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Tensor perturbations are represented by Laplacian eigenfunctions
| ![\begin{displaymath}
\nabla^2 Q_{ij}^{(\pm 2)}= -k^2 Q_{ij}^{(\pm 2)},\end{displaymath}](img140.gif) |
(24) |
which satisfy a transverse-traceless condition
| ![\begin{displaymath}
\delta^{ij} Q^{(\pm 2)}_{ij} = \nabla^i Q^{(\pm 2)}_{ij}{} = 0\, ,\end{displaymath}](img141.gif) |
(25) |
that forbids the construction of scalar and vector objects
such as density and velocity fields. The modes take on
an explicit representation of
| ![\begin{displaymath}
{Q}_{ij}^{(\pm 2)}= - \sqrt{3 \over 8}
(\hat{e}_1 \pm i \h...
...at{e}_1 \pm i \hat{e}_2 )_j
\exp(i \vec{k} \cdot \vec{x})\, .\end{displaymath}](img142.gif) |
(26) |
Notice that
and thus tensors stimulate the
modes in the radiation.
In the following sections, we often only explicitly show
the positive m value with the understanding that its opposite
takes on the same form except where otherwise noted (i.e. in
the B-type polarization where a sign reversal occurs).
Wayne Hu
9/9/1997