and introducing the mass of the scalar field $m^2\equiv-\frac{\phi_0}{3+2\omega_{BD}}V''(\phi_0)$ one finds $$\begin{aligned}
\label{eq:ein:lin}
& R_{\mu\nu}^{(1)} - \frac12g_{\mu\nu}R^{(1)}
 =  -\Phi_{,\mu\nu} + \eta_{\mu\nu}\Box_{\eta}\Phi & \\
\label{eq:BD:lin}
& \Box_{\eta}\Phi  =  m^2\Phi, &\end{aligned}$$ where $\Box_{\eta}\equiv\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$, $\Phi\equiv-\frac{\delta\phi}{\phi_{0}}$ and we denote $R_{\mu\alpha\nu\beta}^{(1)}$, $R_{\mu\nu}^{(1)}$ and $R^{(1)}$ the linearizations of the Riemann tensor, Ricci tensor and Ricci scalar to first order in $h_{\mu\nu}$ respectively (the explicit form of the $R^{(1)}_{\mu\alpha\nu\beta}$ is given in [@MTW] and recalled in the Appendix \[A2\]).

One way to obtain the solutions of the linearized field equations (\[eq:ein:lin\]), (\[eq:BD:lin\]) (see e.g.[@MagNic00]) is to define $$\begin{aligned}
\label{eq:theta}
\theta_{\mu\nu} & \equiv & h_{\mu\nu} - \eta_{\mu\nu}\left(\frac12 h - \Phi\right),\qquad h=\eta^{\mu\nu}h_{\mu\nu}\end{aligned}$$ and to use the gauge freedom $h_{\mu\nu}\rightarrow h'_{\mu\nu}$, $\phi\rightarrow\phi'$, $$\begin{aligned}
\label{eq:gfreedom}
h'_{\mu\nu}(x) & = &  h_{\mu\nu}(x) - \zeta_{(\mu,\nu)}, \qquad |\zeta_{\mu}|\ll 1 \\
\phi'(x) & = &  \phi(x), \nonumber\end{aligned}$$ with the gauge parameter $\zeta_{\mu}$ satisfying $$\begin{aligned}
\Box_{\eta}\zeta_{\mu}=\theta_{\mu\nu}{}^{,\nu},\end{aligned}$$ to impose on $\theta_{\mu\nu}$ the Lorentz gauge condition (we omit primes in the transformed fields) $$\begin{aligned}
\label{eq:glor}
\theta^{\mu\nu}{}_{,\nu} & = & 0.\end{aligned}$$ In this gauge the field equations have the form of the wave and the Klein–Gordon equations in the flat spacetime, $$\begin{aligned}
\label{eq:theta:lor}
\Box_{\eta}\theta_{\mu\nu} & = & 0,  \\
\label{eq:BD:lin:lor}
\Box_{\eta}\Phi & = & m^2\Phi,\end{aligned}$$ describing the (superpositions) of the 