Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 4

Canonical quantization imposesThe two-point correlation function has mode power $P_{\delta \varphi }=|f_k{|}^2/a^2$. After cosmological horizon exit, $|k\tau |\ll 1$, soThereforeThis is the scale-invariant inflationary power spectrum of a light canonical scalar.

Linearizing the curvaton equation and Fourier transforming givesWriting $f_k^{\sigma }=a\delta {\sigma }_k$ yieldsSince $m_{\sigma }\ll H$, the mass term is negligible and this is the equation stated. It has the same Bunch-Davies mode as the inflaton perturbation, so after horizon exitup to a small mass-induced spectral tilt.

When $H\ll m_{\sigma }$ over an oscillation, the background equation reduces toTime averaging gives $⟨{\dot{\bar{\sigma }}}^2⟩=m_{\sigma }^2⟨{\bar{\sigma }}^2⟩$, henceThe oscillating quadratic scalar is therefore pressureless matter. Its continuity equation giveswith the slowly varying amplitude obeying $\Sigma \propto a^{-3/2}$.

After reheating, inflaton decay products are radiation with ${\rho }_{\gamma }\propto a^{-4}$, whereas the oscillating curvaton has ${\rho }_{\sigma }\propto a^{-3}$. Its fractional density therefore grows, so only the curvaton can become dynamically important at late times.

The curvaton mechanism can generate the observed primordial curvature perturbation even though the inflaton drives inflation. A viable pure-curvaton realization requires the field to be light during inflation, to become sufficiently important before decay, to decay into the visible sector before nucleosynthesis, and to avoid excessive residual isocurvature and primordial non-Gaussianity. If it dominates before complete decay, these conditions can be compatible with observations.