Cooling Times and Entropy of Warm-Hot Filament Gas: Why the Phase, Once Heated, Persists

Whether the warm-hot intergalactic medium is a permanent feature of the universe or a transient one depends on a race between its radiative cooling time and the age of the universe. This paper evaluates that race, and the associated entropy index K = k_B T / n_e^(2/3), on the phase's density-temperature grid with stated cooling functions. At electron density 1e-05 per cubic centimetre and 1e6 kelvin, the cooling time is 18.1 billion years — 1.25 times the Hubble time of 14.5 billion years — and the entropy index is 186 keV square centimetres. At 1e7 kelvin and the same density the cooling time stretches to 904 billion years. Only at the grid's dense corner, 1e-04 per cubic centimetre and 1e6 kelvin, does cooling win locally, with 1.81 billion years, 0.125 of the Hubble time. At 1e-06 per cubic centimetre the cooling time reaches 181 billion years and the entropy 862 keV square centimetres. The warm-hot phase at typical filament densities therefore cannot cool in the age of the universe: once shock-heated, it stays hot, and the baryon census's largest reservoir is thermodynamically locked in.