# Horizon curvature and spacetime structure influences on black hole
scalarization

Hong Guo1,2<EMAIL_ADDRESS>Xiao-Mei Kuang1<EMAIL_ADDRESS>(corresponding author) Eleftherios Papantonopoulos3<EMAIL_ADDRESS>Bin
Wang2,1<EMAIL_ADDRESS>1 Center for Gravitation and Cosmology, College
of Physical Science and Technology, Yangzhou University, Yangzhou 225009,
China  2 School of Aeronautics and Astronautics, Shanghai Jiao Tong
University, Shanghai 200240, China  3 Physics Division, National Technical
University of Athens, 15780 Zografou Campus, Athens, Greece.

###### Abstract

Black hole spontaneous scalarization has been attracting more and more
attention as it circumvents the well-known no-hair theorems. In this work, we
study the scalarization in Einstein-scalar-Gauss-Bonnet theory with a probe
scalar field in a black hole background with different curvatures. We first
probe the signal of black hole scalarization with positive curvature in
different spacetimes. The scalar field in AdS spacetime could be formed easier
than that in flat case. Then, we investigate the scalar field around AdS black
holes with negative and zero curvatures. Comparing with negative and zero
cases, the scalar field near AdS black hole with positive curvature could be
much easier to emerge. And in negative curvature case, the scalar field is the
most difficult to be bounded near the horizon.

###### Contents

  1. I Introduction
  2. II Scalar Field Perturbations in Einstein-Scalar-Gauss-Bonnet Theories
  3. III Signal of Scalarization of Black Holes with Positive Curvature
    1. III.1 Asymptotical flat black hole
    2. III.2 Asymptotical AdS black hole
    3. III.3 Asymptotical dS black hole
  4. IV Signal of Scalarization of Black Holes with Negative Curvature
  5. V Signal of Scalarization of Black Holes with Zero Curvature
  6. VI Closing remarks

##  I Introduction

Experimental progress on gravitational waves GW1 ; GW2 ; GW3  and the shadow
of the $M87$ black hole EHT  further demonstrates the great success of
Einstein’s general relativity (GR). Yet it is unabated that the GR theory
should be generalized, and in the generalized theories extra fields or higher
curvature terms are always involved in the action Nojiri:2006ri ;
Clifton:2011jh ; MG3 . So physicists have proposed various modified
gravitational theories which indeed provide richer framework and significantly
help us further understand GR as well as our universe.

Among them, the scalar-tensor theories which introduces a scalar field into
the action attract lots of attention Fujii . When the scalar field backreacts
to the background metric, one could expect that hairy black hole solutions
would be generated. One of the first hairy black hole solution in an
asymptotically flat spacetime was discussed in BBMB  but soon the solution was
argued to be unstable because the scalar field is divergent on the event
horizon bronnikov . However, such irregular behaviour of the scalar field on
the horizon was avoided in asymptotically AdS/dS spacetime with a presence of
a cosmological constant in the gravity theory. It was found that the resulting
hairy black hole solutions have a regular scalar field behaviour and all the
possible divergence could be hidden behind the horizon Martinez:1996gn ;
Banados:1992wn ; Martinez:2004nb ; Zloshchastiev:2004ny ; Martinez:2002ru ;
Dotti:2007cp ; Torii:2001pg ; Winstanley:2002jt ; Martinez:2006an ;
Kolyvaris:2009pc ; Charmousis:2014zaa ; Khodadi:2020jij .

The effects of higher-order curvature terms become more significant as we are
exploring the strong field regime of gravity via detections of gravitational
waves and black hole shadows. It is known that the inclusion of such terms
probably bring in the well-known ghost problem stelle , and Gauss-Bonnet (GB)
corrections is a counter-case which is ghost-free. But it becomes a
topological term in four-dimensional spacetime and has no dynamics in field
equations when it is minimally coupled with Einstein-Hilbert action. A way to
make this term meaningful in four-dimensional spacetimes is to consider its
coupling with a scalar field stringT . As a special scalar-tensor theory with
higher derivatives Horndeski ; Deffayet:2009mn , the Einstein-scalar-Gauss-
Bonnet (ESGB) gravity recently has attracted a lot of attention. Especially,
the introduction of this coupling could lead to hairy black holes. Various
black hole solutions and compact objects in the four-dimensional ESGB theories
were studied in the literature Mignemi_1993 ; Kanti_1996 ; Torrii_1996 ;
Ayzenberg_2014 ; Kleihaus_2011 ; Kleihaus_2016a .

Recently, the spontaneous scalarization with particular coupling function in
ESGB theory is widely investigated. In this setup, besides GR solutions with a
trivial scalar field configuration, the scalarized hairy solutions for black
holes and stars could also exist, which evades the no-hair theorems Bekenstein
; Antoniou_2018 ; Antoniou:2017hxj . It was shown in Doneva:2017bvd  that
below a certain mass the Schwarzschild black hole background may become
unstable in regions of strong curvature, and then when the scalar field
backreacts to the metric, a scalarized hairy black hole emerges and it is
physically favorable. A natural extension of these results was to consider the
case of nonzero black hole charge. A large set of coupling functions between
the Gauss-Bonnet invariant and the scalar field was considered in order to
understand better the behaviour of the scalarized solutions Doneva:2018rou
-Fernandes:2019rez  .

Various generalizations of the spontaneous scalarization procedure were
followed in the literature. The scalarization due to a coupling of a scalar
field to Ricci scalar was studied in Herdeiro:2019yjy  and in the presence of
Chern-Simons invariant was studied in Brihaye:2018bgc . The generalized study
of scalarized black hole solutions and compact objects in asymptotical flat
spacetime of ESGB theories was discussed in Silva:2017uqg -Hunter:2020wkd .
The spontaneous scalarization of asymptotically AdS/dS black holes in ESGB
theory with a negative/positive cosmological constant was extended in
Bakopoulos:2018nui ; Brihaye:2019gla ; Bakopoulos:2019tvc ; Bakopoulos:2020dfg
; Lin:2020asf . Especially, the connections of asymptotically AdS black holes
scalarization with holographic phase transitions in the dual boundary theory
was studied in Brihaye:2019dck ; Guo:2020sdu . Recently the spontaneous
scalarization in f(R) gravity theories in the presence of a scalar field
minimally coupled to gravity with a self-interacting potential was discussed
in Tang:2020sjs . Also in ESGB theory the spin-induced black hole spontaneous
scalarization, which is the outcome of linear tachyonic instability triggered
by rapid rotation, was explored in Collodel:2019kkx ; Dima:2020yac ;
Doneva:2020kfv ; Herdeiro:2020wei ; Berti:2020kgk ; Zhang:2020pko .

The scalarization in asymptotical flat, AdS and dS black holes in ESGB gravity
theory was studied, however detailed information on how different spacetime
structures influence the scalarization process is not disclosed. An
interesting question is how the cosmological constant leaves its imprint on
the black hole scalarization and in which kind of spacetime the scalar field
could be formed more easily. Besides, the studies of spontaneous scalarization
in the existed literatures were focused on the black hole with spherical
horizon, but as known in AdS spacetime, one can have AdS black holes with
different topologies and it is interesting to study how these topologies of
the black hole horizon affect the scalarization procedure111Note that in
Motohashi:2018mql , the authors discussed the influence of the topology in the
matter configuration on the spontaneous scalarization. They found that
comparing to the spherical distribution, the planar symmetric matter
distribution makes the spontaneous scalarization easier to occur.. Thus, the
aim of this work is to probe the signal of scalarization in various black
holes with different horizon curvature and try to understand its effects on
the scalarization processes. To this end, we first consider the test scalar
field perturbation and analyze its effective potential in various black hole
backgrounds. Then we study the tachyonic instabilities of the scalar field
near the horizon by solving the perturbation equation. We find that the scalar
hair around the black hole with spherical horizon is easier to form in AdS
spacetime than in flat case. While in dS case, the regular condition near
cosmological horizon leads to divergence at large distance, so that the test
field approximation breaks down. Moreover, our study on the effect of horizon
curvatures indicates that the scalar hair around AdS black hole with toroidal
horizon could be easier to be formed than that with hyperbolic horizon, but
more difficult than that with spherical horizon. Both the effective potential
analysis and the test field solution conclude the above observations.

The work is organized as follows. We write down the general equation of motion
of a test scalar field in the background of four dimensional black holes with
different topologies in ESGB gravity theory in section II. In section III, we
study the signal of scalarization of a black hole with positive curvature in
different spacetimes. Then we investigate the signal of a black hole
scalarization with negative curvature and zero curvature in section IV and
section V respectively. The last section is devoted to conclusions and
discussions.

##  II Scalar Field Perturbations in Einstein-Scalar-Gauss-Bonnet Theories

We consider four-dimensional ESGB theories with the action

$\displaystyle S=\frac{1}{16\pi G}\int
d^{4}x\sqrt{-g}\Big{[}R-2\Lambda-\nabla_{\mu}\Phi\nabla^{\mu}\Phi-m^{2}\Phi^{2}+\lambda^{2}f(\Phi){\cal
R}^{2}_{GB}\Big{]}~{},$ (1)

where $\Lambda$ is the cosmological constant, $\lambda$ is the coupling
constant between Gauss-Bonnet term and scalar field with the dimension of
length and $\nabla_{\mu}$ denotes the covariant derivative. The scalar field
coupling function $f(\Phi)$ only depends on the neutral scalar field $\Phi$,
and the Gauss-Bonnet term is given by

${\cal
R}^{2}_{GB}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}~{},$
(2)

where $R,R_{\mu\nu}$ and $R_{\mu\nu\alpha\beta}$ are the Ricci scalar, Ricci
and Riemann tensors, respectively. The Einstein equation and Klein-Gordon
equation can be derived from the above action as

$\displaystyle G_{\mu\nu}-\Lambda
g_{\mu\nu}+\Gamma_{\mu\nu}=\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}\nabla_{\alpha}\Phi\nabla^{\alpha}\Phi-\frac{1}{2}g_{\mu\nu}m^{2}\Phi^{2}~{},$
(3)
$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}\partial^{\mu})\Phi=m^{2}\Phi-\frac{\lambda^{2}}{2}\frac{df(\Phi)}{d\Phi}{\cal
R}^{2}_{GB}~{},$ (4)

where the energy-momentum tensor $\Gamma_{\mu\nu}$ is given by

$\displaystyle\Gamma_{\mu\nu}$ $\displaystyle=$
$\displaystyle-R(\nabla_{\mu}\Psi_{\nu}+\nabla_{\nu}\Psi_{\mu})-4\nabla^{\alpha}\Psi_{\alpha}\left(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}\right)+4R_{\mu\alpha}\nabla^{\alpha}\Psi_{\nu}+4R_{\nu\alpha}\nabla^{\alpha}\Psi_{\mu}$
(5)
$\displaystyle-4g_{\mu\nu}R^{\alpha\beta}\nabla_{\alpha}\Psi_{\beta}+\,4R^{\beta}_{\;\mu\alpha\nu}\nabla^{\alpha}\Psi_{\beta}~{},$

with $\Psi_{\mu}=\lambda^{2}\frac{df(\Phi)}{d\Phi}\nabla_{\mu}\Phi$.

It is obvious that different forms of the coupling function $f(\Phi)$ shall
give different properties of the ESGB theory. As addressed in Doneva:2017bvd ,
$f(\Phi)$ could satisfy conditions $\frac{df(\Phi)}{d\Phi}\mid_{\Phi=0}=0$ and
$\frac{d^{2}f(\Phi)}{d\Phi^{2}}\mid_{\Phi=0}=b^{2}>0$, where $b$ is a
constant, in order to admit Schwarzschild black hole as a background solution
and further explore the black hole scalarization in the ESGB theory. Moreover,
one usually assumes that the scalar field vanishes at infinity and normalizes
the constant $b$ to unity. Thus, in this work, we shall choose the form of the
coupling function as $f(\Phi)=\frac{1}{2}(1-e^{-\Phi^{2}})$ which also
satisfies $f(\Phi=0)=0$.

When the scalar field vanishes, i.e., $\Phi=0$, the gravity theory admits
black hole solutions, whose metric takes the general form,

$ds^{2}=-g(r)dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}d\overrightarrow{x}_{k,2}^{2}~{},~{}~{}~{}~{}g(r)=k-\frac{M}{r}-\frac{\Lambda
r^{2}}{3}~{},$ (6)

where $M$ is the integral constant related the black hole mass, and $k$
determines the curvature of two dimensional geometry near the horizon with the
line element $d\overrightarrow{x}_{k,2}^{2}$ given by

$d\overrightarrow{x}_{k,2}^{2}=\left\\{\begin{matrix}d\Omega^{2}_{2}&=&d\theta^{2}+\sin^{2}\theta\,d\psi^{2}\
\ \ &\ \ {\rm for\ }k=+1\ \,,\cr d\ell^{2}_{2}&=&dx^{2}+dy^{2}\ \
\qquad\qquad&\ \ {\rm for\ }k=0\ \ \,,\cr
d\sigma^{2}_{2}&=&d\theta^{2}+\sinh^{2}\theta\,d\psi^{2}&\ \ {\rm for\
}k=-1\,,\end{matrix}\right.$ (7)

where $k=1,0$ and $-1$ correspond to black hole horizon with spherical, planar
and hyperbolic topology, respectively. Then the corresponding GB term can be
easily computed as

${\cal
R}^{2}_{GB}=\frac{4}{r^{2}}\left[g^{\prime}(r)^{2}+(g(r)-k)g^{\prime\prime}(r)\right]~{}.$
(8)

In order to explore the (in)stability of the background black holes, we
consider a small fluctuation in the form $\delta\Phi=e^{-i\omega
t}\phi(r)Y(x_{1},x_{2})$ where $Y(x_{1},x_{2})$ satisfies Laplace-Beltrami
equation $\Box_{2}Y(x_{1},x_{2})=-AY(x_{1},x_{2})$. We consider that
$Y(x_{1},x_{2})$ is spherical harmonics $Y_{\ell,m}$ for positive curvature so
that $A=\ell(\ell+1)$; For zero curvature $Y(x_{1},x_{2})$ is plane wave case
as $e^{i\mathcal{K}(x+y)}$ so that $A=\mathcal{K}^{2}$; and for negative
curvature $Y(x_{1},x_{2})$ is spherical harmonics $Y_{-\frac{1}{2}\pm
i\zeta,\mathfrak{m}}$ and $A=\frac{1}{4}+\zeta^{2}$ Koutsoumbas:2008pw ;
Wang:2001tk . Then we use the tortoise coordinate $dr_{*}=\frac{dr}{g(r)}$,
and bring the Klein-Gordon equation into a Schrodinger-like form by reforming
$\phi(r)=\frac{\varphi(r)}{r}$

$\displaystyle\frac{\partial^{2}\varphi(r)}{\partial
r_{*}^{2}}+(\omega^{2}-V(r))\varphi(r)=0~{},$ (9)

where the effective potential takes the form

$V(r)=g(r)\left(\frac{g^{\prime}(r)}{r}+\frac{A}{r^{2}}+m^{2}-\frac{\lambda^{2}}{2}{\cal
R}^{2}_{GB}\right).$ (10)

Noted that in the Schrodinger-like form (9) and (10) of the perturbation, we
have only kept the leading term of the expansion
$\frac{df(\Phi)}{d\Phi}\simeq\Phi(1-\Phi^{2}+\frac{1}{2}\Phi^{4}+\mathcal{O}(\Phi^{2}))$
in the potential function.

We then analyze the signal of scalarization of the system by computing the
effective potential with the lowest mode $(\ell=\mathcal{K}=\zeta=0)$. For
simplification, we could focus on the static perturbation with $\omega=0$,
thereby we shall solve the radial perturbed equation for $\phi(r)$,

$\displaystyle\phi^{\prime\prime}(r)+\left(\frac{2}{r}+\frac{g^{\prime}(r)}{g(r)}\right)\phi^{\prime}(r)-\left(\frac{A}{r^{2}g(r)}+\frac{m^{2}}{g(r)}\right)\phi(r)+\frac{\lambda^{2}}{2g(r)}{\cal
R}^{2}_{GB}\frac{df(\phi)}{d\phi}=0~{},$ (11)

and see how the scalar field with the lowest mode emerges as the Gauss-Bonnet
coupling increases. To control the variables of the black holes and compare
the results, we shall set $r_{h}=1$ in various cases, namely, we fix the size
of the black holes such that the black hole mass is solved via
$g(r=r_{h})=g(r_{h}=1)=0$ in the following discussion. We would numerically
employ the spectral method to solve the differential equations.

Since the existence of above black hole with different curvatures is dependent
on the type of asymptotic background, i.e., the sign of $\Lambda$, we would
discuss the signal of scalarization of various black holes in terms of the
curvature. Moreover, we shall rewrite $\Lambda=\pm 3/L^{2}$ where $L$ is the
curvature AdS radius. As it was discussed in Guo:2020sdu ; Kolyvaris:2013zfa
because the scalar field is coupled to GB invariant, the effective mass of the
scalar field receives a contribution from this coupling. Then for various
values of the coupling, the effective mass of the scalar field becomes
tachyonic outside the background black hole and the fluctuations are unstable
destabilizing in this way the background metric.

##  III Signal of Scalarization of Black Holes with Positive Curvature

In this section, we focus on the background solution with positive curvature,
i.e., $k=1$ in (6), in which case there exist asymptotical flat $(\Lambda=0)$,
AdS $(\Lambda<0)$ and dS $(\Lambda>0)$ black holes as possible backgrounds.
Consequently, we shall separately study the signal of scalarization in all
these asymptotical backgrounds, and then do comparisons on the emergence of
non-trivial scalar field which is introduced by the Gauss-Bonnet coupling in
these different spacetime backgrounds.

###  III.1 Asymptotical flat black hole

For $\Lambda=0$, the asymptotical flat black hole (AFBH) with spherical
horizon ($k=1$) is just the Schwarzschild black hole. In this case, the
redshift function in the field equation (11) is $g(r)=1-\frac{M}{r}$ where
$M=r_{h}=1$ and ${\cal
R}^{2}_{GB}=\frac{4}{r^{2}}(g^{\prime}(r)^{2}+(g(r)-1)g^{\prime\prime}(r))$.

In this case, the potential (10) for the lowest mode $\ell=0$ of the massless
scalar field $(m=0)$ is

$V(r)=(1-\frac{M}{r})\left(\frac{M}{r^{3}}-\frac{6\lambda^{2}M^{2}}{r^{6}}\right)=(1-\frac{1}{r})\left(\frac{1}{r^{3}}-\frac{6\lambda^{2}}{r^{6}}\right)$
(12)

where we have inserted $M=1$ in the second equality. A sufficient condition
for the existence of an unstable scalar field mode is Buell ; Doneva:2017bvd

$\int^{\infty}_{-\infty}V(r_{*})dr_{*}=\int^{\infty}_{M}\frac{V(r)}{g(r)}dr<0,$
(13)

which leads to $\lambda^{2}>\frac{5}{12}M^{2}=\frac{5}{12}$. This indicates
that the tachyonic instability emerges when the coupling constant satisfies
$\lambda^{2}>\frac{5}{12}$. Especially, near the horizon $r\to 1$, we can
reduce from equation (11) that the scalar field $\phi(1)$ and its derivative
$\phi^{\prime}(1)$ are both regular with a relation

$\displaystyle\phi^{\prime}(1)+6\lambda^{2}\phi(1)e^{-\phi(1)^{2}}=0~{}.$ (14)

It is obvious that the contribution of the GB coupling could control the near
horizon behaviour of the scalar field. While at infinity $r\to+\infty$, the
scalar field behaves as

$\phi(r\to\infty)\to\phi_{\infty}+\frac{\phi_{-}}{r}+\frac{\phi_{+}}{r^{2}}+\mathcal{O}(\frac{1}{r^{3}}).$
(15)

We would set $\phi_{\infty}=0$ as the Dirichlet boundary condition so that the
scalar field converges to zero as $r\to\infty$ which is well behaved. Thus, we
can consider the scalar field as a probed field to study the signal of the
scalarization. Also $\phi_{-}$ could be treated as the scalar charge.

In Fig. 1, we show the effective potential of the scalar field as the function
of radial coordinate. When $\lambda=0$ without GB coupling, there is a
potential barrier. As the GB coupling is turned on and increased, the barrier
is suppressed, and a negative potential well form near the horizon which could
trigger an instability and then lead to the development of scalar hair.

Figure 1: The effective potential of the scalar field in asymptotical flat
schwarzschild black hole with different values of $\lambda$.

We then study the behaviour of scalar field by solving the perturbation
equation (11) and expect to see the signal of scalarization in the probe
limit. By scanning the coupling parameter, it is found that there exists a
threshold value $\lambda_{t}\simeq 0.60234$222Noted that previously we
discussed that for $\lambda^{2}>\frac{5}{12}M^{2}$, i.e. $\lambda\gtrsim
0.645$, the background black holes would be unstable. Here the threshold value
$\lambda_{t}\simeq 0.60234$ is a bit smaller. This is reasonable because (13)
is only a sufficient condition, which was also verified inDoneva:2017bvd . .
When $\lambda$ is smaller than the threshold value, the scalar field is zero
everywhere which is trivial. When the coupling is larger than $\lambda_{t}$,
the scalar hair would emerge and increase as the coupling increases. The
profile of the scalar field with various values of $\lambda$ is shown in the
left plot of Fig. 2. Moreover, in the middle plot of Fig. 2, we present the
value of the scalar field near the event horizon $\phi_{h}$ as the function of
$\lambda$ and corresponding scalar charge $\phi_{-}$ at infinity is shown in
the right figure. The black line on the horizontal axis is the trivial scalar
field. As the coupling parameter increases larger than $\lambda_{t}$, there
emerges the bifurcation dividing the trivial solution, the non-trivial scalar
field formes and grows dramatically.

Figure 2: Left: the profiles of the scalar field in asymptotical flat
schwarzschild black hole with different values of $\lambda$. Middle: the value
of scalar field near the event horizon $\phi_{h}$ as the function of
$\lambda$. Right: the corresponding scalar charge of the black hole $\phi_{-}$
at infinity as the function of $\lambda$.

This phenomenon indicates that due to the instability brought by the GB
coupling, the scalar field may emerge and grow near the horizon of AFBH
background, such that a new branch of scalarized hairy black hole develops. It
is noticed that the scalarized hairy black hole solutions with backreaction
were numerically explored in Doneva:2018rou , and it was found that the
entropy of the scalarized black hole is larger than of AFBH solution which
indicates the hairy solution is physically favorable.

###  III.2 Asymptotical AdS black hole

In the AdS case with $\Lambda=-3/L^{2}<0$, the redshift function in the
solution (6) with spherical horizon is
$g(r)=1-\frac{M}{r}+\frac{r^{2}}{L^{2}}$ ($M=1+\frac{1}{L^{2}}$ as $r_{h}=1$)
and the GB term ${\cal
R}^{2}_{GB}=\frac{4}{r^{2}}\left[g^{\prime}(r)^{2}+(g(r)-1)g^{\prime\prime}(r)\right]$.

As is known, the AdS background could possess a classical instability for the
real scalar field, but provided its effective mass is above the Breitenlohner-
Freedman (BF) bound, the perturbation do not break the AdS behaviorBFbound .
In our case, the effective mass derived from (4) is given by
$m_{eff}^{2}=m^{2}-\frac{\lambda^{2}}{2}{\cal R}^{2}_{GB}$, and so the regular
tachyonic instability could exist as $m_{eff}^{2}>m_{BF}^{2}=-9/4L^{2}$.
Specially, near infinity, the Klein-Gordon equation becomes

$\phi^{\prime\prime}(r)+\frac{4}{r}\phi^{\prime}(r)+\frac{-m^{2}L^{2}+\frac{12\lambda^{2}}{L^{2}}}{r^{2}}\phi(r)=0$
(16)

which gives the boundary behavior of the scalar field as

$\phi(r\rightarrow\infty)=\phi_{-}r^{-\alpha_{-}}+\phi_{+}r^{-\alpha_{+}}.$
(17)

Here $\alpha_{\pm}=\frac{3\pm\sqrt{9+4m_{e}^{2}L^{2}}}{2}$ and
$m_{e}^{2}L^{2}=m^{2}L^{2}-12\frac{\lambda^{2}}{L^{2}}$ is the asymptotical
behavior of $m_{eff}^{2}L^{2}$. It is obvious that
$m_{e}^{2}>m_{BF}^{2}=-9/4L^{2}$ could also ensure the scalar field at
infinity to be convergent to be zero. Thus, in AdS case, for the parameters
satisfied the above analysis, we are safe to solve the probed perturbative
equation (11) to see how exactly the GB coupling and AdS curvature radius
affect the fate of the scalar field.

For a proper comparison in the flat and AdS cases, we set $m_{e}^{2}L^{2}=-2$
and then the asymptotic behaviour becomes
$\phi(r\rightarrow\infty)=\phi_{-}/r+\phi_{+}/r^{2}$, such that the scalar
charge $\phi_{-}$ term has the same decay behavior as that in flat case. In
the numeric, we set $\phi_{+}=0$ as the boundary condition. It is noted that
in general, the asymptotic behaviour of the scalar charge term could be
different in flat and AdS cases. Thus, to compare the signal of scalarization,
we have to carefully control the variables and then we require the decay
behavior of the scalar charge term is coincident. It means that we shall probe
the scalar field with the same scalar charge but not the same scalar mass,
which is reasonable because non-vanishing scalar charge implies the occurrence
of scalarization and modifies the general relativity black holeAntoniou_2018 .
Actually, though the mass of scalar field
$m^{2}L^{2}=m_{e}^{2}L^{2}-12\frac{\lambda^{2}}{L^{2}}$ in AdS case seems to
be very different from $m=0$ in flat case, it is obvious that the mass in AdS
case $m^{2}=-2/L^{2}+12\lambda^{2}/L^{4}$ go to zero as $L\to\infty$, which is
the flat limit. Our numerical result will also support the analysis, We shall
show soon that all properties for very large $L$ in AdS case recover those in
flat case as expected.

To proceed, we first show the potential function (10) in Fig. 3. We see that
the potential near the horizon could be negative for certain parameter regime
and the negative potential may lead to instability of the scalar field and a
small scalar field perturbation may finally destabilizes the AdS black hole
background. In the left plot without the GB coupling, the effective potential
never becomes negative unless the scalar field is charged as addressed in
Gubser1 ; Gubser2 . In the middle plot with $\lambda=0.3$, the potential well
becomes deeper as $L$ decreases, meaning that there exists a critical $L_{*}$
that the potential is negative enough for the scalar field emerging near the
horizon. In the right plot with fixed $L=0.5$, as $\lambda$ increases, the
potential becomes deeper. This behaviour is similar as that in AFBH case,
however, by careful comparison, we find that for the same GB coupling, the
well in the right plot is more deeper than that in AFBH case (see Fig. 1).
This implies that comparing to the flat spacetime, the scalar field in AdS
spacetime would be easier bounded near the horizon, saying with smaller GB
coupling.

Figure 3: The effective potential in asymptotical AdS black hole with positive
curvature.

Considering the scalar field near the horizon

$\phi^{\prime}(1)=\frac{L^{2}}{3+L^{2}}(m^{2}-6\frac{\lambda^{2}}{L^{4}}(3+2L^{2}+L^{4})e^{-\phi(1)^{2}})\phi(1)~{}.$
(18)

and the boundary condition (17) by choosing $\phi_{+}=0$, we scan the
parametric space $(\lambda,L)$ to find the solution of equation (11). We find
a critical line for the threshold of scalar field as shown on the left panel
of Fig. 4. Below the critical line, the scalar field is zero which is trivial,
while above the line the scalar field behaves as a non-trivial solution. We
can extract two properties from the left panel of Fig. 4. Firstly, the
threshold value of $\lambda$ first increases as $L$ becomes larger, and then
tends to be a constant as $L\rightarrow\infty$. The constant is around
$\lambda_{t}\simeq 0.60234$ which matches the aforementioned value in AFBH
case (see the blue dot). This is reasonable because as $L\rightarrow\infty$,
the cosmological constant goes to zero which is the case we studied in the
previous subsection. Secondly, comparing to AFBH case, the threshold value of
$\lambda$ in AdS case is smaller, which implies that weaker GB coupling could
introduce spontaneous scalarization for spherical AdS black hole. This
behaviour supports our previous analysis on the effective potential of Fig. 3.

Figure 4: Left: the threshold curves for AdS black hole with positive
curvature where the blue dot represents the threshold value of Schwarzschild
black hole; Middle: the scalar hair of the black hole as the function of
$\lambda$ with different AdS radius $L$; Right: the corresponding scalar
charge of the black hole $\phi_{-}$ behaves with different AdS radius $L$.

In the middle panel of Fig. 4, we present the scalar field near the event
horizon $\phi_{h}$ as a function of $\lambda$. For larger $L$, the value of
$\lambda$ for the emergence of non-zero $\phi_{h}$ is larger, which is
consistent with the observed on the left of Fig. 4. The scalar hair near the
horizon emerges at the threshold coupling, and the value at the horizon
increases as $\lambda$ increases and then tends to be constant. The scalar
charge of the black hole $\phi_{-}$ shown in the right figure of Fig. 4
supports our results. However, when $\lambda$ is small, $\phi_{h}$ decreases
as $L$ increases, but when $\lambda$ is large enough, $\phi_{h}$ increases as
$L$ increases. Starting from the probe limit, we can obtain certain signal of
spontaneous scalarization of the AdS background brought by GB coupling with
the form $f(\phi)=\frac{1}{2}(1-e^{-\phi^{2}})$. Numerical methods are called
for to handle the fully backreacted system and get the hairy solution, as was
explored for various different $f(\phi)$ in Bakopoulos:2018nui .

###  III.3 Asymptotical dS black hole

In the dS case with $\Lambda=3/L^{2}>0$, only the black hole with spherical
horizon exists, so the redshift function in the solution (6) is
$g(r)=1-\frac{M}{r}-\frac{r^{2}}{L^{2}}$. As $r_{h}=1$, the mass is
$M=1-\frac{1}{L^{2}}$, so that $L>1$ is required for the positive mass.
Moreover, the cosmological horizon is located at
$r_{c}=\frac{1}{2}(-1+\sqrt{4L^{2}-3})>r_{h}=1$, which further requires
$L>\sqrt{3}$ in our setup.

Similar as the above subsections, we first discuss the validity of the probe
approximation considering the scalar field as a test field in the dS
spacetime. Following the analysis of Brihaye:2019gla , we consider the Klein-
Gordon equation Eq.(11) in pure dS background with
$g(r)=1-\frac{r^{2}}{L^{2}}$ for which the cosmological horizon is located at
$r_{c}=L$. By setting $\phi(r)=\frac{\varphi(r)}{r}$ and $x=\frac{r}{r_{c}}$,
the Klein-Gordon equation is reduced as

$(1-x^{2})\varphi^{\prime\prime}(x)-2x\varphi^{\prime}(x)+u(u+1)\varphi(x)=0$
(19)

where $u=\frac{3\chi-1}{2}$, and

$\chi=\sqrt{1-\frac{4}{9}m^{2}r_{c}^{2}+4\frac{\lambda^{2}}{r_{c}^{2}}}\equiv\sqrt{1-\frac{4}{9}\bar{m}^{2}_{eff}r_{c}^{2}}$
(20)

where we define $\bar{m}_{eff}^{2}=m^{2}-\frac{9\lambda^{2}}{r_{c}^{4}}$ in
the second equality. We see that the Gauss-Bonnet coupling contributes to
$\chi$ as a modification of that in Brihaye:2019gla .

The solution of (19) is

$\phi(\frac{r}{r_{c}})=\frac{\varphi(\frac{r}{r_{c}})}{r}=\frac{1}{r}P_{u}\big{(}\frac{r}{r_{c}}\big{)}+\frac{s}{r}Q_{u}\big{(}\frac{r}{r_{c}}\big{)}$
(21)

where $P_{u}$ and $Q_{u}$ are Legendre functions and $s$ is an arbitrary
constant. It is obvious that the solution diverges at $r=0$, and also the
second term diverges at $r=r_{c}$ because of
$Q_{u}\big{(}\frac{r}{r_{c}}\big{)}$ diverges. So the regularity at $r=r_{c}$
requires $s=0$, then the solution near $r_{c}$ is

$\phi(r)\simeq\frac{1}{r_{c}}-\frac{\bar{m}_{eff}^{2}}{2}(r-r_{c})+\mathrm{higher~{}order~{}term},$
(22)

while the solution at $r\gg r_{c}$ behaves as

$\phi(r)\simeq c_{+}r^{-\frac{3}{2}(1+\chi)}+c_{-}r^{-\frac{3}{2}(1-\chi)},\ \
\ \ \mathrm{where}\ \
c_{\pm}=\frac{\Gamma(\mp\frac{3\chi}{2})}{\sqrt{\pi}\Gamma(1\mp\frac{3\chi}{2})}\big{(}\frac{r_{c}}{2}\big{)}^{\frac{1\pm
3\chi}{2}}.$ (23)

The precondition of scalarization is that a scalar field must have a tachyonic
behaviour somewhere in between the BH and cosmological horizon, which requires
$\bar{m}^{2}_{eff}<0$ and $\chi>0$. But this implies that the scalar field
could diverge at $r\gg r_{c}$. Therefore, the regularity of the scalar field
at the cosmological horizon makes the tachyonic test field divergent at large
distance, which breaks the probe approximation. Even though the probe limit
may be valid between the event horizon and cosmological horizon, we can not
check the signal of scalarization in probe limit with the same boundaries as
working in flat and AdS cases. We could improve the numerical skills to
involve the backreaction and do the comparison, and we hope to address this
issue in the future.

##  IV Signal of Scalarization of Black Holes with Negative Curvature

In this section we will study the signal of scalarization if the background
metric is a topological black hole (TBH). For the action

$S=\int d^{4}x\sqrt{-g}\Big{[}\frac{R+6L^{-2}}{16\pi G}\Big{]}~{},$

the presence of a negative cosmological constant allows the existence of black
holes with a topology ${R}^{2}\times\Sigma$, where $\Sigma$ is a two-
dimensional manifold of constant curvature. The simplest solution of this
kind, when $\Sigma$ has negative constant curvature, reads

$ds^{2}=-\left(-1+\frac{r^{2}}{L^{2}}-\frac{2G\mu}{r}\right)dt^{2}+\frac{dr^{2}}{\left(-1+\frac{r^{2}}{L^{2}}-\frac{2G\mu}{r}\right)}+r^{2}d\sigma^{2}_{2}\;,$
(24)

where $d\sigma^{2}_{2}$ is the line element of $\Sigma$ which is just (6) with
$k=-1$, $\Lambda=-3L^{-2}$ and $M=2G\mu$. The line element $d\sigma^{2}_{2}$
is locally isomorphic to the hyperbolic manifold $H^{2}$ and $\Sigma$ must be
of the form

$\Sigma=H^{2}/\Gamma{\quad}\mathrm{{with\quad}\Gamma\subset O(2,1)\;,}$

where $\Gamma$ is a freely acting discrete subgroup (i.e., without fixed
points).

The configurations (24) are asymptotically local AdS spacetimes and these
black holes are known as TBHs mann -Cai:2001dz . It has been shown in
Gibbons:2002pq  that the massless configurations where $\Sigma$ has negative
constant curvature are stable under gravitational perturbations and the
stability of the topological black holes was discussed in Birmingham:2007yv ;
Wang:2001tk . In a series of papers Koutsoumbas:2006xj ; Koutsoumbas:2008pw ;
Koutsoumbas:2008yq  it was shown that there is evidence that a vacuum TBH goes
over to a hairy configuration, the MTZ black hole Martinez:2004nb , through a
second-order phase transition. Therefore it is interesting to see if in the
presence of the coupling between scalar field and the GB term, there is
evidence of TBH scalarization.

In this case, the GB term (8) is evaluated as ${\cal
R}^{2}_{GB}=\frac{4}{r^{2}}(g^{\prime}(r)^{2}+(g(r)+1)g^{\prime\prime}(r))$.
To fulfill $g(r_{h})=0$, the black hole mass should satisfy
$M=r_{h}^{3}/L^{2}-r_{h}$. The positivity of black hole mass forces
$0<L<r_{h}$, while the positivity of Hawking temperature requires
$L<\sqrt{3}r_{h}$. Thus for $r_{h}\leq L<\sqrt{3}r_{h}$, the topological black
hole have a positive Hawking temperature, but the black hole mass could be
zero or negative, and the stability of the these massless and negative mass
black hole has been investigated in Birmingham:2007yv . It is noted that the
topological black holes have interesting structures in their own right, here
for security we focus on topological black hole with positive mass, such that
we have $0<L<1$ as we have set $r_{h}=1$. Then the effective potential is
shown in Fig. 5. Without the GB coupling (see the left plot), the effective
potential is always positive, so there is no signal of tachyonic instability
to induce the scalarization, so that the TBH keeps stable. With $\lambda=0.3$
(see the middle plot), the potential well becomes deeper as $L$ decreases .
When $L$ is small enough, the potential well would be deep enough to allow the
tachyonic instability to happen. In the right plot with fixed $L=0.5$, as
$\lambda$ increases, the potential well becomes deeper. We note that the rule
for negative curvature in AdS case extracted from Fig. 5 is qualitatively the
same as that in the positive curvature case (see Fig. 3). But with the same
parameters, the potential well for $k=-1$ is shallower than that in the case
with $k=1$. This means that the onset of spontaneous scalarization is easier
to happen for AdS black hole with positive curvature than that with negative
curvature.

Figure 5: The effective potential for the AdS black hole with negative
curvature.

Again, to further fix the parameters for the threshold of the nontrivial
scalar field, we need to call back the perturbation equation (11). Near the
event horizon, the scalar field satisfies

$\displaystyle\phi^{\prime}(1)=\frac{L^{2}}{3-L^{2}}(m^{2}-6\frac{\lambda^{2}}{L^{4}}(3-2L^{2}+L^{4})e^{-\phi(1)^{2}})\phi(1)~{}.$
(25)

While in asymptotical infinity, the behavior is also (17). We shall also set
$\phi_{+}=0$ in the numeric.

The critical threshold parameters in $(\lambda,L)$ space is shown on the left
of Fig. 6 where in the green shadow region the scalar field in nontrivial. As
$L$ increases, the critical coupling also increases which is similar as that
in the cases with $k=1$. The curve tends to be divergent as $L\rightarrow 1$
because there is no black hole solution when $L\geq 1$ in our setup. By
comparing with left of Fig. 6 and Fig. 4, we find that the scalar hair is more
difficult to form in the negative curvature than in the positive case. This is
consistent with the conclusion from the comparison of the potential function.

Figure 6: Left: the threshold curves for AdS black hole with negative
curvature; Middle: the scalar hair near the horizon as the function of
$\Lambda$ with different AdS radius $L$; Right: the corresponding scalar
charge of the black hole $\phi_{-}$ as the function of $\lambda$.

We also present the scalar hair near the horizon and the scalar charge at
infinity as the function of $\lambda$ for different values of $L$ in the
middle and right of Fig. 6, respectively. With the increase of $L$, the
threshold value $\lambda$ becomes larger, which matches the observation on the
left of Fig. 6. Moreover, as $L$ becomes larger, the curves become lower and
lower rapidly and the value of $\phi_{h}(\phi_{-})$ decreases. Especially,
when $L$ goes closely to the unity, $\phi_{h}$ becomes very small, which
indicates that the scalarization of the background black hole becomes very
difficult. This behaviour is consistent with that in positive curvature at
small coupling (see the middle plot of Fig. 4) as well as that in zero
curvature shown in next section.

We have observed that scalarizations in hyperbolic AdS black hole backgrounds
depend on two factors, the coupling strength between the scalar field and the
GB term and the cosmological constant through the radius $L$. For the chosen
$L$, once the coupling parameter $\lambda$ increases, the effective potential
in Fig. 5 becomes deeper which can ignite tachyonic instability and start the
hair formation more easily. When we have big enough $L$, in Fig. 6 we see that
the required critical $\lambda$ to start the scalarization becomes bigger. In
the middle panel of Fig. 6, we find that for bigger $L$, the formed scalar
hair is weaker even we have stronger coupling strength $\lambda$. The growth
of $L$ counteracts the effect of the $\lambda$ and hinders the scalarization
process.

##  V Signal of Scalarization of Black Holes with Zero Curvature

AdS black holes with a flat space ($k=0$) is described as toroidal black hole
spacetimes. Thus the corresponding redshift function in the metric (6) is
$g(r)=-\frac{M}{r}+\frac{r^{2}}{L^{2}}$ ($M=\frac{1}{L^{2}}$ as $r_{h}=1$) and
the GB term is ${\cal
R}^{2}_{GB}=\frac{4}{r^{2}}(g^{\prime}(r)^{2}+g(r)g^{\prime\prime}(r))$.

We plot the effective potential in Fig. 7. The potential function with
$\lambda=0$ does not show negative well which insures the stability of the
original toroidal hole. When the coupling between the scalar field and GB term
is turned on, for the chosen AdS radius $L$, the potential well near the
horizon appears and becomes deeper as $\lambda$ increases. For the same
coupling strength $\lambda$, similar potential well behavior will appear when
we decrease the AdS radius $L$. The deep potential well can bound the scalar
field near the horizon and lead to the instability of the original toroidal
black hole. This property is similar to those observed in the case with
positive curvature (Fig. 3) and negative curvature (Fig. 5). Through careful
comparisons, we find that for the same parameters, the potential well in zero
curvature case is deeper than that in the negative curvature case, while
shallower than that in the positive curvature background. This indicates that
in ESGB theory, the horizon curvature influences the scalarization. For AdS
black hole with toroidal horizon the scalar hair can be formed more easily
than the hyperbolic case, but if compared with the spherical AdS black hole,
the scalarization process in toroidal AdS black hole is still more difficult.

Figure 7: The effective potential for AdS black hole with zero curvature.

To see the signal of the scalar field growing near the horizon, we solve the
perturbation equation (11) with the boundary condition near the horizon

$\displaystyle\phi^{\prime}(1)=\frac{L^{2}}{3}(m^{2}-18\frac{\lambda^{2}}{L^{4}}e^{-\phi(1)^{2}})\phi(1)~{},$
(26)

The behavior is also (17) in asymptotical infinity, and we set $\phi_{+}=0$ as
usual in the numeric.

The phase diagram in parameters $(L,\lambda)$ space is shown in the left panel
of Fig. 8 where below the blue line, the scalar field perturbation will
trivially die out, but above this line the black hole scalarization will
happen. The values on this line are critical threshold values to ignite the
scalarization. When the AdS radius $L\rightarrow\infty$, we see that the
threshold coupling $\lambda$ linearly approaches infinity, which means that
the toroidal AdS black hole cannot be scalarized in such limits. In the limit
$L\rightarrow\infty$, the background returns to the flat spacetime and there
is no black hole with the zero curvature.

Figure 8: Left: the threshold curves for AdS black hole with zero curvature;
Middle: the scalar hair near the horizon as the function of $\lambda$ with
different AdS radius $L$; Right: the corresponding scalar charge of the black
hole $\phi_{-}$ as the function of $\lambda$.

With some fixed $L$, we study the scalar field near the horizon and the scalar
charge at infinity as the function of $\lambda$, and the results are shown in
the middle and right of Fig. 8, respectively. Note that for large GB coupling,
the probe limit may be not reasonable and backreaction should be involved. But
in some sense our result for small GB coupling is reliable and we see the
signal of scalarization from the scalar field perturbation. In Guo:2020sdu ,
we constructed the scalarized hairy black hole with planar horizon in AdS
space by considering the full backreaction. There we have set $L=1$ and the
threshold value is $\lambda\simeq 0.64$ which is consistent with that in Fig.
8 via careful comparison.

##  VI Closing remarks

In this paper, we discussed the signal of scalarization for black holes with
different curvature topologies in various spacetimes. We considered the scalar
field as a probe to the general relativity solution in Einstein-scalar-Gauss-
Bonnet theory. Our discussion is based on the analysis of its effective
potential and the onset of nontrivial solution to the perturbed scalar
equation.

We studied the signal of black hole scalarization with positive curvature in
asymptotical flat, AdS and dS spacetime. We first analyzed the validity of the
probe approximation in various spacetimes. We found that in flat and AdS
cases, a regular condition of test scalar field near event horizon could
correspond to convergent (vanishing) condition at the asymptotical infinity,
meaning the test field approximation could be valid; while in dS case, the
regular condition near cosmological horizon leads to divergence at large
distance, so that the test field approximation breaks down. We then compared
the effect of asymptotical flat and AdS spacetime spacetime structures on the
scalarization process. Without the GB coupling $\lambda$, the effective
potential of the scalar field are alway positive and no well emerges. As we
increased the coupling, the negative potential well would show and becomes
deeper which may lead to tachyonic instability. With the same parameters, the
potential well for AdS black hole is deeper than that for Schwarzschild black
hole, meaning that the scalar hair around the spherical AdS black hole may be
easier to form. We solved the perturbed scalar field equation in the parameter
space $(\lambda,L)$, and obtained the critical line above which the non
trivial scalar hair can be found. Fig. 4 shows that the scalar cloud around
AdS black hole could form easier than that around the Schwarzschild black
hole. Moreover, in the limit $L\rightarrow\infty$, the threshold coupling in
AdS case approaches the value in Schwarzschild black hole from below as shown
in the left of Fig. 4.

We also investigated the signal of AdS black hole scalarization process with
different curvatures of the horizon. For AdS black hole with positive $(k=1)$,
negative $(k=-1)$ and zero $(k=0)$ curvatures, the properties of effective
potential is qualitatively the same (see Fig. 3, Fig. 5 and Fig. 7,
respectively), that’s to say, no negative potential well appears with
$\lambda=0$ for instability. The potential well near horizon appears and
becomes deep as $\lambda$ (or $L$) increases (decreases), which could bound
the scalar field near the horizon and lead to instability. However, the
potential well for zero curvature is deeper than that for negative curvature
but shallower than that for positive curvature. Notice that the effect of
horizon curvature on the potential could be roughly analytically evaluated
from the expression (10), i.e,
$V(r)=g(r)\left(\frac{g^{\prime}(r)}{r}+\frac{A}{r^{2}}+m^{2}-\frac{\lambda^{2}}{2}{\cal
R}^{2}_{GB}\right)$. With fixed $L$ and $\lambda$, we can diagnose the
potential as follows. (i) $m^{2}=-2/L^{2}+12\lambda^{2}/L^{4}$ for different
horizon curvatures is the same in our setup. (ii) For the lowest mode, the
topological term $A$ satisfies $A(k=1)=A(k=0)=0<A(k=-1)=1/4$ . (iii) Fixing
horizon radius implies that the black hole mass satisfies
$M(k=1)>M(k=0)>M(k=-1)$, then it is easy to obtain that the group term
$T=\frac{g^{\prime}(r)}{r}-\frac{\lambda^{2}}{2}{\cal}R^{2}_{GB}$ in the
potential satisfies $T(k=1)<T(k=0)<T(k=-1)<0$ for fixed $r$ near horizon. (iv)
The overall factor $g(r)$ fulfills $g(k=1)>g(k=0)>g(k=-1)>0$ for fixed $r$
outside the horizon. Combining (i)-(iv), it is not difficult to reduce that
the effect of horizon curvature on the negative potential well is
$V(k=1)<V(k=0)<V(k=-1)$ near horizon. It is noted that only the angular
eigenvalues term $A$ mentioned in point (ii) is truly the topological effect,
and the terms mentioned in points (iii-iv) are actually geometrical effect.
Thus, in this sense, the effect from the horizon curvature on the potential as
well as the scalarization is reflected by the interplay between the topology
and geometry.

The behavior of potential indicates that in ESGB theory, the scalar hair
around AdS black hole with toroidal horizon may be easier to be formed than
that with hyperbolic horizon but more difficult than that with spherical
horizon. Moreover, we collected the critical threshold curves for different
curvatures in Fig. 9 where the blue dot represents the threshold value of
Schwarzschild black hole in the limit $L\rightarrow\infty$. It is obvious that
for the AdS black hole with fixed $L$, the scalar hair is the easiest to be
formed around spherical horizon, then around the toroidal horizon and it is
the most difficult to be bounded near hyperbolic horizon.

Figure 9: The threshold curves for AdS black hole with different curvatures.
The blue dot represents the threshold value of Schwarzschild black hole.

We shall present two comments on Fig. 9. $i)$ In the limit $L\rightarrow 0$,
the threshold coupling in all three cases is very small in which case the
scalarization could easily realized. Moreover, we know from the effective
potential that the GB coupling should be involved for the tachyonic
instability and it further leads to the spontaneous scalarization. So we may
argue that the threshold value would be tiny as $L\rightarrow 0$, or the
scalarization could cut off at certain tiny $L$. This direction with large
$\Lambda$ deserves further study. $ii)$ In the limit $L\rightarrow\infty$, the
threshold coupling for zero or negative curvature are divergent, suggesting
that the scalar field can not be bounded in these cases. This is
understandable because as $L\rightarrow\infty$, the spacetime become flat and
only Schwarzschild black hole with positive curvature exists as background in
probe limit. Moreover, it is interesting to involve the full backreaction of
the scalar field into the geometry and verify our results.

###### Acknowledgements.

This work is supported by the Natural Science Foundation of China under grant
Nos.11705161, 12075202 and 11690022. Xiao-Mei Kuang is also supported by Fok
Ying Tung Education Foundation under grant No.171006.

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