3.7 21 cm

After recombination, the CMB photons are redshifted and their temperature drops as (1 + z). However, the baryons are prevented from cooling adiabatically since the residual amount of free electrons, that can couple the gas to the radiation through Compton scattering, is too small. It follows that the matter decouples thermally from the radiation at a redshift of order z ∼ 200.

The intergalactic hydrogen atoms after recombination are in their ground state, which hyperfine-structure splits into a singlet and a triple states (1s1∕2 with F = 0 and F = 1 respectively, see Section III.B.1 of FCV [500Jump To The Next Citation Point]). It was recently proposed [284Jump To The Next Citation Point] that the observation of the 21 cm emission can provide a test on the fundamental constants. We refer to [221Jump To The Next Citation Point] 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 Ts defined by the relation

( ) nt- T∗- n = 3 exp − T (117 ) s s
where T∗ ≡ hc∕(λ21kB ) = 68.2 mK is the temperature corresponding to the 21 cm transition and the factor 3 accounts for the degeneracy of the triplet state (note that this is a very simplified description since the assumption of a unique spin temperature is probably not correct [221Jump To The Next Citation Point]. The population of the two states is determined by two processes, the radiative interaction with CMB photons with a wavelength of λ21 = 21.1 cm (i.e., ν21 = 1420 MHz) and spin-changing atomic collision. Thus, the evolution of the spin temperature is dictated by [221Jump To The Next Citation Point].
dTs ( 1 1 ) ( 1 1 ) T γ ---- = 4C10 --− --- T 2s + (1 + z )HA10 ---− --- --- (118 ) dt Ts Tg Ts T γ T ∗
The first term corresponds to the collision de-excitation rate from triplet to singlet and the coefficient C10 is decomposed as
C10 = κHH10 np + κeH10 xenp
with the respective contribution of H-H and e-H collisions. The second term corresponds to spontaneous transition and A10 is the Einstein coefficient. The equation of evolution for the gas temperature Tg is given by Equation (114View Equation) with TM = Tg (we recall that we have neglected the contribution of helium) and the electronic density satisfies Equation (113View Equation).

It follows [284Jump To The Next Citation Point, 285] that the change in the brightness temperature of the CMB at the corresponding wavelength scales as 2 Tb ∝ A12∕ ν21, where the Einstein coefficient A12 is defined below. Observationally, we can deduce the brightness temperature from the brightness Iν, 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 I ν. Thus, k T ≃ I c2∕2ν2 B b ν. It has a mean value, T¯ (z ) b obs at various redshift where today 1 + zobs = ν21 ∕νobs. Besides, as for the CMB, there will also be fluctuation in Tb due to imprints of the cosmological perturbations on np and Tg. It follows that we also have access to an angular power spectrum C ℓ(zobs) at various redshift (see [329Jump To The Next Citation Point] 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 A12 = 23παEM ν321m −e2 ∼ 2.869 × 10− 15 s−1. It follows that it scales as A10 ∝ g3pμ3α13EMme. The brightness temperature depends on the fundamental constant as Tb ∝ gpμ α5 ∕me EM. Note that the signal can also be affected by a time variation of the gravitational constant through the expansion history of the universe. [284] (see also [221] for further discussions), focusing only on αEM, showed that this was the dominant effect on a variation of the fundamental constant (the effect on C10 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 Δ αEM ∕αEM ∼ 0.85%, 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 z ≲ 1000 to z ∼ 20, 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., [329]. It is motivating since C (z ) ℓ obs is expected to depend on the correlators of the fundamental constants, e.g., ′ ⟨αEM (x,zobs)αEM (x ,zobs)⟩ 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).

  Go to previous page Go up Go to next page