Download Temperature–Time Relation

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].