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We use the non-equlibrium statistical field theory for classical particles,recently developed by Mazenko and Das and Mazenko, together with the freegenerating functional we have previously derived for point sets initiallycorrelated in phase space, to calculate the time evolution of power spectra inthe free theory, i.e. neglecting particle interactions. We provide expressionstaking linear and quadratic momentum correlations into account. Up to thispoint, the expressions are general with respect to the free propagator of themicroscopic degrees of freedom.  We then specialise the propagator to that expected for particles in cosmologytreated within the Zel'dovich approximation and show that, to linear order inthe momentum correlations, the linear growth of the cosmological power spectrumis reproduced. Quadratic momentum correlations return a first contribution tothe non-linear evolution of the power spectrum, for which we derive a simpleclosed expression valid for arbitrary wave numbers. This expression is aconvolution of the initial density power spectrum with itself, multiplied by amode-coupling kernel. We also derive the bispectrum expected in this theorywithin these approximations and show that its connected part reproduces almost,but not quite, the bispectrum expected in Eulerian perturbation theory of thedensity contrast.

Non-equilibrium statistical field theory for classical particles: Linear and mildly non-linear evolution of cosmological density power spectra