Evolution of Ideal-Fluid Cosmological Perturbations

We present solutions of cosmological perturbations for an ideal flu id medium with a constant pressure (p) to density (μ) relation in a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) background model. Solutions are presented for six different gauge choices which include most of those conventionally used. Both exact and asymptotic solutions are shown in tabular forms. Solutions span a range in w(≡p/μ) from -1 to 1. On sub- horizon scales except for the uniform-density gauge, for general w, the density perturbations in all gauge choices behave in the same way; also for a pressureless medium the behavior is in accord with the Newtonian results. In the large scale (generally meaning scales where the pressure gradient term is negligible, thus corresponding to superhorizon scales for a medium with nonvanishing pressure, and to general scales for a pressureless medium), the integral-form solutions are known which can cover changing w in time as long as the ideal-fluid assumption is valid; i.e., for general p(μ). Using this we show that on superhorizon scales, except for the zero-shear gauge and the uniform-curvature gauge, the growing mode of the perturbed curvature variable φ remains constant considering general p(μ) and the dominating decaying mode cancels out. A central role will be played by an integration constant denoted C which contains information about spatial structures and of course remains constant. Knowing C we can determine the growing mode part of all the variables. On super-horizon scales C can be identified with φ's in many different gauge choices. We also identify the perturbed density variable in the comoving gauge and the perturbed velocity and potential variables in the zero-shear gauge which have correct correspondences with the Newtonian perturbed density, velocity, and potential variables, respectively. We note that in our approach, if the solution of a variable is known in a given gauge, all the rest of the solutions in all the other gauges can be simply derived unless the gauge is ill-behaved. Using the full solutions as a guide, attempts have been made to clarify relations between solutions known in the literature.