Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 53 / 3

For a statistically homogeneous comoving curvature perturbation, define the connected primordial bispectrum byStatistical isotropy makes $B$ depend only on the three magnitudes, which must form a triangle. A Gaussian random field has zero connected bispectrum. Canonical attractor single-field slow-roll inflation with the usual vacuum produces only slow-roll-sized primordial non-Gaussianity. A measurable signal can arise from additional light fields and nonlinear conversion of isocurvature perturbations, or from enhanced interactions such as a small sound speed, departures from an attractor, or suitable features/excited initial states. The resulting model must still reproduce the observed nearly scale-invariant power spectrum and remain under perturbative control, with acceptable backreaction and late-time conversion to the observed perturbations. Large non-Gaussianity is therefore possible but is not automatic in a viable model.

Put $c=3f_{\mathrm{NL}}/5$ and use the consistent Fourier transform convention with ${\zeta }_G(\mathbf{x})$, rather than the repeated momentum argument printed in the integrand. The quadratic part has transformThe subtraction sets the mean to zero and removes the internal contraction of a single quadratic factor. At first order in $f_{\mathrm{NL}}$, choose one of the three external factors to be quadratic. For example, the two connected Wick contractions at the third leg pair its two fields with the first two legs, givingSumming the three placements provesThis is the leading local primordial bispectrum. The factor $6/5$ is the quadratic coefficient $3/5$ multiplied by the two cross-pairings. Terms of order $f_{\mathrm{NL}}^2$ have an odd Gaussian moment and vanish.

The literal quadratic local model also has a connected cubic-in-$f_{\mathrm{NL}}$ contribution, from one quadratic factor at every leg:This is the loop correction to the local primordial bispectrum; regulators may be needed for idealized spectra. Thus the PDF's displayed formula is the tree/leading-order result, not an exact identity for arbitrary $f_{\mathrm{NL}}$. The weak-non-Gaussian expansion assumes these loop terms are small. The local shape is enhanced in the squeezed bispectrum configuration when a long-wavelength perturbation modulates small-scale power.

Insert the primordial bispectrum into the product of the three linear transfer integrals. Write $L={\ell }_1+{\ell }_2+{\ell }_3$ and abbreviate $g_{{\ell }_i}({\eta }_0,k_i)$ by $g_i$. The observer-position phase is one because the momentum delta function imposes $\sum {\mathbf{k}}_i=0$. Represent that delta function byThe Rayleigh plane-wave expansion and angular orthogonality give, for each momentum,The three $i^{\ell }$ factors cancel the $(-i{)}^L$ in the temperature multipoles. Angular integration over $\hat{x}$ leaves the complex conjugate Gaunt integral. In the conventional complex spherical harmonics, this integral is real, and it vanishes unless the angular momentum triangle, even-parity and $m_1+m_2+m_3=0$ selection rules hold. Therefore its conjugate equals itself.

The radial measure is $x^{2}dx$, and the momentum radial measures are $k_i^{2}d{k}_i$. The combined numerical prefactor is $(4\pi {)}^6/(2\pi {)}^9=8/{\pi }^3=64/(2\pi {)}^3$. Hence the reduced CMB bispectrum isand the angular three-point function factorizes asThis primordial-to-angular bispectrum projection separates dynamics and radial transfer from purely angular geometry. The spatial integration variable is auxiliary, not the observer position. Linear transfer is justified at leading order in the primordial signal; it does not require a large amplitude mathematically, although a signal must exceed measurement uncertainty to be detectable.

Let $b_{{\ell }_1{\ell }_2{\ell }_3}^1$ denote the template at unit $f_{\mathrm{NL}}$, and define $A_{123}=b_{{\ell }_1{\ell }_2{\ell }_3}^1{\mathcal{G}}_{m_1{m}_2{m}_3}^{{\ell }_1{\ell }_2{\ell }_3}$. All sums below are over ordered triples of the retained multipoles, with the monopole removed and $C_{\ell }>0$. A finite maximum multipole makes these expressions ordinary finite sums. Taking the expectation of the cubic numerator and using the template relation givesThus the full-sky cubic bispectrum estimator is unbiased forThe assumed template must have nonzero support, so $F>0$. Using the Gaunt sum rule, writeThe factor $1/6$ belongs to the ordered sum; an unordered triangle sum would instead use multiplicity factors for equal multipoles. This normalization is also the Gaussian Fisher information for the template amplitude.

For the cosmic variance, use the reality condition ${\Theta }_{\ell m}^{\ast }=(-1{)}^m{\Theta }_{\ell ,-m}$ and Gaussian covarianceThere are $15$ pairings of the six multipoles in the squared cubic numerator. Six pairings connect every factor in the first triple to a factor in the second. Each gives $\sum A_{123}^2/(C_{{\ell }_1}{C}_{{\ell }_2}{C}_{{\ell }_3})=6F$. The covariance phases cancel: nonzero Gaunt integrals have $m_1+m_2+m_3=0$, and simultaneous reversal of the three $m$ indices multiplies the Gaunt integral by $(-1{)}^{{\ell }_1+{\ell }_2+{\ell }_3}=1$.

The other nine pairings contain one contraction within each triple. To see why their sums vanish, contract two legs of a Gaunt integral and use the spherical harmonic addition theorem:The template and inverse-covariance weights are independent of $m$, so they do not spoil this cancellation. The only possible surviving unpaired mode is the monopole, and ${\Theta }_{00}=0$ removes it. This is monopole cancellation of internal cubic-estimator contractions. It explains why no linear correction is needed in this ideal full-sky isotropic problem; masks or anisotropic noise would spoil the argument.

The cubic numerator therefore has Gaussian variance $6(6F)=36F$. Dividing by $(6F{)}^2$ givesThe result concerns the Gaussian-limit covariance. Non-Gaussian connected four- and six-point terms can change the variance at finite amplitude. Unbiasedness uses the assumed linear template relation for the observed three-point function.