Varying the scalar action and Fourier transforming the spatial coordinates gives the <Klein-Gordon equation>
$$
\phi_k''+2\frac{a'}a\phi_k'+(k^2+a^2m^2)\phi_k=0.
$$
In <de Sitter spacetime>, $a=-1/(H\eta)$ and $m^2=2H^2$, so
$$
\boxed{\phi_k''-\frac2\eta\phi_k'
+\left(k^2+\frac2{\eta^2}\right)\phi_k=0}.
$$
Direct substitution shows that $\eta e^{-ik\eta}$ and $\eta e^{ik\eta}$ are independent solutions. Hence
$$
\boxed{\phi(\eta,\mathbf k)
=\eta\left[C_+(\mathbf k)e^{-ik\eta}
+C_-(\mathbf k)e^{ik\eta}\right]}.
$$
The mass value is the one that gives a <conformally coupled scalar field> in four-dimensional de Sitter spacetime.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 312 / 1 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-312/1/i/solution