tokens: of a gas parcel being essentially independent of its collapse history – explaining why the asymptotic T − Δ relation holds so tightly despite the varied histories of cosmological fluid elements. (The tightness of this relation is absolutely striking in calculations that do not allow for shocking but allow for varied collapse histories. See the Zeldovich approximation calculations in Fig. 1 .) We further show that Compton cooling off of the CMB has no effect on the asymptotic slope of T (Δ) and that recombination cooling is just weak enough not to break the relation. This paper is organized as follows. We first enumerate the relevant heating and cooling processes in Section 2 . Next, in Section 3 , we derive an analytic expression for the asymptotic T − Δ relation in the limit of only photoheating and adiabatic heating/cooling. These are the processes considered in previous attempts to understand this relation analytically. Section 4 generalizes our solutions to include all relevant cooling processes. We briefly consider three applications of our results in Section 5 : (1) how quickly temperature fluctuations are erased, (2) the importance of shock heating, and (3) the thermal evolution of collapsing gas clouds. Lastly, Section 6 contrasts our results with previous attempts to understand the intergalactic T − Δ relation. Throughout we use the standard phenomenological parametrization for the temperature of the low-density IGM, where Δ is the baryonic density in units of its cosmic mean, T 0 is the gas kinetic temperature at Δ = 1, and γ − 1 is a power-law index. (Sometimes γ is called the ‘equation of state.’) When necessary, we assume a flat ΛCDM Universe with h = 0.7, Ω m = 0.3, Ω b = 0.045, and Y He = 0.25. Following Hui Gnedin ( 1997 ), we use the Zeldovich approximation to follow the density (necessary for computing the temperature) of fluid elements for certain calculations. Our Zeldovich approximation calculations draw matter elements randomly from the probability distribution of collapse eigenvalues, assuming that the gas traces the dark matter. This probability distribution only depends on the variance of the density field, which we set to 1.2. This choice is equal to the variance of a real space top hat filter containing a Lagrangian mass of 10 10 M ⊙ in our reference cosmology at z = 3 (and approximately the Jeans mass near the cosmic mean density at this redshift). 2 In addition, we also use a 20 Mpc, 512 3 gas and 512 3 dark matter particle simulation that was run with the gadget -3 smooth particle hydrodynamic code (Springel 2005 ). Reionization occurs instantaneously in this simulation at z = 9, heating all gas to 20 000 K, and the subsequent heating and cooling takes rates applicable for primordial gas if only the H i and He i are kept ionized with an ultraviolet background that is flat in specific intensity [erg s −1 Hz −1 cm −2 sr −1 ]. 2 RELEVANT HEATING AND COOLING PROCESSES The evolution of the temperature of a Lagrangian
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