Second order perturbations

[Claire]

Hello,

I am expanding a Lagrangian to second order in perturbations, and I'm unsure about the terms I'm getting proportional to delta g^2 (see attached image), namely if these terms should really be kept, as normally we just take the g^1 terms and keep the products of them.

If they should be kept, I'm not sure how to substitute the perturbed matrix into this expression. For example, for the g^1 terms I was using delta g[ LI[1], a_ , b_]-> pmetric[a, b], where pmetric is the perturbed metric. How to I define the substitution for g[ LI[2] ,a_ , b_]?

Kind regards,

Claire