COBE limits on explosive structure formation scenarios

We calculate the distortions to the spectrum of the cosmic background radiation (CBR) that would be produced by explosive models for the formation of large-scale structure; we then compare these predictions to the recent measurements of the spectrum made by the Cosmic Background Explorer (COBE) satellite. These models invoke high-energy explosions at redshifts of 7 or greater to catalyze the collapse of material on large mass scales. In our calculations, we take the injection of energy to be instantaneous, we include the effects of inverse Compton cooling and cosmological expansion on the evolution of the resultant bubbles, and we consider both an early noncosmological phase and a late-time cosmological phase (the largest contribution to the spectral distortion is generated in the first phase). Our primary result is the predicted amplitude of the Compton y-distortion; we find roughly y ≳ 10^-3[R_b0/(5 Mpc)]²h³₀f₀ Ω_B, where R_b0 is the average radius of the bubbles today, h₀ is the Hubble constant in units of 100 km s^-1 Mpc^-1, Ω_B is the ratio of the density in baryons to the critical density required to just close the universe, and finally f₀ is the filling factor of bubbles in the universe. (This approximate expression ignores the complicated redshift dependence, which can increase the predicted distortion by as much as a factor of 10 and is discussed in the paper). Recent COBE observations of the cosmic background radiation find y_obs < 10^-3. This limit rules out explosive models as an explanation for structure on scales ∼15 Mpc or greater in a universe with Ω_Bh³₀f₀ ≳ 0.1. If COBE can achieve sensitivities of y ≈ 10^-4, then either a distortion will be detected, or this scenario will be ruled out for structure formation (a) on scales ∼14 Mpc or greater for values of Ω_Bh³₀f₀ ≳ 0.01, and (b) for galactic scales (a few megaparsecs) with Ω_Bh³₀f₀ ≳ 0.1. We also briefly discuss the (large-angle) anisotropy in the microwave background generated by explosive scenarios to compare with COBE anisotropy measurements.