Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 4 / d / Solution

For a canonical scalar field with positive kinetic energy, the Friedmann equation givesso the first Hubble slow-roll parameter satisfies $ϵ\ge 0$. The tensor tilt is consequentlyA blue primordial spectrum of tensor modes cannot arise in standard canonical single-field slow-roll inflation; it requires changing assumptions, such as the matter content, kinetic structure, initial state, or gravitational dynamics.

The scalar tilt instead obeysIt can be positive in a standard single-field model if $\eta <-2ϵ$, meaning that $ϵ$ decreases sufficiently rapidly with the number of e-folds. Equivalently, in potential slow-roll parameters the first-order condition is $2{\eta }_V>6{ϵ}_V$. Thus a blue scalar spectrum is allowed by the framework, although it is not produced by every slowly rolling potential.