Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 1

Conformal time obeys $dt=ad\eta$, so $\dot{\varphi }=a^{-1}{\varphi }^{\prime }$. ThereforeFor expanding de Sitter spacetime, $a=-1/(H\eta )$ with $-\infty <\eta <0$.

In one standard Schwinger-Keldysh propagator convention, suppressing the momentum delta function,The two bulk-to-boundary propagators to the late-time insertion areInterchanging every $L$ and $R$ gives the equivalent opposite contour-label convention. Direct differentiation of the supplied Bunch-Davies vacuum mode gives

Each differentiated external propagator contributes $H^2\eta /(2k)$, the differentiated internal propagator contributes $H^{2}s\eta {\eta }^{\prime }/2$, and the two vertices contribute $a(\eta )a({\eta }^{\prime })=1/(H^2\eta {\eta }^{\prime })$. The common factor is thereforeThe ordered time integral needed on either same contour branch isWriting $K=p+q$ and integrating the polynomial exponential givesThe two time orderings have $K=E_T$ and respectively $q=E_R,E_L$. Combining the $RR$ and $LL$ contour signs and complex conjugates yieldsThus $C_0={\lambda }^2{H}^8/4$ and

For opposite contour branches the two vertex integrals factorize. Sincetheir product contributes $4/(E_L^3{E}_R^3)$. The two assignments $LR$ and $RL$ then giveConsequently

A late-time connected four-point function of a scale-invariant field in three spatial dimensions, after removing its momentum-conserving delta function, must obeyIn both contributions, $s/(k_1{k}_2{k}_3{k}_4)$ scales as ${\Lambda }^{-3}$ and every energy-denominator term scales as ${\Lambda }^{-6}$. Hence the complete $s$-channel primordial trispectrum scales as ${\Lambda }^{-9}$, as required by scale invariance.