The game of Big Bang Nucleosynthesis (BBN) is to compute the
abundances of light nuclides *X*_{i} (the *yields*)
generated around *T* = 10^{9}
K and to compare them with the observed abundances. To reach this goal
we must face two different tasks: 1) to follow the physics of the Big
Bang forward in time until 10^{9} K, in order to obtain a set of
physical parameters from which the *X*_{i} can be computed,
2) to extrapolate the chemical abundances from the observed data, all
the way back to 10^{9} K.
Then, try to match the calculated and extrapolated abundances
*X*_{i} at the *interface* of BBN, around
10^{9} K.

I discuss first the computations of the yields. In the recent years, we have had important progresses but also some unforeseen and unpleasant complications.

Two new important laboratory experiments have resulted in a decrease
in the number of *effective* parameters needed for the computations. As
discussed before, from the recent LEP results,
(CERN collaboration 1990)
we have obtained the number of neutrino species (3.2 ± 0.2). This
number is of importance in relation with the cosmic energy density and
hence with the rate of expansion of the universe. This rate, in turn,
fixes the value of the decoupling temperature of the weak
interaction. The neutron density at the decoupling temperature fixes
the calculated abundance of helium-4. In the early computation of BBN,
this number was left as a *parameter* to be fitted by comparison with
the observed abundances. For years it had been foreseen that the
number of neutrino species had to be close to the presently measured
value of three
(Yang et al. 1984).
It is worthwhile recalling here
than in BBN computations, *fractional* numbers of relativistic species
are not necessarily meaningless since they could correspond to species
which would have decoupled at earlier times.

From the Grenoble cold neutron laboratory (Mampe et al. 1989) we have an improved value of the neutron lifetime. The value: lifetime of 890 ± 4 sec (corresponding to a half-life of 10.3 ± 0.06 minutes) is accurate enough that one does not have to include the influence of its uncertainty on the yields. This fact effectively reduces again the number of parameters of BBN.

The only remaining parameter of BBN is the density of baryonic
matter. In the past years, it was generally assumed that this density
was spatially homogeneous throughout the universe at the moment of
BBN. In this case (usually called the *standard* BBN), the computed
yields are found to match reasonably well the extrapolated
observations of light nuclides in the density range from 2 to
4 × 10^{-31} gr/cm^{3} (or *h* = 3 to
6 × 10^{-10}: the ratio of the number of
nucleons to the number of photons)
(Yang et al. 1984,
Pagel 1989,
Beaudet and Reeves, 1984).
Over the recent years, we have progressively come to
realise that a realistic computation of BBN should take into account
the possible effects of the quark-hadron phase transition around 200
MeV (Witten 1984,
Satz 1985).
In view of this developments, it seems
that we should no more use the word *standard* for the models assuming
homogeneity in baryonic density nor should we use the words *non
standard Big-Bang* to describe models taking into account the reality
of this phase transition.

The main potential effect of the Q-H transition on BBN is the
possible creation of baryonic density and neutron to proton ratio
inhomogeneities in cosmic matter. If the transition is of first order,
overcooling may take place and the hadronization may occur somewhat
below the critical temperature, just as water may find itself in a
liquid state well below 0^{o}C in a case of fast cooling. Just
as ice forms by nucleation of small crystals, pockets of hadrons will
nucleates in a sea of quarks and gluons and grow until they fill the
space. Statiscal mechanics indicates that the hadron nucleation
centers will have a density lower than the surrounding quark-gluon
sea. This is related to the fact that at the critical temperature (200
MeV) is comparable or smaller than the masses of hadrons (pions and
nucleons) while it is much larger than the *lagrangian* masses of the
quarks (a few Mev) and gluons (massless). From phase space arguments
it is easy to understand that in the first case (hadrons) most of the
available energy goes in generating the masses, therefore less is
available for phase space, while in the second case (quarks and
gluons) all the energy is available to increase the volume of phase
space. When the hadronization is completed we would then have an
inhomogeneous baryonic density distribution of matter.

From the moment of the Q-H transition (approximately 20
microseconds) until about one second, the weak interactions were fast
enough to insure a uniform neutron to proton ratio (given by the
Boltzman factor: (*n*/*p* = exp(*Mn* -
*Mp*) / *kT*)) throughout the expanding
inhomogeneities, despite the fact that the neutrons diffused away from
the density condensations while the protons were kept inside by their
electromagnetic interactions with the photon gas. Around one second,
the weak interactions became too slow to insure this weak interaction
equilibrium any more. The neutrons diffused away faster than the
protons and *n*/*p* inhomogeneities were created. Around 0.1
MeV (one
hundred seconds) primordial nucleosynthesis began, first in the high
density regions. The capture of neutrons to form deuterons decreased
the neutron density and generated neutron back-diffusion from the low
density to the high density regions. The computations have to take
into account this phenomena, in a dynamical way, in order to obtain
realistic yields.

This crucial question of the order of the transition should be
answered by high energy collisions of heavy nuclei. Experiments have
already started at CERN. The preliminary tests, based on the number of
*J*/*Y* particles emitted
(Satz 1987,
Potvin 1989),
indicates that the state of quark-gluon plasma has been reached during the
collisions. However the correct interpretations of the results is a
matter of controversy. It is expected that definite conclusions are
still a long way in the future.

The question can also be studied through QCD calculations on
lattices along a method initiated by Wilson and Polyakoff. The results
already published in the past years are not free of difficulties and
still involve a number of simplifying assumptions
(Ukawa 1988).
Year after years, the situation is improving but we are still far from
having definite answers. The last *news from the lattice*
(Fukujita et al. 1990)
favors a first order phase transition. This result however
is based on so-called *pure gauge fields*, meaning that only the
effects of the gluons have been taken into account but not the effects
of the quarks themselves. Pending better QCD results and forthcoming
experimental data, I will assume that the transition is indeed first
order. In order to cover all uncertainties, the effects of the phase
transition are taken into account by introducing a new set of
parameters. As far as the yields are concerned, we have to face
spatial variations of the baryonic density:
_{b}
becomes
_{b}(*r*). This
space dependance is usually dealt with by the introduction of three
new parameters: *d*, the mean distance between the condensation peaks,
*R*, the density contrast between the maxima and the minima, and
*f*_{v}, a measure of the *clumpiness* of the medium.

The value of the mean distance *d* depends on the surface energy of
the *bubbles* of hadrons nucleating in the sea of quarks during the
transition. If this surface density (*s*) is high, overcooling will be
extended, only large bubbles will form and there will be few of them:
thus the value of *d* will also be large. If *s* is low, many
small bubbles will form early and *d* will be low. We expect the
value of *s* to be given by QCD calculation on a lattice
(Potvin 1989).
In our present
ignorance of the value of *s* we have to face the possibility that d
could extend all the way from zero to the value of the horizon at Q-H
phase transition (ten kilometers).

Qualitatively the situation is the following. In the upper part of
the range (1 < *d* < 10^{4} km at Q-H transition) the
distances between the
bubbles are too large for appreciable neutron diffusion before and
during BBN. In consequence we have density inhomogeneities but no
*p*/*n*
inhomogeneities. The yields are obtained by a summing and averaging
the results of constant density yields over the various density
regions. At the other extreme, if *d* is less than 0.1 m, both the
neutrons and the protons diffuse and the density is homogenize before
BBN. The intermediate case (1 km > *d* > 0.1 m) is the
crucial one where
the neutrons diffuse effectively but not the protons.

The contrast *R* depends, in part, upon the value of the critical
temperature *T*_{c} which is expected to be between 100 and
250 MeV. In the lower part of the range (*T*_{c} < 150
Mev), this computed contrast *R* tends
to be large (several tens). It decreases gradually toward the upper
end of the range, as simple phase space argument would predict. The
value of *T*_{c} should come out from QCD
calculations. There are already
some indications, from perturbative approaches, that it should lie in
the upper part of the range
(Gasser and Leutwyler 1987,
1988).

Finally, the value of *f*_{v} depends in a complicated way
on the hydrodynamic of the hadron bubble growth. The problem is that QCD
calculations on lattice can only deal with statistical equilibrium
situations; they are unable to treat dynamical processes (in these
calculational techniques, the parameter *time* replaced by the
*temperature*). The problem is not with the universal expansion (too
slow to influence the course of events) but with the rate of growth of
the bubbles.

Simplified models have been made of these processes, based on the hypothesis of weakly interacting particles (Miller and Pantano 1989, Fuller et al. 1988). In these models the pressure and the energy density are given by the product of the number of particle species (with appropriate multiplicity factors) and the fourth power of the temperature. The transition then corresponds to an abrupt change of this demographic factor (from 37 for the quark-gluon plasma to 3 in the hadronic phase). However there are reasons, based on QCD calculations to doubt the validity of the weak interaction hypothesis close to the critical temperature.

In view of all these uncertainties in the values of our three parameters, the standard procedure has been to cover the whole parameter space, computing with appropriate averaging the corresponding yields. These calculations, including fine zoning and neutron back-diffusion, have been made by several groups (Mathews et al. 1988, Reeves et al. 1990, Terasawa and Sato 1989, Kurki-Suonio et al. 1990, Applegate et al. 1988). The results are in general agreement.

With all these numbers, one is in position to evaluate the range of
mean baryonic density compatible with the observations. More
specifically we ask the question: in what fraction of the parameter
space of *d*, *R*, *f*_{v}, do we find
appropriate yields? This question
necessarily introduces an element of subjectivity in the decision of
the minimum fraction acceptable. However the expected chaotic
hydrodynamic processes accompanying the bubbling and percolation
probably imply a rather wide dispersion of these effective parameters
around their mean values. Since the yields are often strong function
of these parameters, this dispersion would seem to argue against any
scenario based on a very small fraction of the parameter space (narrow
choice of parameters).

I will discuss the situation with respect to the baryonic density
range (Reeves 1990,
Reeves et al. 1990).
The lower range (from 1 to
2 × 10^{-31} g/cm^{3}) would require heavy *D*
astration (initial *D*/*H* > 3 × 10^{-4})
which is made unlikely by the present upper limit of value of the
((*D* + ^{3}*He*) / *H* > 10^{-4}). In
the range from 2 to 10 × 10^{-31}, we find
acceptable yields in a large part of the parameter space, with a
tendency toward larger *R* values (*R* > 10^{4}) as
we move toward higher densities. In the range from 10 to 20 ×
10^{-31}, the acceptable area in
the space parameter shrinks rapidly. Values of *R* >> 10 and
narrow ranges of *f*_{v} are required in order to fit the
data. Above 20 × 10^{-31}
the required values of *R* reach several thousands. Even at that,
the *He* yields are always larger than 0.25 and the
*Li*/*H* always larger than
10^{-9}. My subjective feeling is to select a compatible mean
baryonic density range between 2 and 10 × 10^{-31}, (3
× 10^{-10} <
< 15 ×
10^{-10}). Rather similar bounds on the mean baryonic
density acceptable have been obtained by
Kurki-Suonio et al. (1990)
and by Pagel (1989).

As the QCD calculations on lattices proceed, quantitative evaluations of the parameters will improve progressively, and the range of acceptable baryonic density will undoubtedly narrow down.

Some authors
(Malaney and Fowler 1989,
Mathews et al. 1989,
Applegate, Hogan and
Scherrer 1987,
Boyd and Kajino 1989)
have considered the possibility of
_{b} = 1,
(corresponding to
_{b} =
10^{-29} for
*H*_{0} is 75 km / sec / Megaparsec). Beside requiring
unrealistically large
values of the primordial *Li* abundance, this scenario corresponds to
very narrow choices of the parameters (which also appears somewhat
unrealistic). These authors have studied the formation of the light
elements *Li*, *Be*, *B* and the r-elements in this
scenario. It already
appears unlikely that, even with these extreme assumptions, they could
produce interesting amounts of these elements. (Recent calculations by
Terasawa and Sato (1990)
implied that the previous calculations have
overestimated the abundance of *Be* by two orders of magnitude.)
Observational data on old stars will be of prime importance to decide
(Rebolo 1990).