The Gaussian cell two-point `energy-like' equation: application to large-scale galaxy redshift and peculiar motion surveys

We introduce a simple linear equation relating the line-of-sight peculiar-velocity and density contrast correlation functions. The relation, which we call the Gaussian cell two-point `energy-like' equation, is valid at the distant-observer limit and requires Gaussian smoothed fields. In the variance case, i.e. at zero lag, the equation is similar in its mathematical form to the Irvine-Layzer cosmic energy equation. β estimation with this equation from the Point Source Catalogue Redshift (PSC) survey and the SEcat catalogue of peculiar velocities is carried out, returning a value of β= 0.44 +/- 0.08. The applicability of the method for the 6dF galaxy redshift and peculiar motions survey is demonstrated with mock data where it is shown that β could be determined with ~10 per cent accuracy. The prospects for constraining the dark energy equation of state with this method from the kinematic and thermal Sunyaev-Zel'dovich cluster surveys are discussed. The equation is also used to construct a non-parametric mass-density power-spectrum estimator from peculiar-velocity data.