&& =\
ł[g5]{} &&= e\^[2y]{}([C\_0]{}\^2 +)\^[1/2]{}\
ł[g6]{} &&()\^2 -[C\_0]{}\^2 e\^[2y]{}=C\_4 .Using these relations, it is straightforward to verify that ł[g7]{} ()\^2 +e\^[-2y]{}=const is independent of $\la$..

The energy-momentum tensor of the bulk gravitons is taken in the form of free radiation of massless particles [@lan2] ł[g0]{} Ţ\^[ij]{}(x) =d\^5 p (p\_i p\^i )(x,p) p\^i p\^j , where $\t{f}(x,p)$ is the phase space density of the distribution function of gravitons. The function $\t{f}(x,p)$ is determined by identifying the interaction term in the rhs of the Boltzmann equation (\[i4\]) and $2\c{T}_{04}$ in the $\{04\}$ component of the Einstein equations (\[22a\]) written in the radiation-dominated period, $p=\r/3$, in terms of the unnormalized energy density $\h{\r}$ [@lan2] $$\f{d\h{\r}}{dt} +4H \h{\r}=2 \c{T}_{04}
.$$

The components of momentum $p^i$ of gravitons subject to condition $p_i p^i =0$ are proportional to the tangent vectors to null geodesics ł[o1]{} p\^0= N(y,t) e\^[2y]{}([C\_0]{}\^2 +)\^[1/2]{},p\^4= N(y,t) [C\_0]{} e\^[y]{} ,p\^a (y) = N(y,t) ,where $N(y,t)$ is a normalization factor determined below. Integrating the geodesic equations (\[g4\])-(\[g6\]) with the initial conditions $$y(\la =0) =0,\qquad t (\la =0)=t_0, \qquad x^a (\la =0)=x^a  
,$$ we obtain $x^i =x^i (\la ,C^i , t_0 , x^a )$. Integrating (\[g6\]), we have $$\int^y_0 dy' (C_4 +C^2_0 e^{2\m y'} 