aainstitutetext: Dipartimento di Fisica, Università di Trento,
Via Sommarive 14, I-38123 Povo (TN), Italybbinstitutetext: Trento Institute
for Fundamental Physics and Applications TIFPA-INFN,
Via Sommarive 14, I-38123 Povo (TN), Italy

# On the stability of scale-invariant black holes

S. Boudet a,b M. Rinaldi a,b and S. Silveravalle<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Quadratic scale-invariant gravity non minimally coupled to a scalar field
provides a competitive model for inflation, characterized by the transition
from an unstable to a stable fixed point, both characterized by constant
scalar field configurations. We provide a complementary analysis of the same
model in the static, spherically symmetric setting, obtaining two
Schwarzschild-de Sitter solutions, which corresponds to the two fixed points
existing in the cosmological scenario. The stability of such solutions is
thoroughly investigated from two different perspectives. First, we study the
system at the classical level by the analysis of linear perturbations. In
particular, we provide both analytical and numerical results for the late-time
behavior of the perturbations, proving the stable and unstable character of
the two solutions. Then we perform a semi-classical, non-linear analysis based
on the Euclidean path integral formulation. By studying the difference between
the Euclidean on-shell actions evaluated on both solutions, we prove that the
unstable one has a meta-stable character and is spontaneously decaying into
the stable fixed point which is always favoured.

##  1 Introduction

Quadratic gravity is a well-known minimal extension of General Relativity
(GR), which adds, to the usual Einstein-Hilbert term $R$ in the action,
quadratic terms obtained by contracting the Riemann tensor
$R_{\alpha\beta\mu\nu}$. The classical and quantum properties of this theory
were thoroughly investigated already in the late 70’s Stelle:1976gc ;
Stelle:1977ry , initially in the context of compact and spherically symmetric
objects (see also Bonanno:2022ibv ; Bonanno:2019rsq ; Bonanno:2021zoy ;
Silveravalle:2022lid  for further developments).

The simplest quadratic Lagrangian, often called Starobinski model, has the
form ${\cal L}\propto\sqrt{-g}(R+R^{2}/m^{2})$ starobinsky_new_1980  and, when
used to describe the inflationary expansion of the Universe, turns out to be
among the most accurate models when compared to observations Planck:2018vyg .
The linear and quadratic terms can also be interpreted as the truncation of a
Taylor expansion of an analytic function $f(R)$ around flat space. The
cosmological and astrophysical implications of $f(R)$ gravity have attracted a
flurry activity in the past decades, see e.g. 2010LRR133D ; Sotiriou:2008rp
for reviews.

All quadratic models of gravity share the property that, when the curvature is
large with respect to a suitable scale, the theory becomes globally scale-
invariant, since the coefficients of the quadratic terms in the action are
dimensionless. Typically, in these scale-invariant models of gravity, scales
are generated by quantum corrections Cooper:1981byv ; RevModPhys.54.729 ;
PhysRevD.7.1888 ; Salvio:2014soa ; Einhorn:2014gfa ; Einhorn:2015lzy ;
Einhorn:2016mws ; Edery:2015wha ; Salvio:2020axm . As for the Starobinski
model, fundamental scale-invariance naturally leads to flat inflationary
potentials Khoze:2013uia ; Kannike:2014mia ; Rinaldi:2014gha ; Salvio:2014soa
; Kannike:2015apa ; Barrie:2016rnv ; Tambalo:2016eqr . Finally, black holes in
the simplest model of quadratic gravity (${\cal L}=\sqrt{g}R^{2}$) were
studied concerning their stability in Dioguardi:2020nxr  and the
thermodynamics in Cognola:2015uva ; Cognola:2015wqa .

As mentioned above, the main mechanism to generate scales in a scale-invariant
model is assumed to be loop quantum corrections. However, mass scales can
emerge naturally also in a classical way. This was first noticed, in the
context of inflationary cosmology, in Rinaldi:2015uvu . Here, it was shown
that the simplest extension of scale-invariant quadratic gravity, obtained by
adding a scalar field non-minimally coupled to gravity and with a quartic
potential, describes a system that dynamically evolves from an unstable fixed
point to a stable one. In both fixed points, the geometry of the two manifolds
is that of a flat de Sitter space with two different values for the
cosmological constants and the scalar field. In particular, at the stable
fixed point the scalar field stabilizes and the coefficient of the linear term
in $R$ of the action can be identified as the Planck mass squared, while the
others become negligible. During the transition between the two fixed points
the Universe expands exponentially, giving rise to a competitive model of
inflation Tambalo:2016eqr ; Ghoshal:2022qxk , which is also robust against
quantum corrections Vicentini:2019etr .

The aim of the present paper is to explore the possibility that the fixed
points mentioned above exist also in the space of spherically symmetric and
static solutions. In fact, it is quite easy to show that the equations of
motion endowed with the spherically symmetric and static symmetry, yield two
asymptotically de Sitter black hole solutions, with arbitrary cosmological
constant. Thus, the same theory yields to static solutions and we would like
to understand their stability and, eventually, how the transition from the
unstable to the stable one happens.

We will show that, indeed, one solution is unstable while the other is stable
by following two complementary paths, based on different notions of stability.

On one hand, from a completely classical perspective, the dynamical stability
of an exact solution of the field equations can be deduced from the behaviour
of linear perturbations. In GR, when we deal with static, spherically
symmetric backgrounds, this amounts to linearize Einstein’s equations on the
given background solution and adopt a harmonic decomposition for the metric
perturbation, resulting in the well-known Regge-Wheeler and Zerilli equations
Maggiore:2018sht . For a scalar-tensor theory instead, one has to consider the
dynamics of the scalar field perturbation as well, which, as we will see,
turns out to be the only one modified with respect to the GR case. We will
then investigate the stability by analyzing the late-time behaviour of the
perturbations, which can either diverge, yielding unstable solutions, or go
asymptotically to zero. In the second case, the black hole is stable against
perturbations, eventually relaxing to a new static configuration.

On the other hand, we will study the semiclassical properties, by means of the
path integral, of the system and determine a criterion to establish which of
the solutions is favoured. The task is not easy since it is notoriously
difficult to interpret an asymptotically de Sitter black hole in terms of a
thermodynamic system in equilibrium, the reason being that the two horizons
have different temperatures. Therefore, we need to resort to different
considerations. While this approach is less rigorous in assessing the
stability of the solutions, it can give relevant insight for the transition
between one solution and the other, and on the role of black holes in the
cosmological evolution.

The paper is organized as follows: in Sec. 2 we recall the main features of
the scale-invariant cosmological solution proposed in Rinaldi:2015uvu . We
examine the equations of motion in the Einstein frame and in a convenient
representation. We also consider an extension of the no-hair theorem in this
context. In Sec. 3 we briefly recall the cosmological solutions and the fixed
point structure of the phase space. We then show that the most general scale-
invariant tensor-scalar action admits asymptotically (anti) de Sitter black
holes with arbitrary mass and arbitrary cosmological constant, together with a
possibly non trivial scalar field value. In Sec. 5 we study the stability of
the solutions in terms of dynamical classical perturbations and, in Sec. 4, in
terms of the Euclidean action. We conclude in Sec. 6 with some conclusions and
remarks.

##  2 Quadratic scale-invariant gravity

We consider the general quadratic scale-invariant action studied in
Rinaldi:2015uvu ; Ghoshal:2022qxk , that is

$\mathcal{I}=\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{\alpha}{36}R^{2}+\frac{\xi}{6}\phi^{2}R-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{\lambda}{4}\phi^{4}\right],$
(1)

where $\alpha,\,\xi$, and $\lambda$ are dimensionless arbitrary positive
constants. Strictly speaking (1) is not the most general quadratic scale-
invariant action with one scalar field, having no terms with quadratic
combinations of the Ricci and Riemann tensors. However, these terms are known
to bring instabilities at quantum level Stelle:1976gc , and greatly complicate
the study of perturbations, having no simple description in the Einstein frame
Hindawi:1995an . Some considerations on the general theory will be made in
Appendix A.
The metric field equations of (1) read

$\displaystyle\left(\frac{\alpha}{18}R+\frac{\xi}{6}\phi^{2}\right)G_{\mu\nu}-\frac{1}{2}\left(\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla_{\rho}\phi\nabla^{\rho}\phi\right)$
$\displaystyle-\left(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box\right)\left(\frac{\alpha}{18}R+\frac{\xi}{6}\phi^{2}\right)+\left(\frac{\alpha}{72}R^{2}+\frac{\lambda}{8}\phi^{4}\right)g_{\mu\nu}=0,$
(2)

where $G_{\mu\nu}$ is the Einstein tensor, while varying the action with
respect to the scalar field yields

$\Box\phi+\left(\frac{\xi}{3}R-\lambda\phi^{2}\right)\phi=0.$ (3)

If we restrict ourselves to solutions with constant $\phi$ and $R$, it is
trivial to see that (2) reduce to the equations of General Relativity with a
cosmological constant. From (3) is instead manifest that in order to have
$R\leq 0$, the scalar field has to be zero, while for $R>0$ it is possible to
have both a zero and a non-zero scalar field. Therefore it will be possible to
have two distinct de Sitter (or asymptotically de Sitter) solutions, and we
will focus our attention to this case.

###  2.1 Einstein frame formulation

To simplify the discussion when dealing with a perturbed system, we first
write (1) in the Einstein frame. This is achieved by introducing the auxiliary
field $\varphi$ and an auxiliary variable $\chi$ defined as

$\chi=\frac{\alpha\varphi}{18}+\frac{\xi\phi^{2}}{6}\ .$ (4)

Then, the Lagrangian in eq. (1) becomes

$\frac{\mathcal{L}}{\sqrt{-g}}=\chi
R-\frac{\alpha\varphi^{2}}{36}-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{\lambda}{4}\phi^{4}\
.$ (5)

The equation of motion for $\varphi$ is simply $\varphi=R$, thus, in terms of
the field variable $\chi$, the above Lagrangian can be recast in the form

$\frac{\mathcal{L}}{\sqrt{-g}}=\chi
R-\frac{1}{2}(\partial\phi)^{2}+\frac{3\xi}{\alpha}\phi^{2}\chi-\left(\frac{\lambda}{4}+\frac{\xi^{2}}{4\alpha}\right)\phi^{4}-\frac{9}{\alpha}\chi^{2}\,.$
(6)

Now let $\omega=M\sqrt{3/2}\ln(2\xi/M^{2})$. Upon the conformal transformation
$g_{\mu\nu}\rightarrow\exp(2\omega/\sqrt{6}M)g_{\mu\nu}$, one can write the
Lagrangian in the Einstein frame, where it reads

$\displaystyle\frac{\mathcal{L}}{\sqrt{-g}}={M^{2}\over
2}R-{3M^{2}\over\mathfrak{f}^{2}}(\partial\mathfrak{f})^{2}-{\mathfrak{f}^{2}\over
2M^{2}}(\partial\phi)^{2}+{3\xi\phi^{2}\mathfrak{f}^{2}\over
2\alpha}-{\Omega\phi^{4}\mathfrak{f}^{4}\over 4\alpha M^{4}}-{9M^{4}\over
4\alpha}\ ,$ (7)

and where we defined $\mathfrak{f}=M\,e^{-{{\omega}\over\sqrt{6}M}}$. Here,
$M$ is an arbitrary parameter with mass dimensions111Note that, despite the
appearance of a mass parameter, the action is still scale-invariant. In fact,
$M$ turns out to be a so-called redundant parameter, see discussion in
Rinaldi:2015uvu .. Finally, upon the scalar fields redefinition

$\displaystyle\zeta$
$\displaystyle=\sqrt{6}M\text{arcsinh}\left(\frac{\mathfrak{f}\phi}{\sqrt{6}M^{2}}\right),$
(8) $\displaystyle\rho$
$\displaystyle=\frac{M}{2}\ln\left(\frac{\phi^{2}}{2M^{2}}+\frac{3M^{2}}{\mathfrak{f}^{2}}\right),$
(9)

we obtain the expression for the Lagrangian that we will use in the following
(for full details see Ghoshal:2022qxk ), namely

$\frac{\mathcal{L}}{\sqrt{-g}}=\frac{M^{2}}{2}R-\frac{1}{2}(\partial\zeta)^{2}-3\cosh^{2}\left(\frac{\zeta}{\sqrt{6}M}\right)(\partial\rho)^{2}-U(\zeta)\,,$
(10)

where

$\displaystyle U(\zeta)=-{9\xi
M^{4}\over\alpha}\sinh^{2}\left(\zeta\over\sqrt{6}M\right)+{9\Omega
M^{4}\over\alpha}\sinh^{4}\left(\zeta\over\sqrt{6}M\right)+{9M^{4}\over
4\alpha}\,.$ (11)

This representation, in the Einstein frame, of the original action is
particularly convenient since the potential now depends on one field only. The
field equations for the metric and scalar fields are

$\displaystyle M^{2}G_{\mu\nu}$
$\displaystyle=6\cosh^{2}{\left(\frac{\zeta}{\sqrt{6}M}\right)}\left(\partial_{\mu}\rho\partial_{\nu}\rho-\frac{1}{2}g_{\mu\nu}\partial_{\lambda}\rho\partial^{\lambda}\rho\right)$
$\displaystyle+\partial_{\mu}\zeta\partial_{\nu}\zeta-\frac{1}{2}g_{\mu\nu}\partial_{\lambda}\zeta\partial^{\lambda}\zeta-
g_{\mu\nu}U(\zeta),$ (12a) $\displaystyle\Box\zeta$
$\displaystyle=\frac{\sqrt{6}}{M}\cosh\left(\frac{\zeta}{\sqrt{6}M}\right)\sinh\left(\frac{\zeta}{\sqrt{6}M}\right)\partial_{\mu}\rho\partial^{\mu}\rho+\frac{dU}{d\zeta},$
(12b) $\displaystyle\Box\rho$
$\displaystyle=-\frac{2}{\sqrt{6}M}\tanh\left(\frac{\zeta}{\sqrt{6}M}\right)\partial_{\mu}\rho\partial^{\mu}\rho.$
(12c)

Note that $\rho=$ const (not necessarily vanishing) is a trivial solution of
the system precisely because the potential depends on $\zeta$ only.

###  2.2 No-hair-like theorem for scale invariant gravity

As can be seen manifestly from the Einstein frame formulation, scale invariant
gravity has two additional degrees of freedom with respect to General
Relativity. In this section we will show that in many relevant cases, as the
ones that we will consider, these degrees of freedom have to satisfy a
constraint, and the deviations from General Relativity can be characterized by
a single scalar field. Thanks to this theorem, to study a Schwarzschild, de
Sitter or Schwarzschild-(anti) de Sitter solution it is not necessary to
impose a constant scalar curvature and constant scalar field separately, as
one follows to the other. The trace of the metric equations of motion in the
Jordan frame can be written as

$\nabla^{\mu}\nabla_{\mu}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)=0,$
(13)

where the quantity in brackets is directly linked to the “spectator” field
$\rho$ of the Einstein frame formulation as

$R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}=\frac{3M^{2}}{\alpha}\mathrm{e}^{2\rho/M}$
(14)

whenever the equations of motion for the auxiliary field are satisfied. The
distinctive form of equation (13) opens the possibility of formulating no-
hair-like theorems for symmetric spacetimes. Let us consider a metric in the
form

$\mathrm{d}s^{2}=\tau(x)^{2}\sigma_{ab}(y)\mathrm{d}y^{a}\mathrm{d}y^{b}+\xi_{ij}(x)\mathrm{d}x^{i}\mathrm{d}x^{j},$
(15)

where $\sigma_{ab}(y)$ is the metric of a maximally symmetric manifold with
dimension $d_{y}<4$ and $\xi_{ij}(x)$ is the metric of a generic manifold of
dimension $4-d_{y}$. If we impose that the scalar field has the same
symmetries of the spacetime, i.e. that it depends only on the $x$ variables,
equation (13) becomes

$D^{i}D_{i}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)+\frac{d_{y}}{\tau}D^{i}\tau
D_{i}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)=0,$ (16)

where the $D_{i}$ are the covariant derivatives defined by the metric
$\xi_{ij}(x)$. Multiplying by
$\tau^{d_{y}}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)$ and integrating
over a submanifold $\Sigma$ defined by the $x$ coordinates we get

$\begin{split}\int_{\Sigma}\mathrm{d}^{4-d_{y}}x\sqrt{|\xi|}\Bigg{[}&\tau^{d_{y}}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)D^{i}D_{i}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)\\\
&+d_{y}\tau^{d_{y}-1}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)D^{i}\tau
D_{i}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)\Bigg{]}=0,\end{split}$
(17)

that, integrated by parts, leads to

$\begin{split}&\left[\tau^{d_{y}}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)D^{i}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)\right]_{\partial\Sigma}\\\
&-\int_{\Sigma}\mathrm{d}^{4-d_{y}}x\sqrt{|\xi|}\left[\tau^{d_{y}}D^{i}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)D_{i}\left(R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}\right)\right]=0.\end{split}$
(18)

If the metric $\xi_{ij}(x)$ is either positive or negative definite in
$\Sigma$ and the boundary term is vanishing, then the combination
$R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}$ is constant in all $\Sigma$. Thanks to
(14) is then clear that in the Einstein frame formulation the $\rho$ field is
not only a spectator field, but in many cases is also trivial.
This general statement becomes meaningful when specific cases are considered:
with a static metric

$ds^{2}=-\tau^{2}(x)\mathrm{d}t^{2}+\xi_{ij}(x)\mathrm{d}x^{i}\mathrm{d}x^{j},$
(19)

the boundary term is vanishing whenever the boundary is taken on event
horizons $\left(\tau^{2}=0\right)$, or on hypersurfaces of constant scalar
curvature and scalar field, and it leads to

$R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}=const.\ \left(\text{in space}\right);$
(20)

with a Friedmann-Lematre-Robertson-Walker (FLRW) metric

$ds^{2}=-\xi(t)\mathrm{d}t^{2}+\tau(t)^{2}\sigma_{ab}(y)\mathrm{d}y^{a}\mathrm{d}y^{b},$
(21)

the boundary term is vanishing whenever the boundary is taken on cosmological
singularities $\left(\tau^{2}=0\right)$, or on times at which the scalar
curvature and scalar field are constant, and it leads to

$R+\frac{3(1+2\xi)}{2\alpha}\phi^{2}=const.\ \left(\text{in time}\right).$
(22)

The spatial constraint (20) is then ensured for the external region of black
holes which are either asymptotically flat or asymptotically anti de Sitter,
and in the region between the black hole and cosmological horizon whenever
both are present; these are precisely the regions that form the Euclidean
sector of the Scharzschild and Schwarzschild-(anti) de Sitter metrics that we
will study in section 5. The time constraint (22) is instead ensured if the
cosmological evolution is between two fixed points, and can be used to keep
track of the value of the effective cosmological constant throughout a
cosmological evolution.

##  3 Solutions of scale-invariant gravity

###  3.1 Quadratic scale-invariant cosmology

Before considering black holes, we briefly summarize the results of
Rinaldi:2015uvu ; Ghoshal:2022qxk . By imposing the flat Robertson-Walker
metric $ds^{2}=-dt^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j}$ in the equations of
motion, one finds two second-order equations in the scalar field $\phi$ and
the Hubble parameter $H=\dot{a}/a$, namely

$\displaystyle\ddot{\phi}+3H\dot{\phi}-2\xi\phi\dot{H}-\phi(4\xi
H^{2}-\lambda\phi^{2})=0\,,$ (23)
$\displaystyle\alpha\left(2H\ddot{H}-\dot{H}^{2}+6H^{2}\dot{H}\right)-{1\over
2}\dot{\phi}^{2}+2\xi\phi\dot{\phi}H+{\phi^{2}\over 4}(4\xi
H^{2}-\lambda\phi^{2})=0\,.$

Despite the apparent complicate form, these equations can be reduced to 4
first-order non linear equations and studied with the means of dynamical
system analysis. The result is that the system has only two fixed points. The
first, at

$H_{u}=\sqrt{\frac{\Lambda_{u}}{3}},\qquad\qquad\phi_{u}=0,$ (24)

where $\Lambda_{u}$ is an arbitrary cosmological constant, is unstable, while
the second, at

$H_{s}=\sqrt{\frac{\Lambda_{s}}{3}},\qquad\qquad\phi_{s}=2\sqrt{\frac{\xi\Lambda_{s}}{3\lambda}},$
(25)

where $\Lambda_{s}$ is still arbitrary, is an attractor. It turns out that the
evolution from the unstable to the stable point yields an inflationary phase,
thoroughly studied in Ghoshal:2022qxk . The interesting feature is that, at
the stable fixed point, the scalar field has a non-trivial value and the
coefficient of $R$ in the action (1) classically stabilizes around at a
certain value that can be identified with the Planck mass squared, while the
other terms become negligible. As a result, a mass term is spontaneously
generated and the scale-invariance is broken. In particular we can use (22) to
relate the two cosmological constants of the two possible de Sitter solutions
with zero and non-zero scalar field:

$\Lambda_{u}=\left(1+\frac{\xi\left(1+2\xi\right)}{2\alpha\lambda}\right)\Lambda_{s}.$
(26)

One can then wonder if a similar situation occurs in the space of static or
stationary and spherically symmetric solutions. In the next section we show
that there are two solutions corresponding to two distinct asymptotically de
Sitter black holes.

###  3.2 Quadratic scale-invariant black holes

The two cosmological solutions discussed in Rinaldi:2015uvu  have a static,
spherically symmetric counterpart in terms of Schwarzschild-de Sitter
spacetimes (SdS), with line element

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\varphi^{2}),$
(27)

where

$f(r)=1-\frac{2m}{r}-\frac{\Lambda}{3}r^{2}.$ (28)

Note that both $m$ and $\Lambda$ are arbitrary integration constants, while in
standard GR the cosmological constant is fixed at the level of the action.
When they satisfy the relation

$9m^{2}\Lambda<1,$ (29)

then the function $f(r)$ has three real roots given by

$\displaystyle r_{b}$
$\displaystyle=\frac{2}{\sqrt{\Lambda}}\cos{\left(\frac{\pi+\eta}{3}\right)},$
(30) $\displaystyle r_{c}$
$\displaystyle=\frac{2}{\sqrt{\Lambda}}\cos{\left(\frac{\pi-\eta}{3}\right)},$
(31) $\displaystyle r_{0}$
$\displaystyle=-\frac{2}{\sqrt{\Lambda}}\cos{\left(\frac{\eta}{3}\right)},$
(32)

where $\cos{\eta}=3m\sqrt{\Lambda}$. The first two are positive and they
represent the black hole and cosmological horizons, respectively. As in the
cosmological case, the two solutions are characterized by constant
configurations for the scalar field, as one can easily verify by substituting
the metric (27) into the equations of motion. Although the very purpose of
this work is the study of the stability of such solutions, from now on we will
refer to them as unstable and stable solution, in analogy with Rinaldi:2015uvu
. The unstable solution is characterized by

$R_{u}=4\Lambda_{u},\qquad\qquad\phi_{u}=0,$ (33)

while the stable one is obtained setting

$R_{s}=4\Lambda_{s},\qquad\qquad\phi_{s}=2\sqrt{\frac{\xi\Lambda_{s}}{3\lambda}},$
(34)

in perfect agreement with the cosmological case. In the Einstein frame, the
two SdS solutions presented above are still characterized by constant scalar
fields whose values are mapped to the new variables as

$\displaystyle\rho_{u}$ $\displaystyle=\text{arbitrary},$ (35a)
$\displaystyle\zeta_{u}$ $\displaystyle=0,$ (35b) $\displaystyle\Lambda_{u}$
$\displaystyle=\frac{9M^{2}}{4\alpha},$ (35c)

for the unstable solution, and

$\displaystyle\rho_{s}$ $\displaystyle=\text{arbitrary},$ (36a)
$\displaystyle\zeta_{s}$
$\displaystyle=\sqrt{6}M\text{arcsin}\left(\sqrt{\frac{\xi}{2(\alpha\lambda+\xi^{2})}}\right),$
(36b) $\displaystyle\Lambda_{s}$
$\displaystyle=\frac{9M^{2}\lambda}{4(\alpha\lambda+\xi^{2})},$ (36c)

for the stable one. Note that in both cases, even if we have to fix its
expression, the cosmological constant is still arbitrary because of its
dependence on $M$.

##  4 Black hole perturbations

To study the dynamical stability of the two SdS solutions, it is convenient to
work in the Einstein frame, where the Lagrangian density is given by (10). We
now proceed to study the dynamics of metric and scalar perturbations around
the two solutions (35) and (36). The fields can be expanded to linear order as

$\displaystyle g_{\mu\nu}$ $\displaystyle=\bar{g}_{\mu\nu}+h_{\mu\nu},$ (37)
$\displaystyle\rho$ $\displaystyle=\bar{\rho}+\delta\rho,$ (38)
$\displaystyle\zeta$ $\displaystyle=\bar{\zeta}+\delta\zeta,$ (39)

where barred quantities represent either the stable or unstable background
solution and we will also use $\bar{\Lambda}$ to denote (35c) or (36c),
indifferently. At first order, the system (12) gives

$\displaystyle M^{2}\delta G_{\mu\nu}+U(\bar{\zeta})h_{\mu\nu}$
$\displaystyle=0,$ (40) $\displaystyle\bar{\Box}\delta\rho$ $\displaystyle=0,$
(41)
$\displaystyle\left(\bar{\Box}-\frac{d^{2}U}{d\zeta^{2}}(\bar{\zeta})\right)\delta\zeta$
$\displaystyle=0,$ (42)

where $\delta G_{\mu\nu}$ is the first order perturbation of the Einstein
tensor and $\bar{\Box}$ denotes the d’Alambert operator built with the
background metric. We see that, at linear order, the three perturbations are
decoupled. Moreover, since $U(\bar{\zeta})=M^{2}\bar{\Lambda}$, the first
equation coincides with the GR equation for the metric perturbation on a SdS
background with cosmological constant $\bar{\Lambda}$. Hence, the dynamics of
$h_{\mu\nu}$ is not modified. The same happens for (41), which is nothing but
the usual Klein-Gordon equation on a SdS background. The dynamics of
$\delta\zeta$ is instead modified by the presence of the mass term in (42).
Therefore, in the following we will only focus on the dynamics of
$\delta\zeta$, since instabilities can only arise from the latter. To proceed,
one can exploit the spherical symmetry of the problem, adopting the following
harmonic expansion for the scalar perturbation:

$\delta\zeta=\frac{Z(r)}{r}Y^{\ell\ell^{\prime}}(\theta,\varphi)e^{-i\omega
t},$ (43)

where $Y^{\ell\ell^{\prime}}$ are the standard spherical harmonics.
Substituting into (42), one finds

$\frac{d^{2}Z}{dr_{*}^{2}}+\left[\omega^{2}-V_{\zeta}(r)\right]Z=0,$ (44)

where $r_{*}$ is the tortoise coordinate defined by

$\frac{dr^{*}}{dr}=\frac{1}{f(r)},$ (45)

and the argument of the potential must be considered as a function of $r^{*}$,
inverting its definition in terms of $r$. The effective potential featuring
(44) can be written in terms of a parameter $\bar{p}$ as

$V_{\zeta}(r)=V_{GR}(r)-\frac{4\bar{\Lambda}}{3}\bar{p}f(r),$ (46)

where $V_{GR}(r)$ represent the effective potential appearing in the GR case,
namely for a Klein-Gordon equation on a SdS background:

$V_{GR}(r)=f(r)\left(\frac{\ell(\ell+1)}{r^{2}}+\frac{2m}{r^{3}}-\frac{2\bar{\Lambda}}{3}\right).$
(47)

For $\bar{p}=0$ one recovers the GR case, while deviations for non-vanishing
$\bar{p}$ can be appreciated in figure 1.

Figure 1: Effective potential within the static region $r_{b}<r<r_{c}$ for
$\ell=0$ and different values of $\bar{p}$. The cosmological constant is set
to $\Lambda=0.001/m^{2}$. The $\bar{p}=0$ curve represents the GR case.

The value of the parameter $\bar{p}$ depends on the specific solution one is
considering. For the unstable solution we have

$\bar{p}=p_{u}\equiv\xi,$ (48)

while the stable one is characterized by

$\bar{p}=p_{s}\equiv
1-\frac{9M^{2}}{4\alpha\Lambda_{s}}(1+2\xi)=-\frac{\xi(2\alpha\lambda+\xi+2\xi^{2})}{\alpha\lambda},$
(49)

where in the last step we used (36c). We will now show that the asymptotic
behaviour of the perturbation crucially depends on the value of $\bar{p}$. The
critical value of $\bar{p}$ separating stable from unstable modes is related
to the form of the potential in the pure de Sitter, $m\rightarrow 0$ limit.
This was shown in PhysRevD.60.064003 , where the case of a non-minimally
coupled scalar field on a SdS background is considered. It turns out that the
effective potential (46) is very similar to the one computed in
PhysRevD.60.064003 . In that case, starting from

$(\Box-\nu R)\phi=0,$ (50)

one gets

$V(r)=f(r)\left(\frac{\ell(\ell+1)}{r^{2}}+\frac{2m}{r^{3}}-\frac{2\Lambda}{3}(1-6\nu)\right).$
(51)

Therefore, the two potentials are mapped into each other via the substitution
$\nu\leftrightarrow-\bar{p}/3$, which allows to obtain the late time behavior
of the scalar perturbation from the results of PhysRevD.60.064003  as

$\delta\zeta\sim e^{-\mu\kappa_{c}t},$ (52)

where $\kappa_{c}$ is the surface gravity of the cosmological horizon, i.e.

$\kappa_{c}=\frac{\Lambda(r_{c}-r_{b})(r_{c}-r_{0})}{6r_{c}},$ (53)

and

$\mu=\ell+\frac{3}{2}-\frac{1}{2}\sqrt{9+16\bar{p}}+O\left(\frac{r_{b}}{r_{c}}\right).$
(54)

We can identify three different behaviours, for different values of the
parameter $\bar{p}$. When $\bar{p}<-9/16$, $\mu$ becomes complex and the
purely exponential decay is replaced by a damped oscillatory regime, as
observed in PhysRevD.60.064003  (see fig. 2):

$\delta\zeta\sim
e^{-(\ell+\frac{3}{2})\kappa_{c}t}e^{\frac{i}{2}\sqrt{\absolutevalue{9+16\bar{p}}}\kappa_{c}t}.$
(55)

Figure 2: Evolution of $|Z|/m^{2}$ as a function of $t/m$ at fixed radius, for
$\ell=0$ and different values of $\bar{p}$, showing the onset of the
oscillatory regime in the late-time region for $\bar{p}<-9/16=-0.5625$. The
cosmological constant is set to $\Lambda=0.001/m^{2}$.

The main difference with respect to PhysRevD.60.064003 , is that now $\mu$
changes sign for $\bar{p}$ greater than the critical value

$\bar{p}_{c}=\frac{(2\ell+3)^{2}-9}{16},$ (56)

so that we have an exponential decay for $-9/16<\bar{p}<\bar{p}_{c}$, with no
oscillations, while the case $\bar{p}>\bar{p}_{c}$ yields unstable diverging
modes.

This is confirmed by numerical integrations. To solve (44) numerically, we
follow the well-established method of PhysRevD.49.883 ; PhysRevD.63.084001 ;
PhysRevD.70.064025 ; Boudet_2022 . First, we can introduce the light-cone
variables $u=t-r_{*}$ and $v=t+r_{*}$, in terms of which the equation reads

$4\frac{\partial^{2}Z}{\partial u\partial v}+V_{\zeta}(r)Z=0,$ (57)

Then, the numerical integration is based on a discretization of the $u-v$
plane by a lattice spacing $\Delta$ where $(u,v)$ range between zero and
$(u_{max},v_{max})$. The discretized version of the equation is

$Z_{N}=Z_{W}+Z_{E}-Z_{S}+\frac{\Delta^{2}}{2}V_{\zeta}(r_{c})\left(Z_{W}+Z_{E}\right),$
(58)

where a subscript means that the function is evaluated at the points
$S=(u,v)$, $W=(u+\Delta,v)$, $E=(u,v+\Delta)$, $N=(u+\Delta,v+\Delta)$, while
the potential is computed at $r_{c}=(u+\Delta/2,v+\Delta/2)$. Beside a first
transient regime, the evolution of the perturbation and especially its late-
time asymptotic behaviour are insensible to the initial conditions. In the
results presented in the following, we assigned boundary conditions on the two
axes $u=0$ and $v=0$, following PhysRevD.81.124021 . In particular, we set
$Z(u,0)=Z(u,0)=0$, and Gaussian initial data on the $v$ axis, i.e.
$Z(0,v)=\exp[-(v-v_{c})^{2}/2\sigma]$. The integration starts by computing the
value of $Z_{N}$ via (58), using the initial conditions at the three points
$S$, $W$, and $E$. Then, one proceeds increasing $v$ and eventually completing
the grid row by row. In order to compute the potential, at every step of the
integration the tortoise coordinate must be inverted obtaining $r$ as a
function of $r^{*}$. We do this numerically solving the explicit definition of
the latter, that starting from (45) can be written as

$r^{*}=\ln{\left(\frac{(r-r_{b})^{\beta_{b}}(r+r_{c}+r_{b})^{\beta_{0}}}{(r_{c}-r)^{\beta_{c}}}\right)},$
(59)

where

$\displaystyle\beta_{c}$
$\displaystyle=\frac{3r_{c}}{\Lambda(r_{c}-r_{b})(2r_{c}+r_{b})},$ (60)
$\displaystyle\beta_{b}$
$\displaystyle=\frac{3r_{b}}{\Lambda(r_{c}-r_{b})(2r_{b}+r_{c})},$ (61)
$\displaystyle\beta_{0}$
$\displaystyle=\frac{3(r_{b}+r_{c})}{\Lambda(2r_{c}+r_{b})(2r_{b}+r_{c})}.$
(62)

At the end of the integration, the perturbation evaluated at some constant
$r^{*}$ as a function of time can be extracted as $Z(t)=Z(t-r_{*},t+r_{*})$.
The results presented here are obtained setting: $u_{max}=v_{max}=500$,
$\Delta=0.1$, $v_{c}=10$, $\sigma=1$ and the function is extracted at
$r_{*}=50m$. We work with adimensional variables rescaling every quantity by
the appropriate power of the black hole mass $m$. The outcome of the numerical
integrations is shown in figure 2 and 3. In particular, figure 2 shows the
transition from an exponential decay to the damped oscillatory regime when
$\bar{p}$ becomes lower than $-9/16$. Figure 3 instead, confirms the stability
behaviour discussed above, showing how the perturbations diverge if
$\bar{p}>\bar{p}_{c}$, while they are stable and decay exponentially to zero
when $\bar{p}<\bar{p}_{c}$.

Figure 3: Evolution of $|Z|/m^{2}$ as a function of $t/m$ at fixed radius, for
$\ell=0$ and different values of $\bar{p}$. The cosmological constant is set
to $\Lambda=0.001/m^{2}$.

Let us now apply these results to the two solutions under consideration. Given
that $\bar{p}_{c}$ is non negative, instabilities can never trigger when the
stable solution is considered, since the latter has $\bar{p}=p_{s}<0$ (see
equation (49)). Regarding the other solution instead, since $p_{u}=\xi$ the
stability condition is provided by

$\xi<\bar{p}_{c}.$ (63)

The value of $\bar{p}_{c}$ is increasing with the angular momentum number $l$,
implying that the first multiple suffering from instabilities would be the
lowest lying mode, namely the monopole, identified by $\ell=0$. Since
$\bar{p}_{c}(\ell=0)=0$ and being $\xi$ non negative, we conclude that the
stability condition (63) is never satisfied and the second solution is always
unstable. Therefore, the stability character of the two solutions agrees with
the one resulting from the investigations pursued in cosmological setting in
Rinaldi:2015uvu .

##  5 Euclidean action and non-linear stability

The linear analysis of the previous section has rigorously shown that the
Schwarzschild-de Sitter solution with zero scalar field is always unstable,
while the one with a non-zero scalar field is always stable. What cannot be
inferred by the linear analysis, however, is whether one solution can have a
transition into the other or not, and what is the role of black holes in the
cosmological evolution shown in subsection 3.1. A non-linear analysis can give
an answer to both these problem, as well as confirm that there is no strong
non-linear effect that spoils the results of the linear analysis. Usually the
non-linear stability of black holes is inferred from their Thermodynamics,
which however is not always well defined. The standard thermodynamical
description of black holes is tightly linked with the path integral
formulation of quantum gravity Gibbons:1976ue ; Hawking:1978jz . The partition
function

$Z=\displaystyle\sum_{g}\langle g|\mathrm{e}^{-\beta H}|g\rangle,$ (64)

can be equivalently formulated as the partition function of an Euclidean path
integral

$Z=\int\mathcal{D}g\,\Big{[}\mathrm{e}^{-\mathcal{I}(g)}\Big{]},$ (65)

with the Euclidean time having periodicity $\beta$. Calculations can be made
resorting to the semiclassical approximation, which reduces the partition
function to

$Z\sim\mathrm{e}^{-\mathcal{I}(g_{0})},$ (66)

where $g_{0}$ is the metric which minimizes the action. Usually, the
thermodynamical properties of Schwarzschild, de Sitter and Schwarzschild-anti
de Sitter solutions can be extracted from (66) Gibbons:1976ue ; Hawking:1978jz
; Gibbons:1977mu ; Hawking:1982dh ; PhysRevD.103.084034 . Unfortunately,
despite the effort of many authors (see as examples Shankaranarayanan:2003ya ;
Robson:2019yzx ), there is still no general consensus on how to define a
global temperature for the Schwarzschild-de Sitter spacetime, and a
thermodynamical description of such type of solution is a much debated
subject. From a thermodynamical point of view this is due to the fact that
Schwarzschild-de Sitter solutions have two horizons with different
temperatures, and the system is not at thermodynamical equilibrium; from a
path integral point of view, it is not possible to define a time periodicity,
which removes the two conical singularities emerging in the Euclidean action
from the presence of two horizons. In this section therefore we will not
indulge in the thermodynamical description of Schwarzschild-de Sitter black
holes, but we will remain in the semiclassical description of Euclidean
quantum gravity. While in a purely classical description any spacetime which
is in a stationary point of the action has equivalent relevance, in the path
integral formulation the absolute value of the action determines which
solution has a dominant contribution to the partition function, opening the
possibility of having meta-stable solutions which will tunnel away with
probability

$\Gamma=A\,\mathrm{e}^{-\left(\mathcal{I}_{ms}-\mathcal{I}_{s}\right)},$ (67)

where $A$ is a prefactor and $\mathcal{I}_{ms}$, $\mathcal{I}_{s}$ are the
Euclidean actions of the meta-stable and stable solutions. This approach has
been used to derive the Hawking-Page transition for Schwarzschild-anti de
Sitter black holes Hawking:1982dh , and we believe it will be of relevance
also in our case.

In order to evaluate the Euclidean action we need to add to the action a
suitable boundary term. In appendix A we compute boundary terms and actions
for a more general scale invariant model with all possible quadratic
combinations of curvature tensors; however for that theory it is not possible
to have a manageable Einstein frame, and the dynamical discussion on the
stability of the solutions cannot be made. In order to have a better
comparison with the classical discussion, we then analyze the euclidean
version of the action (1), for which the boundary term is

$\mathcal{I}_{\partial
M}=\int\mathrm{d}^{3}x\sqrt{-h}\left(\frac{\alpha}{9}\,R+\frac{\xi}{3}\,\phi^{2}\right)K,$
(68)

where $h_{\mu\nu}$ is the metric of the boundary and $K$ is its extrinsic
curvature. It can be proved that for Schwarzschild, de Sitter and
Schwarzschild-(anti) de Sitter spacetimes the Euclidean scale-invariant action
$\mathcal{I}_{SI}$ is

$\mathcal{I}_{SI}=\frac{4}{9}\left(\alpha+\epsilon\frac{\xi^{2}}{\lambda}\right)\frac{\Lambda}{M_{p}^{2}}\mathcal{I}_{GR},$
(69)

where $M_{p}$ is the reduced Planck mass, $\epsilon=0,1$ respectively in the
$\phi=0$ and $\phi\neq 0$ cases, and $\mathcal{I}_{GR}$ is the Euclidean
action of General Relativity. A very similar relation holds for the more
general scale invariant theory, as shown in appendix A, which is a cross-check
of the validity of the calculations that we performed to evaluate the actions.
The particular cases are (dropping the $SI$ subscript)

$\mathcal{I}_{S}=0,$ (70a)
$\mathcal{I}_{dS}=-\frac{32}{3}\pi^{2}\left(\alpha+\epsilon\frac{\xi^{2}}{\lambda}\right),$
(70b) $\mathcal{I}_{SadS}=\frac{32}{27}\pi^{2}\alpha\frac{\Lambda
r_{b}^{2}\left(3+\Lambda r_{b}^{2}\right)}{1-\Lambda r_{b}^{2}},$ (70c)
$\mathcal{I}_{SdS}=-\frac{32}{9}\pi^{2}\left(\alpha+\epsilon\frac{\xi^{2}}{\lambda}\right)\Lambda\left(r_{b}^{2}+r_{c}^{2}\right),$
(70d)

where $\Lambda$ is negative in the Schwarzschild-anti de Sitter case and
positive in the Schwarzschild-de Sitter case. While the Schwarzschild (70a),
de Sitter (70b) and Schwarzschild-anti de Sitter cases (70c) can be evaluated
with the same techniques of General Relativity, the Schwarzschild-de Sitter
case requires some care, due to the presence of two conical singularities
which cannot be simultaneously removed with a suitable period of the Euclidean
time. In Morvan:2022ybp  the Euclidean action for a Schwarzschild-de Sitter
black hole in General Relativity has been calculated as

$\mathcal{I}_{SdS,GR}=-\frac{A_{b}}{4G}-\frac{A_{c}}{4G}=-8\pi^{2}M_{p}^{2}\left(r_{b}^{2}+r_{c}^{2}\right),$
(71)

using the explicit expression for the conical singularities found in
Fursaev:1995ef  in terms of Dirac delta functions. In the same paper
Fursaev:1995ef  it is shown that is not possible to define quadratic curvature
invariants with the same techniques, having to regularize a squared delta
function. However, having no physical singularities in the Euclidean sector it
is sensible to expect a finite value of the Euclidean action also in the scale
invariant case. Moreover, in the calculation of the Euclidean action in
Morvan:2022ybp  it is shown that the action does not depend on the periodicity
of the Euclidean time, which acts as a tool to derive the action in the
standard way and acquires physical meaning only through the thermodynamical
interpretation. To evaluate the action we then exploited this fact to have
more freedom in the bounds of integration of the Euclidean action. In
particular, we considered three manifolds: a Schwarzschild-de Sitter from
which we removed by hand the cosmological horizon, a Schwarzschild-de Sitter
from which we removed the black hole horizon and a Schwarzschild-de Sitter
from which we removed both horizons. We then assumed that the Euclidean action
of Schwarzschild-de Sitter is the sum of the ones of the first two manifolds,
minus the action of the third

$\mathcal{I}_{SdS}=\mathcal{I}_{SdS\setminus\\{c\\}}+\mathcal{I}_{SdS\setminus\\{b\\}}-\mathcal{I}_{SdS\setminus\\{c,b\\}}.$
(72)

Starting from the third manifold, the Euclidean action is

$\begin{split}\mathcal{I}_{SdS\setminus\\{c,b\\}}=&4\pi\beta\Bigg{(}\int_{r_{b}}^{r_{c}}\mathrm{d}r\,r^{2}\left(\frac{\alpha}{36}R^{2}+\frac{\xi}{6}\phi^{2}R-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{\lambda}{4}\phi^{4}\right)+\\\
&+\sqrt{-g_{tt}}\left(\frac{\alpha}{9}\,R+\frac{\xi}{3}\,\phi^{2}\right)K\bigg{\rvert}_{r_{c}}+\sqrt{-g_{tt}}\left(\frac{\alpha}{9}\,R+\frac{\xi}{3}\,\phi^{2}\right)K\bigg{\rvert}_{r_{b}}\Bigg{)}=\\\
=&\frac{16}{9}\pi\left(\alpha+\epsilon\frac{\xi^{2}}{\lambda}\right)\Lambda\beta\Bigg{(}\frac{\Lambda}{3}\left(r_{c}^{3}-r_{b}^{3}\right)+\left(2r_{c}-3M-\Lambda
r_{c}^{3}\right)+\\\ &-\left(2r_{b}-3M-\Lambda r_{b}^{3}\right)\Bigg{)}=\\\
=&\frac{32}{9}\pi\left(\alpha+\epsilon\frac{\xi^{2}}{\lambda}\right)\Lambda\beta\Bigg{(}r_{c}-\frac{\Lambda}{3}r_{c}^{3}-\left(r_{b}-\frac{\Lambda}{3}r_{b}^{3}\right)\Bigg{)}=0,\end{split}$
(73)

where $\beta$ is a generic Euclidean time periodicity, and the last
equivalence is given by the horizons definition. The other two integrals are
then given by

$\begin{split}\mathcal{I}_{SdS\setminus\\{c\\}}=&4\pi\beta_{b}\Bigg{(}\int_{r_{b}}^{r_{c}}\mathrm{d}r\,r^{2}\left(\frac{\alpha}{36}R^{2}+\frac{\xi}{6}\phi^{2}R-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{\lambda}{4}\phi^{4}\right)+\\\
&+\sqrt{-g_{tt}}\left(\frac{\alpha}{9}\,R+\frac{\xi}{3}\,\phi^{2}\right)K\bigg{\rvert}_{r_{c}}\Bigg{)}=-4\pi\beta_{b}\sqrt{-g_{tt}}\left(\frac{\alpha}{9}\,R+\frac{\xi}{3}\,\phi^{2}\right)K\bigg{\rvert}_{r_{b}},\\\
\mathcal{I}_{SdS\setminus\\{b\\}}=&4\pi\beta_{c}\Bigg{(}\int_{r_{b}}^{r_{c}}\mathrm{d}r\,r^{2}\left(\frac{\alpha}{36}R^{2}+\frac{\xi}{6}\phi^{2}R-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{\lambda}{4}\phi^{4}\right)+\\\
&+\sqrt{-g_{tt}}\left(\frac{\alpha}{9}\,R+\frac{\xi}{3}\,\phi^{2}\right)K\bigg{\rvert}_{r_{b}}\Bigg{)}=-4\pi\beta_{b}\sqrt{-g_{tt}}\left(\frac{\alpha}{9}\,R+\frac{\xi}{3}\,\phi^{2}\right)K\bigg{\rvert}_{r_{c}},\end{split}$
(74)

where $\beta_{b}$ and $\beta_{c}$ are the periodicities required to remove
respectively the black hole and cosmological horizon conical singularities,
and the equivalences are given by the result in (73). The two terms evaluated
on the Schwarzschild-de Sitter metric then appear as

$\begin{split}\mathcal{I}_{SdS\setminus\\{c\\}}=&-\frac{32}{9}\pi^{2}\left(\alpha+\epsilon\frac{\xi^{2}}{\lambda}\right)\Lambda
r_{b}^{2},\\\
\mathcal{I}_{SdS\setminus\\{b\\}}=&-\frac{32}{9}\pi^{2}\left(\alpha+\epsilon\frac{\xi^{2}}{\lambda}\right)\Lambda
r_{c}^{2},\end{split}$ (75)

where we considered a strictly positive temperature for the cosmological
horizon, as in the pure de Sitter case. It is then trivial to see that the sum
(72) gives exactly (70d), which satisfies the same relation (69) as all the
other solutions considered, which have been calculated with standard
techniques. Moreover, when applied to the context of General Relativity our
procedure has exactly the same result of Morvan:2022ybp . While it might seem
a crude calculation, this result indicates that the Euclidean action of
Schwarzschild-de Sitter is given entirely by the conical singularities,
exactly as shown for General Relativity in Morvan:2022ybp . The main
difference here is that the singularity is integrated away while defining the
action, and not during the volume integration, opening the possibility of
using the same technique to all the actions for which the boundary terms are
known.

###  5.1 Stability of Schwarzschild-de Sitter black holes

In order to understand whether an unstable SdS black hole undergoes a
transition into a stable one, we have to understand which are the most
convenient parameters to use to compare the two solutions. While in principle
the Euclidean action of a Schwarzschild-de Sitter black hole is characterized
by two parameters, that is its mass and cosmological constant, the scale
invariant nature of the theory removes one degree of freedom by making
everything dependent only from a dimensionless combination of the two
parameters. With the definitions for the black hole and cosmological horizons
(30), the expression (70d) becomes

$\mathcal{I}_{SdS}=-\frac{64}{9}\pi^{2}\left(\alpha+\epsilon\frac{\xi^{2}}{\lambda}\right)\left(2-\cos\left(\frac{2\eta}{3}\right)\right),$
(76)

where $\cos\eta=3m\sqrt{\Lambda}$, and therefore the difference in the actions
with zero and non-zero scalar field is

$\mathcal{I}_{SdS,\,u}-\mathcal{I}_{SdS,\,s}=\frac{64}{9}\pi^{2}\Bigg{(}\frac{\xi^{2}}{\lambda}\left(2-\cos\left(\frac{2\eta_{s}}{3}\right)\right)+\alpha\left(\cos\left(\frac{2\eta_{u}}{3}\right)-\cos\left(\frac{2\eta_{s}}{3}\right)\right)\Bigg{)},$
(77)

with $\cos\eta_{u}=3m_{u}\sqrt{\Lambda_{u}}$ and
$\cos\eta_{s}=3m_{s}\sqrt{\Lambda_{s}}$. The two parameters $\Lambda_{u}$ and
$\Lambda_{s}$, however, are not independent one from the other. If we consider
the black holes as immersed in a large, evolving universe, it is sensible to
expect that the value of the effective cosmological constant will be given by
the global cosmological evolution, and will not be affected by the presence of
the black holes. The two parameters will then be linked by the relation (26),
and the two SdS black holes can be compared using relations between $\eta_{u}$
and $\eta_{s}$ which will depend only from the free parameters of the theory.
We can now consider four different cases: black holes with the same horizons
radii, and then with the same geometry, which satisfy

$\eta_{u}=\eta_{s};$ (78)

black holes with the same mass parameter $m$, that is for which only the
cosmological constant undergoes an evolution, which satisfy

$\eta_{u}=\arccos\left(\sqrt{1+\frac{\xi(1+2\xi)}{2\alpha\lambda}}\cos\eta_{s}\right);$
(79)

black holes with the same cosmological horizon temperature, which satisfy

$\begin{split}\eta_{u}=&\pi-3\arccos\Bigg(\sqrt{1+\frac{\xi(1+2\xi)}{2\alpha\lambda}}\frac{1}{8\cos\left(\frac{\pi-\eta_{s}}{3}\right)}\Bigg{(}-3+4\cos^{2}\left(\frac{\pi-\eta_{s}}{3}\right)+{\\\
}\hfil&+\sqrt{\frac{48}{\sqrt{1+\frac{\xi(1+2\xi)}{2\alpha\lambda}}}\cos^{2}\left(\frac{\pi-\eta_{s}}{3}\right)+\left(-3+4\cos^{2}\left(\frac{\pi-\eta_{s}}{3}\right)\right)^{2}}\,\Bigg{)}\Bigg{missing});\end{split}$
(80)

and, finally, black holes with the same black hole horizon temperature, that
shares also the same absorption cross section Sini:2007zz , which satisfy

$\begin{split}\eta_{u}=&3\arccos\Bigg(\sqrt{1+\frac{\xi(1+2\xi)}{2\alpha\lambda}}\frac{1}{8\cos\left(\frac{\pi+\eta_{s}}{3}\right)}\Bigg{(}-3+4\cos^{2}\left(\frac{\pi+\eta_{s}}{3}\right)+{\\\
}\hfil&+\sqrt{\frac{48}{\sqrt{1+\frac{\xi(1+2\xi)}{2\alpha\lambda}}}\cos^{2}\left(\frac{\pi+\eta_{s}}{3}\right)+\left(-3+4\cos^{2}\left(\frac{\pi+\eta_{s}}{3}\right)\right)^{2}}\,\Bigg{)}\Bigg{missing})-\pi.\end{split}$
(81)

Figure 4: Difference of the Euclidean actions for black holes with the same
temperature at the black hole horizon, varying separately $\alpha\lambda$, in
the first panel, and $\xi$ in the second panel.

While a quantitative discussion on the behaviour of (77) requires fixing the
three free parameters of the theory, the positiveness of the difference is
assessed in all the cases under analysis simply by requiring
$\alpha,\,\xi,\,\lambda>0$. If we compare black holes with the same geometry,
the case is trivial because only the term multiplied by $\xi^{2}/\lambda$,
which is always positive, survives. If we compare black holes with the same
mass parameter or the same cosmological horizon temperature, we have to add
the term multiplied by $\alpha$ which, however, it is also always positive
taking into account that $\eta_{u}<\eta_{s}$ for
$\sqrt{1+\frac{\xi(1+2\xi)}{2\alpha\lambda}}>1$. The latter case requires some
care, considering that in this case $\eta_{u}>\eta_{s}$ for
$\sqrt{1+\frac{\xi(1+2\xi)}{2\alpha\lambda}}>1$, and then the term multiplied
by $\alpha$ is now negative; however as $\alpha$ is increased, the combination
$1+\frac{\xi(1+2\xi)}{2\alpha\lambda}$ gets closer to 1 and the terms which is
multiplied by $\alpha$ gets smaller, and there is no analytical way to assess
whether the difference between the actions is positive or not. In figure 4 we
show numerically that the difference is positive both in the case of $\alpha$
being much larger than $\frac{\xi}{\lambda}$, and in the opposite case.
Therefore, we can state that in all the cases under consideration

$\mathcal{I}_{SdS,\,u}>\mathcal{I}_{SdS,\,s}\qquad\qquad\text{for every
}m,\Lambda_{u}\text{ s.t. }3m\sqrt{\Lambda_{u}}<1,$ (82)

where the constraint on the parameters is given by
$m<\frac{1}{3}\min\left(\frac{1}{\sqrt{\Lambda_{u}}},\frac{1}{\sqrt{\Lambda_{s}}}\right)=\frac{1}{3\sqrt{\Lambda_{u}}}$.
Such relation guarantees that any Schwarzschild-de Sitter black hole prefers
to be immersed in a de Sitter space with a smaller cosmological constant and,
in our specific case, confirms that a Schwarzschild-de Sitter black hole will
have a transition from a solution with zero scalar field to a solution with a
non-zero scalar field.

Figure 5: Decrease of the tunnelling probability between two Schwarzschild-de
Sitter and two de Sitter spacetimes for black holes with the same temperature
at the black hole horizon, in the panels on top, and with the same temperature
at the cosmological horiozn, in the panels in the bottom, varying separately
$\alpha\lambda$, in the panels on the left, and $\xi$ in the panels on the
right.

Having clarified that the unstable Schwarzschild-de Sitter will have a
transition into the stable one, what is left to understand is if the presence
of a black hole immersed in the cosmological setting described in (3.1) can
stabilize the unstable de Sitter space, or aggravates its instability. Taking
into account the difference in the Euclidean actions of the two de Sitter
spacetimes

$\mathcal{I}_{dS,\,u}-\mathcal{I}_{dS,\,s}=\frac{32}{3}\pi^{2}\frac{\xi^{2}}{\lambda},$
(83)

it is clear that, when comparing black holes by their radius, the difference
(77) is always smaller than (83); the tunnel probability (67) will then always
be smaller if a black hole is present, and the unstable de Sitter
configuration will have longer life expectation. The situation is the same for
black holes with the same temperature either at the black hole or cosmological
horizons, as shown in figure 5. Comparing black holes by their mass renders
the discussion more complicated: if the parameters of the theory are such that
$\alpha\lambda\ll\xi^{2}$, then the presence of a black hole will stabilize
the de Sitter space; in the opposite case, when $\alpha\lambda\gg\xi^{2}$ the
presence of a black hole will increase the decaying rate of the unstable de
Sitter space. Intermediate values can present a situation where a small black
hole will stabilize the de Sitter space, while a large one will increase its
instability. As general statement, we have that the presence of a black hole
will always stabilize the de Sitter space whenever the parameters of the
theory satisfy

$\xi\left(1+\frac{\xi^{2}}{\alpha\lambda}\right)>\frac{1}{2}.$ (84)

Taking into account the most recent results obtained in the cosmological
setting Ghoshal:2022qxk , we expect $\alpha$ to be much smaller than
$\frac{\xi^{2}}{\lambda}$, and then the presence of black holes will always
decrease the tunnelling probability of an unstable de Sitter spacetime.

##  6 Conclusions

In this work we examined the static and spherically symmetric solutions of a
scale-invariant tensor-scalar theory of gravity. It is known that, in such a
theory, inflationary solutions exist connecting one unstable fixed point
(characterized by a vanishing scalar field) to a stable one (with non-zero
constant scalar field). Analogously, in the space of static and spherically
symmetric solutions, we have found that there are two asymptotically de Sitter
black hole configurations, corresponding to the two fixed points (one with
zero and the other with non-zero scalar field). In the absence of time
evolution in the system, the problem of understanding the stability of these
solutions has been faced in two complementary ways. First, we studied the
scalar perturbations and we showed that the solution with zero scalar field is
characterized by exponentially growing perturbation modes and is actually
unstable, while these modes are absent in the one with non-zero scalar field,
which is actually stable. Then we analyzed the Euclidean path integral of the
solutions and we found that there is a non-zero transition probability from
the unstable solution to the stable one, which is however always smaller than
the transition probability between two pure de Sitter solutions.

##  Appendix A Euclidean action of extended scale-invariant theory

Here, we slightly generalize the action by including the Weyl and the Gauss
Bonnet terms, that is we include all the possible quadratic invariant of the
curvature tensors. In the cosmological case, these terms would be irrelevant,
at least at the level of background equations of motion. Thus, we now consider
the most general scale-invariant theory with a single scalar field, namely

$\mathcal{I}=\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{\alpha}{36}R^{2}-\frac{\beta}{6}C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}+\frac{\gamma}{6}\mathcal{G}+\frac{\xi}{6}\phi^{2}R-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{\lambda}{4}\phi^{4}\right],$
(85)

where $C_{\mu\nu\rho\sigma}$ is the Weyl tensor, $\mathcal{G}$ is the Gauss-
Bonnet combination, and the Weyl squared term has a negative sign for
consistence with the results found using the Stelle action, i.e. with
$\lambda=0$ and $\phi=$ const. Setting $\beta=\gamma=0$ one recovers the
quadratic scale-invariant model studied in Rinaldi:2015uvu . We note that,
thanks to the topological nature of the Gauss-Bonnet term and the traceless
nature of the Weyl term, the theorems discussed in subsection 2.2 are still
valid.

In order to evaluate the Euclidean action we need to add to the action a
suitable boundary term. Taking into account all the different terms in (85),
the boundary terms result to be

$\begin{split}\mathcal{I}_{\partial
M,\,1}=&\int\mathrm{d}^{3}x\sqrt{-h}\,\frac{\alpha}{9}\,R\,K,\\\
\mathcal{I}_{\partial
M,\,2}=&-\int\mathrm{d}^{3}x\sqrt{-h}\,\frac{\beta}{3}\,C^{\mu\nu\rho\sigma}\left(n_{\mu}n_{\rho}K_{\nu\sigma}-n_{\mu}n_{\sigma}K_{\nu\rho}+n_{\nu}n_{\sigma}K_{\mu\rho}-n_{\nu}n_{\rho}K_{\mu\sigma}\right),\\\
\mathcal{I}_{\partial
M,\,3}=&\int\mathrm{d}^{3}x\sqrt{-h}\,\frac{\gamma}{3}\big{[}R^{\mu\nu\rho\sigma}\left(n_{\mu}n_{\rho}K_{\nu\sigma}-n_{\mu}n_{\sigma}K_{\nu\rho}+n_{\nu}n_{\sigma}K_{\mu\rho}-n_{\nu}n_{\rho}K_{\mu\sigma}\right)+\\\
&-4\left(R^{\mu\nu}K_{\mu\nu}+R^{\mu\nu}n_{\mu}n_{\nu}K-2R^{\mu\nu}n_{\mu}n^{\rho}K_{\rho\nu}\right)+2R\,K\big{]},\\\
\mathcal{I}_{\partial
M,\,4}=&\int\mathrm{d}^{3}x\sqrt{-h}\,\frac{\xi}{3}\,\phi^{2}\,K,\end{split}$
(86)

where the ones for the Weyl and Gauss-Bonnet combinations are derived from
Solodukhin:2015eca . It can be proved that for Schwarzschild, de Sitter and
Schwarzschild-(anti) de Sitter spacetimes the Euclidean scale-invariant action
is

$\mathcal{I}_{SI}=\frac{4}{9}\left(\alpha+\beta+\epsilon\frac{\xi^{2}}{\lambda}\right)\frac{\Lambda}{M_{p}^{2}}\mathcal{I}_{GR}+\frac{32}{3}\pi^{2}\left(\beta-\gamma\right)N_{H},$
(87)

where $N_{H}$ is the number of horizons, $M_{p}$ is the reduced Planck mass,
$\epsilon=0,1$ respectively in the $\phi=0$ and $\phi\neq 0$ cases, and
$\mathcal{I}_{GR}$ is the Euclidean action of General Relativity. The
particular cases are (dropping the SI pedices)

$\mathcal{I}_{S}=\frac{32}{3}\pi^{2}\left(\beta-\gamma\right),$ (88a)
$\mathcal{I}_{dS}=-\frac{32}{3}\pi^{2}\left(\alpha+\beta+\epsilon\frac{\xi^{2}}{\lambda}\right)+\frac{32}{3}\pi^{2}\left(\beta-\gamma\right),$
(88b)
$\mathcal{I}_{SadS}=\frac{32}{27}\pi^{2}\left(\alpha+\beta\right)\frac{\Lambda
r_{b}^{2}\left(3+\Lambda r_{b}^{2}\right)}{1-\Lambda
r_{b}^{2}}+\frac{32}{3}\pi^{2}\left(\beta-\gamma\right),$ (88c)
$\mathcal{I}_{SdS}=-\frac{32}{9}\pi^{2}\left(\alpha+\beta+\epsilon\frac{\xi^{2}}{\lambda}\right)\Lambda\left(r_{b}^{2}+r_{c}^{2}\right)+\frac{64}{3}\pi^{2}\left(\beta-\gamma\right).$
(88d)

The relation (87) is simply related with (69) by a redefinition of
$\alpha\to\alpha+\beta$ and with the addition of a constant term
$\frac{32}{3}\pi^{2}\left(\beta-\gamma\right)N_{H}$; the discussion made in
subsection 5.1 therefore is still valid in the general theory.

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