Is a Classical Language Adequate in Assessing the Detectability of the   Redshifted 21cm Signal from the Early Universe?
  The classical radiometer equation is commonly used to calculate the detectability of the 21cm emission by diffuse cosmic hydrogen at high redshifts. However, the classical description is only valid in the regime where the occupation number of the photons in phase space is much larger than unity and they collectively behave as a classical electromagnetic field. At redshifts z<20, the spin temperature of the intergalactic gas is dictated by the radiation from galaxies and the brightness temperature of the emitting gas is in the range of mK, independently from the existence of the cosmic microwave background. In regions where the observed brightness temperature of the 21cm signal is smaller than the observed photon energy, of 68/(1+z) mK, the occupation number of the signal photons is smaller than unity. Neverethless, the radiometer equation can still be used in this regime because the weak signal is accompanied by a flood of foreground photons with a high occupation number (involving the synchrotron Galactic emission and the cosmic microwave background). As the signal photons are not individually distinguishable, the combined signal+foreground population of photons has a high occupation number, thus justifying the use of the radiometer equation. 
Abraham Loeb (Harvard)