4. Measured signal
The 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 n is the slope of the power spectrum, its normalization, the redshift of the sources and the top-hat smoothing filter radius. The expected effect is at the percent level, but at small scales the non-linear dynamics is expected to increase the signal by a factor of a few (Jain & Seljak 1997). Nevertheless Eq. (8) has the advantage of clearly giving the cosmological dependence of the variance of the shear.
From the unweighted galaxy ellipticities , an estimate of at the position is given by:
The inner summation is performed over the N galaxies located inside the smoothing window centered on , and the outer summation over the ellipticity components. The ensemble average of Eq. (9) is
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 w is the weight as defined in Eq. (7). The variance of the shear is not only the easiest quantity to measure, but it is also fairly weakly sensitive to the systematics provided that they are smaller than the signal. The reason is that any spurious alignment of the galaxies, in addition to the gravitational effect, adds quadratically to the signal and not linearly:
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 7. The shape of this redshift distribution mimics those observed in spectroscopic magnitude-limited samples as well as those inferred from theoretical predictions of galaxy evolution models. Since we did significant selections in our galaxy catalog the final redshift distribution could be modified. We have not quantified this, but we do not think it would significantly change the average redshift of the sample, even if the shape may be modified. The variance of the shear is computed via the formula (see Schneider et al. 1998 for the notations):
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 w ( is the horizon distance), and is the redshift distribution of the sources. The nonlinear mass power spectrum is calculated using a fitting formula (Peacock & Dodds 1996).
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