Gauge Invariant Density and Temperature Perturbations in the Quasi-Newtonian Formulation

We give an improved formalism for calculating the evolution of density fluctuations and temperature perturbations in flat universes. Our equations are general enough to treat the perturbations in collisionless relics like massive neutrinos. We find this formulation to be simpler to use than gauge dependent and other gauge-invariant formalisms. We show how to calculate temperature fluctuations (including multipole moments) and transfer functions, including the case of collisionless relics like massive neutrinos. We call this formalism "quasi-Newtonian" because the equations for the potential and cold matter fluctuation evolution have the same form as the Newtonian gravitational equations in an expanding space. The density fluctuation variable also has the same form inside and outside of the horizon which allows the initial conditions to be specified in a simple intuitive way. Our sample calculations demonstrate how to use these equations in cosmological models which have hot, cold, and mixed dark matter and adiabatic (isentropic) or isocurvature modes. We also give an approximation which may be used to get transfer functions quickly.