## 4. Measured signalThe quantity directly accessible from the galaxy shapes and related to the cosmological model is the variance of the shear . An analytical estimate of it using a simplified cosmological model (power-law power spectrum, sources at a single redshift plane, leading order of the perturbation theory, and no cosmological constant) gives (Kaiser 1992, Villumsen 1996, Bernardeau et al. 1997, Jain & Seljak 1997): where From the unweighted galaxy ellipticities , an estimate of at the position is given by: The inner summation is performed over the The term can be easily removed using a random realization of the galaxy catalogue: each position angle of the galaxies is randomized, and the variance of the shear is calculated again. This randomization allows us to determine and the error bars associated with the noise due to the intrinsic ellipticity distribution. At least 1000 random realizations are required in order to have a precise estimate of the error bars. Note that it is strictly equivalent to use an estimator where the diagonal terms are removed in the sum (9), which suppress automatically the bias. When we take into account the weighting scheme for each galaxy, the estimator Eq. (9) has to be modified accordingly as follows: where Therefore, a systematic of say for a signal of only contributes to in . We investigate in detail in the next sections the term and show that it has a negligible contribution. We will present results on the shear variance measured from the data sets described in Sect. 2. The variance is measured in apertures which are placed on a grid for each of the CCDs. By construction the apertures never cross the CCD boundaries, and if more than of the included objects turns out to be masked objects, this aperture is not used. Fig. 1 shows (thick line) with error bars obtained from 1000 random realizations. The three other thin lines correspond to theoretical predictions obtained from an exact numerical computation for three different cosmological models, in the non-linear regime. We assumed a normalized broad source redshift distribution given by with the parameters are supposed
to match roughly the redshift distribution in our data
sets where is the Fourier transform of a Top-Hat window function, and is the convergence power spectrum, which depends on the projected 3-dimensional mass power spectrum : is the comoving angular diameter
distance out to a distance
We see in Fig. 1 that the measured signal is consistent with the theoretical prediction, both in amplitude and in shape. In order to have a better idea of how significant the signal is we can compare for each smoothing scale the histogram of the shear variance in the randomized samples and the measured signal. This is is shown in Fig. 2, for all the smoothing scales shown in Fig. 1. The signal is significant up to a level of . Note that the measurement points at different scales are correlated, and that an estimate of the overall significance of our signal would require the computation of the noise correlation matrix between the various scales.
© European Southern Observatory (ESO) 2000 Online publication: June 26, 2000 |