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The observed large scale structure of the Universe is thought to be due to the gravitational growth of density fluctuations in the post-inflation era. In this model, the evolving cosmic web is governed by non-linear gravitational growth of the initially weak density fluctuations in the dark energy dominated cosmology. The web is traced by a tiny fraction of luminous baryonic matter. Cosmological shock waves are an essential and often the only way to power the luminous matter by converting a fraction of gravitational power to thermal and non-thermal emissions of baryonic/leptonic matter.

At high redshifts (z > 1100) the pre-galactic medium was hot, relatively dense, ionised, with a substantial pressure of radiation. The cosmic microwave background (CMB) observations constrain the amplitudes of density inhomogeneities to be very small at the last scattering redshift z ~ 1000. Strong non-linear shocks are therefore unlikely at that stage. The universe expands, the matter cools, and eventually recombines, being mostly in neutral phase during the "dark ages" of the universe. At some redshift, 6 < z < 14, hydrogen in the universe is reionised, likely due to UV radiation from the first luminous objects, leaving the intergalactic medium (IGM) highly reionised (see e.g. Fan et al. 2006 for a recent review). The reionisation indicates the formation of the first luminous objects at the end of the "dark ages", either star-forming galaxies or Active Galactic Nuclei (AGN). The compact luminous objects with an enormous energy release would have launched strong (in some cases, relativistic) shock waves in the local vicinity of the energetic sources. At the same evolution stage, formation of strong density inhomogeneities in the cosmic structure occurs. Since then the non-linear dynamical flows in the vicinity of density inhomogeneities would have created large scale cosmic structure shocks of modest strength, thus heating the baryonic matter and simultaneously producing highly non-equilibrium energetic particle distributions, magnetic fields and electromagnetic emission.

Most of the diffuse X-ray emitting matter was likely heated by cosmological shocks of different scales. Accretion and merging processes produce large-scale gas shocks. Simulations of structure formation in the Universe predict that in the present epoch about 40% of the normal baryonic matter is in the Warm-Hot Intergalactic Medium (WHIM) at overdensities delta ~ 5-10 (e.g. Cen & Ostriker 1999, Davé et al. 2001). The WHIM is likely shock-heated to temperatures of 105 - 107 K during the continuous non-linear structure evolution and star-formation processes.

The statistics of cosmological shocks in the large-scale structure of the Universe were simulated in the context of the LambdaCDM-cosmology using PM / Eulerian adiabatic hydrodynamic codes (e.g. Miniati et al. 2000, Ryu et al. 2003, Kang et al. 2007) and more recently with a smoothed particle hydrodynamic code by Pfrommer et al. (2006). They identified two main populations of cosmological shocks: (i) high Mach number "external" shocks due to accretion of cold gas on gravitationally attracting nodes, and (ii) moderate Mach number (2 leq Ms leq 4) "internal" shocks. The shocks are due to supersonic flows induced by relaxing dark matter substructures in relatively hot, already shocked, gas. The internal shocks were found by Ryu et al. (2003) to be most important in energy dissipation providing intercluster medium (ICM) heating, and they were suggested by Bykov et al. (2000) to be the likely sources of non-thermal emission in clusters of galaxies.

Hydrodynamical codes deal with N-body CDM and single-fluid gas dynamics. However, if a strong accretion shock is multi-fluid, providing reduced post-shock ion temperature and entropy, then the internal shocks could have systematically higher Mach numbers.

Space plasma shocks are expected to be collisionless. Cosmological shocks, being the main gas-heating agent, generate turbulent magnetic fields and accelerate energetic particles via collisionless multi-fluid plasma relaxation processes thus producing non-thermal components. The presence of these non-thermal components may affect the global dynamics of clusters of galaxies Ostriker et al. (2005) and the sigmav - T, M - T, LX - T scaling relations Bykov (2005). Detailed discussion of the cosmological simulations of the scaling relations with account of only thermal components can be found in Borgani et al. 2008 - Chapter 13, this volume.

In Sect. 2 we discuss the basic features of the standard collisional shocks. The main part of the review is devoted to physical properties of cosmological shocks with an accent on collisionless shocks and associated non-thermal components. In Sect. 5 we discuss the most important features of multi-fluid collisionless shocks in the cosmological context including the effects of reduced entropy production, energetic particle acceleration and magnetic field amplification in the shocks.

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