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Cosmology with Very Large Gauge Models (1983) |
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Assuming isotropy and homogeneity (and zero or negligible cosmological constant), the Robertson-Walker scale factor 
of the universe expands according to (e.g., Ref. [11])
where 
is the energy density of matter in the Universe, and 
is the Planck mass. In the standard cosmological model, it is usually assumed that at sufficiently high temperatures all matter can be approximated as a weakly interacting gas of ultrarelativistic particles in thermal equilibrium. The resulting energy density is then linearly proportional to the number of particle degrees of freedom 
in equilibrium at a particular temperature; in a very large gauge model, the expansion rate of the early universe thus increases as 
. Such an increase in the expansion rate must be matched by an increase in interaction rates if processes are to remain in thermal equilibrium (see, for example, comments on baryon-number generation below).
The approximation of a weakly interacting ultrarelativistic gas leads to an equation of state 
for the pressure 
in the early universe. As discussed at length elsewhere, this equation of state may be modified by phase transitions associated with the restoration of spontaneously broken symmetries.[12] A component with an equation of state 
corresponding to an effective cosmological term is then introduced. The usual equation of state can also be modified in complicated gauge models by the presence of effective couplings which become strong at high energies and which lead to strong interactions in the gas.
The behavior of effective couplings is governed by a set of coupled renormalization-group equations. In the leading-logarithm approximation, all couplings typically exhibit Landau singularities (e.g. Ref. [13]) at either high or low momenta (or temperatures). For example, QCD in the leading-logarithm approximation yields a Landau singularity at low energies 
, but is asymptotically free at high energies. The leading-logarithm approximation becomes inaccurate near the Landau singularity. Nevertheless, beyond the region of the Landau singularity, a region of strong coupling is reached. QED also exhibits a Landau singularity, but at very high energies 
. Grand unified models may under some circumstances also exhibit Landau singularities at high energies, yielding strong couplings and potentially modifying the equation of state for matter and thus the expansion rate of the early universe.
The effective temperature-dependent gauge coupling 
for a grand unified model in the leading-logarithmic approximation takes the form
where
is the lowest-order coefficient in the 
function and 
are, respectively, the total combinatorial weights associated with all vector, two-component spinor, and real scalar-boson representations whose masses are much less than 
. 
is the temperature at which the coupling is normalized. For a group with an adjoint representation of dimension 
, the 
of a particular representation is given in terms of its dimensionality 
and quadratic Casimir invariant 
by 
. Thus, for example, with the gauge group SU(5), the 
representation has 
, the 
has 
, the 
(adjoint) has 
, the 
has 
, and the 
(popular in some recent supersymmetric models) has 
(e.g., Ref. [14]). The effective gauge coupling of Eq. (2) ceases to be asymptotically free at high temperatures if 
and it exhibits a Landau singularity at the temperature for which its denominator vanishes. Equation (2) shows that these effects occur if the number of fermions and Higgs scalar bosons becomes large compared to the number of gauge vector bosons. The number of vector bosons is, however, always given by the dimensionality of the adjoint representation of the gauge group, and is fixed for a particular gauge group. The number of fermions and particularly Higgs bosons may however usually be chosen in an apparently arbitrary manner.
For an SU(5) model with three families, asymptotic freedom is lost in Eq. (2) when 
, which is attained with 4 or more representations of Higgs scalar bosons or 87 representations. With 
at 
,[1] the SU(5) effective gauge coupling exhibits a Landau singularity below the Planck mass if 
, corresponding to 7 or more representations with mass 
. When many Higgs bosons have masses 
, as in models involving ``intermediate'' mass scales, the effective coupling may start to increase at lower temperatures, and a Landau singularity below the Planck mass is achieved with fewer Higgs bosons. For example, in the SO(10) model of Ref. [15], a Landau singularity appears below 
for some choices of parameters even with the ``minimal'' complement of three 
, a 
, and a complex representation of Higgs bosons. Asymptotic freedom may be lost not only through the presence of large numbers of Higgs bosons, but also by the effects of many fermions. Models based on SO(
) have the special feature that if fermions are restricted to the spinor representations, their number grows like 
, while the dimensionality of the vector-boson adjoint representation grows only like 
so that the contributions of fermions alone destroy asymptotic freedom if 
.
The variation of effective fermion Yukawa couplings and Higgs self-couplings with temperature is analogous to that given in Eq. (2) for the effective gauge coupling. Typically, however, these couplings are smaller then the gauge coupling, so that any Landau singularity appears at the lowest temperature in the gauge coupling. Nevertheless, if there exist sufficiently massive quarks [
(Ref. [16])] the fermion Yukawa couplings are sufficiently large that they alone yield a Landau singularity below the Planck mass. Such massive quarks are, however, supposedly forbidden by the stability of the effective potential.[17]
If a Landau singularity is reached, the matter in the Universe would become strongly interacting at high temperatures, and its equation of state would presumably deviate from the ideal-gas form . There are, however, few reliable indications of the possible equations of states attained. For classical gases with short-range interactions (and potential energy always much less than kinetic energy) one finds 
.[18] These inequalities are preserved if electromagnetic or other conformally invariant interactions are introduced.[19] However, in general it appears that the inequalities may be violated[20] (first-order corrections might be obtained by a virial expansion, but relativistic and hence particle production effects must be included), although there are indications that in most cases 
, where 
is the velocity of sound.[20]
One possibility is that the equilibrium statistical properties of a strongly interacting system of elementary particles may be approximated by those of a weakly interacting gas of bound states of these particles.[20] The resulting effective system always yields 
. The relation between energy and temperature depends on the level density for the bound states. A power-law density is obtained from simple (essentially nonrelativistic) potential models, and is to some extent supported by low-energy hadron spectroscopy. With such a density, increases with temperature faster than 
, leading to a lower increase of temperature at early times after the ``big bang.'' An exponentially rising level density is suggested by the Veneziano model and statistical bootstrap approaches to strong interactions.[21] Such a level density would lead to a maximum temperature, [21,11] beyond which any increase in energy generates particles with higher rest masses, rather than increasing particle kinetic energies. Degrees of freedom which become significant only above the maximum temperature would not appear in the early universe. (Phase transitions associated with restoration of symmetries at higher temperature would not occur, and their associated magnetic monopoles would thus not be produced.) In addition, the expansion of the early universe would be more rapid. [21,11]
Many proofs in general relativity assume either the ``strong'' energy condition 
or the ``dominant'' energy condition 
.[22] If the strong energy condition is violated (as by an effective cosmological constant giving ), then the expansion of the early universe is greatly slowed, and no particle horizons survive, so that causal dynamical processes could give rise to the observed homogeneity and small density fluctuations (galaxies). In general, with an equation of state of the form 
, the Robertson-Walker scale factor of the Universe expands with time according to 
.
``Stiff''' equations of state with 
appear to have little direct effect on the early universe, although it appears that they may lead to a significant increase in the rate of primordial black-hole formation.[23]
Regardless of strong coupling, the presence of many particle species increases the density of the Universe, and reduces the interaction length for all particle species. Ultimately, the interaction length may become shorter than the thermal Compton wavelength of each particle (typically when the number of species exceeds l/
), leading to the possibility of important collective quantum-mechanical effects.
