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.

24 pages including 12 figures. Extensive wording changes for clarification, main results unchanged. To apper in ApJ, November 10, 1999, issue, Volume 525 #2

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