Transport equations for the inflationary trispectrum

We use transport techniques to calculate the trispectrum produced in multiple-field inflationary models with canonical kinetic terms. Our method allows the time evolution of the local trispectrum parameters, tauNL and gNL, to be tracked throughout the inflationary phase. We illustrate our approach using examples. We give a simplified method to calculate the superhorizon part of the relation between field fluctuations on spatially flat hypersurfaces and the curvature perturbation on uniform density slices, and obtain its third-order part for the first time. We clarify how the 'backwards' formalism of Yokoyama et al. relates to our analysis and other recent work. We supply explicit formulae which enable each inflationary observable to be computed in any canonical model of interest, using a suitable first-order ODE solver.Comment: 24 pages, plus references and appendix. v2: matches version published in JCAP; typo fixed in Eq. (54

The δN formula is the dynamical renormalization group

We derive the 'separate universe' method for the inflationary bispectrum, beginning directly from a field-theory calculation. We work to tree-level in quantum effects but to all orders in the slow-roll expansion, with masses accommodated perturbatively. Our method provides a systematic basis to account for novel sources of time-dependence in inflationary correlation functions, and has immediate applications. First, we use our result to obtain the correct matching prescription between the 'quantum' and 'classical' parts of the separate universe computation. Second, we elaborate on the application of this method in situations where its validity is not clear. As a by-product of our calculation we give the leading slow-roll corrections to the three-point function of field fluctuations on spatially flat hypersurfaces in a canonical, multiple-field model.Comment: v1: 33 pages, plus appendix and references; 5 figures. v2: typographical typos fixed, minor changes to the main text and abstract, reference added; matches version published in JCA

The inflationary bispectrum with curved field-space

We compute the covariant three-point function near horizon-crossing for a system of slowly-rolling scalar fields during an inflationary epoch, allowing for an arbitrary field-space metric. We show explicitly how to compute its subsequent evolution using a covariantized version of the separate universe or "delta-N" expansion, which must be augmented by terms measuring curvature of the field-space manifold, and give the nonlinear gauge transformation to the comoving curvature perturbation. Nonlinearities induced by the field-space curvature terms are a new and potentially significant source of non-Gaussianity. We show how inflationary models with non-minimal coupling to the spacetime Ricci scalar can be accommodated within this framework. This yields a simple toolkit allowing the bispectrum to be computed in models with non-negligible field-space curvature.Comment: 22 pages, plus appendix and reference

Inflationary perturbation theory is geometrical optics in phase space

A pressing problem in comparing inflationary models with observation is the accurate calculation of correlation functions. One approach is to evolve them using ordinary differential equations ("transport equations"), analogous to the Schwinger-Dyson hierarchy of in-out quantum field theory. We extend this approach to the complete set of momentum space correlation functions. A formal solution can be obtained using raytracing techniques adapted from geometrical optics. We reformulate inflationary perturbation theory in this language, and show that raytracing reproduces the familiar "delta N" Taylor expansion. Our method produces ordinary differential equations which allow the Taylor coefficients to be computed efficiently. We use raytracing methods to express the gauge transformation between field fluctuations and the curvature perturbation, zeta, in geometrical terms. Using these results we give a compact expression for the nonlinear gauge-transform part of fNL in terms of the principal curvatures of uniform energy-density hypersurfaces in field space.Comment: 22 pages, plus bibliography and appendix. v2: minor changes, matches version published in JCA

Evolution of fNL to the adiabatic limit

We study inflationary perturbations in multiple-field models, for which zeta typically evolves until all isocurvature modes decay--the "adiabatic limit". We use numerical methods to explore the sensitivity of the nonlinear parameter fNL to the process by which this limit is achieved, finding an appreciable dependence on model-specific data such as the time at which slow-roll breaks down or the timescale of reheating. In models with a sum-separable potential where the isocurvature modes decay before the end of the slow-roll phase we give an analytic criterion for the asymptotic value of fNL to be large. Other examples can be constructed using a waterfall field to terminate inflation while fNL is transiently large, caused by descent from a ridge or convergence into a valley. We show that these two types of evolution are distinguished by the sign of the bispectrum, and give approximate expressions for the peak fNL.Comment: v1: 25 pages, plus Appendix and bibliography, 6 figures. v2: minor edits to match published version in JCA

Numerical evaluation of inflationary 3-point functions on curved field space

We extend the public CppTransport code to calculate the statistical properties of fluctuations in multiple-field inflationary models with curved field space. Our implementation accounts for all physical effects at tree-level in the 'in-in' diagrammatic expansion. This includes particle production due to time-varying masses, but excludes scenarios where the curvature perturbation is generated by averaging over the decay of more than one particle. We test our implementation by comparing results in Cartesian and polar field-space coordinates, showing excellent numerical agreement and only minor degradation in compute time. We compare our results with the PyTransport 2.0 code, which uses the same computational approach but a different numerical implementation, finding good agreement. Finally, we use our tools to study a class of gelaton-like models which could produce an enhanced non-Gaussian signal on equilateral configurations of the Fourier bispectrum. We show this is difficult to achieve using hyperbolic field-space manifolds and simple inflationary potentials

Moment transport equations for the primordial curvature perturbation

In a recent publication, we proposed that inflationary perturbation theory can be reformulated in terms of a probability transport equation, whose moments determine the correlation properties of the primordial curvature perturbation. In this paper we generalize this formulation to an arbitrary number of fields. We deduce ordinary differential equations for the evolution of the moments of zeta on superhorizon scales, which can be used to obtain an evolution equation for the dimensionless bispectrum, fNL. Our equations are covariant in field space and allow identification of the source terms responsible for evolution of fNL. In a model with M scalar fields, the number of numerical integrations required to obtain solutions of these equations scales like O(M^3). The performance of the moment transport algorithm means that numerical calculations with M >> 1 fields are straightforward. We illustrate this performance with a numerical calculation of fNL in Nflation models containing M ~ 10^2 fields, finding agreement with existing analytic calculations. We comment briefly on extensions of the method beyond the slow-roll approximation, or to calculate higher order parameters such as gNL.Comment: 23 pages, plus appendices and references; 4 figures. v2: incorrect statements regarding numerical delta N removed from Sec. 4.3. Minor modifications elsewher

Large slow-roll corrections to the bispectrum of noncanonical inflation

Nongaussian statistics are a powerful discriminant between inflationary models, particularly those with noncanonical kinetic terms. Focusing on theories where the Lagrangian is an arbitrary Lorentz-invariant function of a scalar field and its first derivatives, we review and extend the calculation of the observable three-point function. We compute the "next-order" slow-roll corrections to the bispectrum in closed form, and obtain quantitative estimates of their magnitude in DBI and power-law k-inflation. In the DBI case our results enable us to estimate corrections from the shape of the potential and the warp factor: these can be of order several tens of percent. We track the possible sources of large logarithms which can spoil ordinary perturbation theory, and use them to obtain a general formula for the scale dependence of the bispectrum. Our result satisfies the next-order version of Maldacena's consistency condition and an equivalent consistency condition for the scale dependence. We identify a new bispectrum shape available at next-order, which is similar to a shape encountered in Galileon models. If fNL is sufficiently large this shape may be independently detectable.Comment: v1: 37 pages, plus tables, figures and appendices. v2: supersedes version published in JCAP; some clarifications and more detailed comparison with earlier literature. All results unchanged. v3:improvements to some plots; text unchange

Separable and non-separable multi-field inflation and large non-Gaussianity

In this paper we provide a general framework based on $\delta N$ formalism to estimate the cosmological observables pertaining to the cosmic microwave background radiation for non-separable potentials, and for generic \emph{end of inflation} boundary conditions. We provide analytical and numerical solutions to the relevant observables by decomposing the cosmological perturbations along the curvature and the isocurvature directions, \emph{instead of adiabatic and entropy directions}. We then study under what conditions large bi-spectrum and tri-spectrum can be generated through phase transition which ends inflation. In an illustrative example, we show that large $f_{NL}\sim {\cal O}(80)$ and $\tau_{NL}\sim {\cal O}(20000)$ can be obtained for the case of separable and non-separable inflationary potentials.Comment: 21 pages, 6 figure

Numerical evaluation of the bispectrum in multiple field inflation

We present a complete framework for numerical calculation of the power spectrum and bispectrum in canonical inflation with an arbitrary number of light or heavy fields. Our method includes all relevant effects at tree-level in the loop expansion, including (i) interference between growing and decaying modes near horizon exit; (ii) correlation and coupling between species near horizon exit and on superhorizon scales; (iii) contributions from mass terms; and (iv) all contributions from coupling to gravity. We track the evolution of each correlation function from the vacuum state through horizon exit and the superhorizon regime, with no need to match quantum and classical parts of the calculation; when integrated, our approach corresponds exactly with the tree-level Schwinger or 'in-in' formulation of quantum field theory. In this paper we give the equations necessary to evolve all two- and three-point correlation functions together with suitable initial conditions. The final formalism is suitable to compute the amplitude, shape, and scale dependence of the bispectrum in models with |fNL| of order unity or less, which are a target for future galaxy surveys such as Euclid, DESI and LSST. As an illustration we apply our framework to a number of examples, obtaining quantitatively accurate predictions for their bispectra for the first time. Two accompanying reports describe publicly-available software packages that implement the method