$$D_\r D^\r\, \phi_{(m)\n}^\m = m^2_< \phi_{(m)\n}^\m ,$$ and the eigenfunctions $h^<_m (y)$ are determined from the equation ł[en2]{} [h\^&lt;\_m]{}” (y) - 4\^2 h\^&lt;\_m (y) + e\^[2|y|]{} m\^2\_&lt; h\^&lt;\_m (y) +(y)4h\^&lt;\_m (y) =0. In the region $y>\b{y}$ in (\[21a\]) the term with the increasing exponent becomes dominant. The approximate metric is ł[7.5]{} ds\^2 =dy\^2 + e\^[2y]{}(-dt\^2 +()\^2 \_[ab]{}dx\^a dx\^b ) .The equation for the eigenmodes is ł[en15]{} [h\^&gt;\_m]{}” (y)-4\^2 h\^&gt;\_m (y) + e\^[-2|y|]{} m\^2\_&gt; h\^&gt;\_m (y) =0 Eqs. (\[en2\]) and (\[en15\]) are solved in appendix B. We show that the norm of the function $ h^>_m$ is smaller than that $h^<_m$. Effectively, this allows to neglect the contribution from the region $y>\b{y}$. Solving the equations we neglect $t$ dependence of $\b{y}(t)$ which in the final expressions can be effectively sent to infinity. In the radiation-dominated period of late cosmology $e^{2\m |\b{y}|}< 1$ and $(d\b{y}/dt)/\m \b{y} \ll 1$.

We obtain the spectrum ł[en13]{} m\_ne\^[-]{}(n+) and the normalized eigenmode $h_m (0)$ ł[en14]{} h\_m (0)(e\^[-]{})\^[1/2]{} .For the following we need the sum $\sum_n h_{m_n}^2 (0)$, where $m_n$ is determined by (\[en13\]). Because of a narrow spacing between the levels, we change summation to integration and obtain ł[en21]{} \_n h\_[m\_n]{}\^2 (0) e\^[-]{}= .The integral (\[en21\]) 