Large-scale radial velocity correlations as a test of gaussian initial conditions

Here we test for the Gaussian character of the large-scale mass fluctuation spectrum as a function of scale using a relation between higher order velocity correlation functions of Gaussian random fields. An underlying Gaussian distribution should satisfy the following relationship between the two-point second- and fourthorder radial velocity correlation functions 〈u₁² u₂²〉 - 2〈u₁ u₂〉² - 〈u²(0)〉² = 0 , where u₁, u₂ are the radial velocities of mass points in the cosmic microwave background frame separated by a distance r = |r₁ - r₂|. In addition, the two-point third-order radial velocity correlation function should equal zero. Our analysis checks the consistency of the hypothesis that the initial fluctuation spectrum can be considered an isotropic Gaussian random field with random phase and that the present distribution has grown linearly as a result of gravitational instabilities. Utilizing a simulated Gaussian random velocity field, we investigate the effects of distance errors, thermal motion, selection functions, Malmquist corrections, and different weighting schemes on the correlation functions. We show a superior fit to the underlying correlation functions using a Malmquist correction derived from the actual distribution of estimated distances and an even weighting scheme. These methods are then applied to a sample of elliptical galaxies. The results show a consistency with the assumption of a Gaussian field ; however, the inherent limitations of the data set preclude a definitive answer.