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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 21cm lensing observations. Here, we investigate the weak lensing signal in 21cm photons emitted by neutral hydrogen atoms in the dark ages induced by the second-order vector mode by decomposing the deflection angle of the 21cm 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 21cm 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 21cm 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 $\ell_{\rm max} = 10^{5}$ and $10^{6}$ in reconstructing the curl mode from 21cm temperature maps, the signal-to-noise ratios of the 21cm curl mode from the second-order vector mode achieve ${\rm S/N} \approx 0.46$ and $73$, respectively. Observation of 21cm radiation is, in principle, a powerful tool to explore not only the tensor mode but also the cosmological vector mode.