A general metric perturbation can be broken up into the normal modes of scalar (m=0), vector () and tensor ( types,
There remains gauge freedom associated with the coordinate choice for the metric perturbations (see Appendix A2). It is typically employed to eliminate two out of four of these quantities for scalar perturbations and one of the two for vector perturbations. The metric is thus specified by four quantities. Two popular choices are the synchronous gauge, where
The stress energy tensor can likewise be broken up into scalar, vector, and tensor contributions. Furthermore one can separate fluid (f) contributions and seed (s) contributions. The latter is distinguished by the fact that the net effect can be viewed as a perturbation to the background. Specifically where ,and is given by the fluid alone. The fluctuations can be decomposed into the normal modes of §IIA as
A minimally coupled scalar field with Lagrangian
These equations hold equally well for relativistic matter such as the CMB photons and the neutrinos. However in that case they do not represent a closed system of equations (the equation of motion of the anisotropic stress perturbations is unspecified) and do not account for the higher moments of the distribution or for momentum exchange between different particle species. To include these effects, we require the Boltzmann equation which describes the evolution of the full distribution function under collisional processes.