Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 15B (Cosmology)

Write the action density asUnder a compactly supported variation $\varphi \mapsto \varphi +ϵ\eta$,Integrating the first term by parts in time and the second in space, and discarding the boundary terms, givesThe principle of stationary action requires this to vanish for every $\eta$. Sincedivision by $-a^3$ yields the field Euler-Lagrange equationThis is the equation generated by the inflaton action in an expanding universe.

When $V$ is approximately constant, $dV/d\varphi \simeq 0$, and exponential expansion gives the constant Hubble parameter $H=H_{\inf }$. Take the spatial Fourier transformSinceeach independent inflationary scalar Fourier mode satisfies

Conformal time is defined byFor $a=a_0{e}^{H_{\inf }t}$, choose $\tau \to 0$ as $t\to +\infty$. Integration givesas recorded by Conformal time during de Sitter expansion.

Writing a prime for $d/d\tau$, the relationsturn the Fourier-mode equation intoSetA direct substitution givesHere $a^{\prime \prime }/a=2/{\tau }^2$, so the Canonically rescaled de Sitter scalar mode obeys

In the far past, $|ck\tau |\gg 1$, the term $2/{\tau }^2$ is negligible and the rescaled mode is a quantum harmonic oscillator of angular frequency $ck$. The stated choice of its ground state is the Bunch-Davies vacuum.

At late times, after the mode is far outside the Hubble scale, $|ck\tau |\ll 1$. Its variance becomesSince $P_k$ has dimensions of length cubed, the dimensionless power is proportional to $k^3{P}_k$. More precisely, with the standard isotropic Fourier convention,which is independent of the wavenumber $k$. Thus equal logarithmic intervals in $k$ carry equal late-time power. This is the sense in which inflation naturally generates a scale-invariant inflationary power spectrum.