Astroparticle Physics

252 12 InflationOne can now ask what will happen if the vacuum energyor, in general, if any constant term dominates the totalenergy density. Suppose this is the case, i.e., ϱ ≈ ϱ v ,andassumeas well that one can neglect the k/R 2 term; this shouldalways be a good approximation for the early universe. TheFriedmann equation then readsṘ 2R 2 = 8πG3 ϱ v . (12.14)Thus, one finds that the expansion rate H is a constant,exponential increaseof the scale factoracceleration equationH = Ṙ√8πGR = 3 ϱ v . (12.15)The solution to (12.15) for t>t i is[√ ]8πGR(t) = R(t i ) e H(t−ti) = R(t i ) exp3 ϱ v (t − t i )[√ ]Λ= R(t i ) exp3 (t − t i) . (12.16)That is, the scale factor increases exponentially in time.More generally, the condition for a period of acceleratingexpansion can be seen by recalling the accelerationequation from Sect. 8.6,¨RR =−4πG (ϱ + 3P) . (12.17)3This shows that one will have an accelerating expansion, i.e.,¨R >0, as long as the energy density and pressure satisfyϱ + 3P