Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 (Field Theory in Cosmology)

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.

Use the spatial transverse-longitudinal decompositionUnder the U(1) gauge symmetry $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }ϵ$,The assumption ${\nabla }^2ϵ\ne 0$ makes the longitudinal split unambiguous. The scalaris therefore gauge-invariant.

The electric components of the electromagnetic field tensor arewhile $F_{ij}$ depends only on the transverse vector. The scalar-vector cross term integrates to zero because ${\partial }_i{A}_i^V=0$, so the scalar action isVarying the nondynamical scalar gives the constraint equation in field theoryFor every nonzero Fourier momentum this fixes ${\Phi }_A=0$, confirming that a source-free massless vector has no propagating scalar polarization.

Applying $ϵ=b_i(t)x^i$ to $A_{\mu }=0$ givesIts field strength vanishes, so it is a large gauge transformation of the background. To arise as the zero-momentum limit of a physical transverse perturbation, $b_i(t)$ must obey the zero-momentum vector equation of motion. In the variables used in the question this iswith a constant growing solution and a decaying solution proportional to ${\int }^{t}d{t}^{\prime }/a^3(t^{\prime })$. This condition makes the large gauge mode an adiabatic mode that can be continued to small nonzero momentum.

For constant $b_i$, the transformation is $\delta A_i^V=b_i$. Its Noether charge is thereforeup to the Fourier-sign convention. Expand a transverse polarization asOnly the creation term survives on the vacuum, while $A_i^V(\mathbf{q})|0⟩=f_q^{\ast }{a}_i^{†}(-\mathbf{q})|0⟩$. Hence

Hermitian conjugation of part d gives the corresponding charge insertion on the bra. The Ward identity $i⟨[Q,\mathcal{O}]⟩=⟨{\delta }_b\mathcal{O}⟩$ therefore becomesFor a neutral operator, ${\delta }_b\mathcal{O}=0$, so the two soft limits, multiplied by their respective wavefunctional coefficients, are equal. If $\mathcal{O}$ is charged, the right side is nonzero. For a product of fields of charges $e_a$ at positions ${\mathbf{x}}_a$, it is proportional to ${\sum }_a{e}_a{b}_i{x}_a^i⟨\mathcal{O}⟩$; in momentum space this becomes the corresponding momentum derivative. This is a soft-vector Ward-Takahashi identity.

For nonzero momentum, the Fourier transform of the local bias expansion isThe subtracted variance contributes only at $\mathbf{k}=0$. Because $\delta$ is a Gaussian random field, its three-point function vanishes and its four-point function factorizes by Wick theorem. At leading order one of the three galaxy fields supplies the quadratic term and the other two supply linear terms. The two cross-contractions cancel the factor $1/2$, givingThus even Gaussian matter fluctuations acquire a nonzero galaxy bispectrum through local quadratic galaxy bias.

Along the unperturbed photon path, combine the temperature and gravitational-redshift terms and use ${\mathcal{T}}^{\prime }=-\Gamma$:The integrating factor is $e^{-\mathcal{T}}$, and $g=\Gamma e^{-\mathcal{T}}$. Neglecting the exponentially hidden initial boundary term gives the Cosmic microwave background line-of-sight solutionFor instantaneous recombination, $g(\tau )=\delta (\tau -{\tau }_{\mathrm{dec}})$, and the observer potential contributes only an unobservable monopole. ThereforeThe first two terms form the ordinary Sachs-Wolfe effect, the velocity term is the Doppler CMB anisotropy at last scattering, and the integral is the Integrated Sachs-Wolfe effect produced by evolving potentials.

Insert the photon temperature multipole expansion into the Fourier-space transport equation. The Legendre polynomial recurrence relation turns multiplication by $\mu$ into nearest-neighbour multipole couplings, while Orthogonality of Legendre polynomials projects onto a fixed $\ell$. The monopole projection has no collision term because Thomson scattering conserves photon number:The dipole projection receives the electron-velocity source,For every $\ell \ge 2$, the collision term damps the anisotropic multipole and the photon Boltzmann hierarchy isThus the Free-streaming photon Boltzmann equation moves angular structure between neighbouring multipoles, whereas Thomson scattering suppresses all multipoles above the dipole in the tight-coupling approximation.

The displayed kernel is the growing-mode result of standard perturbation theory in cosmology. Its assumptions are a Newtonian, weak-field, subhorizon treatment of pressureless cold matter; a single-stream, irrotational velocity field; and negligible velocity-dispersion tensor of collisionless matter, so ${\sigma }_{ij}=0$. The background is the Einstein-de Sitter universe, which makes the growing mode proportional to $a$ and permits the separated powers $a^n{F}_n$. The density contrast is assumed perturbatively small and the decaying solutions are discarded. Gaussian initial conditions are not needed to derive $F_2$, but they are needed for the loop contractions in the later parts.

For symmetrized standard perturbation theory density kernels, the two one-loop contractions areandThe factors $2$ and $3$ count the relevant Wick contractions. Since the cross-correlation occurs in both orders, the one-loop matter power spectrum is

Put $\mathbf{p}=\mathbf{k}-\mathbf{q}$, $p^2=k^2-2kq\mu +q^2$, and $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{q}}$. Direct substitution into $F_2$ givesUsing $d^{3}q=2\pi q^{2}dqd\mu$ in part ii therefore yieldsFor $q\ll k$, $F_2(\mathbf{q},\mathbf{k}-\mathbf{q})\sim k\mu /(2q)$. Including the equal soft region $|\mathbf{k}-\mathbf{q}|\to 0$ and using ${\int }_{-1}^1{\mu }^{2}d\mu =2/3$ givesFor $q\gg k$, the constant and linear hard-momentum terms cancel, displaying the ultraviolet softness of the second-order density kernel. Since $p\sim q$,The angular integral ${\int }_{-1}^1(3-10{\mu }^2{)}^{2}d\mu =18$ then gives

The leading infrared terms cancel in the observable sum:This infrared cancellation in large-scale structure follows from the Equivalence principle: a sufficiently long-wavelength displacement translates short-scale structure without changing an equal-time power spectrum.

The ultraviolet part of $2P_{13}$ is proportional to $k^{2}P(k)$ and depends on the cutoff $\Lambda$. The leading deterministic effective field theory of large-scale structure counterterm has exactly this shape,Its cutoff-dependent part can be chosen aswhich cancels the stated $2P_{13,\mathrm{UV}}$. The $P_{22}$ ultraviolet contribution begins at $k^4$ and, when cutoff sensitive, is absorbed by higher-derivative or stochastic counterterms.

Physically, coarse graining the matter equations does not justify setting the short-scale stress ${\sigma }_{ij}$ to zero. Unresolved multistreaming and nonlinear motion generate an effective pressure and viscosity. Their leading derivative contribution to the Euler equation is proportional to $-c_{\mathrm{eff}}^2{\nabla }_i\delta$, producing the required $k^2\delta$ correction and making long-distance predictions independent of the arbitrary cutoff.