Scalar perturbations can be expanded in terms of harmonic functions satisfying and the vector and tensor harmonics derived from them, Because we assume that $\K^n$ is compact, $k^2$ takes discrete values starting from zero, and, in particular for $\K^n=S^n$, its spectrum is given by The above definitions of $\SHB_i$ and $\SHB_{ij}$ become meaningless for $k^2=0$. Because the harmonic function for $k^2=0$ is constant, the corresponding perturbation merely represents a change of the background metric with respect to a variation of the mass parameter. Thus, this is a trivial perturbation with respect to the uniqueness problem. Therefore, we only consider modes with $k^2>0$ from this point, unless otherwise stated.

In addition to $k^2=0$, harmonics with $k^2=n$ for $K=1$ are also exceptional, because $\SHB_{ij}$ vanishes for such harmonic functions. Modes corresponding to such harmonics are gauge modes as shown in Ref. , and we do not consider such modes. This implies that $k^2>n$ can be assumed when $K=1$, because for $K=1$, the second smallest eigenvalue of $k^2$ is greater than or equal to $n$.

Ignoring these exceptional modes, stationary scalar perturbations of the metric can be expanded in terms of the harmonics as We adopt the following combinations as a basis for 