KOBE-COSMO-22-10

Stability of Hairy Black Holes in Shift-Symmetric Scalar-Tensor Theories via
the Effective Field Theory Approach

Justin<EMAIL_ADDRESS>Toshifumi
<EMAIL_ADDRESS>Mark<EMAIL_ADDRESS>and
Sam S. C<EMAIL_ADDRESS>
aCenter for Particle Cosmology, Department of Physics and Astronomy,

University of Pennsylvania 209 S. 33rd St., Philadelphia, PA 19104, USA

bDepartment of Physics, Kobe University, Kobe 657-8501, Japan

###### Abstract

Shift-symmetric Horndeski theories admit an interesting class of
Schwarzschild-de Sitter black hole solutions exhibiting time-dependent scalar
hair. The properties of these solutions may be studied via a bottom-up
effective field theory (EFT) based on the background symmetries. This is in
part possible by making use of a convenient coordinate choice — Lemaître-type
coordinates — in which the profile of the Horndeski scalar field is linear in
the relevant time coordinate. We construct this EFT, and use it to understand
the stability of hairy black holes in shift-symmetric Horndeski theories,
providing a set of constraints that the otherwise-free functions appearing in
the Horndeski Lagrangian must satisfy in order to admit stable black hole
solutions. The EFT is analyzed in the decoupling limit to understand potential
sources of instability. We also perform a complete analysis of the EFT with
odd-parity linear perturbations around general spherically symmetric space-
time.

###### Contents

  1. 1 Introduction
  2. 2 Background geometry and Ingredients for the EFT
  3. 3 The effective action for perturbations
    1. 3.1 Tadpole terms
    2. 3.2 Quadratic terms
    3. 3.3 Consistency with diffeomorphisms on the surface $\Sigma_{\tau}$
    4. 3.4 Background isometry and shift symmetry
  4. 4 Second order action in the decoupling limit
    1. 4.1 Gradient (in-)stability for $\pi$
      1. 5 Odd-parity perturbations
        1. 5.1 Schwarzschild-de Sitter limit
          1. 6 Discussion
            1. A Background equations of motion
            2. B The Stückelberg procedure
            3. C Decoupling limit in Schwarszchild-de Sitter

##  1 Introduction

In the more than a hundred years since the discovery of black hole solutions
to the theory of General Relativity (GR), enormous theoretical efforts have
been devoted to understanding these remarkable space-time objects.
Nevertheless, it has only been in the last decade that two stunning
observational results — the measurement of gravitational waves from binary
mergers by the LIGO collaboration [1], and the imaging of super massive black
hole shadows by the Event Horizon Telescope [2] — have provided direct
observational evidence for astrophysical black holes. These observations have
lifted the curtain on the new era of black hole science, and these and other
upcoming precision measurements will provide powerful tools for scrutinizing
theoretical ideas beyond our currently established theories. Among these new
tools, of particular importance for this paper are the measurements of black
hole quasinormal mode frequencies.

A binary black hole merger proceeds through three somewhat distinct phases;
inspiral, merging and ringdown. During the ringdown phase, deviations from the
black hole space-time gradually decay, and therefore can be treated as small
perturbations around the black hole solution [3, 4]. The study of the linear
system of these small perturbations around a black hole is sometimes called
black hole spectroscopy, since the decaying oscillations exhibit a set of
characteristic wave forms called quasinormal modes, with corresponding
frequencies known as quasinormal frequencies [5]. The quasinormal frequency
spectrum of a neutral, spinning black hole in GR is entirely determined by two
parameters, the mass and spin of the black hole. Thus, it is theoretically
possible to probe proposed modifications to pure GR black holes by studying
even just the lowest two ringdown tones. Thus, quasinormal mode analysis
provides a particularly promising way to constrain, rule out or even provide
hints for theoretical constructions beyond GR, and we expect that the
increasingly accurate measurements provided by forthcoming missions, such as
LISA, will yield important new insights through this method.

A particularly well-motivated and well-studied class of modifications to GR is
scalar-tensor theories, which arise in many different settings. While a
general effective field theory (EFT) construction of such theories would
include all operators up to a certain order, if one only allows a restricted
set of operators involving a scalar and the metric, such that they give rise
to second-order equations of motion, then such theories are free of a
particularly pernicious instability known the “Ostragradsky ghost” [6]. (See
[7] for a systematic discussion of the extent to which it is necessary to
include higher-derivative operators in the EFT of general scalar-tensor
theories, and the circumstances under which it is correct to restrict to only
second-order operators.) The resulting theory is known as the Horndeski theory
[8], which can be thought of as a generalization of Galileon theories to
curved space-time [9, 10, 11]. It is also possible to study generalizations of
this theory to GLPV [12] or DHOST [13, 14, 15, 16, 17], in which the equations
of motion are nominally higher order, but which can be shown to be reducible
to second-order ones.

It is well-known that scalar-tensor theories admit numerous hairy black hole
solutions, in which one or more fields take on a nontrivial profile around the
black hole, and for which, as a result, quantum numbers beyond the charge,
spin, and mass are required to characterize them. One class of hairy solutions
admits a radially-dependent scalar profile $\phi(r)$ [18, 19, 20, 21, 22, 23,
24, 25, 26]. The general quasinormal mode analysis for this type of black hole
has been carried out [27, 28] and, remarkably, it has also been shown that the
properties of perturbations around black holes with such a radial hair profile
can be captured by an EFT description [29]. Perhaps more relevant to a
cosmological setting, another class of hairy black hole solutions features a
time-dependent profile $\phi(t,r)$, with a time-like gradient [30, 31, 32, 33,
34, 35, 36, 37, 26, 38]. These black holes exist in a wide class of scalar-
tensor theories satisfying a shift symmetry $\phi\to\phi+c$. They are akin to
the time-dependent scalar profile that drives cosmological expansion in ghost
condensation [39, 40]. These time-dependent hairy black hole solutions require
careful study, since it has been shown in certain circumstances that they may
suffer from a strong coupling problem [41, 33, 42, 43] and/or a gradient
instability [38, 44, 45, 46].

In this work we analyze hairy black holes with time-like hair using an EFT
approach. The construction of the EFT follows the logic of the EFT of
inflation [47] and the EFT of the purely radial hairy solution [48]. The first
example of such a construction was presented in [49]. Here, we further
constrain the EFT coefficient functions using the isometries of the background
space-time and the shift symmetry of the scalar field, with the primary goal
of diagnosing the existence of gradient instabilities. While a complete
analysis of tensor fluctuations in the EFT remains highly technical, it can be
shown that gradient instabilities that persist in the decoupling limit cannot
be cured through mixing with tensor fluctuations away from that limit. Thus,
it is possible to obtain useful results by studying the theory in the
decoupling limit. We analyze effective operators in the EFT individually to
provide constraints and insights about potential instabilities from the
choices of coefficient functions. We also carry out a comprehensive analysis
of odd sector perturbations. We further obtain useful constraints on the EFT
by considering the stealth black hole limit, in which the background geometry
reduces to Schwarzschild-de Sitter, while the scalar profile remains non-
trivial.

##  2 Background geometry and Ingredients for the EFT

We focus on black hole (BH) solutions in general scalar-tensor theories such
that the geometry is a static, spherically symmetric solution,

${\rm d}s^{2}=-f(r){\rm d}t^{2}+\frac{{\rm d}r^{2}}{g(r)}+r^{2}{\rm
d}\Omega^{2}\,,$ (2.1)

with ${\rm d}\Omega^{2}={\rm d}\theta^{2}+\sin^{2}\theta{\rm d}\varphi^{2}$.
We adopt Lemaître-type coordinates for this type of geometry,

$\displaystyle{\rm d}s^{2}=-{\rm d}\tau^{2}+\big{(}1-f(r)\big{)}{\rm
d}\rho^{2}+r^{2}{\rm d}\Omega^{2}\,,$ (2.2)

where {$\tau,\rho$} are related to {$t,r$} through

$\displaystyle{\rm d}t$ $\displaystyle=\frac{1}{f}{\rm
d}\tau+\left(1-\frac{1}{f}\right){\rm d}\rho\,;$ $\displaystyle{\rm d}r$
$\displaystyle=\sqrt{\frac{(1-f)g}{f}}\big{(}-{\rm d}\tau+{\rm
d}\rho\big{)}\,.$ (2.3)

Throughout this paper we use the notation
$(\;)\dot{}\equiv\frac{\partial}{\partial\tau}(\;)$ and
$(\;)^{\prime}\equiv\frac{\partial}{\partial\rho}(\;)$. It is clear from the
metric that this coordinate system is synchronous, and constant
$(\rho,\,\Omega)$ are free-falling time-like geodesics. When the theory enjoys
a shift symmetry for the scalar, $\phi$, it supports a class of interesting
hairy black holes with time-dependent hair of the form[50],111In general, the
gradient of a time-dependent scalar hair does not necessarily align with
$\partial_{\tau}$. In such a case, it is more convenient to use a time
coordinate that aligns with $\nabla_{\mu}\phi$.

$\displaystyle\bar{\phi}(x)=m^{2}\tau\,.$ (2.4)

Furthermore, for the subclass of shift-symmetric theories we are interested
in, the action of fluctuations exhibits the set of symmetries inherited from
the background isometries, as we will see in Sec. 3.

The first step in constructing the EFT of fluctuations about this black hole
is to note that the gradient of the scalar field,

$n^{\mu}=-\frac{\nabla^{\mu}\phi}{\sqrt{-\nabla_{\nu}\phi\nabla^{\nu}\phi}}\,;\quad
n^{\mu}n_{\mu}=-1\,,$ (2.5)

defines a foliation of space-time. This gradient is the vector normal to
surfaces of constant-$\phi$. From here until Sec. 3, we work in unitarity
gauge defined by $\phi=\bar{\phi}$, in which these surfaces reduce to
constant-$\tau$ surfaces, denoted by $\Sigma_{\tau}$. From this one can define
an induced metric on such hypersurfaces,

$h_{\mu\nu}=g_{\mu\nu}+n_{\mu}n_{\nu}\,,$ (2.6)

which also serves as a projector onto these surfaces. For example, the
covariant derivative on the surface $D_{\mu}$ is defined by

$\displaystyle
D_{\mu}V_{\nu}=h^{\alpha}_{\;\mu}h^{\beta}_{\;\nu}\nabla_{\alpha}V_{\beta}\,,$
(2.7)

and the extrinsic curvature of surfaces is given by

$\displaystyle
K_{\mu\nu}=h_{\;\mu}^{\alpha}h_{\;\nu}^{\beta}\nabla_{\alpha}n_{\beta}=h_{\;\mu}^{\alpha}\nabla_{\alpha}n_{\nu}=\nabla_{\mu}n_{\nu}+n_{\mu}n^{\alpha}\nabla_{\alpha}n_{\nu}\,.$
(2.8)

Using the ADM form of the metric,

$\displaystyle{\rm d}s^{2}=-N^{2}{\rm d}\tau^{2}+h_{ij}\left({\rm
d}x^{i}+N^{i}{\rm d}\tau\right)\left({\rm d}x^{j}+N^{j}{\rm d}\tau\right)\,,$
(2.9)

this can be written more explicitly in component form,

$\displaystyle K^{\tau\nu}=0\,;\quad
K_{ij}=\frac{1}{2N}\left(\dot{h}_{ij}-D_{i}N_{j}-D_{j}N_{i}\right)\,,$ (2.10)

where we have written $n^{\mu}$ in terms of the lapse function $N$ and the
shift vector $N_{i}$,

$n^{\mu}=\left(\frac{1}{N},-\frac{N^{i}}{N}\right),\quad
n_{\mu}=\left(-N,\vec{0}\right)\,.$ (2.11)

Another important geometric quantity is the intrinsic curvature of the
hypersurface, $\hat{R}_{\mu\nu\rho\sigma}$, which is defined by

$\displaystyle\hat{R}_{\mu\nu\rho}^{\quad\;\;\sigma}V_{\sigma}=[D_{\mu},D_{\nu}]V_{\rho}\,.$
(2.12)

Together with $K_{\mu\nu}$, this is related to the Riemann tensor of the full
space-time via the Gauss-Codazzi relations

$\displaystyle\hat{R}_{\mu\nu\rho}^{\quad\;\;\sigma}+K_{\mu\rho}K_{\nu}^{\;\;\sigma}-K_{\nu\rho}K_{\mu}^{\;\;\sigma}$
$\displaystyle=h_{\mu}^{\;\;\alpha}h_{\nu}^{\;\;\beta}h_{\rho}^{\;\;\gamma}h^{\sigma}_{\;\;\delta}R_{\alpha\beta\gamma}^{\quad\;\;\delta}\,;$
(2.13) $\displaystyle D_{\mu}K^{\mu}_{\;\;\nu}-D_{\nu}K$
$\displaystyle=R_{\alpha\beta}n^{\alpha}h^{\beta}_{\;\;\nu}\,.$ (2.14)

Evaluated on the background (2.2), the above geometric quantities are

$\displaystyle\bar{K}^{\rho}_{\;\;\rho}$
$\displaystyle=\frac{f_{r}}{2(1-f)}\sqrt{\frac{g}{f}}\,;$
$\displaystyle\bar{K}^{\theta}_{\;\;\theta}$
$\displaystyle=\bar{K}^{\varphi}_{\;\;\varphi}=-\frac{1}{r}\sqrt{\frac{(1-f)g}{f}}\,;$
(2.15) $\displaystyle\bar{\hat{R}}^{\rho}_{\;\;\rho}$
$\displaystyle=\frac{gf_{r}-fg_{r}}{rf^{2}}\,;$
$\displaystyle\bar{\hat{R}}^{\theta}_{\;\;\theta}$
$\displaystyle=\bar{\hat{R}}^{\varphi}_{\;\;\varphi}=\frac{2f^{2}-2fg+rgf_{r}-rfg_{r}}{2r^{2}f^{2}}\,,$
(2.16)

where $(\;)_{r}=\frac{\partial}{\partial r}(\;)$. One can then see that the
traceless part of $\bar{K}_{ij}$ and $\bar{\hat{R}}_{ij}$,

$\displaystyle\bar{K}^{\rm T}_{ij}$
$\displaystyle=\bar{K}_{ij}-\frac{1}{3}\bar{h}_{ij}\bar{K},$
$\displaystyle\bar{\hat{R}}^{\rm T}_{ij}$
$\displaystyle=\bar{\hat{R}}_{ij}-\frac{1}{3}\bar{h}_{ij}\bar{\hat{R}},$
(2.17)

are proportional to each other, $\bar{\hat{R}}^{\rm T}_{ij}\propto\bar{K}^{\rm
T}_{ij}$. Note that when $f(r)=g(r)$, all components of
$\bar{\hat{R}}_{\mu\nu\rho}^{\quad\;\;\sigma}$ on $\Sigma_{\tau}$ vanish. With
all of these three-dimensional covariant quantities at hand, we are now ready
to build the EFT.

##  3 The effective action for perturbations

The most general action for perturbations, including all terms up to second
order in fields and second derivatives that describe scalar tensor theories of
our interest, is given by

$\displaystyle S_{(2)}=\int{\rm d}^{4}x\sqrt{-g}\bigg{[}$
$\displaystyle\frac{M_{1}(x)}{2}R-\Lambda(x)+\alpha(x)g^{\tau\tau}+\beta(x)\bar{K}_{\mu\nu}K^{\mu\nu}$
$\displaystyle+M_{2}(x)(\delta g^{\tau\tau})^{2}+M_{3}(x)\delta
g^{\tau\tau}\delta K+M_{4}(x)\bar{K}_{\mu\nu}\delta g^{\tau\tau}\delta
K^{\mu\nu}$ $\displaystyle+M_{5}(x)(\partial_{\tau}\delta
g^{\tau\tau})^{2}+M_{6}(x)(\partial_{\tau}\delta g^{\tau\tau})\delta
K+M_{7}(x)\bar{K}_{\mu\nu}(\partial_{\tau}\delta g^{\tau\tau})\delta
K^{\mu\nu}+M_{8}(x)(D_{i}\delta g^{\tau\tau})^{2}$
$\displaystyle+M_{9}(x)\delta K^{2}+M_{10}(x)\delta K_{\mu\nu}\delta
K^{\mu\nu}+M_{11}(x)\bar{K}_{\mu\nu}\delta K\delta
K^{\mu\nu}+M_{12}(x)\bar{K}_{\mu\nu}\delta K^{\rho\mu}\delta
K^{\nu}_{\;\;\rho}$ $\displaystyle+M_{13}(x)\delta
g^{\tau\tau}\delta\hat{R}+M_{14}(x)\bar{K}_{\mu\nu}\delta
g^{\tau\tau}\delta\hat{R}^{\mu\nu}\bigg{]}\,,$ (3.1)

where $x\equiv\\{\tau,\,\rho\\}$. As we will soon show, the coefficient
functions $\\{\alpha(x),\,\beta(x),\,\dots,\,M_{i}(x),\,\dots\\}$ will reduce
to functions of $r(\tau,\rho)$ when describing perturbations around (2.1).
However, unless otherwise specified, in the following sections we will use the
slightly more general action above.

We are of course free to perform the analysis in any frame, such as Jordan
frame or Einstein frame, by performing a suitable conformal transformation.
Such transformation would only affect the coupling of $\phi$ to matter fields,
which we ignore in our analysis. Without loss of generality, therefore, we
choose to work in Einstein frame, setting $M_{1}(x)$ to be constant and equal
to the reduced Planck mass: $M_{1}(r)=M_{\rm Pl}^{2}\equiv\frac{1}{8\pi G}$.

The above action can in principle be derived from the expansion in
perturbations of a general function in terms of the building blocks of the
EFT,

$\displaystyle S=\int{\rm d}^{4}x\sqrt{-g}\,{\cal
L}(g^{\mu\nu},R_{\mu\nu\rho\sigma},g^{\tau\tau},K_{\mu\nu},\nabla_{\mu};\tau)\,.$
(3.2)

In the following we will explain the construction of tadpoles and quadratic
terms, and their relation to the most general action (3.2).

###  3.1 Tadpole terms

The tadpole terms of the action are constructed from the building blocks
$\delta g^{\tau\tau}$, $K_{\mu\nu}$, $R_{\mu\nu\rho\sigma}$ and scalar
functions of the background coordinates. Their most general form is

$\displaystyle S_{\rm tad.}=\int{\rm
d}^{4}x\sqrt{-g}\,\Big{[}\Lambda(x)+\alpha(x)g^{\tau\tau}+\beta_{\mu\nu}(x)K^{\mu\nu}+\zeta_{\mu\nu\rho\sigma}(x)R^{\mu\nu\rho\sigma}\Big{]}\,.$
(3.3)

By construction, only the components $K_{ij}$ are non-zero, and hence the
third term reduces to $\beta_{ij}K^{ij}$, where $\beta_{ij}$ is constructed
from the set of tensor functions that obey the background isometries, namely
$\bar{g}_{ij}$ and $\bar{K}_{ij}$.222Since the 2-sphere is maximally-
symmetric, all other tensors compatible with the background isometries should
be linear combinations of $\bar{g}_{ij}$ and $\bar{K}_{ij}$. We have seen in
(2) that $\bar{\hat{R}}_{ij}$ is an example. Note also that, obviously,
$K=\nabla_{\mu}n^{\mu}$ is a total derivative term. Thus the two index tensor
$\beta_{\mu\nu}$ can be chosen as $\beta(x)\bar{K}_{\mu\nu}$. We refer the
reader to [47, 48] for the detailed construction of the term
$\zeta_{\mu\nu\rho\sigma}(x)R^{\mu\nu\rho\sigma}$. The tadpole action gives
rise to background equations of motion for $g^{\mu\nu}$ as shown in appendix
A. When applied to the ansatz (2.2), it generates a set of consistency
relations between the coefficient functions $M_{1}$, $\Lambda$, $\alpha$ and
$\beta$ and the background metric.

###  3.2 Quadratic terms

It is useful to start the analysis by thinking of possible terms in the action
(3.2). There are infinitely many contractions of $K_{\mu\nu}$ that obey
diffeomorphism invariance on $\Sigma$, for instance

$\displaystyle a_{1}(g^{\tau\tau},\partial g^{\tau\tau})(K)^{n_{1}}\,,\quad
a_{2}(g^{\tau\tau},\partial
g^{\tau\tau})(K)^{n_{2}}K_{\mu\nu}K^{\mu\nu}\,,\quad
a_{3}(g^{\tau\tau},\partial
g^{\tau\tau})K_{\mu\nu}K^{\mu}_{\;\;\rho}K^{\rho\nu}\,,$ $\displaystyle\quad
a_{4}(g^{\tau\tau},\partial g^{\tau\tau})\hat{R}\,,\quad
a_{5}(g^{\tau\tau},\partial g^{\tau\tau})K\hat{R}\,,\quad
a_{6}(g^{\tau\tau},\partial
g^{\tau\tau})K_{\mu\nu}\hat{R}^{\mu\nu}\,,~{}\ldots$ (3.4)

where the $a_{i}$ are arbitrary functions of $g^{\tau\tau}$, and where
$n_{1}=0,1,\,2,\,3$ and $n_{2}=0,1$.

Since we have in mind that the EFT describes perturbations of Horndeski or
DHOST theories, the maximal number of derivatives in each operator is three
[51]. For example, the operators $K^{3}$, $K_{\mu\nu}K^{\mu\nu}$ and
$K_{\mu\nu}K^{\nu}_{\;\;\rho}K^{\rho\mu}$ originate from the Horndeski $G_{5}$
or cubic DHOST term. Therefore, the above set of operators are all the
possible operators involving $K_{\mu\nu}$ and $\hat{R}_{\mu\nu\rho\sigma}$.
Expanding these up to and quadratic order in perturbations $\delta
g^{\tau\tau}=g^{\tau\tau}-\bar{g}^{\tau\tau}$, $\delta K=K-\bar{K}$, $\delta
K_{\mu\nu}=K_{\mu\nu}-\bar{K}_{\mu\nu}$, $\delta\hat{R}=\hat{R}-\bar{\hat{R}}$
and $\delta\hat{R}_{\mu\nu}=\hat{R}_{\mu\nu}-\bar{\hat{R}}_{\mu\nu}$, and up
to second-order in derivatives, we arrive at the quadratic terms in (3).

Note that when one specializes to the EFT of perturbations around the
background (2.2) with $n^{\nu}$ defined by (2.2), the quadratic terms in (3)
can actually describe perturbations of additional operators such as
$K_{\mu_{1}\nu_{1}}K^{\nu_{1}}_{\;\;\nu_{2}}\dots K^{\nu_{n}\mu_{1}}$, since
$\bar{K}_{\mu_{1}\nu_{1}}\bar{K}^{\nu_{1}}_{\;\;\nu_{2}}\dots\bar{K}^{\nu_{n-1}}_{\;\;\nu_{n}}$
can be written as a linear combination of $\bar{h}_{\mu_{1}\nu_{n}}$ and
$\bar{K}_{\mu_{1}\nu_{n}}$. In principle, if one seeks to describe an
arbitrary modified gravity theory, more independent operators (up to second-
order in derivatives) can be added to the action, such as
$(\bar{K}^{\mu\nu}\delta K_{\mu\nu})^{2}$ and
$\bar{K}^{\mu\nu}\bar{K}^{\alpha\beta}\delta K_{\mu\alpha}\delta
K_{\nu\beta}$.

###  3.3 Consistency with diffeomorphisms on the surface $\Sigma_{\tau}$

Since the operators $\delta K^{\mu\nu}$ and $\delta\hat{R}^{\mu\nu\rho\sigma}$
do not transform covariantly under diffeomorphisms on $\Sigma_{\tau}$, the
above action for perturbations, with general coefficients, explicitly breaks
diffeomorphism invariance on $\Sigma_{\tau}$. However, as mentioned in [29,
49], the EFT (3) for perturbations derives from the parent action (3.2), which
is constructed to be invariant under all symmetries. As such, the action for
perturbations should respect diffeomorphisms on $\Sigma_{\tau}$. More
explicitly, it has been shown in [49] that these two observations are
reconciled by diffeomorphism invariance of the action implying a set of non-
trivial constraints on the EFT coefficient functions in (3).

###  3.4 Background isometry and shift symmetry

The background (2.2) has four Killing vector fields,

$\displaystyle v_{1}$
$\displaystyle=\partial_{t}=\partial_{\tau}+\partial_{\rho}\,;$ $\displaystyle
J_{1}$
$\displaystyle=-\sin\varphi\partial_{\theta}-\cot\theta\cos\varphi\partial_{\varphi}\,;$
$\displaystyle J_{2}$
$\displaystyle=\cos\varphi\partial_{\theta}-\cot\theta\sin\varphi\partial_{\varphi}\,;$
$\displaystyle J_{3}$ $\displaystyle=\partial_{\varphi}\,,$ (3.5)

associated with the time translation and rotation invariances of the black
hole background. The coefficient functions in the perturbation Lagrangian are
obtained by expanding (3.2). For example:

$\displaystyle\left.\frac{\partial^{2}{\cal L}}{(\partial
g^{\tau\tau})^{2}}\right|_{\bar{g}_{\mu\nu},\bar{n}^{\mu}}(\delta
g^{\tau\tau})^{2}+\left.\frac{\partial^{2}{\cal L}}{\partial
g^{\tau\tau}\partial K}\right|_{\bar{g}_{\mu\nu},\bar{n}^{\mu}}\delta
g^{\tau\tau}\delta K+\left.\frac{\partial^{2}{\cal L}}{\partial
g^{\tau\tau}\partial K^{\mu\nu}}\right|_{\bar{g}_{\mu\nu},\bar{n}^{\mu}}\delta
g^{\tau\tau}\delta K^{\mu\nu}+\ldots$ (3.6)

Since all the coefficients $\left.\frac{\partial^{2}{\cal
L}}{(\partial\dots)^{2}}\right|_{\bar{g}_{\mu\nu},\bar{n}^{\mu}}$ are
evaluated on the background, and since the action ${\cal L}$ is itself
constructed to respect the symmetries, the coefficients must respect all the
isometries generated by (3.4). Therefore they should obey

$\displaystyle{\cal L}_{v_{1}}\left.\frac{\partial^{2}{\cal
L}}{(\partial\dots)^{2}}\right|_{\bar{g}_{\mu\nu},\bar{n}^{\mu}}={\cal
L}_{J_{1}}\left.\frac{\partial^{2}{\cal
L}}{(\partial\dots)^{2}}\right|_{\bar{g}_{\mu\nu},\bar{n}^{\mu}}={\cal
L}_{J_{2}}\left.\frac{\partial^{2}{\cal
L}}{(\partial\dots)^{2}}\right|_{\bar{g}_{\mu\nu},\bar{n}^{\mu}}={\cal
L}_{J_{3}}\left.\frac{\partial^{2}{\cal
L}}{(\partial\dots)^{2}}\right|_{\bar{g}_{\mu\nu},\bar{n}^{\mu}}=0\,,$ (3.7)

where ${\cal L}_{X}$ denotes the Lie derivative. As a result, as we mentioned
earlier, we must demand that all the scalar coefficients
$\\{\alpha(x),\,\beta(x),\,\dots,\,M_{i}(x),\,\dots\\}$ are functions of $r$
only, and that any background tensor of the form $\bar{T}_{AB}$ must be
proportional to $\gamma_{AB}$, where $\gamma_{AB}$ is the metric on the
2-sphere, and $A,B$ are coordinate indices on the sphere.

However, we must take care here, since the time-dependent scalar profile
$\bar{\phi}=m^{2}\tau$ spontaneously breaks the isometry generated by $v_{1}$.
Nevertheless, in addition to the isometries, there is an additional shift
symmetry, $\phi\to\phi+c$, in the class of scalar-tensor theories of interest.
Because of this, the isometry generated by $v_{1}$ remains a good symmetry in
the action for perturbations. In other words, in the particular case we are
interested in, due to the existence of the time isometry, the shift symmetry
is equivalent to an isometry. To see this, we write the background scalar
profile as

$\displaystyle\bar{\phi}=m^{2}\tau=\Lambda^{2}(t+\psi(r))\,.$ (3.8)

A shift in $\bar{\phi}$, due to transforming $\phi\to\phi+c$, can therefore be
absorbed into $t$ by making use of an isometry, $t\to t-\frac{c}{m^{2}}$. In
other words, the spontaneously broken time translation due to $\bar{\phi}$ is
absorbed by a shift in $\phi$, while the “diagonal” part of the time
translation and $\phi$-shift symmetry remains unbroken. Only when the scalar
profile is more complicated, in such a way that the diffeomorphism performed
to absorb the shift in $\phi$ cannot be an isometry, will this lead to more
non-trivial constraints on the EFT coefficients [48].333An explicit example is
the ultra slow-roll regime, where $\bar{\phi}\approx\frac{c}{\tau^{6}}$ in
conformal time $\tau$.

##  4 Second order action in the decoupling limit

Previous studies of a class of black holes with time-dependent hair in scalar-
tensor theories have shown that gradient instabilities are inevitable, since
the radial and angular sound speed squared of perturbations have opposite
signs [38, 44]. We have also learned that it is the scalar modes that are
responsible for this instability [38]. While establishing the stability of a
theory against perturbations requires a complete analysis, proving the
existence of an instability requires much less work.

For this purpose, it is practically useful to analyze stability in the
decoupling limit. In general, our EFT has two characteristic scales. One is
the decoupling scale $\Lambda_{\rm dec}$, at which gravity decouples from the
Nambu-Goldstone mode $\pi$. The other is the cutoff scale $\Lambda_{\rm EFT}$,
beyond which the EFT is not trustworthy, either due to strong coupling or a
breakdown of the derivative expansion. One may study the dispersion relation
of $\pi$ up to the second order derivatives. If a gradient instability appears
at some scale $\Lambda_{\rm grad}$ in the regime $\Lambda_{\rm
dec}<\Lambda_{\rm grad}<\Lambda_{\rm EFT}$, it indicates an instability.
Recall that $\Lambda_{\rm EFT}$ depends on details of higher order terms,
which is beyond the scope our present work. Thus, we study (in)stability in
the decoupling limit based on the dispersion relation up to second-order
derivatives.

One way to study the decoupling limit in the EFT language is to introduce a
Stückelberg field $\pi(x^{\mu})$ to explicitly restore time-translational
invariance in $\tau$ [47]. Under $\tau\to\tilde{\tau}=\tau+\pi(x^{\mu})$, the
important elements are

$\displaystyle\left[\frac{\partial\tilde{x}^{\mu}(x)}{\partial
x^{\nu}}\right]$
$\displaystyle=\begin{pmatrix}1+\dot{\pi}(t,\vec{x})&\partial_{k}\pi(t,\vec{x})\\\
\mathbf{0}&\delta^{i}_{\;j}\end{pmatrix}\,;$
$\displaystyle\left[\frac{\partial
x^{\mu}(\tilde{x})}{\partial\tilde{x}^{\nu}}\right]$
$\displaystyle=\begin{pmatrix}\frac{1}{1+\dot{\pi}(t,\vec{x})}&-\frac{\partial_{k}\pi(t,\vec{x})}{1+\dot{\pi}(t,\vec{x})}\\\
\mathbf{0}&\delta^{i}_{\;j}\end{pmatrix}\,,$ (4.1)

where $\mu$ and $\nu$ are row and column indices, respectively. For instance,
the transformation $\tilde{g}_{\mu\nu}(\tilde{x})=\frac{\partial
x^{\alpha}(\tilde{x})}{\partial\tilde{x}^{\mu}}\frac{\partial
x^{\beta}(\tilde{x})}{\partial\tilde{x}^{\nu}}g_{\alpha\beta}(x)$ leads to

$\displaystyle g_{\tau\tau}$ $\displaystyle\to
g_{\tau\tau}\frac{1}{(1+\dot{\pi})^{2}}\,;$ $\displaystyle g_{\tau i}$
$\displaystyle\to g_{\tau
i}\frac{1}{1+\dot{\pi}}-g_{\tau\tau}\frac{\partial_{i}\pi}{(1+\dot{\pi})^{2}}\,;$
$\displaystyle g_{ij}$ $\displaystyle\to g_{ij}-g_{\tau
i}\frac{\partial_{j}\pi}{1+\dot{\pi}}-g_{\tau
j}\frac{\partial_{i}\pi}{1+\dot{\pi}}+g_{\tau\tau}\frac{\partial_{i}\pi\partial_{j}\pi}{(1+\dot{\pi})^{2}}\,.$
(4.2)

The transformation rule for the corresponding components in the ADM
decomposition is then

$\displaystyle h_{ij}\to
h_{ij}-(1-\dot{\pi})\Big{(}N_{i}\partial_{j}\pi+N_{j}\partial_{i}\pi\Big{)}+\left(-N^{2}+N^{k}N_{k}\right)\partial_{i}\pi\partial_{j}\pi+{\cal
O}(\pi^{3})\,,$ (4.3)

while the spatial Christoffel connection
$\hat{\Gamma}^{l}_{ij}=\frac{1}{2}h^{lk}\left(\partial_{i}h_{jk}+\partial_{j}h_{ik}-\partial_{k}h_{ij}\right)$
used to define $D_{i}$ transforms as

$\hat{\Gamma}^{l}_{ij}\to\hat{\Gamma}^{l}_{ij}-\frac{1}{2}h^{lk}\left(\dot{h}_{ik}\partial_{j}\pi+\dot{h}_{jk}\partial_{i}\pi-\dot{h}_{ij}\partial_{k}\pi\right)+{\cal
O}(\pi^{2})\,.$ (4.4)

Here we have used that $\bar{N}_{i}=0$ on the background geometry. Therefore
the relevant transformations for $\delta K_{ij}$, $\delta K$,
$\delta\hat{R}_{ij}$ and $\delta\hat{R}$, up to first order in $\pi$, become

$\displaystyle\delta K_{ij}$ $\displaystyle\to\delta
K_{ij}-\dot{\bar{K}}_{ij}\pi-ND_{i}D_{j}\pi\,;$ $\displaystyle\delta K$
$\displaystyle\to\delta K-\dot{\bar{K}}\pi-Nh^{ij}D_{i}D_{j}\pi\,;$
$\displaystyle\delta\hat{R}_{ij}$
$\displaystyle\to\delta\hat{R}_{ij}-\dot{\bar{\hat{R}}}_{ij}\pi-\partial_{k}\pi\dot{\hat{\Gamma}}^{k}_{ij}-\frac{1}{2}h^{kl}\dot{\hat{\Gamma}}^{m}_{kl}\left(\partial_{i}\pi
h_{mj}+\partial_{j}\pi h_{mi}\right)$
$\displaystyle\quad\,+\frac{1}{2}\left(\partial_{i}\pi\dot{\hat{\Gamma}}^{m}_{jm}+\partial_{j}\pi\dot{\hat{\Gamma}}^{m}_{im}+\partial^{k}\pi\left(\dot{\hat{\Gamma}}^{m}_{ik}h_{mj}+\dot{\hat{\Gamma}}^{m}_{jk}h_{mi}\right)\right)$
$\displaystyle\quad\,-\frac{1}{2}\left(\dot{h}_{li}D^{l}D_{j}\pi+\dot{h}_{lj}D^{l}D_{i}\pi-h^{kl}\dot{h}_{kl}D_{i}D_{j}\pi-\dot{h}_{ij}\hat{\square}\pi\right);$
$\displaystyle\delta\hat{R}$
$\displaystyle\to\delta\hat{R}-\dot{\bar{\hat{R}}}\pi-2\partial_{k}\pi
h^{ij}\dot{\hat{\Gamma}}^{k}_{ij}+2\partial^{i}\pi\dot{\hat{\Gamma}}^{m}_{im}-h^{ij}\dot{h}_{li}D^{l}D_{j}\pi+h^{kl}\dot{h}_{kl}\hat{\square}\pi\,.$
(4.5)

###  4.1 Gradient (in-)stability for $\pi$

With the scalar mode restored through the Stückelberg trick, we are ready to
study the stability of the theory in the decoupling limit. Given the
symmetries of the background, there are in general two different sound speeds
in the problem: the radial sound speed squared, $c_{\rho}^{2}$, and the
angular sound speed squared, $c_{\theta,\varphi}^{2}$. In the following we
individually analyze quadratic operators in the effective action (3.2). In
each case, we neglect terms that are cubic and higher-order in $\pi$, as well
as terms that involve mixing with $\delta h_{ij}$. We also neglect terms that
are irrelevant to the dispersion relation up to second-order in derivatives,
such as $M_{5,6,7,8}$ as they contribute to higher order derivative terms for
$\pi$. Note that we treat the coefficients as general functions of
$(\tau,\rho)$, except when it becomes convenient to use the specific
dependence of $r(\tau,\rho)$.

  * •

$M_{2}(\tau,\rho)\left(\delta g^{\tau\tau}\right)^{2}$: Using the background
quantities $\bar{g}^{\tau\tau}=-1$ and $\bar{g}^{a\tau}=0$, this term only
contributes to the time-derivative part of $\pi$,

$\displaystyle\int{\rm d}^{4}x\sqrt{-g}\,M_{2}\left(\delta
g^{\tau\tau}\right)^{2}\to\int{\rm d}^{4}x\sqrt{-g}\,4M_{2}\dot{\pi}^{2}\,.$
(4.6)

In the Schwarzschild-de Sitter limit, $\alpha=\beta=0$, therefore the above
term is crucial for generating a $\dot{\pi}^{2}$ term and $M_{2}$ must be
positive.

  * •

$M_{3}(\tau,\rho)\,\delta g^{\tau\tau}\delta K$: The contribution to the $\pi$
action coming from this term is

$\displaystyle\int{\rm d}^{4}x\sqrt{-g}\,M_{3}\delta g^{\tau\tau}\delta
K\to\int{\rm d}^{4}x\sqrt{-g}\Big{[}-(M_{3}h^{ij})\dot{}\,D_{i}\pi
D_{j}\pi+2(D^{i}M_{3})\dot{\pi}D_{i}\pi-m_{3}(\tau,\rho)\pi^{2}\Big{]}\,.$
(4.7)

Here, $m_{3}(\tau,\rho)$ is the mass term, whose explicit form is not
necessary for our purposes. (Henceforth, $m_{i}(\tau,\rho)$ will denote the
mass contribution from the operator with corresponding coefficient
$M_{i}(\tau,\rho)$.) In general, due to the existence of the mixing term
$2(D^{i}M_{3})\dot{\pi}D_{i}\pi$, it is necessary to find a coordinate system
that diagonalizes the kinetic matrix. In the specific case in which $M_{3}$ is
a function of $r(\tau,\rho)$ we have
$\partial_{\rho}M_{3}(r)=-\dot{M_{3}}(r)$, so that the kinetic terms become

$\displaystyle-2\dot{M_{3}}g^{\rho\rho}\dot{\pi}\pi^{\prime}-(M_{3}g^{\rho\rho})\dot{}\,\pi^{\prime
2}-\partial_{\tau}\left(\frac{M_{3}}{r^{2}}\right)\gamma^{AB}\partial_{A}\pi\partial_{B}\pi\,.$
(4.8)

Stability along the angular directions therefore requires

$\displaystyle\partial_{\tau}\left(\frac{M_{3}}{r^{2}}\right)>0\,.$ (4.9)

Given that the $\dot{\pi}^{2}$ term (from other operators) is of the form
$A|g^{\tau\tau}|\dot{\pi}^{2}$, with $A>0$, a straight forward Hamiltonian
analysis 444For a Lagrangian given by  $\displaystyle
L=a\dot{\pi}^{2}+2b\dot{\pi}\pi^{\prime}-c\pi^{\prime 2}\,,$ (4.10) the
corresponding Hamiltonian is  $\displaystyle
H=\frac{(P_{\pi}-2b\pi^{\prime})^{2}}{4a}+c\pi^{\prime 2},\quad
P_{\pi}=\frac{\partial L}{\partial\dot{\pi}}=2a\dot{\pi}+2b\pi^{\prime}.$
(4.11) Therefore, the stability conditions are  $\displaystyle a>0,\quad
c>-\frac{b^{2}}{a}.$ (4.12) give the following stability constraints:

$\displaystyle(M_{3}g^{\rho\rho})\dot{}\,>-\frac{\big{(}\dot{M_{3}}g^{\rho\rho}\big{)}^{2}}{A|g^{\tau\tau}|}.$
(4.13)

There also exist suitable coordinates $(T,\,R)$ that diagonalize the kinetic
matrix, with the corresponding kinetic terms

$\displaystyle\frac{1}{2}\Big{(}A|g^{\tau\tau}|(2+\delta_{3})+\left(M_{3}g^{\rho\rho}\right)\dot{}\,\delta_{3}\Big{)}(\partial_{T}\pi)^{2}$
$\displaystyle-\frac{1}{2}\Big{(}A|g^{\tau\tau}|\delta_{3}+\left(M_{3}g^{\rho\rho}\right)\dot{}\,(2+\delta_{3})\Big{)}(\partial_{R}\pi)^{2}$
$\displaystyle-\partial_{\tau}\left(\frac{M_{3}}{r^{2}}\right)\gamma^{AB}\partial_{A}\pi\partial_{B}\pi\,,$
(4.14)

where

$\delta_{3}\equiv\left[1+\frac{4\left(\dot{M_{3}}g^{\rho\rho}\right)^{2}}{\left(A|g^{\tau\tau}|+\partial_{\tau}\left(M_{3}g^{\rho\rho}\right)\right)^{2}}\right]^{\frac{1}{2}}-1\geq
0\,.$

  * •

$M_{4}(\tau,\rho)\bar{K}_{\mu\nu}\delta g^{\tau\tau}\delta K^{\mu\nu}$: This
operator generates the following contribution:

$\displaystyle\int{\rm d}^{4}x\sqrt{-g}\,M_{4}\bar{K}_{\mu\nu}\delta
g^{\tau\tau}\delta K^{\mu\nu}\to\int{\rm
d}^{4}x\sqrt{-g}\Big{[}-\partial_{\tau}\big{(}M_{4}\bar{K}^{ij}\big{)}D_{i}\pi
D_{j}\pi+2D_{i}\big{(}M_{4}\bar{K}^{ij}\big{)}\dot{\pi}D_{j}\pi-
m_{4}(\tau,\rho)\pi^{2}\Big{]}\,.$ (4.15)

Since $M_{4}$ is a function of $(\tau,\rho)$, and using (2), there is no
$\dot{\pi}\partial_{A}\pi$ mixing. Therefore, without further specifying the
geometry, stability along the angular direction requires

$\displaystyle\partial_{\tau}\left(M_{4}\frac{\partial_{\tau}(r^{2})}{r^{4}}\right)>0\,.$
(4.16)

As mentioned earlier, other contributions to the $\dot{\pi}^{2}$ term are of
the form $A|g^{\tau\tau}|\dot{\pi}^{2}$, with $A>0$. Stability obtained from a
Hamiltonian analysis requires that

$\displaystyle\partial_{\tau}\big{(}M_{4}\bar{K}^{\rho\rho}\big{)}>-\frac{\Big{[}D_{i}\big{(}M_{4}\bar{K}^{i\rho}\big{)}\Big{]}^{2}}{A|g^{\tau\tau}|}$
(4.17)

It follows that the two eigenvalues of the kinetic matrix in the $(\tau,\rho)$
directions are given by

$\displaystyle\lambda_{1}=\frac{1}{2}\Big{(}A|g^{\tau\tau}|(2+\sigma)+\partial_{\tau}\big{(}M_{4}\bar{K}^{\rho\rho}\big{)}\sigma\Big{)}\,;\quad\lambda_{2}=-\frac{1}{2}\Big{(}\partial_{\tau}\big{(}M_{4}\bar{K}^{\rho\rho}\big{)}(2+\sigma)+A|g^{\tau\tau}|\sigma\Big{)}\,,$
(4.18)

where

$\sigma\equiv\left[1+\frac{4\left(D_{i}\big{(}M_{4}\bar{K}^{i\rho}\big{)}\right)^{2}}{\left(A|g^{\tau\tau}|+\partial_{\tau}(M_{4}\bar{K}^{\rho\rho})\right)^{2}}\right]^{\frac{1}{2}}-1\geq
0\,.$

  * •

$M_{9}(\tau,\rho)\delta K^{2}+M_{10}(\tau,\rho)\delta K_{\mu\nu}\delta
K^{\mu\nu}$: For these operators, it is sufficient to restrict our analysis to
the combination

$\displaystyle M_{9}\left(\delta K^{2}-\delta K_{\mu\nu}\delta
K^{\mu\nu}\right)\,,$ (4.19)

since any other choice of relative coefficients differs from this choice only
by higher-derivative terms in $\pi$. We then have

$\displaystyle\int{\rm d}^{4}x\sqrt{-g}\,M_{9}\left(\delta K^{2}-\delta
K_{ij}\delta K^{ij}\right)$ $\displaystyle~{}~{}~{}\rightarrow\int{\rm
d}^{4}x\sqrt{-g}\bigg{[}\left(M_{9}\hat{R}^{ij}+D^{i}D^{j}M_{9}-\left(\hat{\square}M_{9}+2M_{9}\dot{\bar{K}}\right)h^{ij}+2M_{9}\dot{\bar{K}}^{ij}\right)D_{i}\pi
D_{j}\pi-m_{9}(\tau,\rho)\pi^{2}\bigg{]}\,.$ (4.20)

Without any $\dot{\pi}\pi^{\prime}$ mixing, stability of this operator alone
simply requires negativity for the coefficient of $D_{i}\pi D_{j}\pi$,

$\displaystyle\left(M_{9}\hat{R}^{ij}+D^{i}D^{j}M_{9}-\left(\hat{\square}M_{9}+2M_{9}\dot{\bar{K}}\right)h^{ij}+2M_{9}\dot{\bar{K}}^{ij}\right)<0\,.$
(4.21)

  * •

$M_{11}(\tau,\rho)\bar{K}_{\mu\nu}\delta K\delta
K^{\mu\nu}+M_{12}(\tau,\rho)\bar{K}_{\mu\nu}\delta K^{\rho\mu}\delta
K^{\nu}_{\;\;\rho}$: A similar pattern emerges for these operators. Only the
combination

$\displaystyle M_{11}\left(\bar{K}_{\mu\nu}\delta K\delta
K^{\mu\nu}-\bar{K}_{\mu\nu}\delta K^{\rho\mu}\delta K^{\nu}_{\;\;\rho}\right)$
(4.22)

generates second-order equations for $\pi$. The associated contribution to the
action is then

$\displaystyle\int{\rm d}^{4}x\sqrt{-g}M_{11}(r)\left(\bar{K}_{\mu\nu}\delta
K\delta K^{\mu\nu}-\bar{K}_{\mu\nu}\delta K^{\rho\mu}\delta
K^{\nu}_{\;\;\rho}\right)$ $\displaystyle~{}~{}~{}\rightarrow\int{\rm
d}^{4}x\sqrt{-g}\bigg{[}\bigg{(}D_{k}D^{i}\left(M_{11}\bar{K}^{kj}\right)-\frac{1}{2}D_{k}D_{l}\left(M_{11}\bar{K}^{kl}\right)h^{ij}-\frac{1}{2}D_{k}D^{k}\left(M_{11}\bar{K}^{ij}\right)+M_{11}\bar{K}^{k\ell}\hat{R}^{\;\,i\;\,j}_{k\;\ell}$
$\displaystyle\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+M_{11}\dot{\bar{K}}\bar{K}^{\ij}+M_{11}\bar{K}^{kl}\dot{\bar{K}}_{kl}h^{ij}-2M_{11}\bar{K}^{ki}\dot{\bar{K}}_{k}^{\;\;j}\bigg{)}D_{i}\pi
D_{j}\pi-m_{11}(\tau,\rho)\pi^{2}\bigg{]}\,$

Stability condition is also straight forward as there is no
$\dot{\pi}\pi^{\prime}$ mixing.

  * •

$M_{13}(\tau,\rho)\delta g^{\tau\tau}\delta\hat{R}$: This operator gives rise
to the following terms in the $\pi$ action:

$\displaystyle\int{\rm d}^{4}x\sqrt{-g}M_{13}\delta g^{\tau\tau}\delta\hat{R}$
$\displaystyle~{}~{}~{}\rightarrow\int{\rm
d}^{4}x\sqrt{-g}\bigg{[}4\left(M_{13}\left(h^{k\ell}\dot{\hat{\Gamma}}^{m}_{\ell
m}-h^{\ell m}\dot{\hat{\Gamma}}^{k}_{\ell
m}\right)+D_{i}\left(M_{13}\bar{K}^{ik}\right)-D^{k}\left(M_{13}\bar{K}\right)\right)\dot{\pi}D_{k}\pi$
$\displaystyle\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2\partial_{\tau}\left(M_{13}\bar{K}h^{ij}-M_{13}\bar{K}^{ij}\right)D_{i}\pi
D_{j}\pi-m_{13}(\tau,\rho)\pi^{2}\bigg{]}\,,$ (4.23)

where we have used the fact that $\dot{\bar{h}}_{ij}=2\bar{K}_{ij}$. Again, it
is not hard to see that there is no mixing of the form
$\dot{\pi}\partial_{A}\pi$. It can be diagonalized in the usual way but the
resulting expression is lengthy and not informative which we do not display
here. The stability constraints arising from these operators are rather less
informative. We will restrict in appendix C to de Sitter-Schwarszchild limit
to extract explicit constraints.

  * •

$M_{14}(\tau,\rho)\bar{K}_{\mu\nu}\delta g^{\tau\tau}\delta\hat{R}^{\mu\nu}$:
The quadratic $\pi$ action generated from this operator is

$\displaystyle\int{\rm d}^{4}x\sqrt{-g}M_{14}\bar{K}_{\mu\nu}\delta
g^{\tau\tau}\delta\hat{R}^{\mu\nu}$ $\displaystyle~{}~{}~{}\rightarrow\int{\rm
d}^{4}x\sqrt{-g}\bigg{[}2\bigg{(}M_{14}\left(\dot{\hat{\Gamma}}^{m}_{km}\bar{K}^{ik}+\dot{\hat{\Gamma}}^{m}_{\ell
k}\bar{K}^{\ell}_{\;\,m}h^{ki}-\dot{\hat{\Gamma}}^{m}_{k\ell}h^{k\ell}\bar{K}_{m}^{\;\;i}-\dot{\hat{\Gamma}}^{i}_{k\ell}\bar{K}^{k\ell}\right)$
$\displaystyle\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+D_{k}\left(2M_{14}\bar{K}^{i\ell}\bar{K}_{\ell}^{\;\,k}-M_{14}\bar{K}K^{ik}\right)-D^{i}\left(M_{14}\bar{K}^{k\ell}\bar{K}_{k\ell}\right)\bigg{)}\dot{\pi}D_{i}\pi$
$\displaystyle\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\partial_{\tau}\left(M_{14}\bar{K}^{k\ell}\bar{K}_{k\ell}h^{ij}+M_{14}\bar{K}\bar{K}^{ij}-2M_{14}\bar{K}^{i\ell}\bar{K}_{\ell}^{\;\,j}\right)D_{i}\pi
D_{j}\pi-m_{14}(\tau,\rho)\pi^{2}\bigg{]}\,.$ (4.24)

After evaluating the form of $\hat{\Gamma}$’s, one finds that there is only
$\dot{\pi}\pi^{\prime}$ mixing.

In principle, one should consider an action with all these terms together in
order to come up with a constraint. However, such expression is not
particularly illuminating here with many free functions. We stress again that
instability of each term signals potential instability in the entire theory.
In Appendix C, we collect the above constraints in the Schwarzschild-de Sitter
limit.

##  5 Odd-parity perturbations

In this Section we analyze the effective action (3) for odd-parity
perturbations for $\ell\geq 2$. The perturbed metric $\delta g^{\rm
odd}_{\mu\nu}$ is expanded in terms of the standard odd-parity vector and
tensor spherical harmonics as

$\displaystyle\delta g^{\rm odd}_{\mu\nu}=\sum_{\ell
m}\begin{pmatrix}0&0&h_{0}^{\ell m}\epsilon_{A}^{\;\;C}\nabla_{C}\\\
0&0&h_{1}^{\ell m}\epsilon_{A}^{\;\;C}\nabla_{C}\\\ h_{0}^{\ell
m}\epsilon_{B}^{\;\;C}\nabla_{C}&h_{1}^{\ell
m}\epsilon_{B}^{\;\;C}\nabla_{C}&h_{2}^{\ell m}\epsilon_{(B}^{\quad
C}\nabla_{A)}\nabla_{C}\end{pmatrix}Y_{\ell}^{m}(\theta,\varphi)\,,$ (5.1)

where $\nabla_{A}$ is the covariant derivative on the 2-sphere associated with
$\gamma_{AB}$. As usual, we work in the Regge-Wheeler gauge

$\displaystyle h_{2}^{\ell m}=0\,,$ (5.2)

which is justified since the EFT is invariant under diffeomorphisms on the
2-sphere. Note that $h_{2}$ is exactly zero for $\ell=1$.

Odd perturbations on the 2-sphere are only affected by a restricted set of
operators, since odd-parity contributions to $\delta g^{\tau\tau}$ and $\delta
K$ are non-zero only starting at second order. For the same reason, odd sector
perturbations correspond only to tensor perturbations. Therefore, at linear
level the relevant EFT is

$\displaystyle S_{(2)}=\int{\rm d}^{4}x\sqrt{-g}\bigg{[}$
$\displaystyle\frac{M_{\rm
Pl}^{2}}{2}R-\Lambda(\tau,\rho)+\alpha(\tau,\rho)g^{\tau\tau}+\beta(\tau,\rho)\bar{K}_{\mu\nu}K^{\mu\nu}$
$\displaystyle+M_{10}(\tau,\rho)\delta K_{\mu\nu}\delta
K^{\mu\nu}+M_{12}(\tau,\rho)\bar{K}_{\mu\nu}\delta K^{\rho\mu}\delta
K^{\nu}_{\;\,\rho}\bigg{]}\,,$ (5.3)

where once again we have chosen to work in Einstein frame, in which
$M_{1}(\tau,\rho)=M_{\rm Pl}^{2}$. Furthermore note once more that we are
leaving the coefficient functions as general functions of $(\tau,\rho)$,
specializing to functions of $r(\tau,\rho)$ only when the analysis is
restricted to Schwarzschild-de Sitter space-time.

In general, the action for $h_{0}$ and $h_{1}$ for a specific value of
$(\ell,m)$ takes the form

$\displaystyle S=\int{\rm d}\tau{\rm
d}\rho\left[c\left(h^{\prime}_{0}-\dot{h}_{1}+ah_{0}+bh_{1}\right)^{2}+\frac{1}{2}k_{00}h_{0}^{2}+k_{01}h_{0}h_{1}+\frac{1}{2}k_{11}h_{1}^{2}\right]\,,$
(5.4)

where all the coefficient functions $c$, $a$, $b$, $k_{00}$, $k_{01}$ and
$k_{11}$ are functions of $(\tau,\,\rho)$. Restricting to the EFT (5), these
are then explicitly given by

$\displaystyle c$ $\displaystyle=\frac{M_{\rm
Pl}^{2}}{2\sqrt{F}}+\frac{M_{10}}{2\sqrt{F}}+\frac{M_{12}\left(r^{2}F\right)\dot{}}{8F^{3/2}r^{2}}\,;$
$\displaystyle a$ $\displaystyle=-2\frac{r^{\prime}}{r}\,;$ $\displaystyle b$
$\displaystyle=\frac{2F\dot{r}\beta+(2F\dot{r}-r\dot{F})M_{\rm
Pl}^{2}}{4cF^{3/2}r}+\frac{\dot{F}}{2F}+\frac{\dot{r}}{r}\,;$ $\displaystyle
k_{00}$ $\displaystyle=\sqrt{F}(\alpha-\Lambda)+\frac{M_{\rm
Pl}^{2}\left(F^{2}\left(j^{2}+2\dot{r}^{2}\right)-2F\left(r\left(2r^{\prime\prime}-\dot{F}\dot{r}\right)+r^{\prime
2}\right)+2rF^{\prime}r^{\prime}\right)}{F^{3/2}r^{2}}$
$\displaystyle\quad\,+M_{10}\frac{\sqrt{F}\left(j^{2}-2\right)}{r^{2}}+M_{12}\frac{\sqrt{F}\dot{r}\left(j^{2}-2\right)}{r^{3}}\,;$
$\displaystyle k_{01}$ $\displaystyle=\frac{M_{\rm
Pl}^{2}\left(4F\dot{r}^{\prime}-2\dot{F}r^{\prime}\right)}{F^{3/2}r}+\frac{r^{2}\left(\dot{F}\left(\beta
F^{\prime}-F\beta^{\prime}\right)-\beta F\dot{F}^{\prime}\right)+4\beta
F^{2}r^{\prime}\dot{r}-2\beta Fr\dot{F}r^{\prime}}{2F^{5/2}r^{2}}\,;$
$\displaystyle k_{11}$
$\displaystyle=\frac{3r^{2}\dot{F}^{2}-8Fr\dot{F}\dot{r}+12Fr^{\prime
2}-6F^{2}\left(j^{2}+4r\ddot{r}\right)}{6F^{5/2}r^{2}}M_{\rm Pl}^{2}$
$\displaystyle\quad\,+\frac{6Fr(\alpha+\lambda)-8M_{10}\dot{F}\dot{r}+\beta\left(6\dot{F}\dot{r}+3r\ddot{F}\right)+3r\dot{\beta}\dot{F}}{6F^{3/2}r}$
$\displaystyle\quad\,-\frac{\left(2F\dot{r}\beta+(2F\dot{r}-r\dot{F})M_{\rm
Pl}^{2}\right)^{2}}{8cF^{3}r^{2}}+c\left(\frac{2\dot{F}\dot{r}}{3Fr}-\frac{\dot{F}^{2}}{2F^{2}}-\frac{2\dot{r}^{2}}{r^{2}}\right)\,,$
(5.5)

where $F(\tau,\rho)=1-f(r)$ and $j^{2}=\ell^{2}+\ell$. The above expressions
apply to a general case where $\Lambda,\,\alpha,\,\beta,\,M_{10},\,M_{12}$ are
general functions of $(\tau,\,\rho)$.

One can identify the relevant degrees of freedom by integrating in an
auxiliary field $\Psi$ as follows:

$\displaystyle S=\int{\rm d}\tau{\rm d}\rho$
$\displaystyle\bigg{[}c\left(h^{\prime}_{0}-\dot{h}_{1}+ah_{0}+bh_{1}\right)^{2}-c\left(\frac{1}{c}\Psi-(h^{\prime}_{0}-\dot{h}_{1}+ah_{0}+bh_{1})\right)^{2}$
$\displaystyle~{}~{}~{}+\frac{1}{2}k_{00}h_{0}^{2}+k_{01}h_{0}h_{1}+\frac{1}{2}k_{11}h_{1}^{2}\bigg{]}\,.$
(5.6)

The equations of motion for $h_{0}$ and $h_{1}$ become,

$\displaystyle\begin{pmatrix}k_{00}&k_{01}\\\
k_{01}&k_{11}\end{pmatrix}\begin{pmatrix}h_{0}\\\
h_{1}\end{pmatrix}=\begin{pmatrix}2\Psi^{\prime}-2a\Psi\\\
-2\dot{\Psi}-2b\Psi\end{pmatrix},$ (5.7)

which can then be expressed purely in terms of $\Psi$ in the action,

$\displaystyle S$ $\displaystyle=\int{\rm d}\tau{\rm
d}\rho\Bigg{[}-\frac{1}{2}\begin{pmatrix}2\Psi^{\prime}-2a\Psi\\\
-2\dot{\Psi}-2b\Psi\end{pmatrix}^{\rm T}\begin{pmatrix}k_{00}&k_{01}\\\
k_{01}&k_{11}\end{pmatrix}^{-1}\begin{pmatrix}2\Psi^{\prime}-2a\Psi\\\
-2\dot{\Psi}-2b\Psi\end{pmatrix}-\frac{1}{c}\Psi^{2}\Bigg{]}$
$\displaystyle=\int{\rm d}\tau{\rm
d}\rho\Bigg{[}\frac{2}{k_{01}^{2}-k_{00}k_{11}}\left(k_{00}\dot{\Psi}^{2}+2k_{01}\dot{\Psi}\Psi^{\prime}+k_{11}\Psi^{\prime
2}\right)-\frac{1}{2}m_{\Psi}\Psi^{2}\Bigg{]}\,.$ (5.8)

Stability of the Hamiltonian requires that

$\displaystyle k_{00}>0,\quad k_{11}<\frac{k_{01}^{2}}{k_{00}}\,.$ (5.9)

The kinetic terms can be diagonalized as usual, with the corresponding
eigenvalues given by

$\displaystyle\frac{1}{2}\left(k_{00}+k_{11}\pm(k_{00}-k_{11})\left(1+\frac{4k_{01}^{2}}{(k_{00}-k_{11})^{2}}\right)^{\frac{1}{2}}\right).$
(5.10)

However, the resulting stability constraints are unclear in general, due to
the complicated forms of the coefficient functions (5).

###  5.1 Schwarzschild-de Sitter limit

We now restrict the analysis to the Schwarzschild-de Sitter case, in order to
obtain useful constraints. As mentioned in Sec. 3.4, the background isometries
constrain the EFT coefficients to be functions of $r(\tau,\rho)$,

$\displaystyle\Lambda(\tau,\rho)$ $\displaystyle\to\Lambda(r)\,;$
$\displaystyle\alpha(\tau,\rho)$ $\displaystyle\to\alpha(r)\,;$
$\displaystyle\beta(\tau,\rho)$ $\displaystyle\to\beta(r)\,;$ $\displaystyle
M_{10}(\tau,\rho)$ $\displaystyle\to M_{10}(r)\,;$ $\displaystyle
M_{12}(\tau,\rho)$ $\displaystyle\to M_{12}(r)\,.$ (5.11)

One can show that, under the condition $g(r)=f(r)$, together with the
background equations of motion (A),

$\displaystyle k_{01}=0\,.$ (5.12)

This holds even without using the explicit form of $f(r)$. Therefore,
stability requires that

$\displaystyle k_{00}>0\,;\qquad k_{11}<0\,,$ (5.13)

with

$\displaystyle k_{00}$
$\displaystyle=\frac{\sqrt{1-f}\left(j^{2}-2\right)\left(rM_{\rm
Pl}^{2}+rM_{10}-\sqrt{1-f}M_{12}\right)}{r^{3}}\,;$ $\displaystyle k_{11}$
$\displaystyle=\frac{\left[12(1-f)^{2}+rf_{r}\left(3rf_{r}+4(1-f)\right)\right]\left[M_{12}\left(2(1-f)-rf_{r}\right)-4r\sqrt{1-f}\left(M_{10}+M_{\rm
Pl}^{2}\right)\right]}{48(1-f)^{2}r^{3}}$
$\displaystyle\quad-\frac{\left(2\beta(1-f)+\left(2(1-f)+rf_{r}\right)M_{\rm
Pl}^{2}\right)^{2}}{(1-f)r\left(4r\sqrt{1-f}\left(M_{10}+M_{\rm
Pl}^{2}\right)-M_{12}\left(2(1-f)-rf_{r}\right)\right)}$
$\displaystyle\quad+\frac{M_{\rm
Pl}^{2}\left[6(1-f)(2(1-f)-j^{2}+2)+rf_{r}\left(3rf_{r}-8(1-f)\right)\right]+4(1-f)\left(3\beta(1-f)+2rf_{r}M_{10}\right)}{6(1-f)^{3/2}r^{2}}\,,$
(5.14)

where we have defined $j^{2}\equiv\ell(\ell+1)$. In the above we have used the
background equations to eliminate $\Lambda$, $\alpha$ and $\beta_{r}$ in terms
of $f$ and $\beta$. Furthermore, one finds that $\alpha=\beta=0$ and
$\Lambda=2M_{\rm Pl}^{2}\lambda$ in the Schwarzschild-de Sitter limit, with
$f(r)=g(r)=1-\frac{r_{s}}{r}-\frac{\lambda}{3}r^{2}$. Therefore, the radial
sound speed for perturbations in the Schwarzschild-de Sitter limit is

$\displaystyle c_{\rho}^{2}$
$\displaystyle=-\frac{g^{\tau\tau}k_{11}}{g^{\rho\rho}k_{00}}$
$\displaystyle=\frac{1}{48(1-f)^{3/2}(j-2)\left(4r\sqrt{1-f}\left(M_{\rm
Pl}^{2}+M_{10}\right)-M_{12}\left(2(1-f)-rf_{r}\right)\right)\left(r\left(M_{\rm
Pl}^{2}+M_{10}\right)-\sqrt{1-f}M_{12}\right)}$
$\displaystyle\qquad\times\bigg{[}-M_{12}^{2}\left(2(1-f)-rf_{r}\right)^{2}\left(12(1-f)^{2}+3r^{2}f_{r}^{2}+4r(1-f)f_{r}\right)-48M_{10}^{2}(1-f)r^{2}\left(2(1-f)-rf_{r}\right)^{2}$
$\displaystyle\quad\qquad-16(1-f)^{3/2}M_{12}r\left(2(1-f)-rf_{r}\right)\left(6\beta(1-f)+M_{\rm
Pl}^{2}\left(2rf_{r}-3j+6\right)\right)$
$\displaystyle\quad\qquad-192(1-f)^{2}r^{2}\left(\beta^{2}(1-f)+\beta
rf_{r}M_{\rm Pl}^{2}+(j-2)M_{\rm Pl}^{4}\right)$
$\displaystyle\quad\qquad+8M_{10}M_{12}r\sqrt{1-f}\left(24(1-f)^{3}-20rf_{r}(1-f)^{2}-3r^{3}f_{r}^{3}+10r^{2}(1-f)f_{r}^{2}\right)$
$\displaystyle\quad\qquad+192(1-f)^{2}M_{10}r^{2}\left(2\beta(1-f)-M_{\rm
Pl}^{2}\left(j-2-2rf_{r}\right)\right)\bigg{]}\,.$ (5.15)

The dependence on $j^{2}=\ell(\ell+1)$ of the sound speed is due to the
unknown coefficients $M_{10}$ and $M_{12}$. In a Horndeski theory with ${\cal
L}_{2}$ and ${\cal L}_{4}$ only,555We are using the convention that
$\displaystyle{\cal L}_{2}$ $\displaystyle=P(X),$ $\displaystyle{\cal L}_{4}$
$\displaystyle=G_{4}(X)R+G_{4,X}(X)\big{(}(\square\phi)^{2}-\phi_{\mu\nu}\phi^{\mu\nu}\big{)},$
(5.16) where $X=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$.  $P(X)$,
$G_{4}(X)$ and their derivatives evaluated on the background have to satisfy
the following relations on the Schwarzschild-de Sitter geometry,

$\displaystyle\bar{P}+2\lambda\big{(}\bar{G}_{4}-m^{4}\bar{G}_{4,X}\big{)}=0\,;$
$\displaystyle\bar{P}_{,X}+2\lambda\big{(}\bar{G}_{4,X}+m^{4}\bar{G}_{4,XX}\big{)}=0\,.$
(5.17)

After using these relations, one finds that $c_{s}^{2}$ coming from Horndeski
${\cal L}_{2}$ and ${\cal L}_{4}$ is simply

$\displaystyle c_{s}^{2}=\frac{\bar{G}_{4}}{\bar{G}_{4}-m^{4}\bar{G}_{4,X}}.$
(5.18)

Note that a quadratic DHOST gives the same sound speed in the odd sector [44].
The $\ell$ dependence in (5.1) will cancel out and give the exact same
expression as in Horndeski if one chooses $M_{12}=0$ and $M_{10}$ being the
corresponding coefficient from Horndeski ${\cal L}_{4}$. Phenomenologically
speaking, apart from the stability constraints (5.13), there is also stringent
observational restrictions on the propagation speed of gravitational waves
from neutron star mergers [52, 53, 54]. In such cases, the relevant
gravitational wave perturbation is the lowest $\ell=2$ mode, given the current
experimental signal-to-noise ratio.

##  6 Discussion

Effective field theory is a powerful tool for the study of perturbative
systems or of the macroscopic behavior of theories at low energies, which is a
typical situation that arises in many different areas of physics. The
robustness of EFT allows us to investigate a large class of theories, in our
case modified gravity theories, using a unified framework. Furthermore, it has
a wide range of applicability in extreme limits of gravitational systems, such
as in cosmological inflation and in black holes. One particularly interesting
application has been to scalar-tensor theories which, over the past several
decades, have drawn significant attention due to their rich phenomenological
applications in both early and late-time cosmology. While one well-studied
application has been to the physics of inflation, the technique can also be
used to study other interesting approaches to the early universe, including
those in which a time-dependent scalar profile drives cosmological expansion
through a conformal coupling with matter. Scalar profiles of this type provide
an interesting connection between cosmological evolution and astrophysical
black holes, which can emerge as allowed non-trivial solutions with time-like
hair is generic in scalar tensor theories.

In this work we have investigated the EFT of perturbations around such black
holes with time-like scalar hair. Building on the construction in [40], we
argue that the EFT coefficients must satisfy certain constraints imposed by
the background isometries. When the underlying scalar-tensor theory enjoys a
shift symmetry in $\phi$, the combination between this symmetry and time-
translation invariance (in Schwarszchild type coordinates) remains as an
unbroken symmetry, and constrains the EFT coefficient functions, despite the
fact that the time-like scalar profile spontaneously breaks both symmetries
individually. We have constructed a general set of operators, up to second-
order in derivatives, which are compatible with the aforementioned symmetries.
Motivated by the fact that gradient instabilities appear in the scalar sector
in both the Horndeski and DHOST theories [38, 44], we have performed a
stability analysis in the decoupling limit, which we argue should capture the
essential properties of scalar perturbations. In addition, we have shown that
odd sector perturbations can be comprehensively analyzed despite the
appearance of many unknown functions, since only a few terms in the action
contribute to odd-parity perturbations. This analysis also provides hints as
to how we might construct potentially stable black hole/wormhole solutions.

In order to complete the story, a full analysis of even-parity perturbations
is required. Such a task promises to be technically challenging, despite the
fact that a subset of the EFT operators (essentially those corresponding to
the quadratic DHOST terms, for instance $\delta K^{2}$) can be analyzed using
the technique in [44]. However, Inclusion of operators of the type
$\bar{K}_{\mu\nu}\delta K^{\rho\mu}\delta K^{\nu}_{\;\;\rho}$ would require
different techniques in order to identify the correct propagating degrees of
freedom, and to perform the stability analysis.

It is also worth commenting on higher-derivative terms in the EFT. The
inclusion of higher-derivative operators does not necessarily mean higher-
order equations of motion for perturbations. In fact, we expect that a similar
story to the DHOST analysis should play out for perturbations, so that a
degeneracy condition between coefficient functions of higher-derivative
operators can be imposed, reducing the equation of motion to second order.
Furthermore, it is possible for higher derivative operators to serve as
“Scordatura terms” [55, 56, 57], which contribute to a non-vanishing
dispersion relation in such a way as to solve the strong coupling problem for
some solutions.

In future work, we intend to extend this work to the case of rotating black
holes, which comprise the majority of astrophysical black holes observed
through current and future instruments. The EFT for slowly rotating black
holes with time-independent scalar hair has already been constructed [58],
while interesting background solutions for rotating black holes with time-like
hair have found in DHOST models [59, 60]. The EFT description of perturbations
around these objects remains to be constructed, and may be complicated by the
fact that the background construction relies on the use of disformal
transformations. It is an interesting future exercise to investigate whether
similar techniques to those used in this paper can be applied to perturbations
and the construction of the EFT in these rotating backgrounds.

Acknowledgements

While this paper was in preparation, we were aware of the research by
Mukohyama, Takahashi and Yingcharoenrat on a similar subject. Their paper
appeared concurrently on arXiv and we thank them for their correspondence. We
thank Hayato Motohashi, Luca Santoni, Kazufumi Takahashi and Enrico
Trincherini for useful discussions. This work is supported in part by the US
Department of Energy (HEP) Award DE-SC0013528, NASA ATP grant 80NSSC18K0694,
and by the Simons Foundation Origins of the Universe Initiative. T.N. is
supported in part by JSPS KAKENHI Grant No. 20H01902 and No. 22H01220, and
MEXT KAKENHI Grant No. 21H00075, No. 21H05184 and No. 21H05462.

##  Appendix A Background equations of motion

The tadpole terms of (3),

$\displaystyle S_{\rm tadpole}=\int{\rm
d}^{4}x\sqrt{-g}\Big{[}M_{1}R-\Lambda+\alpha
g^{\tau\tau}+\beta\bar{K}_{\mu\nu}K^{\mu\nu}\Big{]}\,,$ (A.1)

generate the background equation of motion,

$\displaystyle\left(\bar{R}_{\mu\nu}-\frac{1}{2}\bar{g}_{\mu\nu}\bar{R}-\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}+\bar{g}_{\mu\nu}\bar{\square}\right)M_{1}=\bar{T}_{\mu\nu}\,,$
(A.2)

where

$\displaystyle\bar{T}_{\mu\nu}$ $\displaystyle=\Big{(}\alpha
g^{\tau\tau}-\Lambda+\beta\bar{K}_{\rho}^{\;\sigma}\bar{K}_{\sigma}^{\;\rho}\Big{)}\bar{g}_{\mu\nu}-2\alpha\delta_{\mu}^{\tau}\delta_{\nu}^{\tau}-2\beta\bar{K}_{\mu}^{\;\rho}\bar{K}_{\rho\nu}+\beta\bar{K}_{\rho}^{\;\sigma}\bar{K}_{\sigma}^{\;\rho}n_{\mu}n_{\nu}$
$\displaystyle\qquad+\bar{\nabla}_{\lambda}\Big{(}\beta\bar{K}_{\mu}^{\;\lambda}n_{\nu}+\beta\bar{K}_{\nu}^{\;\lambda}n_{\mu}-\beta\bar{K}_{\mu\nu}n^{\lambda}\Big{)}\,.$
(A.3)

Here we collect the useful equations of motion evaluated on the ansatz (2.2),
with $r$-dependent coefficient functions and constant $M_{1}=M_{\rm Pl}^{2}$,

$\displaystyle\frac{\delta}{\delta g^{\tau\tau}}:$ $\displaystyle\quad
0=r^{2}(\alpha-\Lambda)+2M_{1}\frac{r(1-f)gf_{r}+f^{2}(1-g)+rfg_{r}}{f^{2}}\,;$
$\displaystyle\frac{\delta}{\delta g^{\tau\rho}}:$ $\displaystyle\quad
0=4M_{1}r\frac{(f-1)\left(gf_{r}-fg_{r}\right)}{f}+gr^{2}f_{r}\beta_{r}$
$\displaystyle\qquad\quad+\beta\left(\frac{1}{2}r^{2}\left(\frac{(2f-1)gf_{r}^{2}}{(1-f)f}+f_{r}g_{r}+2gf_{rr}\right)+2grf_{r}+4(1-f)g\right)\,;$
$\displaystyle\frac{\delta}{\delta g^{\rho\rho}}:$ $\displaystyle\quad
0=M_{1}\left(\frac{4r\left((1-f)fg_{r}-gf_{r}\right)}{f}+4f(1-g)\right)+r^{2}\left(gf_{r}\beta_{r}-2f(\alpha+\Lambda)\right)$
$\displaystyle\qquad\quad+\beta\left(\frac{1}{2}r^{2}\left(\frac{gf_{r}^{2}}{(f-1)f}+f_{r}g_{r}+2gf_{rr}\right)+2rgf_{r}+4(1-f)g\right)\,;$
$\displaystyle\frac{\delta}{\delta g^{\theta\theta}}\mbox{ or
}\frac{\delta}{\delta g^{\phi\phi}}:$ $\displaystyle\quad
0=M_{1}\left(2r^{2}\frac{ff_{r}g_{r}-gf_{r}^{2}+2fgf_{rr}}{f}+4r(f_{r}g+fg_{r})\right)+4r^{2}f(\alpha+\Lambda)$
$\displaystyle\qquad\quad+\beta\left(2r\frac{(1-f)fg_{r}-(1+f)gf_{r}}{f}+\frac{r^{2}gf_{r}^{2}}{(f-1)}+4g(1-f)\right)+4r\beta_{r}(1-f)g\,.$
(A.4)

These are used in our analysis to re-express $\Lambda$, $\alpha$, $\beta$ and
$\beta_{r}$ in terms of $f(r)$, $g(r)$ and their derivatives.

##  Appendix B The Stückelberg procedure

This Appendix contains the complete expressions for the appearance of the
Stückelberg field $\pi$.

$\displaystyle g^{\tau\tau}$ $\displaystyle\to
g^{\tau\tau}(1+2\dot{\pi}+\dot{\pi}^{2})+2g^{a\tau}(\partial_{a}\pi+\dot{\pi}\partial_{a}\pi)+g^{ab}\partial_{a}\pi\partial_{b}\pi,$
$\displaystyle g^{a\tau}$ $\displaystyle\to
g^{a\tau}(1+\dot{\pi})+g^{ab}\partial_{b}\pi,$ $\displaystyle g^{ab}$
$\displaystyle\to g^{ab}.$ (B.1)

The derivatives $\partial_{\mu}$ transform as

$\displaystyle\partial_{\tau}$
$\displaystyle\to(1-\dot{\pi}+\dot{\pi}^{2})\partial_{\tau}+{\cal
O}(\pi^{3})\,;$ $\displaystyle\partial_{i}$
$\displaystyle\to\partial_{i}-(1-\dot{\pi})\partial_{i}\pi\partial_{\tau}+{\cal
O}(\pi^{3})\,.$ (B.2)

To calculate the transformation of $\hat{R}_{ij}$ in (4), we have used

$\displaystyle\hat{R}_{ij}$
$\displaystyle\to\hat{R}_{ij}-\partial_{k}\pi\dot{\hat{\Gamma}}^{k}_{ij}+\partial_{i}\pi\dot{\hat{\Gamma}}^{k}_{jk}+D_{k}\left(\delta\hat{\Gamma}^{k}_{ij}\right)-D_{i}\left(\delta\hat{\Gamma}^{k}_{kj}\right)$
$\displaystyle D_{k}\dot{h}_{li}$
$\displaystyle=\dot{\hat{\Gamma}}^{m}_{li}h_{mi}+\dot{\hat{\Gamma}}^{m}_{ki}h_{lm}\,.$
(B.3)

##  Appendix C Decoupling limit in Schwarszchild-de Sitter

It is convenient to collect the results in section (4.1) restricted to the
Schwarszchild-de Sitter geometry. In this case the metric is given by (2.1)
with $f(r)=g(r)=1-\frac{r_{s}}{r}-\frac{\lambda}{3}r^{2}$, where
$0<\lambda<\frac{4}{9r_{s}^{2}}$ for two horizons $r_{-},r_{+}$ to exist, and
the inner horizon $r_{-}$ satisfies $1<\frac{r_{-}}{r_{s}}<\frac{3}{2}$. The
individual kinetic terms for $\pi$ are:

    * –

$M_{3}(\tau,\rho)\,\delta g^{\tau\tau}\delta K$:

$\displaystyle\frac{2M_{3,r}}{\sqrt{F}}\dot{\pi}\pi^{\prime}+\frac{Fr^{2}M_{3,r}+M_{3}(3r_{s}-2rF)}{r^{2}F^{3/2}}\pi^{\prime
2}+\frac{\sqrt{F}\left(rM_{3,r}-2M_{3}\right)}{r^{3}}\gamma^{AB}\partial_{A}\pi\partial_{B}\pi$
(C.1)

where $F(r)=\frac{r_{s}}{r}+\frac{\lambda}{3}r^{2}$, and
$\left(\frac{3}{2}\right)^{2/3}r_{s}^{2/3}\lambda^{1/3}<F<1$ between two
horizons.

    * –

$M_{4}(\tau,\rho)\bar{K}_{\mu\nu}\delta g^{\tau\tau}\delta K^{\mu\nu}$:

$\displaystyle\quad\frac{2Fr^{2}(3r_{s}-2Fr)M_{4,r}+9r_{s}^{2}M_{4}}{2F^{2}r^{4}}\dot{\pi}\pi^{\prime}+\frac{M_{4}\left(8F^{2}r^{2}-36Frr_{s}+27r_{s}^{2}\right)+2Fr^{2}(3r_{s}-2Fr)M_{4,r}}{4F^{2}r^{4}}\pi^{\prime
2}$
$\displaystyle+\frac{(4Fr+3r_{s})M_{4}-2Fr^{2}M_{4,r}}{2r^{5}}\gamma^{AB}\partial_{A}\pi\partial_{B}\pi$
(C.2)

    * –

$M_{9}\left(\delta K^{2}-\delta K_{\mu\nu}\delta K^{\mu\nu}\right)$:

$\displaystyle\quad\frac{M_{9}\left(-4F^{2}r^{2}+18Frr_{s}-9r_{s}^{2}\right)-2Fr^{3}M_{9,r}}{F^{2}r^{4}}\pi^{\prime
2}$
$\displaystyle-\left(\frac{M_{9}(4Fr-3r_{s})(2Fr+3r_{s})}{2Fr^{6}}+\frac{rM_{9,rr}+M_{9,r}}{r^{3}}\right)\gamma^{AB}\partial_{A}\pi\partial_{B}\pi$
(C.3)

    * –

$M_{11}\left(\bar{K}_{\mu\nu}\delta K\delta K^{\mu\nu}-\bar{K}_{\mu\nu}\delta
K^{\rho\mu}\delta K^{\nu}_{\;\;\rho}\right)$:

$\displaystyle\quad\frac{2Fr^{3}(4Fr-3r_{s})M_{11,r}-3M_{11}\left(8F^{3}r^{3}-20F^{2}r^{2}r_{s}+3(8F+1)rr_{s}^{2}-9r_{s}^{3}\right)}{4F^{5/2}r^{6}}\pi^{\prime
2}$
$\displaystyle+\frac{1}{16F^{5/2}r^{8}}\Big{[}4Fr^{2}\left(\left(4F^{2}r^{2}-9r_{s}^{2}\right)M_{11,r}+Fr^{2}(4Fr-3r_{s})M_{11,rr}\right)$
$\displaystyle+M_{11}\left(-96F^{4}r^{3}+96F^{3}r^{2}r_{s}+36(3-2F)Frr_{s}^{2}+27(2F-3)r_{s}^{3}\right)\Big{]}\gamma^{AB}\partial_{A}\pi\partial_{B}\pi$
(C.4)

    * –

$M_{13}(\tau,\rho)\delta g^{\tau\tau}\delta\hat{R}$:

$\displaystyle\quad\frac{8M_{13,r}}{r}\dot{\pi}\pi^{\prime}+\left[\frac{M_{13}(6r_{s}-8Fr)}{Fr^{3}}+\frac{4M_{13,r}}{r}\right]\pi^{\prime
2}$
$\displaystyle+\left[\frac{(4Fr-3r_{s})}{r^{4}}M_{13,r}+M_{13}\left(-\frac{9r_{s}^{2}}{2Fr^{6}}-\frac{8F}{r^{4}}+\frac{9r_{s}}{r^{5}}\right)\right]\gamma^{AB}\partial_{A}\pi\partial_{B}\pi$
(C.5)

    * –

$M_{14}(\tau,\rho)\bar{K}_{\mu\nu}\delta g^{\tau\tau}\delta\hat{R}^{\mu\nu}$:

$\displaystyle\quad\frac{9r_{s}^{2}\left(2M_{14}+rM_{14,r}\right)-4F^{2}r^{3}M_{14,r}}{2F^{3/2}r^{5}}\dot{\pi}\pi^{\prime}$
$\displaystyle+\frac{M_{14}\left(8F^{2}r^{2}-15Frr_{s}+9r_{s}^{2}\right)+Fr^{2}(3r_{s}-4Fr)M_{14,r}}{F^{3/2}r^{5}}\pi^{\prime
2}$
$\displaystyle+\frac{Fr^{2}\left(-16F^{2}r^{2}+18Frr_{s}-9r_{s}^{2}\right)M_{14,r}+M_{14}\left(32F^{3}r^{3}-42F^{2}r^{2}r_{s}+72Frr_{s}^{2}-27r_{s}^{3}\right)}{4F^{3/2}r^{8}}\gamma^{AB}\partial_{A}\pi\partial_{B}\pi$
(C.6)

## References

    * [1] LIGO Scientific, Virgo , B. Abbott et al., “Binary Black Hole Mergers in the first Advanced LIGO Observing Run,” Phys. Rev. X 6 (2016) no. 4, 041015, arXiv:1606.04856 [gr-qc]. [Erratum: Phys.Rev.X 8, 039903 (2018)].
    * [2] Event Horizon Telescope , K. Akiyama et al., “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. 875 (2019) no. 1, L1, arXiv:1906.11238 [astro-ph.GA].
    * [3] T. Regge and J. A. Wheeler, “Stability of a Schwarzschild singularity,” Phys. Rev. 108 (1957) 1063–1069.
    * [4] F. J. Zerilli, “Effective potential for even parity Regge-Wheeler gravitational perturbation equations,” Phys. Rev. Lett. 24 (1970) 737–738.
    * [5] E. Berti, V. Cardoso, and A. O. Starinets, “Quasinormal modes of black holes and black branes,” Class. Quant. Grav. 26 (2009) 163001, arXiv:0905.2975 [gr-qc].
    * [6] R. P. Woodard, “Ostrogradsky’s theorem on Hamiltonian instability,” Scholarpedia 10 (2015) no. 8, 32243, arXiv:1506.02210 [hep-th].
    * [7] A. R. Solomon and M. Trodden, “Higher-derivative operators and effective field theory for general scalar-tensor theories,” JCAP 02 (2018) 031, arXiv:1709.09695 [hep-th].
    * [8] G. W. Horndeski, “Second-order scalar-tensor field equations in a four-dimensional space,” Int. J. Theor. Phys. 10 (1974) 363–384.
    * [9] C. Deffayet, S. Deser, and G. Esposito-Farese, “Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress-tensors,” Phys. Rev. D 80 (2009) 064015, arXiv:0906.1967 [gr-qc].
    * [10] C. Deffayet, X. Gao, D. Steer, and G. Zahariade, “From k-essence to generalised Galileons,” Phys. Rev. D 84 (2011) 064039, arXiv:1103.3260 [hep-th].
    * [11] T. Kobayashi, M. Yamaguchi, and J. Yokoyama, “Generalized G-inflation: Inflation with the most general second-order field equations,” Prog. Theor. Phys. 126 (2011) 511–529, arXiv:1105.5723 [hep-th].
    * [12] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, “Healthy theories beyond Horndeski,” Phys. Rev. Lett. 114 (2015) no. 21, 211101, arXiv:1404.6495 [hep-th].
    * [13] D. Langlois and K. Noui, “Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability,” JCAP 02 (2016) 034, arXiv:1510.06930 [gr-qc].
    * [14] M. Crisostomi, K. Koyama, and G. Tasinato, “Extended Scalar-Tensor Theories of Gravity,” JCAP 04 (2016) 044, arXiv:1602.03119 [hep-th].
    * [15] J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui, and G. Tasinato, “Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order,” JHEP 12 (2016) 100, arXiv:1608.08135 [hep-th].
    * [16] K. Takahashi and T. Kobayashi, “Extended mimetic gravity: Hamiltonian analysis and gradient instabilities,” JCAP 11 (2017) 038, arXiv:1708.02951 [gr-qc].
    * [17] D. Langlois, M. Mancarella, K. Noui, and F. Vernizzi, “Mimetic gravity as DHOST theories,” JCAP 02 (2019) 036, arXiv:1802.03394 [gr-qc].
    * [18] T. P. Sotiriou and S.-Y. Zhou, “Black hole hair in generalized scalar-tensor gravity,” Phys. Rev. Lett. 112 (2014) 251102, arXiv:1312.3622 [gr-qc].
    * [19] T. P. Sotiriou and S.-Y. Zhou, “Black hole hair in generalized scalar-tensor gravity: An explicit example,” Phys. Rev. D 90 (2014) 124063, arXiv:1408.1698 [gr-qc].
    * [20] E. Babichev, C. Charmousis, and A. Lehébel, “Black holes and stars in Horndeski theory,” Class. Quant. Grav. 33 (2016) no. 15, 154002, arXiv:1604.06402 [gr-qc].
    * [21] R. Benkel, T. P. Sotiriou, and H. Witek, “Black hole hair formation in shift-symmetric generalised scalar-tensor gravity,” Class. Quant. Grav. 34 (2017) no. 6, 064001, arXiv:1610.09168 [gr-qc].
    * [22] E. Babichev, C. Charmousis, and A. Lehébel, “Asymptotically flat black holes in Horndeski theory and beyond,” JCAP 04 (2017) 027, arXiv:1702.01938 [gr-qc].
    * [23] A. Lehébel, E. Babichev, and C. Charmousis, “A no-hair theorem for stars in Horndeski theories,” JCAP 07 (2017) 037, arXiv:1706.04989 [gr-qc].
    * [24] M. Minamitsuji and H. Motohashi, “Stealth Schwarzschild solution in shift symmetry breaking theories,” Phys. Rev. D 98 (2018) no. 8, 084027, arXiv:1809.06611 [gr-qc].
    * [25] J. Ben Achour, H. Liu, and S. Mukohyama, “Hairy black holes in DHOST theories: Exploring disformal transformation as a solution-generating method,” JCAP 02 (2020) 023, arXiv:1910.11017 [gr-qc].
    * [26] M. Minamitsuji and J. Edholm, “Black holes with a nonconstant kinetic term in degenerate higher-order scalar tensor theories,” Phys. Rev. D 101 (2020) no. 4, 044034, arXiv:1912.01744 [gr-qc].
    * [27] T. Kobayashi, H. Motohashi, and T. Suyama, “Black hole perturbation in the most general scalar-tensor theory with second-order field equations I: the odd-parity sector,” Phys. Rev. D 85 (2012) 084025, arXiv:1202.4893 [gr-qc]. [Erratum: Phys.Rev.D 96, 109903 (2017)].
    * [28] T. Kobayashi, H. Motohashi, and T. Suyama, “Black hole perturbation in the most general scalar-tensor theory with second-order field equations II: the even-parity sector,” Phys. Rev. D 89 (2014) no. 8, 084042, arXiv:1402.6740 [gr-qc].
    * [29] G. Franciolini, L. Hui, R. Penco, L. Santoni, and E. Trincherini, “Effective Field Theory of Black Hole Quasinormal Modes in Scalar-Tensor Theories,” JHEP 02 (2019) 127, arXiv:1810.07706 [hep-th].
    * [30] E. Babichev and C. Charmousis, “Dressing a black hole with a time-dependent Galileon,” JHEP 08 (2014) 106, arXiv:1312.3204 [gr-qc].
    * [31] T. Kobayashi and N. Tanahashi, “Exact black hole solutions in shift symmetric scalar–tensor theories,” PTEP 2014 (2014) 073E02, arXiv:1403.4364 [gr-qc].
    * [32] E. Babichev and G. Esposito-Farese, “Cosmological self-tuning and local solutions in generalized Horndeski theories,” Phys. Rev. D 95 (2017) no. 2, 024020, arXiv:1609.09798 [gr-qc].
    * [33] E. Babichev, C. Charmousis, G. Esposito-Farèse, and A. Lehébel, “Stability of Black Holes and the Speed of Gravitational Waves within Self-Tuning Cosmological Models,” Phys. Rev. Lett. 120 (2018) no. 24, 241101, arXiv:1712.04398 [gr-qc].
    * [34] J. Ben Achour and H. Liu, “Hairy Schwarzschild-(A)dS black hole solutions in degenerate higher order scalar-tensor theories beyond shift symmetry,” Phys. Rev. D 99 (2019) no. 6, 064042, arXiv:1811.05369 [gr-qc].
    * [35] H. Motohashi and M. Minamitsuji, “Exact black hole solutions in shift-symmetric quadratic degenerate higher-order scalar-tensor theories,” Phys. Rev. D 99 (2019) no. 6, 064040, arXiv:1901.04658 [gr-qc].
    * [36] K. Takahashi, H. Motohashi, and M. Minamitsuji, “Linear stability analysis of hairy black holes in quadratic degenerate higher-order scalar-tensor theories: Odd-parity perturbations,” Phys. Rev. D 100 (2019) no. 2, 024041, arXiv:1904.03554 [gr-qc].
    * [37] M. Minamitsuji and J. Edholm, “Black hole solutions in shift-symmetric degenerate higher-order scalar-tensor theories,” Phys. Rev. D 100 (2019) no. 4, 044053, arXiv:1907.02072 [gr-qc].
    * [38] J. Khoury, M. Trodden, and S. S. C. Wong, “Existence and instability of hairy black holes in shift-symmetric Horndeski theories,” JCAP 11 (2020) 044, arXiv:2007.01320 [astro-ph.CO].
    * [39] N. Arkani-Hamed, P. Creminelli, S. Mukohyama, and M. Zaldarriaga, “Ghost inflation,” JCAP 04 (2004) 001, arXiv:hep-th/0312100.
    * [40] S. Mukohyama, “Black holes in the ghost condensate,” Phys. Rev. D 71 (2005) 104019, arXiv:hep-th/0502189.
    * [41] H. Ogawa, T. Kobayashi, and T. Suyama, “Instability of hairy black holes in shift-symmetric Horndeski theories,” Phys. Rev. D 93 (2016) no. 6, 064078, arXiv:1510.07400 [gr-qc].
    * [42] E. Babichev, C. Charmousis, G. Esposito-Farèse, and A. Lehébel, “Hamiltonian unboundedness vs stability with an application to Horndeski theory,” Phys. Rev. D 98 (2018) no. 10, 104050, arXiv:1803.11444 [gr-qc].
    * [43] C. de Rham and J. Zhang, “Perturbations of stealth black holes in degenerate higher-order scalar-tensor theories,” Phys. Rev. D 100 (2019) no. 12, 124023, arXiv:1907.00699 [hep-th].
    * [44] K. Takahashi and H. Motohashi, “Black hole perturbations in DHOST theories: master variables, gradient instability, and strong coupling,” JCAP 08 (2021) 013, arXiv:2106.07128 [gr-qc].
    * [45] M. Minamitsuji, K. Takahashi, and S. Tsujikawa, “Linear stability of black holes in shift-symmetric Horndeski theories with a time-independent scalar field,” Phys. Rev. D 105 (2022) no. 10, 104001, arXiv:2201.09687 [gr-qc].
    * [46] M. Minamitsuji, K. Takahashi, and S. Tsujikawa, “Linear stability of black holes with static scalar hair in full Horndeski theories: generic instabilities and surviving models,” arXiv:2204.13837 [gr-qc].
    * [47] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan, and L. Senatore, “The Effective Field Theory of Inflation,” JHEP 03 (2008) 014, arXiv:0709.0293 [hep-th].
    * [48] B. Finelli, G. Goon, E. Pajer, and L. Santoni, “The Effective Theory of Shift-Symmetric Cosmologies,” JCAP 05 (2018) 060, arXiv:1802.01580 [hep-th].
    * [49] S. Mukohyama and V. Yingcharoenrat, “Effective Field Theory of Black Hole Perturbations with Timelike Scalar Profile: Formulation,” arXiv:2204.00228 [hep-th].
    * [50] N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama, “Ghost condensation and a consistent infrared modification of gravity,” JHEP 05 (2004) 074, arXiv:hep-th/0312099.
    * [51] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, “Essential Building Blocks of Dark Energy,” JCAP 08 (2013) 025, arXiv:1304.4840 [hep-th].
    * [52] LIGO Scientific, Virgo , B. Abbott et al., “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys. Rev. Lett. 119 (2017) no. 16, 161101, arXiv:1710.05832 [gr-qc].
    * [53] LIGO Scientific, Virgo, Fermi GBM, INTEGRAL, IceCube, AstroSat Cadmium Zinc Telluride Imager Team, IPN, Insight-Hxmt, ANTARES, Swift, AGILE Team, 1M2H Team, Dark Energy Camera GW-EM, DES, DLT40, GRAWITA, Fermi-LAT, ATCA, ASKAP, Las Cumbres Observatory Group, OzGrav, DWF (Deeper Wider Faster Program), AST3, CAASTRO, VINROUGE, MASTER, J-GEM, GROWTH, JAGWAR, CaltechNRAO, TTU-NRAO, NuSTAR, Pan-STARRS, MAXI Team, TZAC Consortium, KU, Nordic Optical Telescope, ePESSTO, GROND, Texas Tech University, SALT Group, TOROS, BOOTES, MWA, CALET, IKI-GW Follow-up, H.E.S.S., LOFAR, LWA, HAWC, Pierre Auger, ALMA, Euro VLBI Team, Pi of Sky, Chandra Team at McGill University, DFN, ATLAS Telescopes, High Time Resolution Universe Survey, RIMAS, RATIR, SKA South Africa/MeerKAT , B. Abbott et al., “Multi-messenger Observations of a Binary Neutron Star Merger,” Astrophys. J. Lett. 848 (2017) no. 2, L12, arXiv:1710.05833 [astro-ph.HE].
    * [54] LIGO Scientific, Virgo, Fermi-GBM, INTEGRAL , B. Abbott et al., “Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A,” Astrophys. J. Lett. 848 (2017) no. 2, L13, arXiv:1710.05834 [astro-ph.HE].
    * [55] H. Motohashi and S. Mukohyama, “Weakly-coupled stealth solution in scordatura degenerate theory,” JCAP 01 (2020) 030, arXiv:1912.00378 [gr-qc].
    * [56] M. A. Gorji, H. Motohashi, and S. Mukohyama, “Stealth dark energy in scordatura DHOST theory,” JCAP 03 (2021) 081, arXiv:2009.11606 [gr-qc].
    * [57] A. De Felice, S. Mukohyama, and K. Takahashi, “Built-in scordatura in U-DHOST,” Phys. Rev. Lett. 129 (2022) 031103, arXiv:2204.02032 [gr-qc].
    * [58] L. Hui, A. Podo, L. Santoni, and E. Trincherini, “Effective Field Theory for the perturbations of a slowly rotating black hole,” JHEP 12 (2021) 183, arXiv:2111.02072 [hep-th].
    * [59] C. Charmousis, M. Crisostomi, R. Gregory, and N. Stergioulas, “Rotating Black Holes in Higher Order Gravity,” Phys. Rev. D 100 (2019) no. 8, 084020, arXiv:1903.05519 [hep-th].
    * [60] J. Ben Achour, H. Liu, H. Motohashi, S. Mukohyama, and K. Noui, “On rotating black holes in DHOST theories,” JCAP 11 (2020) 001, arXiv:2006.07245 [gr-qc].


configuration of ${\mathbb{N}}^{\mathbb{Z}^{d}}$, it follows that
$\check{\nu}(A_{\eta})=0$ for every
$\eta\in{\mathbb{N}}^{{\mathfrak{Q}}_{N}}$, where $\check{\nu}$ is the product
of $\nu$’s over ${\mathbb{N}}^{\mathbb{Z}^{d}\setminus{\mathfrak{Q}}_{N}}$.
Thus, resorting to the coupling in the previous paragraph, we find that

${\mathbf{P}}^{\mathsf{z}}_{\hat{\mu}_{0}}\big{(}\varpi^{\mathsf{z}}\in
A\big{)}\leq{\mathbf{P}}^{\mathsf{z}}_{\hat{\mu}_{0}}\big{(}\omega^{\mathsf{z}}_{\text{eq}}\in
A\big{)}+{\mathbf{P}}^{\mathsf{z}}_{\hat{\mu}_{0}}\big{(}{\check{\omega}}^{\mathsf{z}}\neq{\check{\omega}}^{\mathsf{z}}_{\text{eq}}\big{)}={\mathbf{P}}^{\mathsf{z}}_{\hat{\mu}_{0}}\big{(}{\check{\omega}}^{\mathsf{z}}\neq{\check{\omega}}^{\mathsf{z}}_{\text{eq}}\big{)},$
(5.36)

since the first probability on the right hand side equals

$\sum_{\eta\in{\mathbb{N}}^{{\mathfrak{Q}}_{N}}}\check{\nu}\big{(}A_{\eta}\big{)}{\mathbf{P}}^{\mathsf{z}}_{\hat{\mu}_{0}}\big{(}\hat{\varpi}^{\mathsf{z}}=\eta\big{)}=0,$
(5.37)

as follows from what was pointed out above in this paragraph. Since the right
hand side of (5.36) vanishes as $N\to\infty$, as pointed out at the end of the
one but previous paragraph, and the left hand side does not depend on $N$, we
have that
${\mathbf{P}}^{\mathsf{z}}_{\hat{\mu}_{0}}\big{(}\varpi^{\mathsf{z}}\in
A\big{)}=0$, as we set out to prove.

##  Appendix A Equivalence of (3.2) and (3.3)

The formulas we will present below may be found in the literature, either
explicitly or from other explicit formulas [17]. For this reason, but also in
an attempt of self-containment, we will deduce them, rather briefly and
sketchily, trusting the reader to be readily able to fill details in (or go to
the literature).

For $n\in{\mathbb{N}}_{*}$, let ${\mathcal{T}}_{n}$ and ${\mathcal{S}}_{n}$
denote ${\mathbb{E}}_{n}(T_{n-1})$ and ${\mathbb{E}}_{n}(T^{2}_{n-1})$,
respectively. Under ${\mathbb{P}}_{n}$, by the Markov property, we have that

$T_{n-1}=1+\mathbb{1}\\{\mathsf{w}_{1}=n+1\\}(T_{n}^{\prime}+T_{n}^{\prime\prime}),$
(A.1)

where $\mathbb{1}\\{\mathsf{w}_{1}=n+1\\},T_{n}^{\prime},T_{n}^{\prime\prime}$
are independent, $T_{n}^{\prime}$ has the same distribution as $T_{n}$ and
$T_{n}^{\prime\prime}$ has the same distribution as $T_{n+1}$ under
${\mathbb{P}}_{n+1}$. It follows that for $n\geq 1$

${\mathcal{T}}_{n}=\frac{1}{q_{n}}+\rho_{n}{\mathcal{T}}_{n+1}\quad\mbox{and}\quad{\mathcal{S}}_{n}=\sigma_{n}+\rho_{n}{\mathcal{S}}_{n+1},$
(A.2)

where $\sigma_{n}=s_{n}/q_{n}$, and
$s_{n}=1+2p_{n}({\mathcal{T}}_{n}+{\mathcal{T}}_{n+1}+{\mathcal{T}}_{n}{\mathcal{T}}_{n+1})$.171717The
finitude of ${\mathcal{T}}_{n}$ follows from the ergodicity of $\mathsf{w}$;
that of ${\mathcal{S}}_{n}$ may be checked by comparison to versions of
$\mathsf{w}$ reflected at large states, using a similar reasoning as in the
present argument for the reflected versions of ${\mathcal{S}}_{n}$, which are
more obviously finite. It then readily follows that

${\mathcal{T}}_{n}=\frac{1}{R_{n-1}}\sum_{\ell\geq
n}\frac{1}{q_{\ell}}R_{\ell-1}\quad\mbox{and}\quad{\mathcal{S}}_{1}=\sum_{\ell\geq
1}{\sigma_{\ell}}R_{\ell-1}.$ (A.3)

Our condiitons on $\mathbf{p},\mathbf{q}$ imply that
${\mathcal{T}}_{n}\asymp\frac{S_{n-1}}{R_{n-1}}$ and
$\sigma_{n}\asymp{\mathcal{T}}_{n}{\mathcal{T}}_{n+1}$. It follows that

${\mathcal{S}}_{1}\asymp\sum_{\ell\geq
1}\frac{S_{\ell-1}S_{\ell}}{R_{\ell}}\asymp\sum_{\ell\geq
1}\frac{S^{2}_{\ell}}{R_{\ell}},$ (A.4)

and the second equivalence is established. For the first one, we have that
${\mathbb{E}}_{\nu}(T_{0})<\infty$ if and only if

$\infty>\sum_{n\geq 0}R_{n}\sum_{i=0}^{n}\frac{S_{i}}{R_{i}}=\sum_{i\geq
0}\frac{S^{2}_{i}}{R_{i}},$ (A.5)

completing the argument.

##  Appendix B Justification of (5.19) and (5.20)

Notice that the distribution of $\varrho_{\ell}$ depends on $\ell$ only
through the sign of $\mathsf{y}_{\vec{\vartheta}_{\ell-1}}$ and equals that of
$\mathbf{T}^{-}_{0}$, if $\mathsf{y}_{\vec{\vartheta}_{\ell-1}}>0$, and that
of $\mathbf{T}^{+}_{0}$, if $\mathsf{y}_{\vec{\vartheta}_{\ell-1}}<0$, where
for $L\geq 0$

$\mathbf{T}^{-}_{L}:=\inf\\{n\geq 1:\,\mathsf{y}_{n}<-L\\}\,\text{ and
}\,\mathbf{T}^{+}_{L}:=\inf\\{n\geq 1:\,\mathsf{y}_{n}>L\\}.$ (B.1)

Also, given $\vec{\vartheta}_{\ell-1}=J$ and
$\mathsf{y}_{\vec{\vartheta}_{\ell-1}}=K>0$, we have that for $m\geq 1$

$\\{\chi_{\ell}>m\\}\subset\big{\\{}\mathsf{y}_{\reflectbox{$\vec{\reflectbox{$\vartheta$}}$}_{\ell-1}}-K>m\big{\\}}\cup\left\\{\big{\\{}\mathsf{y}_{\reflectbox{$\vec{\reflectbox{$\vartheta$}}$}_{\ell-1}}\in(-K,K)\big{\\}}\cap\Big{\\{}\big{\\{}\chi^{\prime}_{\ell}>m\big{\\}}\cup\big{\\{}\chi^{\prime\prime}_{\ell}>m\big{\\}}\Big{\\}}\right\\},$
(B.2)

where
$\chi^{\prime}_{\ell}=\mathsf{y}_{\varrho^{\prime}_{\ell}}-\Upsilon_{\ell}$,
with
$\varrho^{\prime}_{\ell}=\inf\\{n>\reflectbox{$\vec{\reflectbox{$\vartheta$}}$}_{\ell-1}:\,\mathsf{y}_{n}>\Upsilon_{\ell}\\}$,
and
$\chi^{\prime\prime}_{\ell}=|\mathsf{y}_{\varrho^{\prime\prime}_{\ell}}|-\Upsilon_{\ell}$,
with
$\varrho^{\prime\prime}_{\ell}=\inf\\{n>\reflectbox{$\vec{\reflectbox{$\vartheta$}}$}_{\ell-1}:\,\mathsf{y}_{n}<-\Upsilon_{\ell}\\}$.

A similar reasoning holds in the case where $K<0$, and it readily follows
that, in order to justify (5.19) and (5.20), it is enough to show that

$\displaystyle{\mathbf{P}}(\mathbf{T}^{+}_{0}>m),\,{\mathbf{P}}(\mathbf{T}^{-}_{0}>m)\geq\frac{\text{const}}{\sqrt{m}};$
(B.3) $\displaystyle\sup_{L\geq
0}{\mathbf{P}}\big{(}\mathsf{y}_{\mathbf{T}^{+}_{L}}-L>m\big{)},\,\sup_{L\geq
0}{\mathbf{P}}\big{(}|\mathsf{y}_{\mathbf{T}^{-}_{L}}|-L>m\big{)}\leq\frac{\text{const}}{m^{\varepsilon}}.$
(B.4)

(B.4) follows from the known, stronger result that both expressions in it are
asymptotic to const$/\sqrt{m}$ as $m\to\infty$ — see e.g. [12], Theorem 1, and
the remark which follows its statement.

Having not found a direct reference for (B.4), which we suspect may exist, we
develop a strategy based on material of [21], wherefrom that claim may be
argued, as follows.

Our argument goes through an adaptation of a hint to an exercise proposed in
Chapter IV of [21], namely, Exercise 6 in that chapter. The hint, describing
an approach attributed by the author to Harry Kesten (without further
reference) to give a bound on the moments of what in our notation is
$\mathsf{y}_{\mathbf{T}^{+}_{L}}$, can be adapted to yield the following. Let
us argue the former case of (B.4); the latter case is similar.

Following Kesten’s approach, as outlined in the above mentioned hint, we may
write

$\displaystyle{\mathbf{P}}\big{(}\mathsf{y}_{\mathbf{T}^{+}_{L}}-L>m\big{)}$
$\displaystyle=$ $\displaystyle\sum_{k>m}\sum_{z\leq
L}\,g_{(L,\infty)}(0,z){\mathbf{P}}(\xi^{\prime}_{1}=L-z+k)$ (B.5)
$\displaystyle=$ $\displaystyle\sum_{z\leq
L}\,g_{(L,\infty)}(0,z){\mathbf{P}}(\xi^{\prime}_{1}>L-z+m),$

where $g_{(L,\infty)}(0,z)$ is the expected number of visits of $\mathsf{y}$
to $z$ before going above $L$ for the first time.

We claim that

$g_{(L,\infty)}(0,z)\leq\text{const }\\{(L-z+1)\wedge(L+1)\\},\,z\leq L.$
(B.6)

It follows that
${\mathbf{P}}\big{(}\mathsf{y}_{\mathbf{T}^{+}_{L}}-L>m\big{)}$ is bounded
above by constant times

$(L+1)\sum_{w\geq L}{\mathbf{P}}(\xi^{\prime}_{1}>w+m)+\sum_{w\geq
0}(w+1){\mathbf{P}}(\xi^{\prime}_{1}>w+m),$ (B.7)

and the $2+\varepsilon$ moment condition on $\xi_{1}$ readily implies the
former case of (B.4).

To justify (B.6), we first notice that, by a standard use of the Markov
property, we have that, for $0<z\leq L$,

$g_{(L,\infty)}(0,z)\leq\sup_{w\leq 0}g_{(L-z+1,\infty)}(0,w),$ (B.8)

and for $z\leq 0$

$g_{(L,\infty)}(0,z)\leq g_{(0,\infty)}(0,z)+\sup_{w<0}g_{(L-1,\infty)}(0,w).$
(B.9)

Setting ${\mathcal{G}}_{k}=\sup_{w\leq 0}g_{(k,\infty)}(0,w)$, $k\geq 0$, we
find from (B.8,B.9) that

${\mathcal{G}}_{L}\leq(L+1){\mathcal{G}}_{0}\,\text{ and, for }0<z\leq
L,\,g_{(L,\infty)}(0,z)\leq(L-z+1){\mathcal{G}}_{0}.$ (B.10)

It remains thus to show that

${\mathcal{G}}_{0}<\infty.$ (B.11)

In order to do that, we proceed as follows. Let $\Theta_{0}=0$ and, for $n\geq
1$, $\Theta_{n}=\inf\\{\ell>\Theta_{n-1}:\,\mathsf{y}_{\ell}=0\\}$.
$(\Theta_{n})_{n\geq 0}$ are the successive passages of $\mathsf{y}$ by the
origin. They are a.s. all finite under our conditions. Now for $n\geq 1$ and
$z<0$, let
$N_{n}(z)=\sum_{i=\Theta_{n-1}}^{\Theta_{n}}\mathbb{1}\\{\mathsf{y}_{i}=z\\}$
be the number of visits of $\mathsf{y}$ to $z$ between times $\Theta_{n-1}$
and $\Theta_{n}$. Finally, set $\tau^{+}=\inf\\{n\geq
0:\,\mathsf{y}_{\Theta_{n}+1}>0\\}$.

We then have for $z<0$ that

$g_{(0,\infty)}(0,z)\leq{\mathbf{E}}_{0}\Big{\\{}\sum_{n=1}^{\tau^{+}}N_{n}(z)\Big{\\}},$
(B.12)

where $\sum_{n=1}^{0}\cdots=0$ by convention. One can readily check that the
latter expectation equals

${\mathbf{E}}_{0}(\tau^{+})\,{\mathbf{E}}_{0}\big{(}N_{1}(z)|\xi^{\prime}_{1}<0\big{)}.$
(B.13)

Now, from the identity

${\mathbf{E}}_{0}\big{(}N_{1}(z)|\xi^{\prime}_{1}<0\big{)}{\mathbf{P}}_{0}(\xi^{\prime}_{1}<0)+{\mathbf{E}}_{0}\big{(}N_{1}(z)|\xi^{\prime}_{1}>0\big{)}{\mathbf{P}}_{0}(\xi^{\prime}_{1}>0)={\mathbf{E}}_{0}\big{(}N_{1}(z)\big{)}=1,$
(B.14)

where the last equality follows from an application of Proposition 3 in
Chapter 3 of [21], Section 11, we find that the latter expectation in (B.13)
is bounded above by $1/{\mathbf{P}}_{0}(\xi^{\prime}_{1}<0)<\infty$.

The former expectation in (B.13) is quite clearly finite;
$g_{(0,\infty)}(0,0)$ is readily seen to be bounded above by that same
expectation, and (B.11) follows.

##  Appendix C Bounding the upper tail of an auxiliary random variable

######  Lemma C.1.

Let $\omega$ be a $BDP(p,q)$ with $p<\frac{1}{2}$ starting from the
equilibrium distribution, and let $T$ be a random variable, independent of
$\omega$, distributed as the sum of $L^{3}$ iid exponential random variables
of rate $\rho>0$, with $L\geq 1$. Set $M=\max_{0\leq t\leq T}\omega(t)$. Then

$P\big{(}M>(\log L)^{2}\big{)}\leq e^{-c(\log L)^{2}}$ (C.1)

for all large enough $L$, where $c$ is a positive constant.

###### Proof.

We may assume, using the upper bound on $\varphi$, and resorting to standard
large deviation estimates, that $N:=$ the number of jumps of $\omega$ up to
$T$ is bounded above by $H:=\lceil c^{\prime}L^{3}\rceil$, where $c^{\prime}$
is a finite constant.

Let now $W$ be a random variable distributed as the equilibrium distribution
of $\omega$, and, independently, consider $Y_{1},Y_{2},\ldots$, independent
copies of the maximum of a discrete time simple random walk on ${\mathbb{Z}}$
with probability of jumping to the left given by $p$ and starting at the
origin. One now readily checks that $M\preceq W+(Y_{1}\vee\cdots\vee Y_{H})$.
Thus

$P\big{(}M>(\log L)^{2}\big{)}\leq P\big{(}W>(\log
L)^{2}/2\big{)}+HP\big{(}Y_{1}>(\log L)^{2}/2\big{)}$ (C.2)

plus a term that decays expoenentially in $L^{3}$ (which controls the
probability that $N>H$), and the well known exponential tails of the
distributions of $W$ and $Y_{1}$ yield the result. ∎

######  Remark C.2.

Notice that it is enough to have the distribuion $W$ with an exponentially
decaying tail for the argument to hold.

######  Lemma C.3.

Let $\omega$ be a $BDP(p,q)$ with $p<\frac{1}{2}$ starting from a distribution
$\bar{\mu}$ with an exponentially decaying tail as in (4.1). Then, there exist
constants $C$ and $\beta^{\prime}>0$ such that

$P\big{(}\omega(t)>n\big{)}\leq Ce^{-\beta^{\prime}n}$ (C.3)

for all $n$ (uniformly in $t$).

###### Proof.

The strategy follows lines already pursued above. We consider a coupling of
$\omega$ to a copy $\omega^{\prime}$ starting from $\nu$, the equilibrium for
the BDP, and independent till first meeting, after which both processes
coalesce. Let $T$ and $T^{\prime}$ denote the respective times of first visit
to the origin by $\omega$ and $\omega^{\prime}$. We also couple $\omega$ to
$Z$, a simple random walk, so that $\omega=Z$ up to $T$. We readily check that
$T^{\prime}$ (but also $T$) has an expoential tail.

The result will be readily implied from the following remarks.

  * •

On $\\{T\vee T^{\prime}\leq t\\}$, we have that
$\omega(t)=\omega^{\prime}(t)\sim\nu$;

  * •

on $\\{T>t\\}$, we have that $\omega(t)=Z_{t}\preceq\tilde{Y}=\max_{0\leq
s<\infty}Z_{s}$;

  * •

on $\\{T\leq t<T^{\prime}\leq n\\}\cap\\{N^{\prime}\leq 2n\\}$, we have that
$\omega(t)\leq\max_{0\leq s\leq T^{\prime}}\omega(s)\preceq
W+(Y_{1}\vee\cdots\vee Y_{2n})$,

with $N^{\prime},W,\,Y_{1},\,Y_{2}\ldots$ as in the previous proof, with
$N^{\prime}$ defined similarly as $N$, except that
$\omega^{\prime},\,T^{\prime}$ replace $\omega,\,T$, respectively. Thus

$\displaystyle P(\omega(t)>n)$ $\displaystyle\leq$
$\displaystyle\nu([n,\infty))+P(\tilde{Y}>n)+P(T^{\prime}>n)+P(N^{\prime}>2n)$
(C.4) $\displaystyle+P(W>n/2)+2nP(Y_{1}>n/2),$

and the result follows. ∎

### Acknowledgements

LRF warmly thanks Elena Zhizhina for many enjoyable discussions during a visit
of hers to São Paulo, where this project originated. He also thanks Hubert
Lacoin for bringing up the issue addressed in Remark 3.16.

## References

  * [1] L. Avena, F. den Hollander, and F. Redig, _Law of large numbers for a class of random walks in dynamic random environments_ , Electron. J. Probab. 16 (2011), no. 21, 587–617.
  * [2] Antar Bandyopadhyay and Ofer Zeitouni, _Random walks in dynamic markovian random environment_ , Alea 1 (2006), 205–224.
  * [3] Gérard Ben Arous and Jiří Černý, _Scaling limit for trap models on $\mathbb{Z}^{d}$_, Ann. Probab. 35 (2007), no. 6, 2356–2384.
  * [4] Oriane Blondel, Marcelo R. Hilário, and Augusto Teixeira, _Random walks on dynamical random environments with nonuniform mixing_ , Ann. Probab. 48 (2020), no. 4, 2014–2051.
  * [5] C. Boldrighini, R. A. Minlos, and A. Pellegrinotti, _Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment_ , Probab. Theory Related Fields 109 (1997), no. 2, 245–273.
  * [6] C. Boldrighini, R. A. Minlos, A. Pellegrinotti, and E. A. Zhizhina, _Continuous time random walk in dynamic random environment_ , Markov Process. Related Fields 21 (2015), no. 4, 971–1004.
  * [7] C. Boldrighini, A. Pellegrinotti, and E. A. Zhizhina, _Regular and singular continuous time random walk in dynamic random environment_ , Mosc. Math. J. 19 (2019), no. 1, 51–76.
  * [8] Carlo Boldrighini, Robert A. Minlos, and Alessandro Pellegrinotti, _Interacting random walk in a dynamical random environment. II. Environment from the point of view of the particle_ , Ann. Inst. H. Poincaré Probab. Statist. 30 (1994), no. 4, 559–605.
  * [9] F. den Hollander and R. S. dos Santos, _Scaling of a random walk on a supercritical contact process_ , Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 4, 1276–1300.
  * [10] Dmitry Dolgopyat, Gerhard Keller, and Carlangelo Liverani, _Random walk in markovian environment_ , Ann. Probab. 36 (2008), no. 5, 1676–1710.
  * [11] R. Durrett, H. Kesten, and G. Lawler, _Making money from fair games_ , pp. 255–267, Birkhäuser Boston, Boston, MA, 1991.
  * [12] D. J. Emery, _Limiting behaviour of the distributions of the maxima of partial sums of certain random walks_ , J. Appl. Probability 9 (1972), 572–579.
  * [13] L. R. G. Fontes, M. Isopi, and C. M. Newman, _Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension_ , Ann. Probab. 30 (2002), no. 2, 579–604.
  * [14] Marcelo R. Hilário, Daniel Kious, and Augusto Teixeira, _Random walk on the simple symmetric exclusion process_ , Comm. Math. Phys. 379 (2020), no. 1, 61–101.
  * [15] P. L. Hsu and Herbert Robbins, _Complete convergence and the law of large numbers_ , Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25–31.
  * [16] N. James and Y. Peres, _Cutpoints and exchangeable events for random walks_ , Theory of Probability & Its Applications 41 (1997), no. 4, 666–677.
  * [17] Samuel Karlin and James McGregor, _The classification of birth and death processes_ , Trans. Amer. Math. Soc. 86 (1957), 366–400.
  * [18] Thomas M. Liggett, _Interacting particle systems_ , Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276, Springer-Verlag, New York, 1985.
  * [19] Thomas Mountford and Maria E. Vares, _Random walks generated by equilibrium contact processes_ , Electron. J. Probab. 20 (2015), no. 3, 17.
  * [20] Frank Redig and Florian Völlering, _Random walks in dynamic random environments: a transference principle_ , Ann. Probab. 41 (2013), no. 5, 3157–3180.
  * [21] Frank Spitzer, _Principles of random walk_ , second ed., Graduate Texts in Mathematics, Vol. 34, Springer-Verlag, New York-Heidelberg, 1976.
  * [22] Atilla Yilmaz, _Large deviations for random walk in a space-time product environment_ , Ann. Probab. 37 (2009), no. 1, 189–205.