In the region $0<|y|<|\b{y}|$ and for $\r/\m\ll 1$ the functions $a(y,t)$ and $n(y,t)$ can be approximated as ł[22]{} && a(y,t)a(0,t)e\^[-|y|]{}\
&& n(y,t)e\^[-|y|]{}. With (\[22\]) we obtain the approximate form of the metric in the region $0<|y|<|\b{y}|$ ł[24]{} ds\^2dy\^2 +e\^[-2|y|]{}(-dt\^2 + a\^2 (0,t) \_[ab]{}dx\^a dx\^b ) .With the approximate metric we have $a'(0,t)/a(0,t)=-\m $, which for $\r/\m\ll 1$ is close to the exact junction condition $a'(0,t)/a(0,t)=-(\m +\r )$.

In the radiation-dominated period solution of the Friedmann equation ł[1c]{} H\^2 =2+\^2 for the radiation energy density is ł[25]{} (t) =. In the period of late cosmology the leading term is $$\r (t) \simeq\f{1}{8\m t^2}.$$ If $\r(t)\gg \m$, we have $\r (t)\simeq 1/4t$.

Fluctuations of background metric
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To calculate the spectrum of the Kaluza-Klein tower of gravitons, we solve the field equations for fluctuations $h_{ij}$ over the background metric $g_{ij}$. The purpose of this section is to clarify the issue of the gauge conditions in solving the equations for fluctuations. We show that the traceless transverse conditions on $h_{ij}$ cannot be imposed as gauge conditions at the level of the action, but can be imposed on solutions of the equations of motion.

The part of the action quadratic in fluctuations is ł[2f]{} 