The cosmic energy equation

A more useful approach to non-linear correlations would be to derive results from the BBGKY hierarchy which do not depend either on g(1, 2) being small, or on the absence of higher order correlations. In the limit as all correlations become non-linear such results would follow from Liouville's equation itself. Since Liouville's equation is equivalent to the equations of motion such results should also follow from orbit theory.

Conservation of energy applies to non-linear correlations over large regions just as it applies locally. However, in an expanding system we must allow for two modifications. First, expansion causes the kinetic energy of peculiar motions within a given region to decrease. This opposes the growth of irregularities which increase peculiar velocities. Second, correlations extend across the boundaries of any volume. What happens within the volume depends also on clustering around the region. Unlike a perfect gas, an arbitrary subvolume is not self-contained.

The cosmic energy equation describes this situation. It follows directly from the galaxies' equations of motion (Irvine, 1961; Layzer, 1963). Having set up the apparatus of the BBGKY hierarchy in the previous sections we can use it to derive this cosmic energy equation by following the approach of Fall & Severne (1976). From the two lowest equations of the hierarchy we obtain a powerful result valid to all orders for any irregularity in the distribution of galaxies, whether linear or not.