Cosmological Parameter Survey Using the Gravitational Lensing Method
Premana W Premadi , Hugo Martel , Richard Matzner , Toshifumi Futamase, PASA, 18 (2), in press.

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Contents Page: Volume 18, Number 2

Results

The Magnification Distributions

Figure 1 shows the magnification distribution $P(\mu)$ for various models. The top left panel shows the effect of varying $\sigma _8$ . As $\sigma _8$ increases, the peak of the distribution decreases, the low edge of the distribution moves to even lower values, but the right edge is hardly affected. The explanation is the following: a larger $\sigma _8$ implies that (1) the underdense regions are more underdense and the overdense regions are more overdense, and (2) the fraction of the universe occupied by underdense regions (the ``filling factor'') increases while the fraction occupied by overdense regions decreases. In the case of demagnification, these two effects act in the same direction: as $\sigma _8$ increases, the beam is more likely to propagate through an underdense region, and if it does, it will result in stronger demagnification, because these regions are more underdense. In the case of magnification, these effects act in opposite directions, and almost perfectly cancel each other, making the distributions at values of $\mu>1$ appear independent of $\sigma _8$ .

The bottom left panel shows the effect of varying H₀. The distribution is independent of H₀. This results from competing effects: the cosmological distances are shorter in models with large H₀, resulting in a weaker lensing, but this effect is compensated by the fact that at fixed $\Omega _0$ , the mean background density increases like H₀². The top right panel shows the effect of varying $\Omega _0$ . Of all the various dependences, the $\Omega _0$ dependence is the most difficult one to interpret as we are dealing with three concurrent effects. As $\Omega _0$ increases, the dependence upon the mean background density yields stronger lensing effects, while the dependences upon the cosmological distances and the large-scale structure yields weaker lensing effects. The importance of lensing increases with $\Omega _0$ , resulting in a shift of the distribution toward lower values. The dominant effect in this regime is therefore the mean background density.

**Figure 1:** Magnification distributions for various models, showing the effect of varying $\sigma _8$ (top left panel), H₀ (bottom left panel), $\Omega _0$ (top right panel), and $\lambda _0$ (bottom right panel).
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The bottom right panel shows the effect of varying $\lambda _0$ . The presence of a cosmological constant increases the effect of magnification by increasing the cosmological distances. The cosmological constant results in a widening of the distribution, and a shift toward lower magnifications. The high magnification tail depends on the intrinsic properties of the galaxies, but not on their actual locations or level of clustering.

The magnification probability P_M is defined as the probability that a random source is magnified (i.e. $\mu>1$ ). In Figure 2, we plot P_M vs. $\sigma _8$ , for all models. P_M is essentially independent of $\sigma _8$ . This could have been anticipated from Figure 1, which showed that for most models, $P(\mu)$ is independent of $\sigma _8$ for $\mu>1$ .

The Shear Distributions

Figure 3 shows the shear distribution P(a₁/a₂) for various models, where a₁ and a₂ are the major and minor axes of the images, respectively. The top left panel shows the effect of varying $\sigma _8$ . As $\sigma _8$ increases, the peak of the distribution decreases while the high-tail of the distribution increases. This was expected, since the large-scale structure, whose amplitude is measured by $\sigma _8$ , is the primary origin of the shear.

The bottom left panel shows the effect of varying H₀. The curves in each panel are very similar. The absence of dependence upon H₀ results from competing effects. With larger H₀, the mean background density is higher, increasing the effects of lensing, but the cosmological distances are shorter, decreasing the effects of lensing.

The top right panel shows the effect of varying $\Omega _0$ . The dependence upon the mean background density dominates, and consequently the distribution is wider for models with larger $\Omega _0$ .

The bottom right panel shows the effect of varying $\lambda _0$ . As $\lambda _0$ increases, the distributions become wider, indicating that the effect of lensing is stronger. A larger value of $\lambda _0$ results in larger cosmological distances, which is clearly the dominant effect here.

Double Images

For each model, we computed the probability P₂ of finding a double image. The results are plotted in Figure 4. There are 43 points in each panel, corresponding to the 43 different cosmological models considered. There is a strong trend for P₂ to increase with $\lambda _0$ .

**Figure 2:** Magnification probability P_M vs. $\sigma _8$ . The values of $\Omega _0$ and $\lambda _0$ are indicated in each panel. The various symbols correspond to various values of H₀.
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To study the variations of P₂ with the other parameters at fixed $\lambda _0$ , we use different symbols to designate the different values of $\lambda _0$ . The large scatter in the values of P₂ is clearly caused by the dependence of P₂ upon $\lambda _0$ . On all panels, $\lambda _0=0.8$ models (solid circles) are concentrated at the top, while $\lambda _0=0$ models (crosses) are concentrated at the bottom. P₂ is essentially independent of $\Omega _0$ , H₀, and $\sigma _8$ at fixed $\lambda _0$ . These results imply that (i) double images, and multiple images in general, are caused by galaxies and not by the background large-scale structure, and (ii) the strong dependence of P₂ upon $\lambda _0$ indicate that the cosmological distances are the dominant effect in multiple imaging.

The Distribution of Image Separations

**Figure 3:** Shear distributions for various models, showing the effect of varying $\sigma _8$ (top left panel), H₀ (bottom left panel), $\Omega _0$ (top right panel), and $\lambda _0$ (bottom right panel).
$\begin{figure} \psfig{file=figure3.ps,height=12cm,angle=90}\end{figure}$

Figure 5 shows histograms of the angular separations (in arc seconds) of all the double image cases, for models with $\Omega _0=1$ , $\lambda _0=0$ , and $\Omega _0=0.2$ , $\lambda _0=0.8$ . Several trends are apparent. We are considering sources with an angular diameter of 1'', hence the smallest possible image separation is of order 0.5''. Most histograms in Figure 5 show a distribution that rises sharply from 0.5'' to 1'', and then drops slowly at larger separations, with a high-tail that extends to separations of order 4''-6''.

As in the case of the double-image probability P₂, we find no obvious correlation between the shape of the histograms and the value of $\sigma _8$ , again indicating that double images are caused primarily by individual galaxies, and not by the large-scale structure. There is, however, a relationship between the largest angular separations and the value of $\lambda _0$ . For models with $\lambda _0=0$ , the high-tail of the distribution function rarely extends beyond 4'', while for $\lambda _0=0.8$ models, the high-tail often extends to separations of 6''. As for the probability P₂, the shape of the high-tail depends strongly upon the cosmological distances. Increasing these distances results in higher image separations for a given lensing galaxy. This affects the magnification distribution, by extending the high-tail to higher separation, and also the probability P₂, by ``separating'' images that otherwise would have overlapped and been detected as a single image.

For about one of every 4 models, mostly the ones with $\lambda_0>0$ , we see a secondary peak at large separation. Consider for instance the model $\Omega _0=0.2$ , $\lambda _0=0.8$ ,

$H_0=65\rm\,km\,s^{-1}Mpc^{-1}$ , $\sigma_8=0.8$ , which is indicated by an asterisk in Figure 5. There are no double images with separations between 4.00'' and 4.75'', but there are several images with separations larger that 4.75''. This might seem like a very small effect that could be dismissed as a statistical fluctuation, but this feature is found in many histograms, suggesting that it could actually be real. This could possibly result from a coupling between galaxies and large-scale structure. Galaxies are predominantly responsible for multiple imaging. But most galaxies are located inside clusters, where the density of background matter is high. This background matter might amplify the lensing effect of the galaxy, resulting in a peak at high separation angles. This issue requires further investigation.