# Perturbed stress-energy tensor in a cosmological context?

In the theory of cosmological pertubations, we can write the metric of a null-curvature expanding Universe as :

$ds^2 = -c^2\left(1+2\frac{\psi}{c^2}\right)dt^2 + a^2 \left(1-2\frac{\phi}{c^2}\right)\left(dx^2+dy^2+dz^2\right)$

From that metric we can compute the perturbed Einstein tensor $G_{\mu\nu}$. Now, I search what is the form of the stress energy tensor associated to this in order to write the Einstein equation : $G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$.

So, starting from the fact that we consider a perturbed fluid :

$\rho=\bar{\rho}+\delta\rho$

$P=\bar{P}+\delta P$

what is the $T_{\mu\nu}$ associated to that fluid ?

Remark : Here are some indications (see 4.4.1) : http://www.damtp.cam.ac.uk/user/db275/Cosmology/Chapter4.pdf but I do not understand very well why the author write the $T_{\nu}^{\mu}$ instead of the $T_{\mu\nu}$ (so if someone knows why, please explain it to me).

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Comment to the question (v1): It would be good if OP (or somebody else?) could provide a full reference for the link, because (i) the lecturer might be grateful, and because (ii) the post would be able to be reconstructed in case of future link rot. – Qmechanic Apr 7 '13 at 15:55
I haven't studied this but with respect to your remark, the author in the link explains (near 4.5.69) "it is more convenient to work with the mixed components" and then demonstrates that in the mixed case $\delta {\tilde{T}}^{\mu}_{\nu} = \delta T^{\mu}_{\nu} + {\mathcal{L}}_{\xi} {\bar{T}}^{\mu}_{\nu}$ – twistor59 Apr 7 '13 at 16:16