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

The creation and annihilation operators obey the canonical bosonic commutation relationsSince $\delta \varphi =\hat{f}/a$, the vacuum two-point correlation function isForone has $|f_k{|}^2=(1+1/(k^2{\tau }^2))/(2k)$. Comparison with the definition of the dimensionless power spectrum giveswhere $a=-1/(H\tau )$. On superhorizon scales, $k\ll aH$ or $|k\tau |\ll 1$, and therefore

The right-hand side of the slow-roll curvature power spectrum must be evaluated separately for each mode at cosmological horizon exit, $k=aH$. Taking logarithms givesFor $N=\log a$,HenceThis is the scalar spectral index in Hubble slow-roll parameters.

At horizon exit, the primordial tensor power spectrum is proportional to $H^2$. ThereforeThusThe ratio of the stated tensor and scalar spectra isEliminating $ϵ$ gives the single-field inflation consistency relation

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.