Cosmological perturbation theory and relationship to Taylor series?

Question

In cosmological perturbation theory, it's hard to find papers that would expose the general principle to perturb physical quantities (metric, fluid pressure and density, speed...) up to the $n$th order. Given the first and second orders I've found, it seems that the perturbation terms follow a Taylor expansion. The reason behind it is not 100% clear to me and I am looking for an explanation of the link between perturbation theory and Taylor expansion (if that's the case). Also, is there a mathematical demonstration of why it should be of that form (or at least why it's the most optimal)? Links to papers/resources are welcome.

Also, if I write at the first order:

$$G_{\mu\nu}+\delta G_{\mu\nu} = \frac{8\pi G}{c^4}\left(T_{\mu\nu} + \delta T_{\mu\nu}\right).$$

What would be the formula for cosmological perturbations up to the $n$th order?

Answer 1

Weinberg's book on cosmology has explicit details on perturbation theory.

In general, in cosmology, the way perturbed quantities are derived is simple but just computationally laborious. Assume that the background metric is flat and the full metric is a perturbation around it (you can generalize this to a curved background as well). Then, we can write the full metric as

$$ g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} $$

The key point is that the above expression is taken as an exact equality. If this is exactly true, we will see that the inverse metric $g^{\mu\nu}$ is an infinite series in $h$. The first $3$ terms are

$$ g^{\mu\nu} = \eta^{\mu\nu} - h^{\mu\nu} + h^{\mu\alpha} h^{\nu}_{\ \alpha} + \cdots $$

All other quantities (e.g. Ricci scalar, Einstein tensor, etc.) are then written in terms of $g_{\mu\nu}$ and $g^{\mu\nu}$ and then expanded to a desired order in the perturbation $h$.

These days, you can easily get perturbation expressions using tensor algebra software packages. The best one is xAct. Some examples are given in my other answer here.