# Geometrical effective actions for a partially massless spin-2 field

Nahomi Kan<EMAIL_ADDRESS>National Institute of Technology, Gifu College,
Motosu-shi, Gifu 501-0495, Japan  Takuma Aoyama<EMAIL_ADDRESS>Graduate School of Sciences and Technology for Innovation, Yamaguchi
University, Yamaguchi-shi, Yamaguchi 753–8512, Japan  Kiyoshi Shiraishi
<EMAIL_ADDRESS>Graduate School of Sciences and Technology for
Innovation, Yamaguchi University, Yamaguchi-shi, Yamaguchi 753–8512, Japan

###### Abstract

We consider nonlinear effective actions for a spin-2 field, whose ‘decoupling’
limit gives Fierz–Pauli action in $D$ dimensional maximally symmetric
spacetime. We find, especially, the effective action for a partially massless
field can take a concise geometrical form. The exact solution for time
evolution of the background metric in the model using the effective action is
also studied.

###### pacs:

04.20.-q, 04.20.Jb, 04.40.-b, 04.50.-h, 04.50.Kd, 11.90.+t, 12.90.+b,
98.80.Cq, 98.80.Jk.

##  I Introduction

In the field-theoretical treatment, the gravitational interaction is mediated
by a massless graviton with spin 2. There have been a lot of candidates for
modifications of Einstein gravity, motivated by the problems in modern
cosmology, and the possibility of massive gravitons has been proposed. In
their seminal paper FP , Fierz and Pauli presented an action for a free spin-2
field. It has been found that their linear equation of motion for a spin-2
field is unique in the flat spacetime, but there are difficulties both with
the formulation in the general curved spacetime AD1 ; AD2 ; BKP ; BGKP ; BGP
and with the nonlinear extensions BD1 . Recently, consistent massive gravity
and bigravity theories have been found and investigated Hinterbichler ; deRham
; SMvS . The mass terms of symmetric tensor fields in these theories have the
complicated expressions of nonpolynomial and look highly nonlinear. We should
notice that the terms are not uniquely fixed and some degree of arbitrariness
still remains.

On the other hand, almost three decades ago, Grigoryan and Gottlöber presented
a nonlinear model of a symmetric second-rank tensor field in a beautiful,
almost unique, geometric form GG1 ; GG2 ; their main interest was in the
introduction of a new field to cause inflation of the Universe and they did
not seem to consider mass of the field seriously.

In the present paper, we extend their model to explore the classical action of
a massive spin-2 field in a maximally symmetric background spacetime. We have
found the action in a geometrically compact form, which coincides with the
action for a partially massless spin-2 field in a weak-field limit or
‘decoupling’ limit. Here the term ‘decoupling’ means that nonlinear effects
and possibly unphysical modes are decoupled from the free theory in such a
limit.111The use of the term also implies that the background field is
strongly curved. Many difficulties are reported in the nonlinear extension of
the linear spin-2 theory, including some no-go theorems, especially for
partially massless fields dHRT ; GR . Although the model proposed in the
present paper may not be completely healthy, it would be worth studying as an
effective field theory of a complete theory including quantum gravity effects
that will be revealed in the future because there is a simple and interesting
symmetry in the model.

This paper is organized as follows. In the next section, we review the spin-2
action in the spacetime manifold of constant curvature and its possible
symmetries which emerge at specific values of the mass parameter. The
Grigoryan–Gottlöber (GG) theory, which approximates the kinetic part of the
spin-2 field theory, is reviewed in Sec. III. In Sec. IV, we examine the
models for massive, massless, and partially massless spin-2 fields in their
decoupling limit, based on the GG theory. It is shown that the action can be
written in a compact form when its decoupling limit describes the partially
massless spin-2 field. We investigate the exact solution for time evolution of
the background metric in the effective nonlinear theory with the partially
massless spin-2 field in Sec. V. Finally, we summarize the results briefly and
conclude in Sec. VI with outlook.

##  II The Fierz–Pauli theory in the curved spacetime: a review

In this section, we give an overview of the action for a free spin-2 field in
a curved background. The Fierz–Pauli action of a massive symmetric tensor
field $h_{\mu\nu}$ on a $D$-dimensional Einstein spacetime with the constant
scalar curvature $R\equiv g^{\mu\nu}R_{\mu\nu}$, where the Ricci tensor
$R_{\mu\nu}$ equals to $\frac{R}{D}g_{\mu\nu}$, is written by AD1 ; AD2 ; BKP
; BGKP ; BGP

$\displaystyle S_{FP}$ $\displaystyle=$ $\displaystyle\int
d^{D}x\sqrt{-g}\Bigl{[}-\frac{1}{2}\nabla_{\mu}h_{\nu\lambda}\nabla^{\mu}h^{\nu\lambda}+\nabla_{\nu}h_{\mu\lambda}\nabla^{\mu}h^{\nu\lambda}-\nabla_{\mu}h\nabla_{\lambda}h^{\mu\lambda}+\frac{1}{2}\nabla_{\mu}h\nabla^{\mu}h$
(1)
$\displaystyle+\frac{R}{D}\Bigl{(}h^{\mu\nu}h_{\mu\nu}-\frac{1}{2}h^{2}\Bigr{)}-\frac{1}{2}m^{2}(h^{\mu\nu}h_{\mu\nu}-h^{2})\Bigr{]}\,,$

where $h\equiv h^{\rho}_{\rho}=g^{\mu\nu}h_{\mu\nu}$, $\nabla_{\mu}$
represents the covariant derivative in terms of the metric $g_{\mu\nu}$, and
$m$ is the mass parameter. The metric of the background geometry $g_{\mu\nu}$
is assumed to be nondynamical, from this section to Sec. IV.

It is known that there are two special cases when the action becomes invariant
against the specific transformations.

If $m=0$, the action and the equation of motion are invariant under the
transformation

$\delta h_{\mu\nu}=\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}\,,$ (2)

where $\xi_{\mu}(x)$ is a vector gauge parameter. By this symmetry, the degree
of freedom of the symmetric tensor field $h_{\mu\nu}$ is reduced to be
$D(D-3)/2$; the field describes a massless spin-2 particle.

If $m^{2}=\frac{D-2}{D(D-1)}R$, when the Higuchi bound Higuchi1 ; Higuchi2  is
saturated, the action takes the form

$\displaystyle S_{PM}$ $\displaystyle=$ $\displaystyle\int
d^{D}x\sqrt{-g}\Bigl{[}-\frac{1}{2}\nabla_{\mu}h_{\nu\lambda}\nabla^{\mu}h^{\nu\lambda}+\nabla_{\nu}h_{\mu\lambda}\nabla^{\mu}h^{\nu\lambda}-\nabla_{\mu}h\nabla_{\lambda}h^{\mu\lambda}+\frac{1}{2}\nabla_{\mu}h\nabla^{\mu}h$
(3)
$\displaystyle\qquad\qquad\qquad+\frac{R}{2D(D-1)}\left(Dh^{\mu\nu}h_{\mu\nu}-h^{2}\right)\Bigr{]}$
$\displaystyle=$ $\displaystyle\int
d^{D}x\sqrt{-g}\Bigl{[}-\frac{1}{2}\nabla_{\mu}h_{\nu\lambda}\nabla^{\mu}h^{\nu\lambda}+\nabla_{\nu}h_{\mu\lambda}\nabla^{\mu}h^{\nu\lambda}-\nabla_{\mu}h\nabla_{\lambda}h^{\mu\lambda}+\frac{1}{2}\nabla_{\mu}h\nabla^{\mu}h$
$\displaystyle\qquad\qquad\qquad+\frac{m^{2}}{2(D-2)}\left(Dh^{\mu\nu}h_{\mu\nu}-h^{2}\right)\Bigr{]}\,.$

Now, the action $S_{PM}$ is found to be invariant under the deviation

$\delta
h_{\mu\nu}=\Bigl{(}\nabla_{\mu}\nabla_{\nu}+\frac{m^{2}}{D-2}g_{\mu\nu}\Bigr{)}\zeta\,,$
(4)

where $\zeta(x)$ is a scalar gauge parameter. The degree of freedom is then
$D(D-1)/2-2$. In this case, the field is called a partially massless field DN1
; DN2 ; DW2 ; DW3 .

##  III The GG action for a spin-2 field: a review

In this section, we briefly review the GG theory of a symmetric tensor field
GG1 ; GG2 . A symmetric tensor $f_{\mu\nu}=f_{\nu\mu}$ in the $D$-dimensional
spacetime, whose metric $g_{\mu\nu}$, is presumed.

First, we define

$\tilde{\Gamma}^{\lambda}_{\mu\nu}\equiv\frac{1}{2}f^{\lambda\sigma}\left(\nabla_{\mu}f_{\sigma\nu}+\nabla_{\nu}f_{\sigma\mu}-\nabla_{\sigma}f_{\mu\nu}\right)\,,$
(5)

where $f^{\mu\nu}$ is the inverse of $f_{\mu\nu}$, i.e.,
$f^{\mu\sigma}f_{\sigma\nu}=\delta^{\mu}_{\nu}$, and $\nabla_{\mu}$ denotes
the covariant derivative in terms of the metric $g_{\mu\nu}$. Note that the
indices are lowered and raised by using the background metric $g_{\mu\nu}$ and
its inverse $g^{\mu\nu}$, except for $f^{\mu\nu}$ which is the inverse of
$f_{\mu\nu}$, i.e., $f^{\mu\nu}\neq g^{\mu\rho}g^{\nu\sigma}f_{\rho\sigma}$.
Note also that $\tilde{\Gamma}^{\lambda}_{\mu\nu}$ is a tensor. Interestingly,
$\tilde{\Gamma}^{\lambda}_{\mu\nu}$ can be expressed in the form

$\tilde{\Gamma}^{\lambda}_{\mu\nu}=\check{\Gamma}^{\lambda}_{\mu\nu}-{\Gamma}^{\lambda}_{\mu\nu}\,,$
(6)

where

$\check{\Gamma}^{\lambda}_{\mu\nu}=\frac{1}{2}f^{\lambda\sigma}\left(\partial_{\mu}f_{\sigma\nu}+\partial_{\nu}f_{\sigma\mu}-\partial_{\sigma}f_{\mu\nu}\right)\,,$
(7)

and ${\Gamma}^{\lambda}_{\mu\nu}$ is the usual Christoffel symbol (constructed
from $g_{\mu\nu}$).

Next, using this tensor, we define the curvature-like tensor

$\tilde{R}^{\lambda}{}_{\rho\mu\nu}\equiv\nabla_{\mu}\tilde{\Gamma}^{\lambda}_{\nu\rho}-\nabla_{\nu}\tilde{\Gamma}^{\lambda}_{\mu\rho}+\tilde{\Gamma}^{\lambda}_{\mu\sigma}\tilde{\Gamma}^{\sigma}_{\rho\nu}-\tilde{\Gamma}^{\lambda}_{\nu\sigma}\tilde{\Gamma}^{\sigma}_{\rho\mu}\,.$
(8)

Similarly, one can find

$\tilde{R}^{\lambda}{}_{\rho\mu\nu}=\check{R}^{\lambda}{}_{\rho\mu\nu}-{R}^{\lambda}{}_{\rho\mu\nu}\,,$
(9)

where

$\check{R}^{\lambda}{}_{\rho\mu\nu}\equiv\partial_{\mu}\check{\Gamma}^{\lambda}_{\nu\rho}-\partial_{\nu}\check{\Gamma}^{\lambda}_{\mu\rho}+\check{\Gamma}^{\lambda}_{\mu\sigma}\check{\Gamma}^{\sigma}_{\rho\nu}-\check{\Gamma}^{\lambda}_{\nu\sigma}\check{\Gamma}^{\sigma}_{\rho\mu}\,,$
(10)

and ${R}^{\lambda}{}_{\rho\mu\nu}$ is the usual Riemann tensor (constructed
from the Christoffel symbol ${\Gamma}^{\lambda}_{\mu\nu}$). We also consider
the symmetric tensor from the contraction of the tensor defined above:

$\tilde{R}_{\mu\nu}\equiv\tilde{R}^{\rho}{}_{\mu\rho\nu}=\check{R}_{\mu\nu}-{R}_{\mu\nu}\,,$
(11)

where $\check{R}_{\mu\nu}=\check{R}^{\rho}{}_{\mu\rho\nu}$, and
$R_{\mu\nu}={R}^{\rho}{}_{\mu\rho\nu}$ is the usual Ricci tensor (constructed
from the Christoffel symbol ${\Gamma}^{\lambda}_{\mu\nu}$). We adopt the
action with rather geometrical form at last:

$S_{GG}=-\frac{2}{\varepsilon^{2}}\int
d^{D}x\sqrt{-g}g^{\mu\nu}\tilde{R}_{\mu\nu}\,,$ (12)

where $\varepsilon$ is a parameter. When we represent the small fluctuation of
$f_{\mu\nu}$ from $g_{\mu\nu}$ by $\psi_{\mu\nu}$, using the small parameter
$\varepsilon$, i.e.,

$f_{\mu\nu}=g_{\mu\nu}+\varepsilon\psi_{\mu\nu}\,,$ (13)

we find

$\displaystyle\lim_{\varepsilon\rightarrow
0}S_{GG}=-\lim_{\varepsilon\rightarrow 0}\frac{2}{\varepsilon^{2}}\int
d^{D}x\sqrt{-g}g^{\mu\nu}\left(\tilde{\Gamma}^{\rho}_{\rho\sigma}\tilde{\Gamma}^{\sigma}_{\mu\nu}-\tilde{\Gamma}^{\rho}_{\mu\sigma}\tilde{\Gamma}^{\sigma}_{\rho\nu}\right)$
$\displaystyle=\int
d^{D}x\sqrt{-g}\left[-\frac{1}{2}\nabla_{\mu}\psi_{\nu\lambda}\nabla^{\mu}\psi^{\nu\lambda}+\nabla_{\nu}\psi_{\mu\lambda}\nabla^{\mu}\psi^{\nu\lambda}-\nabla_{\mu}\psi\nabla_{\lambda}\psi^{\mu\lambda}+\frac{1}{2}\nabla_{\mu}\psi\nabla^{\mu}\psi\right]\,,$
(14)

where $\psi\equiv\psi^{\mu}_{\mu}=g^{\mu\nu}\psi_{\mu\nu}$ and the boundary
terms are omitted. We can easily verify this equation if we utilize the
expressions

$\tilde{\Gamma}^{\lambda}_{\mu\nu}=\frac{\varepsilon}{2}g^{\lambda\sigma}\left(\nabla_{\mu}\psi_{\sigma\nu}+\nabla_{\nu}\psi_{\sigma\mu}-\nabla_{\sigma}\psi_{\mu\nu}\right)+O(\varepsilon^{2})\,,$
(15)

and

$g^{\mu\nu}\tilde{R}_{\mu\nu}=g^{\mu\nu}\left(\tilde{\Gamma}^{\rho}_{\rho\sigma}\tilde{\Gamma}^{\sigma}_{\mu\nu}-\tilde{\Gamma}^{\rho}_{\mu\sigma}\tilde{\Gamma}^{\sigma}_{\rho\nu}\right)+\nabla_{\rho}\left(g^{\mu\nu}\tilde{\Gamma}^{\rho}_{\mu\nu}-g^{\rho\lambda}\tilde{\Gamma}^{\sigma}_{\lambda\sigma}\right)\,.$
(16)

Since the limit of $\varepsilon\rightarrow 0$ yields the linear theory, we
call the limit as the ‘decoupling’ limit, in which the nonlinear contributions
decouple from the free theory.222Note that the nonlinearity in terms of
$\psi_{\mu\nu}$ lurks in $f^{\mu\nu}$. This is, of course, equivalent to the
weak-field limit in the classical theory.

The action $S_{GG}$ supplies only the kinetic part for the symmetric tensor
field in the decoupling limit. It coincides with the form of the kinetic part
of the Fierz–Pauli action, or that in the flat spacetime. In the next section,
we examine the remaining quadratic terms in the tensor field without
derivatives.

Lastly, we should remember the invariance of the action $S_{GG}$ under the
global scale transformation GG2 333Of course, the reflection symmetry under
$f_{\mu\nu}\leftrightarrow-f_{\mu\nu}$ also exists.

$f_{\mu\nu}\rightarrow\Omega_{f}^{2}f_{\mu\nu}\,,$ (17)

where $\Omega_{f}$ is a constant.

##  IV construction of massive, massless, and partially massless spin-2
theories based on the GG theory

###  IV.1 general massive case

Now, we search the extended version of the GG theory whose decoupling limit
coincides with the massive Fierz–Pauli theory in the curved spacetime. We can
construct arbitrary quadratic terms without derivatives of the field and add
them to the kinetic term in the GG action, which has been shown in the
previous section. For example, we can use the combinations of

$\varepsilon^{2}\psi^{2}=(g^{\mu\nu}f_{\mu\nu}-D)^{2}\,,\quad\varepsilon^{2}\psi_{\mu\nu}\psi^{\mu\nu}=g^{\mu\rho}g^{\nu\sigma}(f_{\mu\nu}-g_{\mu\nu})(f_{\rho\sigma}-g_{\rho\sigma})\,.$
(18)

Notice that the quadratic terms in $\psi_{\mu\nu}$ can also arises from

$g_{\mu\nu}f^{\mu\nu}=D-\varepsilon\psi+\varepsilon^{2}\psi^{\mu\nu}\psi_{\mu\nu}+O(\varepsilon^{3})\,.$
(19)

Because arbitrariness is too large, we will stop pursuing the fixing the form
of the generic quadratic term without derivatives until new findings are made
on its form.

###  IV.2 massless case

Now, we are aware of the following relation:

$\sqrt{|\det
f_{\mu\nu}|}=\sqrt{|f|}=\sqrt{-g}\left[1+\frac{\varepsilon}{2}\psi-\frac{\varepsilon^{2}}{4}\left(\psi^{\mu\nu}\psi_{\mu\nu}-\frac{1}{2}\psi^{2}\right)+O(\varepsilon^{3})\right]\,.$
(20)

Although its quadratic part is proportional to that with massless Fierz–Pauli
Lagrangian, the linear term $\propto\psi$ remains and the subtraction of it is
needed. Rather, it may be possible to adopt the Born–Infeld-type gravitational
action BHOR , inspired by the use of square roots. That is:

$\displaystyle S_{BI}\sim-\int
d^{D}x\Bigl{[}\sqrt{g_{\mu\nu}+\alpha(f_{\mu\nu}-g_{\mu\nu})+\beta\tilde{R}_{\mu\nu}+\gamma
g_{\mu\nu}g^{\rho\sigma}\tilde{R}_{\rho\sigma}}$
$\displaystyle\qquad\qquad\qquad\qquad+\sqrt{g_{\mu\nu}-\alpha(f_{\mu\nu}-g_{\mu\nu})+\beta^{\prime}\tilde{R}_{\mu\nu}+\gamma^{\prime}g_{\mu\nu}g^{\rho\sigma}\tilde{R}_{\rho\sigma}}-2\sqrt{-g}\Bigr{]}\,,$
(21)

where $\alpha$, $\beta$, $\gamma$, $\beta^{\prime}$, and $\gamma^{\prime}$ are
constants. Although this form seems lengthy, nonpolynomial, and less
geometrical, this form has its own unique interests. If we replace
$\tilde{R}_{\mu\nu}\rightarrow\tilde{\Gamma}^{\rho}_{\rho\sigma}\tilde{\Gamma}^{\sigma}_{\mu\nu}-\tilde{\Gamma}^{\rho}_{\mu\sigma}\tilde{\Gamma}^{\sigma}_{\rho\nu}$,444Please
remember that $\tilde{\Gamma}^{\lambda}_{\mu\nu}$ is a tensor the dynamical
model is apparently free from the Ostrogradsky instability Ostrogradsky ,
because there is no second derivatives in the time coordinate. However, on the
other hand, there is also a disadvantage that there is room to select the
second-order terms in $\varepsilon$ in the square root. In addition, the
original scale invariance (17) of $S_{GG}$ is broken in this type of action,
so we will leave the exploration around here as well.

###  IV.3 partially massless case

The combination

$(g_{\rho\sigma}f^{\rho\sigma})(g^{\mu\nu}f_{\mu\nu})=D^{2}+\varepsilon^{2}\left(D\psi^{\mu\nu}\psi_{\mu\nu}-\psi^{2}\right)+O(\varepsilon^{3})\,,$
(22)

is scale invariant under the global transformation (17), and the quadratic
part in $\psi_{\mu\nu}$ is proportional to that of the Lagrangian for a
partially massless spin-2 field (3).555Incidentally, the additional cubic and
quartic terms in the theory in $D=4$ suggested by de Rham et al. (found in
Sec. III C of Ref. dHRT ) is invariant under the scale transformation of their
metric tensor. Alternatively, in the Einstein background metric, i.e.,
$R_{\mu\nu}=\frac{R}{D}g_{\mu\nu}$, we find

$R_{\rho\sigma}f^{\rho\sigma}g^{\mu\nu}f_{\mu\nu}=\frac{R}{D}(g_{\rho\sigma}f^{\rho\sigma})(g^{\mu\nu}f_{\mu\nu})=\frac{R}{D}\left[D^{2}+\varepsilon^{2}\left(D\psi^{\mu\nu}\psi_{\mu\nu}-\psi^{2}\right)+O(\varepsilon^{3})\right]\,.$
(23)

Further removing the restriction and assuming maximally symmetric spacetime,
such as the de Sitter spacetime, where
$R_{\mu\nu\rho\sigma}=\frac{R}{D(D-1)}(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho})$,
we have

$\displaystyle
f_{\mu\nu}R^{\mu}{}_{\rho}{}^{\nu}{}_{\sigma}f^{\rho\sigma}=\frac{R}{D(D-1)}\left(g^{\mu\nu}f_{\mu\nu}g_{\rho\sigma}f^{\rho\sigma}-D\right)$
$\displaystyle=\frac{R}{D(D-1)}\left[D(D-1)+\varepsilon^{2}\left(D\psi^{\mu\nu}\psi_{\mu\nu}-\psi^{2}\right)+O(\varepsilon^{3})\right]\,.$
(24)

From the above facts, we can propose the following candidates for the pre-
decoupled action of the part that does not include the derivative of the field
$f_{\mu\nu}$:

  * •

Type-$I$

$\Delta S_{PMI}=\frac{1}{2\varepsilon^{2}}\int
d^{D}x\sqrt{-g}\,(f_{\mu\nu}R^{\mu}{}_{\rho}{}^{\nu}{}_{\sigma}f^{\rho\sigma}-R)$
(25)

  * •

Type-$II$

$\Delta S_{PMII}=\frac{1}{2(D-1)\varepsilon^{2}}\int
d^{D}x\sqrt{-g}\,(R_{\rho\sigma}f^{\rho\sigma}g^{\mu\nu}f_{\mu\nu}-DR)$ (26)

  * •

Type-$III$

$\Delta S_{PMIII}=\frac{1}{2D(D-1)\varepsilon^{2}}\int
d^{D}x\sqrt{-g}\,R\,(g_{\rho\sigma}f^{\rho\sigma}g^{\mu\nu}f_{\mu\nu}-D^{2})$
(27)

  * •

Type-$III^{\prime}$

$\Delta S_{PMIII^{\prime}}=\frac{\Lambda}{(D-1)(D-2)\varepsilon^{2}}\int
d^{D}x\sqrt{-g}\,(g_{\rho\sigma}f^{\rho\sigma}g^{\mu\nu}f_{\mu\nu}-D^{2})$
(28)

Note that Type-$III^{\prime}$ is merely Type-$III$ with the replacement
$R=\frac{2D}{D-2}\Lambda$, where $\Lambda$ is a constant. Note that Type-$I$,
Type-$II$, and Type-$III$ is transformed as

$\Delta S_{PM*}\rightarrow\Omega_{g}^{D-2}\Delta S_{PM*}\qquad(*=I,II,III)$
(29)

under the global scale transformation of the background geometry

$g_{\mu\nu}\rightarrow\Omega_{g}^{2}g_{\mu\nu}\,,$ (30)

but Type-$III^{\prime}$ is transformed as $\Delta
S_{PMIII^{\prime}}\rightarrow\Omega_{g}^{D}\Delta S_{PMIII^{\prime}}$. Note
also that the kinetic part $S_{GG}$ is transformed as
$S_{GG}\rightarrow\Omega_{g}^{D-2}S_{GG}$. Of course, a linear combination of
these can be the mass term of the partially massless field in the decoupling
limit.

If and only if we consider the fluctuation of the background metric $\delta
g_{\mu\nu}$, then the nonlinear couplings to the field $\psi_{\mu\nu}$ beyond
the decoupling limit appear as in bigravity models, and we can distinguish the
different types of the mass term. To select a unique mass term by the
difference at the fully nonlinear level, we need to consider an extended
partially massless condition. However, it is difficult to analyze the
extention considering general nonlinear and derivative couplings as well as
bigravity666Note also that the kinetic terms for $f_{\mu\nu}$ and $g_{\mu\nu}$
in the present model are not symmetric unlike those in general bigravity
models., and therefore it is one of the future tasks.

Note that only Type-$I$ is independent of the spatial dimension (up to
spacetime integration), and any other linear combinations cannot be
independent of the dimension of the spacetime.

Before closing this section, we place a comment on another possible
combination which leads to the quadratic term of partially massless spin-2
theory in the decoupling limit. That is:

$\sqrt{|f|}(g_{\mu\nu}f^{\mu\nu})^{D/2}=D^{D/2}\sqrt{-g}\left[1+\frac{\varepsilon^{2}}{4}\left(\psi^{\mu\nu}\psi_{\mu\nu}-\frac{1}{D}\psi^{2}\right)+O(\varepsilon^{3})\right]\,.$
(31)

Also from this fact, we can guess that the scale invariance in terms of
$f_{\mu\nu}$ is closely related to the partially masslessness in the
decoupling limit.

##  V A little bit cosmology: An exact solution for time evolution of the
background metric

In this section, we shall consider cosmology based on the model in honor of
the first spirit of Grigoryan and Gottlöber. This time, we examine the last
model in the previous section, that is, the effective nonlinear theory with a
partially massless field as the decoupling limit.

We add the Einstein–Hilbert action with the cosmological constant $\Lambda$ so
that the background spacetime can be maximally symmetric. Then, de Sitter
spacetime is expected to be realized asymptotically. Therefore, we consider
the following action:

$S=S_{GG}+\Delta S_{PMI}+\int d^{D}x\sqrt{-g}\Bigl{[}R-2\Lambda\Bigr{]}\,.$
(32)

Needless to say, it is hard to find exact solutions of the equations derived
from the action (32). In bigravity theories, the strong ansatz that two
metrics are proportional to each other, i.e., $f_{\mu\nu}=c^{2}g_{\mu\nu}$, is
adopted to seek exact solutions in several papers SMvS ; HSS1 ; HSS2 ; KN ;
Katsuragawa . Based on that, we here adopt the ansatz

$f_{\mu\nu}=e^{\sqrt{2}\varepsilon\phi(x)}g_{\mu\nu}\,.$ (33)

This is the slimmest assumption; under this condition, $\Delta S_{PMI}$
vanishes (also, $\Delta S_{PMII}=\Delta S_{PMIII}=\Delta
S_{PMIII^{\prime}}=0$) because of the scale invariance under (17). On the
other hand, the kinetic part $S_{GG}$ results in

$S_{GG}\rightarrow+\int
d^{D}x\sqrt{-g}\Bigl{[}(D-1)(D-2)g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\Bigr{]}\,,$
(34)

which corresponds to the action for a massless phantom scalar field Caldwell ;
CKW ; DSS ; DKS , that is, has the opposite sign to the action for the
canonical scalar field.

Now, let us go for the exact solution. We assume the following metric:

$g_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}\,,$ (35)

where $a(t)$ is the scale factor, $t=x^{0}$, and $i=1,2,3$. Accordingly, we
also assume that $\phi$ is homogeneous and expressed as a function of $t$.
Then, the Friedmann equation is expressed as

$\left(\frac{\dot{a}}{a}\right)^{2}=H_{0}^{2}-\dot{\phi}^{2}\,,$ (36)

where the dot $(\dot{~{}})$ denotes the time derivative and
$H_{0}^{2}\equiv\frac{2\Lambda}{(D-1)(D-2)}$. In addition, $\phi$ obeys the
equation of motion

$\ddot{\phi}+(D-1)\frac{\dot{a}}{a}\dot{\phi}=0\,.$ (37)

From the first integration of this equation, one obtain
$\dot{\phi}^{2}=C^{2}/a^{2D-2}$ with a constant $C$. Now, one finds, from (36)

$\dot{V}=(D-1)H_{0}\sqrt{V^{2}-V_{0}^{2}}\,,$ (38)

with $V\equiv a^{D-1}$, $V_{0}\equiv C/H_{0}$, which can easily be integrated
to yield

$V(t)=a^{D-1}(t)=V_{0}\cosh[(D-1)H_{0}(t-t_{0})]\,,$ (39)

where $t_{0}$ is an integration constant. With this solution, we obtain

$\phi(t)=\phi(t_{0})+\frac{2}{D-1}\arctan\tanh\frac{(D-1)H_{0}(t-t_{0})}{2}\,.$
(40)

Note that $\lim_{t\rightarrow\infty}\phi(t)=\phi(t_{0})+\frac{\pi}{2(D-1)}$.

This analytic solution implies that the background geometry asymptotically
approaches de Sitter spacetime, i.e., the scale factor behaves like $a(t)\sim
e^{H_{0}t}$ as $t\rightarrow\infty$ and, at the same time, the scalar mode
$\phi(t)$ of the tensor field $f_{\mu\nu}$ rapidly approaches a constant.
Recall that the systems connected by the global scale transformation (17) are
equivalent. As for an excitation mode, $\phi$ is considered to be identified
with a decoupled degree of freedom, so this result can be interpreted as a
mechanism by which the nonphysical mode freezes dynamically. However, our true
cosmological interest is undoubtedly the contribution of the spin-2 field as a
matter field to the development of the Universe. For this purpose, we should
try to include some tensor modes by adopt more general ansatz such as
$f_{\mu\nu}=e^{\sqrt{2}\epsilon\phi}(g_{\mu\nu}+\psi\delta_{\mu 0}\delta_{\nu
0})$ etc., though no analytic solution can be found (using either mass term).
In this paper, we only showed analytically the most important character
discussed so far, and we would like to investigate generic features in
cosmology with numerical methods to obtain definite results in future separate
papers.

##  VI Summary and conclusion

In this paper, we have studied effective field theories of a spin-2 field,
especially one of which the decoupling limit yields the partially massless
spin-2 field theory. It has a simple geometrical form and has the symmetry
under the global scale invariance. For instance, we considered the action

$S_{GGPMI}=S_{GG}+\Delta S_{PMI}=\frac{1}{\varepsilon^{2}}\int
d^{D}x\sqrt{-g}\Bigl{[}-2g^{\mu\nu}\tilde{R}_{\mu\nu}+\frac{1}{2}(f_{\mu\nu}R^{\mu}{}_{\rho}{}^{\nu}{}_{\sigma}f^{\rho\sigma}-R)\Bigr{]}\,,$
(41)

and if the background geometry admits maximal symmetry, one finds

$\displaystyle\lim_{\varepsilon\rightarrow 0}S_{GGPMI}$ $\displaystyle=\int
d^{D}x\sqrt{-g}\left[-\frac{1}{2}\nabla_{\mu}\psi_{\nu\lambda}\nabla^{\mu}\psi^{\nu\lambda}+\nabla_{\nu}\psi_{\mu\lambda}\nabla^{\mu}\psi^{\nu\lambda}-\nabla_{\mu}\psi\nabla_{\lambda}\psi^{\mu\lambda}+\frac{1}{2}\nabla_{\mu}\psi\nabla^{\mu}\psi\right]$
$\displaystyle\qquad\qquad\qquad+\frac{R}{2D(D-1)}\left(D\psi^{\mu\nu}\psi_{\mu\nu}-\psi^{2}\right)\Bigr{]}\,,$
(42)

where $\psi_{\mu\nu}=\varepsilon^{-1}(f_{\mu\nu}-g_{\mu\nu})$.

The nonlinear terms in the expansion of our action may include unphysical
modes for a finite $\varepsilon$. Our model is well worth studying for brave
researchers who tolerate phantom fields in cosmology.777It is interesting to
extend our model to include bouncing cosmologies NB  with the phantom degrees
of freedom. Unfortunately, this model as bigravity does not fall into
Hassan–Rosen class SMvS ; HR2  of the bimetric theory888Still, we would like
to point out that GG theory is not a just a double copy of general relativity.
and therefore will have instability in the whole theory with finite
$\varepsilon$. Elimination of unstable degrees of freedom will require further
ingenuity in $O(\varepsilon)$. Further, it would be necessary to consider
higher order terms and derivative terms of the symmetric tensor with a view to
symmetry. Additional auxiliary degrees of freedom may also be required.
Therefore, if it is necessary to append an infinite number of additional terms
in the action, we should rather consider the deep relationship with string
theory LMMS . It is also known that the bimetric theories play an important
role in quantum gravity with the functional renormalization group equation MRS
. Bimetric theories of gravity has been studied also by models in the first
order formalism AS  or in the Plebanski theory Speziale , which is closely
related to spin form models Perez1 ; Perez2  and thus to the loop quantum
cosmology AshSin . In any case, we hope that our model will provide a clue to
the quantum spin-2 or quantum gravity theory.

We presented several candidates for terms with no derivative on the field,
which lead to the same decoupling limit. Among these, $\Delta S_{PMI}$ has no
explicit description of the dimension $D$. Therefore, it is expected that
partial masslessness would be inherited with $S_{GGPMI}$ even if the
dimensional reduction or the Kaluza–Klein compactification of the background
metric is performed. The first candidate for spacetime to consider would be
the Nariai spacetime Nariai  and the generalized higher-dimensional Nariai
spacetimes. In contrast, the study of lower dimensional models is also
interesting.

Finally, we list some other possible future explorations as follows. Further
generalizations can be obtained by considering the mixing terms with the
massless graviton and couplings between the tensor and other matter fields. It
would be interesting to consider supersymmetric and/or Born–Infeld extension
of the present model. Finding solutions for compact objects including black
holes and wormholes is also an interesting topic. Furthermore, a study of the
theory in general background and the quantum cosmological treatment of the
theory would be equally worth pursuing.

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