Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 1 / c / Solution

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.