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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
, where
| ![\begin{displaymath}
\nabla^2 {\bf Q}^{(m)} \equiv \gamma^{ij} {\bf Q}_{\vert ij}^{(m)} =
-k^2 {\bf Q}^{(m)} ,\end{displaymath}](img14.gif) |
(5) |
with ``|'' representing covariant differentiation with respect to
the three metric
.Note that the eigentensor
has |m| indices (suppressed
in the above). Vector and tensor modes also satisfy the auxiliary conditions
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}](img16.gif) |
(6) |
where the presence of
, which forms
a local orthonormal basis with
, 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}](img19.gif) |
(7) |
| ![\begin{displaymath}
Q^{(\pm 1)}_{ij} = -(2k)^{-1}( Q^{(\pm 1)}_{i\vert j} + Q^{(\pm 1)}_{j\vert i} ).\end{displaymath}](img20.gif) |
(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}](img21.gif) |
(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}](img22.gif) |
(10) |
A deceptive aspect of this labelling is that for an open universe
the characteristic scale of
the structure in a mode is
and not
, so all
functions have structure only out to the curvature scale even as
.
We often go between the variable sets
,
and
for convenience.
Next: Perturbation Representation
Up: Metric and Stress-Energy Perturbations
Previous: Metric and Stress-Energy Perturbations
Wayne Hu
9/9/1997