As we are going back in cosmological time, a remark concerning particle
cosmology seems in order. While the temperature of the universe heats up
toward the big bang, it is assumed that matter undergoes a number of phase
transitions. All those happening before the so-called electroweak phase
transition at ~ (100-200) GeV, occur at energies not yet attainable in
the laboratory (accelerator particle physics). All are speculative, as
e.g., the grand unification phase transition at which the strong
interaction unifies with the weak and electromagnetic forces. The
confinement-deconfinement (QCD)-phase transition at 10^{-5} s
after the big bang seems to be the only one in future reach of
accelerator physics. Cosmic inflation preceeds all the
mentioned events; whether it is ending in a phase transition or not, is
debated. After the end of inflation, copious particle creation and then
thermalization is asumed to occur followed by baryogenesis. Cosmological
modeling after inflation is characterized by a change of paradigm when
compared to later eras: while, in principle, the description of matter by a
continuous distribution is retained, in practice matter is
differentiated into
elementary units: elementary particles, nuclei, atoms and their reactions;
they interact, can be produced or anihilated. The interplay of elementary
particle reaction rates and the expansion rate of the universe requires
different equations of state for different particle species at the same
epoch. Nuclear physics comes in much later: the end of primordial
nucleosynthesis is assumed to have happened at ≃ 10^{2} s
after the big bang. Particle physicists are interested in the very early
universe as a testbed for their theories concerning high energies. While
by the later evolution of the cosmos limits are set on such theories
(from CBM), the direct contributions of elementary particle physics to
the early universe are speculative.

Again, cosmological modeling of the early states of the universe is based on a number of hypotheses, simplifying the modeling. A selection would be:

- *C*_{1} Baryogenesis occurs after the end of inflation.

As to *C*_{1}, the end of inflation (reheating) is not well
understood;
it is difficult to reconcile the slow-roll conditions with the known
couplings of particle physics candidates for the inflaton. The origin of
the matter-antimatter asymmetry in the cosmos must and can be explained
(cf.
[89]).
A number of theories have been suggested, some of
them using leptogenesis (sphaleron-interaction)
[90].

- *C*_{2} Both, the reactions and reaction rates of
individual particles, and
collective phenomena are important in the early universe.

The assumed occurence of phase transitions cannot be understood without taking into account collective interactions. For a review of such phase transitions in the early universe cf. [91].

- *C*_{3} Elementary particles do not interact gravitationally;
gravitation acts merely as an external field.

This assumption expresses the subordinate role gravitation plays in the
modeling of the early universe despite the assumption that then matter was
extremely condensed. The gravitational field is assumed to show up only
in the expansion of the universe or, perhaps, in pair production of
elementary particles, if quantum field theory in curved space as we
understand it is applicable (there exists not yet a fully worked out
model for strong curvature). For special aspects cf.
[112],
[113],
[114].
^{22}

- *C*_{4} Temperature and entropy of the universe are well
defined after (local) thermodynamic equilibrium is reached.

- *C*_{5} While, in epochs after inflation, matter is in
thermodynamical equilibrium, different particle species can and will
decouple from the equilibrium distribution.

As to the application of thermodynamics and kinetic theory to the early
universe (*C*_{4}, *C*_{5}), it is known that,
in the FLRW cosmological models, an
exact equilibrium distribution is permitted only in two limiting
cases: the ideal radiative model (rest mass of particles is zero) and
the "heavy mass"-model (infinite rest mass)
[115].
Thermodynamically, the expanding universe is treated as a *quasi-static* system with a relaxation time small with regard to the
expansion (Hubble) time.
^{23}
This is called *local
thermodynamical equilibrium*. Whether such a concept can be valid for
*infinite* volume (open space-sections with *k*=0, -1) seems
questionable. From this perspective, a "small" universe
would be preferable. The time dependence of cosmic temperature implied
by the cosmological model (adiabatic cooling), could be interpreted as a
characteristic sign for the universe being a *non-equilibrium* system.

If the validity of the FLRW-cosmological models is extrapolated
to very early epochs, an inflationary period between ≃
10^{-36}*s* and ≃ 10^{-34} s after the big
bang is assumed to have happened. During it,
all distance scales in the universe must increase by at least 75 e-folds
([13],
p. 239). In connection with the cosmological standard
model, a number of questions then could be answered:

- What makes the universe as isotropic and homogeneous as it
is (horizon problem)?
- Why does the overall density parameter Ω differ from
Ω_{c} = 1 by only by very little (flatness problem)?
- How can the ratio η = η_{b} / η_{γ}
≃ (4-7) ⋅ 10^{-10} be explained (entropy problem)?

In order to answer these questions, the idea of the *inflationary
scenario* was invented
[116],
[117],
[32],
[118].
Its characteristic feature is a scalar field ϕ, the "inflaton"
^{24},
which is supposed to dominate the matter
content at very early epochs. This scalar field must be very weakly
coupled to all other matter fields. Usually, although not necessarily,
ϕ is taken to be the order parameter of a phase transition from a
symmetric phase with high
energy corresponding to ϕ = 0 (false vacuum) to a phase with *broken
symmetry* and ϕ = const ≠ 0 (true vacuum). An analogue would
be the delayed transition from the gaseous to the fluid state with
undercooling. The phase transition is made to start at ≃
10^{-35} seconds after the big bang. Dynamically, it is
tripartite: after the tunneling
of a potential barrier between the false and the true vacuum, a slow
descent ("role-down") toward the true vacuum (supercooling) to a
period of field oscillations, (reheating) must occur. In this last
interval, the inflaton decays into the matter particles/fields we see
today, and by producing heat. The reheating process is non-adiabatic and
claimed to bring an increase in the
entropy (of the universe) by a factor of 10^{130}.
^{25}
The equation of state of the
inflaton field is unusual if compared with materials in the laboratory: its
pressure is negative with *p* = -ρ (*w* =
-1). Gravitational attraction is overwhelmed by repulsion responsible
for the rapid expansion of the universe during the inflationary period.

A reason behind the many inflationary models is the ambiguity in
potential energy of the inflaton field: it may be taylored at will. In
some of the models investigated by now, the phase transition is
pictured as a nucleation of bubbles of the
broken-symmetry phase within a matrix of the symmetric phase. During
supercooling such a bubble can grow exponentially by 40 - 50 orders
of magnitude (of 10) and more within a time of the order of a (few
hundred) ⋅ 10^{-35} seconds. The gravitational field during
the exponential growth is described by de Sitter's solution of the
field equations (with constant Hubble parameter), the space sections of
which are *flat* (*k* = 0). By construction, the inflationary
model can solve both the entropy and the horizon
problems: the presently observable part of the universe lies within a
single inflating bubble. This means that, at the epoch of decoupling of
photons and baryons, the various regions of the universe from which the
cosmic microwave background originated have been causally connected. The
model is said to also remove the flatness problem: inflation drives the
density prameter Ω toward one
[22].
Whether Ω = 1 is desirable or not, seems to
be entirely up to one's private beliefs, though.
^{26}
There are also inflationary models with negative and
positive 3-curvature *k*
[120],
[121].
Hence, it seems questionable whether "the flatness of the universe" is an
unavoidable consequence of inflation
([13],
p. 354).
^{27}
We note that
the inflaton field might be inhomogeneous and yet not violating the
homogeneity and isotropy of the energy-momentum tensor of the cosmological
model; the overall homogeneity would then be lost, however.

Although debates about the inflationary model have not ended (cf.
[122],
[112],
[123],
[124],
by the following result its acceptance became overwhelming: through
quantum fluctuations of the inflaton field, the model
was able to provide the nearly scale invariant spectrum in the growing
mode of (adiabatic) density perturbations which had been required from
observations.
^{28}
To make the amplitudes fit the density fluctuations reflected by
the anisotropy of CMB, fine-tuning is required, though. In this context, it
has been shown that large-angle (low-*l*) correlations of the CMB
(from the 3-year WMAP-data) exhibit statistically significant
anomalies. This is weakening "the agreement of the observations with
the predictions of generic inflation"
([127],
p. 16).

**4.3. ΛCDM-questionaire (implying inflation)**

While the inflationary model needed for the ΛCDM model has solved a number of problems, it created others:

- By what physics are the initial conditions for inflation generated?

- What is the inflaton field?

- What is tested by present observations: the nearly scale-invariant
spectrum of density perturbations, or the inflationary scenario, in toto?

- What is dark energy?

- Why dark energy has become dominant only "recently" in the evolution
of the universe (coincidence problem)?

- Did dark energy play a role in the formation of large scale
structure, or not?

- Is an interaction of dark matter and dark energy excluded?

At present, there seems to be no consent on a fundamental theory for
the very early universe in which the inflationary model is embedded and its
initial conditions fixed. Cf. critical remarks by
[128].
^{29}
The inflaton is not the
Higgs particle (both are not observed). Is it connected to a model of hybrid
inflation (2 scalar fields!) with the s-neutrino as the inflaton?
[130]
Is there a link to the scalar field introduced in a later
epoch and named "quintessence" (Cf.
section 3.4.2). Will
there be a *technically accomplished* model for inflation still
lacking?
^{30}
What determines the high energy of the false vacuum? What kind of traces of
the inflationary period can we *observe*? One such effect following
from inflationary models is a stochastic background of primordial
gravitational waves: metric tensor modes could be seen in the
polarization measurements of CMB. So far, they have not (yet) been
detected. If observed, certain inflationary models with respect to
others could be ruled out. If not found, this also can be reproduced by
some models. Gravitational waves from inflation
are not to be mixed up with "gravitons" eventually generated during the
Planck era, nor with the still different "gravitons" claimed by string
theory.

The coincidence problem is alleviated if cosmic acceleration is modeled by space- and time-dependent fields replacing the cosmological constant; a fine-tuning of their contribution to the energy density needed can always be made such that it is largest late in the evolution of the universe. In view of the merely indirect empirical tests through consistency of the full cosmological model, the inflationary scenario is still rather speculative.

^{22} Of course,
in the very early universe, the gravitational field might not exist on its
own but be united with the other fundamental interactions in a Super Grand
Unified Field.
Back.

^{23} Relaxation time, for massive
particles, is related to mass diffusion or heat transport etc. For
massless particles it may be approximated by the collision time and
does not depend on volume.
Back.

^{24} More precisely, the inflaton is the
field quantum of the inflaton field.
Back.

^{25} During the
inflationary phase, entropy grows linearly with cosmic time *t*,
afterwards only with *ln* *t*
([119],
p. 319).
Back.

^{26} Ω = 1 is an
unstable fixpoint in the phase diagram of the time evolution of the
Friedman models.
Back.

^{27} Also, as noted by R. Penrose, if
theory implies flat space sections, no observation, as small as its
error bar can be made, will be able to exlude nonzero curvature
([35],
p. 772).
Back.

^{28} An admixture of isocurvature
(non-adiabatic) perturbations below 10% (3%) seems to be permitted
[125],
[126].
Back.

^{29} Cf. however
[129]
with a worked out suggestion that quantum geometry lead to inflation.
Back.

^{30} For different inflationary models
including chaotic, double, hybrid, new and eternal inflation cf.
[131],
[132].
Back.