Model of the universe including dark energy accounted for by both a quintessence field and a (negative) cosmological constant

In this work we present a model of the universe in which dark energy is modeled explicitly with both a dynamical quintessence field and a cosmological constant. Our results confirm the possibility of a future collapsing universe (for a given region of the parameter space), which is necessary for a consistent formulation of both string and quantum field theories. The predictions of this model for distance modulus of supernovae are similar to those of the standard model.