We develop the path integral formalism for studying cosmological perturbations in multi-field inflation, which is particularly well suited to study quantum theories with gauge symmetries such as diffeomorphism invariance. We formulate the gauge fixing conditions based on the Poisson brackets of the constraints, from which we derive two convenient gauges that are appropriate for multi-field inflation. We then adopt the in-in formalism to derive the most general expression for the power spectrum of the curvature perturbation including the corrections from the interactions of the curvature mode with other light degrees of freedom. We also discuss the contributions of the interactions to the bispectrum.
Very high energy (VHE; energy $E \gtrsim 100\,$GeV) $\gamma$ rays originating from extragalactic sources undergo pair production with low-energy photons of background radiation fields. These pairs can inverse-Compton scatter background photons, initiating an electromagnetic cascade. The spatial and temporal structure of this secondary $\gamma$-ray signal is altered as the $e^+e^-$ pairs are deflected in an intergalactic magnetic field (IGMF). We investigate how VHE observations with the future Cherenkov Telescope Array (CTA) with its high angular resolution and broad energy range can potentially probe the IGMF. We identify promising sources and simulate $\gamma$-ray spectra over a wide range of values of the IGMF strength and coherence length using the publicly available ELMAG Monte-Carlo code. Combining simulated observations in a joint likelihood approach, we find that current limits on the IGMF can be significantly improved. The projected sensitivity depends strongly on the time a source has been $\gamma$-ray active and on the emitted maximum $\gamma$-ray energy.