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\newcommand{\Title}{GW equation in Horndeski theory} % Assignment title
\newcommand{\AuthorName}{Alberto Roper Pol} % Your name

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\textbf{\Title}\\ 
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\begin{document}

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\maketitle

\section{Gravitational Wave Equation}

We assume tensor-mode metric perturbations $\delta g_{ij}$ over the 
isotropic and homogeneous FLRW metric $\gamma_{ij} = a^2 \delta_{ij}$, 
for signature $- + + + $, where $a(t)$ is the scale factor. 
The resulting spatial components of the metric tensor $g_{ij}$, 
neglecting scalar and vector mode perturbations, are
\begin{equation}
g_{ij} = \gamma_{ij} + \delta g_{ij},
\end{equation}
where $\delta g_{ij} = a^2 h_{ij}$ are the tensor-mode 
perturbations. We will be interested in the wave equation governing $h_{ij}$,
which is obtained by linearization of Einstein field equation
\begin{equation}
G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{ij}.
\end{equation}
Gravitational wave equation for the most general second-order tensor mode
perturbation following Horndeski theory, in physical time and space is given by
\begin{equation}
\boxed{ \ddot{h}_{ij} + (3+\alphaM) H \dot{h}_{ij} - \cT^2 \nabla^2 h_{ij}
= \frac{16 \pi G}{c^2} S_{ij},} 
\label{GW_original}
\end{equation}
where $\dot{h}$ represents derivative with respect to physical time, 
$\nabla$ represents derivatives with respect to physical space 
coordinates, $H = \dot{a}/a$ is the Hubble parameter and $S_{ij}$ is the 
projected traceless and transverse gauge invariant stress energy tensor, 
i.e., $S_{ij} = T_{ij}^{\text{TT}}$.
The Horndeski parameters are $\alphaM$ and $\cT$, where the damping term
$\alphaM$ is
\begin{equation}
    \alphaM=\frac{\dd \ln \Mpl}{\dd \ln a},
\end{equation}
and $\cT$ gives the GW speed of propagation.

\subsection{GW equation for scaled $\mu_{ij} = a h_{ij}$ 
and physical time $t$}

If we scale the metric tensor perturbations $\mu = a h$, we transform GW 
equation. Time derivatives are now
\begin{equation}
\dot{h} = \dot{\mu}/a - \mu \dot{a}/a^2, \quad 
\ddot{h} = \ddot{\mu}/a - 2 \dot{\mu} \dot{a}/a^2 - \mu \ddot{a}/a^2 + 
2 \mu \dot{a}^2/a^3
\end{equation}
Hence, substituting into GW equation (\ref{GW_original}), we get the 
GW equation for scaled metric tensor perturbations $\mu_{ij}$.
\begin{equation}
\boxed{ \ddot{\mu}_{ij} + \frac{\dot{a}}{a} \left(1 + \alphaM\right) \dot{\mu}_{ij} -
\left((1+\alphaM) \frac{\dot{a}^2}{a^2} + \frac{\ddot{a}}{a} + 
\cT^2 \nabla^2\right) \mu_{ij} = \frac{16 \pi G}{c^2} a S_{ij},}
\end{equation}
which in Fourier space it is
\begin{equation}
\boxed{ \ddot{\mu}_{ij} + \frac{\dot{a}}{a} \left(1 + \alphaM\right) \dot{\mu}_{ij} +
\left(\cT^2 k^2 - (1+\alphaM) \frac{\dot{a}^2}{a^2} - \frac{\ddot{a}}{a} \right) 
\mu_{ij} = \frac{16 \pi G}{c^2} a S_{ij}.}
\label{GW_mu}
\end{equation}
In terms of $H=\dot{a}/a$, taking into account
that $\ddot{a}/a=\dot{H}+H^2$, we can write
\begin{equation}
\ddot{\mu}_{ij} + \left(1 + \alphaM\right) H \dot{\mu}_{ij} +
\left(\cT^2 k^2 - (2+\alphaM) H^2 - \dot{H}\right) \mu_{ij} =
\frac{16 \pi G}{c^2} a S_{ij}.
\end{equation}
Using comoving $\hat{k}$ and $\hat{S}_{ij}$ we have
\begin{equation}
\ddot{\mu}_{ij} + \left(1 + \alphaM\right) H \dot{\mu}_{ij} +
\left(\frac{\cT^2 \hat k^2}{a^2}  - (2+\alphaM) H^2 - \dot{H}\right) \mu_{ij} =
\frac{16 \pi G}{a^3 c^2} \hat S_{ij}.
\end{equation}

\subsection{GW equation for physical $h_{ij}$ and conformal time $\eta$}
Derivatives with respect to time will now be expressed as derivatives with 
respect to conformal time $\eta$.
\begin{equation}
\frac{\partial}{\partial t } = \frac{\partial }{\partial \eta} \frac{\partial \eta}
{\partial t} = \frac{1}{a} \frac{\partial }{\partial \eta	} 
\end{equation}
Therefore,
\begin{equation}
\dot{h}_{ij} = \frac{1}{a} h_{ij}', \quad 
\ddot{h}_{ij} = \frac{1}{a^2} h''_{ij} - \frac{a'}{a^3}h'_{ij},
\quad \dot{a} = \frac{a'}{a}.
\label{der_time_eta}
\end{equation}
Introducing these derivatives in eq. (\ref{GW_original}) we obtain
the GW equation expressed with respect to conformal time
\begin{equation}
h_{ij}'' + (2+\alphaM) \frac{a'}{a} h_{ij}' - a^2 \cT^2 \nabla^2 h_{ij} =
\frac{16 \pi G}{c^2} a^2 S_{ij},
\end{equation}
which expressed in Fourier space $k$, comoving $\hat{k}$ and $\hat{S}_{ij}$ and 
comoving Hubble parameter $\mathcal{H}=a'/a=aH$, is
\begin{equation}
\boxed{ h_{ij}'' + (2+\alphaM) \mathcal{H} h_{ij}' +
\cT^2 \hat{k}^2 h_{ij} = \frac{16 \pi G}
{c^2a^2} \hat{S}_{ij} }
\label{GW_eta}
\end{equation}

\subsection{GW equation for scaled $\mu_{ij} = a h_{ij}$ and conformal time $\eta$}
Now we need to express time derivatives of $\mu_{ij}$ in terms of conformal time
to transform GW equation for scaled $\mu_{ij}$, eq. (\ref{GW_mu}), which are analogous 
to expressions in eq. (\ref{der_time_eta}),
\begin{equation}
    \dot{\mu}_{ij} = \frac{1}{a} \mu_{ij}', \quad 
    \ddot{\mu}_{ij}=\frac{1}{a^2}\mu_{ij}''-\frac{a'}{a^3} \mu_{ij}',\quad 
    \dot a=\frac{a'}{a}, \quad
    \ddot a=\frac{a''}{a^2} - \frac{a'^2}{a^3}.
\end{equation}
Introducing these derivatives into eq. (\ref{GW_mu}) we obtain
\begin{equation}
\boxed{ \mu_{ij}'' + \frac{a'}{a} \alphaM \mu_{ij}'
+ \left( \cT^2\hat{k}^2 - \alphaM \frac{a'^2}{a^2} - 
\frac{a''}{a} \right) \mu_{ij} = 
\frac{16 \pi G}{c^2 a} \hat{S}_{ij}}.
\label{GW_mu_eta}
\end{equation}
We can observe that this equation gets rid of the damping term that we have in 
GW equation expressed in eqs. (\ref{GW_original}, \ref{GW_mu}, \ref{GW_eta}),
in GR, when $\alphaM=0$.
For radiation-dominated era, $a''=0$, and we get
\begin{equation}
\mu_{ij}'' + \frac{a'}{a} \alphaM \mu_{ij}'
+ \left( \cT^2\hat{k}^2 - \alphaM \frac{a'^2}{a^2} \right) \mu_{ij} = 
\frac{16 \pi G}{c^2 a} \hat{S}_{ij},
\end{equation}
which can be written in terms of the conformal Hubble parameter
$\HH=a'/a$,
\begin{equation}
\boxed{ \mu_{ij}'' +  \alphaM \HH  \mu_{ij}'
+ \left( \cT^2\hat{k}^2 - \alphaM \HH^2 \right) \mu_{ij} = 
\frac{16 \pi G}{c^2 a} \hat{S}_{ij}}.
\label{GW_mu_eta}
\end{equation}


\subsection{Normalized GW equation for scaled $\mu_{ij} = a h_{ij}$ and 
conformal time $\eta$}

We are going to consider the application of eq. (\ref{GW_mu_eta}) to the 
EWPT epoch. For this purpose we consider $a'' = 0$, which is a valid 
assumption for radiation-dominated era.%(see appendix \ref{RD}).
We write $a=a_* \bar \eta = a_* \eta/\eta_*$, with $\eta_*=1/\HH_*$,
and $\HH=\HH_*/\bar \eta$.
\begin{equation}
\partial_{\bar \eta}^2 \mu_{ij} + \frac{\alphaM}{\bar \eta} \partial_{\bar \eta} 
\mu_{ij} + \left(\frac{\cT^2}{c^2} \hat{\bar k}^2 -  \frac{\alphaM}{\bar \eta^2}
\right) \mu_{ij} = 
\frac{6a_*}{\bar \eta} \hat{\bar S}_{ij},
\end{equation}
where we have defined the normalized and comoving wave numbers
$\hat{\bar k}$ and stress-energy tensor $\hat{\bar S}_{ij}$,
\begin{equation}
    \hat{\bar k} = \frac{c k}{\HH_*}, \quad
    \hat{\bar S}_{ij}=\frac{\hat S_{ij}}{\EErad^*}=\frac{3 H_*^2 c^2}{8\pi G}
    \hat S_{ij},
\end{equation}
with $H_*=\HH_*/a_*$.
We are left to set the normalization of $a$, i.e., at which time
$a=1$.
For simplicity, we set $a_*=1$ at the time of generation, which allows to
write $\HH_*=H_*$, and
\begin{equation}
\boxed{ \partial_{\bar \eta}^2 \mu_{ij} + \frac{\alphaM}{\bar \eta} \partial_{\bar \eta} 
\mu_{ij} + \left(\frac{\cT^2}{c^2} \hat{\bar k}^2 -  \frac{\alphaM}{\bar \eta^2}
\right) \mu_{ij} = 
\frac{6}{\bar \eta} \hat{\bar S}_{ij}}.
\label{GW_EWPT}
\end{equation}
Otherwise, if we set $a_0=1$, then $a_*$ depends on the time of generation and
the solutions are already given as observables at the present time.
However, note that this would assume that the derived GW equation is valid for
all times, while it is only valid during radiation-dominated era.

Note that Friedmann equations might be affected by the Horndeski parameters
and hence, one needs to careful check that the derivations used for the
evolution of $a(t)$ and $H_*$ hold.
Otherwise, the final GW equation might be impacted by the modifications.

\subsection{Horndeski parameters}

Horndeski theories consider the most general action that involves
a scalar field in the presence of gravity omitting derivatives
higher than second order in the equations of
motion \cite{Horndeski:1974wa}.
The combined action can be written as
\begin{equation}
    S= \int \dd^4 x \sqrt{-g} \left[\sum_{i=2}^5 \L_i +
    \L_m [g_{\mu\nu}]\right],
\end{equation}
where the matter Lagrangian is $\L_m$, and
\begin{align}
    \L_2=&\,K(\phi,X), \\
    \L_3=&\,-G_3(\phi, X) \Box \phi,  \\
    \L_4=&\,G_4(\phi, X) R + G_{4X} (\phi, X) \left[(\Box \phi)^2 -
    \phi_{;\mu \nu} \phi^{; \mu \nu} \right], \\
    \L_5=&\,G_5(\phi, X) G_{\mu \nu} \phi^{; \mu \nu} - 
    \frac{1}{6} G_{5X} (\phi, X) \left[(\Box \phi)^3 + 2\phi_{;\mu}^{\ \ \nu} 
    \phi_{;\nu}^{\ \ \alpha} \phi_{;\alpha}^{\ \ \mu} - 3\phi_{;\mu \nu} 
    \phi^{;\mu \nu} \Box \phi \right],
\end{align}
where $\phi$ is the scalar field and $X$ its canonical kinetic term
$X=-\half \phi^{;\mu} \phi_{;\mu}$.
$K(\phi,X)$ and $G_i(\phi, X)$ are arbitrary functions of
$\phi$ and $X$.
The functions $G_{iX}$, $G_{i\phi}$ indicate the derivative of
$G_i$ with respect to $X$ or $\phi$, respectively.
This general actions includes: quintessence, Brans-Dicke models,
$k-$eseence, kinetic gravity baiding, galileons, $f(R)$ and $f(G)$ gravity
\cite{Bellini:2014fua}.
The ones affecting tensor-mode perturbations via $\alphaM$ are
galileon cosmology, BDK, $f(R)$, MSG/Palatini $f(R)$,
and Gauss-Bonet.
From these, only the last modified the value $\alphaT$;
see \Fig{table}.
Not included in Horndeski theories are Dvali-Gabadadze-Porrati (GDP)
gravity or ghost-free massive gravity, although their covariantised
decoupling limits, Galileon cosmology and proxy massive gravity do belong
to Horndeski class \cite{Bellini:2014fua}.

We have considered two Horndeski paramers $\alphaM$, and $\alphaT$ which
affects the tensor speed $\cT^2/c^2=1+\alphaT$, \cite{Kreisch:2017uet,
Bellini:2014fua,Noller:2018wyv}.
The other two Horndeski paramerers $\alphaK$ (kineticity), and $\alphaB$
(braiding) do not affect the tensor-mode perturbations $\hij$.
From Ref.~\cite{Bellini:2014fua} (copied here), the different parameters are:
\begin{itemize}
    \item $\alphaK$, kineticity.
    Kinetic energy of scalar perturbations arising directly from the action.
    It is the only contribution of perfect-fluid models but is not present at all in 
    archetypal ``modified gravity'' models such as $f(R)$ and $f(G)$.
    Large values act to suppress the sound speed of scalar perturbations.
    Contribution from all of $K$, $G_3$, $G_4$, $G_5$.
    \item $\alphaK$, braiding.
    Signifies braiding, or mixing of the kinetic terms of the scalar and metric. Contributes to kinetic energy of scalar perturbations indirectly, by backreacting 
    through gravity.
    Second time derivatives of metric and scalar field to appear in equations of
    motion for both the scalar and the metric.
    Causes dark energy to cluster.
    Contributions from $G_3$, $G_4$, $G_5$.
    \item $\alphaM$, Planck-mass run rate.
    Rate of evolution of the effective Planck mass.
    A constant redefinition of the Planck mass does not affect physics.
    Its time evolution in the Jordan frame of the observer creates anisotropic stress. 
    Contributions from $G_4$ and $G_5$.
    \item $\alphaT$ tensor speed excess.
    Deviation of the speed of gravitational waves from that of light.
    This violation of Lorentz-invariance for tensors also changes the response of the
    Newtonian potential $\Psi$ to matter sources even in the presence of no scalar 
    perturbations, leading to anisotropic stress.
    Contributions from $G_4$ and $G_5$.
\end{itemize}
The GW observation  GW170817 and its electromagnetic counterpart puts limits
on $\alphaT$\cite{Kreisch:2017uet},
\begin{equation}
    -6\times 10^{-15} \leq \alphaT \leq 1.4 \times 10^{-5}.
\end{equation}

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{table_horndeski.png}
    \caption{Table 1 of Ref.~\cite{Bellini:2014fua}}
    \label{table}
\end{figure}

\begin{thebibliography}{}

\bibitem{Horndeski:1974wa}
G.~W.~Horndeski,
``Second-order scalar-tensor field equations in a four-dimensional space,''
Int. J. Theor. Phys. \textbf{10} (1974), 363-384.
%doi:10.1007/BF01807638

\bibitem{Bellini:2014fua}
E.~Bellini and I.~Sawicki,
``Maximal freedom at minimum cost: linear large-scale structure in general modifications of gravity,''
JCAP \textbf{07} (2014), 050.
%doi:10.1088/1475-7516/2014/07/050
%[arXiv:1404.3713 [astro-ph.CO]].

\bibitem{Kreisch:2017uet}
C.~D.~Kreisch and E.~Komatsu,
``Cosmological Constraints on Horndeski Gravity in Light of GW170817,''
JCAP \textbf{12} (2018), 030.
%doi:10.1088/1475-7516/2018/12/030
%[arXiv:1712.02710 [astro-ph.CO]].

\bibitem{Noller:2018wyv}
J.~Noller and A.~Nicola,
``Cosmological parameter constraints for Horndeski scalar-tensor gravity,''
Phys. Rev. D \textbf{99} (2019) no.10, 103502.
%doi:10.1103/PhysRevD.99.103502
%[arXiv:1811.12928 [astro-ph.CO]].

\end{thebibliography}

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