Exact solution of the Lifshitz equations governing the growth of fluctuations in cosmology

The exact solution of the Lifshitz equations governing the cosmological evolution of an initial fluctuation is presented. Lifshitz results valid for c/subs/2 = 0 and c/subs/2 = 1/3 are extended in closed form to any equation of state of the form p = c/subs/2epsilon. The solutions embody all the results found previously for special cases of c/subs/2. It is found that the growth of any initial fluctuation is only like t/subn/ with n < or = 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 approx. = 10-43 sec. At those early times the masses allowed by causality requirements are, however, too small to be of interest in galaxy formation