Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 310 / 2

For a spatially homogeneous canonical scalar field, spatial derivatives vanish. Reading the time and spatial components of its energy-momentum tensor in the comoving frame givesThus the kinetic term contributes equally to energy density and pressure, whereas the scalar potential contributes negative pressure.

Substitution of these expressions into the cosmological perfect-fluid continuity equation givesContinuity through a turning point therefore yields the homogeneous Klein-Gordon equationDuring slow-roll inflation, the acceleration and kinetic-energy terms are negligible. In terms of the reduced Planck mass, the approximate evolution is

This is the potential of Starobinsky inflation. Put $b=\sqrt{2/3}/M_{\mathrm{Pl}}$ and $y=e^{-b\varphi }$, so $V=V_0(1-y{)}^2$. Its potential slow-roll parameter and second potential slow-roll parameter areAt large positive $\varphi$, $y\ll 1$, so both ${ϵ}_V$ and $|{\eta }_V|$ are small and slow-roll inflation is possible. Near the minimum, a Taylor expansion gives $1-y\simeq b\varphi$ and henceThey are large for $|\varphi |\ll M_{\mathrm{Pl}}$, so that region cannot sustain slow roll.

Near the minimum, the Taylor expansion of the potential isNeglecting Hubble friction during one short oscillation reduces the Klein-Gordon equation to the harmonic oscillator equationThe virial theorem gives $⟨{\dot{\varphi }}^2/2⟩=⟨V⟩$, so $⟨P_{\varphi }⟩=0$. Restoring the slow cosmological damping in the averaged cosmological perfect-fluid continuity equation givesThe coherently oscillating inflaton therefore behaves as pressureless matter.

The matter-like scaling is special to a quadratic minimum. More generally, the effective equation of state of an oscillating scalar field in $V\propto |\varphi {|}^n$ isFor example, a quartic minimum has $⟨w⟩=1/3$ and redshifts like radiation in cosmology. Potentials without a stable minimum, oscillations whose period is not short relative to the Hubble time, and significant decay of the inflaton into other particles also invalidate the matter-like argument.