6 Cosmological constraints

As noted elsewhere (e.g. Bernardeau et al. 1997; Jain & Seljak 1997), the parameters which dominate the 2-point shear statistics are the power spectrum normalization $\sigma_8$ and the mean density $\Omega_0$. We investigate below how the statistics measured in Figs. 3 to 6 are consistent with each other when constraining these parameters. Our parameter estimates below rely on some simplifying assumptions; a more detailed analysis over a wider space of parameters will be presented elsewhere. In particular, as discussed later, the choice of the slope of the power spectrum $\Gamma$ is weakly known, and may significantly affect the parameter estimate. As pointed out in Sect. 4, the uncertainty on the redshift distribution is also a concern, but we partialy address this point by constraining the parameters using three different redshift distributions. A complete analysis involving marginalisation over $\Gamma$ and the redshift distribution using tight priors is left for a future work.

We assume that the data follow Gaussian statistics and neglect sample variance since it is a very small contributor to the noise for our survey, as discussed above. We compute the likelihood function $\cal L$:

$$% {\cal L}={1\over (2\pi)^{n/2} \left\vert\vec{S}\right\vert^... ...t)^T \vec{S}^{-1}\left({\rm {\vec d}-{\vec s}}\right)\right]},$$ (18)

where ${\vec d}$ and ${\vec s}$ are the data and model vectors respectively, and $\vec{S}=\langle\left({\rm {\vec d}-{\vec s}}\right)^T\left({\rm {\vec d}-{\vec s}}\right)\rangle$is the noise correlation matrix. $\vec{S}$ was computed for the different statistics from 200 random realizations of the survey, therefore effects associated with the survey geometry are included in our noise matrix. The model ${\vec s}$ was computed for a grid of cosmological models which covers $\Omega_0\in [0,1]$ and $\sigma_8\in [0.2,1.8]$ with a zero cosmological constant. The prior is chosen to be flat over this grid, and zero outside. We also fixed $\Gamma =0.21$ and we first use the redshift distribution given by Eq. (2) with $(z_0,\alpha ,\beta )=(0.8,2,1.5)$. The two other redshift distributions defined in Sect. 4.1 will be discussed at the end of this section. We discuss below the impact of this choice of priors.

Figure 4 (bottom panel) shows that for effective scales smaller than 1' there is a non-vanishing R-mode which could come either from a residual systematic, or from an intrinsic alignment effect. Therefore it is safer to exclude this part from the likelihood calculation: for the top-hat variance, we excluded the point at 1', for the correlation functions the points below 2', and for the $M_{\rm ap}$ statistic the points below 5'. For the correlation function, we also excluded the points at scales larger than 20' because of the small fluctuations in the measured correlations. The constraints on the cosmological parameters are not significantly affected whether these large scale points are excluded or not.

Figure 9: Likelihood contours in the $\Omega _0 -\sigma _8$ plane from the top-hat smoothed variance $\langle \gamma^2\rangle$shown in Fig. 3. The first point in Fig. 3 was not included in the likelihood calculation to avoid the small scale systematic shown in Fig. 4 (bottom panel). The cosmological models have $\Lambda =0$, with a CDM-type power spectrum and $\Gamma =0.21$. The redshift of the sources is given by Eq. (2). with $(z_0,\alpha ,\beta )=(0.8,2,1.5)$. The confidence levels are (0.68, 0.95, 0.999).

Figures 9 to 13 show the $(\Omega_0, \sigma_8)$ constraints for each of the statistics shown in Figs. 3 to 6. The contours show the $99.9\%$, $95.0\%$ and $68.0\%$ confidence levels. The agreement between the contours is excellent, though the $M_{\rm ap}$ statistic and the radial correlation function do not give as tight constraints as the other statistics. The correlation function measurements below 2' may be considered by using error bars that include a possible systematic bias: this is equivalent to adding a systematic covariance matrix $\vec{S}^{\rm sys}$ to the noise covariance $\vec{S}$ matrix in Eq. (18). The new contours computed with the enlarged error bars are shown in Fig. 14. The maximum of the likelihood in the variance and correlation function likelihood plots is at $\sigma_8\simeq 0.9$ and $\Omega_0\simeq 0.3$. Note that the results are in very good agreement with a similar plot in Maoli et al. (2001) (Fig. 8), here the contours are narrower, and are obtained from a homogeneous data set. Moreover, the degeneracy between $\Omega_0$ and $\sigma_8$ is broken.

Figure 10: As in Fig. 9, but using the $M_{\rm ap}$ statistic of Fig. 4 (top panel) instead of the top-hat variance. The first five points in Fig. 4 were not included in the likelihood calculation in order to avoid the small scale systematic shown in Fig. 4 (bottom panel).

Figure 11: Likelihood contours as in Fig. 9, but using the shear correlation function $\langle\gamma\gamma\rangle_\theta$(Fig. 5) instead of the top-hat variance. The first two points and scales larger than 20' in Fig. 5 were not included in the likelihood calculation to avoid the contribution from the small scale systematic shown in Fig. 4 (bottom panel).

Figure 12: As in Fig. 9, but using the tangential shear correlation function $\langle\gamma_{\rm t}\gamma_{\rm t}\rangle_\theta$ (Fig. 6) instead of the top-hat variance. The first two points and scales larger than 20' in Fig. 6 were not included in the likelihood calculation in order to avoid the contribution from the small scale systematic shown in Fig. 4 (bottom panel).

Figure 13: As in Fig. 9, but using the radial shear correlation function $\langle\gamma_{\rm r}\gamma_{\rm r}\rangle_\theta$ (results in Fig. 6) instead of the top-hat variance. The first two points and scales larger than 20' in Fig. 6 were not included in the likelihood calculation in order to avoid the contribution from the small scale systematic shown in Fig. 4 (bottom panel).

The partial breaking of degeneracy between $\Omega_0$ and $\sigma_8$was expected from the fully non-linear calculation of shear correlations (Jain & Seljak 1997). In the non-linear regime the dependence of the 2-points statistics on $\Omega_0$ and $\sigma_8$ becomes sensitive to angular scale. For example, as shown in Jain & Seljak (1997), the shear rms measures $\sigma_8~\Omega_0^{0.5}$ on scale between 2'-5', and $\sigma_8~\Omega_0^{0.8}$ on scales $\gtrsim\!10'$. Therefore a low $\Omega_0$universe should see a net decrease of shear power at large scale compared to a $\Omega_0=1$ universe (for a given shape of the power spectrum), as is evident in Fig. 3. Note that the aperture mass $M_{\rm ap}$ is still degenerate with $\Omega_0$ and $\sigma_8$(Fig. 10) because it probes effective scales up to ${\sim} 2.6'$ only, which is not enough to break the degeneracy.

Figure 14: Likelihood contours as in Fig. 11, but all the points in Fig. 5 on scales smaller than 20' were used. In order to account for the small scale systematic shown in Fig. 4 (bottom panel) the error bars on the first two points were increased to include the systematic amplitude.

It seems that the aperture mass (Fig. 10) gives a slightly larger $\sigma_8$for a large $\Omega_0$ compared to the other statistics, while they all agree for $\Omega_0 < 0.7$. This could be an indication for a low $\Omega_0$ Universe, however in practice, the probability contours for the different statistics cannot be combined in a straightforward way because they are largely redundant. The best strategy here is to concentrate on one particular statistic. We expect the best constraints from the shear correlation function (since it contains all the information by definition), and therefore base our parameter estimates on the likelihood contours obtained from it. The contours in the $\sigma_8-\Omega_0$ plane in Fig. 14 closely follow the curve $\sigma_8\propto \Omega_0^{0.6}$. This allows us to obtain the following measurement of $\sigma_8~\Omega_0^{0.6}$(from this figure alone):

$$% \sigma_8~\Omega_0^{0.6}\ =\ 0.48^{+0.04(0.06)}_{-0.05(0.07)},$$ (19)

where the uncertainties correspond to the $95\%$ ($99.9\%$) confidence levels. The result in Eq. (19) is derived for CDM models only, and fairly robust against different values of $\Gamma$. These errors are statistical only, and that they do not include systematic error on the redshift distribution and on the value of $\Gamma$.

If we choose a strong prior for $\Gamma$, we can constrain the two parameters separately; for $\Gamma =0.21$we get, at the $95\%$ confidence level: $0.22<\Omega_0<0.55$ and $0.65<\sigma_8<1.2$ for open models and $\sigma_8>0.7$ and $\Omega_0<0.4$ for flat ($\Lambda$-CDM) models. However, this result is clearly sensitive to the prior choosen for $\Gamma$. In particular, if we use the relation $\Gamma=\Omega_0 h$ for a cold dark matter model, then some extreme combinations of $\sigma_8$, $\Omega_0$ and $\Gamma$ cannot be ruled out from lensing alone. The degeneracy between $\Omega_0$ and $\sigma_8$ is broken only if we take $\Gamma$ to lie in a reasonable interval. Such interval can be motivated by galaxy surveys for instance, which give $0.19<\Gamma<0.37$ at $68\%$ confidence level for the APM (Eisentsein & Zaldarriaga 2001). For instance the choice $\Gamma=0.7$ would make $\Omega_0=1, \sigma_8=0.5$ consistent with the data. The second source of uncertainty comes from the redshift distribution, kown only approximately. As discussed in Sect. 4.1 and shown in Fig. 2 we have a rough idea of this distribution, but until we obtain the information on the photometric or spectroscopic redshifts (which is in progress) we cannot guarantee a precise cosmological parameter estimation here. Figures 15 and 16 show the confidence contours as calculated in Fig. 14 but with the two other redshift distributions defined in Sect. 4.1. Despite the large differences of the distribution, in particular for the number of galaxies at z>1.5, it is reassuring that the contours are in fact only slightly modified. The detailed analysis involving a marginalisation over $\Gamma$ and over the redshift distribution of the sources (constrained using photometric redshifts) is left for a forthcoming study. However, for the reasonable values of $\Gamma$, the degeneracy-breaking for the high $\Omega_0$ models is not affected by the present uncertainty on the redshift distribution of the sources. Our result is consistent with the rough guide given by the scaling $\sigma_8~\Omega_0^{0.6} \propto z_0^{-0.5}$ (Jain & Seljak 1997).

Figure 15: Likelihood contours as in Fig. 14, but the source redshift distribution is assumed to be lower, with $(z_0,\alpha ,\beta )=(0.7,2,1.5)$.

Figure 16: Likelihood contours as in Fig. 14, but the source redshift distribution is assumed to be higher , with $(z_0,\alpha ,\beta )=(0.9,2,1.5)$.