Quantum fluctuations of geometry in hot Universe

The fluctuations of spacetime geometries at finite temperature are evaluated within the linearized theory of gravity. These fluctuations are described by the probability distribution of various configurations of the gravitational field. The field configurations are described by the linearized Riemann-Weyl tensor without any reference to the metric. The probability distribution of various configurations is described by the Wigner functional of the gravitational field. It has a foam-like structure, dominant configurations are those with large changes of geometry at nearby points. In the high-temperature limit one obtains the Bolzmann distribution that enables one to identify the expression for the total energy of the gravitational field. The appearance of the same expression for the total energy when the gravitational field is treated as a collection of gravitons and as the high-temperature limit of the Wigner functional proves the consistency of the whole procedure. Striking differences are found between the fluctuations of the electromagnetic field and the gravitational field, among them is the divergence in the gravitational case of the probability distribution at zero temperature. This divergence is of the "infrared type" because it occurs in integrals over the wave vector at small $k$.