= Perturbed Klein-Gordon equation with background lapse
{title2=$\delta\Pi=\mathcal D\delta\phi-\bar\Pi\Psi$}
Let $\mathcal D=\bar N^{-1}\partial_t$ and $\bar\Pi=\mathcal D\bar\phi$. For negative-expansion <extrinsic curvature of a spatial hypersurface>, its trace perturbation is $\kappa$. With $\delta\Pi=\mathcal D\delta\phi-\bar\Pi\Psi$, scalar evolution is $\mathcal D\delta\Pi-\Psi\mathcal D\bar\Pi+3H\delta\Pi-\kappa\bar\Pi-\Delta\delta\phi+V_{,\phi\phi}\delta\phi=0$. Combining background acceleration terms produces the standard lapse source $-\bar\Pi(\kappa+\mathcal D\Psi-3H\Psi)+2\Psi V_{,\phi}$.
Back to article page
