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

The Elements of Gravitational Lensing

Here we review the elements of gravitational lensing, in particular the dependence of these elements on the cosmological parameters. This will facilitate the interpretation of the results presented in the following section.

**Table 1:** Parameters of the 43 cosmological models, including, for each model, the independent parameters $\Omega _0$ , $\lambda _0$ , H₀, and $\sigma _8$ , and the dependent parameter n. H₀ is in units of
$\rm km\,s^{-1}Mpc^{-1}$
$\Omega _0$	$\lambda _0$	H₀	$\sigma _8$	n	$\Omega _0$	$\lambda _0$	H₀	$\sigma _8$	n

0.2	0.0	55	0.3	1.2187	0.5	0.5	65	0.8	0.7808
0.2	0.0	65	0.3	1.0966	0.5	0.5	65	1.0	0.8807
0.2	0.0	65	0.5	1.3188	0.5	0.5	75	0.8	0.7049
0.2	0.0	75	0.3	0.9993	0.5	0.5	75	1.0	0.8024
0.2	0.0	75	0.4	1.1228	0.7	0.0	65	0.9	0.8461
0.2	0.0	75	0.5	1.2190	0.7	0.0	65	1.1	0.9346
0.2	0.0	75	0.6	1.2979	0.7	0.0	75	0.9	0.7773
0.2	0.0	75	0.7	1.3648	0.7	0.0	75	1.1	0.8648
0.2	0.0	85	0.3	0.9191	0.7	0.3	65	0.9	0.7720
0.2	0.8	55	0.8	1.2057	0.7	0.3	65	1.1	0.8601
0.2	0.8	65	0.6	0.9326	0.7	0.3	75	0.9	0.7042
0.2	0.8	65	0.7	1.0062	0.7	0.3	75	1.1	0.7912
0.2	0.8	65	0.8	1.0702	1.0	0.0	55	1.0	0.8465
0.2	0.8	65	0.9	1.1269	1.0	0.0	65	0.9	0.7234
0.2	0.8	65	1.0	1.1568	1.0	0.0	65	1.0	0.7698
0.2	0.8	75	0.6	0.8273	1.0	0.0	65	1.1	0.8120
0.2	0.8	75	0.8	0.9629	1.0	0.0	65	1.2	0.8506
0.2	0.8	85	0.8	0.8749	1.0	0.0	65	1.3	0.8861
0.5	0.0	65	0.8	0.9457	1.0	0.0	75	1.0	0.7094
0.5	0.0	65	1.0	1.0439	1.0	0.0	75	1.2	0.7893
0.5	0.0	75	0.8	0.8686	1.0	0.0	85	1.0	0.6605
0.5	0.0	75	1.0	0.9656

The cosmological distances: The angular displacement caused by lensing depends on the angular diameter distances between the source and the observer, $D_{\rm S}$ , the source and the lens, $D_{\rm LS}$ , and the lens and the observer, $D_{\rm L}$ . These distances depend on $\Omega _0$ , $\lambda _0$ , and H₀, but not $\sigma _8$ .
The mean background density: The importance of lensing depends on the mean density of matter between the source and the observer, which is proportional to $\Omega_0H_0^2$ . Hence, the mean density depends on H₀ and $\Omega _0$ , but not $\lambda _0$ and $\sigma _8$ .
The large-scale structure: Most of the matter responsible for lensing is located at intermediate redshift z_L, half-way between the source and the observer. The rms density fluctuation
$\sigma_{8,\rm L}$ at that redshift is given by

$\sigma_{8,\rm L}\approx\sigma_8/{\cal L}(z_{\rm L},0)$ , where

${\cal L}(z_{\rm L},0)$ is the linear growth factor between redshifts $z_{\rm L}$ and z=0, which depends on $\Omega _0$ and $\lambda _0$ . Thus,

$\sigma_{8,\rm L}$ depends on $\sigma _8$ , $\Omega _0$ , and $\lambda _0$ , but not H₀.