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Temperature–Time Relation Fig. 1. An Introduction to Modern Cosmology The temperature–time relation for a cosmological model describes how the Universe's temperature changes as it ages. Since the temperature of a gas of particles is a measure of the energy of the individual particles, this relation also indicates the types of particle interactions that are prevalent at a given epoch. If cosmological models are extrapolated all the way to the instant of the big bang the temperature would be infinite then, though few if any cosmologists believe that such an extrapolation is possible since known physical laws are predicted to break down at high temperatures. The temperature–time relation therefore first becomes valid sometime after the big bang. Moreover, the ‘time’ which appears is the age of the Universe as if its behaviour really could be extrapolated to the instant of the big bang, defined as time zero. The temperature–time relation is most useful during the radiation-dominated era of the Universe's evolution, lasting for the first few thousand years. During that epoch the density of material in the Universe, responsible for driving the expansion via the Friedmann equation, can be directly related to the temperature of that material, giving an accurate temperature–time relation. Where T is the temperature and t the time, it can be written approximately as The constant of proportionality has been written in a suggestive way to make it easy to generate particular examples. We learn that at an age of one second the Universe had a temperature of about 1010 kelvin. By 100 seconds it had fallen to 109 kelvin, whereas at 10-10 seconds old the temperature would have been 1015 kelvin. The above law embodies the redshifting relation T α 1/a (a being the scale factor), since during the radiation era a α t1/2. This relation holds most of the time, but can be briefly violated if the Universe departs from thermal equilibrium. This can happen if, for instance, the cooling of the Universe means that massive particles can no longer be readily produced in interactions, while those already existing decay or annihilate. The most recent example was the annihilation of electrons and positrons when the Universe was around one second old. The characteristic energy of an individual particle in thermal equilibrium at temperature T is given by k B T, where k B is a fundamental constant of Nature known as the Boltzmann constant. This is really just telling us that the macroscopic property of temperature is derived from the individual energies of the constituent particles. Using particle physics units, we can rewrite the temperature–time relation in terms of the characteristic particle energy as where MeV stands for mega-electron-volt. This version is the key to determining the types of physical process happening at different stages of the Universe's evolution. For example, the binding energy of typical light nuclei, such as helium-4, is around 1 MeV per particle. Hence when the Universe was younger than one second and the ambient energy higher than 1 MeV, any nucleus that might form would rapidly be destroyed. After one second they can begin to form. The epoch starting around one second is therefore that of nucleosynthesis, where the elemental content of the young Universe was fixed. By contrast, the binding energy holding electrons in atoms is typically about an electron-volt. According to our equation, electrons therefore cannot settle in atoms until around 1012 seconds (i.e. tens of thousands of years) after the big bang. This is the epoch of recombination, when the Universe converts from a plasma to a neutral atomic state. [Actually the above equation isn't strictly applicable, as it shows that recombination happens after the radiation era ends. However it is not too bad as an estimate.] The highest energy particles created on Earth, using particle accelerators, have energies around 106 MeV (known as a tera-electron-volt). This is the highest energy at which we can consider fundamental physical processes to be well tested and understood, and was achieved when the Universe was about 10-12 seconds old. Any study of earlier epochs necessarily requires some hypotheses as to which physical laws might be appropriate. During the matter-dominated era which follows radiation domination, one can no longer write such a simple relation, since the dominant material driving the expansion is no longer the radiation. Indeed, after decoupling the baryonic material and the radiation begin to develop different temperatures, which they can do as they no longer interact. Perhaps the most useful guide here is to take the accurately measured present temperature of the microwave background, 2.725K, combined with the redshifting law T α 1/a which has held ever since electron–positron annihilation during the radiation era. Fig. 1 illustrates the evolution of temperature with time from the radiation-dominated epoch onwards. The present epoch of dark energy domination is characterized by an accelerating expansion leading to a sharp drop in temperature. Fig. 1. A simple schematic of the temperature–time relation for our Universe, showing the three mains eras of radiation domination, matter domination, and now dark energy domination. A more detailed analysis would reveal some substructure in this simple curve. One can also consider the temperature history during cosmological inflation. Whatever the temperature might have been when inflation started (if indeed it had a start), it is rapidly supercooled by the expansion and indeed predicted to closely approach absolute zero. At the end of inflation, the processes of preheating and reheating recreate a thermal bath of conventional particles, returning the Universe to a high temperature and a radiation-dominated epoch. Bibliography and More Information about temperature–time relation • Liddle, A. R. An Introduction to Modern Cosmology, 2nd ed., John Wiley and Sons, 2003 [undergraduate level].