Gravitational lensing effects in a time-variable cosmological 'constant' cosmology

A scalar field φ with a potential V(φ)is proportional to φ^-α^(α > 0) has an energy density, behaving like that of a time-variable cosmological `constant', that redshifts less rapidly than the energy densities of radiation and matter, and so might contribute significantly to the present energy density. We compute, in this spatially flat cosmology, the gravitational lensing optical depth, and the expected lens redshift distribution for fixed source redshift. We find, for the values of α~4 and baryonic density parameter {OMEGA}~ 0.2 consistent with the classical cosmological tests, that the optical depth is significantly smaller than that in a constant-{LAMBDA} model with the same {OMEGA}. (For {OMEGA} = 0.2 and source redshift z_s_ = 2.5, in the constant-{LAMBDA} model the optical depth is a factor of ~4.3 larger than in the Einstein-de Sitter case, while in the time- variable {LAMBDA} model, for α = 4, it is a factor of ~2.3 larger than in the Einstein-de Sitter model.) We also find that the redshift of the maximum of the lens distribution falls between that in the constant-{LAMBDA} model and that in the Einstein-de Sitter model.