We prove the existence of a general class of rapidly turning two-field inflationary attractors. By only requiring a large, slowly varying turn rate, we solve the system completely without specifying any metric or potential, and prove the linear stability of the solution. Several recently studied turning inflation models, including hyperinflation, side-tracked inflation, and a flat field-space model, turn out to be examples of this general class of attractor solutions. Very rapidly turning models are of particular interest since they can be compatible with the swampland conjectures, and we show that the solutions further simplify in this limit.

In negatively curved field spaces, inflation can be realised even in steep potentials. Hyperinflation invokes the 'centrifugal force' of a field orbiting the hyperbolic plane to sustain inflation. We generalise hyperinflation by showing that it can be realised in models with any number of fields (N f ≥ 2), and in broad classes of potentials that, in particular, don't need to be rotationally symmetric. For example, hyperinflation can follow a period of radial slow-roll inflation that undergoes geometric destabilisation, yet this inflationary phase is not identical to the recently proposed scenario of 'side-tracked inflation'. We furthermore provide a detailed proof of the attractor mechanism of (the original and generalised) hyperinflation, and provide a novel set of characteristic, explicit models. We close by discussing the compatibility of hyperinflation with observations and the recently much discussed 'swampland conjectures'. Observationally viable models can be realised that satisfy either the 'de Sitter conjecture' (V /V 1) or the 'distance conjecture' (∆φ 1), but satisfying both simultaneously brings hyperinflation in some tension with successful reheating after inflation. However, hyperinflation can get much closer to satisfying all of these criteria than standard slow-roll inflation. Furthermore, while the original model is in stark tension with the weak gravity conjecture, generalisations can circumvent this issue. ArXiv ePrint: 1901.08603 arXiv:1901.08603v2 [hep-th] 8 Apr 2019 7 Conclusions 20 A Perturbation theory 21 A.1 Two-field case 22 A.2 Multifield extension 23 1 I.e. L/ √ −g = − 1 2 ∂µφ∂ µ φ − V (φ). 2 This mechanism generalises the idea of 'spinflation' [34].

Inflation can be supported in very steep potentials if it is generated by rapidly turning fields, which can be natural in negatively curved field spaces. The curvature perturbation, ζ, of these models undergoes an exponential, transient amplification around the time of horizon crossing, but can still be compatible with observations at the level of the power spectrum. However, a recent analysis (based on a proposed single-field effective theory with an imaginary speed of sound) found that the trispectrum and other higher-order, non-Gaussian correlators also undergo similar exponential enhancements. This arguably leads to 'hyper-large' non-Gaussianities in stark conflict with observations, and even to the loss of perturbative control of the calculations. In this paper, we provide the first analytic solution of the growth of the perturbations in two-field rapid-turn models, and find it in good agreement with previous numerical and single-field EFT estimates. We also show that the nested structure of commutators of the in-in formalism has subtle and crucial consequences: accounting for these commutators, we show analytically that the naively leading-order piece (which indeed is exponentially large) cancels exactly in all relevant correlators. The remaining non-Gaussianities of these models are modest, and there is no problem with perturbative control from the exponential enhancement of ζ. Thus, rapidturn inflation with negatively curved field spaces remains a viable and interesting class of candidate theories of the early universe.

We construct models of inflation with many randomly interacting fields and use these to study the generation of cosmological observables. We model the potentials as multi-dimensional Gaussian random fields (GRFs) and identify powerful algebraic simplifications that, for the first time, make it possible to access the manyfield limit of inflation in GRF potentials. Focussing on small-field, slow-roll, approximate saddle-point inflation in potentials with structure on sub-Planckian scales, we construct explicit examples involving up to 100 fields and generate statistical ensembles comprising of 164,000 models involving 5 to 50 fields. For the subset of these that support at least sixty e-folds of inflation, we use the 'transport method' and δ N formalism to determine the predictions for cosmological observables at the end of inflation, including the power spectrum and the local non-Gaussianities of the primordial perturbations. We find three key results: i) Planck compatibility is not rare, but future experiments may rule out this class of models; ii) In the manyfield limit, the predictions from these models agree well with, but are sharper than, previous results derived using potentials constructed through non-equilibrium Random Matrix Theory; iii) Despite substantial multifield effects, non-Gaussianities are typically very small: fNLloc ≪ 1. We conclude that many of the 'generic predictions' of single-field inflation can be emergent features of complex inflation models.

Many applications of Gaussian random fields and Gaussian random processes are limited by the computational complexity of evaluating the probability density function, which involves inverting the relevant covariance matrix. In this work, we show how that problem can be completely circumvented for the local Taylor coefficients of a Gaussian random field with a Gaussian (or 'square exponential') covariance function. Our results hold for any dimension of the field and to any order in the Taylor expansion. We present two applications. First, we show that this method can be used to explicitly generate non-trivial potential energy landscapes with many fields. This application is particularly useful when one is concerned with the field locally around special points (e.g. maxima or minima), as we exemplify by the problem of cosmic 'manyfield' inflation in the early universe. Second, we show that this method has applications in machine learning, and greatly simplifies the regression problem of determining the hyperparameters of the covariance function given a training data set consisting of local Taylor coefficients at single point. An accompanying Mathematica notebook is available at https://doi.