Imprints of primordial voids on the cosmic microwave background

We generalize in several ways the results existing in the literature regarding the interaction between primordial voids and the cosmic microwave background: (a) we make use of an exact general relativistic solution for a spherical, nearly empty cavity in the matter-dominated era to evaluate the null geodesics and the Sachs-Wolfe effect; (b) we evaluate the magnitude of the adiabatic fluctuations of the photon-baryon plasma; (c) we study the influence of the shell profile; and (d) we take into account the finite thickness of the last scattering surface (LSS) and the influence of its position with respect to the void centre. We find empirically an analytic approximation to the Sachs-Wolfe effect for all crossing geometries and we derive an upper limit of ≈ 25 h−1 Mpc for the comoving radii of voids sitting on the LSS in order to achieve compatibility with COBE data. As a nearly empty void has an overcomoving expansion of a factor of ≈ 4 between decoupling and the present, the maximum allowed size at present is ≈ 100 h−1 Mpc. On the other hand, the smallness of the comoving size relative to the sound horizon reduces strongly the adiabatic effect by Silk damping and makes it negligible. Most of the signature of primordial voids comes therefore from metric effects and consists of subdegree spots that are blue or red depending on whether their centre lies beyond or within the LSS. In conclusion we refine and confirm earlier constraints on a power-law void spectrum originating in an inflationary phase transition and capable of generating the observed large-scale structure.