Analytical Modeling of the Weak Lensing of Standard Candles I. Empirical Fitting of Numerical Simulation Results

Weak lensing leads to the non-Gaussian magnification distribution of standard candles at given redshift z, p(μ|z). In this paper, we give accurate and simple empirical fitting formulae of the weak lensing numerical simulation results with the generalized Dyer-Roeder prescription. The smoothness parameter α̃ essentially represents the amount of matter that can cause magnification of a given source. Since matter distribution in our universe is inhomogeneous, we can think of our universe as a mosaic of cones centered on the observer, each with a different value of α̃. We define the direction dependent smoothness parameter α̃ via the Dyer-Roeder equation; there is a unique mapping between α̃ and the magnification of a source. We find that the distribution of α̃ at given z, p(α̃|z), is well described by a modified Gaussian distribution. 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 formulae can be conveniently used to calculate the weak lensing effects for observed Type Ia supernovae at arbitrary redshifts.