Energy density of standing sound waves at the radiation-dominated phase of the universe expansion (hydrodynamic derivation)
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In the early Universe up to hydrogen recombination in the Universe, the radiation pressure was much greater than the pressure of baryons and electrons. Moreover, the energy density of cosmic microwave background (CMB) photons was greater than or close to the energy density contained in the rest mass of baryonic matter, i.e., the primordial plasma was a radiated-dominated one and the adiabatic index was close to 4/3. The small density perturbations from which the observed galaxies have grown grew as long as the characteristic perturbation scales exceeded the horizon of the Universe сt at that time. On smaller scales, the density perturbations were standing sound waves. Radiative viscosity and heat conduction must have led to the damping of sound waves on very small scales. After the discovery of the cosmic microwave background, J. Silk calculated the scales of this damping, which is now called Silk damping, knowing the CMBtemperature and assuming the density of baryons and electrons. Observations with the South Pole Telescope, the Atacama Cosmology Telescope, and the Planck satellite have revealed the predicted damping of acoustic peaks in the CMB power spectrum and confirmed one important prediction of the theory. In 1970, R.A. Sunyaev and Ya.B. Zeldovich showed that such energy release in the early Universe should lead to characteristic deviations of the CMB spectrum from the Planck one. The development of the technology of cryogenic detectors of submillimeter and millimeter wavelength radiation has made it possible to measure the CMB spectral distortions at 10−8 of its total intensity (PIXIE). This has sharply increased the interest of theoretical cosmologists in the problem of energy release when smallscale sound waves are damped. We have derived a relativistic formula for the energy of a standing sound wave in a photon–baryon–electron plasma from simple hydrodynamic and thermodynamic relations. This formula is applicable for an arbitrary relation between the energy density of photons and the rest energy density of baryons and their thermal energy density. It continuously describes the transition between the two extreme cases. We obtain the expression for a radiation-dominated plasma in one limit and return to the expression for a gas of classicalmassive particles in the other limit. We have derived the relations that relate the amplitudes of velocity, baryon number density, and temperature perturbations in a radiation-dominated plasma of photons, baryons, and electrons.
Keywordsradiation-dominated Universe Silk damping standing sound waves CosmicMicrowave Background Radiation energy release in early Universe
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