Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 1

The Euler-Lagrange equation isAfter a spatial Fourier transform and the change to conformal time, ${\partial }_t=a^{-1}{\partial }_{\tau }$, this becomesFor the exact de Sitter spacetime scale factor $a=-1/(H\tau )$, one has $a^{\prime \prime }/a=2a^2{H}^2$. Setting $u_{\mathbf{k}}=a{\phi }_{\mathbf{k}}$ therefore cancels both the friction term and the effective mass term, leavingThis is the special cancellation for a conformally coupled scalar field.

The positive-frequency solution for $u_k=a{f}_k$ is proportional to $e^{-ick\tau }$. Since $a=-1/(H\tau )$, its scalar-field mode function has the stated form $f_k=\mathcal{N}\tau e^{-ick\tau }$. The canonical momentum in conformal time is $\pi =a^2{\phi }^{\prime }$, so the canonical commutation relation requires the Wronskian normalizationSubstitution gives $2ick{a}^2|\mathcal{N}{|}^2{\tau }^2=i$. Hence, up to an irrelevant constant phase,The choice $e^{-ick\tau }$ is the Bunch-Davies vacuum condition at early conformal time.

Expanding the free field asand using the vacuum creation and annihilation operators algebra givesThus the dimensional power spectrum is $P_{\phi }(k,\tau )=H^2{\tau }^2/(2ck)$. Unlike a minimally coupled massless inflationary fluctuation, this conformally coupled field decays as $a^{-1}$ and does not freeze at late time.

In conformal time the interaction Hamiltonian isThe first-order in-in formalism formula and Wick theorem give the connected primordial trispectrumThe factor $4!$ from the Wick contractions cancels the vertex factor. Writing $K=k_1+k_2+k_3+k_4$, the powers of $\eta$ cancel because $a^4{\prod }_a{f}_{k_a}$ is constant apart from its phase, and the i-epsilon prescription givesConsequentlyThe full four-point function also contains the three disconnected products of the free two-point function found in part c.

Constant $c$ and $\lambda$ preserve the six spatial Euclidean isometries, namely three spatial translations and three spatial rotations. They also preserve the de Sitter dilation $(\tau ,\mathbf{x})\mapsto (s\tau ,s\mathbf{x})$, under which an equal-time correlator transforms covariantly together with its observation time. For generic $c\ne 1$, the preferred propagation speed breaks the three special conformal transformations. The correlators therefore obey seven of the ten de Sitter isometries. When $c=1$, the action is fully de Sitter invariant and all ten are restored.