# Distinguishing modified gravity with just two tensorial degrees of freedom
from general relativity: Black holes, cosmology, and matter coupling

Aya Iyonaga<EMAIL_ADDRESS>Department of Physics, Rikkyo University,
Toshima, Tokyo 171-8501, Japan  Tsutomu Kobayashi<EMAIL_ADDRESS>Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan

###### Abstract

We consider spatially covariant modified gravity in which the would-be scalar
degree of freedom is made non-dynamical and hence there are just two tensorial
degrees of freedom, i.e., the same number of dynamical degrees of freedom as
in general relativity. Focusing on a class of such modified gravity theories
characterized by three functions of time, we discuss how modified gravity with
two tensorial degrees of freedom can be distinguished observationally or
phenomenologically from general relativity. It is checked that the theory
gives the same predictions as general relativity for weak gravitational fields
and the propagation speed of gravitational waves. We also find that there is
no modification to asymptotically flat black hole solutions. Due to a large
degree of freedom to choose the time-dependent functions in the theory, the
homogeneous and isotropic cosmological dynamics can be made close to or even
identical to that of the $\Lambda$CDM model. We investigate the behavior of
cosmological perturbations in the long and short wavelength limits and show
that in both limits the effects of modified gravity appear only through the
modification of the background evolution. Finally, it is remarked that in the
presence of a galileon field in the matter sector, the scalar degree of
freedom is revived, ruining the essential feature of the theory.

††preprint: RUP-21-16

##  I Introduction

According to Lovelock’s theorem [1, 2], the Einstein tensor plus a
cosmological term is the only possible second-order Euler-Lagrange equation
that is obtained from a diffeomorphism invariant action constructed from the
metric tensor alone in four spacetime dimensions. Modifying gravity amounts to
relaxing the postulates of the theorem. Abandoning full spacetime
diffeomorphism invariance, one can for example consider modified gravity
enjoying only three-dimensional spatial diffeomorphism invariance. Basically,
this way of modifying gravity is equivalent to introducing a new dynamical
scalar degree of freedom (DOF) that spontaneously breaks time diffeomorphism
invariance. This is the underlying idea utilized in the construction of
effective field theories for inflation and dark energy [3, 4, 5, 6]. A general
framework for scalar-tensor theories based on this idea has been developed in
Refs. [7, 8].

An interesting twist is that a scalar DOF incorporated in such a framework is
not necessarily dynamical and it is possible that modified gravity in which
time diffeomorphism invariance is broken has only two tensorial degrees of
freedom as general relativity (GR). The simplest example of such “scalarless”
scalar-tensor theories is the cuscuton theory [9], in which the scalar field
has an infinite propagation speed. An attempt to identify the cuscuton-like
subclass within the Horndeski [10] and beyond Horndeski [11] theories is
presented in Ref. [12]. Other modified gravity theories with two DOFs have
been constructed in Refs. [13, 14, 15, 16, 17], a subclass of which is in fact
equivalent to the cuscuton theory [18]. On the basis of the general framework
of spatially covariant theories of gravity [7, 8], Ref. [19] has derived
through a rigorous Hamiltonian analysis the conditions under which there are
only two DOFs and presented a particular action satisfying the conditions. The
action is composed of two derivative terms built out of three-dimensional
geometrical quantities as in GR, but depends on the lapse function in a
nontrivial manner. A complementary perturbative approach to spatially
covariant modified gravity without a scalar DOF has been proposed in Ref.
[20].

The purpose of this paper is to explore phenomenological aspects of the theory
of modified gravity with two tensorial DOFs proposed in Ref. [19] to see how
the theory can be distinguished observationally from GR. We will see that it
is quite difficult to find differences between the two theories in the weak
gravity regime, gravitational wave propagation, black-hole spacetimes, and
cosmology, when restricted to a certain subset of the theories of Ref. [19].
The theory of modified gravity we study in this paper is somewhat similar to
the infrared limit of Hořava gravity [21] in some respects, and we share the
same motivation, e.g., as Ref. [22].

The structure of this paper is as follows. In the next section, we review
modified gravity theories with two tensorial DOFs [19] which we study in this
paper. In Sec. III, we identify the subclass of the theories that evades
solar-system and gravitational-wave constraints. Then, we study black hole
spacetimes in Sec. IV and cosmological aspects in Sec. V. In Sec. VI, a side
remark concerning a coupling to a galileon field is given. Finally, we draw
our conclusions in Sec. VII.

##  II Spatially covariant gravity with two tensorial degrees of freedom

In this section, we introduce a class of spatially covariant modified gravity
in which there are just two tensorial DOFs [19] and in particular there is no
scalar DOF.

###  II.1 “Scalarless” scalar-tensor theories

Let us begin with scalar-tensor theories with one scalar DOF, $\phi$, on top
of two tensorial DOFs. In the unitary gauge where $\phi$ is homogeneous on
constant time hypersurfaces, ghost-free scalar-tensor theories can in general
be described by the action of the form [7, 8],

$\displaystyle S=\int\mathrm{d}t\mathrm{d}^{3}x\,N\sqrt{\gamma}{\cal
L}(N,\gamma_{ij},K_{ij},R_{ij},D_{i};t),$ (1)

where $N$ is the lapse function, $\gamma_{ij}$ is the spatial metric, $K_{ij}$
is the extrinsic curvature of constant time hypersurfaces,

$\displaystyle
K_{ij}=\frac{1}{2N}\left(\partial_{t}\gamma_{ij}-D_{i}N_{j}-D_{j}N_{i}\right),$
(2)

with $N_{i}$ being the shift vector, $R_{ij}$ is the intrinsic curvature
tensor, and $D_{i}$ is the covariant derivative with respect to $\gamma_{ij}$.
The Horndeski action [10, 23, 24] can be recast in the above form in terms of
these Arnowitt-Deser-Misner (ADM) variables. One may also include the velocity
of the lapse function, $\partial_{t}N-N^{i}D_{i}N$, and explore a larger class
of healthy scalar-tensor theories with three DOFs [25]. Although such a
generalization leads to an interesting class of theories called degenerate
higher-order scalar-tensor theories [26, 27, 28], we focus for simplicity on
the action of the form (1).

Thus, in general, the action (1) describes scalar-tensor theories with three
DOFs. However, in a particular subset of theories, the scalar mode obeys an
elliptic equation and does not propagate. Consequently, such theories have
only two tensorial DOFs as in GR, but the action is certainly different from
that of GR. A particular example is given by [19]

$\displaystyle S$
$\displaystyle=\frac{1}{2}\int\mathrm{d}t\mathrm{d}^{3}x\,N\sqrt{\gamma}\biggl{[}\frac{\beta_{0}N}{\beta_{2}+N}K_{ij}K^{ij}$
$\displaystyle\quad\quad-\frac{\beta_{0}}{3}\left(\frac{2N}{\beta_{1}+N}+\frac{N}{\beta_{2}+N}\right)K^{2}$
$\displaystyle\quad\quad+\alpha_{1}+\alpha_{2}R+\frac{1}{N}\left(\alpha_{3}+\alpha_{4}R\right)\biggr{]},$
(3)

where $\beta_{A}$ and $\alpha_{A}$ are functions of $t$. A rigorous
Hamiltonian analysis has shown that this theory indeed has only two tensorial
DOFs [19].111The case with $\beta_{0}=1$ is presented in Ref. [19], but it is
clear from the footnote of Ref. [19] that one can include this time-dependent
coefficient. The cuscuton theory [9] is reproduced by setting
$\beta_{0}=\alpha_{2}=1$ and $\beta_{1}=\beta_{2}=\alpha_{4}=0$, with
$\alpha_{3}$ coming from the cuscuton term
$\sigma(\phi)\sqrt{-\partial_{\mu}\phi\partial^{\mu}\phi}$, while the extended
cuscuton theories [12, 29] correspond to the case with $\beta_{1}=\beta_{2}$.
The covariant expression for the action (3) in terms of a Stückelberg scalar
field is given in Appendix B. It is instructive to rewrite the kinetic part of
the Lagrangian as

$\displaystyle\frac{\beta_{0}}{2}\left(\frac{N}{\beta_{2}+N}\widetilde{K}_{ij}\widetilde{K}^{ij}-\frac{2}{3}\frac{N}{\beta_{1}+N}K^{2}\right),$
(4)

where $\widetilde{K}_{ij}:=K_{ij}-(1/3)K\gamma_{ij}$ is the traceless part of
the extrinsic curvature. This expression shows the roles of the functions
$\beta_{1}$ and $\beta_{2}$ more clearly. In the following we will explore
various aspects of the gravitational theory with the action (3).

Before proceeding let us comment on other examples of “scalarless” scalar-
tensor theories with two DOFs. Under the assumption that the Lagrangian is
linear in the lapse function (i.e., ${\cal L}$ in Eq. (1) is independent of
$N$), the conditions for a theory to have only two tensorial DOFs have been
addressed in Ref. [13]. References [14, 15, 16] obtained gravitational
theories with two DOFs via canonical transformations from GR in the
Hamiltonian formulation. In the context of the recent claim concerning the
$D\to 4$ limit of $D$-dimensional Gauss-Bonnet gravity [30], a consistent
theory with two DOFs has been proposed [17] that avoids the propagation of the
scalar mode present in the previous derivations [31, 32, 33, 34, 35].
Symmetries that prohibit the scalar mode from propagating have been discussed
in Ref. [36]. See Refs. [37, 38, 39] for further developments in “scalarless”
modified gravity and Refs. [40, 41, 42] for the application to cosmology.

###  II.2 Quasi-Einstein frame

A field redefinition of a metric given by a disformal transformation [43]
retains the number of propagating DOFs as long as the transformation is
invertible. In terms of the ADM variables, it is expressed as

$\displaystyle N\to{\cal N}(t,N),\quad N^{i}\to N^{i},\quad\gamma_{ij}\to{\cal
A}(t,N)\gamma_{ij}.$ (5)

Let us consider a special case of a disformal transformation,

$\displaystyle N\to b_{0}(t)N+b_{1}(t),\quad N^{i}\to
N^{i},\quad\gamma_{ij}\to\gamma_{ij}.$ (6)

Under the transformation (6), the form of the action (3) is invariant, with
$\beta_{A}(t)$ and $\alpha_{A}(t)$ being transformed as

$\displaystyle\beta_{0}\to\beta_{0}/b_{0},\quad\beta_{1,2}\to(\beta_{1,2}+b_{1})/b_{0},$
$\displaystyle\alpha_{1,2}\to
b_{0}\alpha_{1,2},\quad\alpha_{3,4}\to\alpha_{3,4}+b_{1}\alpha_{1,2}.$ (7)

Therefore, via the field redefinition (6) one can always set $\beta_{0}=1$ and
$\beta_{2}=0$. Since the coefficient of $K_{ij}K^{ij}$ is now standard, we dub
this frame as the quasi-Einstein frame. This motivates us to study the action
of the form222As will be clear shortly, we use units in which Newton’s
constant, $G_{N}$, is equal to $(8\pi\alpha_{2})^{-1}$, and we will be
interested mainly in the case with $\alpha_{2}=1$.

$\displaystyle S$
$\displaystyle=\frac{1}{2}\int\mathrm{d}t\mathrm{d}^{3}x\,N\sqrt{\gamma}\biggl{[}K_{ij}K^{ij}-\frac{1}{3}\left(\frac{2N}{\beta+N}+1\right)K^{2}$
$\displaystyle\quad\quad+\alpha_{1}+\alpha_{2}R+\frac{1}{N}\left(\alpha_{3}+\alpha_{4}R\right)\biggr{]},$
(8)

where $\beta:=\beta_{1}$, as the case of particular interest. General
relativity with a cosmological constant is reproduced by setting
$\beta=\alpha_{3}=\alpha_{4}=0$, $\alpha_{1}=\,$ const, and $\alpha_{2}=1$.
Theories with $\beta\neq 0$ are not included in the (extended) cuscuton
family, and hence the impacts of $\beta$ have not been investigated so far in
the literature.

Note that a subtlety arises when matter is present. In this paper we assume
that matter is minimally coupled to gravity in this quasi-Einstein frame.

Here we should point out the similarity and difference between the action (8)
and that of the infrared limit of nonprojectable Hořava gravity [21, 44, 45],

$\displaystyle S$
$\displaystyle=\frac{1}{2}\int\mathrm{d}t\mathrm{d}^{3}x\,N\sqrt{\gamma}\bigl{(}K_{ij}K^{ij}-\lambda
K^{2}$ $\displaystyle\quad\quad+\alpha_{1}+\alpha_{2}R+\eta
a_{i}a^{i}\bigr{)},$ (9)

where $a_{i}:=D_{i}\ln N$ and now all the coefficients are constants. The same
action is also obtained in the context of Einstein-aether theory [46] if the
aether field is restricted to be hypersurface orthogonal [47, 48]. While being
mostly similar, the essential difference can be perceived in the structure of
the coefficient of $K^{2}$. In the case of $\eta=0$, nonprojectable Hořava
gravity has odd dimensionality of the phase space at each spacetime point and
hence is inconsistent [49, 50]. By contrast, the action (8) is free of such a
trouble thanks to the particular structure of the coefficient of $K^{2}$.

##  III Solar-system tests and gravitational waves

Let us study gravitational potentials produced by nonrelativistic matter and
the propagation of gravitational waves in the theory described by the action
(8). We assume that $\alpha_{1},\alpha_{3}\sim H_{0}^{2}$, where $H_{0}$ is
the Hubble scale, so that they are only relevant to the large-scale
cosmological dynamics. We also assume that $\alpha_{2}$ and $\alpha_{4}$ could
vary only on cosmological time scales, because otherwise nearly static
gravitational potentials would not be possible. We can then consider scalar
and tensor perturbations on a Minkowski background, using the approximation
$\alpha_{1},\alpha_{3}\ll\partial_{t}^{2},\Delta:=\delta_{ij}\partial_{i}\partial_{j}$.

Static scalar perturbations of the ADM variables are given by

$\displaystyle N=1+\Phi(\vec{x}),\quad
N_{i}=\partial_{i}\chi(\vec{x}),\quad\gamma_{ij}=[1-2\Psi(\vec{x})]\delta_{ij}.$
(10)

The quadratic action reads

$\displaystyle S$
$\displaystyle=\int\mathrm{d}t\mathrm{d}^{3}x\biggl{[}-(\alpha_{2}+\alpha_{4})\Psi\Delta\Psi+2\alpha_{2}\Phi\Delta\Psi$
$\displaystyle\quad\quad+\frac{\beta}{3(1+\beta)}(\Delta\chi)^{2}-\Phi\rho(\vec{x})\biggr{]},$
(11)

where $\rho(\vec{x})$ is the matter energy density and this matter is assumed
to be minimally coupled to gravity. Note that in deriving Eq. (11) we actually
assumed that $\beta\Delta\gg H_{0}^{2}$. The equations of motion are then
solved to give

$\displaystyle\Delta\Phi=\left(1+\frac{\alpha_{4}}{\alpha_{2}}\right)\frac{\rho}{2\alpha_{2}},\quad\Delta\Psi=\frac{\rho}{2\alpha_{2}},\quad\Delta\chi=0.$
(12)

This result implies that

$\displaystyle\alpha_{2}\simeq\mathrm{const}\quad\mathrm{and}\quad\alpha_{4}\simeq
0$ (13)

are required in order to evade solar-system constraints (light bending, the
Shapiro time delay, and the time variation of Newton’s constant).333According
to the current limits on the time variation of Newton’s constant,
$|\dot{\alpha}_{2}/\alpha_{2}|$ must be much smaller than $H_{0}$ [51].

Tensor perturbations are given by

$\displaystyle N=1,\quad
N_{i}=0,\quad\gamma_{ij}=\delta_{ij}+h_{ij}(t,\vec{x}),$ (14)

for which the quadratic action is

$\displaystyle
S=\frac{1}{8}\int\mathrm{d}t\mathrm{d}^{3}x\left[(\partial_{t}h_{ij})^{2}+(\alpha_{2}+\alpha_{4})h_{ij}\Delta
h_{ij}\right].$ (15)

The observation of GW170817 and GRB 170817A put a tight bound on the speed of
gravitational waves, $c_{\mathrm{GW}}\simeq 1$ [52, 53]. It is therefore
required that

$\displaystyle\alpha_{2}+\alpha_{4}\simeq 1.$ (16)

Summarizing these results, we identify a phenomenologically interesting class
characterized by

$\displaystyle\alpha_{2}=1,\quad\alpha_{4}=0,$ (17)

which evades solar-system and gravitational-wave constraints. Note that in
these noncosmological phenomena we do not see any effects of nonvanishing
$\beta$.

##  IV Black hole solutions

In this section, we study black hole solutions in the theory defined by the
action (8). On the basis of the discussion in the previous section, we are
interested in the phenomenologically viable case with $\alpha_{2}=1$ and
$\alpha_{4}=0$. Black holes under consideration are supposed to be much
smaller than the size of the cosmological horizon. Therefore, we are mostly
interested in asymptotically flat black holes obtained ignoring $\alpha_{1}$
and $\alpha_{3}$. (These two functions are assumed to be characterized by the
cosmological horizon scale.) The remaining unfixed function, $\beta$, is
assumed to be constant in this section, because we seek for stationary black
hole solutions. Our focus is therefore on whether the effects of $\beta$ can
be seen or not in black hole solutions. Nevertheless, having said that, we
allow $\alpha_{1}$ to be a nonvanishing constant and $\alpha_{2}$ to be a
constant different from 1 in the following discussion, because relaxing these
assumptions does not hinder us from presenting an analytic solution below. To
sum up, in this section we have three constant parameters, $\beta$,
$\alpha_{1}$, and $\alpha_{2}$, though we are mostly interested in the case
with $\alpha_{1}=0$ and $\alpha_{2}=1$.

###  IV.1 Static and spherically symmetric solution

The ADM variables for static and spherically symmetric solutions are taken to
be

$\displaystyle N=N(r),\quad N_{i}\mathrm{d}x^{i}=B(r)F(r)\mathrm{d}r,$
$\displaystyle\gamma_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}=F^{2}(r)\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2},$
(18)

where
$\mathrm{d}\Omega^{2}=\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}$.
Since the action is only invariant under spatial diffeomorphisms, at this
stage we do not make a temporal coordinate transformation, $t\to T(t,r)$, to
remove $B$. Such a coordinate transformation would result in an inhomogeneous
configuration of the nondynamical scalar field.

Substituting the ADM variables (18) to the action and varying it with respect
to $N$, $B$, and $F$, we obtain

$\displaystyle\alpha_{1}+\frac{2\alpha_{2}}{r^{2}}\left[\left(\frac{F^{2}-1}{F^{2}}\right)r\right]^{\prime}+\frac{2(rB^{2})^{\prime}}{r^{2}(\beta+N)^{2}F^{2}}$
$\displaystyle-\frac{2r^{2}}{3}\frac{\beta(\beta+2N)}{N^{2}(\beta+N)^{2}F^{2}}\left[\left(B/r\right)^{\prime}\right]^{2}=0,$
(19)
$\displaystyle\frac{r^{2}B}{(\beta+N)^{2}F^{2}}\left[(\beta+N)F\right]^{\prime}+\frac{\beta}{3}\left[\frac{r^{4}(B/r)^{\prime}}{N(\beta+N)F}\right]^{\prime}=0,$
(20)
$\displaystyle\alpha_{1}\frac{N}{F}+\frac{2\alpha_{2}}{r^{2}}\frac{N}{F}\left(\frac{F^{2}-1}{F^{2}}-\frac{2rN^{\prime}}{NF^{2}}\right)$
$\displaystyle+\frac{2(rB^{2})^{\prime}}{r^{2}NF^{3}}-\frac{2\beta}{3r^{4}N(\beta+N)F^{3}}\left[(r^{2}B)^{\prime}\right]^{2}=0,$
(21)

where a dash denotes differentiation with respect to $r$. The system is of
second order for $B$ (provided that $\beta\neq 0$) and of first order for $N$
and $F$.

Since it is difficult to obtain a general solution to Eqs. (19)–(21), we start
with finding a particular solution. One can easily verify that Eqs. (19)–(21)
admit the following solution:

$\displaystyle N=N_{0}\sqrt{f(r)},\quad F=\frac{1}{\sqrt{f(r)}},\quad
B=\frac{N_{0}b_{0}}{r^{2}},$ (22)

where

$\displaystyle
f(r):=1+\frac{\alpha_{1}}{6\alpha_{2}}r^{2}-\frac{\mu_{0}}{r}+\frac{b_{0}^{2}}{\alpha_{2}r^{4}},$
(23)

and $N_{0}$, $\mu_{0}$, and $b_{0}$ are integration constants. Here, $N_{0}$
is physically less important because it may be set to 1 by rescaling the unit
of $t$.

To see the spacetime structure of this solution more clearly, we note that the
four-dimensional metric can be written as

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-(N^{2}-B^{2})\left(\mathrm{d}t-\frac{BF}{N^{2}-B^{2}}\mathrm{d}r\right)^{2}$
$\displaystyle\quad+\frac{N^{2}F^{2}}{N^{2}-B^{2}}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2},$
(24)

where

$\displaystyle N^{2}-B^{2}$
$\displaystyle=N_{0}^{2}\left(1+\frac{\alpha_{1}}{6\alpha_{2}}r^{2}-\frac{\mu_{0}}{r}+\frac{1-\alpha_{2}}{\alpha_{2}}\frac{b_{0}^{2}}{r^{4}}\right),$
(25) $\displaystyle NF$ $\displaystyle=N_{0}.$ (26)

Thus, introducing a new time coordinate defined by
$\mathrm{d}T=N_{0}\left\\{\mathrm{d}t-[BF/(N^{2}-B^{2})]\mathrm{d}r\right\\}$,
one can write the metric in a diagonal form at the expense of a homogeneous
configuration of the nondynamical scalar field (see Appendix B). It is now
obvious that in the phenomenologically interesting case of $\alpha_{2}=1$, the
solution (22) describes Schwarzschild-(anti-)de Sitter spacetime. Importantly,
we see no $\beta$-dependence in this solution. Note that inside the black hole
horizon there is a location at which $N=0$. If $\alpha_{1}<0$, such a location
also exists outside the cosmological horizon. These are called the universal
horizon, which has been studied in Einstein-aether theory and Hořava gravity
[54, 55].

Aside from the cosmological constant term, essentially the same expression for
a black hole solution has been obtained in Einstein-aether theory [56]. This
is not surprising because the solution (22) satisfies the maximal slicing
condition, $K=-(r^{2}B)^{\prime}/r^{2}NF=0$, while the crucial difference
between the two theories is found in the coefficient of $K^{2}$ in the action.
The maximal slicing condition is the very reason why there is no
$\beta$-dependence in the solution. Note that a similar solution was found in
a different modified gravity theory with two tensorial DOFs [57].

###  IV.2 Static perturbations

We have thus obtained a particular solution (22), but at this stage it is not
clear whether or not there exist other black hole solutions with the
appropriate asymptotic behavior. Since $B$ obeys the second-order differential
equations, one might rather expect that there should be another integration
constant in addition to $N_{0}$, $\mu_{0}$, and $b_{0}$. To address this
point, let us consider a slightly deformed solution,

$\displaystyle N=N_{0}\sqrt{f+h_{0}(r)},\quad F=\frac{1}{\sqrt{f+h_{1}(r)}},$
$\displaystyle B=N_{0}\left[\frac{b_{0}}{r^{2}}+h_{2}(r)\right],$ (27)

where $h_{1}$, $h_{2}$, and $h_{3}$ are to be treated as small perturbations.

Equations (19) and (21) reduce, respectively, to

$\displaystyle
f\left[\frac{2b_{0}}{\alpha_{2}}\left(\frac{h_{2}}{r}\right)^{\prime}-(rh_{1})^{\prime}\right]+\frac{3b_{0}^{2}}{\alpha_{2}r^{4}}(h_{0}-h_{1})=0,$
(28) $\displaystyle
f\left[\frac{2b_{0}}{\alpha_{2}}\left(\frac{h_{2}}{r}\right)^{\prime}-(rh_{0})^{\prime}\right]+\left(1+\frac{\alpha_{1}}{2\alpha_{2}}r^{2}\right)(h_{0}-h_{1})=0.$
(29)

Eliminating $h_{2}$ from these two equations, we obtain

$\displaystyle f(h_{0}-h_{1})^{\prime}-f^{\prime}(h_{0}-h_{1})=0,$ (30)

which can be integrated to give

$\displaystyle h_{0}-h_{1}=C_{1}f,$ (31)

where $C_{1}$ is an integration constant. This integration constant can be
absorbed into a redefinition of $N_{0}$. Substituting this to Eq. (28), we
have

$\displaystyle h_{0}$
$\displaystyle=-\frac{\mu_{1}}{r}+C_{1}\left(f-\frac{b_{0}^{2}}{\alpha_{2}r^{4}}\right)+\frac{2b_{0}}{\alpha_{2}}\frac{h_{2}}{r^{2}},$
(32) $\displaystyle h_{1}$
$\displaystyle=-\frac{\mu_{1}}{r}-\frac{C_{1}b_{0}^{2}}{\alpha_{2}r^{4}}+\frac{2b_{0}}{\alpha_{2}}\frac{h_{2}}{r^{2}},$
(33)

where $\mu_{1}$ is an integration constant. This integration constant simply
corresponds to a constant shift of the mass parameter $\mu_{0}$. Finally,
substituting all these results to Eq. (20), we arrive at

$\displaystyle\left(\beta+N_{0}\sqrt{f}\right)\left(h_{2}^{\prime\prime}+\frac{2}{r}h_{2}^{\prime}-\frac{2}{r^{2}}h_{2}\right)$
$\displaystyle-
N_{0}(\sqrt{f})^{\prime}\left(h_{2}^{\prime}+\frac{2}{r}h_{2}\right)=0,$ (34)

where we assumed $\beta\neq 0$. This equation can be integrated to give

$\displaystyle
h_{2}=\frac{b_{1}}{r^{2}}+\frac{\epsilon_{1}}{r^{2}}\int^{r}r^{2}\left(\beta+N_{0}\sqrt{f}\right)\mathrm{d}r,$
(35)

where $b_{1}$ and $\epsilon_{1}$ are integration constants. The former
integration constant corresponds to a constant shift of $b_{0}$, while the
latter cannot be absorbed into a shift of the integration constants of the
solution (22). Let us now focus specifically on the case of a zero
cosmological constant, $\alpha_{1}=0$, where the solution is asymptotically
flat at zeroth order. In this case, we have

$\displaystyle h_{2}\sim\frac{\epsilon_{1}}{3}(\beta+N_{0})r$ (36)

for large $r$, which would invalidate the perturbative approximation unless
$\epsilon_{1}=0$. Therefore, no asymptotically flat solution can be obtained
by a small deformation of the solution (22). Our result agrees with that of
the decoupling limit analysis of the khrononmetric theory [58].

###  IV.3 Numerical solutions

Figure 1: Numerical solution for $r_{h}=1$, $N_{h}=0.1$, and
$B_{1}=-1.9N_{h}$. The theory parameters are given by $\alpha_{1}=0$,
$\alpha_{2}=1$, and $\beta=0.2$.  Figure 2: Numerical solution for $r_{h}=1$,
$N_{h}=0.1$, and $B_{1}=-3N_{h}$. The theory parameters are given by
$\alpha_{1}=0$, $\alpha_{2}=1$, and $\beta=-0.3$.

We perform a numerical analysis that is complementary to and can go beyond the
perturbative study in the previous subsection.

In the vicinity of the horizon, $r=r_{h}$, the ADM variables can be expanded
as

$\displaystyle N$ $\displaystyle=N_{h}+N_{1}(r-r_{h})+\cdots,$ (37)
$\displaystyle F$ $\displaystyle=\frac{1}{N_{h}}+F_{1}(r-r_{h})+\cdots,$ (38)
$\displaystyle B$
$\displaystyle=N_{h}+B_{1}(r-r_{h})+\frac{B_{2}}{2}(r-r_{h})^{2}+\cdots.$ (39)

Here, one may set $F(r_{h})=1/N_{h}$ without loss of generality since doing so
corresponds to fixing the scale of time. For given $(r_{h},N_{h},B_{1})$, one
can numerically integrate Eqs. (19)–(21) from the horizon outwards to
determine the profile of $N$, $F$, and $B$. We thus try to find asymptotically
flat solutions that are not described as a perturbation of the Schwarzschild
spacetime foliated by maximal slices.

We consider again the case without the cosmological constant, $\alpha_{1}=0$.
For

$\displaystyle B_{1}=-\frac{2N_{h}}{r_{h}},$ (40)

one has the asymptotically flat solution given by Eq. (22) (with
$\alpha_{1}=0$). As far as we have investigated numerically, no asymptotically
flat solutions have been found if the condition (40) is not satisfied.444We
have not looked for asymptotically flat solutions in the case of
$\alpha_{1}\neq 0$. Typical examples obtained in the case of detuned
conditions are given in Figs. 1 and 2. The large $r$ behavior of these
numerical solutions is also different from that of asymptotically de Sitter
solutions.

Our numerical analysis thus supports the conclusion that asymptotically flat,
spherically symmetric black holes in the theory defined by the action (8) with
$\beta=\,$const, $\alpha_{2}=1$, and $\alpha_{1}=\alpha_{3}=\alpha_{4}=0$ are
indistinguishable from those in GR.

###  IV.4 Slowly rotating solution

Let us now discuss slowly rotating black hole solutions (see Refs. [59, 60,
61] for rotating black holes in Einstein-aether theory and Hořava gravity). To
first order in the black hole spin, we write [62, 63]

$\displaystyle\delta N=0,\quad\delta N_{i}\mathrm{d}x^{i}$
$\displaystyle=-r^{2}\sin^{2}\theta\,\omega(r,\theta)\mathrm{d}\varphi,\quad\delta\gamma_{ij}=0.$
(41)

The $(r\varphi)$ component of the evolution equations reads

$\displaystyle\partial_{\theta}\left(\sin^{3}\theta\partial_{\theta}\omega\right)=0,$
(42)

which, together with the regularity at the pole, yields $\omega=\omega(r)$.
Then, the $\varphi$ component of the momentum constraints reduces to

$\displaystyle\left(\frac{r^{4}\omega^{\prime}}{NF}\right)^{\prime}=0.$ (43)

This can be integrated to give

$\displaystyle\omega(r)=-\frac{3J}{4\pi}\int^{r}\frac{NF}{r^{4}}\mathrm{d}r+\omega_{0},$
(44)

where $J$ and $\omega_{0}$ are integration constants. Upon using $NF=1$ we
arrive at $\omega=J/4\pi r^{3}+\omega_{0}$. (For simplicity, here we have set
$N_{0}=1$ without loss of generality.) Noting that the time coordinate $t$
that we are using is nonstandard, we perform a coordinate transformation

$\displaystyle\varphi\to\varphi+\int^{r}\frac{BF}{N^{2}-B^{2}}(\omega-\omega_{0})\mathrm{d}r+\omega_{0}t.$
(45)

The four-dimensional metric then becomes

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-(N^{2}-B^{2})\mathrm{d}T^{2}+\frac{\mathrm{d}r^{2}}{N^{2}-B^{2}}+r^{2}\mathrm{d}\Omega^{2}$
$\displaystyle\quad-\frac{J\sin^{2}\theta}{2\pi
r}\mathrm{d}T\mathrm{d}\varphi,$ (46)

which is nothing but the Kerr metric in GR under the slow-rotation
approximation. Thus, we conclude that no effect of $\beta$ can be seen in
slowly rotating black holes.

##  V Cosmology

Our next step is the analysis of cosmology. In this section, we will mainly
consider the phenomenologically most interesting case with the time-dependent
functions

$\displaystyle\beta_{0}=1,\quad\beta_{1}=\beta(t),\quad\beta_{2}=0,$
$\displaystyle\alpha_{1}=\alpha_{1}(t),\quad\alpha_{2}=1,\quad\alpha_{3}=\alpha_{3}(t),\quad\alpha_{4}=0.$
(47)

However, when deriving a quadratic action for cosmological perturbations, we
will work in the general action (3) where no assumptions are made on these
functions, because the general form of the quadratic action will be used for
another purpose in the next section.

To investigate cosmology, we add a (irrotational, barotropic) perfect fluid
which is minimally coupled to gravity. Such a fluid component can be mimicked
by a scalar field whose action is given by

$\displaystyle S_{\mathrm{mat}}=\int\mathrm{d}^{4}x\sqrt{-g}P(Y),\quad
Y:=-\frac{1}{2}(\partial\varphi)^{2}.$ (48)

The matter energy-momentum tensor reads

$\displaystyle T_{\mu\nu}=(\rho+P)u_{\mu}u_{\nu}+Pg_{\mu\nu},$ (49)

where the energy density and the four-velocity are given respectively by

$\displaystyle\rho=2YP_{,Y}-P,\quad
u_{\mu}=-\frac{\nabla_{\mu}\varphi}{\sqrt{2Y}}.$ (50)

The equation of motion for $\varphi$ is equivalent to the conservation law,
$\nabla_{\mu}T_{\nu}^{\mu}=0$. Here, $\nabla_{\mu}$ stands for the usual four-
dimensional covariant derivative. The sound speed of the fluid is given by

$\displaystyle c_{s}^{2}=\frac{P_{,Y}}{P_{,Y}+2YP_{,YY}},$ (51)

where the right hand side is evaluated at the homogeneous background. For
$P\propto Y^{(1+w)/2w}$ with $w=\,$const, we have $c_{s}^{2}=P/\rho=w$.

###  V.1 Homogeneous and isotropic background

The total action is now composed of the gravity sector (8) and the action for
a cosmological fluid (48).

The ADM variables for a homogeneous and isotropic universe are given by

$\displaystyle N=\bar{N}(t),\quad
N_{i}=0,\quad\gamma_{ij}=a^{2}(t)\delta_{ij}.$ (52)

Since time reparametrization symmetry is spontaneously broken, one cannot put
$\bar{N}=1$ in general, but, as we will show below, the lapse function is
determined from the equations of motion. From the Hamiltonian constraint and
the evolution equations, we obtain

$\displaystyle\frac{3H^{2}}{(\beta/\bar{N}+1)^{2}}+\frac{\alpha_{1}}{2}=\rho,$
(53)
$\displaystyle-\frac{3H^{2}}{\beta/\bar{N}+1}-\frac{2}{\bar{N}}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{H}{\beta/\bar{N}+1}\right)-\frac{1}{2}\left(\alpha_{1}+\frac{\alpha_{3}}{\bar{N}}\right)=P,$
(54)

where $H:=\bar{N}^{-1}\mathrm{d}\ln a/\mathrm{d}t$ is the Hubble parameter.
(As far as the spatially flat model is concerned, the coefficients of $R$,
i.e., $\alpha_{2}$ and $\alpha_{4}$, do not appear anyway in the equations for
a homogeneous universe.) The conservation law reads

$\displaystyle\bar{N}^{-1}\dot{\rho}+3H(\rho+P)=0,$ (55)

where a dot stands for differentiation with respect to $t$. Given the equation
of state, this equation can be integrated to give $\rho$ and $P$ as a function
of $a$: $\rho=\rho(a)$, $P=P(a)$. Note that Eq. (55) is not an automatic
consequence of Eqs. (53) and (54). Rather, substitution of Eqs. (53) and (54)
into Eq. (55) yields

$\displaystyle\dot{\alpha}_{1}-3H\alpha_{3}-\frac{6\beta
H}{\beta/\bar{N}+1}\left[\frac{3H^{2}}{\beta/\bar{N}+1}+\frac{2}{\bar{N}}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{H}{\beta/\bar{N}+1}\right)\right]$
$\displaystyle=0.$ (56)

Using Eqs. (53) and (54) again, one can write this equation as the constraint
among the time-dependent functions,

$\displaystyle\dot{\alpha}_{1}-\sqrt{3}\left(\rho-\frac{\alpha_{1}}{2}\right)^{1/2}\left[\alpha_{3}-(\alpha_{1}+2P)\beta\right]=0.$
(57)

In the covariant formulation presented in Appendix B, this follows from the
equation of motion for a Stückelberg scalar field (see Eq. (120)). In the ADM
formulation of usual scalar-tensor theories with two tensorial and one scalar
DOFs, the equation corresponding to Eq. (57) reduces to a first-order
differential equation for the lapse function, which can be used to determine
$\bar{N}=\bar{N}(t)$. In contrast, in the present case Eq. (57) does not
contain $\bar{N}$, which is the crucial point in “scalarless” theories. Given
that $\rho$ and $P$ are now expressed in terms of $a$, Eq. (57) instead allows
us to write $a$ in terms of $t$. Finally, substituting $a=a(t)$ into Eq. (53),
one can determine the lapse function $\bar{N}$ as a function of $t$.

Let us now present an example. Suppose that the universe is filled with a
fluid with a constant equation of state parameter, $w=P/\rho$, and the time-
dependent functions are given by

$\displaystyle\beta$
$\displaystyle=\mathrm{const},\quad\alpha_{2}=1,\quad\alpha_{4}=0,$ (58)
$\displaystyle\alpha_{1}$
$\displaystyle=6h_{0}^{2}\left[\frac{1}{\xi}-\frac{1}{(1+\beta)^{2}}\right]\coth^{2}\left[\frac{3}{2}(1+w)h_{0}t\right]-\frac{6h_{0}^{2}}{\xi},$
(59) $\displaystyle\alpha_{3}$
$\displaystyle=6h_{0}^{2}\left[\frac{1+w(1+\beta)}{(1+\beta)^{2}}-\frac{1+w}{\xi}\right]\mathrm{csch}^{2}\left[\frac{3}{2}(1+w)h_{0}t\right]$
$\displaystyle\quad-\frac{6\beta h_{0}^{2}}{(1+\beta)^{2}},$ (60)

where $\xi$ and $h_{0}$ are constant parameters. This example admits the
following solution:

$\displaystyle\bar{N}=1,\quad
a^{3(1+w)}\propto\sinh^{2}\left[\frac{3}{2}(1+w)h_{0}t\right].$ (61)

Noting that $\alpha_{1}$ can be written as

$\displaystyle\alpha_{1}=6\left[\frac{1}{\xi}-\frac{1}{(1+\beta)^{2}}\right]H^{2}-6\frac{h_{0}^{2}}{\xi},$
(62)

we see that Eqs. (53) and (54) read

$\displaystyle 3H^{2}$ $\displaystyle=\xi\rho+3h_{0}^{2},$ (63)
$\displaystyle-2\dot{H}$ $\displaystyle=\xi(\rho+P).$ (64)

This shows that the cosmological gravitational constant differs from Newton’s
constant $G_{N}$ by the factor $\xi$:

$\displaystyle\frac{G_{\mathrm{cos}}}{G_{N}}=\xi.$ (65)

(Recall that we are working in units where $8\pi G_{N}=1\,(=1/\alpha_{2})$.)
Aside from this modification, the background evolution obeys the standard
Friedmann equations in the presence of a cosmological constant. A similar
situation occurs in Einstein-aether theory and Hořava gravity, where a mild
bound on the difference between $G_{\mathrm{cos}}$ and $G_{N}$ has been
obtained from the measurement of the primordial abundance of He4 as
$|\xi-1|\lesssim 0.1$ [64, 44]. In our example, $\xi$ is a free parameter so
that one may set $\xi=1$, leading to the background evolution that is
completely indistinguishable from that in GR.

In the above example, $\alpha_{1}$ and $\alpha_{3}$ track the cosmic evolution
as $\alpha_{1},\alpha_{3}\sim H^{2}$. We present another example in which
$\alpha_{1}$ and $\alpha_{3}$ instead remain as small as $H_{0}^{2}$ and
simply behave as a dark energy component with a constant equation of state
parameter $w_{\mathrm{DE}}$. In the case where the matter component is a
pressureless fluid, the model is given by

$\displaystyle\beta$
$\displaystyle=\mathrm{const},\quad\alpha_{2}=1,\quad\alpha_{4}=0,$ (66)
$\displaystyle\alpha_{1}$ $\displaystyle=-2\rho_{\mathrm{DE}}(t),$ (67)
$\displaystyle\alpha_{3}$
$\displaystyle=2[1+w_{\mathrm{DE}}(1+\beta)]\rho_{\mathrm{DE}}(t),$ (68)
$\displaystyle\rho_{\mathrm{DE}}$ $\displaystyle\propto
A^{-3(1+w_{\mathrm{DE}})},$ (69)

where $A$ and $t$ are related via

$\displaystyle t$ $\displaystyle=\frac{2\sqrt{1+r_{0}}}{3H_{0}}A^{3/2}$
$\displaystyle\quad\times{}_{2}F_{1}(1/2,-1/2w_{\mathrm{DE}},1-1/2w_{\mathrm{DE}},-r_{0}A^{-3w_{\mathrm{DE}}}).$
(70)

Here, ${}_{2}F_{1}$ is the hypergeometric function and
$r_{0}:=\rho_{\mathrm{DE}}/\rho|_{a=1}$ is the present value of the ratio
between the dark energy density and the matter energy density. One can check
that the solution is given by $\bar{N}=1$ and $a=A(t)$. This example yields
the background evolution subject to

$\displaystyle 3H^{2}$ $\displaystyle=(1+\beta)^{2}(\rho+\rho_{\mathrm{DE}}),$
(71) $\displaystyle-2\dot{H}$
$\displaystyle=(1+\beta)^{2}\left[\rho+(1+w_{\mathrm{DE}})\rho_{\mathrm{DE}}\right],$
(72)

i.e., it mimics the cosmology with $G_{\mathrm{cos}}=(1+\beta)^{2}G_{N}$ and
the dark energy component.

###  V.2 Cosmological perturbations

We go on to the analysis of scalar perturbations around a cosmological
background, with a particular focus on the case of a pressureless fluid. The
perturbed ADM variables are given by

$\displaystyle N=\bar{N}(1+\delta n),\quad
N_{i}=\bar{N}\partial_{i}\chi,\quad\gamma_{ij}=a^{2}e^{-2\psi}\delta_{ij},$
(73)

where we used the spatial gauge degrees of freedom to write $\gamma_{ij}$ in
the above form.

For the moment we work in the general action (3) to derive the quadratic
action for the scalar perturbations, and later we will focus on the specific
case of our interest. The action (3) can be expanded to quadratic order in
perturbations as

$\displaystyle S^{(2)}$
$\displaystyle=\int\mathrm{d}t\mathrm{d}^{3}x\bar{N}a^{3}\biggl{[}-\frac{3\beta_{0}}{\bar{\beta}_{1}}\left(\psi_{n}+\frac{H\delta
n}{\bar{\beta}_{1}}\right)^{2}$
$\displaystyle\quad-\frac{2\beta_{0}}{\bar{\beta}_{1}}\left(\psi_{n}+\frac{H\delta
n}{\bar{\beta}_{1}}\right)\frac{\Delta\chi}{a^{2}}+\frac{\beta_{0}(\bar{\beta}_{1}-\bar{\beta}_{2})}{3\bar{\beta}_{1}\bar{\beta}_{2}}\left(\frac{\Delta\chi}{a^{2}}\right)^{2}$
$\displaystyle\quad-\left(\alpha_{2}+\frac{\alpha_{4}}{\bar{N}}\right)\frac{\psi\Delta\psi}{a^{2}}+2\alpha_{2}\delta
n\frac{\Delta\psi}{a^{2}}\biggr{]},$ (74)

where the notations $\psi_{n}:=\bar{N}^{-1}\partial_{t}\psi$ and
$\bar{\beta}_{1,2}:=\beta_{1,2}/\bar{N}+1$ are used. The matter action (48)
can also be expanded in terms of perturbations as

$\displaystyle S_{\mathrm{mat}}^{(2)}$
$\displaystyle=\int\mathrm{d}t\mathrm{d}^{3}x\bar{N}a^{3}\left(\frac{\rho+P}{c_{s}^{2}}\right)\biggl{[}c_{s}^{2}\frac{\delta\varphi}{\varphi_{n}}\frac{\Delta\chi}{a^{2}}+\frac{\delta
n^{2}}{2}$ $\displaystyle\quad-\frac{\delta\varphi_{n}}{\varphi_{n}}\delta
n-3c_{s}^{2}\frac{\delta\varphi_{n}}{\varphi_{n}}\psi+\frac{1}{2}\left(\frac{\delta\varphi_{n}}{\varphi_{n}}\right)^{2}$
$\displaystyle\quad+\frac{c_{s}^{2}}{2}\frac{\delta\varphi\Delta\delta\varphi}{\varphi_{n}^{2}a^{2}}\biggr{]},$
(75)

where $\delta\varphi$ is the perturbation of $\varphi$. Our total action is
given by $S_{\mathrm{tot}}^{(2)}=S^{(2)}+S_{\mathrm{mat}}^{(2)}$.555We omit
$\delta n\psi$ and $\psi^{2}$ from Eqs. (74) and (75) because upon combining
$S^{(2)}$ and $S^{(2)}_{\mathrm{mat}}$ they can be eliminated in the end by
the use of the background equations. This is the complete expression for the
quadratic action derived from the general action (3) in the presence of a
perfect fluid (or a shift-symmetric k-essence field).

Now we focus on the case of our interest: $\beta_{0}=\alpha_{2}=1$ and
$\beta_{2}=\alpha_{4}=0$ with the notation $\beta_{1}=\beta$. For simplicity,
we consider the background with $\bar{N}=1$. We follow the analysis of density
perturbations in the extended cuscuton theory [29] to derive the reduced
action written solely in terms of a single variable $\delta$ defined by

$\displaystyle\delta=\frac{\rho+P}{\rho
c_{s}^{2}}\left(\frac{\dot{\delta\varphi}}{\dot{\varphi}}-\delta
n\right)-\frac{3(\rho+P)}{\rho}\psi,$ (76)

where note that the density perturbation is given by
$\delta\rho=(\rho+P)/c_{s}^{2}\cdot(\dot{\delta\varphi}/\dot{\varphi}-\delta
n)$. Later at an appropriate point we will carefully take the limit
$c_{s}^{2},P\to 0$ while keeping $\rho$ finite. We start with introducing
$\delta$ as an auxiliary field and write $S_{\mathrm{tot}}^{(2)}$ equivalently
as

$\displaystyle S_{\mathrm{tot}}^{(2)}$
$\displaystyle=S^{(2)}+S_{\mathrm{mat}}^{(2)}-\int\mathrm{d}t\mathrm{d}^{3}xa^{3}\left(\frac{\rho+P}{2c_{s}^{2}}\right)$
$\displaystyle\quad\times\biggl{[}\frac{\dot{\delta\varphi}}{\dot{\varphi}}-\delta
n-c_{s}^{2}\left(\frac{\rho\delta}{\rho+P}+3\psi\right)\biggr{]}^{2}.$ (77)

The additional third term is introduced so that it removes
$\dot{\delta\varphi}{}^{2}$ in $S_{\mathrm{tot}}^{(2)}$. Namely, one ends up
with the action that depends linearly on $\dot{\delta\varphi}$. The equation
of motion for $\delta$ yields Eq. (76). Substituting this to Eq. (77), the
third term vanishes and the action reduces back to the original one,
$S^{(2)}+S_{\mathrm{mat}}^{(2)}$. Thus, the two representations are indeed
equivalent.

Since $S_{\mathrm{tot}}^{(2)}$ is now linear in $\dot{\delta\varphi}$, one can
use the equation of motion to remove $\delta\varphi$ from the action. At this
stage one may substitute $P\propto Y^{(1+w)/2w}$ and express the action in
terms of $\rho$ and $w$, which allows us for a $w\to 0$ limit without any
divergences. Then, $\delta n$ and $\chi$ can also be removed from the action
by the use of their equations of motion. The resulting action is of the form

$\displaystyle S_{\mathrm{tot}}^{(2)}$
$\displaystyle=\int\mathrm{d}t\mathrm{d}^{3}xa^{3}\bigl{[}\dot{\delta}{\cal
A}(t,\Delta)\dot{\delta}+\delta{\cal O}_{1}(t,\Delta)\delta$
$\displaystyle\quad+2\psi{\cal O}_{2}(t,\Delta)\delta+\psi{\cal
O}_{3}(t,\Delta)\psi\bigr{]},$ (78)

where

$\displaystyle{\cal A}=\frac{\rho}{3\rho-2\Delta/a^{2}}.$ (79)

The explicit expressions for ${\cal O}_{1}$, ${\cal O}_{2}$, and ${\cal
O}_{3}$ are messy and not illuminating. Finally, one can eliminate $\psi$ by
using its equation of motion and arrives at the reduced action expressed in
terms of $\delta$ alone:

$\displaystyle S_{\mathrm{tot}}^{(2)}$
$\displaystyle=\int\mathrm{d}t\mathrm{d}^{3}xa^{3}\left[\dot{\delta}{\cal
A}(t,\Delta)\dot{\delta}+\delta{\cal B}(t,\Delta)\delta\right].$ (80)

On the basis of this action we move to study the behavior of the perturbations
in the short and long wavelength limits. In both limits, we will see that the
effect of modification of gravity comes into play only through the underlying
background model.

####  V.2.1 Short wavelength limit

In the short wavelength limit, $\Delta/a^{2}\gg\rho$, we have

$\displaystyle{\cal A}\simeq-\frac{1}{2}a^{2}\rho\Delta^{-1},\quad{\cal
B}\simeq-\frac{1}{4}a^{2}\rho^{2}\Delta^{-1}.$ (81)

The time-dependent function in the action $\beta$ has dropped out. Since
$\rho\propto a^{-3}$, the equation of motion for $\delta$ is found to be

$\displaystyle\ddot{\delta}+2H\dot{\delta}=\frac{1}{2}\rho\delta.$ (82)

Recalling that $4\pi G_{N}=1/(2\alpha_{2})=1/2$, this is identical to the
evolution equation for matter density perturbations in GR. Note, however, that
the evolution of $\delta$ depends on the underlying background model, which
could be different from that of the $\Lambda$CDM model. For example, if
$G_{\mathrm{cos}}/G_{N}\neq 1$ as in the example in the previous subsection,
the actual evolution of the density perturbations would be modified.

The equations of motion for $\delta n$, $\chi$, and $\psi$ allow us to write
these variables in terms of $\delta$. In the short wavelength limit, we obtain

$\displaystyle\delta n$
$\displaystyle\simeq\frac{1}{2}a^{2}\rho\Delta^{-1}\delta,$ (83)
$\displaystyle\chi$
$\displaystyle\simeq-\frac{3a^{4}\rho}{2(1+\beta)}\left[H\Delta^{-2}\delta+(1+\beta)\Delta^{-2}\dot{\delta}\right],$
(84) $\displaystyle\psi$
$\displaystyle\simeq\frac{1}{2}a^{2}\rho\Delta^{-1}\delta.$ (85)

Therefore, the metric potentials that are used in the familiar Newtonian gauge
analysis,

$\displaystyle\Phi$ $\displaystyle:=\delta n+\dot{\chi},$ (86)
$\displaystyle\Psi$ $\displaystyle:=\psi-H\chi,$ (87)

obey the standard relation,

$\displaystyle\Delta\Phi=\Delta\Psi=\frac{a^{2}}{2}\rho\delta,$ (88)

with $\delta\simeq\delta\rho/\rho$. Note that if one ignores the cosmic
expansion, then this reproduces the result obtained in Sec. III.

####  V.2.2 Long wavelength limit

In the long wavelength limit, $\Delta/a^{2}\ll\rho$, we have

$\displaystyle{\cal A}\simeq\frac{1}{3},\quad{\cal
B}\simeq-\frac{1}{9}\frac{\Delta}{a^{2}}.$ (89)

Again, $\beta$-dependence has dropped out. The long wavelength solution to the
equation of motion for $\delta$ is given by

$\displaystyle\delta\simeq\delta_{0}(\vec{x})-\frac{\Delta\delta_{0}(\vec{x})}{3}\int^{t}\frac{\mathrm{d}t^{\prime}}{a^{3}(t^{\prime})}\int^{t^{\prime}}\mathrm{d}t^{\prime\prime}a(t^{\prime\prime}),$
(90)

where $\delta_{0}$ is independent of time and the decaying mode has been
discarded. From the equations of motion for the other variables in the long
wavelength limit, we obtain

$\displaystyle\delta n$ $\displaystyle={\cal O}(\Delta\delta/a^{2}H^{2}),$
(91) $\displaystyle\chi$ $\displaystyle\simeq a^{2}\Delta^{-1}\dot{\delta},$
(92) $\displaystyle\psi$ $\displaystyle\simeq-\frac{\delta}{3},$ (93)

leading to

$\displaystyle\Phi=\Psi=-\frac{\delta_{0}(\vec{x})}{3}\left[1-\frac{H}{a}\int^{t}a(t^{\prime})\mathrm{d}t^{\prime}\right].$
(94)

The solution expressed in this way is the same as the long wavelength solution
for $\Phi$ and $\Psi$ in the presence of a pressureless fluid and a
cosmological constant in GR. In particular, in the matter-dominated era,
$a\propto t^{2/3}$, we have the time-independent metric potentials,
$\Phi=\Psi=-\delta_{0}/5$. Note, however, that Eq. (94) has been derived
without assuming any particular form of $a(t)$ for an accelerated phase of the
cosmic expansion. In our modified theory of gravity, the background evolution
(of the accelerated phase) could be different from that of the $\Lambda$CDM
model depending on the time-dependent functions in the action, so that the
actual time-dependence of $\Phi$ and $\Psi$ could be different from that in
the $\Lambda$CDM model away from the matter-dominated era.

##  VI When matter matters

Finally, we discuss the problem of the coupling to generalized matter fields.
Let us consider the cubic galileon [65] whose action is given by

$\displaystyle
S_{\mathrm{gal}}=\int\mathrm{d}^{4}x\sqrt{-g}\left[-\frac{1}{2}(\partial\varphi)^{2}-\frac{c}{2}(\partial\varphi)^{2}\Box\varphi\right],$
(95)

where $c$ is a nonzero constant. In the following we will show that the extra
dynamical scalar DOF would reappear when one adds to the action (3) the
galileon as a matter field. This fact can be demonstrated by studying a
quadratic action for scalar perturbations around a cosmological background.

Expanding the action (95) to second order in perturbations, we obtain

$\displaystyle S_{\mathrm{gal}}^{(2)}$
$\displaystyle=\int\mathrm{d}t\mathrm{d}^{3}x\bar{N}a^{3}\biggl{[}\left(\varphi_{n}-3cH\varphi_{n}^{2}\right)\delta\varphi\frac{\Delta\chi}{a^{2}}+c\varphi_{n}^{2}\delta\varphi_{n}\frac{\Delta\chi}{a^{2}}-c\varphi_{n}^{3}\delta
n\frac{\Delta\chi}{a^{2}}-c\varphi_{n}^{2}\delta
n\frac{\Delta\delta\varphi}{a^{2}}+\frac{\varphi_{n}^{2}}{2}\left(1-12cH\varphi_{n}\right)\delta
n^{2}$ $\displaystyle\quad-\varphi_{n}\left(1-9cH\varphi_{n}\right)\delta
n\delta\varphi_{n}-3c\varphi_{n}^{3}\delta
n\psi_{n}-3\left(\varphi_{n}-3cH\varphi_{n}^{2}\right)\psi\delta\varphi_{n}+3c\varphi_{n}^{2}\psi_{n}\delta\varphi_{n}+\frac{1}{2}\left(1-6cH\varphi_{n}\right)\delta\varphi_{n}^{2}$
$\displaystyle\quad+\frac{1}{2a^{2}}\left(1-4cH\varphi_{n}-2c\varphi_{nn}\right)\delta\varphi\Delta\delta\varphi\biggr{]},$
(96)

where $\delta\varphi$ is the fluctuation of the galileon field. We add this to
the action (74): $S_{\mathrm{tot}}^{(2)}=S^{(2)}+S^{(2)}_{\mathrm{gal}}$. To
highlight what causes the problem, we do not make any further simplification.

Using the equations of motion for $\delta n$ and $\chi$, one can eliminate
them from the action. The resultant quadratic action is of the form

$\displaystyle
S^{(2)}_{\mathrm{tot}}=\int\mathrm{d}t\mathrm{d}^{3}x\bar{N}a^{3}\left({\cal
K}_{IJ}q^{I}_{n}q^{J}_{n}+\cdots\right),$ (97)

where $q^{I}=\\{\psi,\delta\varphi\\}$ and we wrote only the terms quadratic
in time derivatives. A straightforward calculation yields

$\displaystyle\mathrm{det}\,{\cal K}$
$\displaystyle=-\frac{27[c\beta_{0}(\beta_{1}/\bar{N})H\varphi_{n}^{2}]^{2}}{\Xi},$
(98)

with

$\displaystyle\Xi$
$\displaystyle:=12\beta_{0}^{2}H^{2}-2\beta_{0}\bar{\beta}_{1}^{2}(\bar{\beta}_{1}-\bar{\beta}_{2})\varphi_{n}^{2}$
$\displaystyle\quad+12c\beta_{0}\bar{\beta}_{1}[2\bar{\beta}_{1}(\bar{\beta}_{1}-\bar{\beta}_{2})+\bar{\beta}_{2}]H\varphi_{n}^{3}$
$\displaystyle\quad+3c^{2}\bar{\beta}_{1}^{3}\bar{\beta}_{2}\varphi_{n}^{6}.$
(99)

This result shows that there are two scalar DOFs in the system unless
$c\beta_{1}=0$; the scalar DOF reappears in the presence of a galileon field
if it is minimally coupled to gravity in the frame where $\beta_{1}\neq 0$.

What we have seen here is analogous to the issue pointed out in the context of
degenerate higher-order scalar-tensor theories [66, 67]. In the case of the
degenerate higher-order scalar-tensor theories, the constraint associated with
the degeneracy is lost if a matter field is coupled to the Christoffel symbol.
In the present case, the constraint that removes the scalar DOF in the gravity
sector is lost due to the coupling of the cubic galileon to the Christoffel
symbol.

##  VII Conclusions and outlook

In this paper, we have studied aspects of spatially covariant theories of
gravity with two tensorial degrees of freedom (DOFs), namely, modified gravity
having the same number of DOFs as general relativity (GR). We have mainly
focused on a subset of the general theories developed in Ref. [19] that is
characterized by three time-dependent functions. Two of the three functions
are related to the cuscuton terms [9] expressed in the unitary gauge, while
the remaining one (denoted as $\beta$) is not included even in the extended
cuscuton theory [12]. More specifically, we have worked with the action (8)
with $\alpha_{2}=1$ and $\alpha_{4}=0$ as a particularly interesting subclass
of the general action (3), with the purpose of exploring how the theory can be
distinguished observationally or phenomenologically from GR.

First, we have seen that the theory of modified gravity under consideration
can evade solar-system tests. We have also seen that the speed of
gravitational waves is equal to the speed of light. Therefore, there is no
obvious contradiction with observations and experiments at this point.

Next, we have studied black hole solutions ignoring the cuscuton terms that
are supposed to be relevant only on cosmological scales. Since the only
modification in the action appears in the coefficient of $K^{2}$, where $K$ is
the trace of the extrinsic curvature of constant time hypersurfaces, the
theory admits GR solutions foliated by maximal slices ($K=0$). Therefore, we
have the Schwarzschild solution. The situation here is essentially the same as
that in Einstein-aether theory and the infrared limit of Hořava gravity. We
have considered small deformations of the Schwarzschild solution, showing that
no other asymptotically flat solutions can be obtained by a perturbative
treatment. Our numerical analysis beyond the perturbative treatment supports
the conclusion that the Schwarzschild solution is the only static, spherically
symmetric, and asymptotically flat vacuum solution. We have also considered a
slowly rotating black hole and obtained the Kerr solution in the slow-rotation
approximation.

We have then investigated the cosmological dynamics of the homogeneous and
isotropic background and scalar perturbations. Since we have a large degree of
freedom to choose the time-dependent functions in the action including the
cuscuton part, it is easy to realize the background evolution that is very
close to or even identical to that of the $\Lambda$CDM model in GR. As far as
the long and short wavelength limits are concerned, we have found that the
perturbation dynamics is modified only through the modification of the
background evolution.

Combining these results, we conclude that it is quite difficult to distinguish
modified gravity with the action (8) with $\alpha_{2}=1$ and $\alpha_{4}=0$
observationally or phenomenologically from GR.

Finally, we have noted, as a side remark, that in the presence of a galileon
field in the matter sector, the dynamical scalar degree of freedom eliminated
from the gravitational sector could reappear in a similar way to the case
analyzed in Ref. [66, 67]. The result of our analysis implies that a galileon
field must be coupled nonminimally to gravity in the frame where the action
takes the form of Eq. (8).

Let us comment on several issues that are left for future work.

  * •

Newtonian gravity is reproduced in the weak field regime, and black hole
solutions in GR are obtained in the strong field regime in vacuo. It is then
natural to move on to the study of the structure of relativistic stars.

  * •

It would be interesting to study black hole perturbations and quasinormal
modes, which could help us to distinguish modified gravity with two tensorial
DOFs from GR.

  * •

In the present paper, we have studied cosmological perturbations only in the
long and short wavelength limits. On intermediate scales the perturbation
evolution could differ from that in GR even if the background evolution
coincides with that of the $\Lambda$CDM model. A more detailed investigation
is necessary.

  * •

An application to inflationary cosmology would also be interesting.

These issues will be addressed in our forthcoming papers.

###### Acknowledgements.

The work of AI was supported by the JSPS Research Fellowships for Young
Scientists No. 20J11285. The work of TK was supported by MEXT KAKENHI Grant
Nos. JP20H04745 and JP20K03936.

##  Appendix A Equations of motion

The equations of motion in vacuum derived from the action (8) are given as
follows.

  * •

The Hamiltonian constraint:

$\displaystyle
K_{ij}K^{ij}-\frac{1}{3}\left[2\left(\frac{N}{\beta+N}\right)^{2}+1\right]K^{2}-\alpha_{1}-\alpha_{2}R=0.$
(100)

  * •

The momentum constraints:

$\displaystyle D_{j}\pi^{ij}=0,$ (101)

where

$\displaystyle\pi^{ij}:=K^{ij}-\frac{1}{3}\left(\frac{2N}{\beta+N}+1\right)K\gamma^{ij}.$
(102)

  * •

The evolution equations:

$\displaystyle\frac{1}{N\sqrt{\gamma}}\partial_{t}\left(\sqrt{\gamma}\pi^{kl}\right)\gamma_{ik}\gamma_{jl}-\frac{1}{2}\left(\alpha_{1}+\frac{\alpha_{3}}{N}\right)\gamma_{ij}$
$\displaystyle+2\left[K_{ik}K_{j}^{\;k}-\frac{1}{3}\left(\frac{2N}{\beta+N}+1\right)KK_{ij}\right]$
$\displaystyle-\frac{1}{2}\left[K_{kl}K^{kl}-\frac{1}{3}\left(\frac{2N}{\beta+N}+1\right)K^{2}\right]\gamma_{ij}$
$\displaystyle+\left(\alpha_{2}+\frac{\alpha_{4}}{N}\right)\left(R_{ij}-\frac{1}{2}R\gamma_{ij}\right)$
$\displaystyle+\frac{\alpha_{2}}{N}\left(D^{2}N\gamma_{ij}-D_{i}D_{j}N\right)$
$\displaystyle+\frac{1}{N}\left[D^{k}(\pi_{ik}N_{j})+D^{k}(\pi_{jk}N_{i})-D^{k}(\pi_{ij}N_{k})\right]=0.$
(103)

##  Appendix B Covariant form of the action

In the main text, we use the ADM decomposition and consider the action of the
form (3) which no longer has full diffeomorphism invariance. However, one can
always restore it by introducing a Stückelberg field $\phi(x^{\mu})$. In this
appendix, we derive the covariant form of the action written in terms of the
four-dimensional metric $g_{\mu\nu}$ and $\phi$.

The Stückelberg scalar field is introduced by writing the unit normal to
constant time hypersurfaces as $n_{\mu}=-\phi_{\mu}/\sqrt{2X}$, where
$\phi_{\mu}:=\nabla_{\mu}\phi$ and $X:=-\phi^{\mu}\phi_{\mu}/2$ with
$\nabla_{\mu}$ being the usual four-dimensional covariant derivative. The ADM
variables in the action (3) are then replaced as follows:

$\displaystyle N\to$ $\displaystyle\;\frac{1}{\sqrt{2X}},$ (104)
$\displaystyle K_{ij}\to$ $\displaystyle\;{\cal
K}_{\mu\nu}:=-\frac{\phi_{\mu\nu}}{\sqrt{2X}}+n_{\mu}a_{\nu}+n_{\nu}a_{\mu}$
$\displaystyle\quad\qquad\;+\frac{1}{2X}n_{\mu}n_{\nu}n^{\rho}\nabla_{\rho}X,$
(105) $\displaystyle R\to$ $\displaystyle\;{}^{(4)}{\cal R}-{\cal
K}_{\mu\nu}{\cal K}^{\mu\nu}+{\cal K}^{2}-2\nabla_{\mu}\left({\cal
K}n^{\mu}-a^{\mu}\right),$ (106)

where $\phi_{\mu\nu}:=\nabla_{\mu}\nabla_{\nu}\phi$,
$a_{\mu}:=n^{\rho}\nabla_{\rho}n_{\mu}$, and ${}^{(4)}{\cal R}$ is the four-
dimensional Ricci scalar. One thus obtains

$\displaystyle S$ $\displaystyle=\int\mathrm{d}^{4}x\sqrt{-g}{\cal L},$ (107)
$\displaystyle{\cal L}$
$\displaystyle=\left[\frac{1}{2}\widetilde{B}(\phi,X)-\widetilde{f}(\phi,X)\right]\left({\cal
K}_{\mu\nu}{\cal K}^{\mu\nu}-{\cal K}^{2}\right)$
$\displaystyle\quad+\widetilde{C}(\phi,X){\cal
K}^{2}+\frac{1}{2}\left[\widetilde{\alpha}_{1}(\phi)+\widetilde{\alpha}_{3}(\phi)\sqrt{2X}\right]$
$\displaystyle\quad+\widetilde{f}(\phi,X)\left[{}^{(4)}{\cal
R}-2\nabla_{\mu}\left({\cal K}n^{\mu}-a^{\mu}\right)\right],$ (108)

where

$\displaystyle\widetilde{f}$
$\displaystyle=\frac{1}{2}\left[\widetilde{\alpha}_{2}(\phi)+\widetilde{\alpha}_{4}(\phi)\sqrt{2X}\right],$
(109) $\displaystyle\widetilde{B}$
$\displaystyle=\frac{\widetilde{\beta}_{0}(\phi)}{\widetilde{\beta}_{2}(\phi)\sqrt{2X}+1},$
(110) $\displaystyle\widetilde{C}$
$\displaystyle=\frac{\widetilde{\beta}_{0}(\phi)}{3}\cdot\frac{\bigl{[}\widetilde{\beta}_{1}(\phi)-\widetilde{\beta}_{2}(\phi)\bigr{]}\sqrt{2X}}{\bigl{[}\widetilde{\beta}_{1}(\phi)\sqrt{2X}+1\bigr{]}\bigl{[}\widetilde{\beta}_{2}(\phi)\sqrt{2X}+1\bigr{]}}.$
(111)

The time-dependent functions in the action (3) are given by666The relation
(116) can be used to write the $\phi$-dependent functions explicitly, for
example, in the first concrete model presented in Sec. V.1 (with $w=0$) as
$\displaystyle\widetilde{\alpha}_{1}$
$\displaystyle=6\left\\{\left[\frac{1}{\xi}-\frac{1}{(1+\beta)^{2}}\right]\phi^{2}-\frac{h_{0}^{2}}{\xi}\right\\},$
(112) $\displaystyle\widetilde{\alpha}_{3}$
$\displaystyle=\frac{4}{\phi^{2}-h_{0}^{2}}\left\\{\left[\frac{1}{\xi}-\frac{1}{(1+\beta)^{2}}\right]\phi^{2}-\left(\frac{1}{\xi}-\frac{1}{1+\beta}\right)h_{0}^{2}\right\\},$
(113) $\displaystyle\widetilde{\beta}_{1}$
$\displaystyle=-\frac{2\beta}{3(\phi^{2}-h_{0}^{2})},$ (114)
$\displaystyle\widetilde{\alpha}_{2}$
$\displaystyle=\widetilde{\beta}_{0}=1,\quad\widetilde{\alpha}_{4}=\widetilde{\beta}_{2}=0,$
(115) with the solution for the scalar field
$\phi(t)=h_{0}\coth[(3/2)h_{0}t]$.

$\displaystyle\beta_{0}(t)=\widetilde{\beta}_{0}(\phi(t)),\quad\beta_{1,2}(t)=\dot{\phi}(t)\widetilde{\beta}_{1,2}(\phi(t)),$
$\displaystyle\alpha_{1,2}(t)=\widetilde{\alpha}_{1,2}(\phi(t)),\quad\alpha_{3,4}(t)=\dot{\phi}(t)\widetilde{\alpha}_{3,4}(\phi(t)).$
(116)

With some manipulation, the Lagrangian can be written in a more suggestive
form as

$\displaystyle{\cal L}$
$\displaystyle=\frac{\widetilde{\alpha}_{1}}{2}-\widetilde{\alpha}_{2}^{\prime\prime}(2X-X\ln
X)+\frac{\widetilde{\alpha}_{3}}{2}\sqrt{2X}+\widetilde{\alpha}_{4}^{\prime\prime}(2X)^{3/2}$
$\displaystyle\quad-\left(\frac{\widetilde{\alpha}_{2}^{\prime}}{2}\ln
X+2\widetilde{\alpha}_{4}^{\prime}\sqrt{2X}\right)\Box\phi+\widetilde{f}{}^{(4)}{\cal
R}+{\cal L}_{\mathrm{quad}},$ (117)

where a prime here denotes differentiation with respect to $\phi$ and

$\displaystyle{\cal L}_{\mathrm{quad}}$
$\displaystyle=A_{1}\phi_{\mu\nu}\phi^{\mu\nu}+A_{2}(\Box\phi)^{2}+A_{3}\Box\phi\phi^{\mu}\phi_{\mu\nu}\phi^{\nu}$
$\displaystyle\quad+A_{4}\phi^{\mu}\phi_{\mu\rho}\phi^{\rho\nu}\phi_{\nu}+A_{5}(\phi^{\mu}\phi_{\mu\nu}\phi^{\nu})^{2},$
(118)

with

$\displaystyle A_{1}=\frac{\widetilde{B}}{4X}-\frac{\widetilde{f}}{2X},\quad
A_{2}=-A_{1}+4X^{2}A_{5},$ $\displaystyle A_{3}=-A_{4}+4XA_{5},\quad
A_{4}=\frac{\widetilde{f}_{,X}+A_{1}}{X},$ $\displaystyle
A_{5}=\frac{\widetilde{C}}{8X^{3}}.$ (119)

This is a particular case of the U-degenerate theories, i.e., higher-order
scalar-tensor theories that are degenerate when restricted to the unitary
gauge [68]. It is easy to check that the above Lagrangian does not satisfy the
degeneracy conditions in an arbitrary gauge if $\widetilde{C}\neq 0$. In
general, there is one dynamical scalar DOF in a U-degenerate theory in the
unitary gauge. However, with the above particular form of the Lagrangian, it
turns out that $\phi$ does not propagate.

To demonstrate that $\phi$ does not propagate, let us study the equation of
motion for $\phi$ in a Friedmann‐Lemaître‐Robertson-Walker universe with the
metric $\mathrm{d}s^{2}=-\mathrm{d}t^{2}+a^{2}(t)\mathrm{d}\vec{x}^{2}$. For
simplicity, we consider the case with $\alpha_{2}=\beta_{0}=1$ and
$\alpha_{4}=\beta_{2}=0$. The equation of motion for $\phi$ then reads

$\displaystyle\widetilde{\alpha}_{1}^{\prime}\dot{\phi}-3H\widetilde{\alpha}_{3}\dot{\phi}-\frac{6\widetilde{\beta}_{1}\dot{\phi}H}{\widetilde{\beta}_{1}\dot{\phi}+1}{\cal
P}=0,$ (120)

where

$\displaystyle{\cal
P}:=2\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{H}{\widetilde{\beta}_{1}\dot{\phi}+1}\right)+\frac{3H^{2}}{\widetilde{\beta}_{1}\dot{\phi}+1}.$
(121)

This equation does contain $\ddot{\phi}$ as well as $\dot{H}$. However, the
$(ij)$ components of the field equations have the form,

$\displaystyle{\cal
P}+\frac{1}{2}\left(\widetilde{\alpha}_{1}+\widetilde{\alpha}_{3}\dot{\phi}\right)+\cdots=0,$
(122)

where the ellipsis denotes the pressure of other matter fields. The evolution
equations for the metric and $\phi$ are thus degenerate as in extended
cuscuton theories [12], implying that the scalar DOF is in fact not dynamical.

Let us turn to discuss black hole solutions in the case where

$\displaystyle\widetilde{\beta}_{1}=\mathrm{const},\quad\widetilde{\beta}_{2}=0,\quad\widetilde{\alpha}_{1}=\alpha_{1}=\mathrm{const},$
$\displaystyle\widetilde{\alpha}_{2}=\alpha_{2}=\mathrm{const},\quad\widetilde{\alpha}_{3}=\widetilde{\alpha}_{4}=0.$
(123)

Concerning black hole solutions, working in the covariant action results in
much more involved equations. However, it is not difficult to see that the
field equations admit the following configuration of the metric and $\phi$:

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-h(r)\mathrm{d}T^{2}+\frac{\mathrm{d}r^{2}}{h(r)}+r^{2}\mathrm{d}\Omega^{2},$
(124) $\displaystyle\phi(T,r)$
$\displaystyle=\frac{1}{N_{0}}\left(T+\int^{r}\frac{b_{0}/r^{2}}{h\sqrt{h+b_{0}^{2}/r^{4}}}\mathrm{d}r\right),$
(125)

where

$\displaystyle
h(r):=1+\frac{\alpha_{1}}{6\alpha_{2}}r^{2}-\frac{\mu_{0}}{r}+\frac{1-\alpha_{2}}{\alpha_{2}}\frac{b_{0}^{2}}{r^{4}}.$
(126)

Note that $\phi$ is allowed to have the linear $T$ dependence thanks to the
shift symmetry $\phi\to\phi+\,$const. This is a so-called stealth solution
(when $\alpha_{2}=1$), but, in contrast to the familiar examples in the
literature, we have $X\neq\,$const in the present case. (See Ref. [69] for a
comprehensive discussion on stealth of solutions in quadratic degenerate
higher-order scalar-tensor theories.) Moving to the unitary gauge,
$\phi(T,r)=t$, this solution reproduces the one obtained in Sec. IV.1.

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