On the problems of relativistic laboratory astrophysics and fundamental physics with super powerful lasers

The ways toward modeling of astrophysical processes and extreme field regimes with super-power lasers are discussed. The main attention is paid to the problem of limited similarity in using the dimensionless parameters characterizing the processes in the laser and astrophysical plasmas. As the most typical examples, we address the magnetic reconnection and collisionless shock waves relevant to the problem of ultrarelativistic particle acceleration. In the extreme field limits we consider the regimes of dominant radiation reaction, changing the electromagnetic wave-matter interaction. In these regimes it, in particular, results in a new powerful source of ultra high-brightness gamma-rays and will make possible electron-positron pair creation in vacuum in a multiphoton processes. This will allow modeling under terrestrial laboratory conditions the processes in astrophysical objects and paves the way to experimental verifications using ultra intense lasers as they are currently developed within the ELI project.