# Theory of interacting vector dark energy and fluid

Masroor C. Pookkillath  0000-0002-7199-8037 Centre for Theoretical Physics and
Natural Philosophy, Mahidol University, Nakhonsawan Campus, Phayuha Khiri,
Nakhonsawan 60130, Thailand  Kazuya Koyama  0000-0001-6727-6915 Institute of
Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building,
Burnaby Road, Portsmouth, PO1 3FX, United Kingdom

###### Abstract

In this work, we study interaction between dark energy and dark matter, where
dark energy is described by a massive vector field, and dark matter is
modelled as a fluid. We present new interaction term, which affects only
perturbations and can give interesting phenomenology. Then we present a
general Lagrangian for the interacting vector dark energy with dark matter.
For the dark energy, we choose Proca theory with $G_{3}$ term to study its
phenomenological consequence. For this model, we explore both background and
perturbation dynamics. We also present the no-ghost condition for tensor modes
and scalar modes. Subsequently, we also study the evolution of the
overdensities of both baryon and cold dark matter in the high$-k$ limit. We
show that the effective gravitational coupling is modified for cold dark
matter and baryon. We also choose a simple concrete model and numerically show
a suppression in the growth of cold dark matter overdensity.

##  I Introduction

Our present understanding of the universe apprises us that the universe is
composed of 68.3% of Dark Energy (DE), which is responsible for the
accelerated expansion today, 26.8% of Dark Matter (DM), which is an unknown
pressure less fluid responsible for dynamics of galaxies and structure
formation and 4.9% of the known matter. We can model our universe assuming
General Relativity (GR) as the theory of gravity together with a cosmological
constant $\Lambda$ and a pressureless fluid, called cold dark matter (CDM),
and baryons [1].

Although this model needs only minimal number of (six) parameters to fit the
cosmological observation, it does not explain why the value of the
cosmological constant is very small, known as the cosmological constant
problem [2], and why it starts dominating over the other ingredients in very
low redshift, known as the cosmic coincidence problem [2, 3]. To address these
crucial questions mentioned above, there have been attempts to explain the
phenomenon of DE, with a dynamical origin, where DE is modeled with a scalar
field, such as quintessence [4, 5] and k-essence [6, 7].

The scalar field can be coupled with other matter fields. Hence, the dynamical
DE is extended to the interacting dark sectors. These ideas have been
discussed in the literature, for example, [8, 9, 10]. It is interesting to
note that this interaction will have an impact on the evolution of the
background, as well as on the large-scale structure [11], which can be used to
understand the features of the DE. In [12], a non-linear study of interacting
DE is also carried out.

In recent years, there have been notable tensions/discrepancies in the
estimation of certain parameters of the universe modeled by $\Lambda$CDM. The
most notable one is the Hubble tension $H_{0}$ [13](see references there in
for different solutions). The measured value of $H_{0}$ from the low-redshift
observation and the estimated value from the high-redshift observations, such
as cosmic microwave background radiation, are of $\sim 5\sigma$ tension [14].
The second discrepancy is the mismatch in the estimation of the $S_{8}$
parameter from CMB and large-scale structure observations, which has a tension
around $\sim 3\sigma$ [15]. There is also a report of other anomalies, such as
the $\Omega_{k}$ anomaly [16], the lensing anomaly [17]. Furthermore, the
results from the latest DESI observation, show that, the dark energy could be
dynamical [18].This shows that, assuming there are no systematic errors in the
observations, the $\Lambda$CDM description of our universe could be an
approximation to the realistic universe. The unsatisfactory explanation of the
cosmological constant problem and the coincidence problem together with the
growing anomaly motivates us to explore cosmologies beyond the standard
$\Lambda$CDM cosmology.

To address the $H_{0}$ tension, there have been many proposals that have been
growing recently. In general, all these proposals fall generally into two
categories: 1) high redshift modification, that is, introduction of early DE
(EDE) or modified gravity in high redshift [19, 20], 2) low redshift
modification by introducing dynamical dark energy or modified gravity [21,
22]. Several interacting dark sector scenarios were also considered in both
early and late time modifications [23, 24].

Of these two proposals, EDE is more promising in addressing the Hubble
tension. This also introduces a new coincidence problem that we will have to
explain exactly why at that redshift (matter-radiation equality) we have a
slight domination of the DE. Recently, an interacting early dark energy was
introduced in [25, 24] to address this new coincidence problem. This
particular approach, that is, the interacting EDE can also address the
worsening of the $S_{8}$ tension, while introducing the EDE to resolve the
Hubble tension 111It has been noticed in [26], this increase in the $S_{8}$,
with introducing EDE to solve $H_{0}$ tension is due to the degeneracy in the
$\omega_{\rm cdm}$.. This motivates us to find models of the early interacting
dark sector that can address the tensions of both $H_{0}$ and $S_{8}$ tension
simultaneously.

The modeling of dynamical dark energy and modified gravity has developed
enough from simple scalar field models like quintessence and k-essence or more
complicated theories like Horndeski theories, which adds additional features
like modifying the effective Newton constant $G_{\rm eff}$ at the level of the
perturbation. On top Horndeski theories, there have also been studies to
include interactions with DM fluids, for example [27, 28]. These theories were
also found to have interesting phenomenological features, such as suppressing
$G_{\rm eff}$ to reduce the $S_{8}$ tension on top of Hubble tension [29, 30].

The model of dark energy is not limited to scalar fields. In the literature
there has been exploration to check if DE phenomenology is vector like. It is
reported in [31], that vector DE can have interesting phenomenology like, it
can violate the dominant dark energy conditions, $w_{\rm DE}<-1$, without
introducing any ghostlike kinetic term. It can also have tracking attractors,
similar to the quintessence trackers. A vector like quintessence has been
introduced in [32]. In [33], it is suggested that a time like vector can cause
for accelerated expansion of the universe. The vector field is also used to
explain inflation [34, 35].

The massive vector field is later generalized to include derivative self-
interactions known as the generalized Proca theory [36, 37]. In this theory,
the number of propagating degrees of freedom is still three. The construction
of this theory is motivated by the generalization of linearized massive
gravity theory, the Fierz-Pauli action in [38] 222Massive gravity is another
modified gravity theory which can explain accelerated expansion of the
universe [39]. Viable theories of massive gravity, its extension and its
cosmology are explored in [40, 41, 42, 43, 44]. This derivative self-
interacting action has been further extended to the Proca-Nuevo theory in
[45].

The cosmology of the generalized Proca theory has been explored in [46], and
that of Proca-Nuevo and its extension in [47]. For generalized Proca theory,
it is shown that the effective gravitational coupling for the perturbation can
be modified, and it is confronted with the data and found constraints on the
parameters [48]. Later, a full confrontation with the CMB data and other
large-scale structure data is carried out, and it is shown that it can reduce
the Hubble tension [21]. It has also been noticed in [21], when adding large-
scale structural data, the Hubble tension is not completely removed.

The vector dark energy that interacts with cold dark matter is introduced in
[49]. In [49], the possibility of an interaction of the form $L_{\rm
int}=-Qf(X)\rho_{c}$, where, $X=-1/2A_{\mu}A^{\mu}$, and $A_{\mu}$ is the
massive vector field was explored. This is similar to the interaction that has
been studied in the context of scalar dark energy $L_{\rm
int}=Q\dot{\phi}\rho_{c}$ in [8, 9]. In [49], they have included the
generalized cubic-order Proca term $G_{3}(X)$ and shown that it can allow the
phantom dark energy equation of state. These models have modified the
effective gravitational constant at the linear perturbation due to the
presence of the $G_{3}(X)$ term.

In a vector dark energy model that interacts with momentum transfer, is
developed in [50], where they have considered the interaction term
$Z=-u_{\alpha}A^{\alpha}$. Where, $u_{\alpha}$ is four velocity of the cold
dark matter. They derived a general expression for the effective gravitational
coupling for the baryon and cold dark matter in the high$-k$ limit and found
that there could be a suppression of growth of matter in the universe. This
study is similar to its scalar counterpart, which is studied in [51], with an
interaction term $L_{\rm int}=\nabla_{\alpha}\phi u^{\alpha}$.

In this present work, we ask the question, when we consider the dark energy to
be driven by a massive vector field, what the possible interaction terms with
cold dark matter are? Indeed, there exist interaction terms other than
$Z=-u_{\alpha}A^{\alpha}$, that have been considered in [50]. It is also
noticed that, we can have parity violating interactions between the vector
dark energy and cold dark matter. We then extend our study to a particular
parity invariant interacting term
$\mathcal{E}=-A^{\alpha}F_{\alpha\beta}u^{\beta}$, where the vector dark
energy is described by a generalised Proca action with $G_{3}$ term. We also
show that in the high $k$ limit this particular interaction term can modify
the effective gravitational coupling $G_{\rm eff}$ for cold dark matter. We
also noted that to have this effect we do need the $G_{3}(X)$ term for vector
dark energy action. This was also the case for the interaction $\mathcal{Z}$,
where we need the $G_{3}$ term to have the distinctive features of interaction
in the high $k$ limit.

This paper is organized as follows. Section II we derive the possible
interaction terms including parity violating interaction terms. Then in
section III, we introduce a general interacting Lagrangian, which is not
parity violating, and the matter (including cold dark matter) Lagrangian. In
Section IV, we proceed to study cosmology, to derive background equations of
motion, and find the ghost conditions for the tensor and scalar modes. To
study cosmology, we have chosen a generalised Proca action (up to $G_{3}$
term) to describe dark energy. Then in Section V, we find the linear equation
of motions. Subsequently, we derive the evolution of baryonic and cold dark
matter overdensities in the high $k$ limit in SectionVI, and we show that the
cold dark matter overdensity, $\delta_{c}$ equation and baryonic overdenisty,
$\delta_{b}$ equation are modified because of the interaction. In Section VII,
we choose a simple concrete model, and we show that there is a suppression of
effective gravitational coupling of cold dark matter. We then present our
conclusion in VIII. In Appendix A, we present a fluid description that we used
to model cold dark matter.

##  II Interacting fluid description

The theory of an interacting fluid with the scalar field (dark energy) is
discussed in [51], where the cold dark matter fluid is described by a “pull-
back” formalism (for the details of pull-back formalism, refer to [52]). For
the interacting fluid with a scalar field dark energy, the general action is

$L=L(n_{\alpha\beta\gamma},u^{\mu},\phi,\nabla_{\nu}\phi)\,,$ (1)

where $n_{\alpha\beta\gamma}$ is a full anti-symmetric tensor, which is dual
to the number density of the dark matter fluid $n^{\alpha}$,

$n_{\alpha\beta\gamma}=\epsilon_{\alpha\beta\gamma\delta}n^{\delta}\,,$ (2)

and, $u^{\mu}$ is the fluid velocity and $\phi$ is the scalar field, which
describes the dark energy. In this section, we develop an interaction of the
fluid with the dark energy field, which originated from a broken $U(1)$ gauge
symmetry vector field.

###  II.1 Building blocks for interacting vector dark energy

Since we are interested in a dark energy model as a vector theory, the above
Lagrangian (1) can be rewritten assuming no additional changes in the fluid
sector.

$L=L(n_{\alpha},u^{\mu},A_{\nu},F_{\mu\nu},\epsilon_{\alpha\beta\gamma\delta})\,,$
(3)

where we have included the $\epsilon_{\alpha\beta\gamma\delta}$ since we have
$F_{\mu\nu}$ which is an anti-symmetric term,

$F_{\mu\nu}\equiv\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}\,.$ (4)

From the combination of $\epsilon_{\alpha\beta\gamma\delta}$ and $F_{\mu\nu}$,
we can define the following dual tensor

$\tilde{F}_{\alpha\beta}\equiv\epsilon_{\alpha\beta\mu\nu}F^{\mu\nu}\,.$ (5)

We can also have the dual fluid number density given by Eq. (2).

The idea is to construct invariants from the basic fields that the Lagrangian
is depending on. First, let us look at the following invariant from the
combinations of $A_{\mu}$ and $u^{\mu}$.

$\mathcal{Z}\equiv-A_{\mu}u^{\mu}\,.$ (6)

This is equivalent to the term considered in [51] with a scalar field. Since
we have the tensor $F_{\mu\nu}$ which is anti-symmetric, we can further have
other invariants from the combination of $F_{\mu\nu}$,
$n_{\alpha\beta\gamma}$, and $u^{\alpha}$

$\mathcal{A}_{p}\equiv F^{\beta\gamma}n_{\alpha\beta\gamma}u^{\alpha}\,.$ (7)

The above interaction term is parity violating. Including the $\epsilon$
tensor, we can have the following invariants as well

$\displaystyle\mathcal{A}_{1}$ $\displaystyle\equiv$
$\displaystyle\epsilon_{\beta\gamma\delta\varepsilon}F^{\beta\gamma}n_{\alpha}{}^{\delta\varepsilon}u^{\alpha}=4F_{\alpha\beta}n^{\alpha}u^{\beta},$
(8) $\displaystyle\mathcal{A}_{2}$ $\displaystyle\equiv$
$\displaystyle\epsilon_{\alpha\gamma\delta\varepsilon}F^{\beta\gamma}n_{\beta}{}^{\delta\varepsilon}u^{\alpha}=4F_{\alpha\beta}n^{\alpha}u^{\beta}\,,$
(9) $\displaystyle\mathcal{A}_{3}$ $\displaystyle\equiv$
$\displaystyle\epsilon_{\beta\gamma\delta\varepsilon}F_{\alpha}{}^{\beta}n^{\gamma\delta\varepsilon}u^{\alpha}=-6F_{\alpha\beta}n^{\alpha}u^{\beta}\,.$
(10)

Instead of $F_{\alpha\beta}$ we can consider its dual
$\tilde{F}_{\alpha\beta}$, then we have

$\displaystyle\mathcal{B}$ $\displaystyle\equiv$ $\displaystyle
n_{\alpha\beta\gamma}\tilde{F}^{\beta\gamma}u^{\alpha}=4F_{\alpha\beta}n^{\alpha}u^{\beta}\,.$
(11)

Now adding the $\epsilon$ tensor we get

$\displaystyle\mathcal{B}_{p1}$ $\displaystyle\equiv$
$\displaystyle\epsilon_{\beta\gamma\delta\varepsilon}n^{\gamma\delta\varepsilon}\tilde{F}_{\alpha}{}^{\beta}u^{\alpha}=\epsilon_{\beta\gamma\delta\alpha}F^{\gamma\delta}n^{\alpha}u^{\beta}\,,$
(12) $\displaystyle\mathcal{B}_{p2}$ $\displaystyle\equiv$
$\displaystyle\epsilon_{\beta\gamma\delta\varepsilon}n_{\alpha}{}^{\delta\varepsilon}\tilde{F}^{\beta\gamma}u^{\alpha}=4\epsilon_{\gamma\delta\alpha\beta}F^{\gamma\delta}n^{\alpha}u^{\beta}\,,$
(13) $\displaystyle\mathcal{B}_{p3}$ $\displaystyle\equiv$
$\displaystyle\epsilon_{\alpha\gamma\delta\varepsilon}n_{\beta}{}^{\delta\
\varepsilon}\tilde{F}^{\beta\gamma}u^{\alpha}=2\epsilon_{\gamma\delta\alpha\beta}F^{\gamma\delta}n^{\alpha}u^{\beta}\,,$
(14)

which are parity violating interaction terms.

Taking in to account the terms $F_{\mu\nu}$, $A_{\alpha}$ and $u^{\beta}$ we
can build another invariant

$\mathcal{E}\equiv-A^{\alpha}F_{\alpha\beta}u^{\beta}\,.$ (15)

The above term is not parity violating. Including the anti-symmetric
$\epsilon$ tensor we can have the following invariants

$\mathcal{E}_{p}\equiv
A^{\alpha}\epsilon_{\alpha\beta\gamma\delta}F^{\gamma\delta}u^{\beta}\equiv
A^{\alpha}\tilde{F}_{\alpha\beta}u^{\beta}\,,$ (16)

which is a parity violating interaction term.

Furthermore, we can also have invariant from the combinations from
$F_{\alpha\beta}$, $A_{\mu}$, and $n_{\rho\sigma\delta}$

$\mathcal{G}_{p}=A^{\alpha}F^{\beta\gamma}n_{\alpha\beta\gamma}\,.$ (17)

Now including $\epsilon_{\beta\gamma\delta\varepsilon}$ we have

$\displaystyle\mathcal{G}_{1}$ $\displaystyle\equiv$ $\displaystyle
A^{\alpha}\epsilon_{\beta\gamma\delta\varepsilon}F^{\beta\gamma}n_{\alpha}{}^{\delta\varepsilon}=-4A^{\alpha}F_{\alpha\beta}n^{\beta}\,\,,$
(18) $\displaystyle\mathcal{G}_{2}$ $\displaystyle\equiv$ $\displaystyle
A^{\alpha}\epsilon_{\alpha\gamma\delta\varepsilon}F^{\beta\gamma}n_{\beta}{}^{\delta\varepsilon}=-2A^{\alpha}F_{\alpha\beta}n^{\beta}\,,$
(19) $\displaystyle\mathcal{G}_{3}$ $\displaystyle\equiv$ $\displaystyle
A^{\alpha}\epsilon_{\beta\gamma\delta\varepsilon}F_{\alpha}{}^{\beta}n^{\gamma\delta\varepsilon}=6A^{\alpha}F_{\alpha\beta}n^{\beta}\,.$
(20)

Considering the dual tensor $\tilde{F}_{\alpha\beta}$, then we get

$\mathcal{F}\equiv
A^{\alpha}n_{\alpha\beta\gamma}\tilde{F}^{\beta\gamma}=-4A^{\alpha}F_{\alpha\beta}n^{\beta}\,,$
(21)

and including $\epsilon$ we get

$\displaystyle\mathcal{F}_{p1}$ $\displaystyle\equiv$ $\displaystyle\epsilon
g_{\beta\gamma\delta\varepsilon}n^{\gamma\delta\
\varepsilon}\tilde{F}_{\alpha}{}^{\beta}=6A^{\alpha}\epsilon_{\alpha\gamma\delta\beta}F^{\gamma\
\delta}n^{\beta}\,,$ (22) $\displaystyle\mathcal{F}_{p2}$
$\displaystyle\equiv$ $\displaystyle A^{\alpha}\epsilon
g_{\beta\gamma\delta\varepsilon}n_{\alpha\
}{}^{\delta\varepsilon}\tilde{F}^{\beta\gamma}=-4A^{\alpha}\epsilon_{\gamma\delta\alpha\beta}F^{\gamma\delta}n^{\beta}\,,$
(23) $\displaystyle\mathcal{F}_{p3}$ $\displaystyle\equiv$
$\displaystyle\epsilon g_{\alpha\gamma\delta\varepsilon}n_{\beta}{}^{\delta\
\varepsilon}\tilde{F}^{\beta\gamma}=-2A^{\alpha}\epsilon_{\gamma\delta\alpha\beta}F^{\gamma\
\delta}n^{\beta}\,.$ (24)

The terms with $\epsilon$ are parity violating, so we will not consider them
in the following. Then the possible interactions are

$\mathcal{Z}\equiv-A_{\mu}u^{\mu}\,,\qquad\mathcal{A}\equiv
F_{\beta\gamma}n^{\beta}u^{\gamma}\,,\qquad\text{and}\qquad\mathcal{E}\equiv-A^{\alpha}F_{\alpha\beta}u^{\beta}\,.$
(25)

As for the interaction term $\mathcal{A}$ we can rewrite is as

$\mathcal{A}=F_{\beta\gamma}n^{\beta}u^{\gamma}=nF_{\alpha\gamma}u^{\beta}u^{\gamma}\,\qquad\text{where}\qquad
n^{\alpha}=nu^{\alpha}\,,$ (26)

hence it is zero since $F_{\alpha\beta}$ is antisymmetric. Then the
contributing interactions are $\mathcal{Z}$ and $\mathcal{E}$. The first
interaction term $\mathcal{Z}$ has already been studied in [50], and has shown
that, one can suppress cosmic growth.

Apart from this interaction, we have the standard non-interaction terms as

$\mathcal{X}\equiv
F_{\mu\nu}F^{\mu\nu}\,,\qquad\mathcal{Y}\equiv-\frac{1}{2}A_{\mu}A^{\mu}\,,$
(27)

where $\mathcal{X}$ is the standard kinetic term and $\mathcal{Y}$ is the mass
term in the Proca theory [36].

##  III An interacting Lagrangian

Now we can write the interacting Lagrangian that does not violate the parity
as

$L=L\left(\tilde{n},\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{E}\right)\,,$
(28)

where

$\tilde{n}\equiv
n^{2}=\frac{1}{3!}n^{\alpha\beta\gamma}n_{\alpha\beta\gamma}\,,$ (29)

and is called the fluid number density.

The Lagrangian that we have considered is completely general. In the
following, we assume that the dark energy is described by the $G_{3}$ Proca
theory, together with fluid interaction as described above, and explore the
physical consequences of the new interaction terms. With matter minimal
coupling to gravity, we have the following action.

$S_{\rm tot}=\int d^{4}x\sqrt{-g}\left[\frac{M_{\rm
P}^{2}}{2}R-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}+G_{3}(\mathcal{Y})\nabla_{\mu}A^{\mu}+f(\mathcal{Y},\mathcal{Z},\mathcal{E})\right]+S_{\rm
c}+S_{\rm m}\,,$ (30)

In the above action Eq. (30), the kinetic terms are separated for each
propagating mode, and then an interacting action is also added. Note that we
can generalise the action Eq. (30) to include the terms beyond $G_{3}$ term in
generalized Proca theory [36, 46].

The matter action, we follow the Schutz-Sorkin action [53, 54, 55]

$S_{\text{m}}=-\int\text{d}^{4}x\sqrt{-g}\left[\rho_{\text{m}}\left(n_{\text{m}}\right)+J_{\text{m}}^{\mu}\left(\nabla_{\mu}l_{\text{m}}\right)\right]\,,$
(31)

where $n_{\text{m}}$ are defined as

$\displaystyle n_{\text{m}}$ $\displaystyle=$
$\displaystyle\sqrt{-g_{\mu\nu}J_{\text{m}}^{\mu}J_{\text{m}}^{\nu}}\,,$ (32)

and

$J^{\mu}\equiv n_{\rm m}u_{\rm m}^{\mu}\,,\qquad\text{and}\qquad u_{{\rm
m}\alpha}u_{\rm m}^{\alpha}=-1\,.$ (33)

To model the interacting dark sector, we assume that the dark matter is
described via the dust fluid as introduced in [52], which is also used in [56,
57, 58, 59, 60, 61, 62]

$S_{\rm c}=-\int{\rm
d}^{4}x\sqrt{-g}\left[\rho_{c}(\tilde{n})+n_{c}^{\mu}\nabla_{\mu}K\right]\,,$
(34)

The $\rho_{\rm c}(\tilde{n})$ is the energy density of the dark matter
component that depends on the square of the number density. This action is
different from the action introduced in [52]. We have an additional term,

$n_{\rm c}^{\mu}\nabla_{\mu}K\,,$ (35)

in other words we are imposing the $\nabla_{\nu}n_{\rm c}^{\nu}=0$ with the
Lagrangian multiplier $K$. We found that without this term, the perturbation
gets an inconsistent tensor-mode propagation. In addition, this term will
restrict the interaction to only momentum transfer, since this demands the
conservation of the continuity equation. We show in appendix.A the above
action gives the general perfect fluid equations of motion for the linear
perturbation.

##  IV Cosmology

Now we can study the cosmology of the interacting Lagrangian (30) introduced
above. In this section we derive the background and ghost conditions for the
tensor and scalar modes.

###  IV.1 Background

Let us look at the background cosmology. We assume an isotropic and
homogeneous flat background, Friedmann-Lemaître-Robertson-Walker(FLRW) space-
time with linear perturbations,

$\displaystyle{\rm d}s^{2}=g_{\mu\nu}{\rm d}x^{\mu}{\rm
d}x^{\nu}=a^{2}\left[-(1+2\alpha){\rm d}t^{2}+2\frac{1}{a}\partial_{i}\chi{\rm
d}x^{i}{\rm
d}t+\left\\{(1+2\zeta)\delta_{ij}+2\partial_{i}\partial_{j}\left(E/a^{2}\right)\right\\}{\rm
d}x^{i}{\rm d}x^{j}\right]\,.$ (36)

For the Proca field $A_{\mu}$ we define

$A^{\mu}=\left[\frac{1}{a}(\phi+\delta\phi),\frac{1}{a^{2}}\delta^{ij}\partial_{j}\delta\phi_{s}\right]\,.$
(37)

Now for the cold dark matter fluid part we have the fields $n^{\alpha}$ and
$K$, which can be defined as

$n_{c}^{\alpha}=\left[\frac{1}{a}n_{0}(1+\delta
n_{0}),\frac{1}{a^{2}}\delta^{ij}\partial_{j}n_{s}\right]\,,\qquad
K=K_{0}+\delta K\,.$ (38)

Finally, we have other matter action from baryons and photons

$\displaystyle J_{\text{m}}^{\nu}=\left[\frac{J_{\text{m}}}{a}\left(1+\delta
J_{\text{m}}\right),\frac{1}{a^{2}}\delta^{ij}\partial_{j}\delta
j_{\text{m}}\right]\,,\qquad l_{\text{m}}=\bar{l}_{\text{m}}+\delta
l_{\text{m}}\,.$ (39)

For the cold dark matter sector we have

$n_{0}=\frac{N_{\rm c}}{a^{3}}\,,$ (40)

and

$K_{0}=-2\int a(\eta)n_{0}(\eta)\frac{\partial\rho_{c}}{\partial\tilde{n}}{\rm
d}\eta\,.$ (41)

where $N_{\rm c}$ is constant, which is the total number of dark matter
particles.

For the matter sector (from $S_{\rm m}$)

$\displaystyle J_{\rm m}$ $\displaystyle=$ $\displaystyle\frac{N_{\rm
m}}{a^{3}}$ (42) $\displaystyle\bar{l}_{\text{m}}$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}\rho\left(n_{\rm
m}\right)_{,n_{\text{m}}}a\left(\eta\right)\,\text{d}\eta\,,$ (43)

Having substituted the matter equations in the action we can get the Friedmann
equations. The first Friedmann equation

$\displaystyle 3M_{\rm P}^{2}H^{2}$ $\displaystyle=$ $\displaystyle\rho_{\rm
m}+\rho_{\rm c}-f-3H\phi^{3}G_{3,\mathcal{Y}}-\phi
f_{,\mathcal{Z}}-2\mathcal{Y}f_{,\mathcal{Y}}\,,$ (44)

where $H=\dot{a}/a^{2}$. The second Friedmann equation

$\displaystyle 2M_{\rm P}^{2}\frac{\dot{H}}{a}$ $\displaystyle=$
$\displaystyle-n_{\rm m}\rho_{,n_{\rm
m}}+2n_{0}^{2}\rho_{c,\tilde{n}}+\phi^{2}\frac{\dot{\phi}}{a}G_{3,\mathcal{Y}}-\left(3H\phi^{3}G_{3,\mathcal{Y}}+\phi
f_{,\mathcal{Z}}+2\mathcal{Y}f_{,\mathcal{Y}}\right),$ (45) $\displaystyle=$
$\displaystyle-(\rho_{\rm m}+P_{\rm m})-\rho_{\rm c}-(\rho_{\rm DE}+P_{\rm
DE}),\,$

where we have used the relation

$n_{\rm m}\rho_{,n_{\rm m}}=\rho_{\rm m}+P_{\rm m}\,,\qquad
2n_{0}^{2}\rho_{c,\tilde{n}}=\rho_{c},$ (46)

where $P_{\rm m}$ is the pressure of matter. The sum of DE density and
pressure is

$\rho_{\rm DE}+P_{\rm
DE}=-\phi^{2}\frac{\dot{\phi}}{a}G_{3,\mathcal{Y}}+\left(3H\phi^{3}G_{3,\mathcal{Y}}+\phi
f_{,\mathcal{Z}}+2\mathcal{Y}f_{,\mathcal{Y}}\right).$ (47)

Finally, we have the equation for the field $\phi$, which is given as

$3H\phi^{3}G_{3,\mathcal{Y}}+\phi
f_{,\mathcal{Z}}+2\mathcal{Y}f_{,\mathcal{Y}}=0\,.$ (48)

Notice that from the equations of motion for $\delta\phi$, for the dark energy
content $\rho_{\rm DE}+P_{\rm
DE}=-\phi^{2}G_{3,\mathcal{Y}}\tfrac{\dot{\phi}}{a}$. In other words, the dark
energy equation of state is given by

$w_{\rm DE}=-1+\frac{\phi^{2}G_{3,\mathcal{Y}}}{f}\frac{\dot{\phi}}{a}\,.$
(49)

On top of it, it is interesting to notice that at the background level we do
not see the consequences of the interaction, $\mathcal{E}$. However, we need
to see how the interactions influence the linear perturbation sector.

###  IV.2 Perturbation

In this section we study the ghost instability condition and the Laplace
instability conditions for tensor and scalar modes for the theory we are
considering Eq. (30).

####  IV.2.1 Tensor modes

In this subsection, we study the propagation of the tensor mode in this
theory. Let us consider the tensor mode perturbation around the FLRW
background,

${\rm d}s^{2}=a(t)^{2}\left[-{\rm
d}t^{2}+\left(\delta_{ij}+\sum_{\lambda=+,\times}\epsilon_{ij}^{\lambda}h_{\lambda}\right){\rm
d}x^{i}{\rm d}x^{j}\right]\,,$ (50)

where the tensor mode perturbations obey standard trace-less and divergence
free gauge condition $\delta_{ij}\epsilon_{ij}=0$, and
$\epsilon_{ij}^{\lambda}\delta^{jl}\partial_{l}h_{\lambda}=0$, and also it
obeys the normalisation condition
$\epsilon_{ij}^{+}\epsilon_{kl}^{+}\delta^{ik}\delta^{jl}=\epsilon_{ij}^{\times}\epsilon_{kl}^{\times}\delta^{ik}\delta^{jl}=1$,
with $\epsilon_{ij}^{+}\epsilon_{kl}^{\times}\delta^{ik}\delta^{jl}=0$. Now
expanding the action Eq. (30) with the above metric Eq. (50), and substituting
the background equations of motion, we get the following reduced action.

$S_{\rm T}=\frac{M_{\rm P}^{2}}{4}\sum_{\lambda=+,\times}\int{\rm
d}^{4}x\,a^{4}\left[\frac{\dot{h}_{\lambda}^{2}}{a^{2}}-\frac{k^{2}}{a^{2}}h_{\lambda}^{2}\right]\,,$
(51)

where we have changed the action into Fourier space, and $k$ is the Fourier
mode.

Hence, the above expression proves that the tensor mode propagation does not
change in the tensor mode propagation.

####  IV.2.2 Scalar mode

In this part, we discuss the linear stability of the scalar modes. We expand
the action without choosing any gauge and make the field redefinition,

$\psi_{\phi}\equiv\phi\chi+\delta\phi_{s}\,,$ (52)

which remove the second derivative of the variable $\chi$. We also make the
following field redefinition

$\displaystyle\delta J_{\rm m}=\frac{N_{\rm
m}}{a^{3}}\left(\frac{\rho}{n\rho_{,n}}\delta_{\rm m}-\alpha\right)\,,$ (53)
$\displaystyle\delta n_{0}=\frac{N_{\rm
c}}{a^{3}}\left(\frac{\rho_{c}}{2\tilde{n}\rho_{c,\tilde{n}}}\delta_{\rm
c}-\alpha\right)\,.$ (54)

Now we find the equations of motion for the field $j_{\rm m}$ and $n_{s}$

$\displaystyle\delta l$ $\displaystyle=$ $\displaystyle\rho_{,n_{\rm
m}}\chi+\frac{\rho_{,n_{\rm m}}}{n_{\rm m}}j_{\rm m}\,,$ (55)
$\displaystyle\delta K$ $\displaystyle=$ $\displaystyle\frac{N_{\rm
c}}{a^{3}}\left(2\rho_{c,\tilde{n}}\chi+f_{,\mathcal{E}}\frac{a^{6}}{N_{c}^{2}}\right)+\left(2\rho_{c,\tilde{n}}+f_{,\mathcal{Z}}\frac{a^{6}}{N_{\rm
c}{}^{2}}\right)n_{s}-\frac{a^{3}}{N_{\rm c}}f_{,\mathcal{Z}}\psi_{\phi}$ (56)
$\displaystyle-f_{,\mathcal{E}}\frac{a^{3}}{N_{\rm
c}}\left(\phi\,\delta\phi+2\phi^{2}\,\alpha+\phi\,\frac{\dot{\psi}_{\phi}}{a}\right)\,.$

We replace $\delta l$ and $\delta K$ with the above equations of motions and
also make another field redefinition

$\displaystyle j_{\rm m}$ $\displaystyle=$ $\displaystyle-\frac{N_{\rm
m}}{a^{3}}\frac{a}{k^{2}}\theta_{\rm m}\,,$ (57) $\displaystyle n_{s}$
$\displaystyle=$ $\displaystyle-\frac{N_{\rm
cmd}}{a^{3}}\frac{a}{k^{2}}\theta_{\rm c}\,.$ (58)

After a few integration by parts we get the second order action can be written
as

$\displaystyle S_{\rm Proca}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\,\,a^{4}\left[\chi\left(-w_{1}\frac{k^{2}\alpha}{a^{2}}-w_{2}\frac{k^{2}\delta\phi}{a^{2}\phi}-y_{1}\frac{2k^{2}\dot{\zeta}}{a}\right)+\delta\phi\left(w_{6}\frac{k^{2}\partial_{t}\left(E/a^{2}\right)}{a}-w_{3}\frac{k^{2}\dot{\psi}_{\phi}}{2a^{3}\phi^{2}}+\frac{k^{2}\psi_{\phi}(w_{2}+w_{6}\phi)}{2a^{2}\phi^{2}}-w_{6}\frac{3\dot{\zeta}}{a}\right)\right.$
(59)
$\displaystyle\left.+\alpha\left(\delta\phi\left(-w_{3}\frac{k^{2}}{a^{2}\phi}-\frac{3Hw_{1}-2w_{4}}{\phi}\right)+k^{2}\left(w_{1}\frac{\partial_{t}\left(E/a^{2}\right)}{a}+2y_{1}\zeta\right)-w_{3}\frac{k^{2}\dot{\psi}_{\phi}}{a^{3}\phi}+w_{6}\frac{k^{2}\psi_{\phi}}{a^{2}}-w_{1}\frac{3\dot{\zeta}}{a}\right)\right.$
$\displaystyle+\alpha^{2}\left(w_{4}-w_{3}\frac{k^{2}}{a^{2}}\right)+k^{2}\left(\frac{2M_{\rm
P}^{2}\partial_{t}\left(E/a^{2}\right)\dot{\zeta}}{a^{2}}+\zeta\left(y_{2}\frac{\partial_{t}\left(E/a^{2}\right)}{a}-y_{4}E/a^{2}\right)+y_{2}\frac{E/a^{2}\dot{\zeta}}{a}+y_{1}\zeta^{2}\right)$
$\displaystyle\left.+\delta\phi^{2}\left(w_{5}\frac{1}{\phi^{2}}-w_{3}\frac{k^{2}}{4a^{2}\phi^{2}}\right)-w_{3}\frac{k^{2}\dot{\psi}_{\phi}^{2}}{4a^{4}\phi^{2}}+w_{7}\frac{k^{2}\psi_{\phi}^{2}}{2a^{2}}-\frac{3M_{\rm
P}^{2}\dot{\zeta}^{2}}{a^{2}}+y_{3}k^{4}\left(-E/a^{2}\right)+y_{3}3\zeta^{2}\right]$

$\displaystyle S_{\mathcal{E}}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\,\,a^{4}\left[f_{\mathcal{E}}\left\\{\chi\left[k^{2}\left(-\frac{\delta\phi\phi}{a^{2}}-\frac{\phi\dot{\psi}_{\phi}}{a^{3}}\right)-\frac{2k^{2}\alpha\phi^{2}}{a^{2}}\right]\right.\right.$
(60)
$\displaystyle+\alpha\left[-\frac{2\rho_{c}\phi^{2}\dot{\delta}_{c}}{a(P_{c}+\rho_{c})}+k^{2}\left(\frac{2\phi^{2}\partial_{t}\left(E/a^{2}\right)}{a}+\frac{2\psi_{\phi}\phi}{a^{2}}\right)-\frac{6\phi^{2}\dot{\zeta}}{a}+\frac{6\delta_{c}H\phi^{2}(\delta\rho_{c}P_{c}-\delta
P_{c}\rho_{c})}{\delta\rho_{c}(P_{c}+\rho_{c})}\right]$
$\displaystyle+\dot{\delta}_{c}\left(-\frac{\delta\phi\rho_{c}\phi}{a(P_{c}+\rho_{c})}-\frac{\rho_{c}\phi\dot{\psi}_{\phi}}{a^{2}(P_{c}+\rho_{c})}\right)+\delta_{c}\left(H\phi\frac{3\dot{\psi}_{\phi}(\delta\rho_{c}P_{c}-\delta
P_{c}\rho_{c})}{a\delta\rho_{c}(P_{c}+\rho_{c})}+H\phi\frac{3\delta\phi(\delta\rho_{c}P_{c}-\delta
P_{c}\rho_{c})}{\delta\rho_{c}(P_{c}+\rho_{c})}\right)$
$\displaystyle\left.+k^{2}\left(\phi\frac{\delta\phi\partial_{t}\left(E/a^{2}\right)}{a}+\frac{\delta\phi\psi_{\phi}}{a^{2}}+\phi\frac{\partial_{t}\left(E/a^{2}\right)\dot{\psi}_{\phi}}{a^{2}}-\frac{H\psi_{\phi}^{2}}{2a^{2}}\right)-\phi\frac{3\delta\phi\dot{\zeta}}{a}-\phi\frac{3\dot{\psi}_{\phi}\dot{\zeta}}{a^{2}}\right\\}$
$\displaystyle\left.-\frac{k^{2}\psi_{\phi}^{2}\dot{\phi}f_{,\mathcal{ZE}}}{2a^{3}}-\frac{k^{2}\psi_{\phi}^{2}\phi\dot{\phi}f_{,\mathcal{YE}}}{2a^{3}}\right]$

$\displaystyle S_{\mathcal{Z}}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\,\,a^{4}\left[f_{,\mathcal{Z}}\left\\{\chi\left(\frac{\rho_{c}\phi\dot{\delta}_{c}}{a(P_{c}+\rho_{c})}+k^{2}\left(-\frac{\phi\partial_{t}\left(E/a^{2}\right)}{a}-\frac{\psi_{\phi}}{a^{2}}\right)+\frac{3\phi\dot{\zeta}}{a}+\frac{3\delta_{c}H\phi(\delta
P_{c}\rho_{c}-\delta\rho_{c}P_{c})}{\delta\rho_{c}(P_{c}+\rho_{c})}\right)\right.\right.$
(61)
$\displaystyle+\phi\frac{k^{2}\chi^{2}}{2a^{2}}+\theta_{c}\left(-3aH\phi\frac{\delta_{c}(\delta
P_{c}\rho_{c}-\delta\rho_{c}P_{c})}{k^{2}\delta\rho_{c}(P_{c}+\rho_{c})}-\rho_{c}\phi\frac{\dot{\delta}_{c}}{k^{2}(P_{c}+\rho_{c})}+\phi\partial_{t}\left(E/a^{2}\right)-3\phi\frac{\dot{\zeta}}{k^{2}}\right)+\delta_{c}\left(\frac{\rho_{c}\dot{\psi}_{\phi}}{a(P_{c}+\rho_{c})}+\frac{3H\psi_{\phi}\rho_{c}}{P_{c}+\rho_{c}}\right)$
$\displaystyle\left.+k^{2}\left(-\frac{(E/a^{2})\dot{\psi}_{\phi}}{a}+\frac{\psi_{\phi}^{2}}{2a^{2}\phi}-3(E/a^{2})H\psi_{\phi}\right)+\frac{3\zeta\dot{\psi}_{\phi}}{a}-\alpha\delta\phi-\frac{1}{2}\phi\alpha^{2}-\frac{\delta\phi^{2}}{2\phi}+9H\psi_{\phi}\zeta-\phi\frac{\theta_{c}^{2}}{2k^{2}}\right\\}$
$\displaystyle+f_{,\mathcal{ZZ}}\left\\{\frac{\dot{\phi}}{a}\frac{\delta_{c}\psi_{\phi}\rho_{c}}{(P_{c}+\rho_{c})}-\frac{\dot{\phi}}{a}k^{2}(E/a^{2})\psi_{\phi}+3\frac{\dot{\phi}}{a}\psi_{\phi}\zeta+\phi\,\alpha\delta\phi+\frac{1}{2}\phi^{2}\,\alpha^{2}+\frac{1}{2}\delta\phi^{2}\right\\}$
$\displaystyle\left.+f_{,\mathcal{YZ}}\left\\{\phi\frac{\dot{\phi}}{a}\frac{\delta_{c}\psi_{\phi}\rho_{c}}{(P_{c}+\rho_{c})}-\phi\frac{\dot{\phi}}{a}k^{2}(E/a^{2})\psi_{\phi}+3\phi\frac{\dot{\phi}}{a}\psi_{\phi}\zeta+2\phi^{2}\,\alpha\delta\phi+\phi^{3}\,\alpha^{2}+\phi\,\delta\phi^{2}\right\\}\right]$

$\displaystyle S_{I}$ $\displaystyle=$ $\displaystyle\sum_{I=c,b,r}\int
d^{4}xa^{4}\left[\chi\left(\frac{\rho_{I}\dot{\delta}_{I}}{a}-\frac{k^{2}\partial_{t}\left(E/a^{2}\right)(P_{I}+\rho_{I})}{a}+\frac{3(P_{I}+\rho_{I})\dot{\zeta}}{a}+3\delta_{I}H\left(\frac{\delta
P_{I}\rho_{I}}{\delta\rho_{I}}-P_{I}\right)\right)+\frac{k^{2}\chi^{2}(P_{I}+\rho_{I})}{2a^{2}}\right.$
(62)
$\displaystyle+\theta_{I}\left(\frac{3a\delta_{I}H(\delta\rho_{I}P_{I}-\delta
P_{I}\rho_{I})}{k^{2}\delta\rho_{I}}-\frac{\rho_{I}\dot{\delta}_{I}}{k^{2}}+\partial_{t}\left(E/a^{2}\right)(P_{I}+\rho_{I})-\frac{3(P_{I}+\rho_{I})\dot{\zeta}}{k^{2}}\right)-\alpha\delta_{I}\rho_{I}-\frac{\delta_{I}^{2}\delta
P_{I}\rho_{I}^{2}}{2\delta\rho_{I}(P_{I}+\rho_{I})}$
$\displaystyle\left.+k^{2}(E/a^{2})\zeta(-P_{I}-\rho_{I})+\frac{1}{2}k^{4}(E/a^{2})^{2}(-P_{I}-\rho_{I})+\frac{\theta_{I}^{2}(-P_{I}-\rho_{I})}{2k^{2}}+\frac{3}{2}\zeta^{2}(P_{I}+\rho_{I})+\rho_{r}\delta_{r}\sigma_{r}\right]$

The coefficients $w_{i}$ and $y_{i}$ are given as

$\displaystyle w_{1}$ $\displaystyle=$ $\displaystyle 2M_{\rm
P}^{2}H-\phi^{3}G_{,\mathcal{Y}}$ (63) $\displaystyle w_{2}$ $\displaystyle=$
$\displaystyle 2M_{\rm P}^{2}H-w_{1}$ (64) $\displaystyle w_{3}$
$\displaystyle=$ $\displaystyle-2\phi^{2}$ (65) $\displaystyle w_{4}$
$\displaystyle=$ $\displaystyle-3M_{\rm
P}^{2}H^{2}-\frac{3}{2}H\phi^{3}\left(G_{3,\mathcal{Y}}-\phi^{2}G_{3,\mathcal{YY}}\right)+\frac{1}{2}\phi^{4}f_{,\mathcal{YY}}$
(66) $\displaystyle w_{5}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}H(w_{1}+w_{2})+w_{4}$ (67) $\displaystyle w_{6}$
$\displaystyle=$ $\displaystyle-\phi^{2}G_{3,\mathcal{Y}}$ (68) $\displaystyle
w_{7}$ $\displaystyle=$
$\displaystyle\frac{\dot{\phi}}{a}\frac{w_{2}}{\phi^{3}}$ (69) $\displaystyle
y_{1}$ $\displaystyle=$ $\displaystyle M_{\rm P}^{2}/a^{2}$ (70)
$\displaystyle y_{2}$ $\displaystyle=$ $\displaystyle-M_{\rm P}^{2}H-\phi
G_{3}$ (71) $\displaystyle y_{3}$ $\displaystyle=$ $\displaystyle M_{\rm
P}^{2}\frac{\dot{H}}{a}\frac{1}{2}\phi^{2}w_{7}$ (72) $\displaystyle y_{4}$
$\displaystyle=$ $\displaystyle-3Hy_{2}+3M_{\rm
P}^{2}\frac{\dot{H}}{a}+\frac{\dot{\phi}}{a}G_{3}$ (73)

We choose a flat gauge for simplicity, i.e. we keep the variables $\zeta=0$,
and $E=0$. Then we have the variables, $\alpha$, $\chi$, from the gravity
sector $\delta_{\rm m}$, $\delta_{\rm c}$, $\delta l_{\rm m}$, $\delta K$,
$\delta j_{\rm m}$ and $n_{s}$ in the matter and dark matter sector. Finally
we have $\delta\phi$ and $\psi_{\phi}$ in the dark energy sector.

Now we integrate out the variables $\delta j_{\rm m}$, $n_{s}$, $\delta l_{\rm
m}$, $\delta K$, $\chi$, $\delta\phi$, and $\alpha$. We are remaining with the
variables $\delta_{\rm m}$, $\delta_{\rm c}$, and $\psi_{\phi}$, and notice
that all of them are dynamical. After integration by part we can write the
action in the form

$\int{\rm d}^{3}x{\rm
d}t\,a^{4}\left[\dot{\mathcal{U}}^{T}\,\mathbf{K}\,\dot{\mathcal{U}}-\frac{k^{2}}{a^{2}}\mathcal{U}^{T}\,\mathbf{G}\,\mathcal{U}-\mathcal{U}^{T}\,\mathbf{M}\,\mathcal{U}-\frac{k}{a}\mathcal{U}^{T}\,\mathbf{B}\,\dot{\mathcal{U}}\right]\,,$
(74)

where $\mathbf{K}$, $\mathbf{G}$, $\mathbf{M}$, and $\mathbf{B}$ are
4$\times$4 matrix and the vector $\mathcal{U}^{T}$ is given by

$\mathcal{U}^{T}\equiv(\psi_{\phi},\delta_{c}/k,\delta_{b}/k,\delta_{r}/k)\,.$
(75)

Here, we consider the matter components baryon and radiation.

In the high $k$ limit ($k\rightarrow\infty$) we have

$\displaystyle\mathbf{K}_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{2(w_{1}-2w_{2})^{2}\phi^{2}}\left[3M_{\rm
P}^{2}H^{2}w_{1}^{2}+4M_{\rm P}^{4}H^{2}w_{4}+2M_{\rm
P}^{4}H^{2}\phi\left(-f_{,\mathcal{Z}}+\phi
f_{,\mathcal{ZZ}}+2\phi^{2}f_{,\mathcal{YZ}}\right)\right]\,,$ (76)
$\displaystyle\mathbf{K}_{22}$ $\displaystyle=$
$\displaystyle\frac{a^{2}\rho_{c}^{2}\left[P_{\rm c}+\rho_{c}+\phi\left(-\phi
f_{,\mathcal{E}}+f_{,\mathcal{Z}}\right)\right]}{(\rho_{c}+P_{\rm
c})^{2}}\,,\qquad\mathbf{K}_{33}=\frac{a^{2}\rho_{r}^{2}}{2(\rho_{r}+P_{r})}\,,\qquad\mathbf{K}_{44}=\frac{a^{2}\rho_{b}^{2}}{2(\rho_{b}+P_{b})}\,,$
(77)

and $\mathbf{G}$ in the leading order

$\displaystyle\mathbf{G}_{11}$ $\displaystyle=$
$\displaystyle\mathcal{G}+H\mu+\frac{\dot{\mu}}{a}-\frac{w_{2}{}^{2}\sum_{I}\big{(}\rho_{I}+P_{I}\big{)}}{2\phi^{2}\big{(}w_{1}-2w_{2}\big{)}^{2}}-\frac{M_{\rm
P}^{4}H^{2}f_{,\mathcal{Z}}}{\phi^{2}\big{(}w_{1}-2w_{2}\big{)}^{2}}+\frac{2M_{\rm
P}^{4}f_{,\mathcal{E}}\big{(}-2\phi
G_{3,\mathcal{Y}}+f_{,\mathcal{E}}\big{)}}{\big{(}w_{1}-2w_{2}\big{)}^{2}}\,,$
(78) $\displaystyle\mathbf{G}_{22}$ $\displaystyle=$
$\displaystyle\frac{a^{2}c_{c}^{2}\rho_{c}^{2}}{2(\rho_{c}+P_{c})}\,,\qquad\mathbf{G}_{33}=\frac{a^{2}c_{r}^{2}\rho_{r}^{2}}{2(\rho_{r}+P_{r})}\,,\qquad\mathbf{G}_{44}=\frac{a^{2}c_{b}^{2}\rho_{b}^{2}}{2(\rho_{b}+P_{b})}\,,$
(79)

where,

$\mathcal{G}\equiv-\frac{4M_{\rm
P}^{4}H^{2}w_{2}{}^{2}}{\phi^{2}\big{(}w_{1}-2w_{2}\big{)}^{2}w_{3}}-\frac{w_{2}\dot{\phi}}{2a\phi^{3}}\,,\qquad\text{and}\qquad\mu\equiv\frac{M_{P}^{2}Hw_{2}}{\big{(}w_{1}-2w_{2}\big{)}\phi^{2}}\,.$
(80)

The leading order terms for the anti-symmetric matrix are

$\mathbf{B}_{12}=-\mathbf{B}_{21}=-\frac{M_{\rm
P}^{2}aH\rho_{c}(\phi^{2}G_{3,\mathcal{Y}}f_{,\mathcal{E}}-\phi
f_{,\mathcal{E}}^{2}+f_{,\mathcal{Z}})}{(\rho_{c}+P_{c})(2M_{\rm
P}^{2}H-\phi^{2}G_{3,\mathcal{Y}})}$ (81)

We assume that the fluid obeys the weak energy condition $\rho_{\rm I}+P_{\rm
I}=0$, for ($I=c,b,r$), which follows the standard ghost conditions for
baryons and radiation.

$\mathbf{K}_{33}>0\,,\qquad\mathbf{K}_{44}>0\,.$ (82)

Then we have the following ghost conditions for the fields $\delta\phi$ and
$\delta_{c}$.

$\displaystyle q_{s}$ $\displaystyle\equiv$ $\displaystyle 3w_{1}^{2}+4M_{\rm
P}^{2}w_{4}+2M_{\rm P}^{2}\phi\left(-f_{,\mathcal{Z}}+\phi
f_{,\mathcal{ZZ}}+2\phi^{2}f_{,\mathcal{YZ}}\right)>0\,,$ (83) $\displaystyle
q_{c}$ $\displaystyle\equiv$ $\displaystyle 1+\frac{\phi\left(-\phi
f_{,\mathcal{E}}^{2}+f_{,\mathcal{Z}}\right)}{(\rho_{c}+P_{c})}>0\,,$ (84)

The propagation of the speed for baryons and radiation is not modified by the
$\mathbf{B}$ matrix. Then their speeds of propagation are

$\displaystyle c_{b}^{2}=\frac{\mathbf{G}_{33}}{\mathbf{B}_{33}}\,,\qquad
c_{r}^{2}=\frac{\mathbf{G}_{44}}{\mathbf{B}_{44}}\,.$ (85)

However, the other two dynamical fields $\psi_{\phi}$ and $\delta_{c}$ are
affected by the anti-symmetric metric $\mathbf{B}$ Eq. (81). Now taking the
equations of motion from the action Eq. (74), on assuming a solution

$\psi_{\phi}=\bar{\psi}_{\phi}e^{i(\omega
t-kx)}\,,\qquad\delta_{c}=\bar{\delta}_{c}e^{i(\omega t-kx)}\,,$ (86)

we have

$\displaystyle\omega^{2}\bar{\psi}_{\phi}-\hat{c}_{s}^{2}\frac{k^{2}}{a^{2}}\bar{\psi}_{\phi}-i\omega\frac{k}{a}\frac{\mathbf{B}_{12}}{\mathbf{K}_{11}}\bar{\delta}_{c}$
$\displaystyle\simeq$ $\displaystyle 0$ (87)
$\displaystyle\omega^{2}\bar{\delta}_{c}-\hat{c}_{c}^{2}\frac{k^{2}}{a^{2}}\bar{\delta}_{c}-i\omega\frac{k}{a}\frac{\mathbf{B}_{21}}{\mathbf{K}_{11}}\bar{\psi}_{\phi}$
$\displaystyle\simeq$ $\displaystyle 0$ (88)

where

$\displaystyle\hat{c}_{s}^{2}$ $\displaystyle=$
$\displaystyle\frac{2(w_{1}-2w_{2})^{2}\phi^{2}}{M_{\rm
P}^{2}H^{2}q_{S}}\left[\mathcal{G}+H\mu+\frac{\dot{\mu}}{a}-\frac{w_{2}{}^{2}\sum_{I}\big{(}\rho_{I}+P_{I}\big{)}}{2\phi^{2}\big{(}w_{1}-2w_{2}\big{)}^{2}}-\frac{M_{\rm
P}^{4}H^{2}f_{,\mathcal{Z}}}{\phi^{2}\big{(}w_{1}-2w_{2}\big{)}^{2}}+\frac{2M_{\rm
P}^{4}f_{,\mathcal{E}}\big{(}-2\phi
G_{3,\mathcal{Y}}+f_{,\mathcal{E}}\big{)}}{\big{(}w_{1}-2w_{2}\big{)}^{2}}\right]$
(89) $\displaystyle\hat{c}_{c}^{2}$ $\displaystyle=$
$\displaystyle\frac{\mathbf{G}_{22}}{\mathbf{K}_{22}}=\frac{c_{c}^{2}}{q_{c}}\,.$
(90)

Now we know that $c_{c}^{2}=0$, then it follows that $\hat{c}_{c}^{2}=0$. The
equation of motion Eq. (89) will be reduced to

$\omega^{2}\bar{\delta}_{c}-i\omega\frac{k}{a}\frac{\mathbf{B}_{21}}{\mathbf{K}_{11}}\bar{\psi}_{\phi}\simeq
0$ (91)

There are two solutions

$\displaystyle\omega$ $\displaystyle=$ $\displaystyle 0\,,$ (92)
$\displaystyle\omega\bar{\delta}_{c}$ $\displaystyle=$ $\displaystyle
i\frac{k}{a}\frac{\mathbf{B}_{21}}{\mathbf{K}_{11}}\bar{\psi}_{\phi}\,.$ (93)

The first branch of the solution corresponds to

$c_{c}^{2}=\frac{\omega^{2}a^{2}}{k^{2}}=0\,.$ (94)

The other branch corresponds to the propagation of the longitudinal mode
$\delta\phi$. By substituting the second branch Eq. (93) in to the equation of
motion Eq. (90) we get

$\frac{k^{2}}{a^{2}}c_{s}^{2}=\omega^{2}\,,\qquad
c_{s}^{2}=\hat{c}_{s}^{2}+\Delta c_{s}^{2}\,,$ (95)

where

$\Delta c_{s}^{2}=\frac{1}{q_{s}q_{c}}\left[\frac{M_{\rm
P}^{2}aH\rho_{c}(\phi^{2}G_{3,\mathcal{Y}}f_{,\mathcal{E}}-\phi
f_{,\mathcal{E}}^{2}+f_{,\mathcal{Z}})}{(\rho_{c}+P_{c})(2M_{\rm
P}^{2}H-\phi^{2}G_{3,\mathcal{Y}})}\right]^{2}.$ (96)

##  V Linear perturbation equations of motion

Now let us find the equations of motion for the linear perturbation. The
second order Lagrangian we are considering have the following fields,

$(\alpha,\quad\chi,\quad\zeta,\quad
E,\quad\delta\phi,\quad\psi_{\phi},\quad\delta_{\rm m},\quad\theta_{\rm
m},\quad\delta_{\rm c},\quad\theta_{\rm c}).$ (97)

Notice that we have not made any choice of gauge. Instead, after varying for
each field above, we make a choice of the Newtonian Gauge,

$\alpha=\Psi,\qquad\zeta=-\Phi,\qquad\chi=0,\qquad\text{and}\qquad E=0.$ (98)

###  V.1 Matter sector

Next let us look at the equations of motion for the matter sector:

$\displaystyle E_{\theta_{\rm m}}$ $\displaystyle\equiv$
$\displaystyle\dot{\delta}_{\rm m}+3aH\left(\frac{\delta P_{\rm
m}}{\delta\rho_{\rm m}}-\frac{P_{\rm m}}{\rho_{\rm
m}}\right)\delta+\left(1+\frac{P_{\rm m}}{\rho_{\rm
m}}\right)\theta_{m}-3\left(1+\frac{P_{\rm m}}{\rho_{\rm
m}}\right)\dot{\Phi}=0\,,$ (99) $\displaystyle E_{\delta_{\rm m}}$
$\displaystyle\equiv$ $\displaystyle\dot{\theta}_{\rm
m}+aH\left(1-3\frac{\delta P_{\rm m}}{\delta\rho_{\rm m}}\right)\theta_{\rm
m}-k^{2}\left(\frac{\delta P_{\rm m}/\delta\rho_{\rm m}}{1+P_{\rm m}/\rho_{\rm
m}}\right)\delta_{\rm m}-k^{2}\Psi+k^{2}\sigma=0\,.$ (100)

The above equations are the equations of motion of the standard matter fluids
in the system we are considering, in particular, radiation and baryon.
However, for baryon, we have $P_{\rm m}=0$ and $\delta P_{\rm m}=0$. The field
$\sigma$ denotes the shear of the fluid, for example, for photons.

###  V.2 Dark matter sector

Now we find the following equations of motion for the dark matter fluid,
together with the interaction with the vector dark energy. Here, we can see
the implications of the interaction terms we have considered.

$\displaystyle E_{\theta_{\rm c}}$ $\displaystyle\equiv$
$\displaystyle\dot{\delta}_{\rm c}+\theta_{\rm c}-3\dot{\Phi}=0\,,$ (101)
$\displaystyle E_{\delta_{\rm c}}$ $\displaystyle\equiv$
$\displaystyle\dot{\theta}_{\rm c}+\left[\frac{1}{\rho_{c}+\phi
f_{,\mathcal{Z}}}\left\\{aH(\rho_{c}+4\phi
f_{,\mathcal{Z}})+\dot{\phi}\big{(}f_{,\mathcal{Z}}+\phi(f_{\mathcal{ZZ}}+\phi
f_{,\mathcal{ZY}})\big{)}\right\\}\right]\theta_{\rm c}$ (102)
$\displaystyle+\left[\frac{k^{2}}{\rho_{c}+\phi
f_{,\mathcal{Z}}}\right]\left(f_{,\mathcal{Z}}\frac{\dot{\psi}_{\phi}}{a}-\rho_{c}\Psi\right)+\left[\frac{k^{2}}{a(\rho_{c}+\phi
f_{,\mathcal{Z}})}\big{(}3aHf_{,\mathcal{Z}}+\dot{\phi}(f_{,\mathcal{ZZ}}+\phi
f_{,\mathcal{ZY}})\big{)}\right]\psi_{\phi}$
$\displaystyle+\left[\frac{k^{2}f_{,\mathcal{E}}}{\rho_{c}+\phi
f_{,\mathcal{Z}}}\right]\left(\phi\frac{\delta\dot{\phi}}{a}+2\phi^{2}\frac{\dot{\Psi}}{a}+\phi\frac{\ddot{\psi}_{\phi}}{a^{2}}+\frac{(2aH\phi+\dot{\phi})}{a}\frac{\delta\dot{\phi}_{s}}{a}+2\phi\frac{(3aH\phi+2\dot{\phi})}{a}\Psi+\frac{(3aH\phi+\dot{\phi})}{a}\delta\phi\right)$
$\displaystyle+\left[\frac{k^{2}f_{,\mathcal{EZ}}}{\rho_{c}+\phi
f_{,\mathcal{Z}}}\right]\left(\frac{\phi\dot{\phi}}{a}(\delta\phi+2\Psi+\frac{\dot{\psi}_{\phi}}{a})\right)+\left[\frac{k^{2}f_{,\mathcal{EY}}}{\rho_{c}+\phi
f_{,\mathcal{Z}}}\right]\left(\frac{\phi^{2}\dot{\phi}}{a}\big{(}\delta\phi+2\phi\Psi+\frac{\dot{\psi}_{\phi}}{a}\big{)}\right)=0\,.$

Notice that the continuity equation for the the cold dark matter density
perturbation is not affected by the interaction. However, the evolution
equation for the velocity divergence is modified significantly due to the
interaction that we are considering. This shows that all the interaction terms
that we are considering contribute to the momentum transfer. Also, note that
there is a term in $\ddot{\psi}_{\phi}$, proportional to $f_{\mathcal{E}}$,
which we will replace with the equation for the field $\psi_{\phi}$.

###  V.3 Gravity sector

Finally we will show the equations of motion for the field that comes in the
gravity sector including the vector dark energy. Let us first look at the
equations of motion by varying the Proca fields.

$\displaystyle E_{\delta\phi}$ $\displaystyle\equiv$
$\displaystyle\frac{k^{2}}{a^{2}}\left(\delta\phi+2\phi\Psi-\phi
G_{3,\mathcal{Y}}\psi_{\phi}+\frac{\dot{\psi}_{\phi}}{a}\right)+G_{3,\mathcal{Y}}\left(3H\phi^{2}+\frac{3\phi^{3}\dot{H}}{\dot{\phi}}\right)\Psi-3\phi^{2}G_{3,\mathcal{Y}}\frac{\dot{\Phi}}{a}-3\phi^{2}G_{3,\mathcal{Y}}\frac{\dot{H}}{\dot{\phi}}\delta\phi$
(103)
$\displaystyle+f_{,\mathcal{E}}\left(\frac{k^{2}}{a^{2}}\psi_{\phi}+\frac{\phi}{a}\theta_{\rm
c}\right)=0\,,$ $\displaystyle E_{\psi_{\phi}}$ $\displaystyle\equiv$
$\displaystyle
f_{,\mathcal{E}}\left[-\phi\frac{\ddot{\delta}_{c}}{a^{2}}+k^{2}\left(-\frac{\delta\phi}{a^{2}}+H\frac{\psi_{\phi}}{a^{2}}-2\phi\frac{\Psi}{a^{2}}\right)+2H\phi\frac{\theta_{c}}{a}+3\phi\frac{\ddot{\Phi}}{a^{2}}+\dot{\phi}\frac{\theta_{c}}{a^{2}}\right]$
(104)
$\displaystyle+f_{,\mathcal{ZE}}\left[\dot{\phi}\frac{k^{2}\psi_{\phi}}{a^{3}}+\phi\dot{\phi}\frac{\theta_{c}}{a^{2}}\right]+f_{,\mathcal{YE}}\left[\phi\dot{\phi}\frac{k^{2}\psi_{\phi}}{a^{3}}+\phi^{2}\dot{\phi}\frac{\theta_{c}}{a^{2}}\right]+f_{,\mathcal{Z}}\left[-\frac{k^{2}\psi_{\phi}}{a^{2}\phi}-\frac{\theta_{c}}{a}\right]$
$\displaystyle+k^{2}\left[\frac{\delta\dot{\phi}}{a^{3}}+\delta\phi\left(\frac{\phi
G_{3,\mathcal{Y}}}{a^{2}}+\frac{H}{a^{2}}\right)+\Psi\left(\frac{\phi^{2}G_{3,\mathcal{Y}}}{a^{2}}+\frac{2H\phi}{a^{2}}+\frac{2\dot{\phi}}{a^{3}}\right)+\frac{\psi_{\phi}\dot{\phi}G_{3,\mathcal{Y}}}{a^{3}}+\frac{2\phi\dot{\Psi}}{a^{3}}+\frac{\ddot{\psi}_{\phi}}{a^{4}}\right].$

For the other fields in the metric we have the following equations of motion

$\displaystyle E_{\alpha}$ $\displaystyle\equiv$ $\displaystyle M_{\rm
P}^{2}\left[2\frac{k^{2}}{a^{2}}\Phi+6H\Psi+6\frac{H}{a}\dot{\Phi}\right]+\left(3H\phi^{2}G_{3,\mathcal{Y}}+\frac{3\phi^{2}\dot{H}G_{3,\mathcal{Y}}}{\dot{\phi}}-2\frac{k^{2}}{a^{2}}\phi\right)\,\delta\phi+\frac{k^{2}}{a^{2}}\phi^{2}G_{3,\mathcal{Y}}\psi_{\phi}-2\phi\frac{k^{2}}{a^{2}}\frac{\dot{\psi}_{\phi}}{a}$
(105)
$\displaystyle+3\phi^{3}G_{3,\mathcal{Y}}\frac{\dot{\Phi}}{a}+\left(6H\phi^{3}G_{3,\mathcal{Y}}+\frac{3\phi^{4}\dot{H}G_{3,\mathcal{Y}}}{\dot{\phi}}-4\frac{k^{2}}{a^{2}}\phi^{2}\right)\,\Psi-2\phi
f_{,\mathcal{E}}\left(\frac{k^{2}}{a^{2}}\psi_{\phi}+\frac{\phi}{a}\theta_{\rm
c}\right)+\delta\rho_{\rm m}+\delta\rho_{\rm c}=0\,,$ $\displaystyle E_{\chi}$
$\displaystyle\equiv$ $\displaystyle M_{\rm
P}^{2}k^{2}\left(2H\Psi+2\frac{\dot{\Phi}}{a}\right)-a(\rho_{\rm m}+P_{\rm
m})\theta_{\rm m}-a\left(\rho_{c}+\phi f_{,\mathcal{Z}}\right)\theta_{\rm
c}+k^{2}f_{,\mathcal{Z}}\psi_{\phi}+k^{2}\phi^{2}G_{3,\mathcal{Y}}\delta\phi+k^{2}\phi^{3}G_{3,\mathcal{Y}}\Psi$
(106)
$\displaystyle-k^{2}f_{,\mathcal{E}}\left(\phi\,\delta\phi+2\phi^{2}\Psi+\phi\frac{\dot{\psi}_{\phi}}{a}\right)=0\,,$
$\displaystyle E_{\zeta}$ $\displaystyle\equiv$ $\displaystyle 2M_{\rm
P}^{2}\left[\frac{\ddot{\Phi}}{a^{2}}+2H\frac{\dot{\Phi}}{a}+H\frac{\dot{\Psi}}{a}+\left(3H^{2}+\frac{2\dot{H}}{a}\right)\Psi+\frac{k^{2}}{a^{2}}\left(\frac{\Phi}{3}-\frac{\Psi}{3}\right)\right]+G_{3,\mathcal{Y}}\left(\phi^{2}\frac{\delta\dot{\phi}}{a}+\phi^{3}\frac{\dot{\Psi}}{a}\right)$
(107)
$\displaystyle+\left[G_{3,\mathcal{Y}}\left(3H\phi^{2}+2\phi\frac{\dot{\phi}}{a}\right)+\phi^{3}G_{3,\mathcal{YY}}\frac{\dot{\phi}}{a}\right]\left(\frac{\delta}{\phi}+\Psi\right)-\delta
P_{\rm m}=0\,.$

In the equation $E_{\alpha}$ we have used the equation $E_{\theta_{\rm c}}$ to
remove the term in $\dot{\delta}_{\rm c}$, which is proportional to
$f_{,\mathcal{E}}$. To arrive at the equation $E_{\chi}$ we have used
$E_{\theta_{\rm m}}$, $E_{\theta_{\rm c}}$, and the equation
$E_{\delta\phi_{s}}$ and, we have used the evolution of the velocity
divergence to arrive at the equation $E_{\zeta}$.

From the linear combination of $E_{\zeta}$ and $E_{E}$, i.e.

$E_{\rm shear}\equiv\frac{3}{k^{2}}E_{E}+E_{\zeta}=0\,,$ (108)

we can find the shear equation

$E_{\rm shear}\equiv M_{\rm
P}^{2}k^{2}(\Psi-\Phi)+\frac{3}{2}a^{2}(P_{r}+\rho_{r})\sigma_{r}=0\,.$ (109)

Now we have all the equations of motion given in the Newtonian gauge.

##  VI Quasi Static Approximation and effective gravitational couplings

In this section we look at the equations of motion in the quasi-static limit.
Then in the following subsection we find the effective gravitational coupling
for the dark matter and the baryon.

Let us start from the matter equation.

$\displaystyle E_{\theta_{\rm b}}$ $\displaystyle\equiv$
$\displaystyle\dot{\delta}_{\rm b}+\theta_{\rm b}=0\,,$ (110) $\displaystyle
E_{\delta_{\rm b}}$ $\displaystyle\equiv$ $\displaystyle\dot{\theta}_{\rm
b}+aH\theta_{\rm b}-k^{2}\Psi=0,\,$ (111)

where we have considered that the contribution of matter is from baryon, i.e.
$P_{\rm b}=0$. Taking a time derivative of the (110) and substituting (111),
we find the second order equation for the baryonic over density

$\ddot{\delta}_{\rm b}+aH\dot{\delta}_{\rm b}+k^{2}\Psi=0\,.$ (112)

Following the same procedure for the dark matter sector we have the continuity
equation

$\displaystyle E_{\theta_{\rm c}}$ $\displaystyle\equiv$
$\displaystyle\dot{\delta}_{\rm c}+\theta_{\rm c}\,,$ (113)

which is not affected by the interaction. We notice that in the evolution
equation of the velocity divergence $\theta_{c}$, there is a term proportional
to $\ddot{\psi}_{\phi}$, which is replaced with equation of motion for the
field $\psi_{\phi}$ Eq (104). After doing this replacement the equation
becomes

$\displaystyle\ddot{\delta}_{\rm
c}\left(1-\frac{\phi^{2}f_{,\mathcal{E}}}{\rho_{c}+\phi
f_{,\mathcal{Z}}}\right)-\dot{\psi}_{\phi}\left[\frac{k^{2}\left(\phi^{2}f_{,\mathcal{E}}G_{3,\mathcal{Y}}-\phi
f_{,\mathcal{E}}^{2}+f_{,\mathcal{Z}}\right)}{a\left(\phi
f_{,\mathcal{Z}}+\rho_{c}\right)}\right]+\Psi\frac{k^{2}}{\rho_{c}+\phi
f_{,\mathcal{Z}}}\left[-\rho_{c}+\phi^{3}G_{3,\mathcal{Z}}f_{,\mathcal{E}}\right]$
$\displaystyle-\psi_{\phi}\frac{k^{2}}{\rho_{c}+\phi
f_{,\mathcal{Z}}}\left[a\left(H\left(2\phi^{2}f_{,\mathcal{E}}G_{3,\mathcal{Y}}-3\phi
f_{,\mathcal{E}}^{2}+3f_{,\mathcal{Z}}+f_{,\mathcal{E}}\left(2\phi^{2}f_{,\mathcal{E}}G_{3,\mathcal{Y}}-\phi
f_{,\mathcal{E}}{}^{2}+f_{,\mathcal{Z}}+\phi^{3}\left(-G_{3,\mathcal{Y}}^{2}\right)\right)\right)\right.\right.$
$\displaystyle\left.+\dot{\phi}\left(\phi^{3}f_{,\mathcal{YE}}G_{3,\mathcal{Y}}+\phi^{2}f_{,\mathcal{EZ}}G_{3,\mathcal{Y}}-2\phi
f_{,\mathcal{E}}\left(f_{,\mathcal{EZ}}+\phi f_{,\mathcal{YE}}\right)+\phi
f_{,\mathcal{YZ}}-f_{,\mathcal{E}}^{2}+f_{,\mathcal{ZZ}}\right)\right]$
$\displaystyle-\dot{\delta}_{c}\frac{k^{2}}{\rho_{c}+\phi
f_{,\mathcal{Z}}}\left[a\left(\phi
f_{,\mathcal{E}}\left(\phi^{2}f_{,\mathcal{E}}G_{3,\mathcal{X}}-\phi
f_{,\mathcal{E}}^{2}+f_{,\mathcal{Z}}\right)+H\left(4\phi\left(f_{,\mathcal{Z}}-\phi
f_{,\mathcal{E}}^{2}\right)+\rho_{c}\right)\right)\right.$
$\displaystyle\left.-\dot{\phi}\left(-2\phi^{3}f_{,\mathcal{E}}f_{,\mathcal{YE}}+\phi^{2}\left(f_{,\mathcal{YZ}}-2f_{,\mathcal{E}}f_{,\mathcal{ZE}}\right)+\phi\left(f_{,\mathcal{ZZ}}-2f_{,\mathcal{E}}^{2}\right)+f_{,\mathcal{Z}}\right)\right]$
(114)

In the above expression we have used the fact that, from Eq. (109)

$\Psi=\Phi\,,$ (115)

since baryons do not have any shear. Eq. (103) in the quasi-static limit gives

$\delta\phi\simeq-2\phi\Psi+G_{3,\mathcal{Y}}\phi\psi_{\phi}-\frac{\dot{\psi}_{\phi}}{a}-f_{,\mathcal{E}}\left(\psi_{\phi}+\frac{a\phi\theta_{\rm
c}}{k^{2}}\right)$ (116)

$\theta_{\rm c}$ can be expressed in terms of $\dot{\delta}_{\rm c}$ using Eq.
(113).

Now we want to find the solution of the fields $\Psi(\simeq\Phi)$ and
$\psi_{\phi}$ in the quasi-static limit. We start from the Eq. (105),
$E_{\alpha}$, in the quasi-static limit

$\frac{k^{2}}{a^{2}}\left(-2M_{\rm
P}^{2}\Phi+2\phi\delta\phi+4\phi^{2}\Psi+2\frac{\dot{\phi}}{a}\delta\dot{\psi}_{\phi}-\phi^{2}G_{3,\mathcal{Y}}\right)+\left(2\frac{k^{2}}{a^{2}}\phi\psi_{\phi}+2\frac{\phi^{2}}{a}\theta_{\rm
c}\right)f_{,\mathcal{E}}-\rho_{\rm b}\delta_{\rm b}-\rho_{\rm c}\delta_{\rm
c}\simeq 0\,.$ (117)

Substituting the Eq. (116) to the above equation Eq. (117) we get

$-2M_{\rm
P}^{2}\frac{k^{2}}{a^{2}}\Phi+\frac{k^{2}}{a^{2}}\phi^{2}G_{3,\mathcal{Y}}\psi_{\phi}-\rho_{\rm
b}\delta_{\rm b}-\rho_{c}\delta_{\rm c}\simeq 0\,.$ (118)

We take a time derivative of the Eq. (118)

$\displaystyle 4M_{\rm P}^{2}\frac{k^{2}H\Phi}{a}+3aH\rho_{\rm b}\delta_{\rm
b}+3aH\rho_{c}\delta_{\rm c}-\rho_{\rm b}\dot{\delta}_{\rm
b}-\rho_{c}\dot{\delta}_{\rm c}-\frac{k^{2}}{2}M_{\rm
P}^{2}{a^{2}}\dot{\Phi}+2H\frac{k^{2}}{a^{2}}\phi^{2}G_{3,\mathcal{Y}}\dot{\psi}_{\phi}$
$\displaystyle\frac{k^{2}}{a^{2}}\left(-2aH\phi^{2}G_{3,\mathcal{Y}}+2\phi\dot{\phi}G_{3,\mathcal{Y}}+\phi^{3}\dot{\phi}G_{3,\mathcal{YY}}\right)\psi_{\phi}\simeq
0\,.$ (119)

From the the quasi-static limit of the Eq. (106), $E_{\chi}$, we get

$\displaystyle
k^{2}\left(f_{,\mathcal{E}}\left(-\frac{\phi\dot{\psi}_{\phi}}{a}-\delta\phi\phi-2\Psi\phi^{2}\right)-\psi_{\phi}f_{,\mathcal{Z}}+\delta\phi\phi^{2}\phi
f_{,\mathcal{Z}}-a\rho_{c}\right)+k^{2}M_{\rm
P}^{2}\left(\frac{2\dot{\Phi}}{a}+2H\Psi\right)-a\rho_{b}\theta_{b}\simeq
0\,,$ (120)

From the linear combination of Eq. (VI) and Eq. (120), and replacing
$\Psi(\simeq\Phi)$ with Eq. (118), $\theta_{\rm b}$ with Eq. (110), and
$\theta_{\rm c}$ Eq. (113) respectively we get

$\displaystyle\dot{\delta}_{c}\left(\frac{\phi^{3}f_{,\mathcal{E}}G_{3,\mathcal{Y}}}{a}-\frac{\phi^{2}f_{,\mathcal{E}}^{2}}{a}+\frac{\phi
f_{,\mathcal{Z}}}{a}\right)$
$\displaystyle+\psi_{\phi}\left(-\frac{2k^{2}\phi^{2}f_{,\mathcal{E}}G_{3,\mathcal{Y}}}{a^{2}}+\frac{k^{2}\phi
f_{,\mathcal{E}}^{2}}{a^{2}}-\frac{k^{2}f_{,\mathcal{Z}}}{a^{2}}+\frac{k^{2}\phi^{3}\dot{\phi}G_{3,\mathcal{Y}}}{a^{3}}+\frac{k^{2}H\phi^{2}G_{3,\mathcal{Y}}}{a^{2}}-\frac{k^{2}\phi^{5}G_{3,\mathcal{Y}}^{2}}{2M_{\rm
P}^{2}a^{2}}+\frac{2k^{2}\phi\dot{\phi}G_{3,\mathcal{Y}}}{a^{3}}+\frac{k^{2}\phi^{3}G_{3,\mathcal{Y}}^{2}}{a^{2}}\right)$
$\displaystyle+\frac{\delta_{c}\rho_{c}\phi^{3}G_{3,\mathcal{Y}}}{2M_{\rm
P}^{2}}+\frac{\delta_{b}\rho_{b}\phi^{3}G_{3,\mathcal{Y}}}{2M_{\rm
P}^{2}}\simeq 0\,.$ (121)

Hence, we can use Eq. (118) and Eq. (VI) to solve for $\Psi(\simeq\Phi)$ and
$\psi_{\phi}$, respectively.

###  VI.1 Effective gravitational coupling

The second order equation of motion for the baryonic over density Eq. (112)
substituting for $\Psi(\simeq\Phi)$ from Eq. (118), we get

$\ddot{\delta}_{\rm b}+aH\dot{\delta}_{\rm b}+a\Gamma_{\rm
bc}\dot{\delta}_{\rm c}-\frac{3}{2}\frac{H^{2}}{G}a^{2}(G_{\rm bb}\Omega_{\rm
b}\delta_{\rm b}+G_{\rm bc}\Omega_{\rm c}\delta_{\rm c})=0\,,$ (122)

where

$\displaystyle G_{\rm bb}/G=G_{\rm bc}/G$ $\displaystyle
1-\frac{a\phi^{5}G_{3,\mathcal{Y}}^{2}}{a\left(4M_{\rm
P}^{2}\phi^{2}f_{,\mathcal{E}}G_{3,\mathcal{Y}}-2M_{\rm P}^{2}\phi
f_{,\mathcal{E}}^{2}+2M_{\rm P}^{2}f_{,\mathcal{Z}}-2M_{\rm
P}^{2}H\phi^{2}G_{3,\mathcal{Y}}-2M_{\rm
P}^{2}\phi^{3}G_{3,\mathcal{Y}}^{2}+\phi^{5}G_{3,\mathcal{Y}}^{2}\right)-2M_{\rm
P}^{2}\phi\dot{\phi}\left(\phi^{2}G_{3,\mathcal{YY}}+2G_{3,\mathcal{Y}}\right)}\,,$

and

$\Gamma_{bc}\equiv\frac{a\phi^{3}G_{3,\mathcal{Y}}\left(\phi^{2}f_{\mathcal{E}}G_{3,\mathcal{Y}}-\phi
f_{\mathcal{E}}^{2}+f_{,\mathcal{Z}}\right)}{a\left(4M_{\rm
P}^{2}\phi^{2}f_{\mathcal{E}}G_{3,\mathcal{Y}}-2M_{\rm P}^{2}\phi
f_{\mathcal{E}}^{2}+2M_{\rm P}^{2}f_{,\mathcal{Z}}-2M_{\rm
P}^{2}H\phi^{2}G_{3,\mathcal{Y}}-2M_{\rm
P}^{2}\phi^{3}G_{3,\mathcal{Y}}^{2}+\phi^{5}G_{3,\mathcal{Y}}^{2}\right)-2M_{\rm
P}^{2}\phi\dot{\phi}\left(\phi^{2}G_{3,\mathcal{YY}}+2G_{3,\mathcal{Y}}\right)}$
(124)

In the above equation, we have replaced $\rho_{\rm b}$ with $(3H^{2}M_{\rm
P}^{2})\Omega_{\rm b}$ and $\rho_{c}$ with $(3H^{2}M_{\rm P}^{2})\Omega_{\rm
c}$. It is interesting to notice that when $G_{3,\mathcal{Y}}\rightarrow 0$
and $G_{3,\mathcal{YY}}\rightarrow 0$, $\Gamma_{\rm bc}=0$ and $G_{\rm
bb}=G_{\rm bc}=G$.

Now for the equation of motion for the density evolution of dark matter on
replacing $\Psi(\simeq\Phi)$ by Eq. (118), $\psi_{\phi}$ by Eq. (VI), and
$\dot{\psi}_{\phi}$ by time derivative of Eq. (VI), we get

$\displaystyle\ddot{\delta}_{\rm c}+a\Gamma_{\rm cc}\,\dot{\delta}_{\rm
c}+a\Gamma_{\rm cb}\,\dot{\delta}_{\rm
b}-\frac{3}{2}\frac{H^{2}}{G}a^{2}(G_{\rm cb}\Omega_{\rm b}\delta_{\rm
b}+G_{\rm cc}\Omega_{\rm c}\delta_{\rm c})\,,$ (125)

The expression for $G_{\rm cc}(=G_{\rm cb})$ and $\Gamma_{\rm cc}$ and
$\Gamma_{\rm cb}$ are given in the appendix.

##  VII Concrete model

Until now the interaction term that we considered in the action is completely
general, that is, the interaction is introduced in the Lagrangian as a general
function $f(\mathcal{Y,Z,E})$. In order to study phenomenological consequences
of these interactions, we consider a concrete model given by

$f(\mathcal{Y,Z,E})\equiv
b_{2}\mathcal{Y}^{p_{2}}+\beta\mathcal{Z}+\gamma\mathcal{E}\qquad
G_{3}(\mathcal{Y})\equiv g_{3}\mathcal{Y}^{p_{3}}\,.$ (126)

After choosing the concrete model, we choose $\beta=0$ to see the effect of
the new interaction $\mathcal{E}$.

###  VII.1 Background and ghost conditions

Introducing the density parameters $\Omega_{\rm DE}$, $\Omega_{r}$,
$\Omega_{c}$, and $\Omega_{b}$, the first Friedmann equation Eq. (44) can be
rewritten as

$1-\Omega_{r}-\Omega_{c}-\Omega_{b}-\Omega_{\rm DE}=0\,,$ (127)

where

$\Omega_{i}\equiv\frac{\rho_{i}}{3M_{\rm P}^{2}H^{2}}\,.$ (128)

From the first Friedmann equation we get the energy density of the dark energy
applying the concrete model

$\rho_{\rm DE}=-2^{p_{2}}b_{2}\phi^{2p_{2}}\,.$ (129)

The constraint equation of background, Eq. (48), can be written as

$2^{1-p_{2}}b_{2}p_{2}\phi^{1+2(p_{2}-1)}+3\times
2^{1-p_{3}}g_{3}p_{3}H\phi^{2p_{3}}=0\,.$ (130)

In this concrete model, the relation between the Hubble expansion $H$ and
$\phi$ is given by

$\phi^{p}H=\lambda=\text{constant}\,,$ (131)

where the relation between the powers are given by

$p=2p_{3}-2p_{2}+1\,.$ (132)

Now, taking the time derivative of Eq (48), we get

$\phi^{2}\left(\dot{\phi}f_{,\mathcal{YY}}+3\dot{H}G_{3,\mathcal{Y}}\right)+\phi\dot{\phi}\left(2f_{,\mathcal{YZ}}+3HG_{3,\mathcal{Y}}\right)+\dot{\phi}f_{,\mathcal{ZZ}}-\frac{\dot{\phi}f_{,\mathcal{Z}}}{\phi}+3H\phi^{3}\dot{\phi}G_{3,\mathcal{YY}}=0\,.$
(133)

In the concrete model given by Eq. (126), we get

$\dot{\phi}=-\frac{aH(-3+3\Omega_{DE}-\Omega_{r})\phi}{2(-2+p+2ps+ps\Omega_{\rm
DE})}\,.$ (134)

The above relation can be used to replace $\dot{\phi}$ in the expressions.

Finally, with these concrete model, we can write

$\phi=\phi_{0}\left(\frac{\Omega_{\rm DE}}{\Omega_{{\rm
DE},0}}\right)^{1/[2p(1+s)]}\,,\qquad H=H_{0}\left(\frac{\Omega_{{\rm
DE},0}}{\Omega_{\rm DE}}\right)^{1/[2(1+s)]}\,.$ (135)

The evolution of the density parameters $\Omega_{\rm DE}$, $\Omega_{r}$ and
$\Omega_{\rm c}$ can be found as

$\displaystyle\Omega^{\prime}_{\rm DE}$ $\displaystyle=$
$\displaystyle\frac{(1+s)\Omega_{\rm DE}(3-3\Omega_{\rm
DE}+\Omega_{r})}{1+s\Omega_{\rm DE}}\,,$ (136)
$\displaystyle\Omega^{\prime}_{r}$ $\displaystyle=$
$\displaystyle\frac{(1+(3+4s))\Omega_{\rm
DM}-\Omega_{r})\Omega_{r}}{1+s\Omega_{\rm DE}}\,,$ (137)
$\displaystyle\Omega^{\prime}_{c}$ $\displaystyle=$
$\displaystyle-\frac{(3(1+s)\Omega_{\rm
DE}-\Omega_{r})\Omega_{c}}{1+s\Omega_{\rm DE}}\,,$ (138)

where the prime denotes the derivative with respect to the variable
$\mathcal{N}\equiv\ln(a)$, $s=p_{2}/p$.. The evolution of the background
density parameters are shown in figure (1). The above evolution equation is
first derived in [46].

From the background equation of motion, we can write the equation of state for
the dark energy is,

$w_{\rm DE}=-\frac{3+3s+s\Omega_{r}}{3+3s\Omega_{\rm DE}}\,.$ (139)

The behavior of $w_{\rm DE}$ is given in the figure (2).

Figure 1: Evolution of $\Omega_{\rm DE}$, $\Omega_{r}$, $\Omega_{c}$,
$\Omega_{b}$. Here we chose the value of $s=0.2$. As for the initial condition
we have chosen $\Omega_{{\rm DE},0}=0.68$, $\Omega_{r,0}=10^{-4}$
$\Omega_{c,0}=0.27$, and $\Omega_{b,0}=0.05$, at the redshift $z=0$.  Figure
2: Behaviour of $w_{\rm DE}$. Here also we chose the value of $s=0.2$, and we
use same initial condition to solve the density parameter differential
equations, $\Omega_{{\rm DE},0}=0.68$, $\Omega_{r,0}=10^{-4}$
$\Omega_{c,0}=0.27$, and $\Omega_{b,0}=0.05$, at the redshift $z=0$.

The ghost condition for the dynamical field $\psi_{\phi}$, Eq. (76), becomes

$q_{s}=\frac{3M_{\rm P}^{2}p^{2}sH^{2}\Omega_{\rm DE}(1+s\Omega_{\rm
DE})}{(1-ps\Omega_{\rm DE})^{2}\phi^{2}}\,,$ (140)

and the speed of propagation of the longitudinal mode given by Eq. (96)
becomes

$\displaystyle\hat{c}_{s}^{2}$ $\displaystyle=$
$\displaystyle\frac{2p^{2}s^{2}\Omega_{\rm DE}^{2}\left(2M_{\rm
P}^{2}-\phi^{2}\right)+\Omega_{\rm DE}\left(4M_{\rm
P}^{2}p^{2}s-\left(2p^{2}s+4ps+3p-3\right)\phi^{2}\right)+(p(6s+5)-3)\phi^{2}}{6p^{2}\phi^{2}(s\Omega_{\rm
DE}+1)^{2}}$ (141) $\displaystyle+\frac{\Omega_{r}(2ps+p-1)}{6(ps\Omega_{\rm
DE}+p)^{2}}+\frac{\gamma^{2}\phi^{2}}{6M_{\rm P}^{2}p^{2}sH^{2}\Omega_{\rm
DE}(s\Omega_{\rm DE}+1)}-\frac{2\gamma}{3psH\Omega_{\rm DE}+3pH}.$

The ghost condition for the cold dark matter content is given by Eq. (84) as

$q_{c}=1-\frac{\gamma^{2}\phi^{2}}{3M_{\rm P}^{2}H^{2}\Omega_{c}}\,.$ (142)

We introduce a parameter

$\Omega_{\gamma}\equiv\frac{\gamma^{2}\phi^{2}}{3M_{\rm P}^{2}H^{2}}\,,$ (143)

such that the the ghost condition for the cold dark matter becomes

$q_{c}=\frac{\Omega_{\gamma}}{\Omega_{c}}\,,$ (144)

We introduce another redefinition for the parameter

$r_{\gamma}\equiv\frac{\Omega_{\gamma}}{\Omega_{\rm DE}}\,,$ (145)

which is the ratio between $\Omega_{\gamma}$ and the dark energy density
parameter. With these parameter redefinitions we study the behavior of the
ghost conditions and speed of propagation of the scalar mode and is shown in
the figure (3).

Figure 3: The behaviour of $Q_{\psi_{\phi}}\equiv\mathbf{K}_{11}M_{\rm
P}^{2p}/\phi^{2p}H^{2}$, $q_{c}$, and $c_{s}{}^{2}$. We chose the following
values of the parameter $r_{\gamma}=0.2$, $s=0.2$, $p=5$, $\phi_{0}=0.459$,
where $\phi_{0}$ is the value of the field $\phi$ at redshift $z=0$.

###  VII.2 Dynamics of perturbation

The effect of the interaction $\mathcal{Z}$ was demonstrated in [50]. It has
shown that $\mathcal{Z}$ interaction can suppress the effective Newton
constant, $G_{\rm eff}$. We have also seen that the new interaction that we
have studied in this work can also modify $G_{\rm eff}$. Now we want to see
the phenomenological effect of the interaction $\mathcal{E}$. The behavior of
these quantities in the concrete model is shown in figure (4). We can see that
there is a suppression in $G_{\rm cc}$ and $G_{cb}$.

Figure 4: Behaviour of $G_{\rm cb}$, $G_{\rm cc}$. We chose the following
values of the parameter $r_{\gamma}=0.2$, $s=0.2$, $p=5$, $\Omega_{\rm
DE}=0.68$, $\phi_{0}=0.459$, where $\phi_{0}$ is the value of the field $\phi$
at redshift $z=0$.

We study the evolution of the baryon and cold dark matter overdensities
through the evolution of the universe. This is shown in the figure (5). We can
see that there is small suppression in the growth of cold dark matter. The
total matter evolution given by

$\delta_{\rm
M}=\frac{\Omega_{b}\delta_{b}}{(\Omega_{b}+\Omega_{c})}+\frac{\Omega_{c}\delta_{c}}{(\Omega_{b}+\Omega_{c})}\,,$
(146)

is also shown in the figure. To solve the second order equations of motion, we
convert the time variable to $\mathcal{N}=\ln{a}$. Then we give the initial
conditions $\delta_{c}=\delta_{c}^{\prime}=0.024$, and
$\delta_{b}=\delta_{b}^{\prime}=0.024$, at $\mathcal{N}=-3.7$.

Figure 5: This is the result of numerically solving the equations Eq (90) and
Eq (125). The red line shows the evolution of the baryon overdensity
$\delta_{b}$. The dashed blue line shows the evolution of the cold dark matter
overdensity $\delta_{\rm c}$. The green line shows the evolution of the total
matter overdensity $\delta_{\rm M}$. It is clear that the evolution of the
cold dark matter and total matter overdensities are suppressed compared with
the baryon overdensity. Here, we chose the following values of the parameter
$r_{\gamma}=0.2$, $s=0.2$, $p=5$, $\Omega_{\rm DE}=0.68$, $\phi_{0}=0.459$,
where $\phi_{0}$ is the value of the field $\phi$ at redshift $z=0$.

.

##  VIII Conclusion

In this work, we have constructed a theory of interacting dark energy and dark
matter fluid (Section II). The dark energy is described by a massive vector
field, and the dark matter is described by a perfect fluid. We found a new
interaction, $\mathcal{E}=A^{\alpha}F_{\alpha\beta}u^{\beta}$, which have not
been explored. On top of this interaction, we also find the interactions that
break the parity symmetry. On the basis of the parity-preserving interaction
terms, we construct a general Lagrangian in Section III. Then, we introduced a
toy model to understand the consequence of newly introduced interactions,
where the dark energy action is a cubic generalised Proca theory.

Then we studied the background evolution and linear perturbation equations of
motion in Section IV. We also derived the no ghost condition for the tensor
and scalar mode. Due to the term $\nabla_{\mu}Kn^{\mu}$ in the dark matter
action, this model does not modify the continuity equation, but it modifies
the Euler equation. This indicates that the interaction introduces only a
momentum transfer. Then in Section V, we derived the linear equations of
motion in Newtonian gauge.

To understand the phenomenological features of the theory, we study the
evolution of baryon and cold dark matter in the high $k$ limit in Section VI.
We show that the evolution of cold dark matter is modified as a result of the
interaction. In particular the effective gravitational coupling to the cold
dark matter $G_{\rm cc}$ is modified. In the limit of $G_{3}\rightarrow 0$, we
do not have any modification. For the evolution of baryons, we show that there
are modification similar to that of cold dark matter, particularly effective
gravitational coupling $G_{\rm bb}$ is modified.

We also introduced a simple concrete model for the free function $f$
introduced in the Lagrangian in Section VII, and studied the ghost conditions
and speed of propagation of the dark energy scalar mode. We also showed the
behavior of the modified effective gravitational couplings $G_{\rm bb}$ and
$G_{\rm cc}$. We showed that the effective gravitational coupling $G_{\rm cc}$
was reduced. We also numerically solved the evolution equation of baryon and
cold dark matter over-densities and showed that the evolution of the cold dark
matter was suppressed due to the interaction. The result is qualitatively
similar to [50], however, to understand quantitative differences, we need to
confront this model with cosmological observations.

There are several future directions based on the current work. It is also
interesting to study the implications of the interaction between the vector
dark energy and dark matter with parity-violating interacting terms. Even
though we have shown in this work that there is a suppression of the growth of
cold dark matter over-density in this model, we have to confront this model
with cosmological observations to constrain the parameters. It will also be
interesting to see if this model can address the Hubble and large-scale
structure $S_{8}$ tension simultaneously, modeling this as the early
interacting dark sector. Another interesting direction is to study non-linear
cosmology, which is important in the light of the upcoming cosmological
surveys like Euclid [63], Square Kilometer Array [64], and Rubin observatory’s
LSST [65]. It is also interesting to find new interactions with vector dark
energy models with a disformal coupling to dark matter [66, 67].

###### Acknowledgements.

M.C.P. acknowledges the Mahidol University International Postdoctoral
Fellowship. M.C.P. is also partially supported by Fundamental Fund: fiscal
year 2024 by National Science Research and Innovation Fund (NSRF). M.C.P. also
acknowledges Shinji Tsujikawa for fruitful discussions. K. K. is supported by
STFC grant ST/W001225/1. For the purpose of open access, we have applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising. Supporting research data are available on request from the
corresponding author.

##  Appendix A Equations for fluid

The dark matter action is described by

$S_{\rm fluid}=-\int{\rm
d}^{4}x\sqrt{-g}\left[\rho_{c}(-n_{\mu}n^{\mu})+n^{\mu}\nabla_{\mu}K\right]\,,$
(147)

where $K$ is a Lagrangian multiplier imposing the condition

$\nabla_{\mu}n^{\mu}=0\,.$ (148)

The equation of motion for the field $n^{\mu}$ is given by

$\nabla_{\mu}K-2n_{\mu}\frac{\partial\rho_{c}\bigl{(}\tilde{n}\bigr{)}}{\partial\tilde{n}}.$
(149)

The variation with respect to $g^{\alpha\beta}$ gives the energy-momentum
tensor

$T_{\alpha\beta}\equiv\frac{2}{\sqrt{-g}}\frac{\delta S_{\rm fluid}}{\delta
g^{\alpha\beta}}=g_{\alpha\beta}\rho_{c}-2n_{\alpha}n_{\beta}\frac{\partial\rho_{c}}{\partial\tilde{n}}+2g_{\alpha\beta}n_{\gamma}n^{\gamma}\frac{\partial\rho_{c}}{\partial\tilde{n}}\,.$
(150)

The above expression for $T_{\alpha\beta}$ agrees with $T_{\alpha\beta}$
derived from the action without the constraint term.

Assuming the homogeneous and isotropic background, we define

$K=K_{0}(t)+\delta K(t,x)\,,$ (151)

and $n^{\alpha}$ is defined by (38). Now we expand the action up to linear
order to find the background equations of motion. The background equation of
motion for $\delta K$ sets

$n_{0}=\frac{N}{a^{3}}\,,$ (152)

and equation of motion for $\delta n_{0}$ gives

$K_{0}=-2\int a(\eta)n_{0}(\eta)\frac{\partial\rho_{c}}{\partial\tilde{n}}{\rm
d}\eta\,.$ (153)

To get the equations of motion for perturbation, we expand the action up to
the second order. We make the following field redefinition

$\displaystyle\delta n_{0}$ $\displaystyle=$ $\displaystyle-\alpha
n_{0}+\frac{\rho_{c}}{2n_{0}\frac{\partial\rho_{c}}{\partial\tilde{n}}}\delta\,.$
(154)

The equation of motion for $n_{s}$ is given by

$\delta
K=\frac{2N}{a^{3}}\frac{\partial\rho_{c}}{\partial\tilde{n}}\chi-\frac{2N}{a^{2}}\frac{\partial\rho_{c}}{\partial\tilde{n}}\frac{\theta}{k^{2}}\,,$
(155)

where we used a field redefinition

$n_{s}=-\frac{N}{a^{3}}\frac{a}{k^{2}}\theta\,.$ (156)

The remaining fields are $\delta$ and $\theta$. The equation of motion for
$\delta$ is given by

$E_{\delta}\equiv\dot{\theta}+aH(1-3c_{s}^{2})\theta-\frac{c_{s}^{2}k^{2}\rho}{\rho+P}\delta+\frac{k^{2}}{a}\dot{\chi}+3Hk^{2}c_{s}^{2}\chi-k^{2}\alpha=0\,.$
(157)

Finally the equation of motion for the field $\theta$ is obtained as

$E_{\theta}\equiv\dot{\delta}+3aH\left(c_{s}^{2}-\frac{P}{\rho}\right)\delta+\left(1+\frac{P}{\rho}\right)\theta-k^{2}\left(1+\frac{P}{\rho}\right)\partial_{t}\left(E/a^{2}\right)+3\left(1+\frac{P}{\rho}\right)\dot{\zeta}=0\,.$
(158)

Here we used the following definitions in the above expressions

$\displaystyle c_{s}^{2}$ $\displaystyle\equiv$ $\displaystyle\frac{\partial
P}{\partial\rho}=\frac{\dot{P}}{\dot{\rho}}=1+\frac{2N^{2}\rho_{c}^{\prime\prime}}{a^{6}\rho_{c}^{\prime}}\,.$
(159)

##  Appendix B Expression for $G_{\rm cc}$, $\Gamma_{\rm cc}$, and
$\Gamma_{\rm cb}$

Here we give the full lengthy expression for $G_{\rm cc}$, $\Gamma_{\rm cc}$,
and $\Gamma_{\rm cb}$

$\displaystyle G_{\rm
cc}\equiv\frac{\Delta_{1}}{\Delta_{2}}G\,\qquad\Gamma_{\rm
cc}\equiv\frac{\Xi_{1}}{\Xi_{2}}\qquad\Gamma_{\rm
cb}\equiv\frac{\Upsilon_{1}}{\Upsilon_{2}}$ (160)

Since the expression for $\Xi_{1}$, and $\Xi_{2}$ is very large, the blow
expression is with the limit $f_{\mathcal{Z}}\rightarrow 0$,
$f_{\mathcal{ZZ}}\rightarrow 0$, $f_{\mathcal{YZ}}\rightarrow 0$ ,
$f_{\mathcal{ZE}}\rightarrow 0$.

$\displaystyle\Delta_{1}$ $\displaystyle\equiv$ $\displaystyle 2M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{4}\dot{\phi}f_{,\mathcal{YE}}\phi^{11}+2M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{2}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{10}+4M_{\rm
P}^{2}a^{2}HG_{3,\mathcal{Y}}^{4}f_{,\mathcal{E}}\phi^{10}+2M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{4}\dot{\phi}f_{,\mathcal{ZE}}\phi^{10}$ (161)
$\displaystyle-4M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{10}-4M_{\rm
P}^{2}a^{2}HG_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}^{2}\phi^{9}-8M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{,\mathcal{YY}}^{2}f_{,\mathcal{E}}\phi^{9}+4M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}G_{3,\mathcal{YYY}}f_{,\mathcal{E}}\phi^{9}$
$\displaystyle-2M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{2}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{Z}}\phi^{9}-4M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{ZE}}\phi^{9}-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{4}\dot{\phi}f_{,\mathcal{YE}}\phi^{9}-4M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}G_{,\mathcal{YY}}f_{,\mathcal{YE}}\phi^{9}$
$\displaystyle+2M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{YZ}}\phi^{9}-6M_{\rm
P}^{4}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{4}\phi^{8}+4M_{\rm
P}^{4}\dot{\phi}^{2}G_{,\mathcal{YY}}^{2}f_{,\mathcal{E}}^{2}\phi^{8}+2M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{8}+4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{8}$
$\displaystyle-4M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YYY}}f_{,\mathcal{E}}^{2}\phi^{8}-6M_{\rm
P}^{4}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}G_{,\mathcal{YY}}\phi^{8}-8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{4}f_{,\mathcal{E}}\phi^{8}-16M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{2}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{E}}\phi^{8}$
$\displaystyle+4M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}G_{,\mathcal{YY}}\dot{\phi}^{\prime}f_{,\mathcal{E}}\phi^{8}+4M_{\rm
P}^{2}a^{2}HG_{3,\mathcal{Y}}^{3}f_{,\mathcal{Z}}\phi^{8}-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{4}\dot{\phi}f_{,\mathcal{ZE}}\phi^{8}-4M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}G_{,\mathcal{YY}}f_{,\mathcal{ZE}}\phi^{8}+2M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{ZZ}}\phi^{8}$
$\displaystyle-4M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{YE}}\phi^{8}+8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{8}+8M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{,\mathcal{YY}}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{8}-6M_{\rm
P}^{4}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{3}\phi^{7}$
$\displaystyle-8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{E}}^{3}\phi^{7}+24M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}^{2}\phi^{7}+12M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{7}-4M_{\rm
P}^{4}G_{3,\mathcal{Y}}G_{,\mathcal{YY}}\ddot{\phi}f_{,\mathcal{E}}^{2}\phi^{7}$
$\displaystyle-8M_{\rm
P}^{4}a^{2}H^{2}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}\phi^{7}+12M_{\rm
P}^{4}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}\phi^{7}+4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{3}\dot{H}f_{,\mathcal{E}}\phi^{7}-8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{4}\dot{\phi}f_{,\mathcal{E}}\phi^{7}-16M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}G_{,\mathcal{YY}}f_{,\mathcal{E}}\phi^{7}$
$\displaystyle-4M_{\rm
P}^{4}\dot{\phi}^{2}G_{,\mathcal{YY}}^{2}f_{,\mathcal{Z}}\phi^{7}-4M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{Z}}\phi^{7}+4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{Z}}\phi^{7}+4M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YYY}}f_{,\mathcal{Z}}\phi^{7}$
$\displaystyle-4M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{ZE}}\phi^{7}+8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{ZE}}\phi^{7}+8M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{,\mathcal{YY}}f_{,\mathcal{E}}f_{,\mathcal{ZE}}\phi^{7}-8M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{3}\dot{\phi}^{2}f_{,\mathcal{YE}}\phi^{7}$
$\displaystyle-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{2}f_{,\mathcal{YE}}\phi^{7}+8M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{7}-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{YZ}}\phi^{7}-4M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{,\mathcal{YY}}f_{,\mathcal{YZ}}\phi^{7}$
$\displaystyle+12M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{4}\phi^{6}+4M_{\rm
P}^{4}a\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{E}}^{4}\phi^{6}-24M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{3}\phi^{6}+12M_{\rm
P}^{6}H^{2}\Omega_{c}\dot{\phi}^{2}G_{,\mathcal{YY}}^{2}\phi^{6}+8M_{\rm
P}^{4}a^{2}H^{2}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{6}$
$\displaystyle-6M_{\rm
P}^{4}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{6}-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{H}f_{,\mathcal{E}}^{2}\phi^{6}+28M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{6}+4M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{6}$
$\displaystyle-12M_{\rm
P}^{4}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}\dot{\phi}\phi^{6}+24M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}G_{,\mathcal{YY}}\phi^{6}-36M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}\phi^{6}+8M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{3}\dot{\phi}^{\prime}f_{,\mathcal{E}}\phi^{6}$
$\displaystyle-8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{3}f_{,\mathcal{Z}}\phi^{6}-12M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{Z}}\phi^{6}+4M_{\rm
P}^{4}G_{3,\mathcal{Y}}G_{,\mathcal{YY}}\dot{\phi}^{\prime}f_{,\mathcal{Z}}\phi^{6}+8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{E}}f_{,\mathcal{Z}}\phi^{6}$
$\displaystyle-8M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{3}\dot{\phi}^{2}f_{,\mathcal{ZE}}\phi^{6}-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{2}f_{,\mathcal{ZE}}\phi^{6}+8M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{ZE}}\phi^{6}-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{ZZ}}\phi^{6}$
$\displaystyle-4M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{,\mathcal{YY}}f_{,\mathcal{ZZ}}\phi^{6}+16M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{6}-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{Z}}f_{,\mathcal{YE}}\phi^{6}-4M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{YZ}}\phi^{6}$
$\displaystyle+4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{YZ}}\phi^{6}+8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}f_{,\mathcal{E}}^{4}\phi^{5}+24M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{3}\phi^{5}-32M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{3}\phi^{5}+32M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{5}$
$\displaystyle-8M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\ddot{\phi}f_{,\mathcal{E}}^{2}\phi^{5}+24M_{\rm
P}^{6}aH^{3}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}G_{,\mathcal{YY}}\phi^{5}-48M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}\phi^{5}-32M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{3}\dot{\phi}^{2}f_{,\mathcal{E}}\phi^{5}$
$\displaystyle-48M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{E}}\phi^{5}-8M_{\rm
P}^{4}a^{2}H^{2}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{Z}}\phi^{5}+6M_{\rm
P}^{4}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{Z}}\phi^{5}-8M_{\rm
P}^{4}a\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{E}}^{2}f_{,\mathcal{Z}}\phi^{5}$
$\displaystyle+4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{H}f_{,\mathcal{Z}}\phi^{5}+24M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}f_{,\mathcal{Z}}\phi^{5}+16M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{E}}f_{,\mathcal{ZE}}\phi^{5}-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{Z}}f_{,\mathcal{ZE}}\phi^{5}$
$\displaystyle-4M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{ZZ}}\phi^{5}+4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{ZZ}}\phi^{5}-8M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{YZ}}\phi^{5}+12M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{E}}^{4}\phi^{4}+12M_{\rm
P}^{6}a^{2}H^{4}\Omega_{c}G_{3,\mathcal{Y}}^{2}\phi^{4}$
$\displaystyle+72M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{4}+24M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{E}}^{2}\phi^{4}+24M_{\rm
P}^{6}aH^{2}\Omega_{c}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{4}+4M_{\rm
P}^{4}a\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{Z}}^{2}\phi^{4}$
$\displaystyle+48M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}\dot{\phi}\phi^{4}+48M_{\rm
P}^{6}H^{2}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{,\mathcal{YY}}\phi^{4}-48M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}\phi^{4}-16M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}f_{,\mathcal{E}}^{2}f_{,\mathcal{Z}}\phi^{4}$
$\displaystyle-28M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{Z}}\phi^{4}+8M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{\prime}f_{,\mathcal{Z}}\phi^{4}+28M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{Z}}\phi^{4}-8M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{ZZ}}\phi^{4}$
$\displaystyle-48M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{E}}^{3}\phi^{3}+24M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{E}}^{2}\phi^{3}+8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}f_{,\mathcal{Z}}^{2}\phi^{3}+48M_{\rm
P}^{6}aH^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}\phi^{3}$
$\displaystyle-96M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}\phi^{3}-24M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{Z}}\phi^{3}-16M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{Z}}\phi^{3}$
$\displaystyle-24M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{E}}^{2}f_{,\mathcal{Z}}\phi^{3}-24M_{\rm
P}^{6}aH^{2}\Omega_{c}\dot{\phi}G_{,\mathcal{YY}}f_{,\mathcal{Z}}\phi^{3}+12M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}f_{,\mathcal{E}}^{4}\phi^{2}+48M_{\rm
P}^{6}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}\phi^{2}$
$\displaystyle+48M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{2}+12M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{Z}}^{2}\phi^{2}-24M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{Z}}\phi^{2}+48M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{E}}f_{,\mathcal{Z}}\phi^{2}$
$\displaystyle-24M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}f_{,\mathcal{E}}^{2}f_{,\mathcal{Z}}\phi-48M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{Z}}\phi+12M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}f_{,\mathcal{Z}}^{2}$ $\displaystyle\Delta_{2}$
$\displaystyle\equiv$
$\displaystyle-a^{2}G_{3,\mathcal{Y}}^{4}f_{,\mathcal{E}}^{2}\phi^{12}+a^{2}G_{3,\mathcal{Y}}^{4}f_{3,\mathcal{Z}}\phi^{11}+3M_{\rm
P}^{2}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{4}\phi^{10}+2M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{4}f_{,\mathcal{E}}^{2}\phi^{10}+4M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{10}$
(162) $\displaystyle-4M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}^{3}\phi^{9}+4M_{\rm
P}^{2}a^{2}HG_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}^{2}\phi^{9}-4M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{4}f_{3,\mathcal{Z}}\phi^{9}-4M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{Z}}\phi^{9}$
$\displaystyle-12M_{\rm
P}^{4}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{4}\phi^{8}+2M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{4}\phi^{8}-4M_{\rm
P}^{4}\dot{\phi}^{2}G_{3,\mathcal{YY}}^{2}f_{,\mathcal{E}}^{2}\phi^{8}+8M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{8}$
$\displaystyle-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{8}-12M_{\rm
P}^{4}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}\phi^{8}-4M_{\rm
P}^{2}a^{2}HG_{3,\mathcal{Y}}^{3}f_{,\mathcal{Z}}\phi^{8}+4M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}f_{3,\mathcal{Z}}\phi^{8}$
$\displaystyle-12M_{\rm
P}^{4}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{3}\phi^{7}+8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{3}\phi^{7}-4M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}^{2}\phi^{7}-8M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{7}$
$\displaystyle+24M_{\rm
P}^{4}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}\phi^{7}+4M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}^{4}f_{3,\mathcal{Z}}\phi^{7}+4M_{\rm
P}^{4}\dot{\phi}^{2}G_{3,\mathcal{YY}}^{2}f_{3,\mathcal{Z}}\phi^{7}-4M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}f_{,\mathcal{Z}}\phi^{7}$
$\displaystyle-8M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{Z}}\phi^{7}+8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}f_{3,\mathcal{Z}}\phi^{7}+12M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{4}\phi^{6}-4M_{\rm
P}^{4}a\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{4}\phi^{6}$
$\displaystyle+8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{3}\phi^{6}+12M_{\rm
P}^{6}H^{2}\Omega_{c}\dot{\phi}^{2}G_{3,\mathcal{YY}}^{2}\phi^{6}-4M_{\rm
P}^{4}a^{2}H^{2}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{6}-12M_{\rm
P}^{4}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{6}$
$\displaystyle-8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{6}-16M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{6}+2M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{2}f_{3,\mathcal{Z}}^{2}\phi^{6}-24M_{\rm
P}^{4}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}\dot{\phi}\phi^{6}$
$\displaystyle+24M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}\phi^{6}+8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{3}f_{3,\mathcal{Z}}\phi^{6}+8M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{3,\mathcal{Z}}\phi^{6}-8M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}f_{3,\mathcal{Z}}\phi^{6}$
$\displaystyle-8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}f_{,\mathcal{Z}}\phi^{6}-4M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}f_{,\mathcal{E}}^{4}\phi^{5}+24M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{3}\phi^{5}+16M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{3}\phi^{5}$
$\displaystyle-16M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{5}+24M_{\rm
P}^{6}aH^{3}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}\phi^{5}-48M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}\phi^{5}$
$\displaystyle-48M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}\phi^{5}+4M_{\rm
P}^{4}a^{2}H^{2}G_{3,\mathcal{Y}}^{2}f_{3,\mathcal{Z}}\phi^{5}+12M_{\rm
P}^{4}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{3,\mathcal{Z}}\phi^{5}+4M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}f_{3,\mathcal{Z}}\phi^{5}$
$\displaystyle+8M_{\rm
P}^{4}a\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}f_{,\mathcal{Z}}\phi^{5}+16M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{Z}}\phi^{5}+16M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YY}}f_{3,\mathcal{Z}}\phi^{5}-8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}f_{3,\mathcal{Z}}\phi^{5}$
$\displaystyle-8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{E}}^{4}\phi^{4}+12M_{\rm
P}^{6}a^{2}H^{4}\Omega_{c}G_{3,\mathcal{Y}}^{2}\phi^{4}+72M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{4}-16M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{E}}^{2}\phi^{4}$
$\displaystyle+24M_{\rm
P}^{6}aH^{2}\Omega_{c}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{4}-4M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}^{2}f_{3,\mathcal{Z}}^{2}\phi^{4}-4M_{\rm
P}^{4}a\dot{\phi}G_{3,\mathcal{YY}}f_{3,\mathcal{Z}}^{2}\phi^{4}+48M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}\dot{\phi}\phi^{4}$
$\displaystyle+48M_{\rm
P}^{6}H^{2}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YY}}\phi^{4}-48M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}\phi^{4}+8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}f_{,\mathcal{E}}^{2}f_{3,\mathcal{Z}}\phi^{4}+16M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{3,\mathcal{Z}}\phi^{4}$
$\displaystyle-16M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{Z}}\phi^{4}-48M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{E}}^{3}\phi^{3}+24M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{E}}^{2}\phi^{3}-4M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}f_{3,\mathcal{Z}}^{2}\phi^{3}$
$\displaystyle+48M_{\rm
P}^{6}aH^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}\phi^{3}-96M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}\phi^{3}-24M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{3,\mathcal{Z}}\phi^{3}+16M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{3,\mathcal{Z}}\phi^{3}$
$\displaystyle+16M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{E}}^{2}f_{3,\mathcal{Z}}\phi^{3}-24M_{\rm
P}^{6}aH^{2}\Omega_{c}\dot{\phi}G_{3,\mathcal{YY}}f_{3,\mathcal{Z}}\phi^{3}+12M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}f_{,\mathcal{E}}^{4}\phi^{2}+48M_{\rm
P}^{6}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}\phi^{2}$
$\displaystyle+48M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{2}-8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}f_{3,\mathcal{Z}}^{2}\phi^{2}-24M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{Z}}\phi^{2}+48M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{E}}f_{,\mathcal{Z}}\phi^{2}$
$\displaystyle-24M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}f_{,\mathcal{E}}^{2}f_{3,\mathcal{Z}}\phi-48M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}f_{3,\mathcal{Z}}\phi+12M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}f_{,\mathcal{Z}})^{2}$

$\displaystyle\Upsilon_{1}$ $\displaystyle\equiv$ $\displaystyle 3M_{\rm
P}^{2}aH^{2}\Omega_{b}\phi^{5}f_{,\mathcal{E}}G_{3,\mathcal{Y}}^{2}-3M_{\rm
P}^{2}aH^{2}\Omega_{b}\phi^{4}f_{,\mathcal{E}}^{2}G_{3,\mathcal{Y}}+3M_{\rm
P}^{2}aH^{2}\Omega_{b}\phi^{3}f_{,\mathcal{Z}}G_{3,\mathcal{Y}}$ (163)

$\displaystyle\Upsilon_{2}$ $\displaystyle\equiv$ $\displaystyle-12M_{\rm
P}^{4}aH^{2}\Omega_{c}\phi^{2}f_{,\mathcal{E}}G_{3,\mathcal{Y}}-2M_{\rm
P}^{2}aH\phi^{4}f_{,\mathcal{E}}^{2}G_{3,\mathcal{Y}}+2M_{\rm
P}^{2}aH\phi^{3}f_{,\mathcal{Z}}G_{3,\mathcal{Y}}+2M_{\rm
P}^{2}a\phi^{4}f_{,\mathcal{Z}}G_{3,\mathcal{Y}}^{2}+a\phi^{7}f_{,\mathcal{E}}^{2}G_{3,\mathcal{Y}}^{2}-a\phi^{6}f_{,\mathcal{Z}}G_{3,\mathcal{Y}}^{2}$
(164) $\displaystyle+6M_{\rm P}^{4}aH^{2}\Omega_{c}\phi
f_{,\mathcal{E}}^{2}-6M_{\rm P}^{4}aH^{2}\Omega_{c}f_{,\mathcal{Z}}-2M_{\rm
P}^{2}\phi^{5}\dot{\phi}f_{,\mathcal{E}}^{2}G_{3,\mathcal{YY}}+2M_{\rm
P}^{2}\phi^{4}\dot{\phi}f_{,\mathcal{Z}}G_{3,\mathcal{YY}}-4M_{\rm
P}^{2}\phi^{3}\dot{\phi}f_{,\mathcal{E}}^{2}G_{3,\mathcal{Y}}+4M_{\rm
P}^{2}\phi^{2}\dot{\phi}f_{,\mathcal{Z}}G_{3,\mathcal{Y}}$
$\displaystyle+6M_{\rm
P}^{4}aH^{2}\Omega_{c}\phi^{3}G_{3,\mathcal{Y}}^{2}+6M_{\rm
P}^{4}aH^{3}\Omega_{c}\phi^{2}G_{3,\mathcal{Y}}-3M_{\rm
P}^{2}aH^{2}\Omega_{c}\phi^{5}G_{3,\mathcal{Y}}^{2}+6M_{\rm
P}^{4}H^{2}\Omega_{c}\phi^{3}\dot{\phi}G_{3,\mathcal{YY}}+12M_{\rm
P}^{4}H^{2}\Omega_{c}\phi\dot{\phi}G_{3,\mathcal{Y}}$

$\displaystyle\Xi_{1}$ $\displaystyle\equiv$
$\displaystyle-2a^{2}G_{3,\mathcal{Y}}^{4}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{11}-4a^{3}HG_{3,\mathcal{Y}}^{4}f_{,\mathcal{E}}^{2}\phi^{10}-2a^{2}G_{3,\mathcal{Y}}^{4}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{9}+4M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{4}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{9}$
(165) $\displaystyle+8M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}G_{3,\mathcal{YY}}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{9}+3M_{\rm
P}^{2}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{4}\phi^{8}-4M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{3}\phi^{8}+8M_{\rm
P}^{2}a^{3}HG_{3,\mathcal{Y}}^{4}f_{,\mathcal{E}}^{2}\phi^{8}$
$\displaystyle+16M_{\rm
P}^{2}a^{2}HG_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{8}-4M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}^{2}f_{,\mathcal{YE}}\phi^{8}+8M_{\rm
P}^{2}a^{2}HG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{8}+4M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{4}\phi^{7}$
$\displaystyle-16M_{\rm
P}^{2}a^{3}HG_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}^{3}\phi^{7}+16M_{\rm
P}^{2}a^{3}H^{2}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}^{2}\phi^{7}+8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YY}}^{2}f_{,\mathcal{E}}^{2}\phi^{7}+8M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{4}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{7}$
$\displaystyle+8M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{7}-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}G_{3,\mathcal{YYY}}f_{,\mathcal{E}}^{2}\phi^{7}+16M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{3}\dot{\phi}^{2}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{7}-8M_{\rm
P}^{4}\dot{\phi}^{3}G_{3,\mathcal{YY}}^{2}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{7}$
$\displaystyle-8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}G_{3,\mathcal{YY}}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{7}-12M_{\rm
P}^{4}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{4}\phi^{6}+8M_{\rm
P}^{2}a^{3}HG_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{4}\phi^{6}-8M_{\rm
P}^{4}a\dot{\phi}^{2}G_{3,\mathcal{YY}}^{2}f_{,\mathcal{E}}^{3}\phi^{6}$
$\displaystyle-20M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}^{3}\phi^{6}+8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YYY}}f_{,\mathcal{E}}^{3}\phi^{6}-16M_{\rm
P}^{4}aH\dot{\phi}^{2}G_{3,\mathcal{YY}}^{2}f_{,\mathcal{E}}^{2}\phi^{6}+40M_{\rm
P}^{2}a^{2}HG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{6}$
$\displaystyle-8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{6}-4M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}G_{3,\mathcal{YY}}\ddot{\phi}f_{,\mathcal{E}}^{2}\phi^{6}-12M_{\rm
P}^{4}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}\phi^{6}$
$\displaystyle+8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}f_{,\mathcal{YE}}\phi^{6}-8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{6}-16M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YY}}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{6}$
$\displaystyle+12M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{4}\phi^{5}-4M_{\rm
P}^{4}a\dot{\phi}^{2}G_{3,\mathcal{YYY}}f_{,\mathcal{E}}^{4}\phi^{5}-12M_{\rm
P}^{4}a^{3}H^{4}\Omega_{c}G_{3,\mathcal{Y}}^{3}\phi^{5}+24M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{3}\phi^{5}$
$\displaystyle+8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}G_{3,\mathcal{YY}}\dot{\phi}^{\prime}f_{,\mathcal{E}}^{3}\phi^{5}-16M_{\rm
P}^{4}a^{3}H^{2}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}^{2}\phi^{5}+16M_{\rm
P}^{2}aG_{3,\mathcal{Y}}^{3}\dot{\phi}^{2}f_{,\mathcal{E}}^{2}\phi^{5}-8M_{\rm
P}^{4}\dot{\phi}^{3}G_{3,\mathcal{YY}}^{2}f_{,\mathcal{E}}^{2}\phi^{5}$
$\displaystyle-4M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}^{3}\dot{H}f_{,\mathcal{E}}^{2}\phi^{5}-32M_{\rm
P}^{4}a^{2}H^{2}G_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{5}+24M_{\rm
P}^{4}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}\phi^{5}+8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{2}f_{,\mathcal{YE}}\phi^{5}$
$\displaystyle-16M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{3}\dot{\phi}^{2}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{5}-8M_{\rm
P}^{4}a^{2}H^{2}G_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{5}-32M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{3}G_{3,\mathcal{YY}}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{5}+12M_{\rm
P}^{6}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{4}\phi^{4}$
$\displaystyle-16M_{\rm
P}^{4}a^{2}H\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{4}\phi^{4}-4M_{\rm
P}^{4}aG_{3,\mathcal{YY}}\ddot{\phi}f_{,\mathcal{E}}^{4}\phi^{4}+32M_{\rm
P}^{4}a^{3}H^{2}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{3}\phi^{4}+8M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}^{2}\dot{H}f_{,\mathcal{E}}^{3}\phi^{4}$
$\displaystyle+24M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{3}\phi^{4}+12M_{\rm
P}^{6}aH^{3}\Omega_{c}\dot{\phi}^{2}G_{3,\mathcal{YY}}^{2}\phi^{4}-16M_{\rm
P}^{4}a^{3}H^{3}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{4}-12M_{\rm
P}^{4}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{4}$
$\displaystyle-28M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{4}-80M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{4}-8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{3}\ddot{\phi}f_{,\mathcal{E}}^{2}\phi^{4}-24M_{\rm
P}^{4}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{3}\dot{\phi}\phi^{4}$
$\displaystyle+24M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}\phi^{4}+16M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{E}}^{2}f_{,\mathcal{YE}}\phi^{4}-32M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{4}-16M_{\rm
P}^{4}a^{3}H^{2}G_{3,\mathcal{Y}}f_{,\mathcal{E}}^{4}\phi^{3}$
$\displaystyle-4M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}\dot{H}f_{,\mathcal{E}}^{4}\phi^{3}-24M_{\rm
P}^{4}a\dot{\phi}^{2}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{4}\phi^{3}+24M_{\rm
P}^{6}a^{3}H^{4}\Omega_{c}G_{3,\mathcal{Y}}^{3}\phi^{3}+64M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{3}\phi^{3}+16M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\ddot{\phi}f_{,\mathcal{E}}^{3}\phi^{3}$
$\displaystyle-72M_{\rm
P}^{4}a^{2}H^{2}G_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{3}-32M_{\rm
P}^{4}G_{3,\mathcal{Y}}\dot{\phi}^{3}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{3}+24M_{\rm
P}^{6}a^{2}H^{4}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{Y}}\phi^{3}-48M_{\rm
P}^{6}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}\phi^{3}$
$\displaystyle-48M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}\phi^{3}-32M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{3}f_{,\mathcal{E}}f_{,\mathcal{YE}}\phi^{3}-36M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{E}}^{4}\phi^{2}-8M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\ddot{\phi}f_{,\mathcal{E}}^{4}\phi^{2}$
$\displaystyle+16M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{E}}^{3}\phi^{2}+12M_{\rm
P}^{6}a^{3}H^{5}\Omega_{c}G_{3,\mathcal{Y}}^{2}\phi^{2}+72M_{\rm
P}^{6}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{2}-96M_{\rm
P}^{4}aHG_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{E}}^{2}\phi^{2}$
$\displaystyle+24M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{2}+48M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{3}\dot{\phi}\phi^{2}+48M_{\rm
P}^{6}aH^{3}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YY}}\phi^{2}-48M_{\rm
P}^{6}a^{3}H^{4}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}\phi^{2}$
$\displaystyle-16M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}^{2}f_{,\mathcal{E}}^{4}\phi-48M_{\rm
P}^{6}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{E}}^{3}\phi-32M_{\rm
P}^{4}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{3}f_{,\mathcal{E}}^{2}\phi+24M_{\rm
P}^{6}a^{3}H^{4}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{E}}^{2}\phi+48M_{\rm
P}^{6}a^{2}H^{4}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}\phi$
$\displaystyle-96M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}\phi+12M_{\rm
P}^{6}a^{3}H^{3}\Omega_{c}f_{,\mathcal{E}}^{4}+48M_{\rm
P}^{6}aH^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}+48M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{E}}^{2}$

$\displaystyle\Xi_{2}$ $\displaystyle\equiv$
$\displaystyle-a^{3}G_{3,\mathcal{Y}}^{4}f_{,\mathcal{E}}^{2}\phi^{10}+3M_{\rm
P}^{2}a^{3}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{4}\phi^{8}+2M_{\rm
P}^{2}a^{3}G_{3,\mathcal{Y}}^{4}f_{,\mathcal{E}}^{2}\phi^{8}+4M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{8}-4M_{\rm
P}^{2}a^{3}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}^{3}\phi^{7}$ (166)
$\displaystyle+4M_{\rm
P}^{2}a^{3}HG_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}^{2}\phi^{7}-12M_{\rm
P}^{4}a^{3}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{4}\phi^{6}+2M_{\rm
P}^{2}a^{3}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{4}\phi^{6}-4M_{\rm
P}^{4}a\dot{\phi}^{2}G_{3,\mathcal{YY}}^{2}f_{,\mathcal{E}}^{2}\phi^{6}+8M_{\rm
P}^{2}a^{2}G_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{6}$
$\displaystyle-4M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{6}-12M_{\rm
P}^{4}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}\phi^{6}-12M_{\rm
P}^{4}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{3}\phi^{5}+8M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{3}\phi^{5}$
$\displaystyle-4M_{\rm
P}^{4}a^{3}HG_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}^{2}\phi^{5}-8M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{5}+24M_{\rm
P}^{4}a^{3}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}\phi^{5}+12M_{\rm
P}^{6}a^{3}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{4}\phi^{4}$
$\displaystyle-4M_{\rm
P}^{4}a^{2}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{4}\phi^{4}+8M_{\rm
P}^{4}a^{3}HG_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{3}\phi^{4}+12M_{\rm
P}^{6}aH^{2}\Omega_{c}\dot{\phi}^{2}G_{3,\mathcal{YY}}^{2}\phi^{4}-4M_{\rm
P}^{4}a^{3}H^{2}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{4}$
$\displaystyle-12M_{\rm
P}^{4}a^{3}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{4}-8M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}^{3}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{4}-16M_{\rm
P}^{4}aG_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{4}-24M_{\rm
P}^{4}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}\dot{\phi}\phi^{4}$
$\displaystyle+24M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}G_{3,\mathcal{YY}}\phi^{4}-4M_{\rm
P}^{4}a^{3}HG_{3,\mathcal{Y}}f_{,\mathcal{E}}^{4}\phi^{3}+24M_{\rm
P}^{6}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{3}\phi^{3}+16M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{3}\phi^{3}$
$\displaystyle-16M_{\rm
P}^{4}a^{2}HG_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}^{2}\phi^{3}+24M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}\phi^{3}-48M_{\rm
P}^{6}a^{3}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}f_{,\mathcal{E}}\phi^{3}-48M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}\phi^{3}$
$\displaystyle-8M_{\rm
P}^{4}a^{2}G_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{E}}^{4}\phi^{2}+12M_{\rm
P}^{6}a^{3}H^{4}\Omega_{c}G_{3,\mathcal{Y}}^{2}\phi^{2}+72M_{\rm
P}^{6}a^{3}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}^{2}\phi^{2}-16M_{\rm
P}^{4}aG_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}f_{,\mathcal{E}}^{2}\phi^{2}$
$\displaystyle+24M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}\dot{\phi}G_{3,\mathcal{YY}}f_{,\mathcal{E}}^{2}\phi^{2}+48M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{3}\dot{\phi}\phi^{2}+48M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}^{2}G_{3,\mathcal{YY}}\phi^{2}-48M_{\rm
P}^{6}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}f_{,\mathcal{E}}\phi^{2}$
$\displaystyle-48M_{\rm
P}^{6}a^{3}H^{2}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{E}}^{3}\phi+24M_{\rm
P}^{6}a^{3}H^{3}\Omega_{c}G_{3,\mathcal{Y}}f_{,\mathcal{E}}^{2}\phi+48M_{\rm
P}^{6}a^{2}H^{3}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}\phi-96M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}f_{,\mathcal{E}}\phi$
$\displaystyle+12M_{\rm
P}^{6}a^{3}H^{2}\Omega_{c}f_{,\mathcal{E}}^{4}+48M_{\rm
P}^{6}aH^{2}\Omega_{c}G_{3,\mathcal{Y}}^{2}\dot{\phi}^{2}+48M_{\rm
P}^{6}a^{2}H^{2}\Omega_{c}G_{3,\mathcal{Y}}\dot{\phi}f_{,\mathcal{E}}^{2}$

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