Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 (Field Theory in Cosmology)

The operator ${\dot{\varphi }}^2\varphi$ is not invariant under a scalar-field shift symmetry because one field is undifferentiated. An arbitrary large coefficient would therefore radiatively generate equally unsuppressed nonderivative operators, including a mass and a steep scalar potential, and would spoil the approximate shift symmetry and flat potential needed for single-field slow-roll inflation. In a technically natural slow-roll model its coefficient must consequently be small, so it does not produce parametrically large primordial non-Gaussianity. Treated merely as an effective spectator-field interaction, it does generate the tree-level primordial bispectrum computed below, whose size is controlled by the dimensionless interaction strength at the Hubble scale; taking that strength large abandons the controlled slow-roll premise.

Since $dt=ad\tau$ and $\dot{\varphi }=a^{-1}{\varphi }^{\prime }$, the interaction isto first order in $\lambda$. The tree-level in-in formalism givesThere are two Wick contractions for each choice of the undifferentiated field at the vertex. With $K=k_1+k_2+k_3$, the stated Bunch-Davies vacuum mode obeysThe two powers of $\tau$ from the differentiated modes cancel $a^2=1/(H^2{\tau }^2)$, and the remaining integral isRemoving the momentum-conserving delta function, the bispectrum from a time-derivative cubic scalar interaction is therefore

With the Fourier transform convention $\varphi (\mathbf{k})=\int d^{3}x{e}^{-i\mathbf{k}\cdot \mathbf{x}}\varphi (\mathbf{x})$, scale invariance of the position-space three-point function givesBecause ${\delta }^{(3)}(\mu {\sum }_i{\mathbf{k}}_i)={\mu }^{-3}{\delta }^{(3)}({\sum }_i{\mathbf{k}}_i)$, the reduced primordial bispectrum must satisfyIn the result of part b, the bracket has momentum degree five, while $k_1^3{k}_2^3{k}_3^3{K}^2$ has degree eleven. Its total degree is therefore $-6$, exactly as required.

The constant scalar-field shift symmetry has momentum-space actionFor the canonical commutation relation $[\varphi (\mathbf{x}),\Pi (\mathbf{y})]=i{\delta }^{(3)}(\mathbf{x}-\mathbf{y})$, its Noether charge can be writtenIndeed, $i[Q,\varphi (\mathbf{x})]=c$.

Regulate the zero mode by a soft momentum $\mathbf{q}$ and write the free modes asThe Wronskian normalization $f_q{g}_q^{\ast }-f_q^{\ast }{g}_q=i$ follows from the canonical commutator, while the power spectrum is $P(q)=|f_q{|}^2$. Inserting the creation part of $Q$ on the ket and the annihilation part on the bra, their difference is precisely the Wronskian. For $\mathcal{O}=\varphi (\mathbf{k})\varphi ({\mathbf{k}}^{\prime })$ this gives, after taking $q\to 0$ and setting the irrelevant normalization $c=1$,Equivalently, before removing the regulator, the right-hand side is the soft three-point function divided by $P(q)$.

The variation of the product isIts expectation value vanishes because the background has $⟨\varphi ⟩=0$. The Ward identity and part b therefore give the shift-symmetry soft theorem for a scalar bispectrumThus an exact internal shift creates no leading soft response of the two hard modes.

The single-field inflation squeezed-limit consistency relation for the comoving curvature perturbation isEquivalently, $B_{\mathcal{R}}/P_{\mathcal{R}}(q)=-(3+k{\partial }_k)P_{\mathcal{R}}(k)$. The scalar shift in parts a--c is an internal symmetry and leaves the hard fields unchanged, so its right-hand side vanishes. A long adiabatic curvature mode instead acts as a spatial dilation of the hard coordinates, and its response is the scale dependence measured by the scalar spectral index.

The cosmological continuity equation expresses conservation of dark-matter mass: $\delta$ is the density contrast, $\mathbf{v}$ is the peculiar velocity, and a prime denotes a conformal time derivative. The cosmological Euler equation expresses momentum conservation: $\mathcal{H}=a^{\prime }/a$ is the conformal Hubble rate, $\varphi$ is the peculiar gravitational potential, $\rho$ is the density, and ${\sigma }_{ij}$ is the velocity-dispersion tensor of collisionless matter. The terms are respectively Hubble drag, convective acceleration, gravity, and velocity-dispersion stress.

For curl-free flow, introduce the peculiar-velocity divergence $\theta =\nabla \cdot \mathbf{v}$ and set ${\sigma }_{ij}=0$. A Fourier transform of the nonlinear continuity term giveswith the alpha mode-coupling kernelTaking the divergence of the Euler equation gives the quadratic velocity kernelThis is the beta mode-coupling kernel; its symmetry follows from the two velocity factors.

Expand $\delta ={\delta }^{(1)}+{\delta }^{(2)}+{\delta }^{(3)}+\cdots$, where the standard perturbation theory density kernel $F_n$ convolves $n$ linear fields. Through fourth order in the Gaussian linear field, the only power-spectrum diagrams are the tree contraction $P_{11}$, the loop joining two $F_2$ vertices $P_{22}$, and the two orderings that join an $F_3$ vertex to a linear leg, $P_{13}+P_{31}=2P_{13}$. Their expressions are

The effective field theory of large-scale structure supplements the one-loop prediction by the leading deterministic counterterm and a stochastic term,where the normalization scale may be absorbed into $c_{\mathrm{eff}}^2$ and mass and momentum conservation make $P_{\mathrm{stoch}}=O(k^4)$ at small $k$. The ultraviolet part of the P13 contribution to the one-loop matter power spectrum hasIts cutoff dependence has exactly the $k^2{P}_L(k)$ form of the effective sound-speed counterterm in large-scale structure, so the running of $c_{\mathrm{eff}}^2(\Lambda )$ cancels it. The stochastic and higher-derivative counterterms similarly absorb the allowed analytic ultraviolet dependence of $P_{22}$ and higher orders.

Conservation of tracer number under the map from the Lagrangian coordinate $\mathbf{q}$ to the Eulerian position $\mathbf{x}$ givesMatter conservation gives $[1+{\delta }^{(E)}(\mathbf{x})]d^{3}x=d^{3}q$, and henceAt linear order, ${\delta }_g^{(L)}=b_1^{(L)}\delta$ and thereforeThe Lagrangian-to-Eulerian linear bias relation is

In the absence of scalar anisotropic stress, the traceless spatial Einstein field equations giveThe photon mass-shell condition $g_{\mu \nu }{P}^{\mu }{P}^{\nu }=0$, together with $p^2=g_{ij}{P}^i{P}^j$ and $ϵ=ap$, gives to first orderUse the time component of the geodesic equation, divide it by $d\eta /d\lambda =P^0$, and substitute the stated Christoffel symbols. Keeping the perturbed ratios $P^i/P^0=(1+\Phi +\Psi ){\hat{p}}^i$ where they multiply the background $\mathcal{H}$ terms makes those terms cancel. Comparing the result with the total derivative of $P^0=ϵa^{-2}(1-\Psi )$ yieldsThe first term is the time-dependent gravitational redshift and the second is the gravitational frequency shift along the photon direction.

Write the total derivative of the phase-space distribution function asSince $a\bar{T}$ is constant for the background photon gas, expand the Bose-Einstein distribution asAt first order the four terms are respectivelyThe angular-deflection velocity is already first order and multiplies the first-order angular dependence of $f$, so its contribution is second order. Thus the collisionless left-hand side of the Free-streaming photon Boltzmann equation is

The conventional vertical variable is $D_{\ell }=\ell (\ell +1)C_{\ell }/(2\pi )$ in $\mu {\mathrm{K}}^2$, plotted against the dimensionless angular multipole $\ell$. The Sachs-Wolfe effect supplies a nearly flat large-angle plateau for $\ell \lesssim 30$. The total spectrum has its first acoustic peak near $\ell \simeq 220$ with $D_{\ell }$ of order $5\times 10^3\mu {\mathrm{K}}^2$, followed by further Cosmic microwave background acoustic peaks; the Doppler CMB anisotropy is phase shifted relative to the photon-density oscillation. At $\ell \gtrsim 1000$, Cosmic microwave background diffusion damping lets photons random-walk across perturbations during recombination and produces the rapidly falling damping tail.