The Growth of Primordial Black Holes in a Universe with a Stiff Equation of State

Abstract

It is shown that the Einstein equations permit solutions in which pressure effects cause a black hole to grow as fast as the universe if the equation of state is ‘ stiff ’ (p = µ). This is in contrast to the situation with any softer equation of state |$(p=\alpha \mu ,\,0\leqslant\alpha \lt \text{I})$|⁠, when a black hole cannot grow very much. If the universe is stiff until some time t*, this means that any primordial black holes either formed before then or fed into the universe ab initio will just grow to the horizon size at t*. This implies a lower limit on the mass of a primordial black hole of |${10}^{15}\times ({t}_{*}/{10}^{-23}\,\text{s})\, \text{g}.$| If t* exceeds 10-23s, no primordial black holes could have evaporated by now through the Hawking process. Thus the strong observational limits on the number of 1015 g black holes would no longer exclude the possibility that primordial black holes have a critical density. If the stiff era is extended for as long as possible |$({t}_{*}\,\sim \,{10}^{-4}\text{s})$,| there could be a concentration of primordial black holes with mass around 1 M⊙. This would favour Meszaros' model of galaxy formation, in which galaxies form from the overdensities associated with the statistical excess of 1 M⊙ black holes in some regions.