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Register a new userLarge Slow Roll Parameters in Single Field Inflation
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We initially consider two simple situations where inflationary slow roll parameters are large and modes no longer freeze out shortly after exiting the horizon, treating both cases analytically. We then consider applications to transient phases where the slow roll parameters can become large, especially in the context of the common `fast-roll inflation frequently used as a mechanism to explain the anomalously low scalar power at low $l$ in the CMB. These transient cases we treat numerically. We find when $epsilon$, the first slow roll parameter, and only $epsilon$ is large, modes decay outside the horizon, and when $delta$, the second slow roll parameter, is large, modes grow outside the horizon. When multiple slow roll parameters are large the behavior in general is more complicated, but we nevertheless show in the fast-roll inflation case, modes grow outside the horizon.
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We calculate the conditions required to produce a large local trispectrum during two-field slow-roll inflation. This is done by extending and simplifying the heatmap approach developed by Byrnes et al. The conditions required to generate a large trispectrum are broadly the same as those that can produce a large bispectrum. We derive a simple relation between tauNL and fNL for models with separable potentials, and furthermore show that gNL and tauNL can be related in specific circumstances. Additionally, we interpret the heatmaps dynamically, showing how they can be used as qualitative tools to understand the evolution of non-Gaussianity during inflation. We also show how fNL, tauNL and gNL are sourced by generic shapes in the inflationary potential, namely ridges, valleys and inflection points.
We take a pragmatic, model independent approach to single field slow-roll canonical inflation by imposing conditions, not on the potential, but on the slow-roll parameter $epsilon(phi)$ and its derivatives $epsilon^{prime }(phi)$ and $epsilon^{primeprime }(phi)$, thereby extracting general conditions on the tensor-to-scalar ratio $r$ and the running $n_{sk}$ at $phi_{H}$ where the perturbations are produced, some $50$ $-$ $60$ $e$-folds before the end of inflation. We find quite generally that for models where $epsilon(phi)$ develops a maximum, a relatively large $r$ is most likely accompanied by a positive running while a negligible tensor-to-scalar ratio implies negative running. The definitive answer, however, is given in terms of the slow-roll parameter $xi_2(phi)$. To accommodate a large tensor-to-scalar ratio that meets the limiting values allowed by the Planck data, we study a non-monotonic $epsilon(phi)$ decreasing during most part of inflation. Since at $phi_{H}$ the slow-roll parameter $epsilon(phi)$ is increasing, we thus require that $epsilon(phi)$ develops a maximum for $phi > phi_{H}$ after which $epsilon(phi)$ decrease to small values where most $e$-folds are produced. The end of inflation might occur trough a hybrid mechanism and a small field excursion $Deltaphi_eequiv |phi_H-phi_e |$ is obtained with a sufficiently thin profile for $epsilon(phi)$ which, however, should not conflict with the second slow-roll parameter $eta(phi)$. As a consequence of this analysis we find bounds for $Delta phi_e$, $r_H$ and for the scalar spectral index $n_{sH}$. Finally we provide examples where these considerations are explicitly realised.
We use the long-wavelength formalism to investigate the level of bispectral non-Gaussianity produced in two-field inflation models with standard kinetic terms. Even though the Planck satellite has so far not detected any primordial non-Gaussianity, it has tightened the constraints significantly, and it is important to better understand what regions of inflation model space have been ruled out, as well as prepare for the next generation of experiments that might reach the important milestone of Delta f_NL(local) = 1. We derive an alternative formulation of the previously derived integral expression for f_NL, which makes it easier to physically interpret the result and see which types of potentials can produce large non-Gaussianity. We apply this to the case of a sum potential and show that it is very difficult to satisfy simultaneously the conditions for a large f_NL and the observational constraints on the spectral index n_s. In the case of the sum of two monomial potentials and a constant we explicitly show in which small region of parameter space this is possible, and we show how to construct such a model. Finally, the new general expression for f_NL also allows us to prove that for the sum potential the explicit expressions derived within the slow-roll approximation remain valid even when the slow-roll approximation is broken during the turn of the field trajectory (as long as only the epsilon slow-roll parameter remains small).
We find constraints on inflationary dynamics that yield a large local bispectrum and/or trispectrum during two-field slow-roll inflation. This leads to simple relations between the non-Gaussianity parameters, simplifying the Suyama-Yamaguchi inequality and also producing a new result between the trispectrum parameters tNL and gNL.
We numerically calculate the evolution of second order cosmological perturbations for an inflationary scalar field without resorting to the slow-roll approximation or assuming large scales. In contrast to previous approaches we therefore use the full non-slow-roll source term for the second order Klein-Gordon equation which is valid on all scales. The numerical results are consistent with the ones obtained previously where slow-roll is a good approximation. We investigate the effect of localised features in the scalar field potential which break slow-roll for some portion of the evolution. The numerical package solving the second order Klein-Gordon equation has been released under an open source license and is available for download.
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