Cosmological constant and the time of its dominance

We explore a model in which the cosmological constant L and the density contrast at the time of recombination s rec are random variables, whose range and a priori probabilities are determined by the laws of physics. ~Such models arise naturally in the framework of inflationary cosmology. ! Based on the assumption that we are typical observers, we show that the order of magnitude coincidence among the three time scales, the time of galaxy formation, the time when the cosmological constant starts to dominate the cosmic energy density, and the present age of the universe, finds a natural explanation. We also discuss the probability distribution for s rec . Assuming a power law a priori distribution }s reca we find that for a.3 the most probable values of s rec are near the observationally suggested values, whereas for a,3 the typical s rec would be too large. This may be used to place constraints on inflationary models ~or on any alternative theory of initial conditions!. PACS number~s!: 98.80.Cq, 98.80.Hw During the past year and a half two groups have presented ~independently! strong evidence that the expansion of the universe is accelerating rather than decelerating @1#. This surprising result comes from distance measurements to more than 50 supernovas type Ia ~SNe Ia! in the redshift range z 5 0t oz51.2. While possible ambiguities related to evolution and to the nature of SNe Ia progenitors still exist @2#, the data are consistent with the cosmological constant ~or vacuum energy! contributing to the total energy density about 70% of the critical density ( V L.0.7). At the same time, other methods, and measurements of the anisotropy of the cosmic microwave background, indicate that matter alone contributes about V M.0.3, which when combined with the cosmological constant suggests a flat universe @3#. These findings raise however an extremely intriguing question. It is difficult to understand why we happen to be living in the first and only time in cosmic history in which r M;r L ~where r M is the matter density, and r L the vacuum energy density associated with the cosmological constant!. That is, why t 0;t L , ~1! where t 0 is the present time and t L is the time at which the cosmological constant starts to dominate. Observers living at t!t L would find V M’ 1( V L’0), while observers at t @t L would findV L’ 1( V M’0). There is another, less frequently discussed ‘‘coincidence,’’ which also calls for an explanation. Observationally, the epoch of structure formation, when giant galaxies were