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Register a new userFailure of the stochastic approach to inflation beyond slow-roll
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After giving a pedagogical review we clarify that the stochastic approach to inflation is generically reliable only at zeroth order in the (geometrical) slow-roll parameter $epsilon_1$ if and only if $epsilon_2^2ll 6/epsilon_1$, with the notable exception of slow-roll. This is due to the failure of the stochastic $Delta N$ formalism in its standard formulation. However, by keeping the formalism in its regime of validity, we showed that, in ultra-slow-roll, the stochastic approach to inflation reproduces the power spectrum calculated from the linear theory approach.
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Slow-roll inflation is analyzed in the context of modified gravity within the Palatini formalism. As shown in the literature, inflation in this framework requires the presence of non-traceless matter, otherwise it does not occur just as a consequence of the non-linear gravitational terms of the action. Nevertheless, by including a single scalar field that plays the role of the inflaton, slow-roll inflation can be performed in these theories, where the equations lead to an effective potential that modifies the dynamics. We obtain the general slow-roll parameters and analyze a simple model to illustrate the differences introduced by the gravitational terms under the Palatini approach, and the modifications on the spectral index and the tensor to scalar ratio predicted by the model.
Brief periods of non-slow-roll evolution during inflation can produce interesting observable consequences, as primordial black holes, or an inflationary gravitational wave spectrum enhanced at small scales. We develop a model independent, analytic approach for studying the predictions of single-field scenarios which include short phases of slow-roll violation. Our method is based on Taylor expanding the equations for cosmological fluctuations in a small quantity, which parameterizes the duration of the non-slow-roll eras. The super-horizon spectrum of perturbations is described by few effective parameters, and is characterized by a pronounced dip followed by a rapid growth in its amplitude, as typically found in numerical and analytical studies. The dip position $k_{rm dip}/k_*$ and the maximal enhancement $Pi_{rm max}$ of the spectrum towards small scales are found to be related by the law $k_{rm dip}/k_*propto Pi_{rm max}^{-1/4}$, and we determine the proportionality constant. For a single epoch of slow-roll violation we confirm previous studies, finding that the steepest slope of the spectrum well after the dip has spectral index $n-1,=,4$. On the other hand, with multiple phases of slow-roll violation, the slope of the spectrum is generally enhanced. For example, when two epochs of slow-roll violation take place, separated by a phase of quasi-de Sitter expansion, we find that the spectral index can reach the value $n-1,=,8$. This phenomenon indicates that the slope of the spectrum keeps memory of the history of non-slow-roll phases occurred during inflation.
The primordial power spectra of scalar and tensor perturbations during slow-roll inflation are usually calculated with the method of Bessel function approximation. For constant-roll or ultra slow-roll inflation, the method of Bessel function approximation may be invalid. We compare the numerical results with the analytical results derived from the Bessel function approximation, and we find that they differ significantly on super-horizon scales if the constant slow-roll parameter $eta_H$ is not small. More accurate method is needed for calculating the primordial power spectrum for constant-roll inflation.
We analyse field fluctuations during an Ultra Slow-Roll phase in the stochastic picture of inflation and the resulting non-Gaussian curvature perturbation, fully including the gravitational backreaction of the fields velocity. By working to leading order in a gradient expansion, we first demonstrate that consistency with the momentum constraint of General Relativity prevents the field velocity from having a stochastic source, reflecting the existence of a single scalar dynamical degree of freedom on long wavelengths. We then focus on a completely level potential surface, $V=V_0$, extending from a specified exit point $phi_{rm e}$, where slow roll resumes or inflation ends, to $phirightarrow +infty$. We compute the probability distribution in the number of e-folds $mathcal{N}$ required to reach $phi_{rm e}$ which allows for the computation of the curvature perturbation. We find that, if the fields initial velocity is high enough, all points eventually exit through $phi_{rm e}$ and a finite curvature perturbation is generated. On the contrary, if the initial velocity is low, some points enter an eternally inflating regime despite the existence of $phi_{rm e}$. In that case the probability distribution for $mathcal{N}$, although normalizable, does not possess finite moments, leading to a divergent curvature perturbation.
In models of coupled dark energy, in which a dark energy scalar field couples to other matter components, it is natural to expect a coupling to the inflaton as well. We explore the consequences of such a coupling in the context of single field slow-roll inflation. Assuming an exponential potential for the quintessence field we show that the coupling to the inflaton causes the quintessence field to be attracted towards the minimum of the effective potential. If the coupling is large enough, the field is heavy and is located at the minimum. We show how this affects the expansion rate and the slow-roll of the inflaton field, and therefore the primordial perturbations generated during inflation. We further show that the coupling has an important impact on the processes of reheating and preheating.
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