Symmetries of Vector Perturbations during the de Sitter Epoch

TL;DR

This work demonstrates that the de Sitter isometry acts as the 3D conformal group on super-Hubble scales and uses this symmetry to constrain correlators involving the inflaton and a vector field coupled by $I^2(\phi)F_{\mu\nu}F^{\mu\nu}$. By selecting $I(\tau)=(H\tau)^{-n}$, the authors identify magnetic ($n=2$, $\Delta_V=-n+1$) and electric ($n=-2$, $\Delta_U=n+2$) regimes, deriving explicit, conformally fixed forms for two- and three-point functions such as $\langle \delta\phi A_iA_j \rangle'$. The magnetic and electric cases yield distinct but symmetry-determined templates, e.g. $\langle \delta\phi A_iA_j \rangle' = \beta_M/(k_2^5k_3^5)(\mathbf{k}_2\cdot\mathbf{k}_3\delta_{ij}-k_{2j}k_{3i})$ and $\langle \delta\phi A_iA_j \rangle' = \alpha_E/(k_2k_3)^{3}\Delta_{ij}$ for the electric case (with $n=-2$). Overall, the paper shows that conformal invariance not only fixes the shapes of these correlators but also provides a framework to interpret possible CMB anomalies and cross-correlations with large-scale magnetic fields, with amplitudes potentially carrying a logarithmic dependence on the mode’s horizon-exit history $N(k_t)$.

Abstract

We analyze the class of models where a suitable coupling between the inflaton field and the vector field gives rise to scale-invariant vector perturbations. We exploit the fact that the de Sitter isometry group acts as conformal group on the three-dimensional Euclidean space for the super-Hubble fluctuations in order to characterize the correlators involving the inflaton and the vector fields.