Luminosity distance in “Swiss cheese” cosmology with randomized voids. II. Magnification probability distributions

We study the fluctuations in luminosity distances due to gravitational lensing by large scale ($\gtrsim 35\mathrm{Mpc}$) structures, specifically voids and sheets. We use a simplified “Swiss cheese” model consisting of a $\Lambda \mathrm{CDM}$ Friedman-Robertson-Walker background in which a number of randomly distributed nonoverlapping spherical regions are replaced by mass-compensating comoving voids, each with a uniform density interior and a thin shell of matter on the surface. We compute the distribution of magnitude shifts using a variant of the method of Holz and Wald [2], which includes the effect of lensing shear. The standard deviation of this distribution is $\sim 0.027$ magnitudes and the mean is $\sim 0.003$ magnitudes for voids of radius 35 Mpc, sources at redshift $z_s=1.0$, with the voids chosen so that 90% of the mass is on the shell today. The standard deviation varies from 0.005 to 0.06 magnitudes as we vary the void size, source redshift, and fraction of mass on the shells today. If the shell walls are given a finite thickness of $\sim 1\mathrm{Mpc}$, the standard deviation is reduced to $\sim 0.013$ magnitudes. This standard deviation due to voids is a factor $\sim 3$ smaller than that due to galaxy scale structures. We summarize our results in terms of a fitting formula that is accurate to $\sim 20\%$, and also build a simplified analytic model that reproduces our results to within $\sim 30\%$. Our model also allows us to explore the domain of validity of weak-lensing theory for voids. We find that for 35 Mpc voids, corrections to the dispersion due to lens-lens coupling are of order $\sim 4\%$, and corrections due to shear are $\sim 3\%$. Finally, we estimate the bias due to source-lens clustering in our model to be negligible.

Figure 1

The line (green) is our analytic model (3.24a) of the median width of the distribution of magnitude shifts $\Delta m$, for $N={10}^4$ samples, source redshift $z_s=1.0$, and void radius $R=35\mathrm{Mpc}$, as a function of the fraction of mass $f_0$ on the void shells today. The data points are from our Monte Carlo simulations with the same parameter values, described in Sec. 4 below.

Figure 2

The magnitude shift $\Delta m$ as a function of source redshift $z_s$ for a single run, for voids of radius $R=35\mathrm{Mpc}$, fraction of mass on the shell today $f_0=0.9$, in a $\Lambda \mathrm{CDM}$ cosmology with ${\Omega }_M=0.3$.

Figure 3

The probability distribution of magnitude shifts $\Delta m$ for a simulation in a $\Lambda \mathrm{CDM}$ cosmology with ${\Omega }_M=0.3$, with sources at redshift $z_s=1$, comoving voids radius $R=35\mathrm{Mpc}$, and fraction of void mass on the shell today $f_0=0.9$.

Figure 4

The estimator ${\widehat{\sigma }}_m$ of the standard deviation of the distribution of magnitude shifts $\Delta m$, as a function of number $N$ of runs, for sources at redshift $z_s=1$, comoving voids radius $R=35\mathrm{Mpc}$, and fraction of void mass on the shell today $f_0=0.9$. The plotted quantity is ${log}_{10}|{\widehat{\sigma }}_m/{\sigma }_m-1|$, where ${\sigma }_m=0.03135$ is an estimate of the $N\to \infty$ limit, here taken from our largest run with $N={10}^6$.

Figure 5

Top the mean $⟨\Delta m⟩$ of the distribution of magnitude shifts $\Delta m$ as a function of source redshift $z_s$, for voids of radius $R=35\mathrm{Mpc}$ with fraction of mass on the shell today $f_0=0.9$, for $N={10}^6$ samples. Bottom the same for $R=100\mathrm{Mpc}$.

Figure 6

The probability distributions of magnitude shifts $\Delta m$ for simulations with sources at redshifts of $z_s=1.1$ (top), $z_s=1.6$ (middle), and $z_s=2.1$ (bottom), for comoving voids of radius $R=35\mathrm{Mpc}$ with 90% of the void mass on the shell today.

Figure 7

The probability distributions of magnitude shifts $\Delta m$, at source redshifts $z_s$ of 1.1 (top), 1.6 (middle), and 2.1 (bottom), as in Fig. 6 except with comoving void radius of $R=100\mathrm{Mpc}$.

Figure 8

The probability distributions of magnitude shifts $\Delta m$, at source redshifts $z_s$ of 1.1 (top), 1.6 (middle), and 2.1 (bottom), as in Fig. 6 except with comoving void radius of $R=350\mathrm{Mpc}$.

Figure 9

The standard deviation ${\sigma }_m$ of the distribution of distance modulus shifts $\Delta m$ as a function of void radius $R$, computed using $N={10}^6$ runs for each point. The bottom line (stars) is for sources at $z_s=1.1$, the middle line (squares) is $z_s=1.6$, and the top line (diamonds) is $z_s=2.1$. Void radii range from 35 to 350 Mpc and the fraction of void mass on the shell today is $f_0=0.9$. The lines are fits of the form ${\sigma }_m\propto \sqrt{R}$.

Figure 10

The standard deviation ${\sigma }_m$ as a function of the fraction $f_0$ of the void mass on the shell today, for void radii of $R=35\mathrm{Mpc}$ and source redshift of $z_s=1$, computed using $N={10}^6$ runs for each point. The dashed curve (blue) is a fit of the form ${\sigma }_m=\alpha f_0+\beta f_0^2$. This plot shows that there are nonlinearities present at the level of $\sim 30–40\%$. The solid curve (green) is the analytic model (3.12, 3.13, 3.14), which is accurate to $\sim 30\%$.

Figure 11

The factor $h(z,f_0)$ by which nonlinear evolution corrects the growth function $D_{+}(z)$ of linear perturbation theory, for our void model. The upper curve is for $f_0=0.9$ and the lower curve is for $f_0=0.5$.

Figure 12

The standard deviation ${\sigma }_m$ as a function of the fraction $f_{\mathrm{mid}}$ of the void mass on the shell for voids halfway to the source, for void radii of $R=35\mathrm{Mpc}$ and source redshift of $z_s=1$. The solid line is a fit of the form ${\sigma }_m=\alpha f_{\mathrm{mid}}+\beta f_{\mathrm{mid}}^2$. For this choice of parameterization there is no statistically significant nonlinearity detectable in the data.

Figure 13

A comparison of the probability distributions of magnitude shifts $\Delta m$ in two different cases: fraction of mass on the shell today fixed at $f_0=0.9$ (circles), and $f_0$ drawn from a distribution as described in the text (stars). In both cases void radius is $R=35$ and source redshift is $z_s=1.0$. The spread in the shell surface densities gives rise to a wider distribution of magnitude shifts, by about $\sim 3\%$.

Figure 14

The standard deviation ${\sigma }_m$ as a function of source redshift $z_s$, computed using $N={10}^6$ runs, for voids of radii $R=35\mathrm{Mpc}$ (red, crossed circles), 70 Mpc (green, squares), and 105 Mpc (blue, circles). The lines are fits proportional to the analytic estimate (4.6).

Figure 15

The estimate (a1) of the matter power spectrum $\Delta (k,z{)}^2$ for our void distribution, as a function of comoving wave number $k$, evaluated today at $z=0$. The lower curve includes the correlation correction factor in square brackets in Eq. (a1), and the middle curve omits it. The upper curve is an approximate version of the nonlinear matter power spectrum at $z=0$ obtained from the Millennium $\Lambda \mathrm{CDM}$ $N$-body simulation [15], shown for comparison. The parameter values chosen were $H_0=73\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, ${\Omega }_M=0.3$, $f_0=f(0)=0.9$, $R=35\mathrm{Mpc}$.

Figure 16

The variance of the lensing convergence per unit logarithmic wave number, $d⟨{\kappa }^2⟩/d\ln k$, for a source at redshift $z_s=1$, computed from the spectra shown in Fig. 15. The upper curve is the Millennium simulation, the lower curve is our void model.