The cosmological vector modes from a monochromatic primordial power spectrum

The cosmological background of higher order vector modes can be generated by the first order scalar perturbations. We investigate the second order and the third order vector modes systematically. The explicit expressions of two point functions $\langle V^{(n),\lambda}V^{(n),\lambda'} \rangle$$\left(n=2,3\right)$ and power spectra corresponded are presented. In the case of a monochromatic primordial power spectrum, the second order vector modes do not exist. However, the third order vector modes can be generated by a monochromatic primordial power spectrum. And it is found that the third order vector modes sourced by the second order scalar perturbations dominate the two point function $\langle V^{(3),\lambda}V^{(3),\lambda'} \rangle$ and power spectrum corresponded.