Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 312 (Field Theory in Cosmology)

Under spatial reflection, a scalar field obeys $P\varphi (\mathbf{x})P=\varphi (-\mathbf{x})$, and hence each spatial derivative changes sign. The interaction contains three such derivatives, so changing the integration variable from $\mathbf{x}$ to $-\mathbf{x}$ givesThus this is a parity-odd scalar interaction.

Put $O({\mathbf{k}}_a)={\prod }_{a=1}^n{\varphi }_a({\mathbf{k}}_a)$ and $A({\mathbf{k}}_a)=⟨H_{\mathrm{int}}O({\mathbf{k}}_a)⟩$. Hermitian conjugation and commutativity of equal-time scalar fields giveThe parity-invariant vacuum and the odd parity of $H_{\mathrm{int}}$ imply $A(-{\mathbf{k}}_a)=-A({\mathbf{k}}_a)$. ThereforeThere is no conflict with the usual $2i\mathrm{Im}A$ rule for two Hermitian operators: a momentum-space product at fixed ${\mathbf{k}}_a$ is generally not itself Hermitian, since its adjoint carries momenta $-{\mathbf{k}}_a$.

The first-order in-in formalism formula at observation time ${\tau }_0$ isFourier transforming the three derivatives giveswhere ${\int }_{\mathbf{p}}=\int d^{3}p/(2\pi {)}^3$. Because the four species are distinct, Wick contraction pairs each vertex field with the external field of the same species and introduces no permutation factor. IfthenCombining this with part i gives the requested time integral:The factor $\mathcal{E}$ is a pseudoscalar, so the resulting primordial trispectrum is parity odd and purely imaginary in this momentum-space convention.

The intended oscillatory factor is $e^{-ik\tau }$. Give the early-time endpoint the usual i-epsilon prescription and set $x=-k\tau$. For an integer $p\ge 0$,It is therefore purely imaginary. Equivalently, rotating the contour to the negative imaginary $\tau$ axis turns the remaining integral into a real Gamma integral and leaves one overall factor of $i$.

At late time $f(k,{\tau }_0)\to H/\sqrt{2k^3}$ and $a=-1/(H\tau )$. Define the elementary symmetric polynomialsThen part ii reduces towhere a prime removes the momentum-conserving Dirac delta distribution andExpanding the product givesRepeated integration by parts reduces every negative power to the supplied logarithmic integral. The power divergences are real and disappear when the imaginary part is taken. Writingone obtainsConsequently the late-time parity-odd primordial trispectrum isIts factor of $i$ is required by reality of a momentum-space scalar correlator: reversing all momenta complex-conjugates the correlator, while the scalar triple product changes sign.

Statistical homogeneity forces a scalar two-point function to have momenta $(\mathbf{k},-\mathbf{k})$. Statistical isotropy then makes it a function only of $|\mathbf{k}|$, so it is parity even without using perturbation theory.

For a scalar three-point function, momentum conservation gives ${\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3=0$, so all three vectors lie in one plane. A rotation by $\pi$ around the normal to that plane sends every ${\mathbf{k}}_a$ to $-{\mathbf{k}}_a$. Rotational invariance therefore identifies a triangle with its parity reverse, proving nonperturbatively that the scalar primordial bispectrum is parity even.

A rotationally invariant local parity-odd three-scalar vertex must contain a Levi-Civita symbol contracted with three spatial momenta. Its momentum-space factor is proportional tobecause momentum conservation makes the momenta linearly dependent. The equivalent position-space expression is a total divergence; its spatial integral vanishes under the stated vanishing boundary condition. Thus a parity-odd interaction of three scalar fields contributes nothing to the action.

The background spatial metric is $g_{ij}=a^2{\delta }_{ij}$. For the large spatial diffeomorphismhomogeneity and isotropy imply that the purely spatial background Christoffel symbols vanish. HenceThe shift is constant, transverse and traceless, and is therefore the zero-momentum adiabatic tensor mode.

A scalar transforms by its Lie derivative:Fourier transformation and integration by parts in momentum space giveThe term proportional to ${\omega }_i{}^i$ vanishes because the deformation is traceless.

The equal-time canonical commutation relation for the transverse-traceless graviton and its canonical momentum is the transverse-traceless projector. At zero momentum the supplied polarization completeness relation giveswhere symmetry and tracelessness of ${\omega }_{ij}$ remove the trace term. Thus the charge generates precisely the transformation in part i:This is the soft, field-independent part of the Noether charge associated with the large diffeomorphism.

Let $O=\phi ({\mathbf{k}}_1)\phi ({\mathbf{k}}_2)$. Since $Q_S=-2{\omega }_{ij}{\Pi }_{ij}(0)$,For a soft mode ${\gamma }^s=f_q{a}_s+f_q^{\ast }{a}_s^{†}$ and ${\Pi }^s=g_q{a}_s+g_q^{\ast }{a}_s^{†}$, the vacuum obeysThe mode-function Wronskian normalizationimplieswhere $P_{\gamma }(q)$ is the graviton power spectrum per polarization. The equal-time bispectrum is real in this parity-even configuration, and thereforeThis turns the charge insertion into the soft-graviton insertion used in the Soft graviton theorem.

Applying the scalar transformation from part i to both fields givesThe derivative of the Dirac delta distribution is proportional to ${\delta }_{ij}$ and vanishes after contraction with traceless ${\omega }_{ij}$. Removing that delta function leavesEquating the two sides of the Ward-Takahashi identity and resolving the soft graviton into a polarization $s$ yields the cosmological soft-graviton consistency relationFor an isotropic power spectrum this is equivalentlyThe long-wavelength adiabatic tensor mode acts on the short two-point function as an anisotropic rescaling of its momentum.

Let the photon phase-space distribution be a Planck distribution whose local temperature is $T(\eta )[1+\Theta (\eta ,\mathbf{x},\hat{\mathbf{p}})]$. In the absence of collisions, Liouville theorem says that the distribution is constant along a photon null geodesic. Linearizing $df/d\eta =0$ about the homogeneous distribution and using $d\mathbf{x}/d\eta =\hat{\mathbf{p}}$ givesA spatial Fourier transform sends $\hat{\mathbf{p}}\cdot \nabla$ to $i\hat{\mathbf{p}}\cdot \mathbf{k}$, soThis is the Free-streaming photon Boltzmann equation: the second term transports angular structure, while the last term is the gravitational redshift source.

Write $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{p}}$ and use the Legendre polynomial recurrence relation. The definition in the question is inverted byFor Newtonian-gauge potentials the scanned geodesic equation is the standard identityIts monopole and dipole are $S_0={\Phi }^{\prime }$ and $S_1=k\Psi /3$. Projecting the photon Boltzmann hierarchy onto $P_0$ and $P_1$ therefore givesThe first is the photon continuity equation; the second is the photon Euler equation, with the photon quadrupole providing the anisotropic-stress term. Indeed, with photon density contrast ${\delta }_{\gamma }=4{\Theta }_0$ and velocity divergence ${\theta }_{\gamma }=3k{\Theta }_1$,

The line-of-sight solution for free-streaming photons follows the unperturbed ray $\mathbf{x}(\eta )={\mathbf{x}}_0-\hat{\mathbf{p}}({\eta }_f-\eta )$. If $\Phi =\Psi$ is spatially homogeneous, its gradient vanishes and $d\log ϵ/d\eta ={\Phi }^{\prime }$. Integration along the ray givesThe needed initial condition is the temperature pattern on the initial hypersurface; for a superhorizon adiabatic mode during matter domination one may use ${\Theta }_0({\eta }_i)=-2\Psi ({\eta }_i)/3$.

This calculation contains free streaming of the initial pattern and the gravitational temperature shift caused by the evolving potential, the homogeneous limit of the Integrated Sachs-Wolfe effect. A perfectly homogeneous potential changes only the unobservable sky monopole and therefore contributes no CMB angular power spectrum for $\ell \ge 1$. Direction-dependent Sachs-Wolfe effect, Doppler, acoustic, polarization, lensing, and rescattering contributions require spatially varying perturbations or collision terms and are outside this special calculation.

Linearizing the cosmological Euler equation givesTaking the curl removes the gradient force, so the vorticity $\mathbf{w}=\nabla \times \mathbf{v}$ obeysSince $\mathcal{H}=a^{\prime }/a$, its solution isThus expansion dilutes linear vorticity; in an Einstein-de Sitter universe, $a\propto {\tau }^2$ and $\mathbf{w}\propto {\tau }^{-2}$.

The linearized cosmological continuity equation, the divergence of the cosmological Euler equation, and the cosmological Poisson equation giveEliminating $\theta =-{\delta }^{\prime }$ yields the equation for a linear cosmological density perturbation:In an Einstein-de Sitter universe, ${\Omega }_m=1$ and $\mathcal{H}=2/\tau$, soSubstitution of the power-law ansatz $\delta \propto {\tau }^p$ gives $(p-2)(p+3)=0$. HenceThe leading one of the matter-era growing and decaying density modes is $\delta \propto a\propto {\tau }^2$, as direct substitution confirms, and its velocity divergence is

In standard perturbation theory in cosmology, ${\delta }^{(n)}$ is an $n$-leg vertex carrying the symmetrized standard perturbation theory density kernel $F_n$. For Gaussian random fields, a connected four-point diagram with $L$ loops contains $3+L$ linear power spectra, hence $6+2L$ linear fields.

At tree level the required perturbative-order partitions areThe first is a star with one $F_3$ vertex and three linear external legs. The second has two $F_2$ vertices joined by one internal power-spectrum line, with one linear external leg attached to each vertex. At one loop all five partitions of eight into four positive parts are required:These are conventionally denoted $T_{5111}$, $T_{4211}$, $T_{3311}$, $T_{3221}$, and $T_{2222}$; all inequivalent placements of external legs and all Wick contractions are understood. This list is the complete set of one-loop matter trispectrum topologies.

Let $P_a=P_L(k_a)$ and ${\mathbf{k}}_{ab}={\mathbf{k}}_a+{\mathbf{k}}_b$. With ${\sum }_a{\mathbf{k}}_a=0$, the star contribution iswhere $\{a,b,c\}=\{1,2,3,4\}\setminus \{d\}$. For the exchange contribution, sum over the six unordered choices $\{a,b\}$ of linear external legs and let $\{c,d\}$ be the complementary pair:Thus the connected tree trispectrum is $T_{\mathrm{tree}}=T_{3111}+T_{2211}$.

The pressureless single-stream equations cease to be a complete one-loop prediction because loop momenta probe short nonlinear scales and produce ultraviolet-sensitive terms. A consistent result must include the effective field theory of large-scale structure: effective-stress counterterms, including their insertions into the tree topologies, and stochastic terms at the order allowed by mass conservation and momentum conservation. Their coefficients absorb short-scale dependence and must be fitted or matched. If the observable is a biased tracer rather than matter itself, the corresponding renormalized bias expansion is also required.