Graceful Exit in Extended Inflation and Implications for Density Perturbations

Two qualitatively different modes of ending superluminal expansion are possible in extended inflation. One mode, different from the one envoked in most extended models to date, easily avoids making big bubbles that distort the cosmic microwave background radiation (CMBR). In this mode, the spectrum of density fluctuations is found to be scale-free, P (k) ∝ k, where n might lie anywhere between 0.5 and 1.0 (whereas, previously, it appeared that the range 1.0 > n > ∼ 0.84 was disallowed). Extended inflation was introduced to resolve the fine-tuning problem that plagued previous inflationary models. As in “old inflation,” the scale factor a(t) begins to expand superluminally (a(t) > t) when the universe supercools into a metastable false vacuum phase during a strongly first order phase transition. Unlike old inflation, though, graceful exit can be achieved by bubble nucleation, quantum tunneling through the energy barrier. The added element is modifications to Einstein gravity that slow the expansion from exponential (a(t) = exp(Ht) and constant Hubble parameter H) to polynomial (a(t) > t, but H steadily decreasing). For slowed expansion, the dimensionless parameter ǫ ≡ Γ/H — which measures the competition between the nucleation rate Γ and the expansion rate — steadily increases. Initially, ǫ ≪ 1 and inflation dominates nucleation. In old inflation, ǫ is constant so inflation never stops. In extended inflation, ǫ steadily increases until ǫ > 1, at which point nucleation dominates expansion and the transition to the true vacuum is completed. The purpose of this paper is to show that extended inflation offers two qualitatively distinct modes of “graceful exit.” In the mode considered by most authors, inflation stops when bubbles of true vacuum consume the false vacuum. This mode can create problems of its own, including unacceptable inhomogeneities created by big bubbles. Our point here is to emphasize a second mode in which inflation ends while the universe is still trapped in the false phase. We argue that this mode automatically avoids the big bubble problem. We also examine the implications for the the density perturbation spectrum. In a spectrum characterized by a power-law, P (k) ∝ k, scale invariance (a Harrison-Zel’dovich spectrum) corresponds to n = 1. Previously, avoiding the big bubble problem seemed to require n < 0.84, and, hence, a