  We study the cosmological implications of the Nambu-Jona-Lasinio (NJL model)
when the coupling constant is field dependent. The NJL model has a four-fermion
interaction describing two different phases due to quantum interaction effects
and determined by the strength of the coupling constant g. It describes
massless fermions for weak coupling and a massive fermions and strong coupling,
where a fermion condensate is formed. In the original NJL model the coupling
constant g is indeed constant, and in this work we consider a modified version
of the NJL model by introducing a dynamical field dependent coupling motivated
by string theory. The effective potential as a function of the varying coupling
(aimed to implement a natural phase transition) is seen to develop a negative
divergence, i.e. becomes a "bottomless well" in certain limit region. Although
we explain how an lower unbounded potential is not necessarily unacceptable in
a cosmological context, the divergence can be removed if we consider a mass
term for the coupling-like field. We found that for a proper set of parameters,
the total potential obtained has two minima, one located at the origin (the
trivial solution, in which the fluid associated with the fields behave like
matter); and the other related to the non-trivial solution. This last solution
has three possibilities: 1) if the minimum is positive V_{min}>0, the system
behave as a cosmological constant, thus leading eventually to an accelerated
universe; 2) if the minimized potential vanishes V_{min}=0, then we have matter
with no acceleration ; 3) finally a negative minimum V_{min}<0 leads an
eventually collapsing universe, even though we have a flat geometry.Therefore,
a possible interpretation as Dark Matter or Dark Energy is allowed among the
behaviors implicated in the model.
