The Temperature-Density Relation of Intergalactic Gas with a Uniform Heating Source: Slope Erosion from the Low-Density End

Photoheated intergalactic gas settles onto a tight temperature-density relation, T = T0 (1 + delta)^(gamma-1), whose slope records the balance between photoheating, which deposits energy per atom, and radiative cooling, which removes it per atom pair. This paper computes how that relation deforms when a density-independent volumetric heating term Q = f u_L / t_H — the form any coupling to the uniform surviving vacuum density u_L = 5.25e-09 erg per cubic centimetre must take — is added to the balance at z = 2.5. The deformation begins at the low-density end: at f = 1e-08, gas at half the mean density warms from 8,196 K to 19,128 K while gas at ten times the mean density is unmoved at 42,578 K, and the slope measured between overdensities 1 and 10 has only fallen from 0.55 to 0.46. The slope reaches 0.30 at f = 4.54e-08 and zero — a flat, then inverting, relation — at f = 1.89e-07; at f = 1e-07 gas below the mean density has no equilibrium at all, and the mean-density Jeans length has grown from 268 to 419 kpc. Measured slopes near 0.3-0.6 therefore bound a uniform term at this epoch to f below roughly 5e-08, weaker than the low-redshift normalisation bound but obtained from the relation's shape, which systematics affect differently. Computed from stated assumptions; not a measurement.