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

Abstract

Weak lensing leads to the non-Gaussian magnification distribution of standard candles at given redshift $z$, $p(\mu|z)$. 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 $\tilde{\alpha}$ 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 $\tilde{\alpha}$. We define the {\it direction dependent} smoothness parameter $\tilde{\alpha}$ via the Dyer-Roeder equation; there is a unique mapping between $\tilde{\alpha}$ and the magnification of a source. We find that the distribution of $\tilde{\alpha}$ at given $z$, $p(\tilde{\alpha}|z)$, is well described by a modified Gaussian distribution. For the same matter distribution, i.e., the same $p(\tilde{\alpha}|z)$, different values of $\Omega_m$ and $\Omega_{\Lambda}$ 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.