Inflation is often described through the dynamics of a scalar field, slow-rolling in a suitable potential. Ultimately, this inflaton must be identified as the expectation value of a quantum field, evolving in a quantum effective potential. The shape of this potential is determined by the underlying tree-level potential, dressed by quantum corrections from the scalar field itself and the metric perturbations. Following [1], we compute the effective scalar field equations and the corrected Friedmann equations to quadratic order in both scalar field, scalar metric and tensor perturbations. We identify the quantum corrections from different sources at leading order in slow-roll, and estimate their magnitude in benchmark models of inflation. We comment on the implications of non-minimal coupling to gravity in this context.
We consider the constant-roll condition in the model of the inflaton nonminimal coupling to the Gauss-Bonnet term. By assuming the first Gauss-Bonnet flow parameter $δ_1$ is a constant, we discuss the constant-roll inflation with constant $ε_1$, constant $ε_2$ and constant $η_H$, respectively. Using the Bessel function approximation, we get the analytical expressions for the scalar and tensor power spectrum and derive the scalar spectral index $n_{\mathcal{R}}$ and the tensor to scalar ratio $r$ to the first order of $ε_1$. By using the Planck 2018 observations constraint on $n_{\mathcal{R}}$ …
Using the nonlinear $δN$ formalism, we consider a simple exactly soluble model of multi-component slow-roll inflation in which the nonlinear curvature perturbation can be evaluated analytically.
We study constant-roll inflation using the $β$-function formalism. We show that the constant rate of the inflaton roll is translated into a first order differential equation for the $β$-function which can be solved easily. The solutions to this equation correspond to the usual constant-roll models. We then construct, by perturbing these exact solutions, more general classes of models that satisfy the constant-roll equation asymptotically. In the case of an asymptotic power law solution, these corrections naturally provide an end to the inflationary phase. Interestingly, while from a theoretica…
We study constant-roll inflation in the presence of a gauge field coupled to an inflaton. By imposing the constant anisotropy condition, we find new exact anisotropic constant-roll inflationary solutions which include anisotropic power-law inflation as a special case. We also numerically show that the new anisotropic solutions are attractors in the phase space.
Constant-roll warm inflation is introduced in this work. A novel approach to finding an exact solution for Friedman equations coupled to scalar field equation of motion is presented for cold inflation and is extended to warm inflation with the constant dissipative parameter $Q=\fracΓ{3H}$. The evolution of the primordial inhomogeneities of a scalar field in a thermal bath is also studied. The $1σ$ consistency between the theoretical predictions of the model and observational constraints has been proven for a range of $Q$ and $β=\frac{\ddotϕ}{3Hϕ}$ (constant rate of inflaton roll). In addition,…
We propose a new approach to constant-roll warm inflation as a generalization of constant-roll inflation. Based on the $β$-function formalism, it is shown that constant-roll warm inflation models with a natural end fall into universality classes defined by three different types of $β$-functions, under the assumption that radiation energy density is quasi-stable. Given that warm inflation is completely specified by the $β$-function and dissipation coefficient ratio, we investigate whether or not the inflation can physically be realized for enough number of e-foldings at the background level for…
It is usually supposed that inflation is of the slow-roll variety, and that the inflaton generates the primordial curvature perturbation. According to the curvaton hypothesis, inflation need not be slow-roll, and if it is the inflaton generates a negligible curvature perturbation. We find that the construction of slow-roll inflation models becomes much easier under this hypothesis. Also, thermal inflation followed by fast-roll becomes viable, with no slow-roll inflation at all.
The constant-roll inflation is studied where the inflaton is taken as a tachyon field. Since in this approach the second slow-roll parameter is assumed to be of order one instead of being small, then the perturbation parameters will be considered again. The results are compared with observational data, and it is confirmed that the model could stand as a proper candidate for inflation.
The ultra-slow-roll (USR) inflationary models predict large-amplitude scalar perturbations at small scales which can lead to the primordial black hole production and scalar-induced gravitational waves. In general scalar perturbations in the USR models can only be obtained using numerical method because the usual slow-roll approximation breaks. In this work, we propose an analytical approach to estimate the scalar spectrum which is consistent with the numerical result. We find that the USR inflationary models predict a peak with power-law slopes in the scalar spectrum and energy spectrum of gra…
It is shown that the non-perturbative dynamics of a phase change to the non-trivial phase of $λφ^4$-theory in the early universe can give rise to slow-rollover inflation without recourse to unnaturally small couplings.
We take a pragmatic, model independent approach to single field slow-roll inflation by imposing conditions to the slow-roll parameter $ε$ and its derivative $ε^{\prime }.$ To accommodate the recent (large) values of $r$ reported by the BICEP2 collaboration we advocate for a decreasing $ε$ during most part of inflation. However because at $ϕ_{\mathrm{H}}$, at which the perturbations are produced, some $50$ $-$ $60$ e-folds before the end of inflation, $ε$ is increasing we thus require that $ε$ develops a maximum for $ϕ> ϕ_{\mathrm{H}}$ and then decrease to small values where most e-folds are pr…
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 $ε_1$ if and only if $ε_2^2\ll 6/ε_1$, with the notable exception of slow-roll. This is due to the failure of the stochastic $ΔN$ 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.
Inflation models based on scalar-tensor theories of gravity are formulated in the Jordan frame but most often analysed in the Einstein frame. The transformation between the frames is not always desirable. In this work we formulate slow-roll conditions in the Jordan frame. This is achieved by comparing different background quantities and approximations on the same spatial slice in both frames. We use these approximations to derive simple equations that can be applied to compute inflation model observables in terms of Jordan frame quantities only. Finally, we apply some of the results to analyse…
In constant-roll inflation, the scalar field that drives the accelerated expansion of the Universe is rolling down its potential at a constant rate. Within this framework, we highlight the relations between the Hubble slow-roll parameters and the potential ones, studying in detail the case of a single-field Coleman-Weinberg model characterised by a non-minimal coupling of the inflaton to gravity. With respect to the exact constant-roll predictions, we find that assuming an approximate slow-roll behaviour yields a difference of $Δr = 0.001$ in the tensor-to-scalar ratio prediction. Such a discr…
Constant-roll inflation was recently introduced by Motohashi, Starobinsky and Yokoyama as a phenomenological way to parametrize deviations from the slow-roll scenarios. In this paper, we investigate the dynamics of both the background and the perturbations in this model, without making any slow-roll assumptions. The perturbation spectra are computed with an efficient and accurate novel method that allowed us to quickly scan the parameter space of constant-roll inflation. We derive the constraints on the model parameters from the cosmic microwave background anisotropy measurements provided by t…
We emphasize that the estimation of cosmological parameters from cosmic microwave background (CMB) anisotropy data, such as the recent high resolution maps from BOOMERanG and MAXIMA-1, requires assumptions about the primordial spectra. The latter are predicted from inflation. The physically best-motivated scenario is that of slow-roll inflation. However, very often, the unphysical power-law inflation scenario is (implicitly) assumed in the CMB data analysis. We show that the predicted multipole moments differ significantly in both cases. We identify several misconceptions present in the litera…
Using a recently introduced formalism we discuss slow-roll inflaton from Kaluza-Klein theory without the cylinder condition. In particular, some examples corresponding to polynomic and hyperbolic $ϕ$-potentials are studied. We find that the evolution of the fifth coordinate should be determinant for both, the evolution of the early inflationary universe and the quantum fluctuations.
Slow-roll inflation may simultaneously solve the horizon problem and generate a near scale-free fluctuation spectrum P(k). These two processes are intimately connected via the initiation and duration of the inflationary phase. But a recent study based on the latest Planck release suggests that P(k) has a hard cutoff, k_min > 0, inconsistent with this conventional picture. Here we demonstrate quantitatively that most---perhaps all---slow-roll inflationary models fail to accommodate this minimum cutoff. We show that the small parameter `epsilon' must be > 0.9 throughout the inflationary period t…
We elucidate the counting of the relevant small parameters in inflationary perturbation theory. Doing this allows for an explicit delineation of the domain of validity of the semi-classical approximation to gravity used in the calculation of inflationary correlation functions. We derive an expression for the dependence of correlation functions of inflationary perturbations on the slow-roll parameter $ε= -\dot{H}/H^2$, as well as on $H/M_p$, where $H$ is the Hubble parameter during inflation. Our analysis is valid for single-field models in which the inflaton can traverse a Planck-sized range i…
In this note we point out that, contrary to the standard point of view, slow roll inflation is due to high gravitational friction. We show that the requirement of slow roll coincides with the requirement of a flat scalar field potential in the case of minimally coupled scalar field. In this sense, the search for a successful inflationary theory may be more fruitful by shifting the focus on models with high gravitational friction. We review then a gravitational mechanism, the so called "Gravitationally Enhanced Friction" mechanism, such that high gravitational friction is dynamically generated …
Multiple inflation is a model based on N=1 supergravity wherein there are sudden changes in the mass of the inflaton because it couples to 'flat direction' scalar fields which undergo symmetry breaking phase transitions as the universe cools. The resulting brief violations of slow-roll evolution generate a non-gaussian signal which we find to be oscillatory and yielding f_NL ~ 5-20. This is potentially detectable by e.g. Planck but would require new bispectrum estimators to do so. We also derive a model-independent result relating the period of oscillations of a phase transition during inflati…
We find multi-scalar effective field theories (EFTs) that can achieve a slow inflationary roll despite having a scalar potential that does not satisfy the usual slow-roll condition (d V)^2 << V^2/Mp^2. They evade the usual slow-roll conditions on $V$ because their kinetic energies are dominated by single-derivative terms rather than the usual two-derivative terms. Single derivatives dominate during slow roll and so do not require a breakdown of the usual derivative expansion that underpins calculational control in much of cosmology. The presence of such terms requires some sort of UV Lorentz-s…
We study the generation of helical magnetic fields during single field inflation induced by an axial coupling of the electromagnetic field to the inflaton. During slow roll inflation, we find that such a coupling always leads to a blue spectrum with $B^2(k) \propto k$, as long as the theory is treated perturbatively. The magnetic energy density at the end of inflation is found to be typically too small to backreact on the background dynamics of the inflaton. We also show that a short deviation from slow roll does not result in strong modifications to the shape of the spectrum. We calculate the…
Ultra slow-roll inflation has recently been used to challenge the non-Gaussianity consistency relation. We show that this inflationary scenario belongs to a one parameter class of models and we study its properties and observational predictions. We demonstrate that the power spectrum remains scale-invariant and that the bi-spectrum is of the local type with fnl=5(3-ns)/4 which, indeed, represents a modification of the consistency relation. However, we also show that the system is unstable and suffers from many physical problems among which is the difficulty to correctly WMAP normalize the mode…
We show that in the simplest theories of spontaneous symmetry breaking one can have a stage of a fast-roll inflation. In this regime the standard slow-roll condition |m^2| << H^2 is violated. Nevertheless, this stage can be rather long if |m| is sufficiently small. Fast-roll inflation can be useful for generating proper initial conditions for the subsequent stage of slow-roll inflation in the very early universe. It may also be responsible for the present stage of accelerated expansion of the universe. We also make two observations of a more general nature. First of all, the universe after a l…
In this paper, I present an analytic solution to the equation of motion of the inflaton field in a model of rapid roll inflation with a quartic potential by assuming that the Hubble parameter is a constant during inflation. The result is obtained by using Jacobi elliptic functions. The spectral index $n_s$ and the running spectral index $n_s^\prime$ is obtained without using rapid roll or extended slow roll approximations. The cosmological consequences of the model is discussed.
Ultra-slow-roll (USR) inflation is a new mode of inflation which corresponds to the occasions when the inflaton field must traverse an extremely flat part of the scalar potential, when the usual slow-roll (SR) fails. We investigate USR and obtain an estimate for how long it lasts, given the initial kinetic density of the inflaton. We also find that, if the initial kinetic density is small enough, USR can be avoided and the usual SR treatment is valid. This has important implications for inflection-point inflation.
The next generation of cosmological observations will be sensitive to small deviations from a pure power law in the primordial power spectrum of the curvature perturbations. In the context of slow-roll inflation, these deviations are expected and correspond to the so-called running of the spectral index. Their measurement would bring as much information as the discovery of deviations from scale invariance. However, robust parameter inference requires to marginalize over any possible higher order uncertainties, which have been, up to now, not fully determined. We tackle this issue by deriving t…
We show that a non-minimal coupling to gravity can not only make some inflationary models consistent with cosmological data, similar to the case of Higgs inflation, but can also invoke slow-roll violation to realize graceful exit from inflation. In particular, this is the case in models where a destabilizing mechanism that ends inflation should be assumed when the model is minimally coupled to gravity. As explicit examples, we consider the power-law and inverse monomial inflation models with a non-minimal coupling to gravity. While these models are excluded in the minimally coupled case, we sh…
We consider the impact of quantum diffusion on inflationary dynamics during an ultra-slow-roll phase, which can be of particular significance for the formation of primordial black holes. We show, by means of a fully analytical approach, that the power spectrum of comoving curvature perturbations computed in stochastic inflation matches precisely, at the linear level, the result obtained by solving the Mukhanov-Sasaki equation, even in the presence of an ultra-slow-roll phase. We confirm this result numerically in a model in which the inflaton has a polynomial potential and is coupled quadratic…
Usual inflation is realized with a slow rolling scalar field minimally coupled to gravity. In contrast, we consider dynamics of a scalar with a flat effective potential, conformally coupled to gravity. Surprisingly, it contains an attractor inflationary solution with the rapidly rolling inflaton field. We discuss models with the conformal inflaton with a flat potential (including hybrid inflation). There is no generation of cosmological fluctuations from the conformally coupled inflaton. We consider realizations of modulated (inhomogeneous reheating) or curvaton cosmological fluctuations in th…
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 non-zero 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 wav…
We analyze the weak (2 sigma) evidence for a running spectral index seen in the three-year WMAP dataset and its implications for single field, slow roll inflation. We assume that the running is comparable to the central value found from the WMAP data analysis, and use the Hubble Slow Roll formalism to follow the evolution of the slow roll parameters. For all parameter choices consistent with a large, negative running, single field, slow roll inflation lasts less than 30 efolds after CMB scales leave the horizon. Thus, a definitive observation of a large negative running would imply that any in…
There are many inflationary models in which inflaton field does not satisfy the slow-roll condition. However, in such models, it is always difficult to generate the curvature perturbation during inflation. Thus, to generate the curvature perturbation, one must introduce another component to the theory. To cite a case, curvatons may generate dominant part of the curvature perturbation after inflation. However, we have a question whether it is unrealistic to consider the generation of the curvature perturbation during inflation without slow-roll. Assuming multi-field inflation, we encounter the …
The scenario of brane inflation without using the conventional slow-roll approximations has been investigated. Based on the mechanism of generating the curvature perturbations at the end of inflation, a new brane inflation paradigm was developed. The conditions for making a sufficiently large enough number of e-foldings and for generating the curvature perturbations without producing dangerous relics were also examined. Benefits of our scenario are subsequently discussed in detail.
I describe a general approach to characterizing cosmological inflation outside the standard slow-roll approximation, based on the Hamilton-Jacobi formulation of scalar field dynamics. The basic idea is to view the equation of state of the scalar field matter as the fundamental dynamical variable, as opposed to the field value or the expansion rate. I discuss how to formulate the equations of motion for scalar and tensor fluctuations in situations where the assumption of slow roll is not valid. I apply the general results to the simple case of inflation from an ``inverted'' polynomial potential…
We study an inflationary scenario where thermal inflation is followed by fast-roll inflation. This is a rather generic possibility based on the effective potentials of spontaneous symmetry breaking in the context of particle physics models. We show that a large enough expansion could be achieved to solve cosmological problems. However, the power spectrum of primordial density perturbations from the quantum fluctuations in the inflaton field is not scale invariant and thus inconsistent with observations. Using the curvaton mechanism instead, we can obtain a nearly scale invariant spectrum, prov…
We not only reconstruct the slow-roll parameters for fit to the running spectral index from WMAP three-year data in the usual slow-roll inflation model and noncommutative inflation model, but also investigate the evolution of these slow-roll parameters. Requiring slow-roll inflation lasts more than 20 e-folds after CMB scales leave the horizon suggests that the potential at the last stage of inflation takes the form $V(ϕ)=V_0(1+{η_c\over 2}{(ϕ-ϕ_c)^2\over M_p^2})$, where $η_c$ is a constant. A natural mechanism to end inflation at $ϕ=ϕ_c$ is hybrid type inflation.
We implement Slow Roll Reconstruction -- an optimal solution to the inverse problem for inflationary cosmology -- within ModeCode, a publicly available solver for the inflationary dynamics. We obtain up-to-date constraints on the reconstructed inflationary potential, derived from the WMAP 7-year dataset and South Pole Telescope observations, combined with large scale structure data derived from SDSS Data Release 7. Using ModeCode in conjunction with the MultiNest sampler, we compute Bayesian evidence for the reconstructed potential at each order in the truncated slow roll hierarchy. We find th…
It is generally believed that in single-field slow-roll inflation, a large tensor-to-scalar ratio $r > 0.1$ requires inflaton field values close to or above the Planck scale. Recently, it has been claimed that $r > 0.15$ can be achieved with much smaller inflaton field values $Δϕ< M_{Pl}/10$. We show that in single-field slow-roll inflation, it is impossible to reconcile $r > 0.1$ with such small field values, independently of the form of the potential, and that the recent claim to the contrary is based on an invalid approximation. We conclude that the result of the BICEP2 measurement of $r > …
In this paper we analyze scalar gravitational perturbations on a Robertson-Walker background in the presence of multiple scalar fields that take values on a (geometrically non-trivial) field manifold during slow-roll inflation. For this purpose modified and generalized slow-roll functions are introduced and their properties examined. These functions make it possible to estimate to what extent the gravitational potential decouples from the scalar field perturbations. The correlation function of the gravitational potential is calculated in an arbitrary state. We argue that using the vacuum state…
The ultra-slow-roll (USR) inflation represents a class of single-field models with sharp deceleration of the rolling dynamics on small scales, leading to a significantly enhanced power spectrum of the curvature perturbations and primordial black hole (PBH) formation. Such a sharp transition of the inflationary background can trigger the coherent motion of scalar condensates with effective potentials governed by the rolling rate of the inflaton field. We show that a scalar condensate carrying (a combination of) baryon or lepton number can achieve successful baryogenesis through the Affleck-Dine…
