Magnification Probability Distribution Functions of Standard Candles in a Clumpy Universe

Abstract

Lensing effects on light rays from point light sources, such like Type Ia supernovae, are simulated in a clumpy universe model. In our universe model, it is assumed that all matter in the universe takes the form of randomly distributed objects each of which has finite size and is transparent for light rays. Monte-Carlo simulations are performed for several lens models, and we compute probability distribution functions of magnification. In the case of the lens models that have a smooth density profile or the same degree of density concentration as the spherical NFW (Navarro-Frenk-White) lens model at the center, the so-called gamma distributions fit well the magnification probability distribution functions if the size of lenses is sufficiently larger than the Einstein radius. In contrast, the gamma distributions do not fit the magnification probability distribution functions in the case of the SIS (Singular Isothermal Sphere) lens model. We find, by using the power law cusp model, that the magnification probability distribution function is fitted well using the gamma distribution only when the slope of the central density profile is not very steep. These results suggest that we may obtain information about the slope of the central density profiles of dark matter halo from the lensing effect of Type Ia supernovae. In the case of point mass lens, the magnification probability behaves ∼ μ −3 when μ 1 (see, e.g., Ref. 53) or 54)).

New stochastic approach to cumulative weak lensing

“…In our setup the halos are randomly distributed with a comoving number density n c and all the matter in the universe is within these objects [see Eq. (13)]. We will later generalize this picture for a continuous mass distribution of halos.…”

“…The effect of matter clumping into isolated halos, or isolated cores, was already considered by Kantowski in 1969 [3], and more recent analyses of gravitational lensing by statistically distributed inhomogeneities have been carried out, for example, in Refs. [4][5][6][7][8][9][10][11][12][13][14][15]. Lately many authors have studied exactly solvable models for large-scale inhomogeneities, such as swiss-cheese, onion and meatball models [3,[16][17][18][19][20][21][22][23][24].…”

Effect of gravitational lensing on the distribution of gravitational waves from distant binary black hole mergers

“…Cosmological PDFs of lensing magnifications have been studied using various methods including ray-tracing in numerical simulations (e.g., Wambsganss et al 1997;Hamana, Martel, & Futamase 2000;Takada & Hamana 2003;Hilbert et al 2007Hilbert et al , 2008Takahashi et al 2011;Castro et al 2018) as well as analytical approaches (e.g., Schneider & Weiss 1988;Holz & Wald 1998;Perrotta et al 2002;Wyithe & Loeb 2002;Yoo et al 2008;Lima, Jain, & Devlin 2010;Kainulainen & Marra 2011;Lapi et al 2012;Fialkov & Loeb 2015). These studies showed that the magnification PDF significantly deviates from the Gaussian distribution such that it has a long tail toward high magnifications.…”

Gravitational lensing with three-dimensional ray tracing★

“…This result was originally derived analytically for a random distribution of compact lenses where only one lens dominates; however, Pei (1993) have noted that for higher optical depths, the slope of tail may become shallower. The precise shape of the distribution depends on the assumed density profile of the lenses (Yoo et al 2008). The spread in this distribution increases with source redshift (Babul & Lee 1991).…”