Self-similar cosmological solutions with dark energy. I. Formulation and asymptotic analysis

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 antigravity. 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 quasistatic 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 quasistatic 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.

Figure 1

The conformal diagram of the flat Friedmann solution for $0<\gamma <2/3$. There is a similarity horizon at $z=z_1$ ($>0$), corresponding to the cosmological event horizon, and a regular center at $z=\infty$ and $r=\infty$. The initial singularity at $t=0$ is null, while null infinity is given by $t=\infty$. The dotted and dot-dashed lines denote the singularity and infinity, respectively. The thin solid curves and lines denote similarity surfaces, i.e. orbits of $z=\mathrm{\text{constant}}$.

Figure 2

The conformal diagram of the Kantowski-Sachs solution for $0<\gamma <2/3$. The solution is analytically extendible beyond $z=\pm \infty$ and $r=\pm \infty$. There are two similarity horizons $z=\pm z_1$, both corresponding to cosmological event horizons. The initial singularity at $t=0$ is null and future null infinity is given by $t=\infty$.

Figure 3

$V_{\infty }$ for the asymptotically constant-velocity solution as a function of $\gamma$. Although only the part of the solid curve with $0<\gamma <2/3$ is relevant to the present analysis, it is extended to $0<\gamma <2$ for completeness. The other root for $V_{\infty }$ is also plotted with a dashed line but this may be unphysical.