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The exact time period for baryogenesis is not known but most certainly it is completed no later than the electroweak phase transition. There are of course many mechanisms for baryogenesis which have been proposed in the literature [36], too many to be comprehensive here. All require baryon number violation, C and CP violation, and a departure from thermal equilibrium [37]. The original out-of-equilibrium decay (OOED) scenario [38] is probably still the simplest of all mechanisms. Originally formulated in the context of grand unified theories, OOED involved the decay of a superheavy gauge or Higgs boson with baryon number violating couplings. For example, the X gauge boson of SU(5) couples to both a (ubarubar) pair and (e-d). The decay rate for X will be

Equation 18 (18)

However decays can only begin occurring when the age of the Universe is longer than the X lifetime Gamma-1D, i.e., when GammaD > H

Equation 19 (19)

The out-of-equilibrium condition is that at T = MX, GammaD < H or

Equation 20 (20)

In this case, we would expect a maximal net baryon asymmetry to be produced, nB / s ~ 10-2 epsilon, where epsilon is a measure of the CP violation in the decay.

In the context of inflation, OOED requires either strong reheating (or preheating) [39], or that the decaying particles have masses less than the inflaton mass so that they can be produced (necessarily out of equilibrium) by inflaton decays. For example, if the inflaton potential carries only a single scale which is fixed by the magnitude of density fluctuation as measured in the microwave background radiation, then we can write [40]

Equation 21 (21)


Equation 22 (22)

so that

Equation 23 (23)

In this case a relatively light Higgs is necessary since the inflaton is typically light (meta ~ µ2 / MP ~ O(1011) GeV) and the baryon number violating Higgs would have to be produced during inflaton decay. Clever model building could allow for such a light Higgs, even in the context of supersymmetry [41]. In this case, the baryon asymmetry is given simply by

Equation 24 (24)

where TR is the reheat temperature after inflation, neta = rhoeta / meta ~ Gamma2 MP2 / meta, and Gamma = meta3 / MP2 is the inflaton decay rate.

In the context of supersymmetry, there is an extremely natural mechanism for the generation of the baryon asymmetry utilizing flat directions of the scalar potential [42]. One can show that there are many directions in field space such that the scalar potential vanishes identically when SUSY is unbroken. That is,

Equation 25 (25)

One such example is [42]

Equation 26 (26)

where a, v are arbitrary complex vacuum expectation values. SUSY breaking lifts this degeneracy so that

Equation 27 (27)

where mtilde is the SUSY breaking scale and phi is the direction in field space corresponding to the flat direction. For large initial values of phi, phi0 ~ Mgut, a large baryon asymmetry can be generated [42, 43]. This requires the presence of baryon number violating operators such as O = qqql such that <O> neq 0. The decay of these condensates through such an operator can lead to a net baryon asymmetry.

When combined with inflation, it is important to verify that the AD flat directions remain flat. In general, during inflation, supersymmetry is broken. The gravitino mass is related to the vacuum energy and m3/22 ~ V / MP2 ~ H2, thus lifting the flat directions and potentially preventing the realization of the AD scenario as argued in [44]. To see this, consider a minimal supergravity model whose Kähler potential is defined by

Equation 28 (28)

where z is a Polonyi-like field [45] needed to break supergravity, and we denote the scalar components of the usual matter chiral supermultiplets by phii. W and Wbar are the superpotentials of phii and z respectively. In this case, the scalar potential becomes

Equation 29 (29)

Included in the above expression for V, one finds a mass term for the matter fields phii, eGphii*phii = m3/22phii*phii [46]. As it applies to all scalar fields (in the matter sector), all flat directions are lifted by it as well. The above arguments can be generalized to supergravity models with non-minimal Kähler potentials.

There is however a special class of models called no-scale supergravity models, that were first introduced in [47] and have the remarkable property that the gravitino mass is undetermined at the tree level despite the fact that supergravity is broken. No-scale supergravity has been used heavily in constructing supergravity models in which all mass scales below the Planck scale are determined radiatively [48], [49]. These models emerge naturally in torus [50] or, for the untwisted sector, orbifold [51] compactifications of the heterotic string.

In no-scale models (or more generally in models which possess a Heisenberg symmetry [52]), the Kähler potential becomes

Equation 30 (30)

Now, one can write

Equation 31 (31)

It is important to notice that the cross term |phii*W|2 has disappeared in the scalar potential. Because of the absence of the cross term, flat directions remain flat even during inflation [53]. The no-scale model corresponds to f = - 3 lneta , f '2 = 3f'' and the first term in (31) vanishes. The potential then takes the form

Equation 32 (32)

which is positive definite. The requirement that the vacuum energy vanishes implies <Wi> = <ga> = 0 at the minimum. As a consequence eta is undetermined and so is the gravitino mass m3/2(eta).

The above argument is only valid at the tree level. An explicit one-loop calculation [54] shows that the effective potential along the flat direction has the form

Equation 33 (33)

where Lambda is the cutoff of the effective supergravity theory, and has a minimum around phi appeq 0.5Lambda. Thus, phi0 ~ MP will be generated and in this case the subsequent sfermion oscillations will dominate the energy density and a baryon asymmetry will result which is independent of inflationary parameters as originally discussed in [42, 43] and will produce nB/s ~ O(1). Thus we are left with the problem that the baryon asymmetry in no-scale type models is too large [55, 53, 56].

In [56], several possible solutions were presented to dilute the baryon asymmetry. These included 1) Moduli decay, 2) the presence of non-renormalizable interactions, and 3) electroweak effects. Moduli decay in this context, turns out to be insufficient to bring an initial asymmetry of order nB/s ~ 1 down to acceptable levels. However, as a by-product one can show that there is no moduli problem [57] either. In contrast, adding non-renormalizable Planck scale operators of the form phi2n - 2 / MP2n - 6 leads to a smaller initial value for phi0 and hence a smaller value for nB/s. For dimension 6 operators (n = 4), a baryon asymmetry of order nB/s ~ 10-10 is produced. Finally, another possible suppression mechanism is to employ the smallness of the fermion masses. The baryon asymmetry is known to be wiped out if the net B - L asymmetry vanishes because of the sphaleron transitions at high temperature. However, Kuzmin, Rubakov and Shaposhnikov [58] pointed out that this erasure can be partially circumvented if the individual (B - 3Li) asymmetries, where i = 1, 2, 3 refers to three generations, do not vanish even when the total asymmetry vanishes. Even though there is still a tendency that the baryon asymmetry is erased by the chemical equilibrium due to the sphaleron transitions, the finite mass of the tau lepton shifts the chemical equilibrium between B and L3 towards the B side and leaves a finite asymmetry in the end. Their estimate is

Equation 34 (34)

where the temperature T ~ TC ~ 200 GeV is when the sphaleron transition freezes out (similar to the temperature of the electroweak phase transition) and mtau(T) is expected to be somewhat smaller than mtau(0) = 1.777 GeV. Overall, the sphaleron transition suppresses the baryon asymmetry by a factor of ~ 10-6. This suppression factor is sufficient to keep the total baryon asymmetry at a reasonable order of magnitude in many of the cases discussed above.

Finally, it is necessary to mention one other extremely simple mechanism based on the OOED of a heavy Majorana neutrino [59]. This mechanism does not require grand unification at all. By simply adding to the Lagrangian a Dirac and Majorana mass term for a new right handed neutrino state,

Equation 35 (35)

the out-of-equilibrium decays nuc -> L + H* and nuc -> L* + H will generate a non-zero lepton number L neq 0. The out-out-equilibrium condition for these decays translates to 10-3 lambda2 MP < M and M could be as low as O(10) TeV. (Note that once again in order to have a non-vanishing contribution to the C and CP violation in this process at 1-loop, at least 2 flavors of nuc are required. For the generation of masses of all three neutrino flavors, 3 flavors of nuc are required.) Sphaleron effects can transfer this lepton asymmetry into a baryon asymmetry since now B - L neq 0. A supersymmetric version of this scenario has also been described [40, 60].


As the Inner Space/Outer Space II workshop, and these proceedings are in memory of Dave Schramm, it is fitting to acknowledge Dave's role. Dave's research interests were predominantly concerned with the epoch of Big Bang Nucleosynthesis and the post BBN Universe. His research interests are known to have been extremely diverse covering such areas as chemical evolution, cosmic rays, cosmochronology, dark matter, galaxy formation and mergers, the gamma-ray background, magnetic fields, mass extinctions, neutrinos, (late-time) phase transitions, supernovae, ... In addition, he was a master at synthesizing and finding relationships between these topics. Though pre-BBN cosmology was not Dave's mainstay, he did of course make important contributions in areas such as baryogenesis and topological defects. But Dave's most important contribution was the early recognition of the impact of cosmology on particle physics. He can legitimately be considered a founding father of astro-particle physics. On a more personal note, it is difficult to put into words the immense impact that Dave has had on my research, as an advisor, colleage, and friend. This work was supported in part by DOE grant DE-FG02-94ER40823 at Minnesota.

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