Dependence of cosmological energy-density irregularities on the shape of the scalar-field potential during inflation and "reheating"

Estimates for the baryon-dominated epoch form of the large-scale adiabatic energy-density irregularities generated during an early scalar-field-dominated inflation epoch, in simple inflation-modified hot-big-bang models, are compared to the widely used approximate general expression, which is proportional to the large-scale, gauge-invariant part of $\frac{H^2〈\phi {\phi }^{*}〉}{{\left({\dot{\Phi }}_b\right)}^2}$ evaluated at the first Hubble radius crossing (here ${\Phi }_b$ and $\phi$ are the spatially homogeneous and inhomogeneous parts of the scalar field, $H$ is the Hubble parameter, and an overdot represents a derivative with respect to time). In the de Sitter inflation limit, if the inflation-epoch background scalar-field solution is an "attractor," or if there is sufficient inflation before the scale of interest leaves the Hubble radius, the approximate general expression identically reproduces what we have found. It is also less than an order of magnitude away from our expression in a large fraction of the parameter space of the inflation model we study and is within 2 orders of magnitude of our result in almost all of parameter space. We also show that the more accurate general expression (which the above formula is an approximation of) identically reproduces our results in the simple models studied, provided the inflation-epoch background scalar-field solution is an "attractor" or if there is sufficient inflation. The approximate general formula is used to restudy energy-density inhomogeneities in the quartic-potential scalar-field de Sitter inflation model; the difference between the standard result in this model and our result in related models is traced to a difference in the form of the part of the potential used to model "reheating" and the end of inflation. Very roughly, if (as we have assumed) the inflation-"reheating" transition occurs rapidly on the relevant Hubble time scale, observational data do not unduly constrain the parameters of the inflation model.