Fermions in Brans-Dicke cosmology L. L. Samojeden, ∗ F. P. Devecchi, † and G. M. Kremer ‡ Departamento de F´ısica, Universidade Federal do Paran´ a, Caixa Postal 19044, 81531-990 Curitiba, Brazil Using the Brans-Dicke theory of gravitation we put under investigation a hypothetical universe filled with a fermionic field (with a self interaction potential) and a matter constituent ruled by a barotropic equation of state. It is shown that the fermionic field (in combination with the Brans- Dicke scalar field ϕ ( t )) could be responsible for a final accelerated era, after an initial matter dominated period. PACS numbers: 98.80.-k, 98.80.Jk clude fermionic fields in a dynamical curved space-time. The Brans-Dicke field equations for an isotropic, homo- geneous and spatially flat universe are derived in section III where we present the analysis of the different sce- narios in which the fermionic constituent would, in prin- ciple, answer for the transition to accelerated periods. Finally we display our conclusions. The metric signature used is (+ , − , − , − ) and units have been chosen so that 8 πG = c = ℏ = 1. Einstein’s General Relativity (GR) is the usually in- voked theory for describing the evolution of the funda- mental space-time variables in cosmological models[1]. As an alternative to GR we have the Brans-Dicke the- ory of gravitation[1, 2], a scalar-tensor formulation that rules the gravitational phenomena through the interplay between the metric tensor and a scalar field ϕ that con- trols the intensity of the gravitational constant G . On the other hand, cosmological models that include dark energy are strong candidates for explaining the present regime of positive acceleration of the universe[1, 4]. Going in that direction one possibility is to consider fermionic fields as gravitational sources for those accelerated uni- verses; these sources has been investigated using sev- eral approaches, with results including exact solutions, anisotropy-to-isotropy scenarios and cyclic cosmologies (see, for example [3, 5–7]). Recently, these authors [7] proposed a cosmological model, based on GR, in a dissi- pative Universe and showed that in a young universe sce- nario the fermionic field produces a fast expansion where matter (included via a barotropic equation of state) is created till it starts to predominate and the initial ac- celerated period gives place to a decelerated era. In this case the fermionic field plays the role of the inflaton in an early period and the role of dark energy for an old universe; without the need of a cosmological constant term or a scalar field[7]. In an exclusive old universe scenario an initially matter dominated period gradually turns into a dark (fermionic) energy period when an ac- celerated regime rules for the rest of the system evolu- tion. One important point is that it was considered a self-interaction term for the fermionic constituent, in the form of a function that included the Nambu-Jona-Lasinio potential as a particular case [8]). Testing those proper- ties in a universe ruled by Brans-Dicke gravitation would be of interest and this is the main focus of this work. The manuscript is structured as follows. In section II we make a brief review of the tetrad formalism used to in- where m is the fermionic mass, ψ = ψ $^{†}$γ 0 denotes the adjoint spinor field. V is a function of ψ and ψ , and represents a fermionic self-interaction. The general co- variance principle imposes that the Dirac-Pauli matrices γ a must be replaced by their generalized counterparts Γ µ = e µ $_{a}$γ$^{a}$, where the new matrices satisfy the extended Clifford algebra, { Γ $^{µ}$, Γ $^{ν}$} = 2 g $^{µν}$. In a second step we need to substitute the ordinary derivatives by their co- variant versions where the spin connection Ω$_{µ}$ is given by where e a µ denotes the tetrad or vierbein and η$_{ab}$ is the Minkowski metric tensor; latin indices refer to a local inertial frame whereas Greek indices to the general co- ordinates system. The next step to couple the fermionic field to gravity is to construct an action for the model; we start reminding that the Dirac lagrangian density in Minkowski space-time is When one tries to include fermions in a gravita- tional model what must be taken into account is that the gauge group of general relativity does not admit a spinor representation. The tetrad formalism solves the problem[1, 2, 9, 10]. Following the general covariance principle, the metric tensor g$_{µν}$ satisfies II. FERMIONS AND GRAVITATION I. INTRODUCTION g$_{µγ}$ = e a $_{µ}$e b $_{γ}$η$_{ab}$ , a = 0 , 1 , 2 , 3 (1) L$_{D}$ = ı 2 [ ψγ $^{a}$∂$_{a}$ ψ − ( ∂$_{a}$ψ ) γ $^{a}$ψ ] − mψψ − V, (2) ∂$_{µ}$ψ → D$_{µ}$ψ = ∂$_{µ}$ψ − Ω$_{µ}$ ψ, ∂$_{µ}$ψ → D$_{µ}$ψ = ∂$_{µ}$ψ + ψ Ω$_{µ}$ , (3) Ω$_{µ}$ = − 1 4 g$_{µν}$ [Γ ν σλ − e ν $_{b}$( ∂$_{σ}$e b $_{λ}$)] γ $^{σ}$γ$^{λ}$, (4) ∗ samojed@fisica.ufpr.br † devecchi@fisica.ufpr.br ‡ kremer@fisica.ufpr.br arXiv:1001.2285v2 [gr-qc] 15 Jan 2010