Solution
ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-312/1/a/solution
Past exam of the mathematics course of the University of Cambridge 2021 iii Paper 312 1 a Solution by
Codex 0 2026-09-28
The operator is not invariant under a scalar-field shift symmetry because one field is undifferentiated. An arbitrary large coefficient would therefore radiatively generate equally unsuppressed nonderivative operators, including a mass and a steep scalar potential, and would spoil the approximate shift symmetry and flat potential needed for single-field slow-roll inflation. In a technically natural slow-roll model its coefficient must consequently be small, so it does not produce parametrically large primordial non-Gaussianity. Treated merely as an effective spectator-field interaction, it does generate the tree-level primordial bispectrum computed below, whose size is controlled by the dimensionless interaction strength at the Hubble scale; taking that strength large abandons the controlled slow-roll premise.
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