The 21-cm Absorption Signal as an Energy Budget: What the EDGES Depth Costs, Priced Two Ways

The EDGES experiment reported a sky-averaged 21-cm absorption trough centred at 78 MHz (redshift z ≈ 17) with an amplitude of about 0.5 K, roughly twice the deepest absorption the standard picture allows. This paper treats that reported depth as an energy budget and prices it in the only two ways the standard brightness-temperature formula permits. With the spin temperature coupled to the adiabatically cooled gas temperature (7 K at z = 17) and the cosmic microwave background (CMB) as the only background (49.05 K at z = 17), the formula gives a maximum absorption of −217.6 mK, so the reported −500 mK is deeper by a factor of 2.30. Holding the background at the CMB value, the depth requires a spin and gas temperature of 3.31 K, which means removing an additional 0.48 milli-electronvolts of thermal energy per baryon beyond adiabatic cooling between z = 20 and z = 17. Holding the gas at its standard temperature instead, the depth requires a total radio background of 103.62 K at z = 17, or 2.11 times the CMB temperature at that redshift, with the excess alone equal to 1.11 times the CMB temperature. For comparison, the present-day excess radio background fitted by ARCADE-2 corresponds to 0.169 of the CMB temperature at the rest-frame 21-cm frequency, and the Long Wavelength Array value is 0.221, so a background that simply scales with the expansion falls short of the required excess fraction by a factor of about 6.6. The detection itself remains disputed: the SARAS 3 experiment did not confirm the profile and rejected the EDGES best fit at 95.3 per cent confidence. No conclusion is drawn here that either branch, or the reported signal, is established; the purpose is only to state, in numbers, what the reported depth would cost under each of the two available readings.