# Spatial covariant gravity with two degrees of freedom in the presence of an
auxiliary scalar field: perturbation analysis

Zhi-Chao Wang  Xian Gao<EMAIL_ADDRESS>School of Physics and
Astronomy, Sun Yat-sen University, Zhuhai 519082, China

###### Abstract

We investigate a class of gravity theories respecting only spatial covariance,
termed spatially covariant gravity, in the presence of an auxiliary scalar
field. We examine the conditions on the Lagrangian required to eliminate
scalar degrees of freedom, allowing only two tensorial degrees of freedom to
propagate. Instead of strict constraint analysis, in this work, we employ the
perturbation method and focus on the necessary conditions to evade the scalar
mode at linear order in perturbations around a cosmological background.
Starting from a general action and solving the auxiliary perturbation
variables in terms of the would-be dynamical scalar mode, we derive the
condition to remove its kinetic term, thus ensuring that no scalar mode
propagates. As an application of the general condition, we study a polynomial-
type Lagrangian as a concrete example, in which all the monomials are
spatially covariant scalars containing two derivatives. We find that the
auxiliary scalar field plays a nontrivial role, and new terms in the
Lagrangian are allowed. Our analysis sheds light on constructing gravity
theories with two degrees of freedom in the extended framework of spatially
covariant gravity.

##  I Introduction

Recently, there has been increasing interest in building gravitational
theories that differ from general relativity (GR) while propagating the same
dynamical degrees of freedom of GR. These degrees of freedom arise as
tensorial gravitational waves in a homogeneous and isotropic background, which
we thus dub the two tensorial degrees of freedom (TTDOFs). Apparently, it
seems impossible since Lovelock’s theorem [1, 2] claims that GR is the unique
metric theory that respects spacetime diffeomorphism and possesses second-
order equations of motion in 4 dimensions, which guarantee the propagation of
only TTDOFs. Nevertheless, several theories that propagate only TTDOFs have
been proposed by bypassing the conditions of Lovelock’s theorem.

Theories of TTDOFs can be traced back to the so-called Cuscuton theory [3],
which was extensively studied [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]
and was further extended in the framework of higher derivative scalar-tensor
theory [17, 18]. Previously, such a kind of theory has also been discussed in
a subclass of Hořava gravity [19, 20] (see also [21]), with which the relation
of Cuscuton has been analyzed [22]. Symmetries behind these theories,
including the relation between the TTDOFs (i.e., the transverse and traceless
graviton) and the spacetime symmetry [23, 24], as well as the “scalarless”
symmetry [25, 26], have also been investigated111Another attempt was proposed
in [27] by taking the $D\rightarrow 4$ limit of Einstein-Gauss-Bonnet gravity
in higher dimensions, which yields an arguable theory propagating only the
massless graviton..

An alternative framework of theories propagating only two degrees of freedom
is dubbed minimally modified gravity (MMG) [28]. The so-called type-II MMG,
which differs from GR in the vacuum, is of particular importance [29, 30],
which includes the original Cuscuton theory, the minimal theory of massive
gravity [31, 32] (see also [33, 34, 35, 36]), as well as the aforementioned
$4D$ Einstein-Gauss-Bonnet gravity as special cases. The MMG has been studied
extensively [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52],
and has been generalized in the phase space by introducing auxiliary
constraints [53, 54].

A general framework of spatially covariant gravity (SCG) theories that
propagate TTDOFs with an extra scalar mode was proposed in [55, 56], and was
further generalized by introducing a dynamical lapse function [57, 58] and
with nonmetricity [59]. The SCG can be viewed as an alternative approach to
generalizing the scalar-tensor theories [60, 61, 62, 63], and includes Hořava
gravity [64, 65, 66], the effective field theory of inflation/dark energy [67,
68, 69, 70, 71] as special cases. The SCG has been applied in the study of
cosmology and gravitational waves [72, 73, 74, 75, 76]. Within the framework
of SCG, conditions on the Lagrangian such that only the TTDOFs are propagating
were explored in [77, 78] and with a dynamical lapse function in [79]. Due to
the complexity of the conditions, a systematic construction of the TTDOF
Lagrangians is still to be performed. Nevertheless, a concrete Lagrangian was
found in [77], which has been applied in the study of cosmology [80, 81, 82,
10, 83].

Instead of solving the complicated variation-differential equations for the
TTDOF conditions in [77], an alternative and more straightforward approach is
the perturbation method. The idea is to focus on the kinetic term of the
would-be dynamical scalar mode. By eliminating its kinetic term order by order
in a perturbative expansion around some (typically the cosmological)
background, one can fix terms in the Lagrangian order by order. In particular,
such a perturbative approach will stop at some finite order since there is a
finite number of TTDOF conditions. This perturbation method has been used in
[17] to extend the Cuscuton theory in the framework of higher derivative
scalar-tensor theory. In the framework of SCG, it has also been successfully
applied in [58] to derive the Lagrangian with a single scalar mode in the
presence of a dynamical lapse function, and in [78] to build the Lagrangian
without any scalar mode, i.e., propagating only TTDOFs.

In this work, we will apply the perturbation method to the SCG in the presence
of an auxiliary scalar field and determine the conditions such that the theory
propagates only TTDOFs. The idea of SCG with an auxiliary scalar field first
arises in [84], in which the scalar field was assumed to possess a spacelike
gradient and thus loses its kinetic term in the so-called “spatial gauge”. In
this work, we release this assumption and simply treat it as an auxiliary
field. In the presence of an auxiliary scalar field, one may have much more
room to tune the theory to satisfy the TTDOF conditions and may expect to find
a novel class of TTDOF theories. This paper is thus devoted to this issue.

This paper is organized as follows. In Sec. II, we describe our model of
spatially covariant gravity with an auxiliary scalar field and present the
scalar perturbations around a cosmological background. In Sec. III, we derive
the quadratic action for the scalar perturbations. By solving the auxiliary
perturbation variables and focusing on the kinetic term of the unwanted scalar
mode, we derive the degeneracy condition to evade the scalar mode. In Sec. IV,
we use the polynomial-type Lagrangian of $d=2$ with $d$ being the total number
of derivatives as a concrete example and derive the explicit Lagrangian that
propagates TTDOFs. In Sec. V, we summarize our results.

##  II The perturbation method

###  II.1 Model and cosmological perturbations

Our starting point is the general action of spatial covariant gravity coupled
with a non-dynamical scalar field [84]:

$S=\int\mathrm{d}t\mathrm{d}^{3}xN\sqrt{h}\mathcal{L}\left(h_{ij},K_{ij},R_{ij},N,\phi,\mathrm{D}_{i}\right),$
(1)

where $h_{ij}$ is the spatial metric, $N$ is the lapse function, $R_{ij}$ is
the spatial Ricci tensor, $K_{ij}$ is the extrinsic curvature defined by

$K_{ij}=\frac{1}{2N}\left(\partial_{t}h_{ij}-\mathrm{D}_{i}N_{j}-\mathrm{D}_{j}N_{i}\right),$
(2)

with $N_{i}$ as the shift vector and $\mathrm{D}_{i}$ as the spatial
derivative adapted to $h_{ij}$. The scalar field $\phi$ has no kinetic term
and thus acts as an auxiliary variable. We emphasize that although this
auxiliary scalar field was originally motivated from a scalar field with a
spacelike gradient in the so-called spatial gauge, here (and also in the
analysis in [84]) we do not make such an assumption and merely treat it as an
auxiliary field, i.e., without any time derivative terms. It has been shown in
[84] that generally the theory (1) propagates 3 degrees of freedom, i.e., 2
tensorial and 1 scalar degrees of freedom. The purpose of this work is to find
the condition on the Lagrangian in order to eliminate the scalar mode.

The most strict approach to analyzing the number of degrees of freedom is the
constraint analysis, either in the Hamiltonian or in the Lagrangian formalism.
In the absence of the auxiliary scalar field $\phi$, the exact conditions for
the TTDOFs have been found in [77]. These conditions are variation-
differential equations for the Lagrangian, which are difficult (if not
impossible) to solve to give the concrete Lagrangian systematically. To
address this problem, a perturbation method was employed in [78] (and
previously in [58] with a dynamical lapse function). The advantage of the
perturbation method is that one can determine the conditions on the Lagrangian
order by order in perturbations, which by themselves are much simpler than the
fully nonlinear conditions.

In the following, we will use this perturbation method to determine the
conditions on the action (1) such that no scalar mode propagates. To be
precise, our task is to study the perturbations of the action (1) around a
cosmological background and eliminate the scalar modes at the linear order in
perturbations. The difference of this work from [78, 58] is that here we will
first perform the perturbation analysis on the general form of Lagrangian
instead of a specific Lagrangian (e.g., in the polynomial form). Therefore,
the resulting conditions are generic and can be applied to more general
Lagrangians.

For our purpose of eliminating the unwanted scalar degree of freedom, we focus
solely on the scalar perturbations. We consider perturbations around a
Friedmann-Robertson-Walker background. After fixing the residual gauge freedom
of spatial diffeomorphism, the Arnowitt-Deser-Misner (ADM) variables are
parameterized as follows:

$\displaystyle N$ $\displaystyle=$ $\displaystyle\bar{N}e^{A},$ (3)
$\displaystyle N_{i}$ $\displaystyle=$ $\displaystyle\bar{N}a\partial_{i}B,$
(4) $\displaystyle h_{ij}$ $\displaystyle=$ $\displaystyle
a^{2}e^{2\zeta}\delta_{ij},$ (5)

where $a$ is the scale factor. Here, $A$, $B$, and $\zeta$ are scalar
perturbations of the metric. The auxiliary scalar field is perturbed as usual:

$\phi=\bar{\phi}+\delta\phi.$ (6)

All perturbation variables are set to zero at the background level, i.e.,
$A=B=\zeta=\delta\phi=0$. $\bar{N}$ and $\bar{\phi}$ are the background values
of the lapse function and scalar field, respectively, which are functions of
time. Note that due to the loss of time reparametrization invariance, we
cannot fix the background value of the lapse function $\bar{N}$ to unity. In
this paper, we use the same shorthand as in [72]:

$\dot{X}=\frac{1}{\bar{N}}\frac{\partial X}{\partial t},$ (7)

and

$f^{\prime}=\bar{N}\left.\frac{\partial f}{\partial
N}\right|_{N=\bar{N}},\quad
f^{\prime\prime}=\bar{N}^{2}\left.\frac{\partial^{2}f}{\partial
N^{2}}\right|_{N=\bar{N}},$ (8)

etc.

To obtain the action up to the second order in perturbations, we first expand
all the quantities in the following form

$Q=Q^{(0)}+\delta Q^{(1)}+\delta Q^{(2)},$ (9)

where $Q^{(0)}$ represents the background value of any quantity $Q$, and
$\delta Q^{(n)}$ represents the $n$-th order perturbation of $Q$. Precisely,
we have

$\phi^{(0)}=\bar{\phi},\quad\delta\phi^{(1)}=\delta\phi,$ (10)
$N^{(0)}=\bar{N},\quad\delta N^{(1)}=\bar{N}A,\quad\delta
N^{(2)}=\frac{1}{2}\bar{N}A^{2},$ (11) $N^{i(0)}=0,\quad\delta
N^{i(1)}=\bar{N}a^{-1}\partial^{i}B,$ (12)
$h_{ij}^{(0)}=a^{2}\delta_{ij},\quad\delta h_{ij}^{(1)}=2\zeta
a^{2}\delta_{ij},\quad\delta h_{ij}^{(2)}=2\zeta^{2}a^{2}\delta_{ij},$ (13)
$K_{ij}^{(0)}=a\dot{a}\delta_{ij},\quad\delta
K_{ij}^{(1)}=a\dot{a}\delta_{ij}\left(-A+2\zeta\right)+a^{2}\delta_{ij}\dot{\zeta}-a\partial_{i}\partial_{j}B,$
(14) $\displaystyle\delta K_{ij}^{(2)}$ $\displaystyle=$ $\displaystyle
a\dot{a}\delta_{ij}\left(-2\zeta
A+\frac{1}{2}A^{2}+2\zeta^{2}\right)+a^{2}\delta_{ij}\dot{\zeta}\left(2\zeta-A\right)$
(15)
$\displaystyle+aA\partial_{i}\partial_{j}B+a\left(\partial_{i}\zeta\partial_{j}B+\partial_{j}\zeta\partial_{i}B-\delta_{ij}\partial^{l}B\partial_{l}\zeta\right),$
$R_{ij}^{(0)}=0,\quad\delta
R_{ij}^{(1)}=-\partial_{i}\partial_{j}\zeta-\delta_{ij}\partial^{2}\zeta,$
(16)

and

$\delta
R_{ij}^{(2)}=\partial_{i}\zeta\partial_{j}\zeta-\delta_{ij}\partial_{l}\zeta\partial^{l}\zeta.$
(17)

In the above and in what follows, spatial indices are raised and lowered by
$\delta^{ij}$ or $\delta_{ij}$, respectively. It should be noted that only
linear-order terms are considered for the perturbations of the auxiliary
scalar field $\phi$ and shift vector $N^{i}$. With the above results, we can
proceed to calculate the perturbations of the action in the subsequent
sections.

###  II.2 Background equations of motion

We first derive the background equations of motion, which can be used to
simplify calculations in higher orders. To this end, we need to obtain the
linear-order action $S_{1}$:

$S_{1}=\int\mathrm{d}t\mathrm{d}^{3}x\left.\frac{\delta
S}{\delta\Phi_{I}}\right|_{\mathrm{bg}}\delta\Phi_{I}^{(1)},$ (18)

where

$\Phi_{I}\equiv\left\\{h_{ij},K_{ij},R_{ij},N,\phi\right\\}$ (19)

stands for the basic quantities in our model (1), and “$Q|_{\mathrm{bg}}$”
denotes the background value of any quantity $Q$. We have

$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\int\mathrm{d}t\mathrm{d}^{3}x\left[\bar{N}a^{3}\left.\frac{\delta\mathcal{L}}{\delta
K_{ij}}\right|_{\mathrm{bg}}\delta
K_{ij}^{(1)}+\left(\bar{N}\frac{1}{2}a\delta^{ij}\bar{\mathcal{L}}+\bar{N}a^{3}\left.\frac{\delta\mathcal{L}}{\delta
h_{ij}}\right|_{\mathrm{bg}}\right)\delta
h_{ij}^{(1)}+\bar{N}a^{3}\left.\frac{\delta\mathcal{L}}{\delta\phi}\right|_{\mathrm{bg}}\delta\phi\right.$
(20)
$\displaystyle\left.+\left(a^{3}\bar{\mathcal{L}}+\bar{N}a^{3}\left.\frac{\delta\mathcal{L}}{\delta
N}\right|_{\mathrm{bg}}\right)\delta
N^{(1)}+\bar{N}a^{3}\left.\frac{\delta\mathcal{L}}{\delta
R_{ij}}\right|_{\mathrm{bg}}\delta R_{ij}^{(1)}\right],$

where $\bar{\mathcal{L}}$ represents the background Lagrangian density. We
emphasize that in (20) (and in what follows), when making the variation,
$h_{ij}$, $K_{ij}$, $R_{ij}$, $N$, and $\phi$ are treated as independent
variables222For example, although $K_{ij}$, $R_{ij}$, etc., contain $h_{ij}$
implicitly, when evaluating $\frac{\delta\mathcal{L}}{\delta h_{ij}}$, only
the explicit functional dependence of $\mathcal{L}$ on $h_{ij}$ will be taken
into account.. Note that at the background level, $N=\bar{N}$, $N_{i}=0$,
$h_{ij}=a^{2}\delta_{ij}$, and $\phi=\bar{\phi}$. Plugging (10)-(17) into
(20), after some simplifications, we get

$S_{1}[A,\zeta,\delta\phi]=\int\mathrm{d}t\mathrm{d}^{3}x\bar{N}a^{3}\left(\mathcal{E}_{A}A+\mathcal{E}_{\zeta}\zeta+\mathcal{E}_{\delta\phi}\delta\phi\right),$
(21)

where we define

$\mathcal{E}_{A}=\bar{\mathcal{L}}+\bar{N}\left.\frac{\delta\mathcal{L}}{\delta
N}\right|_{\mathrm{bg}}-a^{2}H\delta_{ij}\left.\frac{\delta\mathcal{L}}{\delta
K_{ij}}\right|_{\mathrm{bg}},$ (22)

with $H\equiv\dot{a}/a\equiv\partial_{t}a/(a\bar{N})$,

$\mathcal{E}_{\zeta}=3\bar{\mathcal{L}}+a^{2}\delta_{ij}\left[2\left.\frac{\delta\mathcal{L}}{\delta
h_{ij}}\right|_{\mathrm{bg}}-3H\left.\frac{\delta\mathcal{L}}{\delta
K_{ij}}\right|_{\mathrm{bg}}-\frac{1}{\bar{N}}\frac{\partial}{\partial
t}\left(\left.\frac{\delta\mathcal{L}}{\delta
K_{ij}}\right|_{\mathrm{bg}}\right)-\frac{\dot{\bar{N}}}{\bar{N}}\left.\frac{\delta\mathcal{L}}{\delta
K_{ij}}\right|_{\mathrm{bg}}\right],$ (23)

and

$\mathcal{E}_{\delta\phi}=\left.\frac{\delta\mathcal{L}}{\delta\phi}\right|_{\mathrm{bg}}.$
(24)

Requiring the linear variation of the background action to be vanishing, i.e.,
$S_{1}[A,\zeta,\delta\phi]=0$, yields:

$\mathcal{E}_{A}=0,\quad\mathcal{E}_{\zeta}=0,\quad\mathcal{E}_{\delta\phi}=0,$
(25)

which are the background equations of motion. At this point, note that since
SCG explicitly breaks the time diffeomorphism, all three of these equations
are independent. This differs from the case of generally covariant theories,
in which only two of the three equations are independent due to the time
diffeomorphism.

##  III Quadratic action and the degeneracy condition

The quadratic action for the perturbations receives contributions from both
first-order and second-order quantities. Its general form is given by:

$S_{2}[A,B,\zeta,\delta\phi]=\int\mathrm{d}t\mathrm{d}^{3}x\left(\left.\frac{\delta
S}{\delta\Phi_{I}}\right|_{\mathrm{bg}}\delta\Phi_{I}^{(2)}+\frac{1}{2}\left.\frac{\delta^{2}S}{\delta\Phi_{I}\delta\Phi_{J}}\right|_{\mathrm{bg}}\delta\Phi_{I}^{(1)}\delta\Phi_{J}^{(1)}\right),$
(26)

where (from now on we suppress the subscript “bg” for simplicity)

$\displaystyle\int\mathrm{d}t\mathrm{d}^{3}x\frac{\delta
S}{\delta\Phi_{I}}\delta\Phi_{I}^{(2)}$ $\displaystyle=$
$\displaystyle\int\mathrm{d}t\mathrm{d}^{3}x\left[\frac{\delta\left(N\sqrt{h}\mathcal{L}\right)}{\delta
K_{ij}}\delta
K_{ij}^{(2)}+\frac{\delta\left(N\sqrt{h}\mathcal{L}\right)}{\delta
h_{ij}}\delta h_{ij}^{(2)}\right.$ (27)
$\displaystyle\qquad\left.+\frac{\delta\left(N\sqrt{h}\mathcal{L}\right)}{\delta
N}\delta N^{(2)}+\frac{\delta\left(N\sqrt{h}\mathcal{L}\right)}{\delta
R_{ij}}\delta R_{ij}^{(2)}\right],$

and

$\displaystyle\int\mathrm{d}t\mathrm{d}^{3}x\frac{\delta^{2}S}{\delta\Phi_{I}\delta\Phi_{J}}\delta\Phi_{I}^{(1)}\delta\Phi_{J}^{(1)}$
(28) $\displaystyle=$
$\displaystyle\int\mathrm{d}t\mathrm{d}^{3}x\Bigg{[}2\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta
h_{mn}\delta K_{ij}}\delta h_{mn}^{(1)}\delta
K_{ij}^{(1)}+\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta
h_{mn}\delta h_{ij}}\delta h_{mn}^{(1)}\delta
h_{ij}^{(1)}+\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta
K_{mn}\delta K_{ij}}\delta K_{mn}^{(1)}\delta K_{ij}^{(1)}$
$\displaystyle+2\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta\phi\delta
K_{ij}}\delta\phi\delta
K_{ij}^{(1)}+2\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta
N\delta K_{ij}}\delta N^{(1)}\delta
K_{ij}^{(1)}+2\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta\phi\delta
h_{ij}}\delta\phi\delta h_{ij}^{(1)}$
$\displaystyle+2\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta
N\delta h_{ij}}\delta N^{(1)}\delta
h_{ij}^{(1)}+2\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta\phi\delta
N}\delta\phi\delta
N^{(1)}+\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta\phi^{2}}\delta\phi\delta\phi$
$\displaystyle+\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta
N^{2}}\delta N^{(1)}\delta
N^{(1)}+2\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta
h_{mn}\delta R_{ij}}\delta h_{mn}^{(1)}\delta
R_{ij}^{(1)}+2\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta
K_{mn}\delta R_{ij}}\delta K_{mn}^{(1)}\delta R_{ij}^{(1)}$
$\displaystyle+2\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta
N\delta R_{ij}}\delta N^{(1)}\delta
R_{ij}^{(1)}+\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta
R_{mn}\delta R_{ij}}\delta R_{mn}^{(1)}\delta
R_{ij}^{(1)}+2\frac{\delta^{2}\left(N\sqrt{h}\mathcal{L}\right)}{\delta\phi\delta
R_{ij}}\delta\phi\delta R_{ij}^{(1)}\Bigg{]}.$

The quadratic action for the scalar perturbations takes the general form

$\displaystyle S_{2}\left[A,B,\zeta,\delta\phi\right]$ $\displaystyle=$
$\displaystyle\int\mathrm{d}t\mathrm{d}^{3}x\bar{N}a^{3}\left(A\hat{\mathcal{O}}_{AA}A+B\hat{\mathcal{O}}_{BB}B+\zeta\hat{\mathcal{O}}_{\zeta\zeta}\zeta+\delta\phi\hat{\mathcal{O}}_{\phi\phi}\delta\phi+A\hat{\mathcal{O}}_{AB}B+A\hat{\mathcal{O}}_{A\zeta}\zeta+A\hat{\mathcal{O}}_{A\phi}\delta\phi\right.$
(29)
$\displaystyle\left.+\delta\phi\hat{\mathcal{O}}_{\phi\zeta}\zeta+\delta\phi\hat{\mathcal{O}}_{\phi
B}B+\zeta\hat{\mathcal{O}}_{\zeta
B}B+\dot{\zeta}\hat{\mathcal{O}}_{\dot{\zeta}A}A+\dot{\zeta}\hat{\mathcal{O}}_{\dot{\zeta}B}B+\dot{\zeta}\hat{\mathcal{O}}_{\dot{\zeta}\zeta}\zeta+\dot{\zeta}\hat{\mathcal{O}}_{\dot{\zeta}\phi}\delta\phi+\dot{\zeta}\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}\dot{\zeta}\right),$

where $\hat{\mathcal{O}}_{XY}$ represents the operator involving perturbation
variables $X$ and $Y$ (e.g., $\hat{\mathcal{O}}_{A\phi}$ represents the
operator involving $A$ and $\delta\phi$). Generally,
$\hat{\mathcal{O}}_{XY}$’s are time-dependent and may contain spatial
derivatives. By plugging (10)-(17) into (27) and (28) and after some tedious
manipulations, we get:

$\hat{\mathcal{O}}_{AA}=\frac{1}{2}\bar{\mathcal{L}}-\frac{1}{2}a\dot{a}\delta_{ij}\frac{\delta\mathcal{L}}{\delta
K_{ij}}+\frac{1}{2}a^{2}\dot{a}^{2}\delta_{mn}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta K_{ij}}+\frac{3}{2}\bar{N}\frac{\delta\mathcal{L}}{\delta
N}-\bar{N}a\dot{a}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta N\delta
K_{ij}}+\frac{1}{2}\bar{N}^{2}\frac{\delta^{2}\mathcal{L}}{\delta N^{2}},$
(30)
$\hat{\mathcal{O}}_{BB}=\frac{1}{2}a^{2}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta K_{ij}}\partial_{m}\partial_{n}\partial_{i}\partial_{j},$ (31)
$\displaystyle\hat{\mathcal{O}}_{\zeta\zeta}$ $\displaystyle=$
$\displaystyle\frac{9}{2}\bar{\mathcal{L}}+8a^{2}\delta_{ij}\frac{\delta\mathcal{L}}{\delta
h_{ij}}+2a^{4}\delta_{mn}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta
h_{mn}\delta h_{ij}}+8a\dot{a}\delta_{ij}\frac{\delta\mathcal{L}}{\delta
K_{ij}}$ (32)
$\displaystyle+4a^{3}\dot{a}\delta_{ij}\delta_{mn}\frac{\delta^{2}\mathcal{L}}{\delta
h_{mn}\delta
K_{ij}}+2a^{2}\dot{a}^{2}\delta_{mn}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta K_{ij}}$
$\displaystyle+\frac{1}{2}\frac{\delta^{2}\mathcal{L}}{\delta R_{mn}\delta
R_{ij}}\left(\partial_{i}\partial_{j}\partial_{m}\partial_{n}+2\delta_{ij}\partial_{m}\partial_{n}\partial^{2}+\delta_{ij}\delta_{mn}\partial^{4}\right)-2a\dot{a}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta R_{ij}}\delta_{mn}\partial_{i}\partial_{j}$
$\displaystyle-2a^{2}\delta_{mn}\frac{\delta^{2}\mathcal{L}}{\delta
h_{mn}\delta
R_{ij}}\left(\partial_{i}\partial_{j}+\delta_{ij}\partial^{2}\right)-2\frac{\delta\mathcal{L}}{\delta
R_{ij}}\left(2\partial_{i}\partial_{j}+\delta_{ij}\partial^{2}\right),$
$\hat{\mathcal{O}}_{\phi\phi}=\frac{1}{2}\frac{\delta^{2}\mathcal{L}}{\delta\phi^{2}},$
(33)
$\hat{\mathcal{O}}_{AB}=\left(-\bar{N}a\frac{\delta^{2}\mathcal{L}}{\delta
N\delta K_{ij}}+a^{2}\dot{a}\delta_{mn}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta K_{ij}}\right)\partial_{i}\partial_{j},$ (34)
$\displaystyle\hat{\mathcal{O}}_{A\zeta}$ $\displaystyle=$ $\displaystyle
3\bar{\mathcal{L}}+2a^{2}\delta_{ij}\frac{\delta\mathcal{L}}{\delta
h_{ij}}+3\bar{N}\frac{\delta\mathcal{L}}{\delta
N}+2\bar{N}a^{2}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta N\delta
h_{ij}}-3a\dot{a}\delta_{ij}\frac{\delta\mathcal{L}}{\delta K_{ij}}$ (35)
$\displaystyle-2a^{3}\dot{a}\delta_{ij}\delta_{mn}\frac{\delta^{2}\mathcal{L}}{\delta
h_{mn}\delta
K_{ij}}+2\bar{N}a\dot{a}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta N\delta
K_{ij}}-2a^{2}\dot{a}^{2}\delta_{mn}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta K_{ij}}$ $\displaystyle-\left(\frac{\delta\mathcal{L}}{\delta
R_{ij}}+\bar{N}\frac{\delta^{2}\mathcal{L}}{\delta N\delta
R_{ij}}\right)\left(\partial_{i}\partial_{j}+\delta_{ij}\partial^{2}\right)+a\dot{a}\delta_{mn}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta R_{ij}}\partial_{i}\partial_{j},$
$\hat{\mathcal{O}}_{A\phi}=\frac{\delta\mathcal{L}}{\delta\phi}+\bar{N}\frac{\delta^{2}\mathcal{L}}{\delta\phi\delta
N}-a\dot{a}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta\phi\delta K_{ij}},$
(36)
$\hat{\mathcal{O}}_{\phi\zeta}=3\frac{\delta\mathcal{L}}{\delta\phi}+2a^{2}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta\phi\delta
h_{ij}}+2a\dot{a}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta\phi\delta
K_{ij}}-\frac{\delta^{2}\mathcal{L}}{\delta\phi\delta
R_{ij}}\left(\partial_{i}\partial_{j}+\delta_{ij}\partial^{2}\right),$ (37)
$\hat{\mathcal{O}}_{\phi B}=-a\frac{\delta^{2}\mathcal{L}}{\delta\phi\delta
K_{ij}}\partial_{i}\partial_{j},$ (38) $\displaystyle\hat{\mathcal{O}}_{\zeta
B}$ $\displaystyle=$ $\displaystyle-5a\frac{\delta\mathcal{L}}{\delta
K_{ij}}\partial_{i}\partial_{j}+a\frac{\delta\mathcal{L}}{\delta
K_{ij}}\delta_{ij}\partial^{2}-2a^{3}\delta_{mn}\frac{\delta^{2}\mathcal{L}}{\delta
h_{mn}\delta
K_{ij}}\partial_{i}\partial_{j}-2a^{2}\dot{a}\delta_{mn}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta K_{ij}}\partial_{i}\partial_{j},$ (39)
$\displaystyle+a\frac{\delta^{2}\mathcal{L}}{\delta K_{mn}\delta
R_{ij}}\left(\partial_{i}\partial_{j}\partial_{m}\partial_{n}+\delta_{ij}\partial^{2}\partial_{m}\partial_{n}\right),$
$\hat{\mathcal{O}}_{\dot{\zeta}A}=-a^{3}\dot{a}\delta_{mn}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta
K_{ij}}+\bar{N}a^{2}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta N\delta
K_{ij}},$ (40)
$\hat{\mathcal{O}}_{\dot{\zeta}B}=-a^{3}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta K_{ij}}\delta_{mn}\partial_{i}\partial_{j},$ (41)
$\hat{\mathcal{O}}_{\dot{\zeta}\zeta}=5a^{2}\delta_{ij}\frac{\delta\mathcal{L}}{\delta
K_{ij}}+2a^{4}\delta_{ij}\delta_{mn}\frac{\delta^{2}\mathcal{L}}{\delta
h_{mn}\delta
K_{ij}}+2a^{3}\dot{a}\delta_{mn}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta K_{ij}}-a^{2}\delta_{mn}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta
R_{ij}}\left(\partial_{i}\partial_{j}+\delta_{ij}\partial^{2}\right),$ (42)
$\hat{\mathcal{O}}_{\dot{\zeta}\phi}=a^{2}\delta_{ij}\frac{\delta^{2}\mathcal{L}}{\delta\phi\delta
K_{ij}},$ (43)

and

$\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}=\frac{1}{2}a^{4}\frac{\delta^{2}\mathcal{L}}{\delta
K_{mn}\delta K_{ij}}\delta_{mn}\delta_{ij}.$ (44)

By deriving the above results, we have made use of integrations by parts to
simplify the expressions.

By varying the quadratic action (29), one can easily obtain the equations of
motion for the perturbation variables $\left\\{A,B,\delta\phi,\zeta\right\\}$
as

$\displaystyle
2\hat{\mathcal{O}}_{AA}A+\hat{\mathcal{O}}_{AB}B+\hat{\mathcal{O}}_{A\phi}\delta\phi+\hat{\mathcal{O}}_{A\zeta}\zeta+\hat{\mathcal{O}}_{\dot{\zeta}A}\dot{\zeta}$
$\displaystyle=$ $\displaystyle 0,$ (45)
$\displaystyle\hat{\mathcal{O}}_{AB}A+2\hat{\mathcal{O}}_{BB}B+\hat{\mathcal{O}}_{\phi
B}\delta\phi+\hat{\mathcal{O}}_{\zeta
B}\zeta+\hat{\mathcal{O}}_{\dot{\zeta}B}\dot{\zeta}$ $\displaystyle=$
$\displaystyle 0,$ (46)
$\displaystyle\hat{\mathcal{O}}_{A\phi}A+\hat{\mathcal{O}}_{\phi
B}B+2\hat{\mathcal{O}}_{\phi\phi}\delta\phi+\hat{\mathcal{O}}_{\phi\zeta}\zeta+\hat{\mathcal{O}}_{\dot{\zeta}\phi}\dot{\zeta}$
$\displaystyle=$ $\displaystyle 0,$ (47)
$\displaystyle\frac{1}{\bar{N}}\left[2\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}\dot{\zeta}\right)+\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}\phi}\delta\phi\right)+\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}B}B\right)+\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}A}A\right)+\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}\zeta}\zeta\right)\right]$
$\displaystyle-a^{3}\left(2\hat{\mathcal{O}}_{\zeta\zeta}\zeta+\hat{\mathcal{O}}_{A\zeta}A+\hat{\mathcal{O}}_{\phi\zeta}\delta\phi+\hat{\mathcal{O}}_{\zeta
B}B\right)$ $\displaystyle=$ $\displaystyle 0.$ (48)

From (45)-(48), it is evident that $A$, $B$, and $\delta\phi$ are all
auxiliary variables, as the equations of motion do not contain their first-
order time derivative terms. In the case of a non-degenerate coefficient
matrix for the perturbation variables $A$, $B$, and $\delta\phi$, i.e.,

$\varOmega\equiv\det\left(\begin{array}[]{ccc}2\hat{\mathcal{O}}_{AA}&\hat{\mathcal{O}}_{AB}&\hat{\mathcal{O}}_{A\phi}\\\
\hat{\mathcal{O}}_{AB}&2\hat{\mathcal{O}}_{BB}&\hat{\mathcal{O}}_{\phi B}\\\
\hat{\mathcal{O}}_{A\phi}&\hat{\mathcal{O}}_{\phi
B}&2\hat{\mathcal{O}}_{\phi\phi}\end{array}\right)\neq 0,$ (49)

they can be formally solved from their equations of motion (45)-(47) as

$\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{1}{2\varOmega}\Big{[}\Big{(}\hat{\mathcal{O}}_{A\zeta}\hat{\mathcal{O}}_{\phi
B}^{2}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{\phi\zeta}-\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{\zeta
B}+2\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\zeta}+2\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\phi\phi}\hat{\mathcal{O}}_{\zeta
B}$ (50)
$\displaystyle\qquad-4\hat{\mathcal{O}}_{A\zeta}\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\phi}\Big{)}\zeta+\Big{(}\hat{\mathcal{O}}_{\dot{\zeta}A}\hat{\mathcal{O}}_{\phi
B}^{2}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{\dot{\zeta}\phi}-\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{\dot{\zeta}B}$
$\displaystyle\qquad+2\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\dot{\zeta}\phi}+2\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\phi\phi}\hat{\mathcal{O}}_{\dot{\zeta}B}-4\hat{\mathcal{O}}_{\dot{\zeta}A}\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\phi}\Big{)}\dot{\zeta}\Big{]},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle-\frac{1}{2\varOmega}\Big{[}\Big{(}\hat{\mathcal{O}}_{\zeta
B}\hat{\mathcal{O}}_{A\phi}^{2}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\phi\zeta}-\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{A\zeta}+2\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\phi\zeta}+2\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\phi\phi}\hat{\mathcal{O}}_{A\zeta}$
(51) $\displaystyle\qquad-4\hat{\mathcal{O}}_{\zeta
B}\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\phi\phi}\Big{)}\zeta+\Big{(}\hat{\mathcal{O}}_{\dot{\zeta}B}\hat{\mathcal{O}}_{A\phi}^{2}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\dot{\zeta}\phi}-\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{\dot{\zeta}A}$
$\displaystyle\qquad+2\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\dot{\zeta}\phi}+2\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\phi\phi}\hat{\mathcal{O}}_{\dot{\zeta}A}-4\hat{\mathcal{O}}_{\dot{\zeta}B}\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\phi\phi}\Big{)}\dot{\zeta}\Big{]},$
$\displaystyle\delta\phi$ $\displaystyle=$
$\displaystyle-\frac{1}{2\varOmega}\Big{[}\Big{(}\hat{\mathcal{O}}_{\phi\zeta}\hat{\mathcal{O}}_{AB}^{2}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{A\zeta}\hat{\mathcal{O}}_{\phi
B}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\zeta
B}+2\hat{\mathcal{O}}_{\phi B}\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\zeta
B}+2\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{A\zeta}$
(52)
$\displaystyle\qquad-4\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\phi\zeta}\Big{)}\zeta+\Big{(}\hat{\mathcal{O}}_{\dot{\zeta}\phi}\hat{\mathcal{O}}_{AB}^{2}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\dot{\zeta}A}\hat{\mathcal{O}}_{\phi
B}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\dot{\zeta}B}$
$\displaystyle\qquad+2\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\dot{\zeta}B}+2\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\dot{\zeta}A}-4\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\dot{\zeta}\phi}\Big{)}\dot{\zeta}\Big{]},$

with

$\varOmega=\hat{\mathcal{O}}_{A\phi}^{2}\hat{\mathcal{O}}_{BB}+\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\phi
B}^{2}+\hat{\mathcal{O}}_{AB}^{2}\hat{\mathcal{O}}_{\phi\phi}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\phi
B}-4\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\phi}.$
(53)

Although we have made the assumption that $\varOmega\neq 0$ to ensure that the
auxiliary variables $A$, $B$, and $\delta\phi$ are solvable (at least
formally), it is important to note that even if $\varOmega\neq 0$ is not
satisfied, it is still possible to obtain conditions to eliminate the unwanted
scalar degree of freedom. We will show this explicitly through a detailed
analysis of a specific example in the next section.

Finally, by plugging the solutions (50)-(52) into the equation of motion for
$\zeta$ (48), we will obtain an equation of motion for the single variable
$\zeta$. If no further conditions are assumed, the effective equation of
motion for $\zeta$ will contain its second-order time derivative term
$\ddot{\zeta}$, indicating a propagating degree of freedom carried by $\zeta$.
For our purpose, in order to eliminate the scalar degree of freedom, we have
to ensure that $\zeta$ is not propagating, at least at linear order in a
cosmological background. To this end, we have to make the coefficient of
$\ddot{\zeta}$ vanish. After some manipulations, the coefficient of
$\ddot{\zeta}$ is found to be

$\displaystyle\varDelta$ $\displaystyle=$ $\displaystyle
4\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}\left(\hat{\mathcal{O}}_{A\phi}^{2}\hat{\mathcal{O}}_{BB}+\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\phi
B}^{2}+\hat{\mathcal{O}}_{AB}^{2}\hat{\mathcal{O}}_{\phi\phi}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\phi
B}-4\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\phi}\right)$
(54) $\displaystyle-\hat{\mathcal{O}}_{\phi
B}^{2}\hat{\mathcal{O}}_{\dot{\zeta}A}^{2}-\hat{\mathcal{O}}_{A\phi}^{2}\hat{\mathcal{O}}_{\dot{\zeta}B}^{2}-\hat{\mathcal{O}}_{AB}^{2}\hat{\mathcal{O}}_{\dot{\zeta}\phi}^{2}+2\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{\dot{\zeta}A}\hat{\mathcal{O}}_{\dot{\zeta}B}+2\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{\dot{\zeta}A}\hat{\mathcal{O}}_{\dot{\zeta}\phi}+2\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{\dot{\zeta}B}\hat{\mathcal{O}}_{\dot{\zeta}\phi}$
$\displaystyle+4\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\phi}\hat{\mathcal{O}}_{\dot{\zeta}A}^{2}+4\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\dot{\zeta}\phi}^{2}+4\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\phi\phi}\hat{\mathcal{O}}_{\dot{\zeta}B}^{2}$
$\displaystyle-4\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\phi
B}\hat{\mathcal{O}}_{\dot{\zeta}B}\hat{\mathcal{O}}_{\dot{\zeta}\phi}-4\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\phi\phi}\hat{\mathcal{O}}_{\dot{\zeta}A}\hat{\mathcal{O}}_{\dot{\zeta}B}-4\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\dot{\zeta}A}\hat{\mathcal{O}}_{\dot{\zeta}\phi}.$

Therefore, we have to require that

$\varDelta=0,$ (55)

which is a necessary condition to eliminate the scalar degree of freedom
(i.e., the TTDOF conditions), at least at the linear order in perturbations
around a cosmological background.

##  IV Theory of $d=2$

The analysis and conditions obtained in the above section are general but
quite formal. In this section, we will apply the condition (54) to a concrete
model as an example. According to the classification of SCG monomials [60,
61], we consider a polynomial-type Lagrangian built of monomials of $d=2$,
where $d$ is the total number of derivatives. Precisely, the action is given
by:

$S_{2}=\int dtd^{3}xN\sqrt{h}\left(\mathcal{L}-\Lambda\right),$ (56)

with

$\mathcal{L}=c_{1}K_{ij}K^{ij}+c_{2}K^{2}+c_{3}R+c_{4}a_{i}a^{i}+d_{1}\mathrm{D}_{i}\phi\mathrm{D}^{i}\phi+d_{2}a_{i}\mathrm{D}^{i}\phi,$
(57)

where $c_{i}$ and $d_{i}$ are general functions of $N$ and $\phi$, and
$\Lambda$ is the cosmological constant. Additionally, the acceleration $a_{i}$
is defined as $a_{i}=\partial_{i}\ln N$. We make the assumption that both
$\phi$ and $N$ are homogeneous and isotropic at the background, i.e.,
$\bar{\phi}=\bar{\phi}(t)$, $\bar{N}=\bar{N}(t)$. To simplify our calculation,
we denote:

$\mathcal{L}_{c}=\mathcal{L}-\Lambda.$ (58)

We will see that a non-vanishing cosmological constant $\Lambda$, which may be
time-dependent, is necessary in order to have a cosmological background
solution.

###  IV.1 The degeneracy condition

Since both $\phi$ and $N$ depend only on time at the background level, the
background values of each quantity in the Lagrangian are given by:

$\bar{K}_{ij}=a\dot{a}\delta_{ij},\quad\bar{K}^{ij}=Ha^{-2}\delta^{ij},\quad\bar{K}=3H,$
(59)

and

$\bar{R}=0,\quad\bar{a}_{i}=0,\quad\partial_{i}\bar{\phi}=0.$ (60)

The background value of the Lagrangian is

$\bar{\mathcal{L}_{c}}=3H^{2}b-\Lambda,$ (61)

where we define

$b=c_{1}+3c_{2}$ (62)

for short. The non-vanishing derivatives of the Lagrangian $\mathcal{L}_{c}$
with respect to various quantities are thus

$\frac{\delta\mathcal{L}_{c}}{\delta N}=\frac{3}{\bar{N}}H^{2}b^{\prime},$
(63) $\frac{\delta\mathcal{L}_{c}}{\delta K_{ij}}=2Ha^{-2}\delta^{ij}b,$ (64)
$\frac{\delta\mathcal{L}_{c}}{\delta\phi}=3H^{2}\frac{\partial
b}{\partial\phi},$ (65) $\frac{\delta^{2}\mathcal{L}_{c}}{\delta
N^{2}}=\frac{3}{\bar{N}^{2}}H^{2}b^{\prime\prime}-\frac{2}{\bar{N}^{2}}c_{4}\partial^{2},$
(66) $\frac{\delta^{2}\mathcal{L}_{c}}{\delta K_{mn}\delta
K_{ij}}=2a^{-4}\left(\frac{1}{2}c_{1}\left(\delta^{im}\delta^{jn}+\delta^{in}\delta^{jm}\right)+c_{2}\delta^{ij}\delta^{mn}\right),$
(67)
$\frac{\delta^{2}\mathcal{L}_{c}}{\delta\phi^{2}}=3H^{2}\frac{\partial^{2}b}{\partial\phi^{2}}-2d_{1}\partial^{2},$
(68) $\frac{\delta^{2}\mathcal{L}_{c}}{\delta N\delta
K_{ij}}=\frac{2}{\bar{N}}Ha^{-2}\delta^{ij}b^{\prime},$ (69)
$\frac{\delta^{2}\mathcal{L}_{c}}{\delta\phi\delta
N}=3H^{2}\frac{\partial^{2}b}{\partial\phi\partial
N}-\frac{d_{2}}{\bar{N}}\partial^{2},$ (70)

and

$\frac{\delta^{2}\mathcal{L}_{c}}{\delta\phi\delta
K_{ij}}=2Ha^{-2}\delta^{ij}\frac{\partial b}{\partial\phi}.$ (71)

We do not present derivatives with respect to $R_{ij}$, as they are irrelevant
to the degeneracy analysis. According to (61)-(71), the background equations
of motion (25) are given by:

$\mathcal{E}_{A}=3H^{2}\left(-b+b^{\prime}\right)-\Lambda=0,$ (72)
$\mathcal{E}_{\zeta}=-9H^{2}b-6\dot{H}b-\frac{6H}{\bar{N}}\dot{\bar{N}}b-3\Lambda=0,$
(73) $\mathcal{E}_{\delta\phi}=3H^{2}\frac{\partial b}{\partial\phi}=0,$ (74)

explicitly. From (74), it is clear that in order to have a homogeneous and
isotropic background, $b$ defined in (62) must be a function of the lapse
function $N$ only. The background equations of motion (72)-(74) can greatly
simplify the calculation of the quadratic action for the perturbations in the
following.

By plugging (61)-(71) into (30)-(44), we can evaluate the operators
$\hat{\mathcal{O}}_{XY}$ that are relevant to eliminating the scalar mode,
which are given by:

$\hat{\mathcal{O}}_{AA}=\frac{3}{2}H^{2}\left(2b-2b^{\prime}+b^{\prime\prime}\right)-c_{4}\partial^{2},$
(75) $\hat{\mathcal{O}}_{BB}=a^{-2}\left(c_{1}+c_{2}\right)\partial^{4},$ (76)
$\hat{\mathcal{O}}_{\phi\phi}=\frac{3}{2}H^{2}\frac{\partial^{2}b}{\partial\phi^{2}}-d_{1}\partial^{2},$
(77) $\hat{\mathcal{O}}_{AB}=2Ha^{-1}\left(-b^{\prime}+b\right)\partial^{2},$
(78)
$\hat{\mathcal{O}}_{A\phi}=3H^{2}\frac{\partial\left(b^{\prime}-b\right)}{\partial\phi}-d_{2}\partial^{2},$
(79) $\hat{\mathcal{O}}_{\phi B}=-2Ha^{-1}\frac{\partial
b}{\partial\phi}\partial^{2},$ (80)
$\hat{\mathcal{O}}_{\dot{\zeta}A}=6H\left(-b+b^{\prime}\right),$ (81)
$\hat{\mathcal{O}}_{\dot{\zeta}B}=-2a^{-1}b\partial^{2},$ (82)
$\hat{\mathcal{O}}_{\dot{\zeta}\phi}=6H\frac{\partial b}{\partial\phi},$ (83)

and

$\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}=3b.$ (84)

Then, by plugging (75)-(84) into the determinant of the coefficient matrix
(53), and keeping in mind that $\frac{\partial b}{\partial\phi}=0$ from the
background equation of motion (74), we find:

$\displaystyle\varOmega$
$\displaystyle=a^{-2}\left(c_{1}+c_{2}\right)\left(d_{2}^{2}-4d_{1}c_{4}\right)\partial^{8}$
$\displaystyle\quad+2H^{2}a^{-2}\left[3\left(2b-2b^{\prime}+b^{\prime\prime}\right)\left(c_{1}+c_{2}\right)-2\left(-b^{\prime}+b\right)^{2}\right]d_{1}\partial^{6}.$
(85)

The degeneracy condition (55) reads

$\displaystyle 0\equiv\varDelta$ $\displaystyle=$ $\displaystyle
8a^{-2}bc_{1}\left(d_{2}^{2}-4c_{4}d_{1}\right)\partial^{8}$ (86)
$\displaystyle+48H^{2}a^{-2}d_{1}c_{1}\left(2bb^{\prime}-2b^{\prime
2}+bb^{\prime\prime}\right)\partial^{6}.$

In deriving (85) and (86), the background equations of motion (72)-(74) have
been used to simplify the expressions.

In the following, we will discuss two cases based on whether $\varOmega$ is
vanishing or not.

###  IV.2 Case $\varOmega\neq 0$

When $\varOmega\neq 0$, all the auxiliary variables $A$, $B$, $\delta\phi$ are
solvable, and thus the degeneracy condition $\varDelta=0$ holds. According to
(86), in order to eliminate the unwanted scalar degree of freedom, we must
require the coefficients of the operators $\partial^{6}$ and $\partial^{8}$ to
be vanishing simultaneously, i.e.,

$\left(2bb^{\prime}-2b^{\prime 2}+bb^{\prime\prime}\right)c_{1}d_{1}=0,$ (87)

and

$bc_{1}\left(d_{2}^{2}-4c_{4}d_{1}\right)=0,$ (88)

which are two constraints among $c_{1}$, $c_{4}$, $d_{1}$, $d_{2}$, and $b$.
Before solving these two equations, let us conduct a brief analysis.

First, (87) and (88) are trivially satisfied if $b\equiv c_{1}+3c_{2}=0$.
However, this conflicts with the background equation of motion (72), which
leads to $\Lambda=0$. As we have mentioned before, a non-vanishing (and
actually positive) cosmological constant is required in order to have a
cosmological background solution. Therefore, we must require $b\neq 0$.
Second, (87) and (88) are also satisfied if $c_{1}=0$. However, although in
this paper we focus on the scalar perturbations only, further calculation
reveals that only the term $c_{1}K_{ij}K^{ij}$ will contribute to the kinetic
term for the tensor perturbations (i.e., the gravitational waves). As a
result, a vanishing $c_{1}$ would also eliminate the gravitational waves,
which is unacceptable in light of the observation of gravitational waves.
Thus, we must also require $c_{1}\neq 0$. After taking into account that
$b,c_{1}\neq 0$, (88) holds only if $d_{2}^{2}-4c_{4}d_{1}=0.$ Finally, if
$d_{1}=0$, it follows that $d_{2}=0$. From (85), it will lead to
$\varOmega=0$, which conflicts with our requirement $\varOmega\neq 0$. Thus,
we must have $d_{1}\neq 0$.

Based on the above arguments, the equations (87)-(88) hold only if:

$\displaystyle d_{2}^{2}-4c_{4}d_{1}$ $\displaystyle=0,$ (89) $\displaystyle
2bb^{\prime}-2b^{\prime 2}+bb^{\prime\prime}$ $\displaystyle=0,$ (90)

are satisfied. The general solutions for $c_{4}$ and $b$ to (89)-(90) are

$c_{4}=\frac{d_{2}^{2}}{4d_{1}},$ (91) $b=\frac{C_{2}N}{1+C_{1}N},$ (92)

where $C_{1}$ and $C_{2}$ are constants. Finally, the Lagrangian containing no
dynamical scalar degree of freedom at linear order in a cosmological
background is

$\mathcal{L}=c_{1}\hat{K}_{ij}\hat{K}^{ij}+\frac{1}{3}\frac{C_{2}N}{1+C_{1}N}K^{2}+c_{3}R+\frac{d_{2}^{2}}{4d_{1}}a_{i}a^{i}+d_{1}\mathrm{D}_{i}\phi\mathrm{D}^{i}\phi+d_{2}a_{i}\mathrm{D}^{i}\phi,$
(93)

where $\hat{K}_{ij}=K_{ij}-\frac{1}{3}Kh_{ij}$ is the traceless part of
$K_{ij}$, and the coefficient $c_{1}$ is a function of $N$ only, while
$c_{3}$, $d_{1}$, and $d_{2}$ are general functions of $N$ and $\phi$.

It is interesting to compare (93) with the result obtained in [78]. In [78]
(see eq. (61)), only the first three terms in (93) are present. In particular,
the acceleration term $a_{i}a^{i}$ is not allowed. Here, due to the presence
of the auxiliary field $\phi$, the acceleration term can be introduced,
together with two other terms involving $\mathrm{D}_{i}\phi$. By taking an
appropriate limit $d_{1},d_{2}\rightarrow 0$, one will return to the result in
[78].

###  IV.3 Case $\varOmega=0$

If $\varOmega=0$, not all of the auxiliary variables $A$, $B$, $\delta\phi$
are solvable. Nevertheless, we can still determine the conditions on the
coefficients in order to eliminate the scalar degree of freedom. First of all,
according to (85), $\varOmega=0$ implies two equations:

$\left(c_{1}+c_{2}\right)\left(d_{2}^{2}-4d_{1}c_{4}\right)=0,$ (94)

and

$\left[3\left(2b-2b^{\prime}+b^{\prime\prime}\right)\left(c_{1}+c_{2}\right)-2\left(-b^{\prime}+b\right)^{2}\right]d_{1}=0.$
(95)

There are several branches of solutions to (94) and (95), which we will
discuss below.

####  IV.3.1 Case 1

The simplest case is to assume $d_{1}=d_{2}=0$. In this case, the Lagrangian
(57) reduces to:

$\mathcal{L}=c_{1}K_{ij}K^{ij}+c_{2}K^{2}+c_{3}R+c_{4}a_{i}a^{i},$ (96)

which is just the case that has already been discussed in [78]. The Lagrangian
that propagates only two tensor degrees of freedom (up to the linear order in
perturbations) is given by

$\mathcal{L}=c_{1}\hat{K}_{ij}\hat{K}^{ij}+\frac{1}{3}\frac{C_{2}N}{1+C_{1}N}K^{2}+c_{3}R,$
(97)

where $c_{1}$ is a function of $N$ (recall that $b\equiv c_{1}+3c_{2}$ has no
$\phi$-dependence) and $c_{3}$ is a general function of $N$ and $\phi$.

####  IV.3.2 Case 2

If $d_{1}\neq 0$, from (95) we must have

$3\left(2b-2b^{\prime}+b^{\prime\prime}\right)\left(b-2c_{2}\right)-2\left(-b^{\prime}+b\right)^{2}=0.$
(98)

As for (94), we assume $c_{1}+c_{2}\neq 0$ and thus it follows that

$d_{2}^{2}-4d_{1}c_{4}=0.$ (99)

Since the background equation of motion (72) must hold, we require
$-b+b^{\prime}\neq 0$ to guarantee a non-vanishing cosmological constant
$\Lambda$ and thus a cosmological background solution. Thus, (98) indicates:

$b-2c_{2}\neq 0,\quad 2b-2b^{\prime}+b^{\prime\prime}\neq 0.$ (100)

Then (98) implies that

$c_{2}=\frac{1}{2}b-\frac{\left(b-b^{\prime}\right)^{2}}{3\left(2b-2b^{\prime}+b^{\prime\prime}\right)},$
(101)

and

$c_{1}=-\frac{1}{2}b+\frac{\left(b-b^{\prime}\right)^{2}}{2b-2b^{\prime}+b^{\prime\prime}},$
(102)

where $b$ is a general function of $N$ only. The Lagrangian now turns to be

$\mathcal{L}=\left[-\frac{1}{2}b+\frac{\left(b-b^{\prime}\right)^{2}}{2b-2b^{\prime}+b^{\prime\prime}}\right]K_{ij}K^{ij}+\left[\frac{1}{2}b-\frac{\left(b-b^{\prime}\right)^{2}}{3\left(2b-2b^{\prime}+b^{\prime\prime}\right)}\right]K^{2}+c_{3}R+c_{4}a_{i}a^{i}+d_{1}\mathrm{D}_{i}\phi\mathrm{D}^{i}\phi+d_{2}a_{i}\mathrm{D}^{i}\phi,$
(103)

where $c_{3}$, $c_{4}$, $d_{1}$ and $d_{2}$ are general functions of $N$ and
$\phi$.

It is not difficult to check that the operators $\hat{\mathcal{O}}_{XY}$ are
the same as in (75)-(84). The equations of motion for the perturbation
variables are:

$\displaystyle
2\hat{\mathcal{O}}_{AA}A+\hat{\mathcal{O}}_{AB}B+\hat{\mathcal{O}}_{A\phi}\delta\phi+\hat{\mathcal{O}}_{A\zeta}\zeta+\hat{\mathcal{O}}_{\dot{\zeta}A}\dot{\zeta}$
$\displaystyle=$ $\displaystyle 0,$ (104)
$\displaystyle\hat{\mathcal{O}}_{AB}A+2\hat{\mathcal{O}}_{BB}B+\hat{\mathcal{O}}_{\zeta
B}\zeta+\hat{\mathcal{O}}_{\dot{\zeta}B}\dot{\zeta}$ $\displaystyle=$
$\displaystyle 0,$ (105) $\displaystyle
2\hat{\mathcal{O}}_{\phi\phi}\delta\phi+\hat{\mathcal{O}}_{A\phi}A$
$\displaystyle=$ $\displaystyle 0,$ (106)
$\displaystyle\frac{1}{\bar{N}}\left[2\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}\dot{\zeta}\right)+\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}B}B\right)+\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}A}A\right)+\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}\zeta}\zeta\right)\right]$
$\displaystyle-a^{3}\left(2\hat{\mathcal{O}}_{\zeta\zeta}\zeta+\hat{\mathcal{O}}_{A\zeta}A+\hat{\mathcal{O}}_{\zeta
B}B\right)$ $\displaystyle=$ $\displaystyle 0.$ (107)

We rewrite the first three equations in the matrix form:

$\bm{D}\bm{\alpha}=\bm{\beta},$ (108)

where

$\bm{D}=\left[\begin{array}[]{ccc}2\hat{\mathcal{O}}_{AA}&\hat{\mathcal{O}}_{AB}&\hat{\mathcal{O}}_{A\phi}\\\
\hat{\mathcal{O}}_{AB}&2\hat{\mathcal{O}}_{BB}&0\\\
\hat{\mathcal{O}}_{A\phi}&0&2\hat{\mathcal{O}}_{\phi\phi}\end{array}\right],$
(109) $\bm{\alpha}=\left[\begin{array}[]{c}A\\\ B\\\
\delta\phi\end{array}\right],\quad\bm{\beta}=\left[\begin{array}[]{c}-\hat{\mathcal{O}}_{A\zeta}\zeta-\hat{\mathcal{O}}_{\dot{\zeta}A}\dot{\zeta}\\\
-\hat{\mathcal{O}}_{\zeta
B}\zeta-\hat{\mathcal{O}}_{\dot{\zeta}B}\dot{\zeta}\\\ 0\end{array}\right].$
(110)

The determinant of coefficient matrix $\bm{D}$ of the auxiliary perturbation
variables $A$, $B$, and $\delta\phi$ is

$\displaystyle\varOmega$ $\displaystyle=\det\bm{D}$
$\displaystyle=\hat{\mathcal{O}}_{A\phi}^{2}\hat{\mathcal{O}}_{BB}+\hat{\mathcal{O}}_{AB}^{2}\hat{\mathcal{O}}_{\phi\phi}-4\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\phi}=0.$
(111)

By multiplying the third row of $\bm{D}$ by
$\hat{\mathcal{O}}_{A\phi}\hat{\mathcal{O}}_{BB}$, the second row by
$\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\phi\phi}$, and the first row by
$-2\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\phi}$, and adding the third
and the second rows to the first row, we arrive at the following matrix:

$\left(\begin{array}[]{ccc}0&0&0\\\
\hat{\mathcal{O}}_{AB}&2\hat{\mathcal{O}}_{BB}&0\\\
\hat{\mathcal{O}}_{A\phi}&0&2\hat{\mathcal{O}}_{\phi\phi}\end{array}\right),$
(112)

from which it is transparent that $\mathrm{rank}\bm{D}=2<3$. The augmented
matrix of the set of equations is defined as

$\bm{E}=\left(\begin{array}[]{cccc}2\hat{\mathcal{O}}_{AA}&\hat{\mathcal{O}}_{AB}&\hat{\mathcal{O}}_{A\phi}&-\hat{\mathcal{O}}_{A\zeta}\zeta-\hat{\mathcal{O}}_{\dot{\zeta}A}\dot{\zeta}\\\
\hat{\mathcal{O}}_{AB}&2\hat{\mathcal{O}}_{BB}&0&-\hat{\mathcal{O}}_{\zeta
B}\zeta-\hat{\mathcal{O}}_{\dot{\zeta}B}\dot{\zeta}\\\
\hat{\mathcal{O}}_{A\phi}&0&2\hat{\mathcal{O}}_{\phi\phi}&0\end{array}\right).$
(113)

By performing the same operations to $\bm{D}$, the augmented matrix becomes

$\left(\begin{array}[]{cccc}0&0&0&\left(2\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\phi}\hat{\mathcal{O}}_{A\zeta}-\hat{\mathcal{O}}_{\zeta
B}\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\phi\phi}\right)\zeta+\left(2\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\phi}\hat{\mathcal{O}}_{\dot{\zeta}A}-\hat{\mathcal{O}}_{\dot{\zeta}B}\hat{\mathcal{O}}_{AB}^{2}\hat{\mathcal{O}}_{\phi\phi}\right)\dot{\zeta}\\\
\hat{\mathcal{O}}_{AB}&2\hat{\mathcal{O}}_{BB}&0&-\hat{\mathcal{O}}_{\zeta
B}\zeta-\hat{\mathcal{O}}_{\dot{\zeta}B}\dot{\zeta}\\\
\hat{\mathcal{O}}_{A\phi}&0&2\hat{\mathcal{O}}_{\phi\phi}&0\end{array}\right),$
(114)

from which it is clear that the crucial term relevant to our analysis is

$\Xi\equiv\left(2\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\phi}\hat{\mathcal{O}}_{A\zeta}-\hat{\mathcal{O}}_{\zeta
B}\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\phi\phi}\right)\zeta+\left(2\hat{\mathcal{O}}_{BB}\hat{\mathcal{O}}_{\phi\phi}\hat{\mathcal{O}}_{\dot{\zeta}A}-\hat{\mathcal{O}}_{\dot{\zeta}B}\hat{\mathcal{O}}_{AB}^{2}\hat{\mathcal{O}}_{\phi\phi}\right)\dot{\zeta}.$
(115)

Fortunately, the explicit expression for $\Xi$ is not needed in the following
analysis.

According to $\Xi$ is identically vanishing or not, we have two case. If

$\Xi\neq 0,$ (116)

we have $\mathrm{rank}\bm{E}=3>\mathrm{rank}\bm{D}=2$, which implies that the
set of equations for the auxiliary variables has no solution.

On the other hand, if

$\Xi=0,$ (117)

we have $\mathrm{rank}\bm{E}=\mathrm{rank}\bm{D}=2<3$, which means that the
set of equations will be solvable. In this case, the equations (104)-(106)
become

$2\hat{\mathcal{O}}_{BB}B+\hat{\mathcal{O}}_{AB}A+\hat{\mathcal{O}}_{\zeta
B}\zeta+\hat{\mathcal{O}}_{\dot{\zeta}B}\dot{\zeta}=0,$ (118)
$2\hat{\mathcal{O}}_{\phi\phi}\delta\phi+\hat{\mathcal{O}}_{A\phi}A=0,$ (119)

from which we can solve $B$ to be

$B=-\frac{\hat{\mathcal{O}}_{AB}A+\hat{\mathcal{O}}_{\zeta
B}\zeta+\hat{\mathcal{O}}_{\dot{\zeta}B}\dot{\zeta}}{2\hat{\mathcal{O}}_{BB}}.$
(120)

Plugging the solution (120) into the last equation of motion (107) yields the
second order time derivative term of $\zeta$,

$\Delta_{1}\ddot{\zeta},$ (121)

where

$\Delta_{1}=\frac{4\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}\hat{\mathcal{O}}_{BB}-\hat{\mathcal{O}}_{\dot{\zeta}B}^{2}}{2\hat{\mathcal{O}}_{BB}}.$
(122)

To eliminate the scalar mode, we must require

$4\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}\hat{\mathcal{O}}_{BB}-\hat{\mathcal{O}}_{\dot{\zeta}B}^{2}=0,$
(123)

which corresponds to

$4a^{-2}b\left[b-3\left(c_{1}+c_{2}\right)\right]\partial^{4}=0.$ (124)

Recall that we have assumed $c_{1}+c_{2}\neq 0$ from the beginning, and thus
the only feasible solution to (124) is $c_{1}=0$. Again, this will lead to the
disappearance of $\hat{K}_{ij}\hat{K}^{ij}$ and thus the kinetic term for the
gravitational wave, which is unacceptable.

To conclude, there is no viable Lagrangian in Case 2.

####  IV.3.3 Case 3

In this case we assume

$c_{1}+c_{2}=0,\quad-b^{\prime}+b=0.$ (125)

As mentioned earlier, this case conflicts with the background equation of
motion (72), and thus is unphysical.

####  IV.3.4 Case 4

In this case we assume

$c_{1}+c_{2}=0,\quad d_{1}=0.$ (126)

The Lagrangian reduces to be

$\mathcal{L}=-\frac{1}{2}bK_{ij}K^{ij}+\frac{1}{2}bK^{2}+c_{3}R+c_{4}a_{i}a^{i}+d_{2}a_{i}\mathrm{D}^{i}\phi.$
(127)

The operators $\hat{\mathcal{O}}_{XY}$ are given by

$\hat{\mathcal{O}}_{AA}=\frac{3}{2}H^{2}\left(2b-2b^{\prime}+b^{\prime\prime}\right)-c_{4}\partial^{2},$
(128) $\hat{\mathcal{O}}_{AB}=2Ha^{-1}\left(-b^{\prime}+b\right)\partial^{2},$
(129) $\hat{\mathcal{O}}_{A\phi}=-d_{2}\partial^{2},$ (130)
$\hat{\mathcal{O}}_{\dot{\zeta}A}=6H\left(-b+b^{\prime}\right),$ (131)
$\hat{\mathcal{O}}_{\dot{\zeta}B}=-2a^{-1}b\partial^{2},$ (132)
$\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}=3b,$ (133)

and
$\hat{\mathcal{O}}_{BB}=\hat{\mathcal{O}}_{\phi\phi}=\hat{\mathcal{O}}_{\phi
B}=\hat{\mathcal{O}}_{\dot{\zeta}\phi}=\hat{\mathcal{O}}_{\phi\zeta}=0$.

Consequently, the quadratic action for the perturbation variables is

$\displaystyle S_{2}[A,B,\zeta,\delta\phi]$
$\displaystyle=\int\mathrm{d}t\mathrm{d}^{3}x\bar{N}a^{3}\left(A\hat{\mathcal{O}}_{AA}A+\zeta\hat{\mathcal{O}}_{\zeta\zeta}\zeta+A\hat{\mathcal{O}}_{AB}B+A\hat{\mathcal{O}}_{A\zeta}\zeta+A\hat{\mathcal{O}}_{A\phi}\delta\phi\right.$
$\displaystyle\quad\left.+\zeta\hat{\mathcal{O}}_{\zeta
B}B+\dot{\zeta}\hat{\mathcal{O}}_{\dot{\zeta}A}A+\dot{\zeta}\hat{\mathcal{O}}_{\dot{\zeta}B}B+\dot{\zeta}\hat{\mathcal{O}}_{\dot{\zeta}\zeta}\zeta+\dot{\zeta}\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}\dot{\zeta}\right).$
(134)

The equations of motion for the perturbation variables are

$\displaystyle
2\hat{\mathcal{O}}_{AA}A+\hat{\mathcal{O}}_{AB}B+\hat{\mathcal{O}}_{A\zeta}\zeta+\hat{\mathcal{O}}_{A\phi}\delta\phi+\hat{\mathcal{O}}_{\dot{\zeta}A}\dot{\zeta}$
$\displaystyle=0,$ (135)
$\displaystyle\hat{\mathcal{O}}_{AB}A+\hat{\mathcal{O}}_{\zeta
B}\zeta+\hat{\mathcal{O}}_{\dot{\zeta}B}\dot{\zeta}$ $\displaystyle=0,$ (136)
$\displaystyle\hat{\mathcal{O}}_{A\phi}A$ $\displaystyle=0,$ (137)
$\displaystyle\frac{1}{\bar{N}}\left[2\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}\dot{\zeta}\right)+\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}B}B\right)+\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}A}A\right)+\partial_{t}\left(a^{3}\hat{\mathcal{O}}_{\dot{\zeta}\zeta}\zeta\right)\right]\quad$
$\displaystyle-a^{3}\left(2\hat{\mathcal{O}}_{\zeta\zeta}\zeta+\hat{\mathcal{O}}_{A\zeta}A+\hat{\mathcal{O}}_{\phi\zeta}\delta\phi+\hat{\mathcal{O}}_{\zeta
B}B\right)$ $\displaystyle=0.$ (138)

In order to have a nontrivial solution for $A$, from (137) we must require
that

$\hat{\mathcal{O}}_{A\phi}=0,$ (139)

which implies

$d_{2}=0.$ (140)

Then we can solve $A$ and $B$ from (135)-(136) to get

$A=-\frac{\hat{\mathcal{O}}_{\zeta
B}\zeta+\hat{\mathcal{O}}_{\dot{\zeta}B}\dot{\zeta}}{\hat{\mathcal{O}}_{AB}},$
(141)

and

$B=\frac{\left(2\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\zeta
B}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{A\zeta}\right)\zeta+\left(2\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\dot{\zeta}B}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\dot{\zeta}A}\right)\dot{\zeta}}{\hat{\mathcal{O}}_{AB}^{2}},$
(142)

where note $\hat{\mathcal{O}}_{AB}$ given in (129) is not vanishing.

Finally, by plugging the solutions (141) and (142) into (138), we obtain an
effective equation of motion for $\zeta$ (with an undetermined $\delta\phi$),
in which the second-order time derivative term can be expressed as:

$\Delta_{2}\ddot{\zeta},$ (143)

with

$\Delta_{2}=2\frac{\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}\hat{\mathcal{O}}_{AB}^{2}+\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\dot{\zeta}B}^{2}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\dot{\zeta}A}\hat{\mathcal{O}}_{\dot{\zeta}B}}{\hat{\mathcal{O}}_{AB}^{2}}.$
(144)

Note that although $\delta\phi$ is left undetermined, it will not contribute
to the $\ddot{\zeta}$ term in its equation of motion. In order to evade the
unwanted scalar mode, we must impose the condition $\Delta_{2}=0$, which is
equivalent to

$\hat{\mathcal{O}}_{\dot{\zeta}\dot{\zeta}}\hat{\mathcal{O}}_{AB}^{2}+\hat{\mathcal{O}}_{AA}\hat{\mathcal{O}}_{\dot{\zeta}B}^{2}-\hat{\mathcal{O}}_{AB}\hat{\mathcal{O}}_{\dot{\zeta}A}\hat{\mathcal{O}}_{\dot{\zeta}B}=0.$
(145)

After plugging (128)-(133) into (145), we get

$6H^{2}a^{-2}b\left(bb^{\prime\prime}+2bb^{\prime}-2b^{\prime
2}\right)\partial^{4}-4a^{-2}b^{2}c_{4}\partial^{6}=0,$ (146)

which implies two equations

$\displaystyle b\left(bb^{\prime\prime}+2bb^{\prime}-2b^{\prime 2}\right)$
$\displaystyle=0,$ (147) $\displaystyle b^{2}c_{4}$ $\displaystyle=0.$ (148)

Since the background equations of motion must be satisfied, there must be
$b\neq 0$. The only solutions to $\Delta_{2}=0$ are thus:

$c_{4}=0,\quad b=\frac{C_{2}N}{1+C_{1}N}.$ (149)

As a result, in this case, the Lagrangian that does not propagate any scalar
degree of freedom at the linear order is

$\mathcal{L}=-\frac{1}{2}\frac{C_{2}N}{1+C_{1}N}\hat{K}_{ij}\hat{K}^{ij}+\frac{1}{3}\frac{C_{2}N}{1+C_{1}N}K^{2}+c_{3}R,$
(150)

where $c_{3}$ is a general function of $N$ and $\phi$.

In this case, if we choose $c_{3}=1$, and let
$\left|C_{2}\right|\rightarrow\infty$, $\left|C_{1}\right|\rightarrow\infty$
while keeping $\frac{C_{2}}{C_{1}}=-2$, the Lagrangian (150) reduces to be

$\mathcal{L}=\hat{K}_{ij}\hat{K}^{ij}-\frac{2}{3}K^{2}+R,$ (151)

which is nothing but GR.

##  V Conclusion

The spatially covariant gravity provides us with a broad playground for
studying modifications of general relativity. According to the discussion in
[84], the spatial covariant gravity theory with non-dynamical auxiliary fields
propagates three dynamical degrees of freedom, i.e., two tensorial and one
scalar degrees of freedom. In light of the detection of gravitational waves,
which correspond to the two tensorial degrees of freedom (TTDOFs), it is
interesting to examine under which conditions the theory propagates only
TTDOFs.

In this work, instead of deriving and solving the fully nonlinear TTDOF
conditions as in [77], we perform a perturbative analysis similar to [58, 78]
and focus on the necessary conditions such that no scalar mode propagates at
the linear order in perturbations around a cosmological background. In Sec.
II, starting from the general action (1), the background equations of motion
(25) are easily obtained from the linear order action (21). These equations of
motion are useful for simplifying the calculations at quadratic order. Then in
Sec. III, we derive the quadratic action for the scalar perturbations (29) and
find that $A$, $B$, and $\delta\phi$ are auxiliary variables. By solving these
auxiliary variables and deriving the effective equation of motion for the
would-be dynamical perturbation variable $\zeta$, this unwanted scalar mode
will be eliminated as long as condition (55) is satisfied.

As an illustration of our analysis, in Sec. IV, we consider a polynomial-type
Lagrangian (56), in which all the monomials are of $d=2$ with $d$ being the
total number of derivatives. Generally, if the condition (53) holds, i.e.,
when all the auxiliary perturbation variables $A$, $B$, $\delta\phi$ are
solvable, the TTDOF condition (55) leads to the Lagrangian (93), which
propagates only TTDOFs at linear order around a cosmological background.
Comparing with the result in [78], terms involving the acceleration are
allowed in the Lagrangian due to the presence of the auxiliary scalar field.
It is interesting that even when the condition (53) does not hold, which
implies that not all of the auxiliary variables can be solved, we can still
derive the degeneracy condition to eliminate the unwanted scalar mode. In this
case, we obtain two viable Lagrangians (97) and (150). Both cases include
general relativity as a special case, as long as appropriate coefficients are
chosen.

We have several comments on the results presented in this work. First, the
degeneracy condition derived in this work is necessary in the sense that the
unwanted scalar mode is removed only at linear order in a cosmological
background, which may reappear at nonlinear orders and/or around a general
background. Therefore, one may perform analysis similar to those in [58, 78]
to derive further constraints on the Lagrangian such that the scalar mode is
fully removed. Second, in the example of $d=2$, the resulting Lagrangian in
the case of $\varOmega\neq 0$ generalizes the one derived in [78]. The
auxiliary scalar field $\phi$ not only affects the coefficients in the
Lagrangian but also makes the acceleration term allowed. Therefore, it is
interesting to explore more general cases (e.g., with $d\geq 3$) where the
auxiliary scalar field may play a nontrivial role, so that a novel class of
TTDOF theories with such auxiliary fields can be constructed.

###### Acknowledgements.

This work was partly supported by the Natural Science Foundation of China
(NSFC) under the grant No. 11975020.

## References

  * [1] D. Lovelock, The Einstein tensor and its generalizations, J.Math.Phys. 12 (1971) 498–501.
  * [2] D. Lovelock, The four-dimensionality of space and the einstein tensor, J. Math. Phys. 13 (1972) 874–876.
  * [3] N. Afshordi, D. J. H. Chung, and G. Geshnizjani, Cuscuton: A Causal Field Theory with an Infinite Speed of Sound, Phys. Rev. D75 (2007) 083513, [hep-th/0609150].
  * [4] N. Afshordi, D. J. H. Chung, M. Doran, and G. Geshnizjani, Cuscuton Cosmology: Dark Energy meets Modified Gravity, Phys. Rev. D75 (2007) 123509, [astro-ph/0702002].
  * [5] M. Mylova and N. Afshordi, Effective Cuscuton Theory, arXiv:2312.06066.
  * [6] S. S. Boruah, H. J. Kim, and G. Geshnizjani, Theory of Cosmological Perturbations with Cuscuton, JCAP 1707 (2017), no. 07 022, [arXiv:1704.01131].
  * [7] S. S. Boruah, H. J. Kim, M. Rouben, and G. Geshnizjani, Cuscuton bounce, JCAP 08 (2018) 031, [arXiv:1802.06818].
  * [8] J. Bhattacharyya, A. Coates, M. Colombo, A. E. Gumrukcuoglu, and T. P. Sotiriou, Revisiting the cuscuton as a Lorentz-violating gravity theory, Phys. Rev. D97 (2018), no. 6 064020, [arXiv:1612.01824].
  * [9] J. Quintin and D. Yoshida, Cuscuton gravity as a classically stable limiting curvature theory, JCAP 02 (2020) 016, [arXiv:1911.06040].
  * [10] N. Bartolo, A. Ganz, and S. Matarrese, Cuscuton inflation, JCAP 05 (2022), no. 05 008, [arXiv:2111.06794].
  * [11] S. A. Hosseini Mansoori and Z. Molaee, Multi-field Cuscuton cosmology, JCAP 01 (2023) 022, [arXiv:2207.06720].
  * [12] P. Channuie, K. Karwan, and J. Sangtawee, Observational constraints and preheating in cuscuton inflation, Eur. Phys. J. C 83 (2023), no. 5 421, [arXiv:2301.07019].
  * [13] K.-i. Maeda and S. Panpanich, Cuscuta-Galileon cosmology: Dynamics, gravitational constants, and the Hubble constant, Phys. Rev. D 105 (2022), no. 10 104022, [arXiv:2202.04908].
  * [14] K. Kohri and K.-i. Maeda, A possible solution to the helium anomaly of EMPRESS VIII by cuscuton gravity theory, PTEP 2022 (2022), no. 9 091E01, [arXiv:2206.11257].
  * [15] S. Panpanich and K.-i. Maeda, Cosmological dynamics of Cuscuta–Galileon gravity, Eur. Phys. J. C 83 (2023), no. 3 240, [arXiv:2109.12288].
  * [16] F. C. E. Lima and C. A. S. Almeida, Topological solitons in the sigma-cuscuton model, Eur. Phys. J. C 83 (2023), no. 9 831, [arXiv:2301.01397].
  * [17] A. Iyonaga, K. Takahashi, and T. Kobayashi, Extended Cuscuton: Formulation, JCAP 1812 (2018), no. 12 002, [arXiv:1809.10935].
  * [18] A. Iyonaga, K. Takahashi, and T. Kobayashi, Extended Cuscuton as Dark Energy, JCAP 07 (2020) 004, [arXiv:2003.01934].
  * [19] T. Zhu, Q. Wu, A. Wang, and F.-W. Shu, U(1) symmetry and elimination of spin-0 gravitons in Horava-Lifshitz gravity without the projectability condition, Phys. Rev. D84 (2011) 101502, [arXiv:1108.1237].
  * [20] T. Zhu, F.-W. Shu, Q. Wu, and A. Wang, General covariant Horava-Lifshitz gravity without projectability condition and its applications to cosmology, Phys. Rev. D85 (2012) 044053, [arXiv:1110.5106].
  * [21] J. Chagoya and G. Tasinato, A new scalar-tensor realization of Hořava-Lifshitz gravity, arXiv:1805.12010.
  * [22] N. Afshordi, Cuscuton and low energy limit of Horava-Lifshitz gravity, Phys. Rev. D80 (2009) 081502, [arXiv:0907.5201].
  * [23] J. Khoury, G. E. Miller, and A. J. Tolley, Spatially Covariant Theories of a Transverse, Traceless Graviton, Part I: Formalism, Phys.Rev. D85 (2012) 084002, [arXiv:1108.1397].
  * [24] J. Khoury, G. E. J. Miller, and A. J. Tolley, How General Relativity and Lorentz Covariance Arise from the Spatially Covariant Effective Field Theory of the Transverse, Traceless Graviton, Int. J. Mod. Phys. D 23 (2014), no. 12 1442012, [arXiv:1405.5219].
  * [25] J. Chagoya and G. Tasinato, A geometrical approach to degenerate scalar-tensor theories, JHEP 02 (2017) 113, [arXiv:1610.07980].
  * [26] G. Tasinato, Symmetries for scalarless scalar theories, Phys. Rev. D 102 (2020), no. 8 084009, [arXiv:2009.02157].
  * [27] D. Glavan and C. Lin, Einstein-Gauss-Bonnet Gravity in Four-Dimensional Spacetime, Phys. Rev. Lett. 124 (2020), no. 8 081301, [arXiv:1905.03601].
  * [28] C. Lin and S. Mukohyama, A Class of Minimally Modified Gravity Theories, JCAP 1710 (2017), no. 10 033, [arXiv:1708.03757].
  * [29] K. Aoki, A. De Felice, C. Lin, S. Mukohyama, and M. Oliosi, Phenomenology in type-I minimally modified gravity, JCAP 1901 (2019), no. 01 017, [arXiv:1810.01047].
  * [30] K. Aoki, C. Lin, and S. Mukohyama, Novel matter coupling in general relativity via canonical transformation, Phys. Rev. D98 (2018), no. 4 044022, [arXiv:1804.03902].
  * [31] A. De Felice and S. Mukohyama, Minimal theory of massive gravity, Phys. Lett. B 752 (2016) 302–305, [arXiv:1506.01594].
  * [32] A. De Felice and S. Mukohyama, Phenomenology in minimal theory of massive gravity, arXiv:1512.04008.
  * [33] N. Bolis, A. De Felice, and S. Mukohyama, Integrated Sachs-Wolfe-galaxy cross-correlation bounds on the two branches of the minimal theory of massive gravity, Phys. Rev. D 98 (2018), no. 2 024010, [arXiv:1804.01790].
  * [34] A. De Felice, F. Larrouturou, S. Mukohyama, and M. Oliosi, Black holes and stars in the minimal theory of massive gravity, Phys. Rev. D 98 (2018), no. 10 104031, [arXiv:1808.01403].
  * [35] A. De Felice, S. Mukohyama, and M. C. Pookkillath, Minimal theory of massive gravity and constraints on the graviton mass, JCAP 12 (2021), no. 12 011, [arXiv:2110.01237].
  * [36] A. De Felice, S. Kumar, S. Mukohyama, and R. C. Nunes, Observational bounds on extended minimal theories of massive gravity: New limits on the graviton mass, arXiv:2311.10530.
  * [37] S. Mukohyama and K. Noui, Minimally Modified Gravity: a Hamiltonian Construction, JCAP 1907 (2019) 049, [arXiv:1905.02000].
  * [38] A. De Felice, A. Doll, and S. Mukohyama, A theory of type-II minimally modified gravity, JCAP 09 (2020) 034, [arXiv:2004.12549].
  * [39] A. De Felice, S. Mukohyama, and M. C. Pookkillath, Addressing $H_{0}$ tension by means of VCDM, Phys. Lett. B 816 (2021) 136201, [arXiv:2009.08718].
  * [40] K. Aoki, A. De Felice, S. Mukohyama, K. Noui, M. Oliosi, and M. C. Pookkillath, Minimally modified gravity fitting Planck data better than $\Lambda$CDM, Eur. Phys. J. C 80 (2020), no. 8 708, [arXiv:2005.13972].
  * [41] M. C. Pookkillath, Minimally Modified Gravity Fitting Planck Data Better Than $\Lambda$CDM, Astron. Rep. 65 (2021), no. 10 1021–1025.
  * [42] A. De Felice, F. Larrouturou, S. Mukohyama, and M. Oliosi, Minimal Theory of Bigravity: construction and cosmology, JCAP 04 (2021) 015, [arXiv:2012.01073].
  * [43] K. Aoki, F. Di Filippo, and S. Mukohyama, Non-uniqueness of massless transverse-traceless graviton, JCAP 05 (2021) 071, [arXiv:2103.15044].
  * [44] A. De Felice, A. Doll, F. Larrouturou, and S. Mukohyama, Black holes in a type-II minimally modified gravity, JCAP 03 (2021) 004, [arXiv:2010.13067].
  * [45] A. De Felice and S. Mukohyama, Weakening gravity for dark matter in a type-II minimally modified gravity, JCAP 04 (2021) 018, [arXiv:2011.04188].
  * [46] A. De Felice, S. Mukohyama, and M. C. Pookkillath, Static, spherically symmetric objects in type-II minimally modified gravity, Phys. Rev. D 105 (2022), no. 10 104013, [arXiv:2110.14496].
  * [47] A. De Felice, K.-i. Maeda, S. Mukohyama, and M. C. Pookkillath, Comparison of two theories of Type-IIa minimally modified gravity, Phys. Rev. D 106 (2022), no. 2 024028, [arXiv:2204.08294].
  * [48] A. F. Jalali, P. Martens, and S. Mukohyama, Spherical scalar collapse in a type-II minimally modified gravity, Phys. Rev. D 109 (2024), no. 4 044053, [arXiv:2306.10672].
  * [49] R. Carballo-Rubio, F. Di Filippo, and S. Liberati, Minimally modified theories of gravity: a playground for testing the uniqueness of general relativity, JCAP 1806 (2018), no. 06 026, [arXiv:1802.02537]. [Erratum: JCAP1811,no.11,E02(2018)].
  * [50] J. Sangtawee and K. Karwan, Inflationary model in minimally modified gravity theories, arXiv:2103.11463.
  * [51] A. Ganz, Dynamical dark energy in minimally modified gravity, JCAP 08 (2022) 074, [arXiv:2203.12358].
  * [52] O. Akarsu, A. De Felice, E. Di Valentino, S. Kumar, R. C. Nunes, E. Ozulker, J. A. Vazquez, and A. Yadav, $\Lambda_{\rm s}$CDM cosmology from a type-II minimally modified gravity, arXiv:2402.07716.
  * [53] Z.-B. Yao, M. Oliosi, X. Gao, and S. Mukohyama, Minimally modified gravity with an auxiliary constraint: A Hamiltonian construction, Phys. Rev. D 103 (2021), no. 2 024032, [arXiv:2011.00805].
  * [54] Z.-B. Yao, M. Oliosi, X. Gao, and S. Mukohyama, Minimally modified gravity with auxiliary constraints formalism, Phys. Rev. D 107 (2023), no. 10 104052, [arXiv:2302.02090].
  * [55] X. Gao, Unifying framework for scalar-tensor theories of gravity, Phys.Rev. D90 (2014) 081501, [arXiv:1406.0822].
  * [56] X. Gao, Hamiltonian analysis of spatially covariant gravity, Phys.Rev. D90 (2014), no. 10 104033, [arXiv:1409.6708].
  * [57] X. Gao and Z.-B. Yao, Spatially covariant gravity with velocity of the lapse function: the Hamiltonian analysis, JCAP 1905 (2019) 024, [arXiv:1806.02811].
  * [58] X. Gao, C. Kang, and Z.-B. Yao, Spatially Covariant Gravity: Perturbative Analysis and Field Transformations, Phys. Rev. D99 (2019), no. 10 104015, [arXiv:1902.07702].
  * [59] Y. Yu, Z. Chen, and X. Gao, Spatially covariant gravity with nonmetricity, arXiv:2402.02565.
  * [60] X. Gao, Higher derivative scalar-tensor monomials and their classification, Sci. China Phys. Mech. Astron. 64 (2021), no. 1 210012, [arXiv:2003.11978].
  * [61] X. Gao and Y.-M. Hu, Higher derivative scalar-tensor theory and spatially covariant gravity: the correspondence, Phys. Rev. D 102 (2020), no. 8 084006, [arXiv:2004.07752].
  * [62] X. Gao, Higher derivative scalar-tensor theory from the spatially covariant gravity: a linear algebraic analysis, JCAP 11 (2020) 004, [arXiv:2006.15633].
  * [63] Y.-M. Hu and X. Gao, Covariant 3+1 correspondence of the spatially covariant gravity and the degeneracy conditions, Phys. Rev. D 105 (2022), no. 4 044023, [arXiv:2111.08652].
  * [64] P. Horava, Membranes at Quantum Criticality, JHEP 0903 (2009) 020, [arXiv:0812.4287].
  * [65] P. Horava, Quantum Gravity at a Lifshitz Point, Phys.Rev. D79 (2009) 084008, [arXiv:0901.3775].
  * [66] D. Blas, O. Pujolas, and S. Sibiryakov, On the Extra Mode and Inconsistency of Horava Gravity, JHEP 0910 (2009) 029, [arXiv:0906.3046].
  * [67] P. Creminelli, M. A. Luty, A. Nicolis, and L. Senatore, Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies, JHEP 0612 (2006) 080, [hep-th/0606090].
  * [68] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan, and L. Senatore, The Effective Field Theory of Inflation, JHEP 0803 (2008) 014, [arXiv:0709.0293].
  * [69] P. Creminelli, G. D’Amico, J. Norena, and F. Vernizzi, The Effective Theory of Quintessence: the w<-1 Side Unveiled, JCAP 0902 (2009) 018, [arXiv:0811.0827].
  * [70] G. Gubitosi, F. Piazza, and F. Vernizzi, The Effective Field Theory of Dark Energy, JCAP 1302 (2013) 032, [arXiv:1210.0201].
  * [71] J. K. Bloomfield, E. E. Flanagan, M. Park, and S. Watson, Dark energy or modified gravity? An effective field theory approach, JCAP 1308 (2013) 010, [arXiv:1211.7054].
  * [72] T. Fujita, X. Gao, and J. Yokoyama, Spatially covariant theories of gravity: disformal transformation, cosmological perturbations and the Einstein frame, JCAP 1602 (2016), no. 02 014, [arXiv:1511.04324].
  * [73] X. Gao and X.-Y. Hong, Propagation of gravitational waves in a cosmological background, Phys. Rev. D 101 (2020), no. 6 064057, [arXiv:1906.07131].
  * [74] T. Zhu, W. Zhao, and A. Wang, Polarized primordial gravitational waves in spatial covariant gravities, Phys. Rev. D 107 (2023), no. 2 024031, [arXiv:2210.05259].
  * [75] T. Zhu, W. Zhao, and A. Wang, Gravitational wave constraints on spatial covariant gravities, Phys. Rev. D 107 (2023), no. 4 044051, [arXiv:2211.04711].
  * [76] T. Zhu, W. Zhao, J.-M. Yan, C. Gong, and A. Wang, Tests of modified gravitational wave propagations with gravitational waves, arXiv:2304.09025.
  * [77] X. Gao and Z.-B. Yao, Spatially covariant gravity theories with two tensorial degrees of freedom: the formalism, Phys. Rev. D101 (2020), no. 6 064018, [arXiv:1910.13995].
  * [78] Y.-M. Hu and X. Gao, Spatially covariant gravity with 2 degrees of freedom: Perturbative analysis, Phys. Rev. D 104 (2021), no. 10 104007, [arXiv:2104.07615].
  * [79] J. Lin, Y. Gong, Y. Lu, and F. Zhang, Spatially covariant gravity with a dynamic lapse function, Phys. Rev. D 103 (2021), no. 6 064020, [arXiv:2011.05739].
  * [80] A. Iyonaga and T. Kobayashi, Distinguishing modified gravity with just two tensorial degrees of freedom from general relativity: Black holes, cosmology, and matter coupling, Phys. Rev. D 104 (2021), no. 12 124020, [arXiv:2109.10615].
  * [81] T. Hiramatsu and T. Kobayashi, Testing gravity with the cosmic microwave background: constraints on modified gravity with two tensorial degrees of freedom, JCAP 07 (2022), no. 07 040, [arXiv:2205.04688].
  * [82] J. Saito and T. Kobayashi, Black hole perturbations in spatially covariant gravity with just two tensorial degrees of freedom, Phys. Rev. D 108 (2023), no. 10 104063, [arXiv:2308.00267].
  * [83] S. Chakraborty, K. Karwan, and J. Sangtawee, Observational predictions of inflationary model in spatially covariant gravity with two tensorial degrees of freedom for gravity, arXiv:2308.09508.
  * [84] X. Gao, M. Yamaguchi, and D. Yoshida, Higher derivative scalar-tensor theory through a non-dynamical scalar field, JCAP 1903 (2019) 006, [arXiv:1810.07434].