This review addresses the issue of whether there are physically realistic self-similar solutions in which a primordial black hole is attached to an exact or asymptotically Friedmann model for an equation of state of the form p = ( \gamma - 1 ) \rho c ^ { 2 } .
In the positive pressure case ( 1 < \gamma < 2 ) , there is no such solution when the black hole is attached to an exact Friedmann background via a sonic point .
However , it has been claimed that there is a one-parameter family of asymptotically Friedmann black hole solutions providing the ratio of the black hole size to the cosmological horizon size is in a narrow range above some critical value .
There are also “ universal ” black holes in which the black hole has an apparent horizon but no event horizon .
It turns out that both these types of solution are only asymptotically quasi -Friedmann , because they contain a solid angle deficit at large distances , but they are not necessarily excluded observationally .
Such solutions may also exist in the 2 / 3 \leq \gamma < \leq 1 case , although this has not been demonstrated explicitly .
In the stiff case ( \gamma = 2 ) , there is no self-similar solution in an exact background unless the matter turns into null dust before entering the event horizon , which is a contrived and probably unphysical situation .
However , there may be asymptotically quasi-Friedmann solutions without a sonic point which contain universal black holes .
In the negative pressure case ( 0 < \gamma < 2 / 3 ) , corresponding to a dark-energy-dominated universe , there is a one-parameter family of black hole solutions which are properly asymptotically Friedmann ( in the sense that there is no angle deficit ) and such solutions may arise naturally in the inflationary scenario .
The ratio of the black hole size to the cosmological horizon size must now be below some critical value , so the range is more extended than in the positive pressure case and one needs less fine-tuning .
If one tries to make a black hole which is larger than this , one finds a self-similar solution which connects two asymptotic regions , one being exactly Friedmann and the other asymptotically quasi-Friedmann .
This might be regarded as a cosmological wormhole solution providing one defines a wormhole throat quasi-locally in terms of a non-vanishing minimal area on a spacelike hypersurface .
The possibility of self-similar black holes in phantom fluids ( \gamma < 0 ) , where the black hole shrinks as the big rip singularity is approached , or tachyonic fluids ( \gamma > 2 ) remains unclear .
We also consider the possibility of self-similar black hole solutions in a universe dominated by a scalar field .
If the field is massless , the situation resembles the stiff fluid case , so any black hole solution is again contrived , although there may still be universal black hole solutions .
The situation is less clear if the scalar field is rolling down a potential and therefore massive , as in the quintessence scenario .
Although no explicit asymptotically Friedmann black hole solutions of this kind are known , they are not excluded and comparison with the 0 < \gamma < 2 / 3 perfect fluid case suggests that they should exist if the black hole is not too large .
This implies that a black hole might grow as fast as the cosmological horizon in a quintessence-dominated universe in some circumstances , supporting the proposal that accretion onto primordial black holes may have played a role in the production of the supermassive black holes in galactic nuclei .