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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
\begin{displaymath}
\nabla^2 {\bf Q}^{(m)} \equiv \gamma^{ij} {\bf Q}_{\vert ij}^{(m)} = 
-k^2 {\bf Q}^{(m)} ,\end{displaymath} (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{displaymath}
\begin{array}
{rcl}Q^{(\pm 1)}_i{}^{\vert i} &=& 0\, , \nonu...
 ...pm 2)}_{ij} &=& Q^{(\pm 2)}_{ij}{}^{\vert i} = 0 \,,\end{array}\end{displaymath}   
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
\begin{displaymath}
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)\,,\end{displaymath} (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
\begin{displaymath}
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)} \,,\end{displaymath} (7)
\begin{displaymath}
Q^{(\pm 1)}_{ij} = -(2k)^{-1}( Q^{(\pm 1)}_{i\vert j} + Q^{(\pm 1)}_{j\vert i} ).\end{displaymath} (8)

The completeness properties of these eigenmodes are discussed in detail in [6], where it is shown that in terms of the generalized wavenumber
\begin{displaymath}
q = \sqrt{k^2+(\vert m\vert+1)K} \, , \qquad \nu= q/\vert K\vert\,,\end{displaymath} (9)
the spectrum is complete for
\begin{displaymath}
\begin{array}
{rll}
\nu&\ge 0, \qquad & K<0\,, \\  & = 3,4,5\ldots, \qquad &K\gt\,.\end{array}\end{displaymath} (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.


next up previous contents
Next: Perturbation Representation Up: Metric and Stress-Energy Perturbations Previous: Metric and Stress-Energy Perturbations
Wayne Hu
9/9/1997