# Spatially covariant gravity with two degrees of freedom:
perturbative analysis

Yu-Min Hu<EMAIL_ADDRESS>School of Physics and Astronomy, Sun Yat-
sen University, Guangzhou 510275, China  Xian Gao<EMAIL_ADDRESS>School of Physics and Astronomy, Sun Yat-sen University, Guangzhou 510275,
China

(April 16, 2021)

###### Abstract

We revisit the problem of building the Lagrangian of a large class of metric
theories that respect spatial covariance, which propagate at most two degrees
of freedom and in particular no scalar mode. The Lagrangians are polynomials
built of the spatially covariant geometric quantities. By expanding the
Lagrangian around a cosmological background and focusing on the scalar modes
only, we find the conditions for the coefficients of the monomials in order to
eliminate the scalar mode at the linear order in perturbations. We find the
conditions up to $d=4$ with $d$ the total number of derivatives in the
monomials and determine the explicit Lagrangians for the cases of $d=2$, $d=3$
as well as the combination of $d=2$ and $d=3$. We also expand the Lagrangian
of $d=2$ to the cubic order in perturbations, and find additional conditions
for the coefficients such that the scalar mode is eliminated up to the cubic
order. This perturbative analysis can be performed order by order, and one
expects to determine the final Lagrangian at some finite order such that the
scalar mode is fully eliminated. Our analysis provides an alternative and
complimentary approach to building spatially covariant gravity with only
tensorial degrees of freedom. The resulting theories can be used as
alternatives to the general relativity to describe the tensorial gravitational
waves in a cosmological setting.

##  I Introduction

Recently, there is revival of the interest of questioning the uniqueness of
general relativity (GR) as the theory of two tensorial degrees of freedom
(TTDOF’s). The Lovelock’s theorem [1] is an answer to this question, which
claims that GR is the unique four dimensional theory for the spacetime metric
with the second order equations of motion, which obeys the general covariance
and locality. As a result, GR is the unique theory for the TTDOF’s if all the
assumptions of Lovelock are preserved.

From the field theoretical point of view, the idea of embedding the
gravitational degrees of freedom in a field theory of metric variables was
also explored. By coupling a massless spin-2 field to the energy–momentum
tensors of matter field(s) as well as of its own, it was arguably believed
that the Einstein-Hilbert action is the unique theory one will arrive at. This
approach can be traced back to the Fierz-Pauli theory [2] and was further
widely developed in [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] (see
[18] for a review)111Recently the “bootstrap” approach also provides new
understandings on the uniqueness of GR [19, 20, 21, 22, 23, 24] (see [25] for
a review)..

In this work, we shall examining the conditions of propagating only TTDOF’s in
a large class of metric theories respecting spatial covariance, which we dub
the spatially covariant gravity (SCG). The SCG can be traced back to the ghost
condensation [26] and was developed in the effective field theory of inflation
[27, 28] as well as in the Hořava gravity [29, 30]. It was further generalized
in [31] in which a large class of SCG theories was proposed and was extended
in [32] by including the dynamical lapse function and in [33] with an
auxiliary scalar field.

Theories different from GR while propagating only TTDOF’s firstly arose in the
so-called “cuscuton” theory [34, 35] and in a sub-class of Hořava gravity [36,
37]. In [38] a class of SCG theories with only TTDOF’s was proposed as the
minimal modification of the GR (MMG). The cuscuton and MMG theories have been
further extended [39, 40, 41, 42, 43, 44, 45] and their applications on
cosmology and black holes have been widely studied [46, 47, 48, 49, 50, 51,
52, 53, 54]. A class of 4-dimension Einstein-Gauss-Bonnet gravity was also
proposed recently as an arguable TTDOF theory [55].

We shall employ the framework of SCG due to the following reasons.

  * •

The Lagrangians of SCG theories are automatically written in the spacetime-
split form, which are convenient for analysing the time evolution and the
degrees of freedom using either the equations of motion or Hamiltonian
constraint analysis.

  * •

The SCG can be viewed as the gauge fixed version of the scalar-tensor theory
with a single timelike scalar field. By choosing the time coordinate as the
scalar field $t=t(\phi)$, which is dubbed the unitary gauge in the literature,
the generally covariant single field scalar-tensor theory can be naturally
recast in the form of a SCG theory. Therefore the SCG can be used as a
“generator” of the scalar-tensor theory, especially when the higher
derivatives are present. The generally covariant scalar-tensor monomials and
SCG monomials have been classified and their correspondence has been
investigated in [56, 57, 58]222The correspondence is subtle when the unitary
gauge is not accessible, see [59] for a discussion.. This may help to build
well-behaved higher derivative scalar-tensor theories after the
(re)construction of the theory of Horndeski [60] in its modern form [61, 62,
63] and the degenerate higher order derivative scalar-tensor theory [64, 65,
66, 67] (see [68, 69] for reviews and references therein).

  * •

Thanks to the spacetime-splitting nature of the SCG, the construction of SCG
with only a single scalar degree of freedom becomes virtually trivial. Indeed,
as being shown in [70, 71, 72], the SCG without the dynamical lapse function
automatically propagates at most three degrees of freedom. When the lapse
function becomes dynamical, more conditions must be imposed [32].

Counting the number of degrees of freedom can be well-performed through
Dirac’s Hamiltonian constraint analysis (see [73] for a comprehensive review).
In the framework of SCG, there are in principle two approaches to finding the
conditions of propagating at most two degrees of freedom.

  * •

The traditional and conservative approach, is to start from the Lagrangian and
perform the Legendre transformation to derive the Hamiltonian, then to find
the conditions for the Lagrangian by performing the Hamiltonian constraint
analysis. In [74], starting from a general local Lagrangian of SCG, conditions
of propagating at most two degrees of freedom have been derived. The analysis
was also generalized in [75] with a dynamical lapse function.

  * •

The other approach is to start with the Hamiltonian directly, and to determine
the conditions for the Hamiltonian instead of the Lagrangian. Indeed,
simplified structure and condition(s) are found at the level of Hamiltonian in
[42] in a class of SCG theories. This approach can be made even more “trivial”
by imposing additional constraint(s) in the phase space through auxiliary
variable(s) [76].

Both approaches have their merits and shortcomings. For the “Lagrangian”
approach, it is more convenient to deal with the _local_ Lagrangian, while the
conditions are functional differential equations for the Lagrangian and
mathematically complicated to be solved. For the “Hamiltonian” approach, one
is able to determine the Hamiltonian in a simple manner, while the
corresponding Lagrangian is involved and typically _non-local_ due to the
presence of extra auxiliary variables.

In view of the above considerations, in this work we employ an alternative
approach to constructing the SCG with only TTDOF’s. We shall deal with the
Lagrangian directly and determine the conditions at the level of equations of
motion. In fact, constraint analysis as well as counting the number of degrees
of freedom can be equivalently performed at the level of the Lagrangian and
the equations of motion [77, 78].

The idea is based on the fact that if the Lagrangian propagates at most two
DOF’s — and in particular, no scalar mode — at the non-perturbative level, the
scalar mode must not show up at any finite order in the perturbative expansion
around a spatially homogeneous and isotropic background. In particular, the
conditions can be determined order by order in a perturbative analysis, which
may be relatively easier to be managed. This is also the approach in [40] to
building the so-called “extended cuscuton” theory. The same idea was also
employed in [79] to find conditions for the SCG Lagrangians quadratic in the
extrinsic curvature and in the velocity of the lapse function to propagate at
most three degrees of freedom.

This work is organized as follows. In Sec. II we briefly review the spatially
covariant gravity and the general conditions to have at most two degrees of
freedom. In Sec. III we describe our perturbative approach and derive the
degeneracy condition, in order to eliminate the scalar mode at linear order in
perturbations. In Sec. IV we use the degeneracy condition to find the
conditions for the Lagrangians up to $d=4$ and give the explicit Lagrangians
for $d=2$, $d=3$ as well as the combination of $d=2,3$. In Sec. V we use the
Lagrangian of $d=2$ as an illustrative example to show how to eliminate the
scalar mode at the next order in perturbations. We summarize our results in
Sec. VI.

##  II Spatially covariant gravity with 2 degrees of freedom

In this section, we make a brief review of the framework of spatially
covariant gravity (SCG) theory, and in particular the classification of the
SCG polynomials. We also briefly summarize the conditions of having only two
degrees of freedom, which are derived in [74].

The action of the spatially covariant gravity theories takes the general form

$S=\int\mathrm{d}t\mathrm{d}^{3}x\,N\sqrt{h}\mathcal{L}\left(t,N,h_{ij},K_{ij},R_{ij},\nabla_{i},\varepsilon_{ijk}\right),$
(1)

where $N$ is the lapse function, $h_{ij}$ is the 3-dimensional spatial metric,
$K_{ij}$ is the extrinsic curvature defined by

$K_{ij}=\frac{1}{2N}\left(\partial_{t}{h}_{ij}-\pounds_{\\!\vec{N}}h_{ij}\right),$
(2)

with $\pounds_{\\!\vec{N}}$ the Lie derivative with respect to the shift
vector $N^{i}$, $R_{ij}$ is the 3-dimensional spatial Ricci tensor,
$\nabla_{i}$ is the covariant derivative compatible with the spatial metric
$h_{ij}$. The spatial Levi-Civita tensor
$\varepsilon_{ijk}=\sqrt{h}\epsilon_{ijk}$ with $\epsilon_{123}=1$ is allowed,
thus the parity-violating terms can be constructed by those building blocks
with $\varepsilon_{ijk}$333The spatially covariant parity violating terms and
their cosmological implications have been widely investigated, see (e.g.) [80,
81, 82, 83, 84, 85, 86, 87].. Note in principle the lapse function $N$ may
also acquire a kinetic term through
$\frac{1}{N}\left(\partial_{t}{N}-N^{i}\nabla_{i}N\right)$, which has been
considered in [32]. The shift vector $N_{i}$ by itself is not a genuine
geometric quantity of the spacetime foliation structure, which merely encodes
the gauge freedom of choosing the spatial coordinates.

In this work, instead of analysing a general Lagrangian as in (1), we
concentrat on polynomial-type Lagrangians, which are linear combinations of
the SCG monomials. The irreducible SCG monomials are exhausted and classified
up to $d=4$ in [57] with $d$ the total number of derivatives in the monomials.
Here we briefly describe the construction with improved notations following
[58]. We assign each SCG model by a set of integers $(c_{0};d_{2},d_{3})$
according to their corresponding monomials in the scalar-tensor theories after
the Stueckelberg mapping. Precisely, $c_{0}$ is the number of spacetime
Riemann curvature tensor, $d_{2}$, $d_{3}$ are numbers of the second, the
third order generally covariant derivatives of the scalar field $\phi$,
respectively. In fact we have the simple correspondences

$\displaystyle K_{ij}\sim a_{i}$ $\displaystyle\sim$ $\displaystyle(0;1,0),$
(3) $\displaystyle R_{ij}$ $\displaystyle\sim$
$\displaystyle\left(1;0,0\right),$ (4)
$\displaystyle\nabla_{k}K_{ij}\sim\nabla_{i}a_{j}$ $\displaystyle\sim$
$\displaystyle\left(0;0,1\right),$ (5)

and thus $d$ can be expressed by

$d=\underset{n=0}{\sum}\left[\left(n+2\right)c_{n}+\left(n+1\right)d_{n+2}\right].$
(6)

We thus classify the various SCG monomials with $d$ and then with the
categories labelled by $(c_{0};d_{2},d_{3})$.

Up to $d=4$, the Lagrangian built of the irreducible SCG monomials is

$\mathcal{L}=\mathcal{L}^{\left(0\right)}+\mathcal{L}^{\left(1\right)}+\mathcal{L}^{\left(2\right)}+\mathcal{L}^{\left(3\right)}+\mathcal{L}^{\left(4\right)}+\tilde{\mathcal{L}}^{\left(3\right)}+\tilde{\mathcal{L}}^{\left(4\right)},$
(7)

where the parity-preserving terms are

$\mathcal{L}^{\left(0\right)}=c_{1}^{\left(0;0,0\right)},$ (8)
$\mathcal{L}^{\left(1\right)}=c_{1}^{\left(0;1,0\right)}K,$ (9)
$\mathcal{L}^{\left(2\right)}=c_{1}^{\left(0;2,0\right)}K_{ij}K^{ij}+c_{2}^{\left(0;2,0\right)}a_{i}a^{i}+c_{3}^{\left(0;2,0\right)}K^{2}+c_{1}^{\left(1;0,0\right)}R,$
(10) $\displaystyle\mathcal{L}^{\left(3\right)}$ $\displaystyle=$
$\displaystyle
c_{1}^{\left(0;3,0\right)}K_{ij}K^{jk}K_{k}^{i}+c_{2}^{\left(0;3,0\right)}K_{ij}a^{i}a^{j}+c_{3}^{\left(0;3,0\right)}K_{ij}K^{ij}K+c_{4}^{\left(0;3,0\right)}Ka_{i}a^{i}+c_{5}^{\left(0;3,0\right)}K^{3}$
(11)
$\displaystyle+c_{1}^{\left(0;1,1\right)}K_{ij}\nabla^{i}a^{j}+c_{2}^{\left(0;1,1\right)}K\nabla_{i}a^{i}$
$\displaystyle+c_{1}^{\left(1;1,0\right)}R^{ij}K_{ij}+c_{2}^{\left(1;1,0\right)}RK,$
$\displaystyle\mathcal{L}^{\left(4\right)}$ $\displaystyle=$ $\displaystyle
c_{1}^{\left(0;4,0\right)}K_{ik}K_{j}^{k}a^{i}a^{j}+c_{2}^{\left(0;4,0\right)}K_{ij}K^{jk}K_{k}^{i}K+c_{3}^{\left(0;4,0\right)}K_{ij}a^{i}a^{j}K+c_{4}^{\left(0;4,0\right)}\left(K_{ij}K^{ij}\right)^{2}$
(12)
$\displaystyle+c_{5}^{\left(0;4,0\right)}K_{ij}K^{ij}a_{k}a^{k}+c_{6}^{\left(0;4,0\right)}\left(a_{i}a^{i}\right)^{2}+c_{7}^{\left(0;4,0\right)}K_{ij}K^{ij}K^{2}+c_{8}^{\left(0;4,0\right)}a_{i}a^{i}K^{2}+c_{9}^{\left(0;4,0\right)}K^{4}$
$\displaystyle+c_{1}^{\left(0;2,1\right)}K_{i}^{k}K_{jk}\nabla^{i}a^{j}+c_{2}^{\left(0;2,1\right)}K_{j}^{i}a^{j}\nabla_{k}K_{i}^{k}+c_{3}^{\left(0;2,1\right)}K_{j}^{i}a^{j}\nabla_{i}K+c_{4}^{\left(0;2,1\right)}K_{ij}\nabla^{i}a^{j}K$
$\displaystyle+c_{5}^{\left(0;2,1\right)}K_{ij}K^{ij}\nabla_{k}a^{k}+c_{6}^{\left(0;2,1\right)}a_{i}a^{i}\nabla_{j}a^{j}+c_{7}^{\left(0;2,1\right)}K^{2}\nabla_{i}a^{i}$
$\displaystyle+c_{1}^{\left(0;0,2\right)}\nabla_{k}K_{ij}\nabla^{k}K^{ij}+c_{2}^{\left(0;0,2\right)}\nabla_{i}K^{ij}\nabla_{k}K_{j}^{k}+c_{3}^{\left(0;0,2\right)}\nabla_{i}K^{ij}\nabla_{j}K$
$\displaystyle+c_{4}^{\left(0;0,2\right)}\nabla_{i}K\nabla^{i}K+c_{5}^{\left(0;0,2\right)}\nabla_{i}a_{j}\nabla^{i}a^{j}+c_{6}^{\left(0;0,2\right)}\left(\nabla_{i}a^{i}\right)^{2}$
$\displaystyle+c_{1}^{\left(1;2,0\right)}R_{ij}K_{k}^{i}K^{jk}+c_{2}^{\left(1;2,0\right)}R_{ij}a^{i}a^{j}+c_{3}^{\left(1;2,0\right)}R_{ij}K^{ij}K+c_{4}^{\left(1;2,0\right)}RK_{ij}K^{ij}+c_{5}^{\left(1;2,0\right)}Ra_{i}a^{i}+c_{6}^{\left(1;2,0\right)}RK^{2}$
$\displaystyle+c_{1}^{\left(2;0,0\right)}R_{ij}R^{ij}+c_{2}^{\left(2;0,0\right)}R^{2}$
$\displaystyle+c_{1}^{\left(1;0,1\right)}R\nabla_{i}a^{i},$

and the parity-violating terms are

$\displaystyle\tilde{\mathcal{L}}^{\left(3\right)}$
$\displaystyle=\tilde{c}_{1}^{\left(0;1,1\right)}$
$\displaystyle\varepsilon_{ijk}K_{l}^{i}\nabla^{j}K^{kl},$ (13)
$\displaystyle\tilde{\mathcal{L}}^{\left(4\right)}$ $\displaystyle=$
$\displaystyle\tilde{c}_{1}^{\left(0;2,1\right)}\varepsilon_{ijk}K^{im}K^{jn}\nabla_{m}K_{n}^{k}+\tilde{c}_{2}^{\left(0;2,1\right)}\varepsilon_{ijk}K^{mn}K_{m}^{i}\nabla^{j}K_{n}^{k}+\tilde{c}_{3}^{\left(0;2,1\right)}\varepsilon_{ijk}K_{l}^{i}a^{j}\nabla^{k}a^{l}+\tilde{c}_{4}^{\left(0;2,1\right)}\varepsilon_{ijk}K_{l}^{i}\nabla^{j}K^{kl}K$
(14)
$\displaystyle+\tilde{c}_{1}^{\left(1;2,0\right)}\varepsilon_{ijk}R_{l}^{i}K^{jl}a^{k}+\tilde{c}_{1}^{\left(1;0,1\right)}\varepsilon_{ijk}R_{l}^{i}\nabla^{j}K^{kl}.$

In the above, $a_{i}\equiv\nabla_{i}\ln N$ is the acceleration. The
coefficients $c_{m}^{\left(c_{0};d_{2},d_{3}\right)}$ and
$\tilde{c}_{m}^{\left(c_{0};d_{2},d_{3}\right)}$ are generally functions of
$t$ and $N$ without spatial derivatives. Note the spatial derivatives of Ricci
tensor, like $\nabla_{i}R_{jk}$, are not included in our model as they are
higher order in $d$.

Without any specific fine tuning of the coefficients, in [31] it has been
shown through a Hamiltonian analysis that the action (1) propagates 3 DOF’s in
general, of which one is a scalar mode and two are tensor modes444In [31] only
the parity-preserving Lagrangian is considered. While from the analysis it is
clear that the presence of $\varepsilon_{ijk}$ would not change the constraint
structure and thus the number of DOF’s.. Therefore, in order to eliminate one
of the three DOF’S, in particular, the scalar mode, additional conditions must
be imposed.

For the action in (1), the general conditions of propagating at most two
tensorial degrees of freedom (TTDOF’s) have been derived in [74], which can be
written as

$\displaystyle\mathcal{S}(\vec{x},\vec{y})\approx
0,\qquad\mathcal{J}(\vec{x},\vec{y})\approx 0,$ (15)

where

$\displaystyle\mathcal{S}\left(\vec{x},\vec{y}\right)$
$\displaystyle\coloneqq$ $\displaystyle\frac{\delta^{2}S}{\delta
N\left(\vec{x}\right)\delta
N\left(\vec{y}\right)}-\int\mathrm{d}^{3}x^{\prime}\int\mathrm{d}^{3}y^{\prime}\,N\left(\vec{x}^{\prime}\right)\frac{\delta}{\delta
N\left(\vec{x}\right)}\left(\frac{1}{N\left(\vec{x}^{\prime}\right)}\frac{\delta
S}{\delta K_{i^{\prime}j^{\prime}}\left(\vec{x}^{\prime}\right)}\right)$ (17)
$\displaystyle\hskip
80.00012pt\times\mathcal{G}_{i^{\prime}j^{\prime},k^{\prime}l^{\prime}}\left(\vec{x}^{\prime},\vec{y}^{\prime}\right)N\left(\vec{y}^{\prime}\right)\frac{\delta}{\delta
N\left(\vec{y}\right)}\left(\frac{1}{N\left(\vec{y}^{\prime}\right)}\frac{\delta
S}{\delta K_{i^{\prime}j^{\prime}}\left(\vec{y}^{\prime}\right)}\right),$

with $\mathcal{G}_{ij,kl}\left(\vec{x},\vec{y}\right)$ the inverse of the
Hessian with respect to $K_{ij}$ satisfying

$\int\mathrm{d}^{3}z\,\mathcal{G}_{ij,mn}\left(\vec{x},\vec{z}\right)\frac{\delta^{2}S}{\delta
K_{mn}\left(\vec{z}\right)\delta
K_{kl}\left(\vec{y}\right)}\equiv\frac{1}{2}\left(\delta_{i}^{k}\delta_{j}^{l}+\delta_{i}^{l}\delta_{j}^{k}\right)\delta^{3}\left(\vec{x}-\vec{y}\right),$
(18)

and

$\displaystyle\mathcal{J}\left(\vec{x},\vec{y}\right)$
$\displaystyle\coloneqq$
$\displaystyle\int\mathrm{d}^{3}x^{\prime}\int\mathrm{d}^{3}y^{\prime}\int\mathrm{d}^{3}x^{\prime\prime}\int\mathrm{d}^{3}y^{\prime\prime}\frac{\delta
C\left(\vec{x}\right)}{\delta
K_{ij}\left(\vec{x}^{\prime}\right)}\mathcal{G}_{i^{\prime}j^{\prime},k^{\prime}l^{\prime}}\left(\vec{x}^{\prime},\vec{x}^{\prime\prime}\right)$
(19) $\displaystyle\qquad\times
N\left(\vec{x}^{\prime\prime}\right)\frac{\delta^{2}S}{\delta
h_{i^{\prime}j^{\prime}}\left(\vec{x}^{\prime\prime}\right)\delta
K_{k^{\prime}l^{\prime}}\left(\vec{y}^{\prime\prime}\right)}\mathcal{G}_{k^{\prime}l^{\prime},kl}\left(\vec{y}^{\prime\prime},\vec{y}^{\prime}\right)\frac{\delta
C\left(\vec{y}\right)}{\delta K_{ij}\left(\vec{y}^{\prime}\right)}$
$\displaystyle-\int\mathrm{d}^{3}x^{\prime}\int\mathrm{d}^{3}y^{\prime}\frac{\delta
C^{\prime}\left(\vec{x}\right)}{\delta
K_{ij}\left(\vec{x}^{\prime}\right)}\mathcal{G}_{ij,kl}\left(\vec{x}^{\prime},\vec{y}^{\prime}\right)N\left(\vec{y}^{\prime}\right)\frac{\delta
C\left(\vec{y}\right)}{\delta
h_{kl}\left(\vec{y}^{\prime}\right)}-\left(\vec{x}\leftrightarrow\vec{y}\right),$

with

$C\left(\vec{x}\right)\coloneqq-\frac{\delta S}{\delta
N\left(\vec{x}\right)}+\frac{1}{N\left(\vec{x}\right)}\frac{\delta S}{\delta
K_{ij}\left(\vec{x}\right)}K_{ij}\left(\vec{x}\right).$ (20)

These two TTDOF conditions, which are dubbed the degenerate condition and the
consistency condition, are the necessary and sufficient conditions for the
action (1) to propagate at most two DOF’s. When the lapse function becomes
dynamical, the generalized conditions have also been derived in [75].

Although these TTDOF conditions are general and conceptually simple, they are
mathematically involved to be solved to yield concrete Lagrangians. This is
one of the motivations of this work to look for an alternative and more
practical approach.

##  III The perturbative approach and degeneracy condition

The unwanted degree of freedom, if not contained in the theory, will never
show up at any order in perturbations around some background. Thus one may
tune the coefficients in the Lagrangian such that the unwanted scalar mode is
eliminated order by order in perturbations. Since there is a finite number of
conditions in the non-perturbative sense, one will stop at some finite order
and get the final Lagrangian in which the scalar mode is fully eliminated. The
perturbative approach can be a possible candidate method to bypass the
mathematical difficulties in deal with the non-perturbative conditions got in
a Hamiltonian analysis. For a class of SCG theories with the dynamical lapse
function, this perturbative analysis has been used to reduce the number of
DOF’s from four to three [79]. It was interesting that even at the cubic order
in perturbations around a cosmological background, one could re-produce the
fully non-perturbative conditions to eliminate the unwanted mode.

We consider perturbations around a Friedmann-Robertson-Walker (FRW)
background. For our purpose, we focus on the scalar perturbations only. After
fixing the gauge freedom of the spatial diffeomorphism, the usual Arnowitt-
Deser-Misner (ADM) variables correspond to

$\displaystyle N$ $\displaystyle=$ $\displaystyle\bar{N}e^{A},$ (21)
$\displaystyle N_{i}$ $\displaystyle=$ $\displaystyle\bar{N}a\,\partial_{i}B,$
(22) $\displaystyle h_{ij}$ $\displaystyle=$ $\displaystyle
a^{2}e^{2\zeta}\delta_{ij},$ (23)

where $a=a(t)$ is the scale factor, $\bar{N}=\bar{N}(t)$ is background value
of the lapse function.

Contrary to what one usually did in generally covariant theories, here we do
not set $\bar{N}=1$, since there is no time-reparametrization symmetry in our
theory in general. In particular, we assume that the Lagrangian depends on the
lapse function $N$ explicitly while not on the time. As a result, generally
the lapse function $N$ has a non-unity background value $\bar{N}(t)$. On the
other hand, setting $\bar{N}=1$ implicitly redefines the time parameter $t$,
which re-introduces the time-dependence of the Lagrangian.

The quadratic action for the scalar perturbations takes the form (we follow
the notation in [79])

$S_{2}\left[\zeta,A,B\right]=\int\\!\mathrm{d}t\mathrm{d}^{3}x\,\bar{N}a^{3}\mathcal{L}_{2}.$
(24)

The quadratic Lagrangian can be split into two parts

$\mathcal{L}_{2}=\mathcal{L}_{2}^{(\mathrm{I})}+\mathcal{L}_{2}^{(\mathrm{II})},$
(25)

in which $\mathcal{L}_{2}^{(\mathrm{I})}$ stands for terms relevant to
counting the number of DOF’s,

$\mathcal{L}_{2}^{(\mathrm{I})}=\dot{\zeta}\hat{\mathcal{C}}_{\dot{\zeta}\dot{\zeta}}\dot{\zeta}+\dot{\zeta}\hat{\mathcal{C}}_{\dot{\zeta}A}A+\dot{\zeta}\hat{\mathcal{C}}_{\dot{\zeta}B}B+A\hat{\mathcal{C}}_{AA}A+A\hat{\mathcal{C}}_{AB}B+B\hat{\mathcal{C}}_{BB}B,$
(26)

and $\mathcal{L}_{2}^{(\mathrm{II})}$ stands for terms irrelevant to counting
the number of DOF’s.

$\mathcal{L}_{2}^{(\mathrm{II})}=\zeta\hat{\mathcal{C}}_{\zeta\zeta}\zeta+\zeta\hat{\mathcal{C}}_{\zeta
A}A+\zeta\hat{\mathcal{C}}_{\zeta B}B.$ (27)

In the above
$\hat{\mathcal{C}}_{\dot{\zeta}\dot{\zeta}},\hat{\mathcal{C}}_{\dot{\zeta}A}$
etc. are time-dependent operators which may contain spatial derivatives.
Following [88], throughout this paper we shall use the shorthand

$\dot{X}\equiv\frac{1}{\bar{N}}\frac{\partial X}{\partial t},\qquad
f^{\prime}\equiv\bar{N}\left.\frac{\partial f}{\partial
N}\right|_{N=\bar{N}},\qquad
f^{\prime\prime}\equiv\bar{N}^{2}\left.\frac{\partial^{2}f}{\partial
N^{2}}\right|_{N=\bar{N}}.$ (28)

At this point, note the quadratic Lagrangian for the scalar modes contain no
parity-violating term. In other words, the parity-violating terms in (13) and
(14) do not contribute to the quadratic order Lagrangian for the scalar modes
and have nothing to do with eliminating the scalar modes at least at the
linear order in perturbations. Mathematically, this is simply because it is
not possible to build a term quadratic in the scalar modes with
$\varepsilon_{ijk}$. If we go to higher order, the parity-violating terms do
contribute to the scalar modes.

It is clear that in the quadratic action (24), $A$ and $B$ act as the
auxiliary variables (i.e., without the time derivatives). We may solve $A$ and
$B$ formally from their equations of motion

$\displaystyle
2\frac{1}{\bar{N}}\partial_{t}\left(a^{3}\hat{\mathcal{C}}_{\dot{\zeta}\dot{\zeta}}\dot{\zeta}\right)+\frac{1}{\bar{N}}\partial_{t}\left(a^{3}\hat{\mathcal{C}}_{\dot{\zeta}A}A\right)+\frac{1}{\bar{N}}\partial_{t}\left(a^{3}\hat{\mathcal{C}}_{\dot{\zeta}B}B\right)-2a^{3}\hat{\mathcal{C}}_{\zeta\zeta}\zeta-a^{3}\hat{\mathcal{C}}_{\zeta
A}A-a^{3}\hat{\mathcal{C}}_{\zeta B}B$ $\displaystyle=$ $\displaystyle 0,$
(29) $\displaystyle
2\hat{\mathcal{C}}_{AA}A+\hat{\mathcal{C}}_{AB}B+\hat{\mathcal{C}}_{\dot{\zeta}A}\dot{\zeta}+\hat{\mathcal{C}}_{\zeta
A}\zeta$ $\displaystyle=$ $\displaystyle 0,$ (30)
$\displaystyle\hat{\mathcal{C}}_{AB}A+2\hat{\mathcal{C}}_{BB}B+\hat{\mathcal{C}}_{\dot{\zeta}B}\dot{\zeta}+\hat{\mathcal{C}}_{\zeta
B}\zeta$ $\displaystyle=$ $\displaystyle 0.$ (31)

The solutions for $A$ and $B$ can be formally written as

$A=\frac{\left(\frac{1}{2}\hat{\mathcal{C}}_{AB}\hat{\mathcal{C}}_{\dot{\zeta}B}-\hat{\mathcal{C}}_{BB}\hat{\mathcal{C}}_{\dot{\zeta}A}\right)\dot{\zeta}+\left(\frac{1}{2}\hat{\mathcal{C}}_{AB}\hat{\mathcal{C}}_{\zeta
B}-\hat{\mathcal{C}}_{BB}\hat{\mathcal{C}}_{\zeta
A}\right)\zeta}{2\hat{\mathcal{C}}_{AA}\hat{\mathcal{C}}_{BB}-\frac{1}{2}\hat{\mathcal{C}}_{AB}\hat{\mathcal{C}}_{AB}},$
(32)

and

$B=\frac{\left(\frac{1}{2}\hat{\mathcal{C}}_{AB}\hat{\mathcal{C}}_{\dot{\zeta}A}-\hat{\mathcal{C}}_{AA}\hat{\mathcal{C}}_{\dot{\zeta}B}\right)\dot{\zeta}+\left(\frac{1}{2}\hat{\mathcal{C}}_{AB}\hat{\mathcal{C}}_{\zeta
A}-\hat{\mathcal{C}}_{AA}\hat{\mathcal{C}}_{\zeta
B}\right)\zeta}{2\hat{\mathcal{C}}_{AA}\hat{\mathcal{C}}_{BB}-\frac{1}{2}\hat{\mathcal{C}}_{AB}\hat{\mathcal{C}}_{AB}},$
(33)

with
$2\hat{\mathcal{C}}_{AA}\hat{\mathcal{C}}_{BB}-\frac{1}{2}\hat{\mathcal{C}}_{AB}\hat{\mathcal{C}}_{AB}\neq
0$. Since
$\hat{\mathcal{C}}_{\dot{\zeta}\dot{\zeta}},\hat{\mathcal{C}}_{\dot{\zeta}A}$
etc. may contain spatial derivatives, the above solutions may be better
understood in the Fourier space. Plugging the above solutions into (29) yields
the equation of motion for the single variable $\zeta$. If in the equation of
motion $\zeta$ acquires a second derivative term $\ddot{\zeta}$, $\zeta$ is
dynamical. Therefore in order to have no scalar mode propagating at the linear
order, we have to “kill” the coefficient of $\ddot{\zeta}$ in its equation of
motion. After some manipulations, one find that this implies

$\Delta\coloneqq
2\hat{\mathcal{C}}_{\dot{\zeta}\dot{\zeta}}\left(2\hat{\mathcal{C}}_{AA}\hat{\mathcal{C}}_{BB}-\frac{1}{2}\hat{\mathcal{C}}_{AB}^{2}\right)+\hat{\mathcal{C}}_{AB}\hat{\mathcal{C}}_{\dot{\zeta}A}\hat{\mathcal{C}}_{\dot{\zeta}B}-\hat{\mathcal{C}}_{BB}\hat{\mathcal{C}}_{\dot{\zeta}A}^{2}-\hat{\mathcal{C}}_{AA}\hat{\mathcal{C}}_{\dot{\zeta}B}^{2}=0.$
(34)

We may refer to (34) the degeneracy condition. The main task in this work is
thus to use (34) as our starting point to find the conditions for the various
coefficients $c_{m}^{\left(c_{0};d_{2},d_{3}\right)}$ and
$\tilde{c}_{m}^{\left(c_{0};d_{2},d_{3}\right)}$ such that such that no scalar
mode is propagating at the linear order in perturbations.

In the above, we derive the degeneracy condition (24) by solving the auxiliary
variables and looking at the coefficient of the kinetic term in the effective
Lagrangian for the single variable $\zeta$, which is the standard operation in
calculating the cosmological perturbations. We emphasize that it is not
trivial to count the number of dynamical degree of freedom even for the
quadratic Lagrangian in (24). In the appendix A, we make a thorough analysis
of a point particle model.

##  IV Eliminate the scalar mode at the linear order

In this section, we shall find the conditions for the coefficients in the
Lagrangian such that the degeneracy condition (24) is satisfied, and thus no
scalar mode propagates at the linear order in perturbations.

###  IV.1 $d=2$

We consider the model constructed by all the terms of $d=2$:

$S=\int\mathrm{d}t\mathrm{d}^{3}x\,N\sqrt{h}\left(\mathcal{L}^{\left(2\right)}-\Lambda\right),$
(35)

where $\mathcal{L}^{(2)}$ is given in (10). We have introduced a positive
cosmological constant $\Lambda>0$ in order to have an expanding background
solution. Equivalently, the above Lagrangian can be regarded as the linear
combination of $\mathcal{L}^{(0)}$ and $\mathcal{L}^{(2)}$ with
$c_{1}^{(0;0,0)}=\Lambda$.

Expanding the action to the first order in perturbations yields

$S_{1}=\int\mathrm{d}t\mathrm{d}^{3}x\,\mathcal{L}_{1},$ (36)

with

$\displaystyle\mathcal{L}_{1}$ $\displaystyle\simeq$
$\displaystyle\bar{N}a^{3}\left[-3H^{2}\left(b_{2}-b_{2}^{\prime}\right)-\Lambda\right]A$
(37)
$\displaystyle+3\bar{N}a^{3}\left[-\left(3H^{2}+2\dot{H}\right)b_{2}-\Lambda-2H\dot{b}_{2}\right]\zeta,$

where we define

$b_{2}\equiv c_{1}^{(0,2,0)}+3c_{3}^{(0,2,0)},$ (38)

for short. The background equations of motion are determined by requiring
$S_{1}=0$, which are

$\displaystyle-3H^{2}\left(b_{2}-b_{2}^{\prime}\right)-\Lambda$
$\displaystyle=$ $\displaystyle 0,$ (39)
$\displaystyle-\left(3H^{2}+2\dot{H}\right)b_{2}-\Lambda-2H\dot{b}_{2}$
$\displaystyle=$ $\displaystyle 0,$ (40)

for $A$ and $\zeta$, respectively. At this point, keep in mind that $\dot{f}$
and $f^{\prime}$ are defined as in (28). The Hubble parameter is defined to be
$H\coloneqq\frac{\dot{a}}{a}\equiv\frac{1}{\bar{N}a}\frac{\partial a}{\partial
t}$. From (39) it is transparent that we get a generally expanding background
only with a non-vanishing cosmological constant $\Lambda$.

Expanding the action to the second order in perturbations yields

$S_{2}=\int\mathrm{d}t\mathrm{d}^{3}x\,\bar{N}a^{3}\mathcal{L}_{2}.$ (41)

According to (24), the relevant coefficients are

$\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}\dot{\zeta}}^{(2)}$
$\displaystyle=$ $\displaystyle 3b_{2},$
$\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}A}^{(2)}$ $\displaystyle=$
$\displaystyle 6H\left(b_{2}^{\prime}-b_{2}\right),$ (42)
$\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}B}^{(2)}$ $\displaystyle=$
$\displaystyle-2b_{2}\frac{\partial^{2}}{a},$ (43)
$\displaystyle\hat{\mathcal{C}}_{AA}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{3}{2}H^{2}\left(b_{2}^{\prime\prime}-2b_{2}^{\prime}+2b_{2}\right)-\tilde{b}_{2}\frac{\partial^{2}}{a^{2}},$
(44) $\displaystyle\hat{\mathcal{C}}_{AB}^{(2)}$ $\displaystyle=$
$\displaystyle 2H\left(-b_{2}^{\prime}+b_{2}\right)\frac{\partial^{2}}{a},$
(45) $\displaystyle\hat{\mathcal{C}}_{BB}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(2w_{2}+b_{2}\right)\frac{\partial^{4}}{a^{2}},$
(46)

where we denote

$\tilde{b}_{2}\coloneqq c_{2}^{(0,2,0)},$ (47)

and

$w_{2}\coloneqq c_{1}^{(0,2,0)},$ (48)

for short. For later convenience, we also have

$\displaystyle\hat{\mathcal{C}}_{\zeta A}^{(2)}$ $\displaystyle=$
$\displaystyle-4\left(h_{2}^{\prime}+h_{2}\right)\frac{\partial^{2}}{a^{2}},$
(49)

with

$h_{2}\coloneqq c_{1}^{(1,0,0)},$ (50)

for short, and

$\hat{\mathcal{C}}_{\zeta B}^{(2)}=0.$ (51)

In the above we have made use of the background equations of motion (39) and
(40) to eliminate $\Lambda$ and to simplify the expressions of the
coefficients. From (32) and (33), the solutions for $A$ and $B$ are

$A=\frac{4\left[3a^{2}H(b_{2}-b_{2}^{\prime})w_{2}\dot{\zeta}+(h_{2}+h_{2}^{\prime})(b_{2}+2w_{2})\partial^{2}\zeta\right]}{3a^{2}H^{2}\left[-2(b_{2}^{\prime})^{2}+b_{2}\left(2b_{2}^{\prime}+b_{2}^{\prime\prime}\right)+2\left(2b_{2}-2b_{2}^{\prime}+b_{2}^{\prime\prime}\right)w_{2}\right]-2\tilde{b}_{2}(b_{2}+2w_{2})\partial^{2}},$
(52)

and

$B=\frac{3a\left\\{4H(b_{2}-b_{2}^{\prime})(h_{2}+h_{2}^{\prime})\partial^{2}\zeta-3a^{2}H^{2}\left(-2(b_{2}^{\prime})^{2}+b_{2}(2b_{2}^{\prime}+b_{2}^{\prime\prime})\right)\dot{\zeta}+2b_{2}\tilde{b}_{2}\partial^{2}\dot{\zeta}\right\\}}{\left\\{-3a^{2}H^{2}\left[-2(b_{2}^{\prime})^{2}+b_{2}\left(2b_{2}^{\prime}+b_{2}^{\prime\prime}\right)+2\left(2b_{2}-2b_{2}^{\prime}+b_{2}^{\prime\prime}\right)w_{2}\right]+2\tilde{b}_{2}(b_{2}+2w_{2})\partial^{2}\right\\}\partial^{2}}.$
(53)

By plugging the above results in (34), we get the degeneracy condition

$\Delta^{(2)}=12H^{2}w_{2}\left[-2\left(b_{2}^{\prime}\right)^{2}+b_{2}\left(2b_{2}^{\prime}+b_{2}^{\prime\prime}\right)\right]\frac{\partial^{4}}{a^{2}}-8b_{2}\tilde{b}_{2}w_{2}\frac{\partial^{6}}{a^{4}}.$
(54)

At this point, note we need to require that $w_{2}\neq 0$, otherwise there
will be no kinetic term for the gravitational waves [87]. In order to have
$\Delta^{(2)}=0$ so that there is no scalar mode propagating, one special
solution is

$b_{2}\equiv c_{1}^{(0,2,0)}+3c_{3}^{(0,2,0)}=0.$ (55)

However, this choice is conflict with the background equation of motion (39).
Therefore we must have $b_{2}\neq 0$, which is also the case of GR. Then first
we have to require that

$\tilde{b}_{2}\equiv c_{2}^{(0,2,0)}=0,$ (56)

so that $\sim\partial^{6}$ term in (54) vanishes, since we have assumed
$w_{2}\neq 0$. This indicates that the acceleration $a_{i}$ should not appear
explicitly. We also need to require

$-2\left(b_{2}^{\prime}\right)^{2}+b_{2}\left(2b_{2}^{\prime}+b_{2}^{\prime\prime}\right)=0.$
(57)

Note (57) must hold for any value of $\bar{N}$. Therefore (57) is regarded as
a homogeneous differential equation for $b_{2}$ as a function of $N$, in which
$b_{2}^{\prime}$ and $b_{2}^{\prime\prime}$ are defined as in (28). For later
convenience, the solutions for $A$ and $B$ (52) in (53) under the conditions
(56) and (57) get simplified to be

$A=\frac{b_{2}}{H\left(b_{2}-b_{2}^{\prime}\right)}\dot{\zeta}+\frac{b_{2}(h_{2}+h_{2}^{\prime})(b_{2}+2w_{2})}{3a^{2}H^{2}\left(b_{2}-b_{2}^{\prime}\right)^{2}w_{2}}\partial^{2}\zeta,$
(58)

and

$B=-\frac{b_{2}\left(h_{2}+h_{2}^{\prime}\right)}{aH\left(b_{2}-b_{2}^{\prime}\right)w_{2}}\zeta,$
(59)

which involves no $\dot{\zeta}$. In (58) and (59) we have made use of (57) to
replace $b_{2}^{\prime\prime}$ in terms of $b_{2}$ and $b_{2}^{\prime}$.

The general solution for $b_{2}$ to (57) is

$b_{2}=\frac{C_{1}N}{1+C_{2}N},$ (60)

where $C_{1},C_{2}$ are two constants. This solution is also consistent with
the analysis in [74] (see eqs. (110)-(111) therein). Obviously,
$b_{2}=\mathrm{const.}$ is a trivial solution, which corresponds to the limit
$C_{1},C_{2}\rightarrow\infty$ by keeping $\frac{C_{1}}{C_{2}}$ finite. To
conclude, the Lagrangian

$\mathcal{L}^{\left(2\right)}=w_{2}\hat{K}_{ij}\hat{K}^{ij}+\frac{1}{3}\frac{C_{1}N}{1+C_{2}N}K^{2}+h_{2}R,$
(61)

with $w_{2}$ and $h_{2}$ being general functions of $N$, contains no dynamical
scalar degree of freedom at linear order in a cosmological background. Here
$\hat{K}_{ij}$ is the traceless part of $K_{ij}$ defined by

$\hat{K}_{ij}\coloneqq K_{ij}-\frac{1}{3}Kh_{ij}.$ (62)

With the form of the Lagrangian (61), GR is a special case with the choice

$c_{1}^{\left(0;2,0\right)}=c_{1}^{\left(1;0,0\right)}=1,$ (63)

and thus corresponds to

$\left|C_{1}\right|,\left|C_{2}\right|\rightarrow\infty,\qquad\text{keeping}\qquad\frac{C_{1}}{C_{2}}=-2.$
(64)

###  IV.2 $d=3$

Next we consider

$S=\int\mathrm{d}t\mathrm{d}^{3}x\,N\sqrt{h}\left(\mathcal{L}^{\left(3\right)}-\Lambda\right),$
(65)

where $\mathcal{L}^{(3)}$ is given in (11), and again we include a positive
cosmological constant $\Lambda$ in order to have a non-vanishing $H$.
Expanding the action to the first order in perturbations yields

$S_{1}=\int\mathrm{d}t\mathrm{d}^{3}x\,\mathcal{L}_{1},$ (66)

with

$\displaystyle\mathcal{L}_{1}$ $\displaystyle=$
$\displaystyle\bar{N}a^{3}\left[3H^{3}\left(b_{3}^{\prime}-2b_{3}\right)-\Lambda\right]A$
(67)
$\displaystyle+3\bar{N}a^{3}\left[-6H\left(H^{2}+\dot{H}\right)b_{3}-\Lambda-3H^{2}\dot{b}_{3}\right]\zeta,$

where we define

$b_{3}\equiv c_{1}^{(0,3,0)}+3c_{3}^{(0,3,0)}+9c_{5}^{(0,3,0)},$ (68)

for short. Thus the background equations of motion are

$\displaystyle-3H^{3}\left(2b_{3}-b_{3}^{\prime}\right)-\Lambda$
$\displaystyle=$ $\displaystyle 0,$ (69)
$\displaystyle-6H\left(H^{2}+\dot{H}\right)b_{3}-\Lambda-3H^{2}\dot{b}_{3}$
$\displaystyle=$ $\displaystyle 0,$ (70)

for $A$ and $\zeta$, respectively. Note we must have $b_{3}\neq 0$ in order to
make (69) consistent with a non-vanishing $\Lambda$.

Expanding the action to the second order in perturbations yields

$S_{2}=\int\mathrm{d}t\mathrm{d}^{3}x\,\bar{N}a^{3}\mathcal{L}_{2}.$ (71)

According to (24), the relevant coefficients are

$\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}\dot{\zeta}}^{(3)}$
$\displaystyle=$ $\displaystyle 9Hb_{3},$ (72)
$\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}A}^{(3)}$ $\displaystyle=$
$\displaystyle-9H^{2}\left(-b_{3}^{\prime}+2b_{3}\right)+f_{3}\frac{\partial^{2}}{a^{2}},$
(73) $\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}B}^{(3)}$ $\displaystyle=$
$\displaystyle-6Hb_{3}\frac{\partial^{2}}{a},$ (74)
$\displaystyle\hat{\mathcal{C}}_{AA}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{3}{2}H^{3}\left(b_{3}^{\prime\prime}-4b_{3}^{\prime}+6b_{3}\right)+H\left(f_{3}^{\prime}-\tilde{b}_{3}\right)\frac{\partial^{2}}{a^{2}},$
(75) $\displaystyle\hat{\mathcal{C}}_{AB}^{(3)}$ $\displaystyle=$
$\displaystyle
3H^{2}\left(-b_{3}^{\prime}+2b_{3}\right)\frac{\partial^{2}}{a}-\tilde{f}_{3}\frac{\partial^{4}}{a^{3}},$
(76) $\displaystyle\hat{\mathcal{C}}_{BB}^{(3)}$ $\displaystyle=$
$\displaystyle H\left(2w_{3}+b_{3}\right)\frac{\partial^{4}}{a^{2}},$ (77)

where we denote

$\displaystyle f_{3}$ $\displaystyle\coloneqq$ $\displaystyle
c_{1}^{(0,1,1)}+3c_{2}^{(0,1,1)},$ (78) $\displaystyle\tilde{b}_{3}$
$\displaystyle\coloneqq$ $\displaystyle c_{2}^{(0,3,0)}+3c_{4}^{(0,3,0)},$
(79) $\displaystyle\tilde{f}_{3}$ $\displaystyle\coloneqq$ $\displaystyle
c_{1}^{(0,1,1)}+c_{2}^{(0,1,1)},$ (80) $\displaystyle w_{3}$
$\displaystyle\coloneqq$ $\displaystyle c_{1}^{(0,3,0)}+c_{3}^{(0,3,0)},$ (81)

for shorthand. We have made use of the background equations of motion (69) and
(70) to simply the coefficients.

After some manipulations, the degeneracy condition (34) now becomes

$\displaystyle\Delta^{(3)}$ $\displaystyle=$ $\displaystyle
54H^{5}w_{3}\left[-3\left(b_{3}^{\prime}\right)^{2}+2b_{3}\left(2b_{3}^{\prime}+b_{3}^{\prime\prime}\right)\right]\frac{\partial^{4}}{a^{2}}$
(82)
$\displaystyle+36H^{3}w_{3}\left[-b_{3}^{\prime}f_{3}+2b_{3}\left(f_{3}+f_{3}^{\prime}-\tilde{b}_{3}\right)\right]\frac{\partial^{6}}{a^{4}}$
$\displaystyle-H\left[b_{3}\left(f_{3}-3\tilde{f}_{3}\right)^{2}+2\left(f_{3}\right)^{2}w_{3}\right]\frac{\partial^{8}}{a^{6}}.$

Similar to the case of $d=2$, we require $w_{3}\neq 0$, otherwise there will
be no gravitational waves [87]. Thus the degeneracy condition $\Delta^{(3)}=0$
yields a set of three equations

$\displaystyle-3\left(b_{3}^{\prime}\right)^{2}+2b_{3}\left(2b_{3}^{\prime}+b_{3}^{\prime\prime}\right)$
$\displaystyle=$ $\displaystyle 0,$ (83) $\displaystyle-
b_{3}^{\prime}f_{3}+2b_{3}\left(f_{3}+f_{3}^{\prime}-\tilde{b}_{3}\right)$
$\displaystyle=$ $\displaystyle 0,$ (84) $\displaystyle
b_{3}\left(f_{3}-3\tilde{f}_{3}\right)^{2}+2\left(f_{3}\right)^{2}w_{3}$
$\displaystyle=$ $\displaystyle 0.$ (85)

Recall that there are nine free coefficients in the original Lagrangian
$\mathcal{L}^{(3)}$ (11), which are subject to the above three equations in
order to eliminate the scalar degree of freedom at the linear order. We can
solve $b_{3}$ from (83) to be

$b_{3}=\frac{D_{1}N^{2}}{\left(1+D_{2}N\right)^{2}},$ (86)

with $D_{1},D_{2}$ being two constants. At this point, note in order to make
the background equation of motion (69) consistent, which now reads

$6H^{3}\frac{D_{1}D_{2}N^{3}}{\left(1+D_{2}N\right)^{3}}+\Lambda=0,$ (87)

we have to require that (since $N>0$)

$\frac{D_{1}D_{2}}{\left(1+D_{2}N\right)^{3}}<0.$ (88)

By using the solution (86), we then solve $\tilde{b}_{3}$ from (84) to be

$\tilde{b}_{3}=\frac{D_{2}N}{1+D_{2}N}f_{3}+f_{3}^{\prime}.$ (89)

As for the last equation (85), according to $f_{3}$ is vanishing or not, we
discuss two cases.

####  IV.2.1 Case 1

If $f_{3}=0$, since we assume $b_{3}\neq 0$, from (85) we must also have
$\tilde{f}_{3}=0$, which implies that

$c_{1}^{(0;1,1)}=0,\qquad c_{2}^{(0;1,1)}=0,$ (90)

and $w_{3}$ can be chosen freely (but non-vanishing) in general. In this case,
there is no $\nabla a$ term in the Lagrangian. As a result, $\tilde{b}_{3}=0$
and thus we may solve

$c_{4}^{(0;3,0)}=-\frac{1}{3}c_{2}^{(0;3,0)}.$ (91)

In this case, the Lagrangian is given by

$\displaystyle\mathcal{L}^{\left(3\right),\mathrm{I}}$ $\displaystyle=$
$\displaystyle
c_{1}^{\left(0;3,0\right)}\hat{K}_{ij}\hat{K}^{jk}\hat{K}_{k}^{i}+w_{3}\,\hat{K}_{ij}\hat{K}^{ij}K+\frac{1}{9}\frac{D_{1}N^{2}}{\left(1+D_{2}N\right)^{2}}K^{3}$
(92)
$\displaystyle+c_{2}^{\left(0;3,0\right)}\hat{K}_{ij}a^{i}a^{j}+c_{1}^{\left(1;1,0\right)}R^{ij}K_{ij}+c_{2}^{\left(1;1,0\right)}RK,$

where the coefficients $c_{1}^{\left(0;3,0\right)}$, $w_{3}$ etc. are
generally functions of $N$.

####  IV.2.2 Case 2

If $f_{3}\neq 0$, we can solve $w_{3}$ or more conveniently $c_{3}^{(0;3,0)}$
from (85) to be

$c_{3}^{(0;3,0)}=-c_{1}^{(0;3,0)}-2\frac{D_{1}N^{2}}{\left(1+D_{2}N\right)^{2}}\left(\frac{c_{1}^{(0;1,1)}}{f_{3}}\right)^{2},$
(93)

As a result, by making use of (68) and (86), we may solve

$c_{5}^{(0;3,0)}=\frac{2}{9}c_{1}^{(0;3,0)}+\frac{2}{3}\frac{D_{1}N^{2}}{\left(1+D_{2}N\right)^{2}}\left(\frac{c_{1}^{(0;1,1)}}{f_{3}}\right)^{2}+\frac{1}{9}\frac{D_{1}N^{2}}{\left(1+D_{2}N\right)^{2}}.$
(94)

In this case the Lagrangian reduces to be

$\displaystyle\mathcal{L}^{\left(3\right),\mathrm{II}}$ $\displaystyle=$
$\displaystyle
c_{1}^{\left(0;3,0\right)}\hat{K}_{ij}\hat{K}^{jk}\hat{K}_{k}^{i}+\frac{1}{9}\frac{D_{1}N^{2}}{\left(1+D_{2}N\right)^{2}}K^{3}$
(95)
$\displaystyle-2\frac{D_{1}N^{2}}{\left(1+D_{2}N\right)^{2}}\left(\frac{c_{1}^{(0;1,1)}}{f_{3}}\right)^{2}\hat{K}_{ij}\hat{K}^{ij}K$
$\displaystyle+c_{2}^{\left(0;3,0\right)}\hat{K}_{ij}a^{i}a^{j}+\frac{1}{3}\left(\frac{D_{2}N}{1+D_{2}N}f_{3}+f_{3}^{\prime}\right)Ka_{i}a^{i}$
$\displaystyle+c_{1}^{\left(0;1,1\right)}\hat{K}_{ij}\nabla^{i}a^{j}+\frac{1}{3}f_{3}K\nabla_{i}a^{i}$
$\displaystyle+c_{1}^{\left(1;1,0\right)}R^{ij}K_{ij}+c_{2}^{\left(1;1,0\right)}RK,$

which contains the spatial derivative terms of the acceleration $\nabla a$.

We thus conclude that (92) and (95) are two viable Lagrangians that do not
propagate any scalar mode at the linear order in a cosmological background.

###  IV.3 $d=2$ with $d=3$

In the above we have determined the viable Lagrangians when only $d=2$ or
$d=3$ terms are present. It should not be surprising that although the scalar
mode is eliminated for $d=2$ and $d=3$ individually, the scalar mode will
reappear if we naively combine them. This also happens in the investigation of
degenerate higher order scalar-tensor theories [68, 66, 89]. Fortunately in
our case, viable Lagrangians with the combination of $d=2$ and $d=3$ terms do
exist, after imposing additional conditions on the coefficients.

We consider the combined Lagrangian

$S=\int\mathrm{d}t\mathrm{d}^{3}x\,N\sqrt{h}\left(\mathcal{L}^{\left(2\right)}+\mathcal{L}^{\left(3\right)}-\Lambda\right),$
(96)

in which $\mathcal{L}^{\left(2\right)}$ and $\mathcal{L}^{\left(3\right)}$ are
given in (10) and (11), respectively.

The analysis is completely parallel to the above. Expanding the action to the
first order in perturbations yields

$S_{1}=\int\mathrm{d}t\mathrm{d}^{3}x\,\mathcal{L}_{1},$ (97)

where it follows from (37) and (67) that

$\displaystyle\mathcal{L}_{1}$ $\displaystyle=$
$\displaystyle\bar{N}a^{3}\left[-3H^{2}\left(b_{2}-b_{2}^{\prime}\right)+3H^{3}\left(b_{3}^{\prime}-2b_{3}\right)-\Lambda\right]A$
(98)
$\displaystyle+3\bar{N}a^{3}\left[-\left(3H^{2}+2\dot{H}\right)b_{2}-2H\dot{b}_{2}-6H\left(H^{2}+\dot{H}\right)b_{3}-\Lambda-3H^{2}\dot{b}_{3}\right]\zeta.$

The coefficients in the quadratic Lagrangian for the scalar modes read

$\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}\dot{\zeta}}^{(2)+(3)}$
$\displaystyle=$
$\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}\dot{\zeta}}^{(2)}+\hat{\mathcal{C}}_{\dot{\zeta}\dot{\zeta}}^{(3)},$
$\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}A}^{(2)+(3)}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}A}^{(2)}+\hat{\mathcal{C}}_{\dot{\zeta}A}^{(3)},$
(99) $\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}B}^{(2)+(3)}$
$\displaystyle=$
$\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}B}^{(2)}+\hat{\mathcal{C}}_{\dot{\zeta}B}^{(3)},$
(100) $\displaystyle\hat{\mathcal{C}}_{AA}^{(2)+(3)}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{C}}_{AA}^{(2)}+\hat{\mathcal{C}}_{AA}^{(3)},$
(101) $\displaystyle\hat{\mathcal{C}}_{AB}^{(2)+(3)}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{C}}_{AB}^{(2)}+\hat{\mathcal{C}}_{AB}^{(3)},$
(102) $\displaystyle\hat{\mathcal{C}}_{BB}^{(2)+(3)}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{C}}_{BB}^{(2)}+\hat{\mathcal{C}}_{BB}^{(3)}.$
(103)

The degeneracy condition thus becomes

$\displaystyle\Delta^{(2)+(3)}$ $\displaystyle=$
$\displaystyle\Big{\\{}12H^{2}\left[-2(b_{2}^{\prime})^{2}+b_{2}(2b_{2}^{\prime}+b_{2}^{\prime\prime})\right]w_{2}+54H^{5}\left[-3(b_{3}^{\prime})^{2}+2b_{3}(2b_{3}^{\prime}+b_{3}^{\prime\prime})\right]w_{3}$
(104)
$\displaystyle\quad+H^{3}\left[12\left(3b_{3}\left(b_{2}^{\prime\prime}+2b_{2}^{\prime}\right)+b_{2}\left(2b_{3}^{\prime}+b_{3}^{\prime\prime}\right)-6b_{2}^{\prime}b_{3}^{\prime}\right)w_{2}+36\left(-2(b_{2}^{\prime})^{2}+b_{2}(2b_{2}^{\prime}+b_{2}^{\prime\prime})\right)w_{3}\right]$
$\displaystyle+H^{4}\left[18\left(-3(b_{3}^{\prime})^{2}+2b_{3}(2b_{3}^{\prime}+b_{3}^{\prime\prime})\right)w_{2}+36\left(3b_{3}\left(b_{2}^{\prime\prime}+2b_{2}^{\prime}\right)+b_{2}\left(2b_{3}^{\prime}+b_{3}^{\prime\prime}\right)-6b_{2}^{\prime}b_{3}^{\prime}\right)w_{3}\right]\Big{\\}}\frac{\partial^{4}}{a^{2}}$
$\displaystyle+\Big{\\{}-8b_{2}\tilde{b}_{2}w_{2}+36H^{3}\left[-b_{3}^{\prime}f_{3}+2b_{3}(f_{3}+f_{3}^{\prime}-\tilde{b}_{3})\right]w_{3}$
$\displaystyle\quad+H\left[8\left(-b_{2}^{\prime}f_{3}-3b_{3}\tilde{b}_{2}+b_{2}(f_{3}+f_{3}^{\prime}-\tilde{b}_{3})\right)w_{2}-24b_{2}\tilde{b}_{2}w_{3}\right]$
$\displaystyle\quad+H^{2}\left[12\left(-b_{3}^{\prime}f_{3}+2b_{3}(f_{3}+f_{3}^{\prime}-\tilde{b}_{3})\right)w_{2}+24\left(-b_{2}^{\prime}f_{3}-3b_{3}\tilde{b}_{2}+b_{2}(f_{3}+f_{3}^{\prime}-\tilde{b}_{3})\right)w_{3}\right]\Big{\\}}\frac{\partial^{6}}{a^{4}}$
$\displaystyle+\left\\{-\frac{1}{3}\left(b_{2}(f_{3}-3\tilde{f}_{3})^{2}+2(f_{3})^{2}w_{2}\right)-H\left[b_{3}(f_{3}-3\tilde{f}_{3})^{2}+2(f_{3})^{2}w_{3}\right]\right\\}\frac{\partial^{8}}{a^{6}}.$

Note we have used the background equations of motion to simply the expressions
for the coefficients. We require that the degeneracy condition should be
satisfied for any power of $\partial$ and $H$, thus we get the set of
constraints

$\displaystyle-2(b_{2}^{\prime})^{2}+b_{2}\left(2b_{2}^{\prime}+b_{2}^{\prime\prime}\right)$
$\displaystyle=$ $\displaystyle 0,$ (105)
$\displaystyle-3(b_{3}^{\prime})^{2}+2b_{3}\left(2b_{3}^{\prime}+b_{3}^{\prime\prime}\right)$
$\displaystyle=$ $\displaystyle 0,$ (106) $\displaystyle
3b_{3}\left(b_{2}^{\prime\prime}+2b_{2}^{\prime}\right)+b_{2}\left(2b_{3}^{\prime}+b_{3}^{\prime\prime}\right)-6b_{2}^{\prime}b_{3}^{\prime}$
$\displaystyle=$ $\displaystyle 0,$ (107) $\displaystyle\tilde{b}_{2}$
$\displaystyle=$ $\displaystyle 0,$ (108) $\displaystyle-
b_{3}^{\prime}f_{3}+2b_{3}(f_{3}+f_{3}^{\prime}-\tilde{b}_{3})$
$\displaystyle=$ $\displaystyle 0,$ (109) $\displaystyle-
b_{2}^{\prime}f_{3}-3b_{3}\tilde{b}_{2}+b_{2}(f_{3}+f_{3}^{\prime}-\tilde{b}_{3})$
$\displaystyle=$ $\displaystyle 0,$ (110) $\displaystyle
b_{2}(f_{3}-3\tilde{f}_{3})^{2}+2(f_{3})^{2}w_{2}$ $\displaystyle=$
$\displaystyle 0,$ (111) $\displaystyle
b_{3}(f_{3}-3\tilde{f}_{3})^{2}+2(f_{3})^{2}w_{3}$ $\displaystyle=$
$\displaystyle 0.$ (112)

Compared with the constraints in $d=2$ and $d=3$ respectively, we find that
additional constraints should be imposed in the combined case to ensure the
degeneracy.

First of all, (105) and (106) are exactly (57) and (83) in the case of $d=2$
and $d=3$ individually. The general solutions for $b_{2}$ and $b_{3}$ are
given in (60) and (86), respectively, from which we may solve

$c_{3}^{(0,2,0)}=-\frac{1}{3}c_{1}^{(0,2,0)}+\frac{1}{3}\frac{C_{1}N}{1+C_{2}N},$
(113)

and

$c_{5}^{(0;3,0)}=-\frac{1}{9}c_{1}^{(0;3,0)}-\frac{1}{3}c_{3}^{(0;3,0)}+\frac{1}{9}\frac{D_{1}N^{2}}{\left(1+D_{2}N\right)^{2}}.$
(114)

If both (105) and (106) are satisfied, (107) reduces to be

$\frac{3}{2b_{2}b_{3}}\left(2b_{3}b_{2}^{\prime}-b_{2}b_{3}^{\prime}\right)^{2}=0,$
(115)

which yields a constraint between $b_{2}$ and $b_{3}$. By plugging the
solutions (60) and (86), we may solve

$C_{2}=D_{2},$ (116)

which implies that $b_{3}$ is determined by $b_{2}$ through

$b_{3}=\frac{D_{1}}{C_{1}^{2}}b_{2}^{2}.$ (117)

As a result, (114) reduces to be

$c_{5}^{(0;3,0)}=-\frac{1}{9}c_{1}^{(0;3,0)}-\frac{1}{3}c_{3}^{(0;3,0)}+\frac{1}{9}\frac{D_{1}N^{2}}{\left(1+C_{2}N\right)^{2}}.$
(118)

(108) implies

$c_{2}^{(0,2,0)}=0.$ (119)

With (117), the left-hand-side of (109) becomes

$\mathrm{L.H.S.}=2\frac{D_{1}}{C_{1}^{2}}b_{2}\left[-b_{2}^{\prime}f_{3}+b_{2}\left(f_{3}+f_{3}^{\prime}-\tilde{b}_{3}\right)\right],$
(120)

while due to (108), the left-hand-side of (110) becomes

$\mathrm{L.H.S.}\equiv-
b_{2}^{\prime}f_{3}+b_{2}\left(f_{3}+f_{3}^{\prime}-\tilde{b}_{3}\right),$
(121)

which is proportional to (109). Generally, we look for solutions with
$b_{2}\neq 0$ (since GR belongs to the case), thus (109) and (110) are
satisfied only if

$-b_{2}^{\prime}f_{3}+b_{2}\left(f_{3}+f_{3}^{\prime}-\tilde{b}_{3}\right)=0.$
(122)

We thus solve $\tilde{b}_{3}$ to be

$\tilde{b}_{3}=\frac{C_{2}N}{1+C_{2}N}f_{3}+f_{3}^{\prime},$

which also implies

$c_{4}^{(0;3,0)}=-\frac{1}{3}c_{2}^{(0;3,0)}+\frac{1}{3}\left(\frac{C_{2}N}{1+C_{2}N}f_{3}+f_{3}^{\prime}\right).$
(123)

In order to make (111) and (112) have solutions, $f_{3}$ and
$f_{3}-3\tilde{f}_{3}$ must be either non-vanishing or vanishing
simultaneously. Therefore we have two cases.

####  IV.3.1 Case 1

If $f_{3}=0$ and $f_{3}-3\tilde{f}_{3}=0$ (and thus $\tilde{f}_{3}=0$), both
(111) and (112) are automatically satisfied. There is no restriction on
$b_{2},b_{3},w_{2},w_{3}$. In this case, since $f_{3}=\tilde{f}_{3}=0$ we have

$c_{1}^{(0;1,1)}=0,\qquad c_{2}^{(0;1,1)}=0.$ (124)

As a result, (123) reduces to be

$c_{4}^{(0;3,0)}=-\frac{1}{3}c_{2}^{(0;3,0)}.$ (125)

In this case, the Lagrangian is given by

$\displaystyle\mathcal{L}^{(2)+(3),\mathrm{I}}$ $\displaystyle=$
$\displaystyle
c_{1}^{\left(0;2,0\right)}\hat{K}_{ij}\hat{K}^{ij}+\frac{1}{3}\frac{C_{1}N}{1+C_{2}N}K^{2}+c_{1}^{\left(1;0,0\right)}R$
(126)
$\displaystyle+c_{1}^{\left(0;3,0\right)}\hat{K}_{ij}\hat{K}^{jk}\hat{K}_{k}^{i}+w_{3}\,\hat{K}_{ij}\hat{K}^{ij}K+\frac{1}{9}\frac{D_{1}N^{2}}{\left(1+C_{2}N\right)^{2}}K^{3}$
$\displaystyle+c_{2}^{\left(0;3,0\right)}\hat{K}_{ij}a^{i}a^{j}+c_{1}^{\left(1;1,0\right)}R^{ij}K_{ij}+c_{2}^{\left(1;1,0\right)}RK,$

which contains no spatial derivative of the acceleration $\nabla a$.

####  IV.3.2 Case 2

If $f_{3}\neq 0$ and $f_{3}-3\tilde{f}_{3}\neq 0$, from (111) we solve

$w_{2}=-\frac{1}{2}b_{2}\left(1-3\frac{\tilde{f}_{3}}{f_{3}}\right)^{2},$
(127)

i.e.,

$c_{1}^{(0,2,0)}=-2\frac{C_{1}N}{1+C_{2}N}\left(\frac{c_{1}^{(0;1,1)}}{f_{3}}\right)^{2},$
(128)

and from (112) we solve

$w_{3}=-\frac{1}{2}b_{3}\left(1-3\frac{\tilde{f}_{3}}{f_{3}}\right)^{2},$
(129)

i.e.,

$c_{3}^{(0;3,0)}=-c_{1}^{(0;3,0)}-2\frac{D_{1}N^{2}}{\left(1+C_{2}N\right)^{2}}\left(\frac{c_{1}^{(0;1,1)}}{f_{3}}\right)^{2}.$
(130)

As a result, (118) reduces to be

$c_{5}^{(0;3,0)}=\frac{2}{9}c_{1}^{(0;3,0)}+\frac{1}{9}\frac{D_{1}N^{2}}{\left(1+C_{2}N\right)^{2}}\left[1+6\left(\frac{c_{1}^{(0;1,1)}}{f_{3}}\right)^{2}\right].$
(131)

In this case, the Lagrangian is given by

$\displaystyle\mathcal{L}^{(2)+(3),\mathrm{II}}$ $\displaystyle=$
$\displaystyle-2\frac{C_{1}N}{1+C_{2}N}\left(\frac{c_{1}^{(0;1,1)}}{f_{3}}\right)^{2}\hat{K}_{ij}\hat{K}^{ij}+\frac{1}{3}\frac{C_{1}N}{1+C_{2}N}K^{2}+c_{1}^{\left(1;0,0\right)}R$
(132)
$\displaystyle+c_{1}^{\left(0;3,0\right)}\hat{K}_{ij}\hat{K}^{jk}\hat{K}_{k}^{i}-2\frac{D_{1}N^{2}}{\left(1+C_{2}N\right)^{2}}\left(\frac{c_{1}^{(0;1,1)}}{f_{3}}\right)^{2}\hat{K}_{ij}\hat{K}^{ij}K+\frac{1}{9}\frac{D_{1}N^{2}}{\left(1+C_{2}N\right)^{2}}K^{3}$
$\displaystyle+c_{2}^{\left(0;3,0\right)}\hat{K}_{ij}a^{i}a^{j}+\frac{1}{3}\left(\frac{C_{2}N}{1+C_{2}N}f_{3}+f_{3}^{\prime}\right)Ka_{i}a^{i}$
$\displaystyle+c_{1}^{\left(0;1,1\right)}\hat{K}_{ij}\nabla^{i}a^{j}+\frac{1}{3}f_{3}K\nabla_{i}a^{i}$
$\displaystyle+c_{1}^{\left(1;1,0\right)}R^{ij}K_{ij}+c_{2}^{\left(1;1,0\right)}RK,$

which contains spatial derivatives of the acceleration $\nabla a$.

We conclude that (126) and (132) are two viable Lagrangians in which both
$d=2$ and $d=3$ terms are present, and propagate no scalar mode at the linear
order in perturbations around a cosmological background.

###  IV.4 $d=4$

Now we consider the most involved case of $d=4$. The action is

$S=\int\mathrm{d}t\mathrm{d}^{3}x\,N\sqrt{h}\left(\mathcal{L}^{\left(4\right)}+\Lambda\right),$
(133)

with $\mathcal{L}^{\left(4\right)}$ given in (12). Expanding the action to the
first order in perturbations yields

$S_{1}=\int\mathrm{d}t\mathrm{d}^{3}x\,\mathcal{L}_{1},$ (134)

with

$\displaystyle\mathcal{L}_{1}$ $\displaystyle=$
$\displaystyle\bar{N}a^{3}\left[-9H^{4}\left(-b_{4}^{\prime}+3b_{4}\right)+\Lambda\right]A$
(135)
$\displaystyle+3\bar{N}a^{3}\left[-9H^{2}\left(3H^{2}+4\dot{H}\right)b_{4}+\Lambda-12H^{3}\dot{b}_{4}\right]\zeta,$

where we denote

$b_{4}\coloneqq
c_{2}^{(0,4,0)}+c_{4}^{(0,4,0)}+3c_{7}^{(0,4,0)}+9c_{9}^{(0,4,0)},$ (136)

for short. The background equations of motion are

$\displaystyle-9H^{4}\left(-b_{4}^{\prime}+3b_{4}\right)+\Lambda$
$\displaystyle=$ $\displaystyle 0,$ (137)
$\displaystyle-9H^{2}\left(3H^{2}+4\dot{H}\right)b_{4}+\Lambda-12H^{3}\dot{b}_{4}$
$\displaystyle=$ $\displaystyle 0,$ (138)

for $A$ and $\zeta$, respectively.

The relevant coefficients in the quadratic Lagrangian for the scalar modes are

$\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}\dot{\zeta}}^{(4)}$
$\displaystyle=$ $\displaystyle 54H^{2}b_{4}-d_{4}\frac{\partial^{2}}{a^{2}},$
(139) $\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}A}^{(4)}$ $\displaystyle=$
$\displaystyle-36H^{3}\left(-b_{4}^{\prime}+3b_{4}\right)+H\left(2d_{4}+2f_{4}-\tilde{f}_{4}\right)\frac{\partial^{2}}{a^{2}},$
(140) $\displaystyle\hat{\mathcal{C}}_{\dot{\zeta}B}^{(4)}$ $\displaystyle=$
$\displaystyle-36H^{2}b_{4}\frac{\partial^{2}}{a}+2\tilde{d}_{4}\frac{\partial^{4}}{a^{3}},$
(141) $\displaystyle\hat{\mathcal{C}}_{AA}^{(4)}$ $\displaystyle=$
$\displaystyle\frac{9}{2}H^{4}\left(b_{4}^{\prime\prime}-6b_{4}^{\prime}+12b_{4}\right)$
(142)
$\displaystyle+H^{2}\left(f_{4}^{\prime}-d_{4}-\tilde{b}_{4}-f_{4}+\tilde{f}_{4}\right)\frac{\partial^{2}}{a^{2}}+\bar{d}_{4}\frac{\partial^{4}}{a^{4}},$
$\displaystyle\hat{\mathcal{C}}_{AB}^{(4)}$ $\displaystyle=$ $\displaystyle
12H^{3}\left(-b_{4}^{\prime}+3b_{4}\right)\frac{\partial^{2}}{a}+H\left(-2\tilde{d}_{4}-\hat{f}_{4}\right)\frac{\partial^{4}}{a^{3}},$
(143) $\displaystyle\hat{\mathcal{C}}_{BB}^{(4)}$ $\displaystyle=$
$\displaystyle
2H^{2}\left(w_{4}+3b_{4}\right)\frac{\partial^{4}}{a^{2}}-\hat{d}_{4}\frac{\partial^{6}}{a^{4}},$
(144)

where we denote

$\displaystyle d_{4}$ $\displaystyle\coloneqq$ $\displaystyle
3c_{1}^{(0,0,2)}+c_{2}^{(0,0,2)}+3c_{3}^{(0,0,2)}+9c_{4}^{(0,0,2)},$ (145)
$\displaystyle f_{4}$ $\displaystyle\coloneqq$ $\displaystyle
c_{1}^{(0,2,1)}+3c_{4}^{(0,2,1)}+3c_{5}^{(0,2,1)}+9c_{7}^{(0,2,1)},$ (146)
$\displaystyle\tilde{f}_{4}$ $\displaystyle\coloneqq$ $\displaystyle
c_{2}^{(0,2,1)}+3c_{3}^{(0,2,1)}.$ (147) $\displaystyle\tilde{d}_{4}$
$\displaystyle\coloneqq$ $\displaystyle
c_{1}^{(0,0,2)}+c_{2}^{(0,0,2)}+2c_{3}^{(0,0,2)}+3c_{4}^{(0,0,2)}.$ (148)
$\displaystyle\tilde{b}_{4}$ $\displaystyle\coloneqq$ $\displaystyle
c_{1}^{(0,4,0)}+3c_{3}^{(0,4,0)}+3c_{5}^{(0,4,0)}+9c_{8}^{(0,4,0)},$ (149)
$\displaystyle\bar{d}_{4}$ $\displaystyle\coloneqq$ $\displaystyle
c_{5}^{(0,0,2)}+c_{6}^{(0,0,2)}.$ (150) $\displaystyle\hat{f}_{4}$
$\displaystyle\coloneqq$ $\displaystyle
2c_{1}^{(0,2,1)}-c_{2}^{(0,2,1)}-c_{3}^{(0,2,1)}+4c_{4}^{(0,2,1)}+2c_{5}^{(0,2,1)}+6c_{7}^{(0,2,1)}.$
(151) $\displaystyle w_{4}$ $\displaystyle\coloneqq$ $\displaystyle
3c_{2}^{(0,4,0)}+2c_{4}^{(0,4,0)}+3c_{7}^{(0,4,0)},$ (152)
$\displaystyle\hat{d}_{4}$ $\displaystyle\coloneqq$ $\displaystyle
c_{1}^{(0,0,2)}+c_{2}^{(0,0,2)}+c_{3}^{(0,0,2)}+c_{4}^{(0,0,2)},$ (153)

for shorthand.

After some manipulation, the degeneracy condition (34) is found to be

$\Delta^{(4)}=H^{8}\Delta_{4}^{(4)}\frac{\partial^{4}}{a^{2}}+H^{6}\Delta_{6}^{(4)}\frac{\partial^{6}}{a^{4}}+H^{4}\Delta_{8}^{(4)}\frac{\partial^{8}}{a^{6}}+H^{2}\Delta_{10}^{(4)}\frac{\partial^{10}}{a^{8}}+\Delta_{12}^{(4)}\frac{\partial^{12}}{a^{10}},$
(154)

where

$\Delta_{4}^{(4)}=648\left[-4\left(b_{4}^{\prime}\right)^{2}+3b_{4}\left(2b_{4}^{\prime}+b_{4}^{\prime\prime}\right)\right]w_{4},$
(155) $\displaystyle\Delta_{6}^{(4)}$ $\displaystyle=$
$\displaystyle+36\left[-\left(2b_{4}^{\prime}+b_{4}^{\prime\prime}\right)d_{4}+12b_{4}\left(f_{4}+f_{4}^{\prime}-\tilde{b}_{4}\right)+4b_{4}^{\prime}\left(-2f_{4}+\tilde{f}_{4}\right)\right]w_{4}$
(156)
$\displaystyle+36\left[4(b_{4}^{\prime})^{2}-3b_{4}(2b_{4}^{\prime}+b_{4}^{\prime\prime})\right]\left(d_{4}+9\hat{d}_{4}-6\tilde{d}_{4}\right),$
$\displaystyle\Delta_{8}^{(4)}$ $\displaystyle=$
$\displaystyle-18\left(2b_{4}^{\prime}+b_{4}^{\prime\prime}\right)\left(\tilde{d}_{4}\right)^{2}+24d_{4}\left[-b_{4}\left(f_{4}+f_{4}^{\prime}-\tilde{b}_{4}\right)+b_{4}^{\prime}\hat{f}_{4}\right]$
(157)
$\displaystyle-6b_{4}\left(-2f_{4}+3\hat{f}_{4}+\tilde{f}_{4}\right)^{2}+24\tilde{d}_{4}\left[6b_{4}\left(f_{4}+f_{4}^{\prime}-\tilde{b}_{4}\right)+b_{4}^{\prime}\left(-2f_{4}-3\hat{f}_{4}+\tilde{f}_{4}\right)\right]$
$\displaystyle+18\hat{d}_{4}\left[\left(2b_{4}^{\prime}+b_{4}^{\prime\prime}\right)d_{4}-12b_{4}\left(f_{4}+f_{4}^{\prime}-\tilde{b}_{4}\right)-4b_{4}^{\prime}\left(-2f_{4}+\tilde{f}_{4}\right)\right]$
$\displaystyle+2\left[-4d_{4}\left(f_{4}+f_{4}^{\prime}-\tilde{b}_{4}\right)+216b_{4}\bar{d}_{4}-\left(-2f_{4}+\tilde{f}_{4}\right)^{2}\right]w_{4},$
$\displaystyle\Delta_{10}^{(4)}$ $\displaystyle=$
$\displaystyle-4\left(f_{4}+f_{4}^{\prime}-\tilde{b}_{4}\right)\left(\tilde{d}_{4}\right)^{2}+d_{4}\left[-24b_{4}\bar{d}_{4}+\left(\hat{f}_{4}\right)^{2}\right]+2\tilde{d}_{4}\left[72b_{4}\bar{d}_{4}+\hat{f}_{4}\left(-2f_{4}+\tilde{f}_{4}\right)\right]$
(158)
$\displaystyle+\hat{d}_{4}\left[4d_{4}\left(f_{4}+f_{4}^{\prime}-\tilde{b}_{4}\right)-216b_{4}\bar{d}_{4}+\left(-2f_{4}+\tilde{f}_{4}\right)^{2}\right]-8d_{4}\bar{d}_{4}w_{4},$

and

$\Delta_{12}^{(4)}=4\bar{d}_{4}\left[d_{4}\hat{d}_{4}-(\tilde{d}_{4})^{2}\right].$
(159)

It is interesting that $\tilde{b}_{4}$ always arises in terms of
$\left(f_{4}+f_{4}^{\prime}-\tilde{b}_{4}\right)$, and $\tilde{f}_{4}$ always
arises in terms of $\left(-2f_{4}+\tilde{f}_{4}\right)$.

Following the analysis in the above, we may solve the coefficients such that
$\Delta^{(4)}=0$. For example, from (155) and (156) we have two constraints

$-4\left(b_{4}^{\prime}\right)^{2}+3b_{4}\left(2b_{4}^{\prime}+b_{4}^{\prime\prime}\right)=0,$
(160)

and

$-\left(2b_{4}^{\prime}+b_{4}^{\prime\prime}\right)d_{4}+12b_{4}\left(f_{4}+f_{4}^{\prime}-\tilde{b}_{4}\right)+4b_{4}^{\prime}\left(-2f_{4}+\tilde{f}_{4}\right)=0,$
(161)

combining which yields

$-\left(b_{4}^{\prime}\right)^{2}d_{4}+9\left(b_{4}\right)^{2}\left(f_{4}+f_{4}^{\prime}-\tilde{b}_{4}\right)+3b_{4}b_{4}^{\prime}\left(-2f_{4}+\tilde{f}_{4}\right)=0.$
(162)

The full treatment of the case of $d=4$, however, is involved and out of the
scope of the present work.

##  V Eliminate the scalar mode at the second order: $d=2$

In the previous section, we have eliminate the scalar mode at linear order in
perturbations by making use of the degeneracy condition (34). Clearly the
conditions for the coefficients derived in the above are merely necessary
conditions, which means that the scalar mode will reappear if we go to higher
orders. Thus one need to find the conditions for the coefficients such that
the scalar mode is eliminated order by order. In this section we take the case
of $d=2$ as an illustrative example.

Expanding the action (35) to the cubic order in perturbations yields

$\displaystyle\mathcal{L}_{3}$ $\displaystyle=$ $\displaystyle
9\bar{N}a^{3}b_{2}\zeta\dot{\zeta}^{2}-\bar{N}a^{2}Hb_{2}\zeta^{2}\partial^{2}B-2\bar{N}ah_{2}\zeta^{2}\partial^{2}\zeta+a\bar{N}\tilde{b}_{2}\zeta\partial_{i}A\partial^{i}A-2a^{2}H\bar{N}b_{2}\zeta\partial_{i}\zeta\partial^{i}B$
(163)
$\displaystyle-2a\bar{N}h_{2}\zeta\partial_{i}\zeta\partial^{i}\zeta+\frac{1}{3}a\bar{N}\left(-b_{2}+w_{2}\right)\zeta\left(\partial^{2}B\right)^{2}+\frac{2}{3}a\bar{N}\left(b_{2}+2w_{2}\right)\partial_{i}\zeta\partial^{i}B\partial^{2}B$
$\displaystyle-2a\bar{N}w_{2}\partial_{i}\partial_{j}B\partial^{i}B\partial^{j}\zeta-2a\bar{N}w_{2}\partial^{i}B\partial_{j}\partial_{i}B\partial^{j}\zeta-a\bar{N}w_{2}\zeta\partial_{j}\partial_{i}B\partial^{j}\partial^{i}B$
$\displaystyle+27\bar{N}a^{3}Hb_{2}\zeta^{2}\dot{\zeta}-2a^{2}b_{2}\zeta\partial^{2}B\partial_{t}\zeta-2\bar{N}a^{2}b_{2}\partial_{i}\zeta\partial^{i}B\dot{\zeta}+2a^{2}H\bar{N}\left(b_{2}-b_{2}^{\prime}\right)A\zeta\partial^{2}B$
$\displaystyle+2a^{2}H\bar{N}\left(b_{2}-b_{2}^{\prime}\right)A\partial_{i}\zeta\partial^{i}B-18a^{3}HA\zeta\partial_{t}\zeta\left(b_{2}-b_{2}^{\prime}\right)+2\bar{N}a^{2}\left(b_{2}-b_{2}^{\prime}\right)A\partial^{2}B\dot{\zeta}$
$\displaystyle-\frac{27}{2}\bar{N}a^{3}H^{2}\left(-2b_{2}+b_{2}^{\prime}\right)\zeta^{3}+3\bar{N}a^{3}\left(-b_{2}+b_{2}^{\prime}\right)A\dot{\zeta}^{2}+a\bar{N}\left(\tilde{b}_{2}+\tilde{b}_{2}^{\prime}\right)A\partial_{i}A\partial^{i}A$
$\displaystyle-4a\bar{N}\left(h_{2}+h_{2}^{\prime}\right)A\zeta\partial^{2}\zeta-2a\bar{N}\left(h_{2}+h_{2}^{\prime}\right)A\partial_{i}\zeta\partial^{i}\zeta+\frac{1}{3}a\bar{N}\left(-b_{2}+w_{2}+b_{2}^{\prime}-w_{2}^{\prime}\right)A\left(\partial^{2}B\right)^{2}$
$\displaystyle+a\bar{N}\left(-w_{2}+w_{2}^{\prime}\right)A\partial_{j}\partial_{i}B\partial^{j}\partial^{i}B+\frac{9}{2}\bar{N}a^{3}H^{2}\left(2b_{2}-2b_{2}^{\prime}+b_{2}^{\prime\prime}\right)A^{2}\zeta$
$\displaystyle-\bar{N}a^{2}H\left(b_{2}-b_{2}^{\prime}+b_{2}^{\prime\prime}\right)A^{2}\partial^{2}B+3\bar{N}a^{3}H\left(b_{2}-b_{2}^{\prime}+b_{2}^{\prime\prime}\right)A^{2}\dot{\zeta}$
$\displaystyle-2\bar{N}a\left(h_{2}+3h_{2}^{\prime}+h_{2}^{\prime\prime}\right)A^{2}\partial^{2}\zeta+\frac{1}{2}\bar{N}a^{3}H^{2}b_{2}^{(3)}A^{3},$

where we have used the background equations of motion to simply the
coefficients. No integration by parts has been performed at this point.

By making use of the degeneracy conditions (56) and (57), and plugging the
solutions for $A$ and $B$ in (58) and (59), after some manipulations, we get
the induced cubic action $S_{3}[\zeta]$ for the single variable $\zeta$. We
tend not to present the full and explicit expression of $S_{3}[\zeta]$ due to
its length. We pay special attention to the terms which are relevant to
eliminating the scalar mode (i.e., $\zeta$), and have the following
observations.

  * •

First, we found that there is no $\dot{\zeta}^{3}$, i.e., there is no terms
with 3 time derivatives.

  * •

Second, there is one “dangerous” term with two time derivatives:

$-2\bar{N}a\frac{b_{2}^{2}\left(2h_{2}^{\prime}+h_{2}^{\prime\prime}\right)}{H^{2}\left(b_{2}-b_{2}^{\prime}\right)^{2}}\dot{\zeta}^{2}\partial^{2}\zeta,$
(164)

thus we need to require (since $b_{2}\neq 0$)

$2h_{2}^{\prime}+h_{2}^{\prime\prime}=0,$ (165)

with $b_{2}-b_{2}^{\prime}\neq 0$. The general solution for $h_{2}$ is

$h_{2}\left(N\right)=C_{3}+\frac{C_{4}}{N},$ (166)

where $C_{3},C_{4}$ are constants.

  * •

Third, there is one “dangerous” term with one time derivative:

$\frac{2b_{2}\left(h_{2}+h_{2}^{\prime}\right)^{2}\left[-2b_{2}w_{2}^{2}+2w_{2}^{2}b_{2}^{\prime}+b_{2}^{2}\left(-w_{2}+w_{2}^{\prime}\right)\right]}{3aH^{3}w_{2}^{2}\bigl{(}b_{2}-b_{2}^{\prime}\bigr{)}^{3}}\bar{N}\dot{\zeta}\left(\partial^{2}\zeta\right)^{2},$
(167)

which implies that $w_{2}$ must be related to $b_{2}$ by

$\frac{1}{w_{2}^{2}}\left(w_{2}-w_{2}^{\prime}\right)=-2\frac{1}{b_{2}^{2}}\left(b_{2}-b_{2}^{\prime}\right).$
(168)

With the solution for $b_{2}$ in (60), the general solution for $w_{2}$ is

$w_{2}=\frac{C_{5}C_{1}N}{1-2C_{5}C_{2}N},$ (169)

where $C_{5}$ is another constant.

By combining (56), (60), (166) and (169), the Lagrangian (61) is further
reduced to be

$\displaystyle\mathcal{L}^{\left(2\right)}$ $\displaystyle=$
$\displaystyle\frac{C_{5}C_{1}N}{1-2C_{5}C_{2}N}\hat{K}_{ij}\hat{K}^{ij}+\frac{1}{3}\frac{C_{1}N}{1+C_{2}N}K^{2}+\left(C_{3}+\frac{C_{4}}{N}\right)R.$
(170)

We conclude that the Lagrangian (170) propagates no scalar degree of freedom
up to the second order in perturbations on a cosmological background.

##  VI Conclusion

In this work, we revisited the problem of propagating at most two tensorial
degrees of freedom in a large class of spatially covariant gravity theories,
of which the Lagrangian are polynomials built of spatial geometric quantities.
Although the general conditions have been derived in [74, 75], these
conditions are mathematically involved to be solved to yield concrete
Lagrangians.

We thus take an alternative and complimentary approach in this work based on a
perturbative analysis. The idea is simple: if the Lagrangian has no scalar
degree of freedom in a fully nonlinear sense, the scalar mode must not show up
at any finite order if we perturbatively expand the Lagrangian around a
cosmological background. This perturbative analysis allows us to determine the
coefficients in the Lagrangian order by order. Since at the fully nonlinear
level, there are finite number of conditions imposed on the functional form of
the Lagrangian, this perturbative analysis must stop at some _finite_ order.
In other words, there must be a finite order up to which we “kill” the scalar
mode, and then the scalar mode is eliminated at fully nonlinear order. In
fact, as being shown in [79] in a specific example, it is sufficient to tune
the coefficients up to the cubic order in the Lagrangian such that the
unwanted scalar mode is fully removed.

In this work, we mainly focus on the linear cosmological perturbations. In
Sec. III we shown that in order to eliminate the unwanted scalar mode at the
linear order, the degeneracy condition (34) must be imposed. This is also
supported by a more rigorous Lagrangian constraint analysis in Appendix A. We
then use (34) as the starting point to determine the coefficients of the
Lagrangians for $d=2,3,4$ in Sec. IV, where $d$ is the total number of
derivatives in a SCG monomial. In particular, we have determined the concrete
form of the Lagrangians for $d=2$ in (61) and for $d=3$ in (92) in the absence
of $\nabla a$ terms and in (95) in the presence of $\nabla a$ terms,
respectively. We thus conclude that (61), (92) and (95) propagate no scalar
mode at the linear order in perturbations around the cosmological background.
The scalar mode will re-arise in the naive combination of the scalar-mode-free
Lagrangians for $d=2$ and $d=3$. Therefore one needs more restrictions on the
coefficients in order to eliminate the scalar mode, if $d=2$ and $d=3$
Lagrangians are present simultaneously. The final results are given in (126)
in the absence of $\nabla a$ terms and in (132) in the presence of $\nabla a$
terms, respectively.

It is not surprising that although the scalar mode has been eliminated at
linear order, it may re-appear at nonlinear orders. In Sec. V we expand the
Lagrangian up to the cubic order for $d=2$ and find the conditions for the
coefficients to eliminate the scalar mode up to the cubic order. The result is
given in (170). In principle this procedure can be performed order by order,
and one expects to determine the Lagrangian at some _finite_ order, such that
the scalar mode has been fully eliminated.

###### Acknowledgements.

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

##  Appendix A Classical mechanics with one dynamical and two auxiliary
variables

In this appendix, we make a thorough analysis of a classical mechanics system
with one dynamical and two auxiliary variables. We shall classify various
cases according to the number of DOF’s as well as the nature of constraints
and gauge identities.

The most general quadratic Lagrangian for three variables
$\left\\{q^{1},q^{2},q^{3}\right\\}$, of which one is dynamical and two are
auxiliary variables, takes the form

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{1}{2}g_{11}\left(\dot{q}^{1}\right)^{2}+f_{12}\dot{q}^{1}q^{2}+f_{13}\dot{q}^{1}q^{3}+\frac{1}{2}w_{22}\left(q^{2}\right)^{2}+\frac{1}{2}w_{33}\left(q^{3}\right)^{2}+w_{23}q^{2}q^{3}$
(171)
$\displaystyle+\frac{1}{2}w_{11}\left(q^{1}\right)^{2}+w_{12}q^{1}q^{2}+w_{13}q^{1}q^{3}.$

The coefficients $g_{11},f_{12}$ etc. are assumed to be constants for
simplicity. We assume $g_{11}\neq 0$ so that $q^{1}$ acquires an apparent
kinetic term. While $q^{2}$ and $q^{3}$ do not have explicit time derivatives
and act as the auxiliary variables. Our task is to search for cases in which
there is no dynamics in the Lagrangian (171).

Varying the Lagrangian yields

$\delta L\simeq-\mathcal{E}_{i}^{(0)}\delta q^{i},$ (172)

where the equations of motion take the form

$\mathcal{E}_{i}^{(0)}\coloneqq W_{ij}^{(0)}\ddot{q}^{j}+V_{j}^{(0)}\approx
0,$ (173)

with

$W_{ij}^{(0)}=\left(\begin{array}[]{ccc}g_{11}&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right),$ (174)

and

$\displaystyle V_{j}^{(0)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}f_{12}\dot{q}^{2}+f_{13}\dot{q}^{3}-w_{11}q^{1}-w_{12}q^{2}-w_{13}q^{3}\\\
-f_{12}\dot{q}^{1}-w_{12}q^{1}-w_{22}q^{2}-w_{23}q^{3}\\\
-f_{13}\dot{q}^{1}-w_{13}q^{1}-w_{23}q^{2}-w_{33}q^{3}\end{array}\right).$
(178)

In this appendix, for clarity we use “$\approx$” to denote on-shell
equalities, i.e., those hold only when the equations of motion are satisfied.
Here and in what follows the superscript “(n)” stands for “level-$n$”, of
which the meaning will be clear soon.

###  A.1 Level-0

Since $\mathrm{rank}\left(W_{ij}^{(0)}\right)=1$, there are 2 null-
eigenvectors for $W_{ij}^{(0)}$:

$u_{1,i}^{(0)}=\left(\begin{array}[]{c}0\\\ 1\\\ 0\end{array}\right),\qquad
u_{2,i}^{(0)}=\left(\begin{array}[]{c}0\\\ 0\\\ 1\end{array}\right),$ (179)

contracting which with $\mathcal{E}_{i}^{(0)}$ yields

$\displaystyle u_{1}^{(0)i}\mathcal{E}_{i}^{(0)}$ $\displaystyle=$
$\displaystyle-
f_{12}\dot{q}^{1}-w_{12}q^{1}-w_{22}q^{2}-w_{23}q^{3}\equiv\mathcal{E}_{2}^{(0)},$
(180) $\displaystyle u_{2}^{(0)i}\mathcal{E}_{i}^{(0)}$ $\displaystyle=$
$\displaystyle-
f_{13}\dot{q}^{1}-w_{13}q^{1}-w_{23}q^{2}-w_{33}q^{3}\equiv\mathcal{E}_{3}^{(0)}.$
(181)

According to the algorithm of detecting constraints in the Lagrangian
formalism, at each level, we have to examine whether the contractions lead to
constraints or identities. We have 3 cases according to how many
constraints/identities we get.

####  A.1.1 Case 1: two identities

If both contractions are vanishing identically, we get two gauge identities
$G_{1}^{(0)}\coloneqq\mathcal{E}_{2}^{(0)}\equiv 0$ and
$G_{2}^{(0)}\coloneqq\mathcal{E}_{3}^{(0)}\equiv 0$ at “level-0”. In this
appendix “$\equiv$” stands for off-shell identities, i.e., which always hold
no matter the equations of motion are satisfied or not. This requires
$f_{12}=f_{13}=w_{12}=w_{13}=w_{22}=w_{23}=w_{33}=0$. This case, however, is
trivial since terms invovling $q^{2},q^{3}$ in the Lagrangian (171) completely
drop out and the Lagrangian reduces to that of a single variable $q^{1}$. Then
the algorithm ends. We include this case merely for completeness.

####  A.1.2 Case 2: one constraint and one identity

Without loss of generality, we assume at least one of
$\left\\{f_{12},w_{12},w_{22},w_{23}\right\\}$ is not vanishing, and denote
the constraint at “level-0” as

$\phi^{(0)}\coloneqq u_{1}^{(0)i}\mathcal{E}_{i}^{(0)}\approx 0.$ (182)

Then that $u_{2}^{(0)i}\mathcal{E}_{i}^{(0)}$ leads to an identity implies
that

$u_{2}^{(0)i}\mathcal{E}_{i}^{(0)}=\lambda\,u_{1}^{(0)i}\mathcal{E}_{i}^{(0)},$
(183)

with some constant $\lambda$, i.e.,

$\displaystyle f_{13}$ $\displaystyle=$ $\displaystyle\lambda\,f_{12},$ (184)
$\displaystyle w_{13}$ $\displaystyle=$ $\displaystyle\lambda\,w_{12},$ (185)
$\displaystyle w_{23}$ $\displaystyle=$ $\displaystyle\lambda\,w_{22},$ (186)
$\displaystyle w_{33}$ $\displaystyle=$
$\displaystyle\lambda\,w_{23}\equiv\lambda^{2}w_{22}.$ (187)

We then get one gauge identity at “level-0”:

$G^{(0)}\coloneqq\lambda\,\mathcal{E}_{2}^{(0)}-\mathcal{E}_{3}^{(0)}\equiv
0.$ (188)

Note $\lambda=0$ is trivial, since in this case $q^{3}$-sector completely
drops out, and the original Lagrangian reduces to that of two variables
$q^{1}$ and $q^{2}$.

####  A.1.3 Case 3: two constraints

This is of course the general case. As long as at least one of (184)-(187) is
not satisfied, we get two constraints at “level-0”:

$\displaystyle\phi_{1}^{(0)}$ $\displaystyle\coloneqq$
$\displaystyle\mathcal{E}_{2}^{(0)}\approx 0,$ (189)
$\displaystyle\phi_{2}^{(0)}$ $\displaystyle\coloneqq$
$\displaystyle\mathcal{E}_{3}^{(0)}\approx 0.$ (190)

###  A.2 Case 2: level-1

Using (184)-(187) to replace $\left\\{f_{13},w_{13},w_{23},w_{33}\right\\}$ in
terms of $\left\\{f_{12},w_{12},w_{22},w_{23}\right\\}$, the constraint
$\phi^{(0)}$ in (182) becomes

$\phi^{(0)}\rightarrow-f_{12}\dot{q}^{1}-w_{12}q^{1}-w_{22}q^{2}-\lambda
w_{22}q^{3}.$ (191)

According the the standard algorithm, we build the enlarged “vector” of
equations of motion as

$\mathcal{E}_{i_{1}}^{(1)}\coloneqq\left(\begin{array}[]{c}\mathcal{E}_{i}^{(0)}\\\
\dot{\phi}^{(0)}\end{array}\right)=W_{i_{1}j}^{(1)}\ddot{q}^{j}+V_{i_{1}}^{(1)},$
(192)

with

$W_{i_{1}j}^{(1)}=\left(\begin{array}[]{ccc}g_{11}&0&0\\\ 0&0&0\\\ 0&0&0\\\
-f_{12}&0&0\end{array}\right),$ (193)

and

$V_{i_{1}}^{(1)}=\left(\begin{array}[]{c}f_{12}\dot{q}^{2}+\lambda
f_{12}\dot{q}^{3}-w_{11}q^{1}-w_{12}q^{2}-\lambda w_{12}q^{3}\\\
-f_{12}\dot{q}^{1}-w_{12}q^{1}-w_{22}q^{2}-\lambda w_{22}q^{3}\\\ -\lambda
f_{12}\dot{q}^{1}-\lambda w_{12}q^{1}-\lambda
w_{22}q^{2}-\lambda^{2}w_{22}q^{3}\\\
-w_{12}\dot{q}^{1}-w_{22}\dot{q}^{2}-\lambda
w_{22}\dot{q}^{3}\end{array}\right).$ (194)

There are two trivial null-eigenvectors for $W_{i_{1}j}^{(1)}$, which are
merely $u_{1,i}^{(0)},u_{2,i}^{(0)}$ in (179) augmented by zero’s. On the
other hand, there is a non-trivial null-eigenvector

$u_{i_{1}}^{(1)}=\left(\begin{array}[]{c}f_{12}\\\ 0\\\ 0\\\
g_{11}\end{array}\right),$ (195)

which is valid no matter $f_{12}=0$ or not. Contracting $u_{i_{1}}^{(1)}$ with
$\mathcal{E}_{i_{1}}^{(1)}$ yields

$\displaystyle u^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ $\displaystyle=$
$\displaystyle-
g_{11}w_{12}\dot{q}^{1}+\left(f_{12}^{2}-g_{11}w_{22}\right)\dot{q}^{2}+\lambda\left(f_{12}^{2}-g_{11}w_{22}\right)\dot{q}^{3}$
(196) $\displaystyle-f_{12}w_{11}q^{1}-f_{12}w_{12}q^{2}-\lambda
f_{12}w_{12}q^{3}.$

We need to check whether $u^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ leads to an
identity or a new constriant. There are two sub-cases.

####  A.2.1 Case 2.1: one identity

Since up to “level-1”, we have only one constraint $\phi^{(0)}$ given in
(191). If $u^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ is not an independent
constraint, it implies that

$u^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}\propto\phi_{1}^{(0)},$ (197)

which puts restrictions on the coefficients. After some manipulation, the
necessary and sufficient condition for (197) can be written as

$w_{12}=\sqrt{w_{11}}\,\frac{f_{12}}{\sqrt{g_{11}}},\qquad
w_{22}=\frac{f_{12}^{2}}{g_{11}},$ (198)

with $f_{12}\neq 0$. Note we must have $f_{12}\neq 0$ since if $f_{12}=0$,
(197) implies $w_{12}=w_{22}=w_{23}=0$, which is conflict with the assumption
that at least one of $\left\\{f_{12},w_{12},w_{22},w_{23}\right\\}$ is not
vanishing in order to have the constraint $\phi^{(0)}$. Then we get one gauge
identity at “level-1”:

$\displaystyle G^{(1)}$ $\displaystyle:=$ $\displaystyle
u^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}-g_{11}w_{11}\,\phi_{1}^{(0)}$ (199)
$\displaystyle=$ $\displaystyle
f_{12}\mathcal{E}_{1}^{(0)}+g_{11}\frac{\mathrm{d}\mathcal{E}_{2}^{(0)}}{\mathrm{d}t}-g_{11}w_{11}\,\mathcal{E}_{2}^{(0)}\equiv
0.$

Then the algorithm ends.

In this case, we have one constraint $\phi_{1}^{(0)}$ in (182), two gauge
identities $G^{(0)}$ and $G^{(1)}$ in (188) and (199), respectively. It is
easy to show that in this case there is no dynamical degree of freedom.

####  A.2.2 Case 2.2: one constraint

As long as (198) (at least one of the two equalities) is not satisfied,
$u^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ leads to a new independent constraint:

$\phi^{(1)}:=u^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}\approx 0.$ (200)

Then we go to the next level.

###  A.3 Case 2.2: level-2

By appending $\dot{\phi}^{(1)}$ (with $\phi^{(1)}$ in (200)) to
$\mathcal{E}_{i_{1}}^{(1)}$, we build the enlarged vector of equations of
motion:

$\mathcal{E}_{i_{2}}^{(2)}=W_{i_{2}j}^{(2)}\ddot{q}^{j}+V_{i_{2}}^{(2)},$
(201)

with

$W_{i_{2}j}^{(2)}=\left(\begin{array}[]{ccc}g_{11}&0&0\\\ 0&0&0\\\ 0&0&0\\\
-f_{12}&0&0\\\
-g_{11}w_{12}&f_{12}^{2}-g_{11}w_{22}&\lambda\left(f_{12}^{2}-g_{11}w_{22}\right)\end{array}\right),$
(202)

and

$V_{i_{2}}^{(2)}=\left(\begin{array}[]{c}f_{12}\dot{q}^{2}+\lambda
f_{12}\dot{q}^{3}-w_{11}q^{1}-w_{12}q^{2}-\lambda w_{12}q^{3}\\\
-f_{12}\dot{q}^{1}-w_{12}q^{1}-w_{22}q^{2}-\lambda w_{22}q^{3}\\\ -\lambda
f_{12}\dot{q}^{1}-\lambda w_{12}q^{1}-\lambda
w_{22}q^{2}-\lambda^{2}w_{22}q^{3}\\\
-w_{12}\dot{q}^{1}-w_{22}\dot{q}^{2}-\lambda w_{22}\dot{q}^{3}\\\
-f_{12}w_{11}\dot{q}^{1}-f_{12}w_{12}\dot{q}^{2}-\lambda
f_{12}w_{12}\dot{q}^{3}\end{array}\right).$ (203)

Since we have assumed $\lambda\neq 0$, then we have two sub-cases.

####  A.3.1 Case 2.2.1: $w_{22}\neq\frac{f_{12}^{2}}{g_{11}}$

In this case, $W_{i_{2}j}^{(2)}$ do not possesses further non-trivial left
null-eigenvector. The algorithm therefore ends.

In this case, we have two constraints $\phi^{(0)},\phi^{(1)}$ and one gauge
identity $G^{(0)}$. One can show that there is one dynamical degree of
freedom.

####  A.3.2 Case 2.2.2: $w_{22}=\frac{f_{12}^{2}}{g_{11}}$ while
$w_{12}\neq\sqrt{w_{11}}\,\frac{f_{12}}{\sqrt{g_{11}}}$

Since $w_{22}=\frac{f_{12}^{2}}{g_{11}}$, we have to require that the other
equality in (198) is not satisfied, i.e.,

$w_{12}\neq\sqrt{w_{11}}\,\frac{f_{12}}{\sqrt{g_{11}}}.$ (204)

In this case, $W_{i_{2}j}^{(2)}$ possesses a new non-trivial null-eigenvector

$u_{i_{2}}^{(2)}=\left(\begin{array}[]{c}w_{12}\\\ 0\\\ 0\\\ 0\\\
1\end{array}\right).$ (205)

Contracting $u_{i_{2}}^{(2)}$ with $\mathcal{E}_{i_{2}}^{(2)}$ yields

$u^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}=-f_{12}w_{11}\dot{q}^{1}-w_{12}w_{11}q^{1}-w_{12}^{2}q^{2}-\lambda
w_{12}^{2}q^{3}.$ (206)

Recall that we have two constraints $\phi_{1}^{(0)}$ in (191) and $\phi^{(1)}$
in (200), which in our case reduce to be

$\displaystyle\phi^{(0)}$ $\displaystyle\rightarrow$ $\displaystyle-
f_{12}\dot{q}^{1}-w_{12}q^{1}-\frac{f_{12}^{2}}{g_{11}}q^{2}-\lambda\,\frac{f_{12}^{2}}{g_{11}}q^{3},$
(207) $\displaystyle\phi^{(1)}$ $\displaystyle\rightarrow$ $\displaystyle-
g_{11}w_{12}\dot{q}^{1}-f_{12}w_{11}q^{1}-f_{12}w_{12}q^{2}-\lambda
f_{12}w_{12}q^{3}.$ (208)

One can show that $u^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}$ is linearly
independent of $\phi^{(0)}$ and $\phi^{(1)}$ as long as (204) is satisfied.
Therefore $u^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}$ leads to a new constraint

$\phi^{(2)}:=u^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}\approx 0.$ (209)

###  A.4 Case 2.2.2: level-3

Appending $\dot{\phi}^{(2)}$ (with $\phi^{(2)}$ given in (209)) to
$\mathcal{E}_{i_{2}}^{(2)}$ yields

$\mathcal{E}_{i_{3}}^{(3)}=W_{i_{3}j}^{(3)}\ddot{q}^{j}+V_{i_{3}}^{(3)},$
(210)

with

$W_{i_{3}j}^{(3)}=\left(\begin{array}[]{ccc}g_{11}&0&0\\\ 0&0&0\\\ 0&0&0\\\
-f_{12}&0&0\\\ -g_{11}w_{12}&0&0\\\ -f_{12}w_{11}&0&0\end{array}\right),$
(211)

and

$V_{i_{3}}^{(3)}=\left(\begin{array}[]{c}f_{12}\dot{q}^{2}+\lambda
f_{12}\dot{q}^{3}-w_{11}q^{1}-w_{12}q^{2}-\lambda w_{12}q^{3}\\\
-f_{12}\dot{q}^{1}-w_{12}q^{1}-\frac{f_{12}^{2}}{g_{11}}q^{2}-\lambda\frac{f_{12}^{2}}{g_{11}}q^{3}\\\
-\lambda f_{12}\dot{q}^{1}-\lambda
w_{12}q^{1}-\lambda\frac{f_{12}^{2}}{g_{11}}q^{2}-\lambda^{2}\frac{f_{12}^{2}}{g_{11}}q^{3}\\\
-w_{12}\dot{q}^{1}-\frac{f_{12}^{2}}{g_{11}}\dot{q}^{2}-\lambda\frac{f_{12}^{2}}{g_{11}}\dot{q}^{3}\\\
-f_{12}w_{11}\dot{q}^{1}-f_{12}w_{12}\dot{q}^{2}-\lambda
f_{12}w_{12}\dot{q}^{3}\\\
-w_{11}w_{12}\dot{q}^{1}-w_{12}^{2}\dot{q}^{2}-\lambda
w_{12}^{2}\dot{q}^{3}\end{array}\right).$ (212)

$W_{i_{3}j}^{(3)}$ possesses a non-trivial null-eigenvector

$u_{i_{3}}^{(3)}=\left(\begin{array}[]{c}f_{12}w_{11}\\\ 0\\\ 0\\\ 0\\\ 0\\\
g_{11}\end{array}\right),$ (213)

contracting which with $\mathcal{E}_{i_{3}}^{(3)}$ yields

$\displaystyle u^{(3)i_{3}}\mathcal{E}_{i_{3}}^{(3)}$ $\displaystyle=$
$\displaystyle-
g_{11}w_{11}w_{12}\dot{q}^{1}+\left(w_{11}f_{12}^{2}-g_{11}w_{12}^{2}\right)\dot{q}^{2}+\lambda\left(w_{11}f_{12}^{2}-g_{11}w_{12}^{2}\right)\dot{q}^{3}$
(214) $\displaystyle-f_{12}w_{11}^{2}q^{1}-f_{12}w_{11}w_{12}q^{2}-\lambda
f_{12}w_{11}w_{12}q^{3}.$

Comparing with the previous constaints $\phi^{(0)},\phi^{(1)},\phi^{(2)}$ in
(182), (200) and (209), clearly $u^{(3)i_{3}}\mathcal{E}_{i_{3}}^{(3)}$ leads
to a new constraint

$\phi^{(3)}:=u^{(3)i_{3}}\mathcal{E}_{i_{3}}^{(3)}\approx 0,$ (215)

since ${\normalcolor w_{11}f_{12}^{2}-g_{11}w_{12}^{2}\neq 0}$.

###  A.5 Case 2.2.2: level-4

By appending $\dot{\phi}^{(3)}$ (with $\phi^{(3)}$ given in (215)) to
$\mathcal{E}_{i_{3}}^{(3)}$ yields

$\mathcal{E}_{i_{4}}^{(4)}=W_{i_{4}j}^{(4)}\ddot{q}^{j}+V_{i_{4}}^{(4)},$
(216)

with

$W_{i_{4}j}^{(4)}=\left(\begin{array}[]{ccc}g_{11}&0&0\\\ 0&0&0\\\ 0&0&0\\\
-f_{12}&0&0\\\ -g_{11}w_{12}&0&0\\\ -f_{12}w_{11}&0&0\\\
-g_{11}w_{12}w_{11}&w_{11}f_{12}^{2}-g_{11}w_{12}^{2}&\lambda\left(w_{11}f_{12}^{2}-g_{11}w_{12}^{2}\right)\end{array}\right),$
(217)

and

$V_{i_{4}}^{(4)}=\left(\begin{array}[]{c}f_{12}\dot{q}^{2}+\lambda
f_{12}\dot{q}^{3}-w_{11}q^{1}-w_{12}q^{2}-\lambda w_{12}q^{3}\\\
-f_{12}\dot{q}^{1}-w_{12}q^{1}-\frac{f_{12}^{2}}{g_{11}}q^{2}-\lambda\frac{f_{12}^{2}}{g_{11}}q^{3}\\\
-\lambda f_{12}\dot{q}^{1}-\lambda
w_{12}q^{1}-\lambda\frac{f_{12}^{2}}{g_{11}}q^{2}-\lambda^{2}\frac{f_{12}^{2}}{g_{11}}q^{3}\\\
-w_{12}\dot{q}^{1}-\frac{f_{12}^{2}}{g_{11}}\dot{q}^{2}-\lambda\frac{f_{12}^{2}}{g_{11}}\dot{q}^{3}\\\
-f_{12}w_{11}\dot{q}^{1}-f_{12}w_{12}\dot{q}^{2}-\lambda
f_{12}w_{12}\dot{q}^{3}\\\
-w_{11}w_{12}\dot{q}^{1}-w_{12}^{2}\dot{q}^{2}-\lambda
w_{12}^{2}\dot{q}^{3}\\\
-f_{12}w_{11}^{2}\dot{q}^{1}-f_{12}w_{11}w_{12}\dot{q}^{2}-\lambda
f_{12}w_{11}w_{12}\dot{q}^{3}\end{array}\right).$ (218)

Clearly, since $w_{11}f_{12}^{2}-g_{11}w_{12}^{2}\neq 0$, $W_{i_{4}j}^{(4)}$
possesses no non-trivial null-eigenvector. The algorithm ends.

To summarize, in this case, we have four constraints:
$\phi^{(0)},\phi^{(1)},\phi^{(2)},\phi^{(3)}$ given in (207), (208), (209) and
(215), and one gauge identity $G^{(0)}$ given in (188), respectively. As a
result, there is no dynamical degree of freedom.

###  A.6 Case 3: level-1

From (189) and (190), the two independent constaints at “level-0” are:

$\displaystyle\dot{\phi}_{1}^{(0)}$ $\displaystyle=$ $\displaystyle-
f_{12}\ddot{q}^{1}-w_{12}\dot{q}^{1}-w_{22}\dot{q}^{2}-w_{23}\dot{q}^{3},$
(219) $\displaystyle\dot{\phi}_{2}^{(0)}$ $\displaystyle=$ $\displaystyle-
f_{13}\ddot{q}^{1}-w_{13}\dot{q}^{1}-w_{23}\dot{q}^{2}-w_{33}\dot{q}^{3}.$
(220)

In the “case-3”, the enlarged vector of equations of motion is

$\mathcal{E}_{i_{1}}^{(1)}=\left(\begin{array}[]{c}\mathcal{E}_{i}^{(0)}\\\
\dot{\phi}_{1}^{(0)}\\\
\dot{\phi}_{2}^{(0)}\end{array}\right)=W_{i_{1}j}^{(1)}\ddot{q}^{j}+V_{i_{1}}^{(1)},$
(221)

with

$W_{i_{1}j}^{(1)}=\left(\begin{array}[]{ccc}g_{11}&0&0\\\ 0&0&0\\\ 0&0&0\\\
-f_{12}&0&0\\\ -f_{13}&0&0\end{array}\right),$ (222)

and

$V_{i_{1}}^{(1)}=\left(\begin{array}[]{c}f_{12}\dot{q}^{2}+f_{13}\dot{q}^{3}-w_{11}q^{1}-w_{12}q^{2}-w_{13}q^{3}\\\
-f_{12}\dot{q}^{1}-w_{12}q^{1}-w_{22}q^{2}-w_{23}q^{3}\\\
-f_{13}\dot{q}^{1}-w_{13}q^{1}-w_{23}q^{2}-w_{33}q^{3}\\\
-w_{12}\dot{q}^{1}-w_{22}\dot{q}^{2}-w_{23}\dot{q}^{3}\\\
-w_{13}\dot{q}^{1}-w_{23}\dot{q}^{2}-w_{33}\dot{q}^{3}\end{array}\right).$
(223)

In this case $W_{i_{1}j}^{(1)}$ possesses 2 non-trivial null-eigenvectors:

$u_{1,i_{1}}^{(1)}=\left(\begin{array}[]{c}f_{12}\\\ 0\\\ 0\\\ g_{11}\\\
0\end{array}\right),\qquad u_{2,i_{1}}^{(1)}=\left(\begin{array}[]{c}f_{13}\\\
0\\\ 0\\\ 0\\\ g_{11}\end{array}\right).$ (224)

Contracting $u_{1,i_{1}}^{(1)}$ and $u_{2,i_{1}}^{(1)}$ with
$\mathcal{E}_{i_{1}}^{(1)}$ yields

$\displaystyle u_{1}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ $\displaystyle=$
$\displaystyle-
g_{11}w_{12}\dot{q}^{1}+\left(f_{12}^{2}-g_{11}w_{22}\right)\dot{q}^{2}+\left(f_{12}f_{13}-g_{11}w_{23}\right)\dot{q}^{3}$
(225) $\displaystyle-f_{12}w_{11}q^{1}-f_{12}w_{12}q^{2}-f_{12}w_{13}q^{3},$

and

$\displaystyle u_{2}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ $\displaystyle=$
$\displaystyle-
g_{11}w_{13}\dot{q}^{1}+\left(f_{13}f_{12}-g_{11}w_{23}\right)\dot{q}^{2}+\left(f_{13}^{2}-g_{11}w_{33}\right)\dot{q}^{3}$
(226) $\displaystyle-f_{13}w_{11}q^{1}-f_{13}w_{12}q^{2}-f_{13}w_{13}q^{3}.$

We need to check whether $u_{1}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ and
$u_{2}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ lead to new constraints or
identities. To this end, together with $\phi_{1}^{(0)}$ and $\phi_{2}^{(0)}$
in (189) and (190), we write

$\left(\begin{array}[]{c}\phi_{1}^{(0)}\\\ \phi_{2}^{(0)}\\\
u_{1}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}\\\
u_{2}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}\end{array}\right)=\bm{M}^{(1)}\left(\begin{array}[]{c}\dot{q}^{1}\\\
\dot{q}^{2}\\\ \dot{q}^{3}\\\ q^{1}\\\ q^{2}\\\ q^{3}\end{array}\right),$
(227)

with the $4\times 6$ matrix

$\bm{M}^{(1)}:=\left(\begin{array}[]{cccccc}-f_{12}&0&0&-w_{12}&-w_{22}&-w_{23}\\\
-f_{13}&0&0&-w_{13}&-w_{23}&-w_{33}\\\
-g_{11}w_{12}&f_{12}^{2}-g_{11}w_{22}&f_{12}f_{13}-g_{11}w_{23}&-f_{12}w_{11}&-f_{12}w_{12}&-f_{12}w_{13}\\\
-g_{11}w_{13}&f_{12}f_{13}-g_{11}w_{23}&f_{13}^{2}-g_{11}w_{33}&-f_{13}w_{11}&-f_{13}w_{12}&-f_{13}w_{13}\end{array}\right).$
(228)

Then the question is equivalent to checking the rank of $\bm{M}^{(1)}$. For
later convenience, we define the sub-matrix

$\bm{\Delta}:=\left(\begin{array}[]{cc}f_{12}^{2}-g_{11}w_{22}&f_{12}f_{13}-g_{11}w_{23}\\\
f_{12}f_{13}-g_{11}w_{23}&f_{13}^{2}-g_{11}w_{33}\end{array}\right),$ (229)

of which the determinant is

$\det\bm{\Delta}=g_{11}\left[-f_{12}^{2}w_{33}+2f_{12}f_{13}w_{23}-f_{13}^{2}w_{22}+g_{11}\left(w_{22}w_{33}-w_{23}^{2}\right)\right].$
(230)

####  A.6.1 2 identities (impossible)

First we shall show that it is impossible to have 2 identities. In fact, in
order to have 2 new identities, we have to require that the entries of the
sub-matrix $\bm{\Delta}$ in (229) are vanishing identically. This can be also
understood that since there are neither $\dot{q}^{2}$ nor $\dot{q}^{3}$ terms
in $\phi_{1}^{(0)},\phi_{2}^{(2)}$, as long as at least one of the
coefficients of $\dot{q}^{2}$ or $\dot{q}^{3}$ in
$u_{1}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ and
$u_{2}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ is not vanishing, we get a new
constraint. To conclude, the necessary conditions to have two new identities
are

$w_{22}=\frac{f_{12}^{2}}{g_{11}},\qquad
w_{23}=\frac{f_{12}f_{13}}{g_{11}},\qquad w_{33}=\frac{f_{13}^{2}}{g_{11}}.$
(231)

On the other hand, comparing with (184)-(187), we have to require

$f_{12}w_{13}-f_{13}w_{12}\neq 0,$ (232)

in order not to go back to “case 2”.

With these considerations, $u_{1}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ and
$u_{2}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$ reduce to be

$\displaystyle u_{1}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$
$\displaystyle\rightarrow$ $\displaystyle-
g_{11}w_{12}\dot{q}^{1}-f_{12}w_{11}q^{1}-f_{12}w_{12}q^{2}-f_{12}w_{13}q^{3},$
(233) $\displaystyle u_{2}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}$
$\displaystyle\rightarrow$ $\displaystyle-
g_{11}w_{13}\dot{q}^{1}-f_{13}w_{11}q^{1}-f_{13}w_{12}q^{2}-f_{13}w_{13}q^{3}.$
(234)

In order to have two identities among the 4 equalities
$\left\\{\phi_{1}^{(0)},\phi_{2}^{(0)},u_{1}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)},u_{2}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}\right\\}\approx
0$ we have to make sure that the rank of the $4\times 4$ matrix

$\left(\begin{array}[]{cccc}-f_{12}&-w_{12}&-\frac{f_{12}^{2}}{g_{11}}&-\frac{f_{12}f_{13}}{g_{11}}\\\
-f_{13}&-w_{13}&-\frac{f_{12}f_{13}}{g_{11}}&-\frac{f_{13}^{2}}{g_{11}}\\\
-g_{11}w_{12}&-f_{12}w_{11}&-f_{12}w_{12}&-f_{12}w_{13}\\\
-g_{11}w_{13}&-f_{13}w_{11}&-f_{13}w_{12}&-f_{13}w_{13}\end{array}\right)$
(235)

is 2. However, the determinant of the above $4\times 4$ matrix is

$-\left(f_{13}w_{12}-f_{12}w_{13}\right)^{3}\neq 0,$ (236)

since we must have (232). Therefore it is impossible to to have two
independent constraints $\left\\{\phi_{1}^{(0)},\phi_{2}^{(0)}\right\\}$ at
“level-0”, and in the meanwhile to have two new gauge identities
$\left\\{G_{1}^{(1)},G_{2}^{(1)}\right\\}$ at “level-1”.

####  A.6.2 Case 3.1: one constraint and one identity

The necessary condition is that the sub-matrix $\bm{\Delta}$ defined in (229)
is degenerate but not identically vanishing, i.e.,

$\mathrm{rank}\bm{\Delta}=1.$ (237)

This is because:

  * •

if the $\mathrm{rank}\bm{\Delta}=2$, there will be 2 new constraints.

  * •

if the $\mathrm{rank}\bm{\Delta}=0$ (then $\bm{\Delta}\equiv 0$), according to
the analysis in Sec. A.6.1, either there are still two new constraints when
(232) is satisfied, or we go back to “case 2.1”.

Without loss of generality, the sub-matrix $\bm{\Delta}$ can be written in the
form

$\bm{\Delta}=\omega\left(\begin{array}[]{cc}1&\eta\\\
\eta&\eta^{2}\end{array}\right),$ (238)

with $\omega,\eta$ being constants. Note we require $\omega\neq 0$ in order to
have $\mathrm{rank}\bm{\Delta}=1$. With (238), $\bm{M}^{(1)}$ reduces to be

$\bm{M}^{(1)}\rightarrow\left(\begin{array}[]{cccccc}-f_{12}&0&0&-w_{12}&\frac{\omega-
f_{12}^{2}}{g_{11}}&\frac{\eta\omega-f_{12}f_{13}}{g_{11}}\\\
-f_{13}&0&0&-w_{13}&\frac{\eta\omega-
f_{12}f_{13}}{g_{11}}&\frac{\eta^{2}\omega-f_{13}^{2}}{g_{11}}\\\
-g_{11}w_{12}&\omega&\omega\eta&-f_{12}w_{11}&-f_{12}w_{12}&-f_{12}w_{13}\\\
-g_{11}w_{13}&\omega\eta&\omega\eta^{2}&-f_{13}w_{11}&-f_{13}w_{12}&-f_{13}w_{13}\end{array}\right),$
(239)

our question thus reduces to checking if it is possible to have
$\mathrm{rank}\bm{M}^{(1)}=3$.

One can show that the necessary condition to have
$\mathrm{rank}\bm{M}^{(1)}=3$ is to require

$\mathcal{D}^{(1)}\coloneqq-\omega\left(f_{13}w_{12}-f_{12}w_{13}\right)^{2}+\omega^{2}\left[\left(\eta
w_{12}-w_{13}\right)^{2}-\frac{w_{11}}{g_{11}}\left(\eta
f_{12}-f_{13}\right)^{2}\right]=0.$ (240)

If

$f_{13}w_{12}-f_{12}w_{13}=0,$ (241)

then we need to require

$f_{13}-\eta\,f_{12}\neq 0,$ (242)

and (240) yields

$w_{12}=\frac{f_{12}\sqrt{w_{11}}}{\sqrt{g_{11}}}.$ (243)

On the other hand, if

$f_{13}w_{12}-f_{12}w_{13}\neq 0,$ (244)

(240) also implies one constraint among the coefficients. In both cases we
have $\mathrm{rank}\bm{M}^{(1)}=3$.

To conclude, it is possible to have $\mathrm{rank}\bm{M}^{(1)}=3$ so that we
get one constraint and one gauge identity on “level-1”. Without loss of
generality, we may choose the new constraint to be

$\phi^{(1)}:=u_{1}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}\approx 0.$ (245)

The gauge identity must be the form

$G^{(1)}:=a\,\phi_{1}^{(0)}+b\,\phi_{2}^{(1)}-\eta\,\phi_{1}^{(1)}+u_{2}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)},$
(246)

where the constants $a,b$ are not vanishing simultaneously. $a$ and $b$ are
determined by the concrete form of the null-eigenvector of $\bm{M}^{(1)}$,
which we do not show here explicitly.

####  A.6.3 Case 3.2: two new constraints

Generally, either

  1. 1.

$\det\bm{\Delta}\neq 0$, or

  2. 2.

$\mathrm{rank}\bm{\Delta}=1$, $\mathcal{D}^{(1)}\neq 0$ (with
$\mathcal{D}^{(1)}$ defined in (240)), or

  3. 3.

$\bm{\Delta}=0$ and $f_{13}w_{12}-f_{12}w_{13}\neq 0$,

we have two constraints at “level-1”:

$\phi_{1}^{(1)}:=u_{1}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}\approx 0,$ (247)

and

$\phi_{2}^{(1)}:=u_{2}^{(1)i_{1}}\mathcal{E}_{i_{1}}^{(1)}\approx 0.$ (248)

###  A.7 Case 3.1: level-2

By appending $\dot{\phi}^{(1)}$ (with $\phi^{(1)}$ given in (245)) to
$\mathcal{E}_{i_{1}}^{(1)}$ we have

$\mathcal{E}_{i_{2}}^{(2)}=W_{i_{2}j}^{(2)}\ddot{q}^{j}+V_{i_{2}}^{(2)},$
(249)

with

$W_{i_{2}j}^{(2)}=\left(\begin{array}[]{ccc}g_{11}&0&0\\\ 0&0&0\\\ 0&0&0\\\
-f_{12}&0&0\\\ -f_{13}&0&0\\\
-g_{11}w_{12}&f_{12}^{2}-g_{11}w_{22}&f_{12}f_{13}-g_{11}w_{23}\end{array}\right).$
(250)

Since

$f_{12}^{2}-g_{11}w_{22}\neq 0,$

$W_{i_{2}j}^{(2)}$ has no non-trivial null-eigenvector. The algorithm ends.

In this case, we have three constraints $\phi_{1}^{(0)},\phi_{2}^{(0)}$ and
$\phi^{(1)}$ given in (189), (190) and (245), one gauge identity $G^{(1)}$
given in (246), respectively. Therefore there is no dynamical degree of
freedom.

###  A.8 Case 3.2: level-2

From (247) and (248), by appending $\dot{\phi}^{(1)}$ and $\dot{\phi}^{(2)}$
to $\mathcal{E}_{i_{1}}^{(1)}$ we get

$\mathcal{E}_{i_{2}}^{(2)}=W_{i_{2}j}^{(2)}\ddot{q}^{j}+V_{i_{2}}^{(2)},$
(251)

with

$W_{i_{2}j}^{(2)}=\left(\begin{array}[]{ccc}g_{11}&0&0\\\ 0&0&0\\\ 0&0&0\\\
-f_{12}&0&0\\\ -f_{13}&0&0\\\
-g_{11}w_{12}&f_{12}^{2}-g_{11}w_{22}&f_{12}f_{13}-g_{11}w_{23}\\\
-g_{11}w_{13}&f_{13}f_{12}-g_{11}w_{23}&f_{13}^{2}-g_{11}w_{33}\end{array}\right),$
(252)

and

$V_{i_{2}}^{(2)}=\left(\begin{array}[]{c}f_{12}\dot{q}^{2}+f_{13}\dot{q}^{3}-w_{11}q^{1}-w_{12}q^{2}-w_{13}q^{3}\\\
-f_{12}\dot{q}^{1}-w_{12}q^{1}-w_{22}q^{2}-w_{23}q^{3}\\\
-f_{13}\dot{q}^{1}-w_{13}q^{1}-w_{23}q^{2}-w_{33}q^{3}\\\
-w_{12}\dot{q}^{1}-w_{22}\dot{q}^{2}-w_{23}\dot{q}^{3}\\\
-w_{13}\dot{q}^{1}-w_{23}\dot{q}^{2}-w_{33}\dot{q}^{3}\\\
-f_{12}w_{11}\dot{q}^{1}-f_{12}w_{12}\dot{q}^{2}-f_{12}w_{13}\dot{q}^{3}\\\
-f_{13}w_{11}\dot{q}^{1}-f_{13}w_{12}\dot{q}^{2}-f_{13}w_{13}\dot{q}^{3}\end{array}\right).$
(253)

We have to examine whether $W_{i_{2}j}^{(2)}$ possesses new non-trivial null-
eigenvectors. According to the rank of the matrix $\bm{\Delta}$ in (229), we
have to discuss three sub-cases.

####  A.8.1 Case 3.2.1: $\det\Delta\neq 0$

In this case, clearly there is no non-trivial eigenvectors of
$W_{i_{2}j}^{(2)}$. The algorithm ends.

####  A.8.2 Case 3.2.2: $\mathrm{rank}\Delta=1$ and $\mathcal{D}^{(1)}\neq 0$

In this case $\det\bm{\Delta}=0$ but $\bm{\Delta}\neq 0$. Similar to the
discussion in Sec. A.6.2, we make use of the form (238) and keep in mind that
$\omega\neq 0$. Then $W_{i_{2}j}^{(2)}$ redueces to

$W_{i_{2}j}^{(2)}\rightarrow\left(\begin{array}[]{ccc}g_{11}&0&0\\\ 0&0&0\\\
0&0&0\\\ -f_{12}&0&0\\\ -f_{13}&0&0\\\ -g_{11}w_{12}&\omega&\omega\,\eta\\\
-g_{11}w_{13}&\omega\,\eta&\omega\,\eta^{2}\end{array}\right).$ (254)

$W_{i_{2}j}^{(2)}$ possesses a single non-trivial null-eigenvector, which can
be chosen to be

$u_{i_{2}}^{(2)}:=\left(\begin{array}[]{c}-\eta\,w_{12}+w_{13}\\\ 0\\\ 0\\\
0\\\ 0\\\ -\eta\\\ 1\end{array}\right).$ (255)

Contracting $u_{i_{2}}^{(2)}$ with $\mathcal{E}_{i_{2}}^{(2)}$ yields

$\displaystyle u^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}$ $\displaystyle=$
$\displaystyle\left(\eta\,f_{12}w_{11}-f_{13}w_{11}\right)\dot{q}^{1}+\left(f_{12}w_{13}-f_{13}w_{12}\right)\dot{q}^{2}+\eta\left(f_{12}w_{13}-f_{13}w_{12}\right)\dot{q}^{3}$
(256)
$\displaystyle+w_{11}\left(\eta\,w_{12}-w_{13}\right)q^{1}+w_{12}\left(\eta\,w_{12}-w_{13}\right)q^{2}+w_{13}\left(\eta\,w_{12}-w_{13}\right)q^{3}.$

Comparing with $\phi_{1}^{(0)},\phi_{2}^{(0)}$ in (189)-(190) and
$\phi_{1}^{(1)},\phi_{2}^{(1)}$ in (247)-(248), after some manipulations, one
find that in this case $u^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}$ always leads to
a new constraint at “level-2”:

$\phi^{(2)}:=u^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}\approx 0.$ (257)

####  A.8.3 Case 3.2.3: $\Delta=0$

In this case clearly there are two new non-trivial null-eigenvectors for
$W_{i_{2}j}^{(2)}$:

$u_{1,i_{2}}^{(2)}=\left(\begin{array}[]{c}w_{12}\\\ 0\\\ 0\\\ 0\\\ 0\\\ 1\\\
0\end{array}\right),\qquad u_{2,i_{2}}^{(2)}=\left(\begin{array}[]{c}w_{13}\\\
0\\\ 0\\\ 0\\\ 0\\\ 0\\\ 1\end{array}\right).$ (258)

Contracting $u_{1,i_{2}}^{(2)}$ and $u_{2,i_{2}}^{(2)}$ with
$\mathcal{E}_{i_{2}}^{(2)}$ yields

$u_{1}^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}=-f_{12}w_{11}\dot{q}^{1}+\left(w_{12}f_{13}-f_{12}w_{13}\right)\dot{q}^{3}-w_{12}w_{11}q^{1}-w_{12}^{2}q^{2}-w_{12}w_{13}q^{3},$
(259)

and

$\displaystyle u_{2}^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}$ $\displaystyle=$
$\displaystyle-
f_{13}w_{11}\dot{q}^{1}+\left(w_{13}f_{12}-f_{13}w_{12}\right)\dot{q}^{2}-w_{13}w_{11}q^{1}-w_{13}w_{12}q^{2}-w_{13}^{2}q^{3}.$
(260)

Since we have already assumed $\bm{\Delta}=0$, (232) must be satisfied, i.e.,
$w_{12}f_{13}-f_{12}w_{13}\neq 0$.

Comparing with the constraints $\phi_{1}^{(0)},\phi_{2}^{(0)}$ in (189) and
(190), $\phi_{1}^{(1)},\phi_{2}^{(1)}$ in (247) and (248), since the
determinant of the $6\times 6$ matrix (after making use of (231))

$\left(\begin{array}[]{cccccc}-f_{12}&0&0&-w_{12}&-\frac{f_{12}^{2}}{g_{11}}&-\frac{f_{12}f_{13}}{g_{11}}\\\
-f_{13}&0&0&-w_{13}&-\frac{f_{12}f_{13}}{g_{11}}&-\frac{f_{13}^{2}}{g_{11}}\\\
-g_{11}w_{12}&0&0&-f_{12}w_{11}&-f_{12}w_{12}&-f_{12}w_{13}\\\
-g_{11}w_{13}&0&0&-f_{13}w_{11}&-f_{13}w_{12}&-f_{13}w_{13}\\\
-f_{12}w_{11}&0&w_{12}f_{13}-f_{12}w_{13}&-w_{12}w_{11}&-w_{12}^{2}&-w_{12}w_{13}\\\
-f_{13}w_{11}&w_{13}f_{12}-f_{13}w_{12}&0&-w_{13}w_{11}&-w_{13}w_{12}&-w_{13}^{2}\end{array}\right)$
(261)

is

$-\left(w_{12}f_{13}-f_{12}w_{13}\right)^{5}\neq 0,$ (262)

$u_{1}^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}$ and
$u_{2}^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}$ lead to two new constraints at
“level-2”:

$\displaystyle\phi_{1}^{(2)}:=u_{1}^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}$
$\displaystyle\approx$ $\displaystyle 0,$ (263)
$\displaystyle\phi_{2}^{(2)}:=u_{2}^{(2)i_{2}}\mathcal{E}_{i_{2}}^{(2)}$
$\displaystyle\approx$ $\displaystyle 0.$ (264)

###  A.9 Case 3.2.2: level-3

From (257), we have

$\mathcal{E}_{i_{3}}^{(3)}=W_{i_{3}j}^{(3)}\ddot{q}^{j}+V_{i_{3}}^{(3)},$
(265)

with

$W_{i_{3}j}^{(3)}=\left(\begin{array}[]{ccc}g_{11}&0&0\\\ 0&0&0\\\ 0&0&0\\\
-f_{12}&0&0\\\ -f_{13}&0&0\\\ -g_{11}w_{12}&\omega&\omega\,\eta\\\
-g_{11}w_{13}&\omega\,\eta&\omega\,\eta^{2}\\\ \eta
f_{12}w_{11}-f_{13}w_{11}&f_{12}w_{13}-f_{13}w_{12}&\eta\left(f_{12}w_{13}-f_{13}w_{12}\right)\end{array}\right),$
(266)

and

$V_{i_{3}j}^{(3)}=\left(\begin{array}[]{c}f_{12}\dot{q}^{2}+f_{13}\dot{q}^{3}-w_{11}q^{1}-w_{12}q^{2}-w_{13}q^{3}\\\
\frac{q^{2}\left(\omega-f_{12}^{2}\right)}{g_{11}}+\frac{q^{3}(\eta\omega-
f_{12}f_{13})}{g_{11}}-f_{12}\dot{q}^{1}-w_{12}q^{1}\\\
\frac{q^{2}(\eta\omega-f_{12}f_{13})}{g_{11}}+\frac{q^{3}\left(\eta^{2}\omega-
f_{13}^{2}\right)}{g_{11}}-f_{13}\dot{q}^{1}-w_{13}q^{1}\\\
\frac{\dot{q}^{2}\left(\omega-
f_{12}^{2}\right)}{g_{11}}+\frac{\dot{q}^{3}(\eta\omega-
f_{12}f_{13})}{g_{11}}-w_{12}\dot{q}^{1}\\\ \frac{\dot{q}^{2}(\eta\omega-
f_{12}f_{13})}{g_{11}}+\frac{\dot{q}^{3}\left(\eta^{2}\omega-
f_{13}^{2}\right)}{g_{11}}-w_{13}\dot{q}^{1}\\\
-f_{12}w_{11}\dot{q}^{1}-f_{12}w_{12}\dot{q}^{2}-f_{12}w_{13}\dot{q}^{3}\\\
-f_{13}w_{11}\dot{q}^{1}-f_{13}w_{12}\dot{q}^{2}-f_{13}w_{13}\dot{q}^{3}\\\
w_{11}\dot{q}^{1}(\eta w_{12}-w_{13})+w_{12}\dot{q}^{2}(\eta
w_{12}-w_{13})-w_{13}\dot{q}^{3}(w_{13}-\eta w_{12})\end{array}\right).$ (267)

There is one non-trivial null-eigenvector for $W_{i_{3}j}^{(3)}$, which can be
chosen to be

$u_{i_{3}}^{(3)}=\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\
\frac{g_{11}w_{12}}{f_{12}}\left(f_{12}w_{13}-f_{13}w_{12}\right)+w_{11}\omega\left(\eta-\frac{f_{13}}{f_{12}}\right)\\\
0\\\ f_{13}w_{12}-f_{12}w_{13}\\\ 0\\\ \omega\end{array}\right)$ (268)

for $f_{12}\neq 0$, and to be

$u_{i_{3}}^{(3)}=\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\ 0\\\
g_{11}\frac{w_{12}}{f_{13}}\left(f_{12}w_{13}-f_{13}w_{12}\right)+\omega\frac{w_{11}}{f_{13}}\left(\eta
f_{12}-f_{13}\right)\\\ f_{13}w_{12}-f_{12}w_{13}\\\ 0\\\
\omega\end{array}\right)$ (269)

for $f_{13}\neq 0$, and to be

$u_{i_{3}}^{(3)}=\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\ 0\\\ 0\\\ 0\\\ 0\\\
1\end{array}\right),$ (270)

for $f_{12}=f_{13}=0$. In all cases, the contraction of $u_{i_{3}}^{(3)}$ with
$\mathcal{E}_{i_{3}}^{(3)}$ can be shown to be a new constraint at “level-3”

$\phi^{(3)}:=u^{(3)i_{3}}\mathcal{E}_{i_{3}}^{(3)}\approx 0,$ (271)

since the $6\times 6$ matrix $\bm{M}^{(3)}$ defined by

$\left(\begin{array}[]{c}\phi_{1}^{(0)}\\\ \phi_{2}^{(0)}\\\ \phi_{1}^{(1)}\\\
\phi_{2}^{(1)}\\\ \phi^{(2)}\\\
u^{(3)i_{3}}\mathcal{E}_{i_{3}}^{(3)}\end{array}\right)=\bm{M}^{(3)}\left(\begin{array}[]{c}\dot{q}^{1}\\\
\dot{q}^{2}\\\ \dot{q}^{3}\\\ q^{1}\\\ q^{2}\\\ q^{3}\end{array}\right),$
(272)

always possesses a non-vanishing determinant.

###  A.10 Case 3.2.2: level-4

There is no non-trivial null-eigenvector of $W_{i_{4}j}^{(4)}$. The algorithm
ends.

To summarize, we have six constraints:
$\phi_{1}^{(0)},\phi_{2}^{(0)},\phi_{1}^{(1)},\phi_{2}^{(1)},\phi^{(2)},\phi^{(3)}$,
and thus there is no dynamical degree of freedom.

###  A.11 Case 3.2.3: level-3

From (263) and (264), by appending $\dot{\phi}_{1}^{(2)}$ and
$\dot{\phi}_{2}^{(2)}$ to $\mathcal{E}_{i_{2}}^{(2)}$, we get

$\mathcal{E}_{i_{3}}^{(3)}=W_{i_{3}j}^{(3)}\ddot{q}^{j}+V_{i_{3}}^{(3)},$
(273)

with (we have used (231) to replace $w_{22},w_{23}$ and $w_{33}$)

$W_{i_{3}j}^{(3)}=\left(\begin{array}[]{ccc}g_{11}&0&0\\\ 0&0&0\\\ 0&0&0\\\
-f_{12}&0&0\\\ -f_{13}&0&0\\\ -g_{11}w_{12}&0&0\\\ -g_{11}w_{13}&0&0\\\
-f_{12}w_{11}&0&w_{12}f_{13}-f_{12}w_{13}\\\
-f_{13}w_{11}&w_{13}f_{12}-f_{13}w_{12}&0\end{array}\right),$ (274)

and

$V_{i_{2}}^{(2)}=\left(\begin{array}[]{c}f_{12}\dot{q}^{2}+f_{13}\dot{q}^{3}-w_{11}q^{1}-w_{12}q^{2}-w_{13}q^{3}\\\
-f_{12}\dot{q}^{1}-w_{12}q^{1}-\frac{f_{12}^{2}}{g_{11}}q^{2}-\frac{f_{12}f_{13}}{g_{11}}q^{3}\\\
-f_{13}\dot{q}^{1}-w_{13}q^{1}-\frac{f_{12}f_{13}}{g_{11}}q^{2}-\frac{f_{13}^{2}}{g_{11}}q^{3}\\\
-w_{12}\dot{q}^{1}-\frac{f_{12}^{2}}{g_{11}}\dot{q}^{2}-\frac{f_{12}f_{13}}{g_{11}}\dot{q}^{3}\\\
-w_{13}\dot{q}^{1}-\frac{f_{12}f_{13}}{g_{11}}\dot{q}^{2}-\frac{f_{13}^{2}}{g_{11}}\dot{q}^{3}\\\
-f_{12}w_{11}\dot{q}^{1}-f_{12}w_{12}\dot{q}^{2}-f_{12}w_{13}\dot{q}^{3}\\\
-f_{13}w_{11}\dot{q}^{1}-f_{13}w_{12}\dot{q}^{2}-f_{13}w_{13}\dot{q}^{3}\\\
-w_{12}w_{11}\dot{q}^{1}-w_{12}^{2}\dot{q}^{2}-w_{12}w_{13}\dot{q}^{3}\\\
-w_{13}w_{11}\dot{q}^{1}-w_{13}w_{12}\dot{q}^{2}-w_{13}^{2}\dot{q}^{3}\end{array}\right).$
(275)

Since $w_{12}f_{13}-f_{12}w_{13}\neq 0$, there is no non-trivial eigenvector
for $W_{i_{3}j}^{(3)}$. The algorithm ends.

To summarize, in this case, we have six constraints:
$\phi_{1}^{(0)},\phi_{2}^{(0)}$ in (189) and (190),
$\phi_{1}^{(1)},\phi_{2}^{(1)}$ in (247) and (248),
$\phi_{1}^{(2)},\phi_{2}^{(2)}$ in (263)-(264), respectively. Therefore there
is no dynamical degree of freedom.

###  A.12 Summary

The classification of Lagrangians with one dynamical and two auxiliary
variables is summarized in Tab. 1. According to the types of
constraints/identities as well as to that at which level these
constraints/identities arise, there are in total 8 cases (in some sense 8
types of theories). Counting the number of degrees of freedom in the
Lagrangian approach is discussed in [90, 91, 92, 93], which is given by the
simple formula [94]

$\\#_{\mathrm{D.O.F.}}=N-\frac{1}{2}(l+g+e),$ (276)

in which $N$, $l$ and $g$ are the total numbers of the variables, the
Lagrangian constraints and the gauge identities, respectively. $e$ is the
total number of the gauge parameters plus its successive derivatives.

| level-0 | level-1 | level-2 | level-3 | $\\#_{\mathrm{D.O.F.}}$
---|---|---|---|---|---
case 1 | $G_{1}^{(0)},G_{2}^{(0)}$ | - | - | - | 1
case 2.1 | $\phi^{(0)},G^{(0)}$ | $G^{(1)}$ | - | - | 0
case 2.2.1 | $\phi^{(1)}$ | - | - | 1
case 2.2.2 | $\phi^{(2)}$ | $\phi^{(3)}$ | 0
case 3.1 | $\phi_{1}^{(0)},\phi_{2}^{(0)}$ | $\phi^{(1)},G^{(1)}$ | - | - | 0
case 3.2.1 | $\phi_{1}^{(1)},\phi_{2}^{(1)}$ | - | - | 1
case 3.2.2 | $\phi^{(2)}$ | $\phi^{(3)}$ | 0
case 3.2.3 | $\phi_{1}^{(2)},\phi_{2}^{(2)}$ | - | 0
Table 1: Classification of the quadratic Lagrangians with one dynamical and
two auxiliary variables.

With this classification, it is transparent that the linear scalar
perturbations in GR belong to “case 3.1”. While what we explored in this work
corresponds to “case 3.2.2” and “case 3.2.3” (together with “case 3.1”).

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