Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 55 / 5 / a / Solution

Differentiate the scalar-field Friedmann equation on the expanding branch $H>0$:The scalar equation of motion gives $V^{\prime }+\ddot{\Phi }=-3H\dot{\Phi }$. Canceling $H$ yieldsThus the Hubble parameter cannot increase for a canonical homogeneous scalar field. The mass scale here is the reduced Planck mass.