Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 312 (Field Theory in Cosmology)

Varying the scalar action and Fourier transforming the spatial coordinates gives the Klein-Gordon equationIn de Sitter spacetime, $a=-1/(H\eta )$ and $m^2=2H^2$, soDirect substitution shows that $\eta e^{-ik\eta }$ and $\eta e^{ik\eta }$ are independent solutions. HenceThe mass value is the one that gives a conformally coupled scalar field in four-dimensional de Sitter spacetime.

Forthe specified cosmological bulk-to-boundary propagator isThe right-right cosmological bulk-to-bulk propagator is time ordered:The mixed Wightman function in the order stated in the question isThe suppressed factor in each expression is $(2\pi {)}^3{\delta }^{(3)}(\mathbf{p}+{\mathbf{p}}^{\prime })$.

The exchange graph has one quartic vertex attached to ${\mathbf{k}}_1,{\mathbf{k}}_2,{\mathbf{k}}_3$, a second attached to ${\mathbf{k}}_4,{\mathbf{k}}_5,{\mathbf{k}}_6$, and an internal line of momentumSchematically,The Schwinger-Keldysh conjugation relation gives

Set ${\eta }_0=0$ only in integration limits and phases, retaining its leading explicit power from the six external propagators. The scale factors, six external propagators, and one internal propagator leave the common coefficientFor two right-branch vertices, split the time-ordered integral into ${\eta }_1>{\eta }_2$ and ${\eta }_2>{\eta }_1$:The two right-vertex factors contribute the compensating minus sign, soFor one vertex on each branch, the two integrals factorize:Thus the quantities requested in the question areAt late time these expressions are real. Summing all four in-in formalism assignments gives

The massless field equation isFor a homogeneous solution this becomesThereforeSince $\dot{\bar{\varphi }}={\bar{\varphi }}^{\prime }/a$, the cosmic-time velocity decays as $a^{-3}$.

The second Friedmann equation gives the first Hubble slow-roll parameterBecause $\dot{\bar{\varphi }}\propto a^{-3}$,ThusAs $a\to \infty$, the kinetic energy decays as $a^{-6}$ and the constant $\Lambda$ dominates. Hence $H$ approaches a constant,This non-attractor background is ultra-slow-roll inflation.

Varying the quadratic action givesFor a Fourier mode,At late times $ϵ\propto a^{-6}$, soThe equation becomes ${\mathcal{R}}_k^{\prime \prime }+4{\mathcal{R}}_k^{\prime }/\eta +k^2{\mathcal{R}}_k=0$. At $k=0$, its independent solutions are a constant and ${\eta }^{-3}$. The leading late-time solution in the form given in the question is thereforeThis is the growing curvature perturbation in ultra-slow-roll inflation.

Expand the quantized field using the normalized modeThe canonical commutation relation fixes the Wronskian $M_{\mathrm{Pl}}^2{z}^2(u_k{u}_k^{\ast \prime }-u_k^{\ast }{u}_k^{\prime })=i$. Writing $x=k\eta$, the primed correlators areEach unprimed expression includes $(2\pi {)}^3{\delta }^{(3)}(\mathbf{k}+{\mathbf{k}}^{\prime })$. The imaginary part of the mixed correlator givesConsequently the classicality parameter of a cosmological perturbation isIt tends to zero on superhorizon scales, so the perturbation becomes effectively classical even while its amplitude grows as $a^3$. In ordinary slow-roll inflation, $\mathcal{R}$ instead freezes to a constant; its analogous normalized commutator also vanishes, but behaves as $|k\eta |$ for the leading de Sitter mode rather than $|k\eta {|}^3$.

Let $\theta =\nabla \cdot \mathbf{v}$. Vanishing vorticity implies in Fourier spaceFourier transforming $\nabla \cdot (\delta \mathbf{v})$ in the continuity equation giveswith the alpha mode-coupling kernelTaking the divergence of $(\mathbf{v}\cdot \nabla )\mathbf{v}$ and symmetrizing its two velocity arguments giveswhere the beta mode-coupling kernel is

The ultraviolet softness of the second-order density kernel requiresfor fixed $\mathbf{k}$ as $q\to \infty$. In this limit the two hard vectors are opposite,The constant part of the stated kernel is thereforeIt must vanish, givingThis recovers the standard standard perturbation theory density kernel $F_2$.

For Gaussian linear perturbations, a connected tree on five external density insertions has four Wick-contraction edges and therefore containsThus $n=4$. The three unlabeled tree-level connected matter five-point function topologies correspond to the perturbative-order partitionsThey are respectively an $F_4$ star with four $F_1$ leaves, an $F_3$ vertex joined to an $F_2$ vertex with three $F_1$ leaves, and a chain with three $F_2$ vertices and two endpoint $F_1$ vertices.

For one labeling of the last topology, let ${\mathbf{k}}_{12}={\mathbf{k}}_1+{\mathbf{k}}_2$ and ${\mathbf{k}}_{123}={\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3$. Its algebraic contribution isThe factor $2^3=8$ counts the interchange of the two linear legs at each symmetric $F_2$ vertex. The full correlator adds all distinct permutations of external labels and the overall momentum-conserving delta function.

Let $\hat{\mathbf{n}}$ point from the observer toward the emission point, so the photon propagation direction is $\hat{\mathbf{p}}=-\hat{\mathbf{n}}$ andAlong this path $D=d/d\eta ={\partial }_{\eta }-\hat{\mathbf{n}}\cdot \nabla$. Substituting $\hat{\mathbf{p}}=-\hat{\mathbf{n}}$ into the Photon Boltzmann equation with Thomson scattering givesBecause ${\partial }_{\eta }{e}^{-\tau }=\Gamma e^{-\tau }$, multiplication by the integrating factor giveswhereIntegrating along the ray and using $e^{-\tau }\to 0$ at sufficiently early time yields the Cosmic microwave background line-of-sight solution

The cosmological visibility function is the probability density for last scattering, so the stipulated two instantaneous populations giveSince $g={\partial }_{\eta }{e}^{-\tau }$ and $e^{-\tau }$ vanishes before the first transparent epoch,Its plot is a nondecreasing step function: it jumps from zero to $p$ at recombination and from $p$ to one at ${\eta }_1$. The cosmological optical depth interpretation is that $e^{-\tau (\eta )}$ is the probability that a photon present at time $\eta$ reaches the observer without any later scattering.

Write ${\mathbf{x}}_j={\mathbf{x}}_0+({\eta }_0-{\eta }_j)\hat{\mathbf{n}}$. The delta functions in $g$ evaluate the local last-scattering sources, while the piecewise $e^{-\tau }$ weights the integrated gravitational source. ThusThe first two lines are weighted Sachs-Wolfe combination and Doppler sources at the two last-scattering surfaces; the final lines are the corresponding Integrated Sachs-Wolfe effect.

To predict these quantities from primordial perturbations, one solves the coupled linear Einstein field equations and Boltzmann equation hierarchy for photons, baryons, cold dark matter, and neutrinos, together with the ionization history that determines $\Gamma$ and hence $g$. The solutions provide ${\Theta }_0$, ${\mathbf{v}}_e$, $\Phi$, and $\Psi$ at both surfaces and along the line of sight.