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Adams, P.J., and V. Canuto, 1975: Exact solution of the Lifshitz equations governing the growth of fluctuations in cosmology. Phys. Rev. D, 12, 3793-3799, doi:10.1103/PhysRevD.12.3793.
We present the exact solution of the Lifshitz equations governing the cosmological evolution of an initial fluctuation. Lifshitz results valid for cs2=0 and cs2=1/3 are extended in closed form to any equation of state of the form p=cs2 ε. The solutions embody all the results found previously for special cases of cs2. It is found that the growth of any initial fluctuation is only like tn with n ≤ 4/3 and hence insufficient to produce galaxies unless the initial fluctuation is very large. A possible way to produce very large initial fluctuations by modifying the equation of state by including gravitational interactions, as originally suggested by Sakharov, is also examined. It is found that a phase transition can occur at baryonic density of 1 nucleon per cubic Planck length, or equivalently at a time t ~ 10-43 sec. At those early times the masses allowed by causality requirements are, however, too small to be of interest in galaxy formation.
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