Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 57 / 3

Let $X={\partial }^{-2}\zeta$, with an inverse spatial Laplacian defined by a boundary or zero-mode prescription. It depends on $\zeta$, not on its time derivative. Differentiation gives the canonical momentumCount $\zeta ,\pi ,X$ as first order. Inverting perturbatively givesFor the Legendre transform in mechanics, write $v=\dot{\zeta }=\pi +\Delta$ with $\Delta$ second order. The quadratic kinetic contribution is $\pi v-v^2/2={\pi }^2/2-{\Delta }^2/2$, whose correction is fourth order; in the cubic terms one may consequently replace $v$ by $\pi$. The Hamiltonian density is thereforeThe cubic interaction Hamiltonian density isIn the free interaction picture, ${\pi }_I={\dot{\zeta }}_I$. Thus here ${\mathcal{H}}_{\mathrm{int}}=-{\mathcal{L}}_3$ with the time derivatives interpreted as free fields. This equality follows from the cubic Legendre transform for scalar derivative interactions; at higher orders additional terms can appear.

For a statistically homogeneous isotropic dimensionless field, write its two-point autocorrelation function of a random field as $C(r)$ andThe formal real-space scale invariance condition is $C(\lambda \mathbf{r})=C(\mathbf{r})$ for all $\lambda >0$. Changing variable $\mathbf{q}=\lambda \mathbf{k}$ in the first expression and equating the nonzero-mode Fourier transforms gives ${\lambda }^{-3}P(q/\lambda )=P(q)$, equivalently $P(\lambda k)={\lambda }^{-3}P(k)$. Taking $\lambda =k/k_0$ yieldsThus a scale-invariant inflationary power spectrum has equal variance per logarithmic wavenumber interval, rather than equal power per Fourier volume.

There is an important mathematical qualification to the literal covariance condition. A nonzero $k^{-3}$ power spectrum over all scales has $C(0)=A\int dk/k/(2{\pi }^2)$, divergent at both endpoints; it is not the covariance of a finite-variance ordinary field. Moreover, exact dilation invariance of a continuous isotropic $C(r)$ makes it constant for $r>0$ and, by continuity, at zero: its spectrum can then consist only of a zero-mode delta measure. The nontrivial cosmological statement consequently concerns nonzero modes, with cutoffs or subtraction of an unobservable constant. For example the finite subtracted covarianceis invariant under simultaneous rescaling of $r,r_0$. This is the precise infrared qualification of a scale-invariant covariance; a regulated unsubtracted $C(r)$ generally shifts by an additive constant under dilation. The formal derivation above gives the intended nonzero-mode scaling, with this qualification rather than an impossible finite-variance premise.

Put $k_1=q\ll k$ and $k_2,k_3\to k$ in the squeezed bispectrum configuration. Then $K\to 2k$ and the three symmetric combinations in the supplied expression have limitsThe first two terms inside the $ϵ$ bracket cancel, leaving $4k^3$. The full bracket is therefore $(\eta +2ϵ)k^3/4$, andThe $(2\pi {)}^3$ is kept in the normalization of $F$ used by the paper; it should not be introduced a second time by changing the primordial bispectrum convention.

Momentum conservation makes the three vectors close into a triangle. As $q/k\to 0$, the two short-wavelength vectors approach equal magnitude and opposite direction, producing a thin, nearly collapsed triangle. In real space the $q$ mode varies slowly over a region containing many short wavelengths, so it modulates their local statistics. For a single adiabatic inflationary clock this is a local spatial dilation, the basis of the single-field inflation squeezed-limit consistency relation. If $\eta$ is the Hubble-flow convention $\dot{ϵ}/(Hϵ)$, the scalar spectral index in Hubble slow-roll parameters gives $\eta +2ϵ=1-n_s$ to first order. This identification must not be made with the potential slow-roll $\eta$ without converting conventions.

Let $G={\Phi }_G$ and $f=f_{NL}^{\mathrm{local}}$. The Gaussian random field covariance in Fourier space isThe Fourier transform of the centered quadratic term isThe Gaussian three-point function vanishes. At first order in $f$, the only terms are $f⟨Q_1{G}_2{G}_3⟩$ and its two cyclic placements. For the first, Wick theorem pairs four Gaussian factors in three ways. Pairing the two factors inside $Q_1$ together is canceled by the mean subtraction. Pairing one of those factors with $G_2$ and the other with $G_3$ gives $P(k_2)P(k_3)$ and the overall momentum delta; exchanging the two factors gives the same contribution. ThusAdding the three placements givesThere is no second-order term because it would contain five centered Gaussian factors. The two-point function receives its first correction at $f^2$, whereas the displayed local-type primordial non-Gaussianity gives a three-point correlation already at $f$. For $P\propto k^{-3}$ the squeezed limit is $B_{\Phi }(q,k,k)\simeq 4fP(q)P(k)$; this explains the strong correlation between a long potential and short-scale power. A finite variance or regulator is needed to define the subtracted $⟨G^2⟩$, as noted in part (b).