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Is it possible that we mistake some other wavelength emission to be 21 cm because of redshift?

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No, not really. The first thing is that we know that ${H}$ is far more abundant than other elements or simple molecules in the universe. The next thing is that the 21 cm line comes from a relatively unusual hyperfine splitting, and there just aren't any other sources near that wavelength and in intensity levels that can easily be detected.

In most cases, astronomers have other methods of estimating objects' speeds, so any measurement of 21cm apparent redshift that was inconsistent with those estimates would be an indication that the wrong line was being observed.

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