Eigenmodes

  In linear theory, each eigenmode of the Laplacian for the perturbation evolves independently, and so it is useful to decompose the perturbations via the eigentensor ${\bf Q}^{(m)}$, where

$$\nabla^2 {\bf Q}^{(m)} \equiv \gamma^{ij} {\bf Q}_{\vert ij}^{(m)} = -k^2 {\bf Q}^{(m)} ,$$ (5)

with ``|'' representing covariant differentiation with respect to the three metric $\gamma_{ij}$.Note that the eigentensor ${\bf Q}^{(m)}$ has |m| indices (suppressed in the above). Vector and tensor modes also satisfy the auxiliary conditions

$$\begin{array} {rcl}Q^{(\pm 1)}_i{}^{\vert i} &=& 0\, , \nonu... ...pm 2)}_{ij} &=& Q^{(\pm 2)}_{ij}{}^{\vert i} = 0 \,,\end{array}$$

which represent the divergenceless and transverse-traceless conditions respectively, as appropriate for vorticity and gravity waves. In flat space, these modes are particularly simple and may be expressed as

$$Q_{i_1 \ldots i_{m}}^{(\pm m)} \propto (\hat{e}_1 \pm i \h... ...{i_m} \exp(i \vec{k} \cdot \vec{x})\,, \qquad (K=0, m\ge 0)\,,$$ (6)

where the presence of $\hat{e}_i$, which forms a local orthonormal basis with $\hat{e}_3=\hat{k}$, ensures the divergenceless and transverse-traceless conditions.

It is also useful to construct (auxiliary) vector and tensor objects out of the fundamental scalar and vector modes through covariant differentiation

$$Q_i^{(0)} = -k^{-1} Q_{\vert i}^{(0)}\,, \qquad Q_{ij}^{(0)}... ...k^{-2} Q_{\vert ij}^{(0)} + {1 \over 3} \gamma_{ij} Q^{(0)} \,,$$ (7)

$$Q^{(\pm 1)}_{ij} = -(2k)^{-1}( Q^{(\pm 1)}_{i\vert j} + Q^{(\pm 1)}_{j\vert i} ).$$ (8)

The completeness properties of these eigenmodes are discussed in detail in [6], where it is shown that in terms of the generalized wavenumber

$$q = \sqrt{k^2+(\vert m\vert+1)K} \, , \qquad \nu= q/\vert K\vert\,,$$ (9)

the spectrum is complete for

$$\begin{array} {rll} \nu&\ge 0, \qquad & K<0\,, \\ & = 3,4,5\ldots, \qquad &K\gt\,.\end{array}$$ (10)

A deceptive aspect of this labelling is that for an open universe the characteristic scale of the structure in a mode is $2\pi/k$ and not $2\pi/q$, so all functions have structure only out to the curvature scale even as $q \rightarrow 0$. We often go between the variable sets $(k,\eta)$, $(q,\eta)$ and $(\nu,\chi)$ for convenience.