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 in 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 Layzer-Irvine cosmic energy equation. \beta estimation with this equation from the PSC$z$ redshift galaxy survey and the SEcat catalogue of peculiar velocities is carried out, returning a value of \beta = 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 beta could be determined with ~5% 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 nonparametric mass density power spectrum estimator from peculiar velocity data