Stress Energy Tensor

The stress energy tensor can be broken up into fluid (f) contributions and seed (s) contributions (see e.g. [18]). The latter is distinguished by the fact that the net effect can be viewed as a perturbation to the background. Specifically $T_{\mu\nu} = \bar T_{\mu\nu} + \delta T_{\mu\nu}$ where $\bar T^0_{\hphantom{0}0} = -\rho_f$,$\bar T^0_{\hphantom{0}i} = \bar T_0^{\hphantom{i}i} =0$and $\bar T^i_{\hphantom{i}j} = p_f \delta^i {}_j$is given by the fluid alone. The fluctuations can be decomposed into the normal modes of §IIC as  

$$\begin{array} {lcl} \delta T^0_{\hphantom{0}0} &=& - [\rho_... ...om{i}j} \vphantom{\displaystyle{\dot a \over a}}\, .\end{array}$$ (31)

for the scalar components,  

$$\begin{array} {rcl} \delta T^i_{\hphantom{0}0} & = & -[(\rh... ...tom{i}j}\vphantom{\displaystyle{\dot a \over a}}\, ,\end{array}$$ (32)

for the vector components, and  

$$\delta T^i_{\hphantom{0}j} = [p_f\pi_f^{{(2)}} + \pi_s^{(2)}] \, Q^{(2)}{}^i_{\hphantom{i}j} \, ,$$ (33)

for the tensor components.