 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 310

Linearizing the <curvaton> equation and Fourier transforming gives
$$
\delta\sigma_k''+2\frac{a'}a\delta\sigma_k'
+(k^2+a^2m_\sigma^2)\delta\sigma_k=0.
$$
Writing $f_k^\sigma=a\delta\sigma_k$ yields
$$
(f_k^\sigma)''+\left(k^2+a^2m_\sigma^2-\frac{a''}{a}\right)f_k^\sigma=0.
$$
Since $m_\sigma\ll H$, 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 exit
$$
\boxed{\Delta_{\delta\sigma}^2(k)=\left(\frac H{2\pi}\right)^2}
$$
up to a small mass-induced spectral tilt.

Solution

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

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

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

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

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)