Vacuum Zero Point Energy and its Statistical Correlations in dS Background

We study the vacuum zero point energy associated to a scalar field with an arbitrary mass and conformal coupling in a dS background. Employing dimensional regularization scheme, we calculate the regularized zero point energy density, pressure and the trace of the energy momentum tensor. It is shown that the classical relation $\langle T \rangle =-4 \langle \rho \rangle$ for the vacuum stress energy tensor receives anomalous quantum correction which depends on the mass and the conformal coupling while the relation $\langle \rho \rangle = - \langle P \rangle$ does hold. We calculate the density contrast associated to the vacuum zero point energy and show that $\delta \rho \sim \langle \rho \rangle$ indicating an inhomogeneous and non-perturbative distribution of the zero point energy. Finally, we calculate the skewness associated to the distribution of the zero point energy and pressure and show that they are highly non-Gaussian.