Likely values of the cosmological constant

In theories in which the cosmological constant takes a variety of values in different "subuniverses," the probability distribution of its observed values is conditioned by the requirement that there be someone to measure it. This probability is proportional to the fraction of matter that is destined to condense out of the background into mass concentrations large enough to form observers. We calculate this "collapsed fraction" with a simple, pressure-free, spherically symmetric, nonlinear model for the growth of density fluctuations in a flat universe with arbitrary value of the cosmological constant, applied in a statistical way to the observed spectrum of density fluctuations at recombination. From this the probability distribution for the vacuum energy density rho(v) for Gaussian random density fluctuations is derived analytically. (The conventional quantity lambda(0) is the vacuum energy density in units of the critical density at present, lambda(0) = rho(v)/rho(crit,0,) where rho(crit,0,) = 3H(0)(2)/8 pi G.) It is shown that the results depend on only one quantity, sigma(3-)rho, where sigma(2) and (rho) over bar> are the variance and mean value of the fluctuating matter density field at recombination, respectively. To calculate sigma, we adopt the flat, cold dark matter model with a nonzero cosmological constant and fix the amplitude and shape of the primordial power spectrum in accordance with data on cosmic microwave background anisotropy from the COBE satellite DMR experiment. A comparison of the results of this calculation of the likely values of rho(v) with present observational bounds on the cosmological constant indicates that the small, positive values of rho(v) (from 1 to 3 times greater than the present cosmic mass density) suggested recently by several lines of evidence are not very unlikely values to observe, even if there is nothing in the a priori probability distribution that favors such relatively small values.