Weak lensing observables in the halo model

The halo model (HM) describes the inhomogeneous universe as a collection of halos. The full nonlinear power spectrum of the universe is well approximated by the HM, whose prediction can be easily computed without lengthy numerical simulations. This makes the HM a useful tool in cosmology. Here we explore the lensing properties of the HM by use of the stochastic gravitational lensing method. We obtain for the case of point sources exact and simple integral expressions for the expected value and variance of the lensing convergence, which encode detailed information about the internal halo properties. In particular a wide array of observational biases can be easily incorporated and the dependence of lensing on cosmology is properly taken into account. This simple setup should be useful for a quick calculation of the power spectrum and the related lensing observables, which can play an important role in the extraction of cosmological parameters from, e.g., supernovae observations. Finally, we discuss the probability distribution function of the HM which encodes more information than the first two moments and can more strongly constrain the large-scale structures of the universe. To check the accuracy of our modeling we compare our predictions to the results from the Millennium Simulation.

Figure 1

Shown is the variance per mass bin for a source at redshift $z=1$. The variance is computed using Eq. (16) restricted to the mass bin $\Delta M=]{10}^n,{10}^{n+1}]h^{-1}{M}_{\odot }$. In the plot $n$ labels the halo mass according to $M_n={10}^n{h}^{-1}{M}_{\odot }$. For halos of mass $M\lesssim {10}^9{h}^{-1}{M}_{\odot }$ the contribution to the total variance is negligible. See Sec. 4 for more details.

Figure 2

Shown is the present-day power spectrum of the halo model (orange solid line and Eq. (1)) for the $\Lambda$CDM universe of the Millennium Simulation, together with the halo (blue dashed line and Eq. (4)) and linear (green dot-dashed line and Eq. (2)) components. Note how the halo component gives a constant contribution at large scales (shot noise). Also shown for comparison is the power spectrum (black dotted line) relative to the Millennium Simulation [24].

Figure 3

Shown is the redshift dependence of the dispersions due to the linear (green dot-dashed line and Eq. (14)) and halo (blue dashed line and Eq. (16)) components, for the $\Lambda \mathrm{CDM}$ model of the Millennium Simulation. Also plotted is the sum (orange solid line) of the variances and the dispersion (black dotted line) relative to the Millennium Simulation [35, 36]. The labeling is as in Fig. 2.

Figure 4

Shown are the lensing PDFs for a source at (from taller to shorter graph) $z=0.83$, 1.08 and 1.50 for the $\Lambda \mathrm{CDM}$ model of the Millennium Simulation. The dotted lines are the lensing PDFs generated by shooting rays through the MS [35, 36], while the histograms are the corresponding lensing PDFs obtained with the sGL method [13, 14, 15].

Figure 5

Shown is the redshift dependence of the convergence dispersion with (red dot-dashed line) and without (blue dashed line) the selection effects of Eq. (18), for the $\Lambda \mathrm{CDM}$ model of the Millennium Simulation. Also plotted is the mean convergence (red solid line) with selection effects (without selection effects it is zero).

Figure 6

Shown is the lensing PDF for a source at $z=1.08$ with (taller histogram) and without (shorter histogram) the selection effects of Eq. (18), for the $\Lambda \mathrm{CDM}$ model of the Millennium Simulation.