Renormalization group and critical behaviour in gravitational collapse

We present a general framework for understanding and analyzing critical behaviour in gravitational collapse. We adopt the method of renormalization group, which has the following advantages. (1) It provides a natural explanation for various types of universality and scaling observed in numerical studies. In particular, universality in initial data space and universality for different models are understood in a unified way. (2) It enables us to perform a detailed analysis of time evolution beyond linear perturbation, by providing rigorous controls on nonlinear terms. Under physically reasonable assumptions we prove: (1) Uniqueness of the relevant mode around a fixed point implies universality in initial data space. (2) The critical exponent ${\beta }_{BH}$ and the unique positive eigenvalue $\kappa$ of the relevant mode is exactly related by ${\beta }_{BH}=\beta \mathrm{/}\kappa$, where $\beta$ is a scaling exponent. (3) The above (1) and (2) hold also for discretely self-similar case (replacing fixed point'' with limit cycle’'). (4) Universality for diffent models holds under a certain condition. According to the framework, we carry out a rather complete (though not mathematically rigorous) analysis for perfect fluids with pressure proportional to density, in a wide range of the adiabatic index $\gamma$. The uniqueness of the relevant mode around a fixed point is established by Lyapunov analyses. This shows that the critical phenomena occurs not only for the radiation fluid but also for perfect fluids with $1<\gamma \lesssim 1.88$. The accurate values of critical exponents are calculated for the models.