Minimal Microwave Anisotropy from Perturbations Induced at Late Times

Aside from primordial gravitational instability of the cosmological fluid, various mechanisms have been proposed to generate large-scale structure at relatively late times, including, e.g., “late-time” cosmological phase transitions. In these scenarios, it is envisioned that the universe is nearly homogeneous at the time of last scattering and that perturbations grow rapidly sometime after the primordial plasma recombines. On this basis, it was suggested that large inhomogeneities could be generated while leaving relatively little imprint on the cosmic microwave background (MBR) anisotropy. In this paper, we calculate the minimal anisotropies possible in any “late-time” scenario for structure formation, given the level of inhomogeneity observed at present. Since the growth of the inhomogeneity involves time-varying gravitational fields, these scenarios inevitably generate significant MBR anisotropy via the Sachs-Wolfe effect. Moreover, we show that the large-angle MBR anisotropy produced by the rapid post-recombination growth of inhomogeneity is generally greater than that produced by the same inhomogeneity grown via gravitational instability. In “realistic” scenarios one can decrease the anisotropy compared to models with primordial adiabatic fluctuations, but only on very small angular scales. The value of any particular measure of the anisotropy can be made small in late-time models, but only by making the time-dependence of the gravitational field sufficiently “pathological”.