Christian T. Byrnes

Primordial trispectrum from inflation

We use the delta N-formalism to describe the leading order contributions tothe primordial power spectrum, bispectrum and trispectrum in multiple-fieldmodels of inflation at leading order in a perturbative expansion. In slow-rollmodels where the initial field fluctuations at Hubble-exit are nearly Gaussian,any detectable non-Gaussianity is expected to come from super-Hubble evolution.We show that the contribution to the primordial trispectrum can be described bytwo non-linearity parameters, tau_{NL} and g_{NL}, which are dependent upon thesecond and third derivatives of the local expansion with respect to the fieldvalues during inflation.

Review of local non-Gaussianity from multi-field inflation

We review models which generate a large non-Gaussianity of the local form. We first briefly consider three models which generate the non-Gaussianity either at or after the end of inflation; the curvaton scenario, modulated (p)reheating, and an inhomogeneous end of inflation. We then focus on ways of generating the non-Gaussianity during inflation. We derive general conditions which a product or sum separable potential must satisfy in order to generate a large local bispectrum during slow-roll inflation. As an application, we consider two-field hybrid inflation. We then derive a formalism not based on slow roll which can be applied to models in which the slow-roll parameters become large before inflation ends. An exactly soluble two-field model is given in which this happens. Finally, we also consider further non-Gaussian observables, a scale dependence of fNL and the trispectrum.

Scale dependence of local fNL

We consider possible scale-dependence of the non-linearity parameter fNL in local and quasi-local models of non-Gaussian primordial density perturbations. In the simplest model where the primordial perturbations are a quadratic local function of a single Gaussian field then fNL is scale-independent by construction. However scale-dependence can arise due to either a local function of more than one Gaussian field, or due to non-linear evolution of modes after horizon-exit during inflation. We show that the scale dependence of fNL is typically first order in slow-roll. For some models this may be observable with experiments such as Planck provided that fNL is close to the current observational bounds.

Large non-Gaussianity from two-component hybrid inflation

We study the generation of non-Gaussianity in models of hybrid inflation with two inflaton fields, (2-brid inflation). We analyse the region in the parameter and the initial condition space where a large non-Gaussianity may be generated during slow-roll inflation which is generally characterised by a large fNL, τNL and a small gNL. For certain parameter values we can satisfy τNL >> fNL2. The bispectrum is of the local type but may have a significant scale dependence. We show that the loop corrections to the power spectrum and bispectrum are suppressed during inflation, if one assume that the fields follow a classical background trajectory. We also include the effect of the waterfall field, which can lead to a significant change in the observables after the waterfall field is destabilised, depending on the couplings between the waterfall and inflaton fields.

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 renormalized 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.

Non-Gaussianity beyond slow roll in multi-field inflation

We study the non-Gaussianity generated during multiple-field inflation. We provide an exact expression for the bispectrum parameter fNL which is valid beyond the slow-roll regime, valid for certain classes of inflationary models. We then study a new, exact multi-field inflationary model considering a case where the bispectrum grows to observable values at the end of inflation. We show that in this case the trispectrum is also large and may even provide the dominant signal of non-Gaussianity

Curvature and isocurvature perturbations from two-field inflation in a slow-roll expansion

We calculate the power spectra of primordial curvature and isocurvature perturbations from a general two-field inflation model at next-to-leading order correction in a slow-roll expansion. In particular we calculate the spectral indices to second order in slow-roll parameters. We show that the cross correlation of the curvature and isocurvature perturbations at the time of Hubble exit during inflation is nonzero at first order in slow-roll parameters. We apply our results to different classes of inflation, including inflaton and curvaton scenarios. The spectrum of primordial gravitational waves, curvature and isocurvature perturbations obey generalized consistency relations in two-field inflation models. We give the first two consistency relations in an infinite hierarchy.

Scale-dependent non-Gaussianity probes inflationary physics

We calculate the scale dependence of the bispectrum and trispectrum in (quasi) local models of non-Gaussian primordial density perturbations, and characterize this scale dependence in terms of new observable parameters. They can help to discriminate between models of inflation, since they are sensitive to properties of the inflationary physics that are not probed by the standard observables. We find consistency relations between these parameters in certain classes of models. We apply our results to a scenario of modulated reheating, showing that the scale dependence of non-Gaussianity can be significant. We also discuss the scale dependence of the bispectrum and trispectrum, in cases where one varies the shape as well as the overall scale of the figure under consideration. We conclude providing a formulation of the curvature perturbation in real space, which generalises the standard local form by dropping the assumption that fNL and gNL are constants.

CMB contraints on primordial non-Gaussianity from the bispectrum (fNL) and trispectrum (gNL and τNL) and a new consistency test of single-field inflation

We outline the expected constraints on non-Gaussianity from the cosmic microwave background with current and future experiments, focusing on both the third (f{sub NL}) and fourth-order (g{sub NL} and {tau}{sub NL}) amplitudes of the local configuration or non-Gaussianity. The experimental focus is the skewness (two-to-one) and kurtosis (two-to-two and three-to-one) power spectra from weighted maps. In addition to a measurement of {tau}{sub NL} and g{sub NL} with WMAP 5-year data, our study provides the first forecasts for future constraints on g{sub NL}. We describe how these statistics can be corrected for the mask and cut-sky through a window function, bypassing the need to compute linear terms that were introduced for the previous-generation non-Gaussianity statistics, such as the skewness estimator. We discus the ratio A{sub NL}={tau}{sub NL}/(6f{sub NL}/5){sup 2} as an additional test of single-field inflationary models and discuss the physical significance of each statistic. Using these estimators with WMAP 5-Year V+W-band data out to l{sub max}=600 we constrain the cubic order non-Gaussianity parameters {tau}{sub NL}, and g{sub NL} and find -7.4<g{sub NL}/10{sup 5}<8.2 and -0.6<{tau}{sub NL}/10{sup 4}<3.3 improving the previous COBE-based limit on {tau}{sub NL}<10{sup 8} nearly 4 orders of magnitude with WMAP.

CMB contraints on primordial non-Gaussianity from the bispectrum (f(NL)) and trispectrum (g(NL) and tau(NL)) and a new consistency test of single-field inflation

We outline the expected constraints on non-Gaussianity from the cosmic microwave background with current and future experiments, focusing on both the third (f{sub NL}) and fourth-order (g{sub NL} and {tau}{sub NL}) amplitudes of the local configuration or non-Gaussianity. The experimental focus is the skewness (two-to-one) and kurtosis (two-to-two and three-to-one) power spectra from weighted maps. In addition to a measurement of {tau}{sub NL} and g{sub NL} with WMAP 5-year data, our study provides the first forecasts for future constraints on g{sub NL}. We describe how these statistics can be corrected for the mask and cut-sky through a window function, bypassing the need to compute linear terms that were introduced for the previous-generation non-Gaussianity statistics, such as the skewness estimator. We discus the ratio A{sub NL}={tau}{sub NL}/(6f{sub NL}/5){sup 2} as an additional test of single-field inflationary models and discuss the physical significance of each statistic. Using these estimators with WMAP 5-Year V+W-band data out to l{sub max}=600 we constrain the cubic order non-Gaussianity parameters {tau}{sub NL}, and g{sub NL} and find -7.4<g{sub NL}/10{sup 5}<8.2 and -0.6<{tau}{sub NL}/10{sup 4}<3.3 improving the previous COBE-based limit on {tau}{sub NL}<10{sup 8} nearly 4 orders of magnitude with WMAP.