Analytical Modeling of the Weak Lensing of Standard Candles

Weak lensing leads to the non-Gaussian magnification distribution of standard candles at given redshift z, p(μ, z). We derive an accurate and convenient analytical method for computing p(μ, z). Our method is based on the unique mapping between the magnification of a source and the local smoothness parameter α̃, the ratio of the density of matter smoothly distributed in the beam connecting the observer and the source and the average matter density in the universe. We find that the distribution of α̃ at given z, p(α̃, z), is well described by a modified Gaussian distribution; it can in principle be measured directly from observations via the weak lensing studies of field galaxies. For the same matter distribution, i.e., the same p(α̃, z), different values of Ωm and ΩΛ can lead to very different magnification distributions. Our analytical formulae can be used to calculate the weak lensing effects for observed Type Ia supernovae at arbitrary redshifts. Our generalization of the smoothness parameter α̃ to a local variable may have other interesting applications.