The intergalactic hydrogen atoms after recombination are in their ground state, which hyperfine-structure splits into a singlet and a triple states ( with and respectively, see Section III.B.1 of FCV ). It was recently proposed  that the observation of the 21 cm emission can provide a test on the fundamental constants. We refer to  for a detailed review on 21 cm.
The fraction of atoms in the excited (triplet) state versus the ground (singlet) state is conventionally related by the spin temperature defined by the relation. The population of the two states is determined by two processes, the radiative interaction with CMB photons with a wavelength of (i.e., ) and spin-changing atomic collision. Thus, the evolution of the spin temperature is dictated by .
It follows [284, 285] that the change in the brightness temperature of the CMB at the corresponding wavelength scales as , where the Einstein coefficient is defined below. Observationally, we can deduce the brightness temperature from the brightness , that is the energy received in a given direction per unit area, solid angle and time, defined as the temperature of the black-body radiation with spectrum . Thus, . It has a mean value, at various redshift where . Besides, as for the CMB, there will also be fluctuation in due to imprints of the cosmological perturbations on and . It follows that we also have access to an angular power spectrum at various redshift (see  for details on this computation).
Both quantities depend on the value of the fundamental constants. Beside the same dependencies of the CMB that arise from the Thomson scattering cross section, we have to consider those arising from the collision terms. In natural units, the Einstein coefficient scaling is given by . It follows that it scales as . The brightness temperature depends on the fundamental constant as . Note that the signal can also be affected by a time variation of the gravitational constant through the expansion history of the universe.  (see also  for further discussions), focusing only on , showed that this was the dominant effect on a variation of the fundamental constant (the effect on is much complicated to determine but was argued to be much smaller). It was estimated that a single station telescope like LWA9 or LOFAR10 can lead to a constraint of the order of , improving to 0.3% for the full LWA. The fundamental challenge for such a measurement is the subtraction of the foreground.
The 21 cm absorption signal in a available on a band of redshift typically ranging from to , which is between the CMB observation and the formation of the first stars, that is during the “dark age”. Thus, it offers an interesting possibility to trace the constraints on the evolution of the fundamental constants between the CMB epoch and the quasar absorption spectra.
As for CMB, the knowledge of the cosmological parameters is a limitation since a change of 1% in the baryon density or the Hubble parameter implies a 2% (3% respectively) on the mean bolometric temperature. The effect on the angular power spectrum have been estimated but still require an in depth analysis along the lines of, e.g., . It is motivating since is expected to depend on the correlators of the fundamental constants, e.g., and thus in principle allows to study their fluctuation, even though it will also depend on the initial condition, e.g., power spectrum, of the cosmological perturbations.
In conclusion, the 21 cm observation opens a observational window on the fundamental at redshifts ranging typically from 30 to 100, but full in-depth analysis is still required (see [206, 286] for a critical discussion of this probe).
Living Rev. Relativity 14, (2011), 2
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