^{3}+3.11041r^{2}r_{0}-1.13326rr_{0}^{2}+0.0010564r_{0}^{3})-0.0225954r^{3}$
$\displaystyle+0.0327384r^{2}r_{0}-0.0128335rr_{0}^{2},$

$\displaystyle\mathcal{D}^{2}_{\text{odd}3}$
$\displaystyle=p^{9}r(1.r^{2}-2.rr_{0}+1.r_{0}^{2})+p^{8}r(-15.624r^{2}+31.3197rr_{0}-15.6957r_{0}^{2})+p^{7}r(75.8827r^{2}$
(A14)
$\displaystyle-153.129rr_{0}+77.2427r_{0}^{2})+p^{6}r(-135.885r^{2}+279.413rr_{0}-143.469r_{0}^{2})+p^{5}r(72.0953r^{2}-161.504rr_{0}$
$\displaystyle+89.0785r_{0}^{2})+p^{4}r(47.4153r^{2}-75.3996rr_{0}+28.8021r_{0}^{2})+p^{3}r(-61.6661r^{2}+111.092rr_{0}-50.4318r_{0}^{2})$
$\displaystyle+p^{2}r(18.923r^{2}-32.9427rr_{0}+14.6226r_{0}^{2})+pr(-2.11497r^{2}+3.11041rr_{0}-1.13289r_{0}^{2})+r(-0.0225954r^{2}$
$\displaystyle+0.0327384rr_{0}-0.0128335r_{0}^{2}).$

###  A.3 Quartic GB-Coupling Functional: $f(\varphi)=\varphi^{4}$

$g_{tt}(r)\approx\frac{\mathcal{N}^{1}_{eve4}}{\mathcal{D}^{1}_{eve4}}(1-\frac{r_{0}}{r}),$
(A15)
$\sqrt{g_{tt}(r)g_{rr}(r)}\approx\frac{\mathcal{N}^{2}_{eve4}}{\mathcal{D}^{2}_{eve4}},$
(A16)

where

$\displaystyle\mathcal{N}^{1}_{\text{eve}4}$
$\displaystyle=p^{15}(1.r^{4}-2.089r^{3}r_{0}+1.089r^{2}r_{0}^{2}+0.0889959rr_{0}^{3}-0.0889959r_{0}^{4})+p^{14}(-6.89482r^{4}$
(A17)
$\displaystyle+14.7896r^{3}r_{0}-7.90886r^{2}r_{0}^{2}-0.757057rr_{0}^{3}+0.771153r_{0}^{4})+p^{13}(12.5106r^{4}-29.4208r^{3}r_{0}$
$\displaystyle+17.0451r^{2}r_{0}^{2}+2.92063rr_{0}^{3}-3.05555r_{0}^{4})+p^{12}(19.4917r^{4}-29.885r^{3}r_{0}+9.84773r^{2}r_{0}^{2}$
$\displaystyle-6.69766rr_{0}^{3}+7.24323r_{0}^{4})+p^{11}(-115.113r^{4}+224.275r^{3}r_{0}-107.925r^{2}r_{0}^{2}+9.97888rr_{0}^{3}$
$\displaystyle-11.206r_{0}^{4})+p^{10}(198.927r^{4}-410.675r^{3}r_{0}+210.013r^{2}r_{0}^{2}-9.86857rr_{0}^{3}+11.562r_{0}^{4})$
$\displaystyle+p^{9}(-164.194r^{4}+364.121r^{3}r_{0}-198.46r^{2}r_{0}^{2}+6.32619rr_{0}^{3}-7.79975r_{0}^{4})+p^{8}(45.2779r^{4}$
$\displaystyle-130.919r^{3}r_{0}+85.2r^{2}r_{0}^{2}-2.38976rr_{0}^{3}+3.15916r_{0}^{4})+p^{7}(28.2324r^{4}-33.1842r^{3}r_{0}$
$\displaystyle+4.40023r^{2}r_{0}^{2}+0.342238rr_{0}^{3}-0.525779r_{0}^{4})+p^{6}(-21.3691r^{4}+38.4559r^{3}r_{0}-16.3563r^{2}r_{0}^{2}$
$\displaystyle+0.0898093rr_{0}^{3}-0.104246r_{0}^{4})+p^{5}(-0.197567r^{4}-1.57336r^{3}r_{0}+1.45736r^{2}r_{0}^{2}-0.0324933rr_{0}^{3}$
$\displaystyle+0.0403357r_{0}^{4})+p^{4}(2.04375r^{4}-3.45917r^{3}r_{0}+1.43513r^{2}r_{0}^{2}-0.00124132rr_{0}^{3}+0.00429355r_{0}^{4})$
$\displaystyle+p^{3}(0.276656r^{4}-0.423294r^{3}r_{0}+0.159108r^{2}r_{0}^{2}+0.0000460001rr_{0}^{3}+0.000142526r_{0}^{4})+p^{2}(0.00912579r^{4}$
$\displaystyle-0.0124531r^{3}r_{0}+0.0040693r^{2}r_{0}^{2}-5.79588\times
10^{-6}rr_{0}^{3}+5.6289\times 10^{-6}r_{0}^{4})$
$\displaystyle+p(-0.0000509404r^{4}+0.000117355r^{3}r_{0}-0.0000607436r^{2}r_{0}^{2}+4.44458\times
10^{-8}rr_{0}^{3}$ $\displaystyle-6.14644\times
10^{-8}r_{0}^{4})-2.63557\times 10^{-7}r^{4}+1.63201\times
10^{-8}r^{3}r_{0}+1.79379\times 10^{-7}r^{2}r_{0}^{2},$

$\displaystyle\mathcal{D}^{1}_{\text{eve}4}$
$\displaystyle=p^{15}r^{2}(1.r^{2}-2.rr_{0}+1.r_{0}^{2})+p^{14}r^{2}(-6.89482r^{2}+13.948rr_{0}-7.05321r_{0}^{2})+p^{13}r^{2}(12.5106r^{2}$
(A18)
$\displaystyle-26.1315rr_{0}+13.6209r_{0}^{2})+p^{12}r^{2}(19.4917r^{2}-36.7725rr_{0}+17.2808r_{0}^{2})+p^{11}r^{2}(-115.113r^{2}$
$\displaystyle+232.465rr_{0}-117.345r_{0}^{2})+p^{10}r^{2}(198.927r^{2}-415.683rr_{0}+216.716r_{0}^{2})+p^{9}r^{2}(-164.194r^{2}+364.547rr_{0}$
$\displaystyle-200.316r_{0}^{2})+p^{8}r^{2}(45.2779r^{2}-129.549rr_{0}+84.468r_{0}^{2})+p^{7}r^{2}(28.2324r^{2}-33.7905rr_{0}+4.977r_{0}^{2})$
$\displaystyle+p^{6}r^{2}(-21.3691r^{2}+38.3498rr_{0}-16.341r_{0}^{2})+p^{5}r^{2}(-0.197567r^{2}-1.50422rr_{0}+1.40334r_{0}^{2})$
$\displaystyle+p^{4}r^{2}(2.04375r^{2}-3.44431rr_{0}+1.42689r_{0}^{2})+p^{3}r^{2}(0.276656r^{2}-0.422662rr_{0}+0.158858r_{0}^{2})$
$\displaystyle+p^{2}r^{2}(0.00912579r^{2}-0.0124564rr_{0}+0.00407385r_{0}^{2})+pr^{2}(-0.0000509404r^{2}+0.000117338rr_{0}$
$\displaystyle-0.0000607602r_{0}^{2})+r^{2}(-2.63557\times
10^{-7}r^{2}+1.63201\times 10^{-8}rr_{0}+1.79379\times 10^{-7}r_{0}^{2}),$

$\displaystyle\mathcal{N}^{2}_{\text{eve}4}$
$\displaystyle=p^{10}(1.r^{3}-2.r^{2}r_{0}+1.rr_{0}^{2})+p^{9}(-13.6797r^{3}+27.4429r^{2}r_{0}-13.6766rr_{0}^{2}-0.0865879r_{0}^{3})$
(A19)
$\displaystyle+p^{8}(67.4124r^{3}-135.988r^{2}r_{0}+67.7236rr_{0}^{2}+0.852114r_{0}^{3})+p^{7}(-148.278r^{3}+302.545r^{2}r_{0}-151.49rr_{0}^{2}$
$\displaystyle-2.77603r_{0}^{3})+p^{6}(146.99r^{3}-308.586r^{2}r_{0}+157.352rr_{0}^{2}+4.22482r_{0}^{3})+p^{5}(-36.5144r^{3}+91.761r^{2}r_{0}$
$\displaystyle-51.9412rr_{0}^{2}-3.22666r_{0}^{3})+p^{4}(-41.1576r^{3}+69.4003r^{2}r_{0}-29.5381rr_{0}^{2}+1.13594r_{0}^{3})+p^{3}(28.9496r^{3}$
$\displaystyle-53.2293r^{2}r_{0}+24.5671rr_{0}^{2}-0.117809r_{0}^{3})+p^{2}(-5.11947r^{3}+9.28066r^{2}r_{0}-4.24613rr_{0}^{2}-0.00574379r_{0}^{3})$
$\displaystyle+p(0.391241r^{3}-0.618344r^{2}r_{0}+0.245794rr_{0}^{2}-0.0000557057r_{0}^{3})+0.00525892r^{3}-0.00809504r^{2}r_{0}$
$\displaystyle+0.00330483rr_{0}^{2},$

$\displaystyle\mathcal{D}^{2}_{\text{eve}4}$
$\displaystyle=p^{10}r(1.r^{2}-2.rr_{0}+1.r_{0}^{2})+p^{9}r(-13.6797r^{2}+27.4429rr_{0}-13.7632r_{0}^{2})+p^{8}r(67.4124r^{2}$
(A20)
$\displaystyle-135.988rr_{0}+68.5757r_{0}^{2})+p^{7}r(-148.278r^{2}+302.545rr_{0}-154.267r_{0}^{2})+p^{6}r(146.99r^{2}-308.586rr_{0}$
$\displaystyle+161.585r_{0}^{2})+p^{5}r(-36.5144r^{2}+91.761rr_{0}-55.1917r_{0}^{2})+p^{4}r(-41.1576r^{2}+69.4003rr_{0}-28.3697r_{0}^{2})$
$\displaystyle+p^{3}r(28.9496r^{2}-53.2293rr_{0}+24.4286r_{0}^{2})+p^{2}r(-5.11947r^{2}+9.28066rr_{0}-4.24704r_{0}^{2})+pr(0.391241r^{2}$
$\displaystyle-0.618344rr_{0}+0.245858r_{0}^{2})+r(0.00525892r^{2}-0.00809504rr_{0}+0.00330483r_{0}^{2}).$

###  A.4 Inverse GB-Coupling Functional: $f(\varphi)=\varphi^{-1}$

$g_{tt}(r)\approx\frac{\mathcal{N}^{1}_{inv}}{\mathcal{D}^{1}_{inv}}(1-\frac{r_{0}}{r}),$
(A21)
$\sqrt{g_{tt}(r)g_{rr}(r)}\approx\frac{\mathcal{N}^{2}_{inv}}{\mathcal{D}^{2}_{inv}},$
(A22)

where

$\displaystyle\mathcal{N}^{1}_{\text{inv}}$
$\displaystyle=p^{13}(0.00630915r^{3}{r_{0}}-0.00630915r^{2}{r_{0}}^{2}-0.00630915r{r_{0}}^{3}+0.00630915{r_{0}}^{4})$
(A23)
$\displaystyle+p^{12}(-0.288247r^{3}{r_{0}}+0.289835r^{2}{r_{0}}^{2}+0.288247r{r_{0}}^{3}-0.289835{r_{0}}^{4})$
$\displaystyle+p^{11}(1.r^{4}+1.44067r^{3}{r_{0}}-2.51152r^{2}{r_{0}}^{2}-2.56824r{r_{0}}^{3}+2.6391{r_{0}}^{4})$
$\displaystyle+p^{10}(-33.5247r^{4}+55.6895r^{3}{r_{0}}-21.3738r^{2}{r_{0}}^{2}+8.14436r{r_{0}}^{3}-8.93465{r_{0}}^{4})$
$\displaystyle+p^{9}(139.634r^{4}-272.59r^{3}{r_{0}}+130.873r^{2}{r_{0}}^{2}-10.712r{r_{0}}^{3}+12.9426{r_{0}}^{4})$
$\displaystyle+p^{8}(-211.343r^{4}+445.224r^{3}{r_{0}}-232.351r^{2}{r_{0}}^{2}+3.40437r{r_{0}}^{3}-5.11853{r_{0}}^{4})$
$\displaystyle+p^{7}(104.307r^{4}-239.683r^{3}{r_{0}}+134.352r^{2}{r_{0}}^{2}+4.79121r{r_{0}}^{3}-5.8472{r_{0}}^{4})$
$\displaystyle+p^{6}(36.8094r^{4}-69.4903r^{3}{r_{0}}+37.49r^{2}{r_{0}}^{2}-4.15303r{r_{0}}^{3}+5.6248{r_{0}}^{4})$
$\displaystyle+p^{5}(-48.322r^{4}+99.557r^{3}{r_{0}}-57.1551r^{2}{r_{0}}^{2}+0.840865r{r_{0}}^{3}-0.70925{r_{0}}^{4})$
$\displaystyle+p^{4}(12.9333r^{4}-21.1041r^{3}{r_{0}}+8.6512r^{2}{r_{0}}^{2}-0.0658137r{r_{0}}^{3}-0.217336{r_{0}}^{4})$
$\displaystyle+p^{3}(0.768694r^{4}-0.492765r^{3}{r_{0}}+1.99322r^{2}{r_{0}}^{2}+0.0218888r{r_{0}}^{3}-0.0839553{r_{0}}^{4})$
$\displaystyle+p^{2}(-2.1015r^{4}+1.63385r^{3}{r_{0}}-0.221489r^{2}{r_{0}}^{2}+0.0142061r{r_{0}}^{3}-0.0117263{r_{0}}^{4})$
$\displaystyle+p(-0.155143r^{4}+0.0901895r^{3}{r_{0}}-0.0282534r^{2}{r_{0}}^{2}+0.000352495r{r_{0}}^{3}-0.000442358{r_{0}}^{4})$
$\displaystyle-0.005224r^{4}+0.00352893r^{3}{r_{0}}-0.000651281r^{2}{r_{0}}^{2},$

$\displaystyle\mathcal{D}^{1}_{\text{inv}}$
$\displaystyle=p^{11}r^{2}(1.r^{2}-2.r{r_{0}}+1.{r_{0}}^{2})+p^{10}r^{2}(-33.5247r^{2}+67.3011r{r_{0}}-33.7764{r_{0}}^{2})$
(A24)
$\displaystyle+p^{9}r^{2}(139.634r^{2}-287.439r{r_{0}}+147.805{r_{0}}^{2})+p^{8}r^{2}(-211.343r^{2}+449.189r{r_{0}}-237.795{r_{0}}^{2})$
$\displaystyle+p^{7}r^{2}(104.307r^{2}-233.152r{r_{0}}+127.187{r_{0}}^{2})+p^{6}r^{2}(36.8094r^{2}-74.6521r{r_{0}}+43.2038{r_{0}}^{2})$
$\displaystyle+p^{5}r^{2}(-48.322r^{2}+100.669r{r_{0}}-57.9912{r_{0}}^{2})+p^{4}r^{2}(12.9333r^{2}-20.9337r{r_{0}}+8.67754{r_{0}}^{2})$
$\displaystyle+p^{3}r^{2}(0.768694r^{2}-0.696371r{r_{0}}+1.94788{r_{0}}^{2})+p^{2}r^{2}(-2.1015r^{2}+1.61854r{r_{0}}-0.227976{r_{0}}^{2})$
$\displaystyle+pr^{2}(-0.155143r^{2}+0.0896651r{r_{0}}-0.0284235{r_{0}}^{2})+r^{2}(-0.005224r^{2}+0.00352893r{r_{0}}$
$\displaystyle-0.000651281{r_{0}}^{2}),$

$\displaystyle\mathcal{N}^{2}_{\text{inv}}$
$\displaystyle=p^{9}(1.r^{3}-2.16356r^{2}{r_{0}}+1.01501r{r_{0}}^{2}+0.148553{r_{0}}^{3})+p^{8}(-8.63116r^{3}+19.3261r^{2}{r_{0}}$
(A25)
$\displaystyle-9.37725r{r_{0}}^{2}-1.31772{r_{0}}^{3})+p^{7}(25.4073r^{3}-61.4392r^{2}{r_{0}}+31.2754r{r_{0}}^{2}+4.7492{r_{0}}^{3})$
$\displaystyle+p^{6}(-25.4485r^{3}+80.1206r^{2}{r_{0}}-45.5637r{r_{0}}^{2}-9.03094{r_{0}}^{3})+p^{5}(-18.958r^{3}-9.26834r^{2}{r_{0}}$
$\displaystyle+18.1535r{r_{0}}^{2}+9.73502{r_{0}}^{3})+p^{4}(67.5176r^{3}-88.9673r^{2}{r_{0}}+28.0939r{r_{0}}^{2}-5.8473{r_{0}}^{3})$
$\displaystyle+p^{3}(-62.3959r^{3}+97.8017r^{2}{r_{0}}-38.225r{r_{0}}^{2}+1.71881{r_{0}}^{3})+p^{2}(25.6081r^{3}-41.9346r^{2}{r_{0}}$
$\displaystyle+17.3586r{r_{0}}^{2}-0.140684{r_{0}}^{3})+p(-4.42145r^{3}+6.8935r^{2}{r_{0}}-2.85028r{r_{0}}^{2}-0.0149354{r_{0}}^{3})$
$\displaystyle+0.322118r^{3}-0.368892r^{2}{r_{0}}+0.119768r{r_{0}}^{2},$

$\displaystyle\mathcal{D}^{2}_{\text{inv}}$
$\displaystyle=p^{9}r(1.r^{2}-2.16356r{r_{0}}+1.16356{r_{0}}^{2})+p^{8}r(-8.63116r^{2}+19.3261r{r_{0}}-10.695{r_{0}}^{2})$
(A26)
$\displaystyle+p^{7}r(25.4073r^{2}-61.4392r{r_{0}}+36.0281{r_{0}}^{2})+p^{6}r(-25.4485r^{2}+80.1206r{r_{0}}-54.6246{r_{0}}^{2})$
$\displaystyle+p^{5}r(-18.958r^{2}-9.26834r{r_{0}}+27.9908{r_{0}}^{2})+p^{4}r(67.5176r^{2}-88.9673r{r_{0}}+22.0663{r_{0}}^{2})$
$\displaystyle+p^{3}r(-62.3959r^{2}+97.8017r{r_{0}}-36.3316{r_{0}}^{2})+p^{2}r(25.6081r^{2}-41.9346r{r_{0}}+17.1297{r_{0}}^{2})$
$\displaystyle+pr(-4.42145r^{2}+6.8935r{r_{0}}-2.84697{r_{0}}^{2})+r(0.322118r^{2}-0.368892r{r_{0}}+0.119768{r_{0}}^{2}).$

###  A.5 Logarithmic GB-Coupling Functional: $f(\varphi)=\ln(\varphi)$

$g_{tt}(r)\approx\frac{\mathcal{N}^{1}_{log}}{\mathcal{D}^{1}_{log}}(1-\frac{r_{0}}{r}),$
(A27)
$\sqrt{g_{tt}(r)g_{rr}(r)}\approx\frac{\mathcal{N}^{2}_{log}}{\mathcal{D}^{2}_{log}},$
(A28)

where

$\displaystyle\mathcal{N}^{1}_{\text{log}}$
$\displaystyle=p^{13}(-0.0626084r^{3}{r_{0}}+0.0626084r^{2}{r_{0}}^{2}+0.0626084r{r_{0}}^{3}-0.0626084{r_{0}}^{4})$
(A29)
$\displaystyle+p^{12}(1.r^{4}-0.782889r^{3}{r_{0}}-0.230599r^{2}{r_{0}}^{2}-1.21711r{r_{0}}^{3}+1.2306{r_{0}}^{4})$
$\displaystyle+p^{11}(-18.2212r^{4}+29.3638r^{3}{r_{0}}-10.8868r^{2}{r_{0}}^{2}+7.28478r{r_{0}}^{3}-7.54055{r_{0}}^{4})$
$\displaystyle+p^{10}(96.59r^{4}-177.29r^{3}{r_{0}}+79.5585r^{2}{r_{0}}^{2}-19.3886r{r_{0}}^{3}+20.5203{r_{0}}^{4})$
$\displaystyle+p^{9}(-234.614r^{4}+457.171r^{3}{r_{0}}-221.897r^{2}{r_{0}}^{2}+23.902r{r_{0}}^{3}-24.2931{r_{0}}^{4})$
$\displaystyle+p^{8}(282.436r^{4}-551.051r^{3}{r_{0}}+275.112r^{2}{r_{0}}^{2}-7.20824r{r_{0}}^{3}-1.91936{r_{0}}^{4})$
$\displaystyle+p^{7}(-133.991r^{4}+164.055r^{3}{r_{0}}-46.0945r^{2}{r_{0}}^{2}-13.4629r{r_{0}}^{3}+41.7195{r_{0}}^{4})$
$\displaystyle+p^{6}(-32.4983r^{4}+341.82r^{3}{r_{0}}-300.797r^{2}{r_{0}}^{2}+12.9375r{r_{0}}^{3}-53.4003{r_{0}}^{4})$
$\displaystyle+p^{5}(34.5271r^{4}-420.143r^{3}{r_{0}}+404.167r^{2}{r_{0}}^{2}+0.0115351r{r_{0}}^{3}+31.9943{r_{0}}^{4})$
$\displaystyle+p^{4}(28.0229r^{4}+178.063r^{3}{r_{0}}-241.727r^{2}{r_{0}}^{2}-4.86976r{r_{0}}^{3}-8.90541{r_{0}}^{4})$
$\displaystyle+p^{3}(-30.9869r^{4}-12.6459r^{3}{r_{0}}+69.661r^{2}{r_{0}}^{2}+2.21647r{r_{0}}^{3}+0.507243{r_{0}}^{4})$
$\displaystyle+p^{2}(8.38109r^{4}-10.7164r^{3}{r_{0}}-6.17049r^{2}{r_{0}}^{2}-0.251471r{r_{0}}^{3}+0.12598{r_{0}}^{4})$
$\displaystyle+p(-1.03581r^{4}+2.49996r^{3}{r_{0}}-0.826428r^{2}{r_{0}}^{2}-0.0167726r{r_{0}}^{3}+0.0233143{r_{0}}^{4})$
$\displaystyle+0.389358r^{4}-0.281216r^{3}{r_{0}}+0.0686144r^{2}{r_{0}}^{2},$

$\displaystyle\mathcal{D}^{1}_{\text{log}}$
$\displaystyle=p^{12}r^{2}(1.r^{2}-2.r{r_{0}}+1.{r_{0}}^{2})+p^{11}r^{2}(-18.2212r^{2}+36.6579r{r_{0}}-18.4367{r_{0}}^{2})$
(A30)
$\displaystyle+p^{10}r^{2}(96.59r^{2}-197.003r{r_{0}}+100.413{r_{0}}^{2})+p^{9}r^{2}(-234.614r^{2}+483.3r{r_{0}}-248.622{r_{0}}^{2})$
$\displaystyle+p^{8}r^{2}(282.436r^{2}-565.092r{r_{0}}+281.412{r_{0}}^{2})+p^{7}r^{2}(-133.991r^{2}+161.605r{r_{0}}-19.988{r_{0}}^{2})$
$\displaystyle+p^{6}r^{2}(-32.4983r^{2}+345.678r{r_{0}}-336.459{r_{0}}^{2})+p^{5}r^{2}(34.5271r^{2}-417.829r{r_{0}}+424.172{r_{0}}^{2})$
$\displaystyle+p^{4}r^{2}(28.0229r^{2}+175.038r{r_{0}}-246.175{r_{0}}^{2})+p^{3}r^{2}(-30.9869r^{2}-11.7993r{r_{0}}+69.4436{r_{0}}^{2})$
$\displaystyle+p^{2}r^{2}(8.38109r^{2}-10.8149r{r_{0}}-6.01301{r_{0}}^{2})+pr^{2}(-1.03581r^{2}+2.54012r{r_{0}}-0.815274{r_{0}}^{2})$
$\displaystyle+r^{2}(0.389358r^{2}-0.281216r{r_{0}}+0.0686144{r_{0}}^{2}),$

$\displaystyle\mathcal{N}^{2}_{\text{log}}$
$\displaystyle=p^{9}(1.r^{3}-1.5934r^{2}{r_{0}}+0.9878r{r_{0}}^{2}-0.394399{r_{0}}^{3})+p^{8}(-11.4341r^{3}+18.4084r^{2}{r_{0}}$
(A31)
$\displaystyle-9.70115r{r_{0}}^{2}+2.72685{r_{0}}^{3})+p^{7}(52.5869r^{3}-85.6446r^{2}{r_{0}}+40.9887r{r_{0}}^{2}-7.89466{r_{0}}^{3})$
$\displaystyle+p^{6}(-129.875r^{3}+213.791r^{2}{r_{0}}-96.5185r{r_{0}}^{2}+12.2873{r_{0}}^{3})+p^{5}(190.394r^{3}-316.075r^{2}{r_{0}}$
$\displaystyle+137.73r{r_{0}}^{2}-10.895{r_{0}}^{3})+p^{4}(-170.683r^{3}+284.439r^{2}{r_{0}}-121.325r{r_{0}}^{2}+5.25315{r_{0}}^{3})$
$\displaystyle+p^{3}(92.159r^{3}-152.461r^{2}{r_{0}}+64.1216r{r_{0}}^{2}-1.0636{r_{0}}^{3})+p^{2}(-28.3648r^{3}+45.1052r^{2}{r_{0}}$
$\displaystyle-18.6363r{r_{0}}^{2}-0.0535942{r_{0}}^{3})+p(4.63158r^{3}-6.37023r^{2}{r_{0}}+2.46462r{r_{0}}^{2}+0.0339455{r_{0}}^{3})$
$\displaystyle-0.413973r^{3}+0.400205r^{2}{r_{0}}-0.112264r{r_{0}}^{2},$

$\displaystyle\mathcal{D}^{2}_{\text{log}}$
$\displaystyle=p^{9}r(1.r^{2}-1.5934r{r_{0}}+0.593401{r_{0}}^{2})+p^{8}r(-11.4341r^{2}+18.4084r{r_{0}}-6.9743{r_{0}}^{2})$
(A32)
$\displaystyle+p^{7}r(52.5869r^{2}-85.6446r{r_{0}}+33.0761{r_{0}}^{2})+p^{6}r(-129.875r^{2}+213.791r{r_{0}}-84.1127{r_{0}}^{2})$
$\displaystyle+p^{5}r(190.394r^{2}-316.075r{r_{0}}+126.509{r_{0}}^{2})+p^{4}r(-170.683r^{2}+284.439r{r_{0}}-115.594{r_{0}}^{2})$
$\displaystyle+p^{3}r(92.159r^{2}-152.461r{r_{0}}+62.6652{r_{0}}^{2})+p^{2}r(-28.3648r^{2}+45.1052r{r_{0}}-18.518{r_{0}}^{2})$
$\displaystyle+pr(4.63158r^{2}-6.37023r{r_{0}}+2.46727{r_{0}}^{2})+r(-0.413973r^{2}+0.400205r{r_{0}}-0.112264{r_{0}}^{2}).$

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