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.