)^{-1/2}=\la
.$$ From this relation is determined $\la =\la (y,C^i )$. Inverting this relation, we obtain $y=y(\la ,C^i )$. Because of the initial conditions, $y(0,C^i ,x^a )=0$.

The phase space density of non-interacting massless particles $\t{f}(x^i , p^i )$ as a function of $\la$ is constant along a null geodesic ł[ge]{} = p\^k -\^k\_[rs]{}p\^r p\^s =0 .Because the function $\t{f}(x^i (\la ,C^i , t_0 , x^a ),\, p^i (\la ,C^i , t_0 ,
x^a ))\equiv \h{f}(\la, C^i, t_0 ,x^a ) $ is constant along geodesics, we have $$\t{f}(x^i (\la ,C^i , t_0 , x^a ),\, p^i (\la ,C^i , t_0 ,
x^a ))=
\t{f}(0,t_0 , x^a, p^i (0, C^i ,t_0 ,x^a ))\equiv f(C^i ,t_0 ,x^a ).$$

Let us consider the $\{ 04\}$ component of the energy-momentum tensor $$\c{T}^{04}(y,t,x^a )=\int\,d^5 p \sqrt{-g}\d (p_i p^i )f(x,p) p^0 p^4 
.$$ Changing integration variables $p^i$ to $C^i$, we obtain ł[f2]{} Ţ\^[04]{}(y,t)=d\^5 C\^i e\^[-4y]{}a\^3 (0,t) || (N\^2 C\_4 ) f(C\^i ,t\_0 ,x\^a )\
N\^2 (y,t)e\^[2y]{}([C\_0]{}\^2 +)\^[1/2]{} ([C\_0]{}\^2 e\^[2y]{} +C\_4 )\^[1/2]{} .The Jacobian of the transformation is ł[f3]{} ()\^3 .We traced the distribution function along the geodesics to $\la =0$, i.e. to the hypersurface $y=0$. Separating the $y$-dependence, we have ł[f4]{} Ţ\^[04]{}(y,t)=Ţ\^[04]{}(0,t )()\^5 e\^[6y]{} .Following [@lan2], 