So I'm referring to these lecture notes for the gauge-invariant variables $\zeta$ and $\mathcal{R}$ (around p. 48), the curvature perturbation on uniform-density hypersurfaces and comoving curvature perturbation respectively in the context of cosmological inflation.
In my calculations, I've been using the (+---) signature and these notes use the (-+++) signature, and I've been finding that while the notes state:
$-\zeta = \Psi + \frac{H}{\dot{\bar{\rho}}} \delta \rho$,
I've been finding that:
$\zeta = \Psi - \frac{H}{\dot{\bar{\rho}}} \delta \rho$.
My question is if a different choice of metric signature really is enough for this difference to manifest.
Checking by Mathematica I've found different transformation rules (compared to those in the notes) for the parameters of the perturbed metric but that the gauge-invariant variables in the notes are indeed gauge-invariant.
I've been letting my perturbed line element be the following:
$ds^2 = (1 - 2\Phi) dt^2 + 2aB_i dx^i dt - a^2((1-2\Psi)\delta_{ij} + E_{ij}) dx^i dx^j$,
whereas the notes have
$ds^2 = -(1 + 2\Phi) dt^2 + 2aB_i dx^i dt + a^2((1-2\Psi)\delta_{ij} + E_{ij}) dx^i dx^j$.
