A question from a paper on perturbation theory

[Safinaz]

TL;DR

I try to read a paper on general relativity and perturbation theory:

https://arxiv.org/pdf/2109.01398.pdf#page16

My question is about equation (3.16) in section (3.2):

$\delta \phi = -\sqrt{\frac{2}{\epsilon}} \left( \Phi+ \frac{\Phi’}{H} \right)$

Where $\delta \phi$ is the first-order perturbation of a scalar field, $\Phi$ is the first-order perturbation of the space-time metric, and $H$ is the universe’s scale factor. It’s mentioned that this relation is given in reference:

https://arxiv.org/pdf/1002.0600.pdf

But I can't find such a relation in that reference. So I appreciate if someone had more experience in perturbation theory help me find out how this equation can be derived or proved from : https://arxiv.org/pdf/1002.0600.pdf

 

[Mordred]

ok it may help to look at the equation in regards to the constraints to a single scalar field. Now while the second paper doesn't show that equation specifically it does provides the momentum constraints in section B see 33,34,35
further detail can be referenced by the second papers reference "Non-Gaussian features of primordial fluctuations in single field inflationary models" reference 3 of the second paper see section 2.4
https://arxiv.org/pdf/astro-ph/0210603.pdf

hope that helps
edit the H is likely the Hamilton in that equation as opposed to the Hubble parameter.