Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 55 / 2

In the Spatially flat FLRW metric, neglect metric perturbations and write $\varphi =\bar{\varphi }+\delta \varphi$. The kinetic invariant is then exactly quadratic in field derivatives:Subtracting $\bar{X}={\dot{\bar{\varphi }}}^2/2$ gives the second-order expansionIn the derivative-dominated approximation, $\delta \varphi =-\dot{\bar{\varphi }}\zeta /H$ with $\dot{\bar{\varphi }}/H$ treated as slowly varying. Thus $\delta \dot{\varphi }\simeq -\dot{\bar{\varphi }}\dot{\zeta }/H$ andThis uses the linear flat-to-comoving slicing relation. Differentiating its slowly varying coefficient would add terms involving undifferentiated $\zeta$; it is not an exact nonlinear gauge transformation, and those neglected terms do not alter the leading pure ${\dot{\zeta }}^3$ coefficient computed here.

Only $P_{,XX}\delta X_1\delta X_2$ and $P_{,XXX}\delta X_1^3/6$ contribute to the requested three-time-derivative term. Therefore the directly evaluated cubic coefficient isThis is the derivative-cubic coefficient of a noncanonical scalar. For a physical $P(X,\varphi )$ scalar field theory, differentiating $\rho =2X{P}_{,X}-P$ at fixed $\varphi$ gives ${\rho }_{,X}=P_{,X}+2X{P}_{,XX}$. Its sound speed is consequentlyThe minus sign printed in the denominator of $c_s^2$ reverses the asserted coefficient identity; it must be a plus. The printed background equation also lacks a power: it is $3H^2=\rho$ in the stated reduced-Planck units. A stable propagating scalar has positive time-kinetic coefficient $P_{,X}+2X{P}_{,XX}$ and positive gradient coefficient $P_{,X}$, as summarized by noncanonical scalar kinetic stability.

The unfactored expression for $C$ stays meaningful if $P_{,XX}=0$, although $\mathcal{A}$ separately then need not exist. For a canonical kinetic term with no higher kinetic derivatives, $C=0$. The cubic curvature action for a P(X, phi) scalar field can include other vertices and gravitational terms; they are outside this requested derivative coefficient.

For a centred Gaussian random field, Wick theorem expresses each even correlation function as a sum over all pairings of two-point functions; odd moments vanish. For example,For six fields there are $5\cdot 3\cdot 1=15$ pairings. In a tree-level connected three-point calculation with one cubic vertex, each of the three external fields must contract with a different vertex field. This gives $3!=6$ connected Wick contractions. Pairings connecting two external fields instead are disconnected tadpole contributions, omitted from the primordial bispectrum.

Put $\mathcal{F}=(1-c_s^2)\mathcal{A}/c_s^2$. Using $dt=ad\tau$ and $\dot{\zeta }={\zeta }^{\prime }/a$, the integrated interaction Hamiltonian becomesThe unequal-time Wick contraction required by the displayed operator order isThe conjugate is essential: the equal-time power spectrum alone does not specify the erroneous unstarred replacement printed in the target expression. The in-in bispectrum conjugation rule fixes the vacuum convergence and final sign.

Fourier transforming the vertex gives $(2\pi {)}^3{\delta }^{(3)}({\mathbf{p}}_1+{\mathbf{p}}_2+{\mathbf{p}}_3)$ with three measures $d^3{p}_j/(2\pi {)}^3$. The six pairings supply the product of three external-vertex delta functions, each carrying $(2\pi {)}^3$. Integrating all three internal momenta leaves one overall $(2\pi {)}^3{\delta }^{(3)}({\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3)$. Thus the normalized connected contraction formula isThis spells out what the printed permutations and delta functions must represent: all six permutations, full Fourier normalization, and conjugated vertex modes. Three cyclic permutations alone would miss a factor of two.

Use exactly the stipulated unit-sound-speed mode functions for this integral. Their late-time values are real and $u_k^{\prime }{}^{\ast }(\tau )=H{k}^2\tau e^{ik\tau }/\sqrt{4ϵk^3}$. ConsequentlyThe vacuum contour lies above the negative real time axis, so $e^{iK\tau }$ decays at its past endpoint. With $s=iK+0^{+}$, repeated differentiation of ${\int }_{-\infty }^0{e}^{s\tau }d\tau =1/s$ givesSubstitution gives the bispectrum with consistent operator ordering and the stated Hamiltonian:The sign is positive in front of $\mathcal{A}$, contrary to the target's negative sign. The printed negative result is obtained algebraically if one uses the unstarred derivatives, $e^{-iK\tau }$, and the conjugate past contour, which gives $-2i/K^3$, while retaining $-2i$ outside. That combination is not the stated vacuum contour or the stated external-before-vertex contraction. A consistently conjugated representation must also conjugate the external prefactor and has the positive result above. The unit-speed cubic time-derivative bispectrum records these conventions explicitly.

The approximation treats $H$, $ϵ$ and the interaction coefficients as constant, uses the de Sitter approximation, retains only the stipulated vertex at tree level, and takes $k_i|{\tau }_{\mathrm{final}}|\to 0$ after doing the convergent early-time integral. The $i0$ prescription must be kept while imposing the past boundary condition; an arbitrary early-time cutoff leaves spurious oscillatory boundary terms. For a physical noncanonical quadratic action with $c_s\ne 1$, the mode functions also change to $H(1+i{c}_{s}k\tau )e^{-i{c}_{s}k\tau }/\sqrt{4ϵc_s{k}^3}$. They were not supplied here; the displayed calculation is conditional on the paper's specified modes, rather than a full self-consistent noncanonical power-spectrum calculation.

With the stipulated power spectrum $P_{\zeta }(k)=H^2/(4ϵk^3)$, an equilateral bispectrum configuration gives $B(k,k,k)/P(k{)}^2=2\mathcal{F}/9$. For the local-style normalization evaluated only at that triangle, $B(k,k,k)=(18/5)f_{\mathrm{NL}}^{\mathrm{effective}}P(k{)}^2$, this meansThis is a normalization convention for an equilateral amplitude, not a claim that the shape is local. In a squeezed bispectrum configuration, the ratio to the long-short power product is suppressed by $(k_L/k_S{)}^2$. Small sound speed or large higher-kinetic derivatives can make $|\mathcal{F}|$ large enough for potentially observable primordial non-Gaussianity, whereas $c_s\to 1$ with finite $\mathcal{A}$ suppresses this vertex. Detectability is conditional on amplitudes, the other cubic interactions, and perturbative control; it cannot be asserted from the given constants alone.