Stability of the wq Const regime

Let us study the stability of the solution wq = Const found above. To do so, we perturb the Klein-Gordon equation around this solution. Here, we need only to consider the case 0 = 4 WQ=Const + where 6 fi is a function of time only. After simple manipulations, we find 

Stability is insured when the exponents of power law solutions to this equation all have a negative real part. This is the case as soon as 

From Eq. (8.9), it is clear that wq v b, so that for the special solution wq = Const, the field starts with an negligible energy density which subsequently decays less fast than that of the background, so that ultimately the quintessence field will dominate the energy density of the universe. Since we have today Qde G,lial. today s epoch represents the transition between the matter dominated era and the quintessence dominated era (as observations favor an Qq only slightly larger than Qmat). 

So far, we studied only the case where the quintessence field was subdominant. The subsequent evolution of the scalar field (when it becomes the main form of energy) is slightly more complicated to compute, but one can show 

An example of the temporal evolution of a quintessence field is given in Fig.8.1. 

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