# Did the universe have a temperature during inflation?

I've heard it said that inflation was not an equilibrium process. But I've also heard it said that during inflation, the temperature of the universe was much cooler than before or after. If the universe was not in equilibrium during inflation, then how could it have had a temperature? Isn't temperature a property that only applies to systems at equilibrium?

• Temperature may or may not be a useful term for non-equilibrium processes. The Clausius formulation of the second law basically defines temperature as the driving force behind heat flow: Unless something else happens, heat only flows from hot to cold. That's temperature defined by a non-equilibrium process. Where did you see a temperature chart that shows the universe during inflation being much cooler than after inflation? – CuriousOne Feb 17 '16 at 21:14
• Here's a paragraph from the Wikipedia page on cosmic inflation: "Inflation is a period of supercooled expansion, when the temperature drops by a factor of 100,000 or so. (The exact drop is model dependent, but in the first models it was typically from 10^27 K down to 10^22 K.) This relatively low temperature is maintained during the inflationary phase. When inflation ends the temperature returns to the pre-inflationary temperature; this is called reheating or thermalization" en.wikipedia.org/wiki/Inflation_(cosmology) – reductionista Feb 17 '16 at 21:33
• I see what you mean. I would say that "temperature" is not a useful parameter during that time because the system is in extreme disequilibrium. Consider it like you would consider the temperature of a battery. Is it the temperature the battery has before or after you short the leads and all of the energy goes into heating the battery? – CuriousOne Feb 17 '16 at 21:44

## 1 Answer

I asked this question 8 months ago, hoping someone on here knew the answer. Since nobody had an answer, I recently started wondering about it again. After reading a few review articles on inflation and reheating, I think I have a good understanding of it and can answer it now.

What I had heard about inflation being a non-equilibrium process was incorrect, or at best--only partially correct. It's not the inflationary epoch itself which departs from thermodynamic equilibrium, but the epoch directly following inflation known as "reheating".

Most of the evolution of the universe, both in inflation and in the traditional hot Big Bang cosmology, is close enough to equilibrium that it's treated by cosmologists as being "adiabatic". The use of adiabatic here is similar to that of its use in quantum mechanics: slow enough so that it does not generate any new entropy by departing from equilibrium. A better word is probably "isentropic" as the term "adiabatic expansion" can also mean fast expansion (in classic thermodynamics) which drastically increases entropy. In practice, of course, new entropy is always being generated but on cosmological scales it's at a fairly negligible rate compared to the overall entropy of the universe, most of which is contained in the cosmic background radiation (the gas of photons which fills the entire universe, on average containing about 400 photons per cubic centimeter) and a similar gas of neutrinos filling the universe. The entropy density drops as space expands and the temperature drops, but the entropy in any comoving volume remains constant. This can be interpreted as entropy simply spreading out due to expansion, rather than new entropy being generated.

During the earlier inflationary epoch, space is expanding more rapidly, but the adiabatic approximation still holds, and the entropy per comoving volume remains constant. The scale factor of the universe $a(t)$ grows exponentially in time ($a(t) \sim e^{H t}$), which makes the temperature drop more quickly. But it's not so quick that it departs from equilibrium--the energy density is just becoming more and more diluted so the temperature is dropping.

Where the universe does depart from equilibrium is in the reheating epoch, which comes in between the inflationary epoch and the standard "hot big bang". During reheating, space expands at a rate that's similar to a matter-dominated universe ($a(t) \sim t^{2/3}$), even though there isn't any matter in the universe yet. The energy is stored in coherent oscillations of the inflaton field (a scalar field similar to the Higgs boson) near the bottom of its potential energy curve. In early inflation models, this energy thermalizes instantly at the end of the reheating epoch, as soon as the Hubble rate of expansion has dropped enough to match the decay width for these inflaton oscillations. So often, the temperature after reheating is determined almost entirely by the decay width of the inflaton. In more recent models, sometimes the decay into radiation particles is more spread out, and starts happening earlier (this is called "preheating"). Either way, the reheating epoch results in an enormous increase in entropy... even entropy per comoving volume, which otherwise tends to remain roughly constant.

By the end of reheating, the universe has returned to thermal equilibrium and the temperature is once again well-defined, but now it is much higher than it was during inflation. This begins the standard "hot big bang" story where expansion happens according to a radiation-dominated energy density ($a(t) \sim t^{1/2}$), which eventually shifts to matter-dominated ($a(t) \sim t^{2/3}$ again).

On a tangential note, it appears that the use of the word "supercooled" in the Wikipedia quote I cited in the comments of the question is also incorrect. Reheating occurs both in "old inflation" (where the inflaton tunnels from a metastable vacuum to a stable one) and in "new inflation" (where it simply rolls gently down a hill) but only the former is analogous to a supercooled liquid. I've suggested to Wikipedia someone update this.

Curious to hear comments or corrections, if anyone out there knows if I've gotten any of this wrong.