G +X} v E_G>
,$$ where in the radiation-dominated period of the evolution of the Universe $n_a ,n_b \sim T^3$ and $ E_G \sim T$. This yields $$\f{d\h{\r}}{dt}\sim -\k^2 T^8
.$$ Here $T$ is temperature of the Universe and $\k^2 =8\pi/M^3$, where $M$ is the 5D Planck mass, is the 5D gravitational constant. Although at the late stages of the cosmological evolution the energy loss to the bulk is small, at high temperatures this effect can be significant.

In the present paper we consider the problem of graviton production and emission to the bulk in a model with one 3-brane embedded in the bulk with one extra dimension. We perform our calculations in a picture in which the metric is time-dependent and the brane is located at a fixed position in the extra dimension.

The Einstein equations including the terms due to graviton production are solved in the perturbative approach. In the leading order we neglect the graviton production, and include it in the next order. The classical background metric is the leading-order solution of the 5D Einstein equations. Background metric is a warped extension of the metric on the brane to the bulk with the warping factor $e^{\pm\m
y}$ 