\section{Introduction and summary}




The experimental observation of an accelerated expansion of the universe has been interpreted as evidence for a cosmological
term in the gravitational action of the usual form
\begin{equation}\label{usual}
 S_{\rm cosm}=-\frac{\Lambda}{8 \pi G} \int d^4x (^{(4)}g)^{1/2}~~~,
\end{equation}
with $\Lambda=3H_0^2\Omega_{\Lambda}$  in terms of the Hubble constant $H_0$ and the cosmological fraction $\Omega_{\Lambda}\simeq 0.72$.
This functional form incorporates the usual assumption that gravitational physics is four-space general coordinate invariant,
with no frame dependence in the fundamental action.

In a series of papers \cite{adler1}-\cite{adler3}, motivated by the frame dependence of the cosmological background radiation,  we have studied the implications of the assumption that there is an induced
gravitation effective action that is three-space general coordinate and Weyl scaling invariant, but is not four-space general coordinate invariant.   For the special class of diagonal metrics for which $g_{0i}=0$, these assumptions imply that the term in the induced effective action with no
 metric derivatives has the  form
\begin{equation}\label{effact2}
 S_{\rm eff}=A_0 \int d^4x (^{(4)}g)^{1/2} (g_{00})^{-2}~~~,
\end{equation}
with $A_0$ a constant.
When $A_0$ is given the value
\begin{equation}\label{a0}
A_0=-\frac{\Lambda}{8\pi G}~~~,
\end{equation}
this effective action becomes
\begin{equation}\label{effact3}
 S_{\rm eff}=-\frac{\Lambda}{8\pi G} \int d^4x (^{(4)}g)^{1/2} (g_{00})^{-2}~~~,
\end{equation}
and in Robertson-Walker (RW) spacetimes where $g_{00}=1$ exactly mimics the cosmological constant
effective action of Eq. \eqref{usual}.



The  paper \cite{adler2} studied the implications of the effective action of Eq. \eqref{effact3} for spherically symmetric solutions of
the Einstein equations, and showed that in a static, spherically symmetric Schwarzschild-like geometry it modifies the black hole horizon structure within microscopic distances of the nominal horizon, in such a way that $g_{00}$ never vanishes.  This could have important implications,
yet to be investigated, for the black hole ``information paradox''.  In the present  paper we turn to studying the implications of the effective action
of Eq. \eqref{effact3} for the equations governing small perturbations around a spatially flat RW geometry.  We find that the equations for tensor perturbations governing gravitational waves are unchanged, as are the equations for vector perturbations.  However, the equations governing scalar perturbations receive Hubble scale
corrections, which could have implications, again yet to be investigated, for structure formation in the early universe.


To set up a phenomenology for testing for the difference between the cosmological actions of Eq. \eqref{usual} and Eq. \eqref{effact3}, we make
the Ansatz that the observed cosmological constant arises from a linear combination of these two actions of the form
\begin{equation} \label{ansatz}
S_{\Lambda} = (1-f)  S_{\rm cosm}+ f  S_{\rm eff}
= -\frac{\Lambda}{8\pi G} \int d^4x (^{(4)}g)^{1/2} [1-f + f(g_{00})^{-2}]~~~~,
\end{equation}
so that $f=0$ corresponds to only a standard cosmological constant, and $f=1$ corresponds to only an apparent cosmological constant
arising from a frame dependent effective action.  Our results for the modifications to the equations governing scalar perturbations will
thus contain the parameter $f$.

As noted in \cite{adler1} and again in \cite{adler3},  while the Einstein-Hilbert action and
the particulate matter action  are four-space general coordinate invariant, the frame-dependent effective action $ S_{\rm eff}$ is
invariant only under the subset of general coordinate transformations that act on the spatial coordinates $\vec x$, while leaving
the time coordinate $t$ invariant.  Consequently, the stress-energy tensor obtained by varying $ S_{\Lambda}$ with respect
to the full metric $g_{\mu\nu}$  will not satisfy the covariant conservation condition, and thus cannot be used as a source
for the full spacetime Einstein equations.  However, it is consistent to include $ S_{\Lambda}$ in the source for the spatial
components of the Einstein equations in the preferred rest frame of the action of Eq. \eqref{effact3}, which we identify
as the rest frame of the cosmological background radiation, giving the following rules:

\begin{enumerate}
\item  The spatial components of the Einstein equations are obtained by varying the full action with respect to $g_{ij}$, giving
\begin{equation}\label{einstein}
G^{ij}+8\pi G ( T_{\Lambda}^{ij} + T^{ij}_{\rm pm})=0  ~~~,
\end{equation}
with $T^{ij}_{\rm pm}$ the spatial components of the usual particulate matter stress-energy tensor, and with
$ T_{\Lambda}^{ij}$ given by
\begin{equation}\label{deltat}
\delta S_{\Lambda }=-\frac{1}{2} \int d^4x (^{(4)}g)^{1/2}  T_{\Lambda}^{ij} \delta g_{ij}~~~.
\end{equation}
\item  The components of the Einstein tensor $G^{0i}=G^{i0}$ and $G^{00}$ are obtained from the
Bianchi identities with $G^{ij}$ as input, and from them we can infer the conserving extensions $ T_{\Lambda}^{i0}$ and
$ T_{\Lambda}^{00}$ of the spatial stress-energy tensor components  $ T_{\Lambda}^{ij}$.
\end{enumerate}

Equivalently, we can infer these conserving extensions by imposing
the covariant conservation condition on the tensor $ T_{\Lambda}^{\mu \nu}$,  with $ T_{\Lambda}^{ij}$ as input, and this is
how we proceed in Section 2.  In Sec. 3 we analyze the residual gauge invariance when there is a frame dependent effective action.  In Sec. 4
we give the modified equations for the scalar perturbations, and in Sec. 5 prove that they do not lead to scalar wave propagation. Sec. 6
gives a brief conclusion.   In Appendix
A we summarize our notational conventions, and give their relation to those of the text of Weinberg \cite{wein}.  In Appendix B we give the
formulas for the zeroth and first order inverse metric, affine connections, and Ricci tensor components. In  Appendix C we show the equivalence between
two different forms of the gauge invariance constraint on the matter perturbations, and derive formulas used in setting up the scalar perturbation equations.  In Appendix D we show that our dark energy model cannot be tested  using  the effective field theory framework of Gubitosi et al. \cite{gub}.
In Appendix E, we show that using the more general effective action that applies when the metric is not diagonal (as is the case for RW perturbations) does not alter the first order perturbation equations.

\section{Conserving extension $ T_{\Lambda}^{\mu \nu}$ of $ T_{\Lambda}^{ij}$}

Adding a small perturbation $h_{\mu\nu}$ to the zeroth order spatially flat RW metric, we have for the total metric $g_{\mu\nu}$
\begin{align}\label{totalmetric}
g_{00}=&1+h_{00}~~~,\cr
g_{i0}=&g_{0i}=h_{i0}~~~,\cr
g_{ij}=&-a^2(t)\delta_{ij}+h_{ij}~~~.\cr
\end{align}
The inverse metric, affine connection, and Ricci tensor corresponding to Eq. \eqref{totalmetric}  are given through  first order terms
in $h_{\mu\nu}$ in Appendix B.

Varying the spatial metric components $g_{ij}$, we find from Eqs. \eqref{deltat} and \eqref{ansatz} that
\begin{align}\label{vary1}
T_{\Lambda}^{ij}= &\frac{\Lambda}{8\pi G}[(1-f) g^{ij}+ f \frac{g^{ij}}{g_{00}^2}] \cr
=&\frac{\Lambda}{8\pi G}[(1-f) g^{ij}+ f \frac{g^{ij}}{(1+h_{00})^2}]~~~.\cr
\end{align}
Expanding through first order in $h_{00}$ and using $g^{ij}=-\delta_{ij}/a^2(t)$ in the term proportional to $h_{00}$, this becomes
\begin{align}\label{vary2}
T_{\Lambda}^{ij}=& \frac{\Lambda}{8\pi G}[ g^{ij}+f t^{ij}]~~~,\cr
t^{ij}=&\frac{2\delta_{ij} h_{00}}{a^2(t)}~~~.
\end{align}
Since the metric is covariantly conserved, $D_{\nu}g^{\mu\nu}=0$, with both $D_{\mu}$ and
$g^{\mu\nu}$ accurate to first order in the perturbation $h_{\mu\nu}$,  the conserving extension of $g^{ij}$ is  $g^{\mu\nu}$, including both zeroth and
first order terms.  Thus our task is to find a conserving extension $t^{\mu\nu}$ of $t^{ij}$ that obeys the covariant conservation condition
$D_{\nu}t^{\mu\nu}=0$. Since $t^{ij}$ is already first order in the perturbation $h_{00}$, it suffices to solve this equation using the zeroth
order covariant derivative constructed by using the affine connections of Eq. \eqref{affineconn} with the first order perturbation terms
omitted.

Expanding $D_{\nu}t^{\mu\nu}=\partial_{\nu}t^{\mu\nu}+\Gamma^{\mu}_{\nu\alpha}t^{\alpha\nu}+\Gamma^{\nu}_{\nu\alpha}t^{\mu\alpha}$ in terms of
temporal and spatial index values, we get
\begin{align} \label{covcons}
0=&D_{\nu}t^{\ell\nu}=\partial_0 t^{\ell0}+\partial_jt^{\ell j}+ \Gamma^{\ell}_{00}t^{00}+2\Gamma^{\ell}_{0j}T^{0j}+\Gamma^{\ell}_{mn}t^{mn}
+\Gamma^{\nu}_{\nu 0}t^{\ell 0}+\Gamma^{\nu}_{\nu j}t^{\ell j}~~~,\cr
0=&D_{\nu}t^{0\nu}=\partial_0 t^{00}+\partial_jt^{0 j}+ \Gamma^{0}_{00}t^{00}+2\Gamma^0_{0j}T^{0j}+\Gamma^{0}_{mn}t^{mn}
+\Gamma^{\nu}_{\nu 0}t^{0 0}+\Gamma^{\nu}_{\nu j}t^{0 j}~~~,\cr
\end{align}
which on substituting the zeroth order affine connections from  Eq. \eqref{affineconn} becomes
\begin{align} \label{covcons1}
0=&D_{\nu}t^{\ell\nu}=\partial_0 t^{\ell 0}+\partial_jt^{\ell j}+2\frac{\dot{a}}{a}t^{0\ell}+3\frac{\dot{a}}{a}t^{\ell 0}\cr
=&\partial_0 t^{\ell 0}+\partial_jt^{\ell j}+5\frac{\dot{a}}{a}t^{0\ell}~~~,\cr
0=&D_{\nu}t^{0\nu}=\partial_0 t^{00}+\partial_jt^{0 j}+ a\dot{a}t^{mm}+3\frac{\dot{a}}{a}t^{00} ~~~.\cr
\end{align}

These differential equations are readily integrated, to give
\begin{align}\label{integrate}
t^{ij}(\vec x,t)=&2\delta_{ij}a^{-2}(t)h_{00}(\vec x,t)~~~,\cr
t^{\ell 0}(\vec x, t)=&-2a^{-5}(t)\int_{L_1}^t du a^3(u)  \partial_{\ell}h_{00}(\vec x,u) ~~~,\cr
t^{00}(\vec x,t)=&-a^{-3}(t)\int_{L_2}^t du a^3(u) \left[\partial_{\ell}t^{0\ell}(\vec x, u)+6 \frac{\dot{a}(u)}{a(u)}h_{00}(\vec x, u)\right]~~~.
\end{align}
The lower limits $L_1$ and $L_2$ are arbitrary constants of integration; if we add an initial condition that all perturbations
should be bounded at the initial time $t_{\rm init}$ where $a(t_{\rm init})=0$,  then we should take $L_1=L_2= t_{\rm init}$.

To write the Einstein equations in terms of the Ricci tensor, as in Eq. \eqref{ricciform}, we need $t_{ij}$, $t_{i0}$, $t_{00}$, and
the trace $t_\alpha^\alpha$.  These can be written as (taking the integration limits now as $t_{\rm init})$
\begin{align}\label{integrate1}
t_{ij}(\vec x,t)=&2\delta_{ij}a^{2}(t)h_{00}(\vec x,t)~~~,\cr
t_{\ell 0}(\vec x, t)=&t_{0 \ell}(\vec x, t)=2a^{-3}(t)\int_{t_{\rm init}}^t du a^3(u)  \partial_{\ell}h_{00}(\vec x,u) ~~~,\cr
t_{00}(\vec x,t)=&a^{-3}(t)\int_{t_{\rm init}}^t du  [a(u)\partial_{\ell}t_{0\ell}(\vec x, u)-6 a^2(u)\dot{a}(u)h_{00}(\vec x, u)]~~~,\cr
\partial_{\ell}t_{\ell 0}(\vec x,t)=&2a^{-3}(t)\int_{t_{\rm init}}^t du a^3(u) \nabla^2 h_{00}(\vec x, u)~~~,\cr
t_{\alpha}^{\alpha}(\vec x, t)=& t_{00}(\vec x, t)-6h_{00}(\vec x,t)~~~.\cr
\end{align}
Substituting these equations into Eq. \eqref{ricciform} gives our final result for the modified Einstein equations, in Ricci tensor form.

Let us now separate all terms of Eq. \eqref{ricciform} into zeroth and first order parts.  Writing
\begin{equation}\label{riccisep}
R_{\mu\nu}=R_{\mu\nu}^{(0)}+R_{\mu\nu}^{(1)}~~~,
\end{equation}
we read off from Eq. \eqref{riccitensor} that
\begin{align}\label{riccisep1}
R_{00}^{(0)}=&3 \frac{\ddot{a}}{a}~~~,\cr
R_{0i}^{(0)}=&0~~~,\cr
R_{ij}^{(0)}=&-\delta_{ij}[a\ddot{a}+ 2(\dot{a})^2]~~~.\cr
\end{align}
We make a similar splitting for $T_{\mu \nu{\rm pm}}$,  taking the zeroth order stress energy
tensor to be
\begin{align}\label{zerostress}
T^{(0)}_{\mu \nu{\rm pm}}=&(p+\rho)u_{\mu}u_{\nu}-pg_{\mu \nu}~~~,\cr
T^{\alpha(0)}_{\alpha{\rm pm}}=&\rho-3p~~~,\cr
\end{align}
with $\rho$ and $p$ the particulate matter
density and pressure, and $u_{\mu}$ the four velocity with $u_{0}=1,\, u_{i}=0$. The zeroth order part of Eq. \eqref{ricciform} gives the standard
equations governing RW cosmology,
\begin{align}\label{zerothordercosm}
\frac{\ddot{a}}{a}+ 2(\frac{\dot{a}}{a})^2=&\Lambda +8\pi G\frac{1}{2}(\rho-p)~~~,\cr
3 \frac{\ddot{a}}{a}=&\Lambda-8 \pi G \frac{1}{2}(\rho+3p)~~~.\cr
\end{align}
The first order part of Eq. \eqref{ricciform} is
\begin{equation}\label{firstorder}
R^{(1)}_{\mu \nu}-\Lambda h_{\mu\nu}=-8 \pi G [T_{\mu \nu{\rm pm}}-\frac{1}{2} g_{\mu\nu} T^\alpha_{\alpha {\rm pm}}]^{(1)}
-\Lambda f[ t_{\mu \nu}-\frac{1}{2} g^{(0)}_{\mu\nu}  t^\alpha_{\alpha }]~~~,
\end{equation}
with
\begin{equation}\label{combo}
[T_{\mu \nu{\rm pm}}-\frac{1}{2} g_{\mu\nu} T^\alpha_{\alpha {\rm pm}}]^{(1)}=
T^{(1)}_{\mu \nu{\rm pm}}-\frac{1}{2}[ g^{(0)}_{\mu\nu} T^{\alpha(1)}_{\alpha {\rm pm}}+h_{\mu\nu}  T^{\alpha(0)}_{\alpha {\rm pm}}]~~~.
\end{equation}
Rewriting Eq. \eqref{firstorder} with all terms on the same side of the equation,
it is
\begin{equation}\label{firstorder1}
0= R^{(1)}_{\mu \nu}-\Lambda h_{\mu\nu}+8 \pi G [T_{\mu \nu{\rm pm}}-\frac{1}{2} g_{\mu\nu} T^\alpha_{\alpha {\rm pm}}]^{(1)}
+\Lambda f[ t_{\mu \nu}-\frac{1}{2} g^{(0)}_{\mu\nu}  t^\alpha_{\alpha }]~~~.
\end{equation}

\section{Residual gauge invariance with a frame dependent effective action}


Since the effective action of Eq. \eqref{effact3} is not four-space general coordinate invariant, but only three-space invariant,
the gauge invariance group of the first order perturbation equations will be reduced.  Let us consider the infinitesimal transformation
\begin{equation}\label{gauge1}
x^{\alpha}=x^{\prime\alpha}-\epsilon^{\alpha}(x^{\prime})~~~,
\end{equation}
with $\epsilon^0=0$, so that $t=t^{\prime}$.\footnote{With a nonzero $\epsilon^0(\vec x)$ that is independent of $t$, $\delta_gh_{00}$ remains zero \big(see \cite{wein}, Eq. (5.3.7)\big).   This additional gauge invariance can be used to impose a condition at only one time, and we do not use
  it in what follows.}  The metric tensor tensor transforms according to
\begin{align}\label{gauge2}
g^{\prime}_{\mu^{\prime}\nu^{\prime}}(x^{\prime})=&g_{\mu\nu}(x)\frac{\partial x^{\mu}}{\partial x^{\prime \mu^{\prime}}}\frac{\partial x^{\nu}}{\partial x^{\prime \nu^{\prime}}}\cr
=&g_{\mu\nu}(x)[\delta^{\mu}_{\mu^{\prime}}-\frac{\partial \epsilon^{\mu}}{\partial x^{\prime \mu^{\prime}}}]
[\delta^{\nu}_{\nu^{\prime}}-\frac{\partial \epsilon^{\nu}}{\partial x^{\prime \nu^{\prime}}}]~~~.\cr
\end{align}
Treating $\epsilon$ as a first order perturbation, dropping second order terms, and using the fact that because of spatial homogeneity of the
RW metric the zeroth order metric has no dependence on the difference between $x^{\prime}$ and $x$, Eq. \eqref{gauge2} reduces to a gauge
transformation formula for the difference $\delta_gh_{\mu \nu}\equiv g^{\prime}_{\mu\nu}-g_{\mu\nu}$~~~,
\begin{align}\label{metricgauge}
\delta_g h_{ij}=&a^2(t) (\partial_j\epsilon^i+\partial_i\epsilon^j)~~~,\cr
\delta_g h_{i0}=&a^2(t)\partial_0\epsilon ^i~~~~,\cr
\delta_g h_{00}=&0~~~.\cr
\end{align}
Since the zeroth order particulate matter stress-energy tensor is also spatially homogeneous, the same
reasoning applies to calculating the gauge variation of the first order particulate matter stress-energy tensor, and we find using Eq. \eqref{zerostress} that (see \cite{wein} for further details)
\begin{equation}\label{stressengauge}
\delta_g T_{\mu\nu{\rm pm}}^{(1)}=-p\,\delta_g h_{\mu \nu}
\end{equation}
Comparing this with $\delta_g$ applied to  Eq. \eqref{stresspert}, we learn that the perturbed quantities $p^{(1)}$, $u^{(1)}$, $\rho^{(1)}$, $\pi^S$, $\pi_i^V$, $\pi_{ij}^T$, $u_i^V$ are all invariant under the gauge transformation of Eq. \eqref{metricgauge}.

Let us now calculate the variations of the first order Ricci tensor components given in Eq. \eqref{riccitensor} under the gauge transformation of Eq.
\eqref{metricgauge}.  After a lengthy calculation, in which many terms cancel,  we find for the ${\mu\nu}={ij},{i0},{00}$ cases
\begin{equation}\label{riccigauge}
\delta_g  R_{\mu\nu}=[\frac{\ddot{a}}{a}+2(\frac{\dot{a}}{a})^2]\delta_g h_{\mu\nu}~~~.
\end{equation}
Since $\delta_g t_{\mu\nu}=0$, the gauge variation of Eq. \eqref{ricciform} becomes, using Eq. \eqref{zerothordercosm}
\begin{equation}\label{ricciform1}
\delta_g[R^{(1)}_{\mu \nu}-\Lambda h_{\mu\nu}]=[\frac{\ddot{a}}{a}+2(\frac{\dot{a}}{a})^2-\Lambda]\delta_g h_{\mu\nu}
 =8\pi G\frac{1}{2}(\rho-p)\delta_g h_{\mu\nu}=-8 \pi G \delta_g[T_{\mu \nu{\rm pm}}-\frac{1}{2} g_{\mu\nu} T^\alpha_{\alpha {\rm pm}}]^{(1)}~~~.
\end{equation}
Thus gauge invariance of the first order perturbation equations requires
\begin{equation}\label{gaugeident}
\frac{1}{2}(p-\rho)\delta_g h_{\mu\nu}=\delta_g[T_{\mu \nu{\rm pm}}-\frac{1}{2} g_{\mu\nu} T^\alpha_{\alpha {\rm pm}}]^{(1)}~~~.
\end{equation}
In Appendix C we show that Eq. \eqref{gaugeident} is implied by Eq. \eqref{stressengauge}, and so the first order perturbation equations are
gauge invariant.

Having established gauge invariance of the perturbed equations, we are free to make a choice of gauge to simplify the subsequent calculations.
Taking
\begin{equation}\label{bfix}
\epsilon^i=\frac{1}{2} \partial_i B,
\end{equation}
we get $\delta_g h_{ij}= a^2 \partial_i \partial_j B$, which cancels the $B$ term in Eq.  \eqref{split}, giving what one might
term ``restricted Newtonian gauge''.  However, one cannot also gauge $F$ to zero as in full Newtonian gauge, since this requires use
of $\epsilon_0$.  We also cannot attain synchronous gauge, since this again requires  use of $\epsilon_0$.

\section{The modified scalar perturbation equations}

Combining Eqs. \eqref{zerothordercosm}, \eqref{firstorder1}, \eqref{split}, \eqref{riccitensor}, and \eqref{gaugeident1}, and choosing $B=0$ gauge, we get the following results
for the modified scalar perturbation equations.   They correspond to Eqs. (5.1.44)-(5.1.47) of Weinberg \cite{wein}, with the omission of the $B$ terms,
and the addition of the $\Lambda f t_{\mu\nu}$ terms arising from the frame dependent effective action.\footnote{Note that $\Lambda$ terms not multiplied by $f$ have cancelled when Eqs. \eqref{zerothordercosm} and \eqref{gaugeident1} were used. That is why our $f=0$ equations in $B=0$ gauge are identical to those
of \cite{wein}, which omits a cosmological constant.} The $ij$ part of the scalar perturbation can be written as
\begin{align}\label{ijpart}
0=&\delta_{ij}X+\partial_i\partial_jY~~~,\cr
X=&4\pi G a^2(\rho^{(1)}-p^{(1)}-\nabla^2 \pi^S)\cr
+&[a \ddot{a} + 2(\dot{a})^2]E+\frac{1}{2}a \dot{a}\dot{E}+\dot{a}\nabla^2F-3a\dot{a}\dot{A}-\frac{1}{2}a^2\ddot{A}+\frac{1}{2}\nabla^2 A\cr
+&\Lambda f a^2 (\frac{1}{2} t_{00}- E)~~~,\cr
Y=&8\pi G a^2 \pi^S\cr
+&2\dot{a}F+a\dot{F}+\frac{1}{2}(E+A)~~~.\cr
\end{align}
The $i0$ part of the scalar perturbation is
\begin{align}\label{i0part}
0=&-8 \pi G (p+\rho) \partial_iu^{(1)}\cr
- &\frac{\dot{a}}{a}\partial_iE +\partial_i\dot{A}\cr
+&\Lambda f t_{0i}~~~,\cr
\end{align}
and the $00$ part of the scalar perturbation is
\begin{align}\label{00part}
0=&4 \pi G(\rho^{(1)}+3p^{(1)}+\nabla^2\pi^S)\cr
-&3\frac{\ddot{a}}{a}E+3\frac{\dot{a}}{a}\dot{A}+\frac{3}{2}\ddot{A}-\frac{3}{2}\frac{\dot{a}}{a} \dot{E}-\frac{\dot{a}}{a^2}\nabla^2 F
-\frac{1}{a}  \nabla^2\dot{F}-\frac{1}{2a^2} \nabla^2 E\cr
+&\Lambda f (\frac{1}{2}t_{00}+3E)~~~.\cr
\end{align}
We have displayed these equations with separate lines giving the matter perturbation source terms, the metric terms, and the additional pieces proportional to
$\Lambda f t_{\mu \nu}$.  The latter are given, we recall, by Eq. \eqref{integrate1} of Sec. 2, with $h_{00}=E$.


Since the particulate matter stress-energy tensor $T_{\mu \nu{\rm pm}}$ and the added stress-energy tensor  $t_{\mu\nu}$ are separately covariantly conserved, $D^{\mu}T_{\mu \nu{\rm pm}}=0$, $D^{\mu}t_{\mu\nu}=0$, the momentum and energy conservation equations given in  Eqs. (5.1.48) and (5.1.49) of \cite{wein} are
unmodified.  Also, since $t_{\mu\nu}$ involves only the scalar $h_{00}$,  the vector and tensor perturbation equations are unmodified.

\section{Absence of propagating scalar gravitational waves}

We turn now to the question of whether the modified scalar perturbation equations of the preceding section admit propagating  wave solutions.
Thus, we investigate whether the homogeneous equations, obtained by dropping the matter source terms, have solutions.  Since these equations have
space-independent coefficients, we can Fourier analyze with respect to the coordinate $\vec x$, and it suffices to keep a generic mode
$e^{i\vec k \cdot \vec x}$.  Once this is done, the $X$ and $Y$ terms in Eq. \eqref{ijpart} decouple, since the tensors $\delta_{ij}$ and $k_ik_j$ are
linearly independent.  So we are left with {\it four} coupled equations to solve.  We can now anticipate the answer, before doing detailed
arithmetic:  Since gauge invariance, which was used to set $B=0$,  has reduced the number of unknowns  from four to the set of {\it three} comprising  $A$, $F$, and $E$, the homogeneous equations are overdetermined.  So unless there is a hidden linear dependence (which we shall see there is not) there are no solutions, other than the trivial solution $A=F=E=0$.  This
turns out to be the case independent of the coefficient $f$ of the added term $\Lambda f t_{\mu\nu}$.

To start our calculation, we rescale $F \to F/a$, after which all terms in the scalar perturbation equations with equal numbers of spatial
derivatives have the same power of $a$ as coefficient.
Since the gravitational waves of interest for binary pulsars or black hole mergers have much shorter periods than the Hubble time scale
on which $a(t)$ changes, we make the approximation of treating $a$, $\dot{a}$, and $\ddot{a}$ as constants.  The homogeneous scalar equations then
become differential equations for the evolution of $A$, $F$, and $E$ with constant coefficients, which we solve by making an Ansatz of $e^{-i\omega t}$
time dependence (where $\omega$ has a small negative imaginary part so the wave vanishes at $t=-\infty$) and $e^{i\vec k \cdot a\vec x}$ spatial dependence (with $a$ treated here as time-independent).\footnote{Regarding the $a$ in the spatial wave as $a(t_0)$ for some late time $t_0$, our approximation thus
consists of neglecting terms of order $H(t-t_0)$, $[H(t-t_0)]^2$, etc. in the Taylor expansion of the coefficient functions $a(t)$, $\dot{a}(t)$, etc.
at $t=t_0$.  When applied to the tensor perturbation equation of Eq. (5.1.53) of \cite{wein}, this procedure leads to the equation
$\nabla^2 D_{ij}-\ddot{D}_{ij}-3H \dot{D}_{ij}=0$, which describes a weakly damped propagating wave.}
We write  $\dot{a}\equiv H a$, $\ddot{a} \equiv H^2 Q a$, with constant $H$, $Q$, and $a$, replace $\partial_j$ by $iak_j$,  $\partial_t$ by $-i\omega$, and $\int dt$ by $(-i\omega)^{-1}$, and factor away
the uniform powers of $a$ in all terms of the equations (which is equivalent to setting $a=1$).
The time integrations of Eq. \eqref{integrate1} now give
\begin{align}\label{teqs}
t_{i 0}=& -2 \frac{k_i}{\omega}E~~~,\cr
t_{00}=&\left[2\frac{(\vec k)^2}{\omega^2}-6i\frac{H}{\omega}\right]E~~~\cr
\end{align}
Using these, we then  get the following
set of four coupled equations for the now constant unknowns $A$, $F$, and $E$:  From $Y$
in the $ij$ equation of Eq. \eqref{ijpart} we get
\begin{equation}\label{ypart}
0=HF-i\omega F+\frac{1}{2}(E+A)~~~;
\end{equation}
from the $i0$ equation of Eq. \eqref{i0part} we get (after
factoring out $k_i$),
\begin{equation}\label{i0part1}
0=-iHE+\omega A-\frac{2\Lambda f}{\omega}E~~~;
\end{equation}
from $X$ in the $ij$ equation of  Eq. \eqref{ijpart} we get
\begin{equation}\label{xpart}
0=(Q+2)H^2E-\frac{1}{2}i\omega H E-(\vec k)^2 H F+3i\omega H A+\frac{1}{2}\omega^2 A-\frac{1}{2} (\vec k)^2 A
+\Lambda f \left[\frac{(\vec k)^2}{\omega^2}-3i\frac{H}{\omega}-1\right]E~~~;
\end{equation}
and from the $00$ equation of Eq. \eqref{00part} we get
\begin{equation}\label{00part1}
0=-3QH^2 E-3i\omega H A-\frac{3}{2}\omega^2 A+\frac{3}{2}i\omega H E
-i\omega (\vec k)^2 F+\frac{1}{2}(\vec k)^2 E+\Lambda f \left[\frac{(\vec k)^2}{\omega^2}-3i\frac{H}{\omega}+3\right]E~~~.
\end{equation}


Eqs. \eqref{ypart} and \eqref{i0part1} can be solved for $F$ and $A$ in terms of $E$, giving
\begin{align}\label{fesoln}
F=&\frac{1}{2i\omega} \frac{1+\frac{iH}{\omega}+\frac{2\Lambda f}{\omega^2}}{1+\frac{iH}{\omega}} E~~~,\cr
A=&\left(\frac{iH}{\omega}+\frac{2\Lambda f}{\omega^2}\right) E~~~.\cr
\end{align}
Substituting these into Eqs. \eqref{xpart} and \eqref{00part1}, we can write the results respectively in the form
\begin{align}\label{results}
0=&E[(\vec k)^2 \alpha(\omega) + \beta(\omega)]~~~,\cr
0=&E[(\vec k)^2 \gamma(\omega) + \delta(\omega)]~~~,\cr
\end{align}
with
\begin{align}\label{alphetc}
\alpha(\omega)=&\gamma(\omega)= (\frac{i\Lambda f H}{\omega^3})/(1+\frac{iH}{\omega})~~~~,\cr
\beta(\omega)=&(Q-1)H^2+3\frac{i\Lambda fH}{\omega}~~~,\cr
\delta(\omega)=&-3\beta(\omega)~~~.\cr
\end{align}
Evidently the two equations in Eq. \eqref{results} are inconsistent, and so can only be solved by $E=0$.

One caveat to this analysis is that the algebra leading to Eqs. \eqref{alphetc} involves multiple cancellations
of leading terms, and so there could be significant corrections when the time dependence of $a$, $\dot{a}$, $\ddot{a}$ is
taken into account.  But there is no reason for these corrections to conspire to make the two equations of Eq. \eqref{results}
consistent.

\section{Conclusion}

We have shown that  cosmological action of Eq. \eqref{ansatz} has potentially testable consequences for the equations governing scalar
perturbations around the RW metric.  This may make it possible to distinguish between the standard and the frame dependent actions, both of which
give rise to an effective cosmological constant.

\section{Acknowledgements}
I wish to thank a number of people for raising the question of whether the frame-dependent effective action affects gravitational wave propagation:
Clifford Burgess and Andrei Barvinsky (at the Galiano Island workshop), Edward Witten (in conversations) and Jeremy Bernstein (in emails).  I also wish
to thank Matias Zaldarriaga for email correspondence calling my attention to the effective field theory framework of \cite{gub}, and to thank
Paul Steinhardt for a helpful conversation.
