$h'(y,x) =C(x)b^{-1}(y)$, where $C(x)$ is an arbitrary function. Because of the reflection symmetry $x\leftrightarrow -x$, the metric components $g_{\m\n}(y,x)$ and $h_{\m\n}(y,x)$ are even functions of $y$, and thus $h' (y,x)$ is an odd function of $y$. Because $b(y)$ is even in $y$, it follows that $C(x)=0$ and $h' (y,x)=0$. We obtain that on the equations of motion $h(y,x)$ is independent of $y,\,\,\,h(y,x)=h(x)$.

After imposing the gauge conditions $h_{4\m}=0$ (\[gc2\]), there remain residual gauge transformations $\xi_\m (y,x) = b (y)C_\m (x)$. Under these transformations the trace $h(x)$ transforms as $\d h(x)=\b{g}^{\m\n} ({D}_\m C_\n (x)+{D}_\n C_\m (x) )$, and by a suitable choice of $C_\m (x)$ can be transformed to zero.

From the Eq. (\[m4\]) with $h=0$ it follows that $D^\m h'_{\m\n}(y,x) =0$, or $D^\m 
h_{\m\n}(y,x)=k_\n (x)$. Using the remaining residual gauge transformations with the functions $C_\m 
(x)$ such that ${D}^\m C_\m =0$, we can impose the condition $D^\m h_{\m\n}(y,x) =0$.

Eigenmodes and eigenvalues
==========================

In the background of the approximate metric (\[ap\]), in the gauge $D^\m h_{\m\n}(y,x) =0,\,\,\,h_\m^\m (y,x)=0$, the ($\m\n$) components of the field equations for fluctuations are ł[e1]{} [h]{}\_” - 4\^2 h\_ + b\^[-1]{}(y) D\_D\^h\_ +(y)4h\_ =0 .Expanding the functions $h^\m_\n (x,y)$ $$h^\m_\n (x,y) =\sum_m\,\phi_{(m)\n}^\m (x) h_m (y),$$ where 