CBR anisotropy from inflation-induced gravitational waves in mixed radiation and dust cosmology

We examine stochastic temperature fluctuations of the cosmic background radiation (CBR) arising via the Sachs-Wolfe effect from gravitational wave perturbations produced in the early universe. We consider spatially flat, perturbed FRW models that begin with an inflationary phase, followed by a mixed phase containing both radiation and dust. The scale factor during the mixed phase takes the form a(η)=${\mathit{c}}_1$ ${\mathrm{\eta }}^2$+${\mathit{c}}_2$η+${\mathit{c}}_3$, where ${\mathit{c}}_{\mathit{i}}$ are constants. During the mixed phase the universe smoothly transforms from being radiation to dust dominated. We find analytic expressions for the graviton mode function during the mixed phase in terms of spheroidal wave functions. This mode function is used to find an analytic expression for the multiple moments 〈${\mathit{a}}_{\mathit{l}}^2$〉 of the two-point angular correlation function C(γ) for the CBR anisotropy. The analytic expression for the multipole moments is written in terms of two integrals, which are evaluated numerically. The results are compared to multipoles calculated for models that are completely dust dominated at last scattering. We find that the multipoles 〈${\mathit{a}}_{\mathit{l}}^2$〉 of the CBR temperature perturbations for l>10 are significantly larger for a universe that contains both radiation and dust at last scattering. We compare our results with recent, similar numerical work and find good agreement. The spheroidal wave functions may have applications to other problems of cosmological interest.