LIKELY VALUES OF THE COSMOLOGICAL CONSTANT

In theories in which the cosmological constant takes a variety of valu es in different ''subuniverses,'' the probability distribution of its observed values is conditioned by the requirement that there be someon e to measure it. This probability is proportional to the fraction of m atter that is destined to condense out of the background into mass con centrations large enough to form observers. We calculate this ''collap sed fraction'' with a simple, pressure-free, spherically symmetric, no nlinear model for the growth of density fluctuations in a flat univers e with arbitrary value of the cosmological constant, applied in a stat istical way to the observed spectrum of density fluctuations at recomb ination. From this the probability distribution for the vacuum energy density rho(v) for Gaussian random density fluctuations is derived ana lytically. (The conventional quantity lambda(0) is the vacuum energy d ensity 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 sig ma, we adopt the flat, cold dark matter model with a nonzero cosmologi cal constant and fix the amplitude and shape of the primordial power s pectrum in accordance with data on cosmic microwave background anisotr opy from the COBE satellite DMR experiment. A comparison of the result s of this calculation of the likely values of rho(v) with present obse rvational bounds on the cosmological constant indicates that the small , positive values of rho(v) (from 1 to 3 times greater than the presen t 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 smal l values.