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The matter power spectrum describes the density contrast of the universe (the difference between the local density and the mean density) as a function of scale. It is the Fourier transform of the matter correlation function. On large scales, gravity competes with cosmic expansion, and structures grow according to linear theory. In this regime, the density contrast field is Gaussian, Fourier modes evolve independently, and the power spectrum is sufficient to completely describe the density field. On small scales, gravitational collapse is non-linear, and can only be computed accurately using N-body simulations. Higher-order statistics are necessary to describe the full field at small scales.
