Gravitational instability and 2-D galaxy surveys

Perturbation theory allows to relate the variance of cosmological density fluctuations on a given spatial smoothing scale to their gravitationally induced skewness. This prediction can be used for observational tests of the gravitational instability theory and as a constraint on models with strongly non-Gaussian initial conditions. Because of a huge number of objects contained in angular surveys and absence of uncertainties connected with the redshift-distance relation, a very important goal is to obtain results applicable to 2-D surveys of galaxies. Using the small angle approximation, we calculate variance and skewness of the projected density field. Spatial smoothing is performed by a 2-D counterpart of a top-hat filter, which is a cone filter. We obtain a semi-analytic formula for skewness for arbitrary power spectra of density fluctuations thus extending the results of Bernardeau (1995) applicable for pure power-law power spectra only. The observed power spectrum in the APM survey cannot be fitted by a pure power-law. Consequently, our formula with an observationally motivated model for the power spectrum allows to obtain much better agreement between the predicted value of skewness and that measured in the APM survey.