Write $\phi=\bar\phi+\varphi$ and evaluate derivatives of $P$ on the homogeneous background
$$
\bar X=\frac{\bar\phi'^2}{2a^2}.
$$
The first- and second-order changes in $X$ are
$$
\delta_1X=\frac{\bar\phi'\varphi'}{a^2},
\qquad
\delta_2X=\frac{\varphi'^2-(\nabla\varphi)^2}{2a^2}.
$$
After using the background equation to remove the linear action, the quadratic action of the <P(X, phi) scalar field theory> is
$$
S_2=\frac12\int d\tau\,d^3x\left\lbrace
a^2A\varphi'^2-a^2P_X(\nabla\varphi)^2
+2a^2P_{X\phi}\bar\phi'\varphi\varphi'
+a^4P_{\phi\phi}\varphi^2\right\rbrace,
$$
where
$$
A=P_X+2\bar XP_{XX}.
$$
Varying gives
$$
(a^2A\varphi')'-a^2P_X\nabla^2\varphi
+\left[(a^2P_{X\phi}\bar\phi')'-a^4P_{\phi\phi}\right]\varphi=0.
$$

The <Sound speed of a P(X, phi) scalar perturbation> is
$$
\boxed{c_s^2=\frac{P_X}{P_X+2\bar XP_{XX}}=\frac{P_X}{A}}.
$$
At leading slow variation, take $H$, $c_s$, and the kinetic coefficients as nearly constant and neglect the effective mass and their logarithmic derivatives. The Fourier equation then becomes
$$
\varphi_k''+2\frac{a'}a\varphi_k'+c_s^2k^2\varphi_k=0.
$$
For $u_k=a\varphi_k$ and <de Sitter spacetime> $a''/a=2/\tau^2$, this is
$$
u_k''+\left(c_s^2k^2-\frac2{\tau^2}\right)u_k=0.
$$
Two independent solutions are
$$
u_k^{\pm}=\left(1\pm\frac{i}{c_sk\tau}\right)e^{\pm ic_sk\tau}.
$$
The <Bunch-Davies vacuum> selects the positive-frequency behavior $e^{-ic_sk\tau}$ as $-c_sk\tau\to\infty$. Canonical normalization of $v=a\sqrt A\,\varphi$ gives the general amplitude; with the field normalization $P_X\simeq1$, so $A\simeq c_s^{-2}$, it reduces, up to an overall phase, to
$$
\boxed{\varphi_k(\tau)=
\frac{H}{\sqrt{2c_sk^3}}
(1+ic_sk\tau)e^{-ic_sk\tau}}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 1 / i / Solution

{c}
{title2=$P(X,\phi)$ theory}
{wiki=K-essence}

A $P(X,\phi)$ scalar theory has Lagrangian density $P$ depending on the field and $X=-g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi/2$. It includes canonical scalar theory and models with noncanonical kinetic interactions.


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