next up previous contents
Next: Perturbation Evolution Up: Perturbation Classification Previous: Vector Perturbations

Tensor Perturbations

Tensor perturbations are represented by Laplacian eigenfunctions  
\nabla^2 Q_{ij}^{(\pm 2)}= -k^2 Q_{ij}^{(\pm 2)},\end{displaymath} (24)
which satisfy a transverse-traceless condition
\delta^{ij} Q^{(\pm 2)}_{ij} = \nabla^i Q^{(\pm 2)}_{ij}{} = 0\, ,\end{displaymath} (25)
that forbids the construction of scalar and vector objects such as density and velocity fields. The modes take on an explicit representation of
{Q}_{ij}^{(\pm 2)}= - \sqrt{3 \over 8} 
 (\hat{e}_1 \pm i \h...{e}_1 \pm i \hat{e}_2 )_j 
 \exp(i \vec{k} \cdot \vec{x})\, .\end{displaymath} (26)
Notice that $n^i n^j Q_{ij}^{(\pm 2)}= G_2^{\pm 2}$ and thus tensors stimulate the $m = \pm 2$ 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