= \frac{1}{\kappa^2}\int{\rm d} ^3x\sqrt{g} \, \Big\{
        &   -\frac12 \bar{\sigma} h^{\mu\nu}\mathcal{G}_{\mu\nu}(h) +k^{\mu\nu}\mathcal{G}_{\mu\nu}(h)
           +2b_2f^{\mu\nu}\mathcal{G}_{\mu\nu}(f) \\
        &   +(2b_2/\ell^2+\beta)(f^{\mu\nu}f_{\mu\nu}-f^2)-(f^{\mu\nu}k_{\mu\nu}-fk)  \Big\} \,,
\end{split}$$ with $\bar{\sigma}=\sigma+3b_2/(2\ell^4)+\beta/(2\ell^2)$. Newton’s constant $G$ is in $\kappa^2=16\pi G$ and the parameters $\beta$ and $b_2$ have mass dimension minus two and minus four respectively, but are otherwise arbitrary parameters of the model. The linearized Einstein tensor $\mathcal{G}_{\mu\nu}$ — without imposing any gauge condition — takes the form[^1] $$\hskip 1cm
  2 \mathcal{G}_{\mu\nu}(h) = -\nabla^2h_{\mu\nu}-\nabla_\mu\nabla_\nu h+2\nabla_{(\mu}\nabla^\beta h_{\nu)\beta}
                            -\frac{2}{\ell^2} h_{\mu\nu}-g_{\mu\nu}(\nabla_\rho\nabla_\sigma h^{\rho\sigma}-\nabla^2h) \,.$$

All fluctuations $h_{\mu\nu}$, $k_{\mu\nu}$ and $f_{\mu\nu}$ can be split into a transverse-traceless (TT), a trace, and a vector (or gauge) part: $$\begin{split}\label{tensorsplit}\hskip 1cm
 &h_{\mu\nu}(h^{TT},h,\xi) = h_{\mu\nu}^{TT}+\frac13 g_{\mu\nu}h +\nabla_{(\mu}\xi_{\nu)} \\
 &k_{\mu\nu}(k^{TT},\bar{K},v) = k_{\mu\nu}^{TT}+\frac13 g_{\mu\nu}\bar{K} +\nabla_{(\mu}v_{\nu)} \\
 &f_{\mu\nu}(f^{TT},\bar{F},u) = f_{\mu\nu}^{TT}+\frac13 g_{\mu\nu}\bar{F} +\nabla_{(\mu}u_{\nu)} \,.
\end{split}$$ Note that by definition $g^{\mu\nu}h_{\mu\nu}^{TT}=\nabla^\mu h_{\mu\nu}^{TT}=0$ and that the vector parts contribute to the traces of $g^{\mu\nu}k_{\mu\nu}=\bar{K}+2\nabla_\mu v^\mu$ and $g^{\mu\nu}f_{\mu\nu}=\bar{F}+2\nabla_\mu u^\mu$.

Each such decomposition of the fluctuations produces a ’ghost’ factor [@Gaberdiel:2010xv] in the measure of the path-integral (\[pathint\]). $$\hskip 1cm
\mathcal{D}h_{\mu\nu}=Z_{\rm gh}\,\mathcal{D}h_{\mu\nu}^{TT}\mathcal{D}\xi_\mu\mathcal{D}h$$

Another useful split is $$\label{vectorsplit}\hskip 1cm
 u^\mu=u^\mu_T + \nabla^\mu \delta \,,$$ where $\nabla_\mu u^\mu_T=0$. This yields the ghost factor $J_1$. $$\hskip 1cm
\mathcal{D}u_\mu=J_1\,\mathcal{D}u_\mu^T\mathcal{D}\delta$$ The values of $Z_{\rm gh}$ and $J_1$ are given by [@Gaberdiel:2010xv] $$\label{Zghost}\hskip 1cm
 Z_{\rm gh}= 