A scalar field as dark energy

As we have seen during the course on inflation, a scalar field can behave as a cosmological constant when its kinetic term becomes negligible in front of its potential term. However, the features of the scalar field we are interested in differ significantly from an inflationary scalar field in the former case, we want a field that is negligible at early times and which dominates afterwards, whereas in the latter case, it is the contrary. Historically, the first scalar field dark energy model was aimed to address the possibility to have some components with a constant equation of state parameter w other than 0 (matter), 1/3 (radiation), —1/3 (curvature) and —1 (cosmological constant) (Ratra Peebles 1988). 

By definition of a homogeneous scalar field, one has the following relation between the kinetic and the potential terms  

Assuming that one is in an era where the background equation of state parameter wb is constant, one can derive a relation between II and H2 with the held of the Friedmann equations, so that the three relations above can be translated into 

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