Alexander Polnarev Queen Mary, University of London MG12, Paris, 17 July 2009

1. The Problem of Initial Conditions for Primordial Black Hole Formation and Asymptotic Quasi-Homogeneous Solution. Alexander Polnarev Queen Mary, University of London MG12, Paris, 17 July 2009

2. This presentation is using the results of the following papers: 1. 2009 PhRvD..79d4006. Hidalgo, J. C.; Polnarev, A. G.Probability of primordial black hole formation and its dependence on the radial profile of initial configurations 2. 2007 CQGra..24.1405. Polnarev, A G.; Musco, ICurvature profiles as initial conditions for primordial black hole formation 3. 1989 LNP...332..369. Polnarev, A. G.Primordial black holes and their cosmological consequences 4. 1985 SvPhU.145..369. Polnarev, A. G.; Khlopov, M. Iu.Cosmology, primordial black holes, and supermassive particles1 5. 1979 A&A....80..104. Novikov, I. D.; Polnarev, A. G.; Starobinskii, A. A.; Zeldovich, Ia. B. Primordial black holes 6. 1978 SvA....22..129. Nadezhin, D. K.; Novikov, I. D.; Polnarev, A. G.The hydrodynamics of primordial black hole formation

3. The absence of PBHs can be used as a tool to probe the early Universe The last 30 years attention was concentrated on different aspects of PBH formation Calculating the amount of Hawking radiation emitted by sufficiently small PBHs, with a mass less than 10^15 g, one can obtain important constraints on parameters that characterize the different epochs and processes of the universe

4. The formation of PBHs was considered within different scenarios, for example during phase transitions , with a soft equation of state , by collapse of cosmic loops or from bubble collisions In general, the study of PBHs provides a unique probe for different areas of physics: the early universe, quantum gravity, gravitational collapse and high energy physics

5. Analogy between Collapse to black hole and H-bomb explosion When to set initial conditions?

6. The asymptotic quasi-homogeneous solution in Cosmology is valid when t → 0 An arbitrary curvature profile K(r) corresponding to a large amplitude perturbation of the metric does not depend on time when t → 0 Deviations of energy density and velocity from homogeneous vanish asymptotically as t → 0.

7. We can solve analytically the Eistein equations for the small velocity and density deviations near cosmological singularity This is the self-consistent way (without violation the Bianchi identity) to set all of the initial conditions corresponding to a moment in time when quasi-homogeneous solution of a certain order is valid.

8. The curvature profile K(r) appears as the source in the right-hand side of the relevant equations and we can say that the density and velocity deviations from homogeneity are generated by inhomogeneity of the curvature Then we use the computer code to follow the subsequent nonlinear evolution of the initial configuration (by the configuration we mean the region of strong metric perturbation)

9. To avoid confusion we should emphasize that the small perturbations predicted by any cosmological model are relevant in this context only when one calculates the probability of finding a configuration with a high amplitude perturbation of the metric. In fact, the small perturbations of density and velocity which we discuss here entirely determined by the curvature profile within the initial configuration.

10. Assuming spherical symmetry, it is convenient to divide the collapsing matter into a system of concentric spherical shells and to label each shell with a Lagrangian comoving radial coordinate which we denote as r. Then the metric can be written in the form used by Misner and Sharp :

11. The system of equations

12. The initial perturbations are defined as

13. where

17. Example: No shocks! • Conclusion: Proper set of initial conditions is extremely important