  We use the recently developed Kinetic Field Theory (KFT) for cosmic structure
formation to show how non-linear power spectra for cosmic density fluctuations
can be calculated in a mean-field approximation to the particle interactions.
Our main result is a simple, closed and analytic, approximate expression for
this power spectrum. This expression has two parameters characterising
non-linear structure growth which can be calibrated within KFT itself. Using
this self-calibration, the non-linear power spectrum agrees with results
obtained from numerical simulations to within typically $\lesssim10\,\%$ up to
wave numbers $k\lesssim10\,h\,\mathrm{Mpc}^{-1}$ at redshift $z = 0$. Adjusting
the two parameters to optimise agreement with numerical simulations, the
relative difference to numerical results shrinks to typically $\lesssim 5\,\%$.
As part of the derivation of our mean-field approximation, we show that the
effective interaction potential between dark-matter particles relative to
Zel'dovich trajectories is sourced by non-linear cosmic density fluctuations
only, and is approximately of Yukawa rather than Newtonian shape.
