Ultra-slow-roll inflation

Ultra-slow-roll inflation occurs on an exceptionally flat potential. The homogeneous field velocity decays as $\dot{\varphi }\propto a^{-3}$, so $ϵ\propto a^{-6}$ and the second Hubble slow-roll parameter approaches $-6$.

In ultra-slow-roll inflation, $z=a\sqrt{2ϵ}\propto a^{-2}\propto {\eta }^2$. The comoving curvature mode equation isand its Bunch-Davies mode is proportional to $(1+ik\eta )e^{-ik\eta }/{\eta }^3$. Unlike ordinary slow roll, the superhorizon curvature perturbation grows as $a^3$.

For a Hermitian perturbation mode $X$ and its time derivative $\dot{X}$, subtract their expectations and use the symmetrized covariance to define variances ${\sigma }_X^2,{\sigma }_{\dot{X}}^2$. A commutator-to-fluctuation diagnostic isFor zero-mean modes this reproduces the usual expression with $⟨XX⟩$ and $⟨\dot{X}\dot{X}⟩$ in the denominator. A finite real Fourier mode or a suitably smeared field must be specified; unregulated coincident field products need not have finite variance.

The Robertson uncertainty relation gives $\mathcal{C}\le 2$ when the denominator is nonzero. Small $\mathcal{C}$ means the commutator is small relative to the marginal fluctuation scales, which can justify classical approximations for some symmetrized correlators. It does not alone show that the quantum state is mixed or that quantum decoherence has occurred. For example, a correlated Gaussian wave packet in a pure state can have a large product of marginal uncertainties while retaining quantum coherence. Which observables and accuracy are being approximated must therefore accompany a claim of effective classicality.