Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 2

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.