Manuscript Em02-E@1 · Em02-E-v1.tex · section · f3795050

Linear sign and convention check

Complete classical lapse expansion › Linear sign and convention check · lines 798–832

Em02-E@1#eq:linear-explicit

In cosmic coordinates, $h^{00} = h_{00}$ and $g_{0}^{00} = - 1$ . Thus

n_{1} = - \frac{h _{00}}{2 v} - \frac{φ ˙}{v ^{2}} + \frac{v ˙ φ}{v ^{3}} = \frac{1}{v} (- \frac{h _{00}}{2} - \partial_{t} \frac{φ}{v}) .

To compare with conformal time, let $d t = a d τ$ . Then $ϕ_{0}^{'} = a v$ , $H = a H$ , and the conventional scalar lapse perturbation satisfies $A = - h_{00} /2$ . Consequently

\frac{a}{ϕ _{0}^{'}} [A - (\partial_{τ} + H) \frac{φ}{ϕ _{0}^{'}}] = \frac{1}{v} [A - \partial_{t} \frac{φ}{v}] .

This matches the off-shell first-order relational lapse construction in BFHPR (equation (36)). No on-shell Bardeen-potential identity is needed here.

Displayed formulas in this passage

Em02-E@1#eq:linear-explicit · line 802
$n_{1} = - \frac{h _{00}}{2 v} - \frac{φ ˙}{v ^{2}} + \frac{v ˙ φ}{v ^{3}} = \frac{1}{v} (- \frac{h _{00}}{2} - \partial_{t} \frac{φ}{v}) .$
unlabeled · line 818
$\frac{a}{ϕ _{0}^{'}} [A - (\partial_{τ} + H) \frac{φ}{ϕ _{0}^{'}}] = \frac{1}{v} [A - \partial_{t} \frac{φ}{v}] .$