# Instability of de-Sitter black hole with massive scalar field coupled to
Gauss-Bonnet invariant and the scalarized black holes

Zhen-Hao Yang<EMAIL_ADDRESS>Guoyang Fu<EMAIL_ADDRESS>Xiao-Mei Kuang<EMAIL_ADDRESS>(corresponding author) Jian-Pin Wu
<EMAIL_ADDRESS>Center for Gravitation and Cosmology, College of Physical
Science and Technology, Yangzhou University, Yangzhou 225009, China

###### Abstract

The black hole scalarization in a special Einstein-scalar-Gauss-Bonnet (EsGB)
gravity has been widely investigated in recent years. Especially, the
spontaneous scalarization of scalar-free black hole in de-Sitter (dS)
spacetime possesses interesting features due to the existence of cosmological
horizon. In this work, firstly, we focus on the massive scalar field
perturbation on Schwarzschild dS (SdS) black hole in a special EsGB theory. By
analyzing the fundamental QNM frequency and time evolution of the scalar field
perturbation, we figure out the unstable/stable regions in
$(\Lambda,\alpha)$-plane as well as in $(m,\alpha)$-plane for various
perturbation modes, where $\Lambda$, $\alpha$ and $m$ denote the cosmological
constant, the GB coupling strength and the mass of scalar field, respectively.
Then by solving the static perturbation equation, we analyze the bifurcation
point at which the SdS black hole supports spherical scalar clouds, and we
find that the bifurcation points match well with $\alpha_{c}$ on the border of
unstable/stable region. Finally, after addressing that the scalarised
solutions could only emerge from the scalar could with node $k\geq 1$. we
explicitly construct the scalarized hairy solutions for different scalar
masses and compare the profile of scalar field to the corresponding scalar
clouds.

###### Contents

  1. I Introduction
  2. II Model
  3. III Analysis on the (in)stability from massive scalar field perturbation
    1. III.1 Preliminary preparation
      1. III.1.1 Frequency domain analysis
      2. III.1.2 Time domain analysis
    2. III.2 The QNM frequencies and instability
      1. III.2.1 $\ell-$dependence
      2. III.2.2 $\Lambda-$dependence
      3. III.2.3 $m-$dependence
  4. IV Scalarized hairy black hole solution
    1. IV.1 Bifurcation points and scalar clouds
    2. IV.2  Construction of scalarized hairy black hole
  5. V Conclusion and discussion

##  I Introduction

It is widely accepted that the effects of higher-order curvature terms are
significant as we are exploring the strong gravity regime via detections of
gravitational waves and black hole shadows. In theoretical framework, the
inclusion of such terms usually involves the well-known ghost problem stelle .
A counterexample which can be ghost-free is including the Gauss-Bonnet (GB)
correction, however, it becomes a topological term in four-dimensional
spacetime and has no dynamics when minimally coupled with Einstein-Hilbert
action. One way to make this term contribute to the dynamic in four-
dimensional spacetime is to introduce a coupling between the GB term and
scalar field stringT . The theory which includes this kind of coupling is
dubbed Einstein-scalar-Gauss-Bonnet (EsGB) gravity, which has attracted plenty
of attention as it admits hairy black holes. Various black hole solutions and
compact objects in four-dimensional EsGB theories were studied in the
literatures Mignemi_1993 ; Kanti_1996 ; Torrii_1996 ; Guo:2008eq ;
Maeda:2010bj ; Ohta:2013lsa ; Ayzenberg_2014 ; Kleihaus_2011 ; Kleihaus_2016a
and therein. More recently, the spontaneous scalarization of scalar-free black
hole with particular coupling functions in EsGB theory was proposed. It was
addressed that below a certain mass the Schwarzschild black hole background
may become unstable in regions of strong curvature, and then a scalarized
hairy black hole emerges when the scalar field backreacts to the geometric.
The precess evades the well-known no-hair theorems Antoniou:2017acq ;
Silva:2017uqg ; Doneva:2017bvd . This proposal on spontaneous scalarization
has inspired wide generalizations in the literatures Herdeiro:2019yjy ;
Brihaye:2018bgc ; Minamitsuji:2018xde ; Silva:2018qhn ; Andreou:2019ikc ;
Minamitsuji:2019iwp ; Peng:2020znl ; Liu:2020yqa ; Doneva:2020qww ;
Astefanesei:2020qxk ; Canate:2020kla ; Hunter:2020wkd ; Bakopoulos:2019tvc ;
Bakopoulos:2020dfg ; Lin:2020asf ; Brihaye:2019dck ; Guo:2020sdu ;
Tang:2020sjs ; Collodel:2019kkx ; Dima:2020yac ; Doneva:2020kfv ;
Herdeiro:2020wei ; Berti:2020kgk ; Guo:2020zqm .

For various reasons, physicists have great interest in de Sitter (dS)
spacetime. On one hand, the theoretical model of our expansion universe
consists on assuming positive cosmological constant, implying that the
physical universe is asymptotically dS. Besides, dS spacetime plays an
important role in primordial inflation theory, which is now part of the
standard cosmological model SupernovaCosmologyProject:1998vns ;
SupernovaSearchTeam:1998fmf . On the other hand, the holographic duality
between quantum gravity in dS spacetime and a conformal field theory on it
boundary sheds remarkable application on the asymptotically dS spacetime
Strominger:2001pn ; Witten:2001kn .

Due to the above consideration, the spontaneous scalarization in EsGB theory
has been soon extensively studied in dS spacetime Bakopoulos:2018nui ;
Brihaye:2019gla . It was found that the positive cosmological constant does
not change the local conditions for a tachyonic instability of the background
black hole, Schwarzschild dS (SdS) in this case, to emerge. In details, in
Bakopoulos:2018nui , the authors claimed that a regular black hole horizon
with a non-trivial hair may be always formed after an analysis in the near-
horizon asymptotic regime. But the complete hairy solution was absent, the
deep reason of which is not clear. Later in Brihaye:2019gla , it was addressed
if the scalar field is confined between the black hole and cosmological
horizons, then it is not likely to form scalarised black hole solution; while
a new hairy black hole was numerically constructed if the scalar field is
permitted to extend beyond the cosmological horizon. Thus, it is obvious that
the existence of cosmological horizon introduces particular situation in the
black hole scalarization in dS spacetime, which deserves further study.

It is noted that though the analysis of (in)stability on the background black
hole under the massless scalar field perturbation was present in
Bakopoulos:2018nui ; Brihaye:2019gla , however, the quasi-normal mode (QNM)
frequencies and the dynamical evolution of the scalar field in the SdS
background is still missing. This paper intends to fill this gap. We shall
consider a massive scalar field as a probe field on the SdS black hole and
admit the scalar field can extend beyond the cosmological horizon. By
computing its fundamental QNM frequencies and the linear time domain dynamical
evolution, we fix the unstable/stable parameter regions in
$(\Lambda,\alpha)$-plane and $(m,\alpha)$-plane, respectively for various
angular momentum modes. Then we analyze the possible existence of scalar cloud
by solving the static perturbation equation. We show that the bifurcation
points from the lowest sate of scalar field with node $k=0$ match well with
the border of unstable/stable region obtained in the fundamental frequency for
various modes massless/massive scalar perturbation. Finally, we analyze the
possible constraint on the models that can have scalarised solutions and
address that it could emerge from the scalar could with node $k\geq 1$.
Considering the backreaction, we construct the scalarized hairy black hole
solution for different scalar masses from the complete non-linear differential
equations, and we also see how the scalar cloud evolves into the hairy
solution by comparing their profiles.

The remaining of this paper is organized as follows. In section II, we briefly
present the EsGB model with a positive cosmological constant, and the
equations of motion. In section III, we analyze the (in)stability of the SdS
black hole under the perturbation of massive scalar field via investigating
its fundamental QNM frequencies and the time evolution. We investigate the
bifurcation points at which the SdS black hole supports spherical scalar
clouds with different modes, and then construct the scalarized hairy black
hole solutions for different scalar masses in section IV. The last section
contributes to our conclusion and discussion. We shall work with the units
$\hbar=G=c=1$.

##  II Model

The action of EsGB gravity in dS spacetime with the scalar field coupled with
the GB invariant is given by

$S=\frac{1}{16\pi}\int
d^{4}x\sqrt{-g}\left[R-2\Lambda-2\nabla_{\mu}\phi\nabla^{\mu}\phi-m^{2}\phi^{2}-f(\phi)\mathscr{L}_{\text{GB}}\right],$
(1)

where $R$ is the Ricci scalar, $\Lambda$ is a positive cosmological constant,
$\phi$ is scalar field with mass $m$, $f(\phi)$ is the coupling function, and

$\begin{split}\mathscr{L}_{\text{GB}}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.\end{split}$
(2)

Noted that different forms of $f(\phi)$ shall give different properties of the
EsGB theory. As mentioned in Doneva:2017bvd , to admit Schwarzschild black
hole as background solution, $f(\phi)$ could satisfy the conditions
$\frac{df(\phi)}{d\phi}\mid_{\phi=0}=0$ and
$\frac{d^{2}f(\phi)}{d\phi^{2}}\mid_{\phi=0}=\mathfrak{b}^{2}>0$, where
$\mathfrak{b}$ is a constant. Moreover, one usually assumes that the scalar
field vanishes at infinity and normalizes the constant $\mathfrak{b}$ to be
unity. Thus, to fulfill the requirement, we shall follow Brihaye:2019gla  and
choose the simplest form of the coupling function as

$f(\phi)=a_{0}-\alpha\phi^{2},$ (3)

where $\alpha_{0}$ can be an arbitrary value, and $\alpha$ is the coupling
parameter with dimension $[length]^{2}$ such that we will use the
dimensionless parameter $\alpha/M^{2}\to\alpha$ (Here $M$ is kind of black
hole mass with length dimension as we will see soon.).

Then the equations of motion for the scalar field and the gravity field are

$\displaystyle\nabla_{\mu}\nabla^{\mu}\phi$ $\displaystyle=$
$\displaystyle\frac{1}{2}m^{2}\phi-\frac{\alpha\phi\mathscr{L}_{\text{GB}}}{2},$
(4) $\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}$
$\displaystyle=$ $\displaystyle 2(T_{\mu\nu}^{\phi}+T_{\mu\nu}^{GB})$ (5)

where

$\displaystyle T_{\mu\nu}^{\phi}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{4}g_{\mu\nu}m^{2}\phi^{2}-\frac{1}{2}g_{\mu\nu}\partial_{\rho}\phi\partial^{\rho}\phi,$
(6) $\displaystyle T_{\mu\nu}^{GB}$ $\displaystyle=$
$\displaystyle-2\alpha(-\frac{1}{4}\varepsilon_{\mu\sigma\gamma\tau}R^{\gamma\tau\epsilon\delta}\varepsilon_{\nu\rho\epsilon\delta})\nabla^{\rho}\nabla^{\sigma}f(\phi).$
(7)

In the above sector, we shall firstly treat the scalar field as perturbation
and study the (in)stability of the sector, and then we involve the
backreaction of the scalar field to the geometry and construct the scalarized
hairy black hole solution.

##  III Analysis on the (in)stability from massive scalar field perturbation

With $\phi=0$, the above equations admit the Schwarzschild de-Sitter (SdS)
black hole solution

$\displaystyle ds^{2}$
$\displaystyle=-g(r)dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}\left(\text{sin}^{2}\theta
d\theta^{2}+d\phi^{2}\right)$ (9)
$\displaystyle\mathrm{with}~{}~{}~{}~{}g(r)=1-\frac{2M}{r}-\frac{\Lambda
r^{2}}{3}$

where the constant $M$ is the black hole mass. In this background, the GB term
is evaluated as

$\mathscr{L}_{\text{GB}}=\frac{48M^{2}}{r^{6}}+\frac{8\Lambda^{2}}{3}.$ (10)

Depending on the model parameters, $g(r)=0$ could have two positive real
roots, $(r_{e},r_{c})$, and one negative real root, $r_{o}$. The positive
roots, $r_{e}$ and $r_{c}$, represent the black hole event horizon and
cosmological horizon, respectively, of the SdS black hole. Note that these two
horizons only hold when $3M\sqrt{\Lambda}<1$, known as the Nariai limit,
within which their analytic formulas are

$r_{e}=\frac{2}{\sqrt{\Lambda}}\cos\left(\frac{1}{3}\cos^{-1}\left(3M\sqrt{\Lambda}\right)+\frac{\pi}{3}\right)\;,\quad
r_{c}=\frac{2}{\sqrt{\Lambda}}\cos\left(\frac{1}{3}\cos^{-1}\left(3M\sqrt{\Lambda}\right)-\frac{\pi}{3}\right).$
(11)

Approaching the Nariai limit, the event horizon and the cosmological horizon
tend to merge into one horizon; and beyond the limit, no black hole horizon
exists.

Considering $r_{c}>r_{e}>r_{o}$, the metric function $g(r)$ can be further
expressed as,

$g(r)=\frac{\Lambda}{3r}(r-r_{e})(r_{c}-r)(r-r_{o}).$ (12)

Introducing the surface gravity $\kappa_{i}=\frac{1}{2}\mid
g^{\prime}(r)\mid_{r=r_{i}}$ associated with each root $r=r_{i}~{}(i=e,c,o)$,
one could calculate the tortoise coordinate $r_{*}=\int g^{-1}(r)\,dr$ in an
analytic form

$r_{*}(r)=\frac{1}{2\kappa_{e}}\ln\left(\frac{r}{r_{e}}-1\right)-\frac{1}{2\kappa_{c}}\ln\left(1-\frac{r}{r_{c}}\right)+\frac{1}{2\kappa_{o}}\ln\left(\frac{r}{r_{o}}-1\right)$
(13)

which will play an important role in our later numerical study.

Then we consider a small scalar field perturbation on the background of SdS
black hole in the linear regime, which is governed by the covariant equation

$\square\phi=\frac{\partial
f(\phi)}{\partial\phi}\frac{\mathscr{L}_{\text{GB}}}{4}+\frac{m^{2}}{2}\phi\equiv\mu_{\text{eff}}^{2}\,\phi.$
(14)

Here the box denotes the d’Alembertian operator and the effective mass is

$\mu_{\text{eff}}^{2}=\frac{m^{2}}{2}-\frac{\alpha}{2}\left(\frac{8\Lambda^{2}}{3}+\frac{48M^{2}}{r^{6}}\right).$
(15)

The tachyonic instability may occur only when $\mu_{\text{eff}}^{2}<0$, which
requires $\alpha>0$. As addressed in Brihaye:2019gla , this instability may
trigger the emergence of a hairy solution on the SdS background via the
spontaneous scalarization process, though the authors only considered the
massless $r-$dependent scalar field. The possible parameter regions which give
unstable situation and how the scalar field grows up were not present. Here we
shall answer those questions by studying the dynamical perturbation of scalar
field with various modes in both frequency domain and time domain.

###  III.1 Preliminary preparation

####  III.1.1 Frequency domain analysis

To analyze the (in)stability in frequency domain, one usually decompose the
scalar field as

$\phi(t,r,\theta,\psi)=\sum_{\ell\mathfrak{m}}\frac{R_{\ell\mathfrak{m}}(r)}{r}Y_{\ell\mathfrak{m}}(\theta,\psi)e^{-i\omega
t}$ (16)

where $Y_{\ell\mathfrak{m}}(\theta,\psi)$ is the spherical harmonic function
with angular momentum $l$ and azimuthal number $\mathfrak{m}$. Working under
the tortoise coordinate $r_{*}$, we obtain that each wave function $R$
satisfies

$\frac{\partial^{2}R(r)}{\partial
r_{*}^{2}}+\left(\omega^{2}-V_{eff}(r)\right)R(r)=0,$ (17)

where the effective potential is

$V_{eff}(r)=g(r)\left[\frac{\ell(\ell+1)}{r^{2}}+\frac{g^{{}^{\prime}}(r)}{r}+\mu_{\text{eff}}^{2}\right],$
(18)

and it is not dependent of the azimuthal number $\mathfrak{m}$ because of the
spherical symmetry. This potential could present a negative well between
$r_{e}$ and $r_{c}$, which plays an important role in the instability of the
SdS black hole under this perturbation as we will see soon. The asymptotic
behavior of the perturbation near the horizons are

$R(r\to r_{e})\sim e^{-i\omega
r_{*}}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}R(r\to r_{c})\sim e^{i\omega r_{*}}$
(19)

which correspond to ingoing and outgoing boundary condition near the event
horizon and cosmological horizon, respectively. It is known that only discrete
eigenfrequencies $\omega=\omega_{R}+i\omega_{I}$ (QNM frequency), where
$\omega_{R}$ and $\omega_{I}$ respectively denote the real part and imaginary
part of the QNM frequency, satisfy the perturbation equation and the boundary
conditions. Once $\omega_{I}>0$, the amplitude of the perturbation will grow
up, implying that the black hole is unstable under this perturbation. There
are many methods developed to compute the QNM frequencies, for instance, WKB
method, shooting method, Horowtiz-Hubeny method, AIM method, spectral method,
etc., and readers could refer to Berti:2009kk ; Konoplya:2011qq  and therein
for nice reviews. In this work, we will employ the spectral method, which has
been well described in Jansen:2017oag . Moreover, we shall also testify our
results with the time domain evolution.

####  III.1.2 Time domain analysis

To study the dynamical evolution of the perturbed scalar field, we decompose
the scalar field as

$\phi(t,r,\theta,\psi)=\sum_{\ell\mathfrak{m}}\frac{R_{\ell\mathfrak{m}}(t,r)}{r}Y_{\ell\mathfrak{m}}(\theta,\psi).$
(20)

In the tortoise coordinate, the perturbation equation reduces to

$\left(-\frac{\partial^{2}}{\partial t^{2}}+\frac{\partial^{2}}{\partial
r_{*}^{2}}-V_{eff}(r)\right)R(t,r)=0$ (21)

where $V_{eff}(r)$ has been defined in (18).

We have to solve the wave equation numerically since there is no analytic form
of this time-dependent wave equation. We adopt the discretization method
proposed in Zhu:2014sya , and discretize the wave equation (21) by defining
$R(r_{*},t)=R(j\Delta r_{*},i\Delta t)=R_{j,i}$ ,
$V\left(r(r_{*})\right)=V(j\Delta r_{*})=V_{j}$. Then it can be deformed as

$-\frac{(R_{j,i+1}-2R_{j,i}+R_{j,i-1})}{\Delta
t^{2}}+\frac{(R_{j+1,i}-2R_{j,i}+R_{j-1,i})}{\Delta
r_{*}}-V_{j}R_{j,i}+O(\Delta t^{2})+O(\Delta r^{2}_{*})=0.$ (22)

With the initial Gaussian distribution
$R(r_{*},t=0)=\exp[-\frac{(r_{*}-a)^{2}}{2b^{2}}]$ and $R(r_{*},t<0)=0$, we
can derive the evolution of $R$ by

$R_{j,i+1}=-R_{j,i-1}+\frac{\Delta t^{2}}{\Delta
r^{2}_{*}}R_{j+1,i}+R_{j-1,i}+\left(2-2\frac{\Delta t^{2}}{\Delta
r^{2}_{*}}-\Delta t^{2}V_{j}\right)R_{j,i}.$ (23)

For the sake of the numerical precision, we shall fix $\frac{\Delta
t^{2}}{\Delta r^{2}_{*}}=0.5$ to fit the von Neumann stability conditions, and
set $a=10,b=3$ in the profile of Guassian wave.

###  III.2 The QNM frequencies and instability

We will study how the coupling parameter and the mass of scalar field affect
the (in)stability of SdS black hole. We will also consider different
cosmological constants and angular momentums of the perturbation modes. To
this end, we fix $M=1$ without loss of generality.

####  III.2.1 $\ell-$dependence

In this subsection, we compute the _fundamental (with overtone $n=1$)_ QNM
frequency of different $\ell-$modes perturbation around the SdS black hole. We
focus on the effect of $\ell$ on the GB coupling dependent QNM frequencies by
fixing $m=0$ and $\Lambda=0.1$.

The results are shown in FIG. 1. For the $\ell=0$ mode, both $\omega_{R}$ and
$\omega_{I}$ are zero in minimal coupling case implying that the mode is prone
to instability. As the coupling parameter, $\alpha$, increases, the black
lines shows that $\omega_{I}$ becomes positive while $\omega_{R}$ is still
zero. This phenomena means that the $\ell=0$ mode is always unstable. For the
$\ell\neq 0$ modes, the left plot shows that as $\alpha$ increases,
$\omega_{I}$ first decreases and then increases. There exists a critical
value, $\alpha_{c}$, at which $\omega_{I}=0$ for each mode. When
$\alpha>\alpha_{c}$, $\omega_{I}$ becomes positive implying that the
corresponding perturbation could grow up to destabilize the background SdS
black hole. $\alpha_{c}$ is larger for modes with larger $\ell$ and the
samples are listed in table 1, which implies that under larger $\ell$ mode
perturbation, stronger GB coupling is called for to trigger the instability.

Moreover, comparing the two plots in FIG. 1, we see that the cases with
decreasing $\omega_{I}$ have non-vanishing $\omega_{R}$, while the cases with
increasing $\omega_{I}$ possess purely imaginal QNM frequency. As addressed in
Aragon:2020tvq , the former cases are dubbed photon sphere modes while the
latter are dS modes. In our study, since the critical $\alpha$ from photon
sphere mode to dS mode is always smaller than $\alpha_{c}$, so the unstable
perturbation modes are all dS modes, which is an interesting phenomena
deserving further study.

Figure 1: The fundamental QNM frequency as a function of the GB coupling for different $\ell$ modes. $\ell$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---|---
$\alpha_{c}$ | $-$ | $4.63$ | $12.75$ | $23.78$ | $37.84$ | $54.93$ | $75.09$
Table 1: The critical coupling for different momentum angular.

We also directly calculate the time evolution of the perturbation field and
further reveal the instability of SdS in EsGB gravity. The pedagogy on
evolutionary analysis has been shown in subsection III.1.2. The evolutions of
the perturbation with different $\ell$ in log plot are shown in FIG. 2. For
$\ell=0$ mode, non-vanishing GB coupling makes the perturbation grow as time,
meaning that the system is unstable. For $\ell\neq 0$, when
$\alpha<\alpha_{c}$, the perturbation will decay as the time evolves, but it
grows up when $\alpha>\alpha_{c}$. The phenomenon indicates that the system
will become unstable under the perturbed modes once the GB coupling is larger
than the corresponding $\alpha_{c}$. These findings in time domain are
consistent with those in frequency domain analysis. The effect of the GB
coupling on the dynamical evolution is different from that of the non-
minimally coupled to curvature studied in Brady:1999wd  which always makes the
evolution decay in SdS black hole.

Figure 2: The time evolution of the perturbation modes with $\ell=1$ ,
$\ell=2$ and $\ell=3$ from left to right.

The effect of the GB coupling on the (in)stability of the scalar perturbation
could be explained by analyzing the effective potential (18). It is
straightforward to obtain that the effective potential with a fixed radius
near the event horizon could be suppressed by larger $\alpha$ but enhanced by
larger $\ell$. Then in FIG. 3, we explicitly show the profile of the potential
between the event horizon and cosmological horizon. It is obvious that in each
case, when $\alpha$ is smaller than a certain value, the effective potential
is always positive. As $\alpha$ increases, a negative potential well would
form, and becomes more deeper as $\alpha$ is further enlarged. It is noted
that comparing $\alpha_{c}$ in the frequency as well as time domain analysis,
the critical value of $\alpha$ for the emergence of negative potential well in
FIG. 3 is smaller in all cases. This is because the negative potential well is
not a sufficient condition for the instability. Only deep enough potential
well could help the scalar to collect near the event horizon and finally
trigger the instability of system.

Figure 3: The profile of the effective potential between the event horizon and
cosmological horizon for $\ell=0,1,2$ from left to right.

####  III.2.2 $\Lambda-$dependence

We then study the effect of the cosmological constant $\Lambda$ on the
fundamental QNM frequency. The results are shown in FIG. 4, which shows that
the effect of $\Lambda$ on various $\ell$-modes is similar. Specifically, for
small $\alpha$, $\omega_{I}$ is negative and larger for larger $\Lambda$. It
means that the perturbation in the spacetime with larger $\Lambda$ can live
longer and then decays, since the lifetime is connected with the QNM frequency
via $\tau\sim 1/|\omega_{I}|$. As $\alpha$ increases, $\omega_{I}$ increases
and there exists an intersection for different $\Lambda$. As $\Lambda$
increases, the critical value $\alpha_{c}$ at which $\omega_{I}$ transits from
negative to positive increases. Then for large enough $\alpha$, $\omega_{I}$
become positive and is smaller for larger $\Lambda$. This indicates that in
SdS black hole with larger $\Lambda$, the perturbation has longer relax time
and then grows up. The above picture can also explicitly reflected in the time
domain analysis, see FIG. 5 for the $\ell=2$ mode with $\alpha=10$ as an
example.

Figure 4: The fundamental QNM frequency as a function of GB coupling for
different cosmological constant. The modes from left to right are $\ell=1,2$
and $3$, respectively.  Figure 5: The time evolution of perturbation mode with
$\ell=2$. Here the GB coupling parameters is fixed as $\alpha=10$.

It is worthwhile to point out that for $\alpha=0$, i.e., in Einstein gravity,
our QNM frequencies with different $\ell$ and $\Lambda$ calculated via
spectral method match well with the results computed by WKB method and Prony
method in Abdalla:2003db ; Zhidenko:2003wq . Then in EsGB theory with
$\alpha\neq 0$, by scanning the cosmological constant inside the Nariai limit,
we extract $\alpha_{c}$ and figure out the unstable/stable region in
$\Lambda-\alpha$ plane under massless scalar perturbation. The results for
samples of $\ell$ are shown in FIG. 6. Under the perturbation mode with each
$\ell$, the instability can be triggered in the region above each line (the
shading region), below which the system is stable. It is obvious that for the
modes with larger $\ell$, the system could be stable in a wider parameters
region in $\Lambda-\alpha$ plane.

Figure 6: The unstable/stable region of SdS black hole under massless scalar
perturbation with $\ell=0,1,2$ and $3$ from fundamental QNM frequency.

####  III.2.3 $m-$dependence

In this subsection, with fixed $\Lambda=0.1$, we shall turn on the mass of the
scalar field and study its effect. The fundamental QNM frequencies for
$\ell=1$ mode with different $m$ is shown in FIG. 7. It shows that the rule is
similar to that for massless case, namely, as the GB coupling increases,
$\omega_{I}$ first decreases and then increases. And for larger $m$, the
turning value of $\alpha$ from decreasing to increasing is larger, so is
$\alpha_{c}$ at which $\omega_{I}$ crosses the horizonal axis. This indicates
that for the scalar field with larger mass, stronger GB coupling is required
to destabilize the SdS black hole.

Figure 7: The fundamental QNM frequency for $\ell=1$ mode as a function of GB
coupling for different mass of scalar perturbation.

This conclusion is also testfied by the dynamical evolution of the massive
scalar field in FIG. 8. Each plot shows that when $\alpha$ is smaller than a
certain $\alpha_{c}$, the scalar field would decay as the time evolves; while
the scalar field will grow up when $\alpha$ is larger than $\alpha_{c}$. The
growth of scalar field could finally destabilize the SdS black hole and
trigger spontaneous scalarization. Comparing the value of $\alpha_{c}$ in each
plot, it is obvious larger $m$ corresponds to larger $\alpha_{c}$ which
matches the findings in FIG. 7. More explicit effect of $m$ on the dynamical
evolution with fixed $\alpha$ is shown in FIG. 9. This mass dependence is
reasonable because even in Einstein gravity, increasing $m$ would enhance
$\omega_{I}$ which is always negative and decrease the scalar field damping
rate, such that the corresponding modes live longer Toshmatov:2017qrq .

Figure 8: The dynamical evolution of massive scalar field in log plot. The
upper-left, upper-right, lower-left and lower-right plots correspond to
$m=0.2,0.3,0.4$ and $0.5$ respectively.  Figure 9: The time evolution of the
$\ell=1$ mode with fixed $\alpha=5.49$.

For other modes , similar phenomena could be seen as for $\ell=1$ mode. So
instead of repeating the analysis, we collect $\alpha_{c}$ with different $m$
for $\ell=0,1,2$ and $3$ modes, and then draw the unstable/stable region in
$m-\alpha$ plane in FIG. 10. For each mode, the SdS black hole could be
unstable when the parameters are in the regime above each line.

Figure 10: The unstable/stable region of SdS black hole under massive scalar
perturbation with $\ell=0,1,2$ and $3$ from fundamental QNM frequency.

##  IV Scalarized hairy black hole solution

In this section, we shall fix the existence of scalar clouds and the
scalarized black hole, whose profile functions between the black hole horizon
and cosmological horizon will be explicitly constructed.

###  IV.1 Bifurcation points and scalar clouds

The tachyonic instability provides the possibility of the scalar field to be
non-trivial between the black hole and the cosmological horizon. Then we shall
go on to seek the bound states (scalar clouds) of the test scalar around the
SdS black hole. To obtain the scalar clouds with various modes, we only need
to solve the perturbation equation (17) in static case with appropriate
boundary conditions. With the two horizons $r_{e}$ and $r_{c}$, the equation
is rewritten as

$\phi^{\prime\prime}(r)+\frac{r_{c}\left(r_{e}-2r\right)(r_{c}+r_{e})-2rr_{e}^{2}+4r^{3}}{r\left(r-r_{c}\right)\left(r-r_{e}\right)\left(r_{c}+r_{e}+r\right)}\phi^{\prime}(r)+\frac{\left(r_{c}r_{e}+r_{c}^{2}+r_{e}^{2}\right)\left(\ell^{2}+\ell+r^{2}\mu_{eff}\right)}{r\left(r-r_{c}\right)\left(r-r_{e}\right)\left(r_{c}+r_{e}+r\right)}\phi^{\prime}(r)=0$
(24)

where we recover $\phi(r)=R(r)/r$ and the effective mass is

$\mu_{eff}=\frac{m^{2}}{2}-\frac{6\alpha\left(r_{c}^{2}r_{e}^{2}\left(r_{c}+r_{e}\right){}^{2}+2r^{6}\right)}{r^{6}\left(r_{c}r_{e}+r_{c}^{2}+r_{e}^{2}\right){}^{2}}.$
(25)

The regularity near the black hole horizons give us

$\displaystyle\phi(r\to
r_{e})=\phi_{h}+\tilde{\phi}_{h}(r-r_{e})+\mathscr{O}(r-r_{e}),$ (26)
$\displaystyle\phi(r\to
r_{c})=\phi_{c}+\tilde{\phi}_{c}(r-r_{c})+\mathscr{O}(r-r_{c})$ (27)

where $\tilde{\phi}_{h}(\tilde{\phi}_{c1})$ can be solved to be dependent on
$\phi_{h}(\phi_{c})$ .

It is known that in the construction of scalar cloud, solving (24) is in
essence an eigenvalue problem Silva:2017uqg ; Cunha:2019dwb . For given
$\ell$, $m$ and $\Lambda$, by imposing smoothness for the scalar filed at the
horizons, one can select a discrete set of $\alpha/M^{2}$, which give a
discrete set scalar field profile characterized by different nodes $k$ in
between the two horizons. Those selected $\alpha/M^{2}$ could be known as
bifurcation points at which SdS black hole supports a spherical scalar could.
In order to give an explicit picture, in FIG.11 we show typical profiles of
scalar field with the selected $\alpha/M^{2}$ for the lowest state with $k=0$
and two other $\alpha/M^{2}$ with $m=0$, $\Lambda=0.1$ and $\ell=0,1$. It is
obvious that only the selected $\alpha/M^{2}$ corresponds to a smooth scalar
profile between the event horizon and cosmological horizon, giving the scalar
clouds which satisfy the perturbed equation and the imposed boundary
condition.

Figure 11: The scalar field profile of the lowest state for
$\alpha=bifurcation~{}point~{}\alpha_{c}$ (black), $a=\alpha_{c}+0.5$ (red)
and $a=\alpha_{c}-0.5$ (blue) with $m=0$ , $\Lambda=0.1$. The left panel is
for $\ell=0$ with $\alpha_{c}=0$ while the right panel is for $\ell=1$ with
$\alpha_{c}=4.63$. Here we have set $M=1$ and $\phi(r_{e})=1$.

To study the bifurcation points for various modes and masses of scalar field,
we focus on the lowest state with $k=0$ for all cases. We show the results for
$\ell=0,1,2,3$ modes in FIG. 12. In each panel, the green dashed curve denotes
the border of unstable/stable region we obtained in the analysis from
fundamental frequencies (see the corresponding curves in FIG.6). In FIG.13, we
exhibits the effect of scalar mass on the bifurcation point and the scalar
field value at the cosmological horizon for scalar clouds, and again the green
dashed curve in each plot denotes the border of unstable/stable region
obtained from fundamental frequencies (see the corresponding curves in
FIG.10). As we expect, the bifurcation points for various mode scalar clouds
match well with the critical value $\alpha_{c}$ for the unstable/stable region
obtained from the fundamental frequency and time domain.

Figure 12: The coupling parameter against the cosmological constant for the
critical SdS black hole that admits a spherical massless scalar could with the
lowest state with $k=0$ for various modes, and the insets exhibit the
corresponding values of scalar field at $r_{c}$. Here we have set $M=1$ and
$\phi(r_{e})=1$.

Figure 13: The coupling parameter against the mass of scalar field for the
critical SdS black hole that admits a spherical massive scalar could with the
lowest state with $k=0$ for various modes, and the insets exhibit the
corresponding values of scalar field at $r_{c}$. Here we have set $M=1$ and
$\phi(r_{e})=1$.

###  IV.2  Construction of scalarized hairy black hole

The existence of instability and smooth configurations cannot really guarantee
the existence of the scalarized hairy solutions, but the authors of
Brihaye:2019gla  gave possible constraints on this model in the case with
massless zero mode scalar field. As aforementioned, the requirement of
effective mass $\mu^{2}_{\text{eff}}<0$ should be fulfilled for tachyonic
instability. Morevover, following Brihaye:2019gla , we integrate (14) along a
hypersurface $V$ bounded by $r_{e}$ and $r_{c}$, then considering no
contribution from the boundary terms for smooth configurations could lead to
the identity

$\int_{V}d^{4}x\sqrt{-g}\mu^{2}_{\text{eff}}\phi=0$ (28)

which implies that $\phi$ must change sign in the integration interval
$r_{e}<r<r_{c}$ for possible non-trivial scalar fields. Thus, the node $k$ of
the scalar field have to be $k\geq 1$, such that for dS case, the lowest state
of the scalar field from which a scalarized black hole may emerge is the first
excited state with node $k=1$, independent of the mass of scalar field and the
modes with different $\ell$.

Then here we shall focus on $\ell=0$ mode for convenience and mainly study the
effect of the scalar mass on the scalarized black hole.

We firstly reproduce the results of Brihaye:2019gla  on the bifurcation point
$\alpha$ against $\Lambda$ for $\ell=0$ massless scalar clouds with node
$k=1$, see the left panel of FIG. 14. In the middle panel, we show the
bifurcation point for different scalar mass with fixed $\Lambda=0.1$, which
shows that for large $m$, the ratio $\alpha/M^{2}$ could increase for the
bifurcation. Again the green dashed lines are the unstable/stable border from
the analysis of QNM frequency with overtone $n=2$, which is consistent with
the bifurcation points as expected. A typical profile of such scalar cloud is
shown in right panel.

Figure 14: The bifurcation points against $\Lambda$ (left panel) and scalar
mass (middle panel) at which it could emerge a scalarized black hole from zero
mode with $k=1$. The right panel shows a typical profile of such scalar cloud.
We have set $M=1$.

We proceed to construct the scalarized black hole solution and we will fix the
profiles between the black hole horizon and cosmological horizon. So we
consider the scalar field as $\phi=\phi(r)$111 To construct the scalarized
solutions bifurcating from higher modes, one could set
$\phi=\sum_{\ell\mathfrak{m}}\phi_{\ell\mathfrak{m}}(r)Y_{\ell\mathfrak{m}}(\theta,\psi)$,
and the next steps are straightforward. and take the ansatz of metric as

$ds^{2}=-e^{-2\delta(r)}N(r)dt^{2}+N^{-1}(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (29)

where $N(r)$ can be parameterized as

$N(r)=1-\frac{2\mathscr{M}(r)}{r}-\frac{\Lambda r^{2}}{3}.$ (30)

Then, recalling the equations of motion (4) and (5), we have the $tt$ and $rr$
components of Einstein equation as

$\displaystyle m^{2}r^{2}\varphi^{2}-16\alpha
N^{2}\left(\varphi\varphi^{\prime\prime}+\varphi^{\prime
2}\right)+8\alpha\varphi N^{\prime}\varphi^{\prime}+2rN^{\prime}+2\Lambda
r^{2}-2$
$\displaystyle+2N\left(\varphi\left(8\alpha\varphi^{\prime\prime}-12\alpha
N^{\prime}\varphi^{\prime}\right)+\left(8\alpha+r^{2}\right)\varphi^{\prime
2}+1\right)=0,$ (31) $\displaystyle m^{2}r^{2}\varphi^{2}+48\alpha
N^{2}\varphi\varphi^{\prime}\delta^{\prime}+8\alpha\varphi
N^{\prime}\varphi^{\prime}+2rN^{\prime}+2\Lambda r^{2}-2$
$\displaystyle-2N\left(12\alpha\varphi
N^{\prime}\varphi^{\prime}+r^{2}\varphi^{\prime
2}+2\delta^{\prime}\left(4\alpha\varphi\varphi^{\prime}+r\right)-1\right)=0,$
(32)

and the scalar equation

$\displaystyle\varphi^{\prime\prime}+\frac{\left(rN^{\prime}+N\left(2-r\delta^{\prime}\right)\right)}{Nr}\varphi^{\prime}$
$\displaystyle+\frac{r^{2}\left(-2m^{2}\phi\right)+8\alpha\phi\left((N-1)\left(N^{\prime\prime}+2N\left(\delta^{\prime
2}-\delta^{\prime\prime}\right)\right)+(3-5N)N^{\prime}\delta^{\prime}+N^{\prime
2}\right)}{4Nr^{2}}=0.$ (33)

To simplify the equations in numeric, we follow the skills in Brihaye:2019gla
and combine the above equations into two first order equations for the metric
functions and a second order equation for scalar field as

$\displaystyle\delta^{\prime}$ $\displaystyle=F_{1}(N,\phi,\phi^{\prime}),$
(34) $\displaystyle N^{\prime}$ $\displaystyle=F_{2}(N,\phi,\phi^{\prime}),$
(35) $\displaystyle\phi^{\prime\prime}$
$\displaystyle=F_{3}(N,\phi,\phi^{\prime})$ (36)

where the formulas of $F_{i}$ are complex and we will not present here. Then
we will numerically integrate the equation groups (35)-(36) between the event
horizon and cosmological horizon. To do so, we should analyze the approximate
form of the solutions at the boundary of our domain of integration. We first
solve (35) and (36) by requiring that $N(r)$ vanishes both at $r_{e}$ and
$r_{c}$, and imposing that the scalar field smoothly goes between $r_{e}$ and
$r_{c}$. Then we substitute the solutions of $N(r)$ and $\phi(r)$ into (34) by
imposing $\delta$ vanishing at cosmological horizon. With this process, we can
numerically construct a branch of scalarized black holes at the bifurcation
points from the zero modes. The typical solutions for different scalar masses
are shown in FIG. 15. We see that the scalar mass has slight effect on the
profile of $N(r)$ but the effect on the profile of scalar field is
significant. Moreover, from the right panel, we can see the difference between
the profile for the scalar field for the scalarized solution (solid curves)
and the corresponding scalar clouds (dashed curves).

Figure 15: Profile functions for typical scalarized solutions with different
scalar masses. Here we have set $\Lambda=0.1$ and $\phi(r_{e})=0.02$. In the
right plot, the solid curves denote the profile for the scalar field after
scalarization while the dashed curves denote the corresponding scalar clouds.

In the numeric, similar to $\Lambda=0$ case Silva:2017uqg ; Doneva:2017bvd ;
Antoniou:2017acq  and dS case with massless scalar field Brihaye:2019gla ,
there exists critical configurations that $\phi^{\prime}(r_{e})$ becomes
imaginary such that the numerical iterations fail to converge, which is
reflected in the existence of critical value of $\phi(r_{e})$. Then with fixed
$\Lambda=0.1$ we show the ‘mass’ $\mathscr{M}_{c}=\mathscr{M}(r_{c})$ for the
scalarized black hole as a function of the $\phi(r_{e})$ in FIG. 16 where the
effect of the scalar mass is explicit. Here $\phi(r_{e})\to 0$ corresponds to
the SdS limit while the rightmost dot represents the aforementioned critical
configuration where the branches stop to exist.

Figure 16: The ‘mass’ $\mathscr{M}_{c}=\mathscr{M}(r_{c})$ for the scalarized
black hole as a function of $\phi(r_{e})$ for different scalar masses.

##  V Conclusion and discussion

In this paper, we firstly studied the dynamics of the massive scalar field
perturbation on Schwarzschild de-Sitter black hole in a special EsGB theory.
In both frequency and time domains, we analyzed the (in)stability of scalar-
free dS black hole from the fundamental QNM frequency. To make sure the
precision, we first repeat the results in Einstein theory without the GB
coupling. In frequency domain, for various $\ell$ perturbation modes, there
exists a critical GB coupling $(\alpha_{c})$ at which the imaginal part of QNM
frequency, $\omega_{I}$, is zero. When $\alpha<\alpha_{c}$, $\omega_{I}$ is
negative indicating that the system is stable; while when $\alpha>\alpha_{c}$,
$\omega_{I}$ turns to be positive implying that the system would undergo a
tachyonic instability. Our calculation showed that larger angular momentum,
cosmological constant and the mass of scalar field correspond to larger
$\alpha_{c}$, which means that in those cases, the SdS black hole is more
difficult to be destabilized. The physical reason is that larger $\alpha$
always give more deeper negative potential well which triggers the tachyonic
instability and finally destroy the SdS black hole, while larger $\ell$,
$\Lambda$ and $m$ provide positive values into the effective potential and
enhance the negative potential well. In time domain, for the case with
$\alpha<\alpha_{c}$, the scalar field perturbation would finally decay as time
evolves while it would grow up for the case with $\alpha>\alpha_{c}$. The
growth of the perturbation could trigger the scalar-free black hole unstable
and a scalarized hairy solution may emerge. y scanning the model parameters,
we figured out the unstable/stable region in the $(\Lambda,\alpha)$-plane and
also in the $(m,\alpha)$-plane for various perturbation modes. It is noted
that though the scalarization of SdS black hole with backreaction has been
investigated in Bakopoulos:2018nui ; Brihaye:2019gla , there are still many
open issues as their authors addressed. Here we analyzed the (in)stability in
probe limit, but our findings could be helpful to further understand the black
hole scalarization in dS spacetime. For example, it is only possible to
construct the scalarized hairy solution for the parameters in the shading
region in FIG. 6 and FIG. 10.

Then we investigated the bifurcation points at which the SdS black hole
supports spherical scalar clouds. We found that the bifurcation points from
the lowest sate of scalar field with node $k=0$ match well with the border of
unstable/stable region obtained in the fundamental frequency for
$\ell=0,1,2,3$ modes massless/massive scalar perturbation. And for $\ell=0$,
the bifurcation points from the scalar field with node $k=1$ match well with
the border of unstable/stable region obtained in the frequency with overtone
$n=2$ massless/massive scalar perturbation. Finally, by including the
backreaction, we constructed the scalarized hairy black hole solutions for
different scalar masses. We also showed how the scalar cloud evolves into the
hairy solution through the backreaction by comparing its scalar field with the
corresponding scalar clouds. It is worthwhile to mention that allthrough this
work, we fix our study in the region between the black hole horizon and
cosmological horizon, two boundaries for dS black hole which are mostly
concerned.

As a simple way to constraint the model parameters of possible spontaneous
scalarization, it is interesting to extended our dynamical analysis into many
cases. A straightforward case is the Schwarzschild or AdS black holes in EsGB
theory as well as other modified gravities mentioned in the introduction. The
second case is the dynamic of massive scalar filed perturbed on Kerr black
hole in EsGB. Though this direction has been investigated in Zhang:2020pko ,
but there the authors only considered the $\ell=1$ mode of massless scalar
field. Since besides the tachyonic instability, the superradiant instability
may occur in rotating black hole, so considering the mass of the scalar field
and different modes will introduce more physics on the fate of scalar-free
background. The next but not the last case is to consider the dynamic of
charged black hole before it was scalarized. The scalarization of charged
black hole has been firstly proposed in Herdeiro:2018wub  and soon been
generalized in Myung:2018vug ; Fernandes:2019rez ; Brihaye:2019kvj ;
Myung:2019oua ; Konoplya:2019goy ; Fernandes:2019kmh ; Guo:2021zed , and a
complete dynamical analysis on the scalar-free charged black hole still
deserves to be present.

###### Acknowledgements.

We appreciate Xi-Jing Wang for helpful discussion. This work is partly
supported by Natural Science Foundation of China under Grant No. 11775036, Fok
Ying Tung Education Foundation under Grant No. 171006 and Natural Science
Foundation of Jiangsu Province under Grant No.BK20211601. Guoyang Fu is
supported by the Postgraduate Research & Practice Innovation Program of
Jiangsu Province (KYCX20 2973). Jian-Pin Wu is supported by Top Talent Support
Program from Yangzhou University.

## References

  * (1) K. S. Stelle, “Renormalization of Higher Derivative Quantum Gravity,” Phys. Rev. D 16, 953 (1977).
  * (2) J. Polchinski, String Theory, Vol.$1\&2$ (Cambridge University Press, Cambridge, 2001).
  * (3) S. Mignemi and N. R. Stewart, “Charged black holes in effective string theory,” Phys. Rev. D 47, 5259 (1993) [hep-th/9212146].
  * (4) P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis and E. Winstanley, “Dilatonic black holes in higher curvature string gravity,” Phys. Rev. D 54, 5049 (1996) [hep-th/9511071].
  * (5) T. Torii, H. Yajima and K. i. Maeda, “Dilatonic black holes with Gauss-Bonnet term,” Phys. Rev. D 55, 739 (1997) [gr-qc/9606034].
  * (6) Z. K. Guo, N. Ohta and T. Torii, “Black Holes in the Dilatonic Einstein-Gauss-Bonnet Theory in Various Dimensions II. Asymptotically AdS Topological Black Holes,” Prog. Theor. Phys. 121 (2009), 253-273 [arXiv:0811.3068 [gr-qc]].
  * (7) K. i. Maeda, N. Ohta and Y. Sasagawa, “AdS Black Hole Solution in Dilatonic Einstein-Gauss-Bonnet Gravity,” Phys. Rev. D 83 (2011), 044051 [arXiv:1012.0568 [hep-th]].
  * (8) N. Ohta and T. Torii, “Asymptotically AdS Charged Black Holes in String Theory with Gauss-Bonnet Correction in Various Dimensions,” Phys. Rev. D 88 (2013), 064002 [arXiv:1307.3077 [hep-th]];“Black Holes in the Dilatonic Einstein-Gauss-Bonnet Theory in Various Dimensions. III. Asymptotically AdS Black Holes with $k=\pm 1$,” Prog. Theor. Phys. 121 (2009), 959 [arXiv:0902.4072 [hep-th]].
  * (9) D. Ayzenberg and N. Yunes, “Slowly-Rotating Black Holes in Einstein-Dilaton-Gauss-Bonnet Gravity: Quadratic Order in Spin Solutions,” Phys. Rev. D 90, 044066 (2014) Erratum: [Phys. Rev. D 91, no. 6, 069905 (2015)] [arXiv:1405.2133 [gr-qc]].
  * (10) B. Kleihaus, J. Kunz and E. Radu, “Rotating Black Holes in Dilatonic Einstein-Gauss-Bonnet Theory,” Phys. Rev. Lett. 106, 151104 (2011) [arXiv:1101.2868 [gr-qc]].
  * (11) B. Kleihaus, J. Kunz, S. Mojica and M. Zagermann, “Rapidly Rotating Neutron Stars in Dilatonic Einstein-Gauss-Bonnet Theory,” Phys. Rev. D 93, no. 6, 064077 (2016) [arXiv:1601.05583 [gr-qc]].
  * (12) G. Antoniou, A. Bakopoulos and P. Kanti, “Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories,” Phys. Rev. Lett. 120, no.13, 131102 (2018) [arXiv:1711.03390 [hep-th]].
  * (13) H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou and E. Berti, “Spontaneous scalarization of black holes and compact stars from a Gauss-Bonnet coupling,” Phys. Rev. Lett. 120 (2018) no.13, 131104 [arXiv:1711.02080 [gr-qc]].
  * (14) D. D. Doneva and S. S. Yazadjiev, “New Gauss-Bonnet Black Holes with Curvature-Induced Scalarization in Extended Scalar-Tensor Theories,” Phys. Rev. Lett. 120, no.13, 131103 (2018) [arXiv:1711.01187 [gr-qc]].
  * (15) C. A. R. Herdeiro and E. Radu, “Black hole scalarization from the breakdown of scale invariance,” Phys. Rev. D 99 (2019) no.8, 084039 [arXiv:1901.02953 [gr-qc]].
  * (16) Y. Brihaye, C. Herdeiro and E. Radu, “The scalarised Schwarzschild-NUT spacetime,” Phys. Lett. B 788 (2019), 295-301 [arXiv:1810.09560 [gr-qc]].
  * (17) M. Minamitsuji and T. Ikeda, “Scalarized black holes in the presence of the coupling to Gauss-Bonnet gravity,” Phys. Rev. D 99 (2019) no.4, 044017 [arXiv:1812.03551 [gr-qc]].
  * (18) H. O. Silva, C. F. B. Macedo, T. P. Sotiriou, L. Gualtieri, J. Sakstein and E. Berti, “Stability of scalarized black hole solutions in scalar-Gauss-Bonnet gravity,” Phys. Rev. D 99 (2019) no.6, 064011 [arXiv:1812.05590 [gr-qc]].
  * (19) N. Andreou, N. Franchini, G. Ventagli and T. P. Sotiriou, “Spontaneous scalarization in generalised scalar-tensor theory,” Phys. Rev. D 99 (2019) no.12, 124022 [erratum: Phys. Rev. D 101 (2020) no.10, 109903] [arXiv:1904.06365 [gr-qc]].
  * (20) M. Minamitsuji and T. Ikeda, “Spontaneous scalarization of black holes in the Horndeski theory,” Phys. Rev. D 99 (2019) no.10, 104069 [arXiv:1904.06572 [gr-qc]].
  * (21) Y. Peng, “Spontaneous scalarization of Gauss-Bonnet black holes surrounded by massive scalar fields,” Phys. Lett. B 807 (2020), 135569 [arXiv:2004.12566 [gr-qc]].
  * (22) H. S. Liu, H. Lu, Z. Y. Tang and B. Wang, “Black hole scalarization in Gauss-Bonnet extended Starobinsky gravity,” Phys. Rev. D 103 (2021) no.8, 084043 [arXiv:2004.14395 [gr-qc]].
  * (23) D. D. Doneva, K. V. Staykov, S. S. Yazadjiev and R. Z. Zheleva, “Multiscalar Gauss-Bonnet gravity: Hairy black holes and scalarization,” Phys. Rev. D 102 (2020) no.6, 064042 [arXiv:2006.11515 [gr-qc]].
  * (24) D. Astefanesei, C. Herdeiro, J. Oliveira and E. Radu, “Higher dimensional black hole scalarization,” JHEP 09 (2020), 186 [arXiv:2007.04153 [gr-qc]].
  * (25) P. Cañate and S. E. Perez Bergliaffa, “Novel exact magnetic black hole solution in four-dimensional extended scalar-tensor-Gauss-Bonnet theory,” Phys. Rev. D 102 (2020) no.10, 104038 [arXiv:2010.04858 [gr-qc]].
  * (26) C. L. Hunter and D. J. Smith, “Novel Hairy Black Hole Solutions in Einstein-Maxwell-Gauss-Bonnet-Scalar Theory,” [arXiv:2010.10312 [gr-qc]].
  * (27) A. Bakopoulos, P. Kanti and N. Pappas, “Existence of solutions with a horizon in pure scalar-Gauss-Bonnet theories,” Phys. Rev. D 101 (2020) no.4, 044026 [arXiv:1910.14637 [hep-th]].
  * (28) A. Bakopoulos, P. Kanti and N. Pappas, “Large and ultracompact Gauss-Bonnet black holes with a self-interacting scalar field,” Phys. Rev. D 101 (2020) no.8, 084059 [arXiv:2003.02473 [hep-th]].
  * (29) K. Lin, S. Zhang, C. Zhang, X. Zhao, B. Wang and A. Wang, “No static regular black holes in Einstein-complex-scalar-Gauss-Bonnet gravity,” Phys. Rev. D 102 (2020) no.2, 024034 [arXiv:2004.04773 [gr-qc]].
  * (30) Y. Brihaye, B. Hartmann, N. P. Aprile and J. Urrestilla, “Scalarization of asymptotically anti–de Sitter black holes with applications to holographic phase transitions,” Phys. Rev. D 101 (2020) no.12, 124016 [arXiv:1911.01950 [gr-qc]].
  * (31) H. Guo, S. Kiorpelidi, X. M. Kuang, E. Papantonopoulos, B. Wang and J. P. Wu, “Spontaneous holographic scalarization of black holes in Einstein-scalar-Gauss-Bonnet theories,” Phys. Rev. D 102, no. 8, 084029 (2020) [arXiv:2006.10659 [hep-th]].
  * (32) Z. Y. Tang, B. Wang, T. Karakasis and E. Papantonopoulos, “Curvature scalarization of black holes in f(R) gravity,” Phys. Rev. D 104 (2021) no.6, 064017 [arXiv:2008.13318 [gr-qc]].
  * (33) L. G. Collodel, B. Kleihaus, J. Kunz and E. Berti, “Spinning and excited black holes in Einstein-scalar-Gauss–Bonnet theory,” Class. Quant. Grav. 37 (2020) no.7, 075018 [arXiv:1912.05382 [gr-qc]].
  * (34) A. Dima, E. Barausse, N. Franchini and T. P. Sotiriou, “Spin-induced black hole spontaneous scalarization,” Phys. Rev. Lett. 125 (2020) no.23, 231101 [arXiv:2006.03095 [gr-qc]].
  * (35) D. D. Doneva, L. G. Collodel, C. J. Krüger and S. S. Yazadjiev, “Spin-induced scalarization of Kerr black holes with a massive scalar field,” Eur. Phys. J. C 80 (2020) no.12, 1205 [arXiv:2009.03774 [gr-qc]].
  * (36) C. A. R. Herdeiro, E. Radu, H. O. Silva, T. P. Sotiriou and N. Yunes, “Spin-induced scalarized black holes,” Phys. Rev. Lett. 126 (2021) no.1, 011103 [arXiv:2009.03904 [gr-qc]].
  * (37) E. Berti, L. G. Collodel, B. Kleihaus and J. Kunz, “Spin-induced black-hole scalarization in Einstein-scalar-Gauss-Bonnet theory,” Phys. Rev. Lett. 126 (2021) no.1, 011104 [arXiv:2009.03905 [gr-qc]].
  * (38) H. Guo, X. M. Kuang, E. Papantonopoulos and B. Wang, “Horizon curvature and spacetime structure influences on black hole scalarization,” Eur. Phys. J. C 81 (2021) no.9, 842 [arXiv:2012.11844 [gr-qc]].
  * (39) S. Perlmutter et al. [Supernova Cosmology Project], “Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae,” Astrophys. J. 517 (1999), 565-586 [arXiv:astro-ph/9812133 [astro-ph]].
  * (40) A. G. Riess et al. [Supernova Search Team], “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” Astron. J. 116, 1009-1038 (1998) [arXiv:astro-ph/9805201 [astro-ph]].
  * (41) A. Strominger, “The dS / CFT correspondence,” JHEP 10, 034 (2001) [arXiv:hep-th/0106113 [hep-th]].
  * (42) E. Witten, “Quantum gravity in de Sitter space,” [arXiv:hep-th/0106109 [hep-th]].
  * (43) A. Bakopoulos, G. Antoniou and P. Kanti, “Novel Black-Hole Solutions in Einstein-Scalar-Gauss-Bonnet Theories with a Cosmological Constant,” Phys. Rev. D 99 (2019) no.6, 064003 [arXiv:1812.06941 [hep-th]].
  * (44) Y. Brihaye, C. Herdeiro and E. Radu, “Black Hole Spontaneous Scalarisation with a Positive Cosmological Constant,” Phys. Lett. B 802 (2020), 135269 [arXiv:1910.05286 [gr-qc]].
  * (45) E. Berti, V. Cardoso and A. O. Starinets, “Quasinormal modes of black holes and black branes,” Class. Quant. Grav. 26, 163001 (2009) [arXiv:0905.2975 [gr-qc]].
  * (46) R. A. Konoplya and A. Zhidenko, “Quasinormal modes of black holes: From astrophysics to string theory,” Rev. Mod. Phys. 83, 793-836 (2011) [arXiv:1102.4014 [gr-qc]].
  * (47) A. Jansen, “Overdamped modes in Schwarzschild-de Sitter and a Mathematica package for the numerical computation of quasinormal modes,” Eur. Phys. J. Plus 132, no.12, 546 (2017) [arXiv:1709.09178 [gr-qc]].
  * (48) Z. Zhu, S. J. Zhang, C. E. Pellicer, B. Wang and E. Abdalla, “Stability of Reissner-Nordström black hole in de Sitter background under charged scalar perturbation,” Phys. Rev. D 90, no.4, 044042 (2014) [arXiv:1405.4931 [hep-th]].
  * (49) A. Aragón, P. A. González, E. Papantonopoulos and Y. Vásquez, “Anomalous decay rate of quasinormal modes in Schwarzschild-dS and Schwarzschild-AdS black holes,” JHEP 08, 120 (2020) [arXiv:2004.09386 [gr-qc]].
  * (50) P. R. Brady, C. M. Chambers, W. G. Laarakkers and E. Poisson, “Radiative falloff in Schwarzschild-de Sitter space-time,” Phys. Rev. D 60, 064003 (1999) [arXiv:gr-qc/9902010 [gr-qc]].
  * (51) E. Abdalla, C. Molina and A. Saa, “Field propagation in the Schwarzschild-de Sitter black hole,” [arXiv:gr-qc/0309078 [gr-qc]].
  * (52) A. Zhidenko, “Quasinormal modes of Schwarzschild de Sitter black holes,” Class. Quant. Grav. 21, 273-280 (2004) [arXiv:gr-qc/0307012 [gr-qc]].
  * (53) B. Toshmatov and Z. Stuchlík, “Slowly decaying resonances of massive scalar fields around Schwarzschild-de Sitter black holes,” Eur. Phys. J. Plus 132, no.7, 324 (2017) [arXiv:1707.07419 [gr-qc]].
  * (54) P. V. P. Cunha, C. A. R. Herdeiro and E. Radu, “Spontaneously Scalarized Kerr Black Holes in Extended Scalar-Tensor–Gauss-Bonnet Gravity,” Phys. Rev. Lett. 123 (2019) no.1, 011101 [arXiv:1904.09997 [gr-qc]].
  * (55) S. J. Zhang, B. Wang, A. Wang and J. F. Saavedra, “Object picture of scalar field perturbation on Kerr black hole in scalar-Einstein-Gauss-Bonnet theory,” Phys. Rev. D 102 (2020) no.12, 124056 [arXiv:2010.05092 [gr-qc]].
  * (56) C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual and J. A. Font, “Spontaneous Scalarization of Charged Black Holes,” Phys. Rev. Lett. 121, no.10, 101102 (2018) [arXiv:1806.05190 [gr-qc]].
  * (57) Y. S. Myung and D. C. Zou, “Instability of Reissner–Nordström black hole in Einstein-Maxwell-scalar theory,” Eur. Phys. J. C 79, no.3, 273 (2019) [arXiv:1808.02609 [gr-qc]].
  * (58) P. G. S. Fernandes, C. A. R. Herdeiro, A. M. Pombo, E. Radu and N. Sanchis-Gual, “Spontaneous Scalarisation of Charged Black Holes: Coupling Dependence and Dynamical Features,” Class. Quant. Grav. 36, no.13, 134002 (2019) [erratum: Class. Quant. Grav. 37, no.4, 049501 (2020)] [arXiv:1902.05079 [gr-qc]].
  * (59) Y. Brihaye and B. Hartmann, “Spontaneous scalarization of charged black holes at the approach to extremality,” Phys. Lett. B 792, 244-250 (2019) [arXiv:1902.05760 [gr-qc]].
  * (60) Y. S. Myung and D. C. Zou, “Stability of scalarized charged black holes in the Einstein–Maxwell–Scalar theory,” Eur. Phys. J. C 79, no.8, 641 (2019) [arXiv:1904.09864 [gr-qc]].
  * (61) R. A. Konoplya and A. Zhidenko, “Analytical representation for metrics of scalarized Einstein-Maxwell black holes and their shadows,” Phys. Rev. D 100, no.4, 044015 (2019) [arXiv:1907.05551 [gr-qc]].
  * (62) P. G. S. Fernandes, C. A. R. Herdeiro, A. M. Pombo, E. Radu and N. Sanchis-Gual, “Charged black holes with axionic-type couplings: Classes of solutions and dynamical scalarization,” Phys. Rev. D 100, no.8, 084045 (2019) [arXiv:1908.00037 [gr-qc]].
  * (63) G. Guo, P. Wang, H. Wu and H. Yang, “Scalarized Einstein–Maxwell-scalar black holes in anti-de Sitter spacetime,” Eur. Phys. J. C 81 (2021) no.10, 864 [arXiv:2102.04015 [gr-qc]].