Observable cosmological vector mode in the dark ages

The second-order vector mode is inevitably induced from the coupling of first-order scalar modes in cosmological perturbation theory and might hinder a possible detection of primordial gravitational waves from inflation through 21 cm lensing observations. Here, we investigate the weak lensing signal in 21 cm photons emitted by neutral hydrogen atoms in the dark ages induced by the second-order vector mode by decomposing the deflection angle of the 21 cm lensing signal into the gradient and curl modes. The curl mode is a good tracer of the cosmological vector and tensor modes since the scalar mode does not induce the curl one. By comparing angular power spectra of the 21 cm lensing curl mode induced by the second-order vector mode and primordial gravitational waves whose amplitude is parametrized by the tensor-to-scalar ratio $r$, we find that the 21 cm curl mode from the second-order vector mode dominates over that from primordial gravitational waves on almost all scales if $r\lesssim {10}^{-5}$. If we use the multipoles of the power spectrum up to ${\ell }_{\max }=10^5$ and $10^6$ in reconstructing the curl mode from 21 cm temperature maps, the signal-to-noise ratios of the 21 cm curl mode from the second-order vector mode achieve $\mathrm{S}/\mathrm{N}\approx 0.46$ and 73, respectively. Observation of 21 cm radiation is, in principle, a powerful tool to explore not only the tensor mode but also the cosmological vector mode.

Figure 1

The angular power spectra of the curl mode induced by PGWs with $r=0.1$ (left) and the second-order vector mode (right) for redshifts from $z=200$ to 0.6 as indicated in the figures. The curl mode from PGWs is substantially suppressed on small scales compared with that from the second-order vector mode.

Figure 2

The angular power spectrum of the curl mode from the second-order vector mode and the coadded reconstruction noise by using ${\ell }_{\max }=10^5$ (top left) and ${\ell }_{\max }=10^6$ (top right). Bottom: The signal-to-noise ratio estimated by Eq. (23). For reference, we also show the curl mode signal and signal-to-noise ratio induced by PGWs with $r={10}^{-5}$ and which tensor-to-scalar ratio corresponds to the same amplitude of the second-order curl mode at $\ell =2$.