Einstein Equations

  In this Appendix, we complete the Boltzmann equations of §III. by giving the Einstein equations for the metric and the matter. We begin with the background evolution and then proceed to the fluctuations. It is occasionally convenient to shift between different representations or gauges and thus we first discuss the transformations that link them. We then derive and present the Einstein equations for scalar, vector and tensor perturbations in a universe with constant comoving curvature in the synchronous and Newtonian gauges (see also [7]).