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The game of Big Bang Nucleosynthesis (BBN) is to compute the abundances of light nuclides Xi (the yields) generated around T = 109 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 109 K, in order to obtain a set of physical parameters from which the Xi can be computed, 2) to extrapolate the chemical abundances from the observed data, all the way back to 109 K. Then, try to match the calculated and extrapolated abundances Xi at the interface of BBN, around 109 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/cm3 (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 0oC 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: rhob becomes rhob(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 fv, 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 < 104 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 Tc which is expected to be between 100 and 250 MeV. In the lower part of the range (Tc < 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 Tc 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 fv 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, fv, 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/cm3) 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 + 3He) / 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 > 104) 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 fv 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 < eta < 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 Omegab = 1, (corresponding to rhob = 10-29 for H0 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).

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