Metric Perturbations

The computation of perturbations requires the choice of a gauge, which also means that metric-perturbations are present. On this page, we give a brief overview of our naming convention for the perturbations of a metric $g^{\mu\nu}$. We use the scalar–vector–tensor decomposition into a scalar part $g^{00}$, a vector part $g^{0i}$ and a tensor part $g^{ij}$, with the perturbations given by

$$g^{00}=\frac{-1}{a^2}\left(1-2\Psi\right)$$ $$g^{0i}=\frac{1}{a^2}\left(B_{,i}+b_i^{(v)}\right)$$ $$g^{ij}=\frac{1}{a^2}\left(\eta^{ij}-2\left(-\Phi\eta^{ij}+E_{,i|j}+c^{(v)}_{(i|j)}+c^{(t)}_{ij}\right)\right).$$

• $i$ and $j$ indices run from 1 to 3, $\mu$ and $\nu$ from 0 to 3,
• a comma represents a partial derivative (e.g. $B_{,i}$ is the gradient of the scalar $B$),
• a vertical bar represents a covariant derivative with respect to the 3-metric $\eta_{ij}$,
• $(i|j)$ is a symmetrisation (e.g. $c^{(v)}_{(i|j)}=\left(c^{(v)}_{i|j}+c^{(v)}_{j|i}\right)/2$)
• the tensor $c^{(t)}_{ij}$ is symmetric ($c^{(t)}_{ij}=c^{(t)}_{ji}$), traceless ($c^{(t)i}_{i}=0$), and divergence free ($c^{(t)j}_{i|j}=0$)
• the vector $b_i^{(v)}$ is divergence free ($b_{|i}^{(v)i}=0$)
• In the Newtonian gauge, the scalars $E$ and $B$ are chosen to vanish.

Please note that in our Mathematica notebook xPand.nb a different naming scheme is used than in CLASS. What is $\Psi$ in CLASS is $\phi$ in xPand, and what is $\phi$ in CLASS is $\Psi$ in xPand.