2.2 Gravitinos
2.2.4 Cosmology: the gravitino problem
So far we have considered the theory of supergravity and how the gravitino emerges in this framework. We conclude this chapter by discussing the role of this particle in cosmology, which constitutes the motivation of this thesis. The gravitino is relevant for several aspects. Not only it has a primary role on theoretical grounds, since it is the gauge field of local supersymmetry. Either stable or unstable gravitinos, when embedded in cosmological backgrounds, can generate enormous problems to theories which are in agreement with observations. This is known in the literature as the gravitino problem.
The production of gravitinos in the early universe occurs through a variety of ways, via thermal or non-thermal mechanisms. In the first case, in the reheating era at the end of inflation (see Chapter 1) a thermal bath is generated via the decay of the inflaton field. The particles in the plasma scatter off each other, and this process generates gravitinos as the final states of 2 → 2 hard scatterings [13, 53, 55, 56, 57, 58, 59]. It has been shown that an important amount of gravitinos was produced right through these reactions [60, 78]. This is the subject of paper [1], where we have found an unexpected effect when the gauge bosons which scatter in the primordial bath are massive.
Non-thermal production is perhaps a more complicated topic. It can follow from several sources: from the decay of the next to lightest SUSY particle (NLSP)
2.2. GRAVITINOS 49 [61, 62, 63, 64], of the moduli fields [65, 66, 67, 68], of the inflaton [69, 70, 71, 72] and of the SUSY breaking field [73, 74, 75, 76, 77]. The latter mechanism is discussed in the article [2], where it is shown that the reheating temperature can be related by the parameters of the model considered, if a right amount of gravitino Dark Matter is required.
The gravitino mass mGe can range in principle from the eV scale up to the TeV scale and beyond [41]. It is strongly model dependent, since it is proportional to the SUSY breaking scale and therefore to the condensation value of the F field, as we have discussed in this chapter.
In general, gauge mediation predicts the gravitino to be the lightest supersym-metric particle, or LSP [46]. Then, if R-parity is conserved, the gravitino is stable and it can be a very attractive candidate for Dark Matter [54]. It can however generate problems: since too many gravitinos might overclose the universe, in the standard Big Bang cosmology it is set an upper limit of mGe ∼< 1 keV [55]. This constraint can be anyway relaxed if we assume inflation, since it dilutes the initial abundance of gravitinos5. However, the problem persists also in this case. The ther-mal scatterings reproduce the particle after reheating. The number density of the secondary gravitinos is proportional to the reheating temperature, thus TR should be constrained in order to avoid particle overproduction.
In other classes of theories, for instance in gravity mediation, the gravitino is unstable and if mGe is smaller than 10 TeV, it has a lifetime τGe which is usually longer than 1 sec. This means that it decays after the beginning of the Big Bang Nu-cleosynthesis (BBN) and it generates hadronic and electromagnetic showers which can be very energetic [37]. This implies disintegration of the primordial light ele-ments [56]. Since cosmological observations have so far verified the BBN predictions to a very high precision, one must impose constraints on the scenario, in order to preserve the agreement between theory and observations [79, 80, 81]. It is therefore evident that the gravitino generates several problems in cosmology. In the literature lots of effort had been devoted to an extensive study of these issues (see for example, [55, 56, 57]).
In particular, BBN constraints on both unstable and stable gravitinos have been recently derived in Ref.[80], which contains an updated analysis of the gravitino problem in this context. We refer to this paper to complete the above generic discussion. For unstable gravitinos, let us consider the decay into a neutralino LSP.
It is found that the upper bound strongly depends on mGe. In order to preserve the light elements 3He, 4He, 6Li and D, the upper limit on the reheating temperature has a mild dependence on the mass spectrum of the MSSM particles. However, it
5Historically, the first bound was calculated to be betweenO(1 MeV) andO(100 GeV) [78].
strongly depends on the gravitino mass:
106GeV ∼< TR <
∼ 1010GeV if 300 GeV ∼< mGe ∼< 30 TeV. (2.141) The above range for TR is interesting, since it provides with a very stringent con-straint on theories of thermal leptogenesis, which actually require a reheating tem-perature that is at least of the order 109GeV [82, 83, 84, 85, 86, 87, 88, 89].
In the case of stable LSP gravitinos, the bounds depend on which particle is the NLSP (the authors have considered the bino, the stau and the sneutrino). Very stringent constraints have been found if the bino or the stau are the NLSP. Namely, mGe ∼> 10 GeV is excluded if the mass of the NLSP is lighter than 1 TeV. If the sneutrino is the lightest supersymmetric particle, the BBN constraints are sensibly weaker since the sneutrino decays mainly into gravitino and neutrino (i.e. weakly interacting particles). In any case, the constraints are generally very restrictive. The decay of the NLSP alone cannot provide with the correct amount of gravitino Dark Matter. Therefore the largest contribution should be produced by other mechanisms, for instance through thermal scatterings or by decay of the scalar condensate.
Both of these possibilities are investigated in the research papers [1] and [2], which are discussed respectively in Chapters 3 and 4 of this thesis.
Chapter 3
Gauge boson scattering and gravitinos
In this chapter, the phenomenology of the Standard Model and of the MSSM will be discussed. Topics like gauge invariance and the MSSM mass spectrum are closely related to the results of paper [1], that concerns the scattering of two massive W bosons
Wa+Wb −→fWc+G ,e W++W−−→χei0+G ,e (3.1) with a gravitino and a gaugino in the final state, both in the gauge and matter eigenstates. In the next section we analyse the massless limit of the above process, namely the gluon scattering in supergravity.