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The FLRW metric is used to model our Universe on large cosmological scales. It is a conformally flat metric and the form of stress-energy tensor that we get from Einstein's equations is often equated to the stress-energy tensor for an ideal fluid. This ideal fluid is interpreted to describe a combination of matter fields (dust) and radiations. Now, it is generally the case that any local source of energy and matter distribution would produce a non-zero conformal curvature. So is it really justified to interpret the stress-energy tensor in FLRW cosmology to represent a general matter-energy distribution (dust + radiation)? This model has predicted that the percentage of ordinary matter is only 4%, dark matter 24%, and dark energy 72%. Should we need to revisit these claims if the interpretation of the stress-energy tensor is inaccurate?

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    $\begingroup$ "Now, it is generally the case that any local source of energy and matter distribution would produce a non-zero conformal curvature." It seems like the existence of the FLRW metric is a counterexample to this claim. Do you have any proof or source backing it up? $\endgroup$
    – Javier
    Commented Jun 8, 2021 at 18:55
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    $\begingroup$ I do not have a proof as such, only some observations: standard examples like - Schwarzschild, Kerr. Reissner-Nordstorm solutions or any solutions describing radiations, or take any standard Lagrangians from QFT (non-conformal fields only). In each of these cases , the stress tensor have a definite physical interpretation and have non-zero conformal curvature. So I really want to confirm if there are any physical Lagrangians which can produce a conformally flat space-time. EFEs are in this sense arbitrary, you can assume any metric, but that doesn't necessarily corresponds to a physical field. $\endgroup$
    – paul230_x
    Commented Jun 8, 2021 at 19:15
  • $\begingroup$ Well, yes, the dust and electromagnetic radiation used in cosmology will produce a conformally flat geometry, as you already know. $\endgroup$
    – Javier
    Commented Jun 8, 2021 at 19:24
  • $\begingroup$ Yes I have seen how they are treated in cosmology. But conformal curvature described by radiation and dust are always algebraically special and not just Petrov type O (flat), so are we making any approximations in cosmology? I don't know what it means to have a radiation with Petrov type O Weyl curvature. $\endgroup$
    – paul230_x
    Commented Jun 8, 2021 at 19:38
  • $\begingroup$ I do not understand your assumption regarding flatness: "The FLRW metric is used to model our Universe on large cosmological scale. It is a conformally flat metric ..." Depending on a value for k, the model is flat if and only if k=0. Other k values are for universe models that have curvature. $\endgroup$
    – Buzz
    Commented Jun 9, 2021 at 14:44

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In the FLRW model, you assume isometry and homogeneity of spacetime. Imposing these conditions on the stress tensor forces you to consider a perfect fluid stress tensor. You can show this rigorously by considering a general stress tensor and imposing that the necessary Lie derivatives of the tensor vanish.

Alternatively, pick a generic FLRW metric and compute the Einstein tensor. Notice it can only be matched by a perfect fluid.

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