CMB temperature fluctuations of the order of a few parts per million imply matter density fluctuations of the same order at the time of recombination.
But the Cosmological Principle implies a Cosmological metric since recombination that depends only on time. Under these assumptions GR then reduces to an equation for the time development of a purely scalar functional DOF, precluding therefore accommodation of these initial but small anisotropies.
My question: how best to modify / refine FRW cosmology to accommodate these anisotropies - assuming for simplicity that they have been frozen since recombination - into a present era 'cosmological' metric.
That is, how should the present era FRW metric be modified to accommodate BAO frozen at the time of recombination?
To incorporate matter density fluctuations implied by cosmic microwave background (CMB) temperature fluctuations into the Friedmann-Robertson-Walker (FRW) metric, you would typically need to consider perturbations in the metric itself as well as in the matter density distribution.
One common approach is to use linear perturbation theory, where you start with the background FRW metric and then introduce small perturbations to the metric and matter density. These perturbations can be characterized using quantities like the Newtonian gauge metric perturbations, scalar, vector, and tensor perturbations, etc.
These perturbations can then be evolved using equations such as the linearized Einstein equations and fluid equations, taking into account the effects of both gravity and the matter content of the universe.
Numerical simulations and analytical techniques are often used to study the evolution of these perturbations and their impact on observable quantities like the CMB temperature fluctuations, galaxy clustering, and large-scale structure.
It's worth noting that incorporating these fluctuations accurately can be complex, and various approximations and numerical methods are often employed in practice. Additionally, the results may depend on assumptions about the nature of dark matter and dark energy, as well as other cosmological parameters.
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