**4.4. Vacuum displacement**

Another important possibility is the existence of relics which were never in thermal equilibrium. An example of these will be discussed later in this lecture: the production of topological defects at phase transitions. Let's discuss another kind of non-thermal relic, which derives from what we might call "vacuum displacement." Consider the action for a real scalar field in curved spacetime:

(121) |

If we assume that
is spatially homogeneous
(_{i}
= 0), its
equation of motion in the Robertson-Walker metric (5) will be

(122) |

where an overdot indicates a partial derivative with respect
to time, and a prime indicates a derivative with respect to
.
For a free massive scalar field,
*V*() = 1/2
*m*_{}^{2}
^{2},
and (122) describes a harmonic oscillator with
a time-dependent damping term. For *H* >
*m*_{} the
field will be overdamped, and stay essentially constant at
whatever point in the potential it finds itself. So let us
imagine that at some time in the very early universe (when *H*
was large) we had such an overdamped homogeneous scalar field,
stuck at a value
=
_{*}; the total energy density in the
field is simply the potential energy
1/2 *m*_{}^{2}
_{*}^{2}.
The Hubble parameter *H* will decrease to approximately
*m*_{} when the temperature reaches
*T*_{*} =
(*m*_{}
*M*_{p})^{2}, after which the field
will be able to evolve and will begin to oscillate in its
potential. The vacuum energy is converted to a combination of
vacuum and kinetic energy which will redshift like matter, as
_{}
*a*^{-3}; in a particle interpretation, the
field is a Bose condensate of zero-momentum particles. We
will therefore have

(123) |

which leads to a density parameter today

(124) |

A classic example of a non-thermal relic produced by vacuum
displacement is the QCD axion, which has a typical primordial value
<> ~
*f*_{PQ} and a mass
*m*_{} ~
_{QCD} /
*f*_{PQ}, where *f*_{PQ} is the
Peccei-Quinn symmetry-breaking scale and
_{QCD} ~
0.3 GeV is the QCD scale
[1].
In this case, plugging in numbers reveals

(125) |

The Peccei-Quinn scale is essentially a free parameter from a
theoretical point of view, but experiments and astrophysical
constraints have ruled out most values except for a small window
around *f*_{PQ} ~ 10^{12} GeV. The axion therefore
remains a viable dark matter candidate
[115,
116].
Note that, even though dark matter axions are very light
(_{QCD}^{2} /
*f*_{PQ} ~ 10^{-4} eV), they are
extremely non-relativistic, which can be traced to the non-thermal
nature of their production process. (Another important way to
produce axions is through the decay of axion cosmic strings
[1,
117].)