Accurate modeling of weak lensing with the stochastic gravitational lensing method

We revise and extend the stochastic gravitational lensing method (the sGL method) first introduced by Kainulainen and Marra [Phys. Rev. D 80, 123020 (2009)]. Here we include a realistic halo-mass function and density profiles to model the distribution of mass between and within galaxies, galaxy groups, and galaxy clusters. We also introduce a modeling of the filamentary large-scale structures and a method to embed halos into these structures. We show that the sGL method naturally reproduces the weak lensing results for the Millennium simulation. The strength of the sGL method is that a numerical code based on it can compute the lensing probability distribution function (PDF) for a given inhomogeneous model universe in a few seconds. This makes it a useful tool to study how lensing depends on cosmological parameters and its impact on observations. The sGL method can also be used to simulate the effect of a wide array of systematic biases on the observable lensing PDF. As an example we show how simple selection effects may reduce the variance of the observed PDF, which could possibly mask opposite effects from very large-scale structures. We also show how a survey like the Joint Dark Energy Mission could constrain the lensing PDF relative to a given cosmological model. The updated turboGL code is available at turboGL.org.

Figure 1

Shown is an illustrative-only projection of the random matter density within a $100h^{-1}\mathrm{Mpc}$ thick slice generated by our stochastic model. The shaded disks represent clusters and the shaded cylinders the filamentary dark matter structures. Only large clusters are displayed.

Figure 2

Shown is a comoving segment of a photon geodesic intercepting a number of halos represented by shaded disks.

Figure 3

This illustration shows how we project the cylindrical filaments on the plane perpendicular to the geodesic. The triangular section on the right is moved to the left end of the filament so that the projected profile is independent of $l_{\theta }$.

Figure 4

Sketch of a cylindrical filament projected on a plane perpendicular to the geodesic. The effective length of the cylinder is $L_{\theta }$ of Eq. (77) as shown in Fig. 3.

Figure 5

Shown are the lensing PDFs for a source at $z=1.5$ for the $\Lambda \mathrm{CDM}$ model of the Millennium simulation [14]. The dashed line is the lensing PDF generated by shooting rays through the MS [11, 12], while the histograms are obtained with the sGL method. The first (orange) histogram from the top corresponds to a model universe with halos but no filaments, the second (green) is relative to a universe with halos and cylindrical filaments randomly placed, and the third (blue) is relative to the case in which the halos are confined within massless filaments. The fourth (red) histogram is the most realistic case with halos confined within randomly placed massive filaments. The filaments are modeled according to Table  (parameterization I). See Sec. 7a for details.

Figure 6

Shown is the redshift dependence of mode and dispersion $v=\mathrm{FWHM}/2.35$ of the lensing PDF for the $\Lambda \mathrm{CDM}$ model of the Millennium simulation [14]. The data points were obtained with the sGL method (parameterization I of Table ), while the dashed line refers to lensing PDFs generated by shooting rays through the MS [11, 12].

Figure 7

Lensing PDF for a source at $z=1.5$ for the filament parameterization II of Table . The solid (green) histogram is without selection effects, while the dotted (blue) histogram is with the selection effects toy-modeled by Eq. (88). For comparison with Fig. 5, the lensing PDF relative to the Millennium simulation is plotted again as a dashed line. See Sec. 7 for details.

Figure 8

Shown is the redshift dependence of $\sigma$ (top panel), $v=\mathrm{FWHM}/2.35$ (middle panel), and mode (bottom panel) for the filament parameterization II of Table  with (dotted line) and without (dashed line) the selection effects of Eq. (88). The solid line is relative to the filament parameterization I without selection effects.

Figure 9

Lensing PDF for a source at $z=1.5$ for the filament parameterization I of Table  (without selection effects). The histogram simulates a possible PDF obtained with a data set of 200 perfect standard candles. The $2\sigma$ errors are represented as rectangles. The PDF relative to the Millennium simulation is plotted as a dashed line. See Sec. 7c for details.

Figure 10

The same as in Fig. 9 but for a data set of 1000 perfect standard candles.