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Vector Perturbations

Vector perturbations can be decomposed into harmonic modes $Q^{(\pm 1)}_i$ of the Laplacian in the same manner as the scalars,  
\nabla^2 Q^{(\pm 1)}_{i} = -k^2 Q^{(\pm 1)}_{i} ,\end{displaymath} (20)
which satisfy a divergenceless condition
{\nabla}^i {Q}_i^{(\pm 1)}= 0 \, .\end{displaymath} (21)
A velocity field based on vector perturbations thus represents vorticity, whereas scalar objects such as density perturbations are entirely absent. The corresponding symmetric tensor is constructed out of derivatives as
Q_{ij}^{(\pm 1)}= - {1 \over 2k}
 (\nabla_i Q_j^{(\pm 1)}+ \nabla_j Q_i^{(\pm 1)})\, .\end{displaymath} (22)
A convenient representation is
{Q}_i^{(\pm 1)}= - {i \over \sqrt{2}} 
 (\hat{e}_1 \pm i \hat{e}_2)_i \exp(i\vec{k}
 \cdot \vec{x}) \, .\end{displaymath} (23)
Notice that $n^i Q_i^{(\pm 1)}= G_1^{\pm 1}$ and $n^i n^j Q_{ij}^{(\pm 1)}\propto G_2^{\pm 1}$. Thus vector perturbations stimulate the $m = \pm 1$ modes in the radiation.

Wayne Hu