The curvature perturbation at second order

We give an explicit relation, up to second-order terms, between scalar-field fluctuations defined on spatially-flat slices and the curvature perturbation on uniform-density slices. This expression is a necessary ingredient for calculating observable quantities at second-order and beyond in multiple-field inflation. We show that traditional cosmological perturbation theory and the `separate universe' approach yield equivalent expressions for superhorizon wavenumbers, and in particular that all nonlocal terms can be eliminated from the perturbation-theory expressions

Non Gaussianity of General Multiple-Field Inflationary Models

Using the "$\delta N$-formalism", We obtain the expression of the non-Gaussianity of multiple-field inflationary models with the nontrivial field-space metric. Further, we rewritten the result by using the slow-rolling approximation.Comment: 10 pages, no figure, references is added; mistakes is correcte

Non-Gaussianity in Axion N-flation Models

We study perturbations in the multifield axion N-flation model, taking account of the full cosine potential. We find significant differences from previous analyses which made a quadratic approximation to the potential. The tensor-to-scalar ratio and the scalar spectral index move to lower values, which nevertheless provide an acceptable fit to observation. Most significantly, we find that the bispectrum non-Gaussianity parameter fNL may be large, typically of order 10 for moderate values of the axion decay constant, increasing to of order 100 for decay constants slightly smaller than the Planck scale. Such a non-Gaussian fraction is detectable. We argue that this property is generic in multifield models of hilltop inflation

Non-gaussianity of inflationary field perturbations from the field equation

We calculate the tree-level bispectrum of the inflaton field perturbation directly from the field equations, and construct the corresponding f_NL parameter. Our results agree with previous ones derived from the Lagrangian. We argue that quantum theory should only be used to calculate the correlators when they first become classical a few Hubble times after horizon exit, the classical evolution taking over thereafter.Comment: 16 pages, uses iopart.sty. v2: replaced with version accepted by JCAP; minor changes of wording only. v3: supersedes version published by journal; typo fixed in Eq. (20) and updated references. v4: sign errors in Eqs. (32) and (38) correcte

The inflationary trispectrum

We calculate the trispectrum of the primordial curvature perturbation generated by an epoch of slow-roll inflation in the early universe, and demonstrate that the non-gaussian signature imprinted at horizon crossing is unobservably small, of order tau_NL < r/50, where r < 1 is the tensor-to-scalar ratio. Therefore any primordial non-gaussianity observed in future microwave background experiments is likely to have been synthesized by gravitational effects on superhorizon scales. We discuss the application of Maldacena's consistency condition to the trispectrum.Comment: 23 pages, 2 diagrams drawn with feynmp.sty, uses iopart.cls. v2, replaced with version accepted by JCAP. Estimate of maximal tau_NL refined in Section 5, resulting in smaller numerical value. Sign errors in Eq. (44) and Eq. (48) corrected. Some minor notational change

Generating the curvature perturbation at the end of inflation

The dominant contribution to the primordial curvature perturbation may be generated at the end of inflation. Taking the end of inflation to be sudden, formulas are presented for the spectrum, spectral tilt and non-gaussianity. They are evaluated for a minimal extension of the original hybrid inflation model.Comment: 5 pages. v3: as it will appear in JCA

Diagrammatic approach to non-Gaussianity from inflation

We present Feynman type diagrams for calculating the n-point function of the primordial curvature perturbation in terms of scalar field perturbations during inflation. The diagrams can be used to evaluate the corresponding terms in the n-point function at tree level or any required loop level. Rules are presented for drawing the diagrams and writing down the corresponding terms in real space and Fourier space. We show that vertices can be renormalised to automatically account for diagrams with dressed vertices. We apply these rules to calculate the primordial power spectrum up to two loops, the bispectrum including loop corrections, and the trispectrum.Comment: 17 pages, 13 figures. v2: Comments and references added, v3: Introduction expanded, subsection on evaluating loop diagrams added, minor errors corrected, references adde

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

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