Based on the asymptotic analysis of ordinary differential equations , we classify all spherically symmetric self-similar solutions to the Einstein equations which are asymptotically Friedmann at large distances and contain a perfect fluid with equation of state p = ( \gamma - 1 ) \mu with 0 < \gamma < 2 / 3 .
This corresponds to a “ dark energy ” fluid and the Friedmann solution is accelerated in this case due to anti-gravity .
This extends the previous analysis of spherically symmetric self-similar solutions for fluids with positive pressure ( \gamma > 1 ) .
However , in the latter case there is an additional parameter associated with the weak discontinuity at the sonic point and the solutions are only asymptotically “ quasi-Friedmann ” , in the sense that they exhibit an angle deficit at large distances .
In the 0 < \gamma < 2 / 3 case , there is no sonic point and there exists a one-parameter family of solutions which are genuinely asymptotically Friedmann at large distances .
We find eight classes of asymptotic behavior : Friedmann or quasi-Friedmann or quasi-static or constant-velocity at large distances , quasi-Friedmann or positive-mass singular or negative-mass singular at small distances , and quasi-Kantowski-Sachs at intermediate distances .
The self-similar asymptotically quasi-static and quasi-Kantowski-Sachs solutions are analytically extendible and of great cosmological interest .
We also investigate their conformal diagrams .
The results of the present analysis are utilized in an accompanying paper to obtain and physically interpret numerical solutions .