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 gravitational waves, and we derive the expression of the spectral indexes in terms of the inflationary potential. In turn, the inflationary potential near the USR regime can be reconstructed from the negative spectral index of the gravitational wave energy spectrum.
We consider Einstein-Gauss-Bonnet (EGB) inflationary models using the effective potential approach. We present evolution equations in the slow-roll regime using the effective potential and the tensor-to-scalar ratio. The choice of the effective potential is related to an expression of the spectral index in terms of e-folding number $N_e$. The satisfaction of the slow-roll regime is mostly related to the form of the tensor-to-scalar ratio $r$. The case of $r\sim1/N^2_e$ leads to a generalization of $α$-attractors inflationary parameters to Einstein-Gauss-Bonnet gravity with exponential effectiv…
We study the entire coupled evolution of the inflaton and the scale factor for general initial conditions at a given initial time. The generic early universe evolution has three stages: decelerated fast-roll followed by inflationary fast roll and then inflationary slow-roll. This evolution is valid for all regular inflaton potentials. In addition, we find a special (extreme) slow-roll solution starting at t = -infty in which the fast-roll stages are absent. At some time t = t_*, the generic evolution backwards in time reaches a mathematical singu- larity where a(t) vanishes and Hubble becomes …
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…
In this paper we investigate the implications of having a varying second slow-roll index on the canonical scalar field inflationary dynamics. We shall be interested in cases that the second slow-roll can take small values and correspondingly large values, for limiting cases of the function that quantifies the variation of the second slow-roll index. As we demonstrate, this can naturally introduce a smooth transition between slow-roll and constant-roll eras. We discuss the theoretical implications of the mechanism we introduce and we use various illustrative examples in order to better understa…
In this paper, with the presence of swampland conjecture, we use the complex form of a scalar field and investigate the anisotropic constant-roll of an inflationary scenario. Recently, the complex quintessence field has been used to describe the accelerating expansion of the universe which has had interesting results connecting with various conditions. Therefore, the Lagrangian density of the quintessence field leads to obtaining the equations of the complex scalar field. Also, we use the anisotropic constant-roll conditions and the field equation and calculate the exact solutions for some cos…
During the inflationary epoch,geometry of the universe may be described by quasi-de Sitter space. On the other hand,maximally extended de Sitter metric in the comoving coordinates accords with a special FLRW model with positive spatial curvature,so in this article we focus on the positively curved inflationary paradigm.For this purpose,first we derive the power spectra of comoving curvature perturbation and primordial gravitational waves in a positively curved FLRW universe according to the slowly rolling inflationary senario. It can be shown that the curvature spectral index in this model aut…
Recently a class of inflationary models satisfying the constant rate of roll constraint, $\ddotϕ+(3+α)H\dotϕ=0$, has been studied and compared with the latest cosmological observational data. We consider the broader class of constraints $ \ddotϕ+\frac{α_1}{2}\dotϕ+α_2 H\dotϕ=0$ and find an exact particular solution, without initial singularity, which is an attractor of the dynamics governed by a positive periodic potential $V(ϕ)=E+ F \cos (Ωϕ)+G \cos (2 Ωϕ)$. The spectral index $n_s$ and the tensor to scalar ratio $r$ are consistent with the recent observational data.
We propose a new single field inflation model in which the usual slow-roll inflation is joined to a new period of slow-climb and slow-roll inflation through a short intermediate non-inflationary phase. We then show that primordial curvature perturbations can be enhanced at small scales, a sizable amount of primordial black holes (PBHs) can be produced which make up most of dark matter, the gravitational waves (GWs) induced by scalar metric perturbations that accompany with the formation of PBHs can be detectable by future GW experiments, and last but not least, our model is compatible with the…
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…
We analyse slow-roll inflationary cosmologies that are holographically dual to a three-dimensional conformal field theory deformed by a nearly marginal scalar operator. We show the cosmological power spectrum is inversely proportional to the spectral density associated with the 2-point function of the trace of the stress tensor in the deformed CFT. Computing this quantity using second-order conformal perturbation theory, we obtain a holographic power spectrum in exact agreement with the expected inflationary power spectrum to second order in slow roll.
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 construct an analytically solvable simplified model that captures the essential features for primordial black hole (PBH) production in most models of single-field inflation. The construction makes use of the Wands duality between the constant-roll (or slow-roll) and the preceding ultra-slow-roll phases and can be realized by a simple inflaton potential of two joined parabolas. Within this framework, it is possible to formulate explicit inflationary scenarios consistent with the CMB observations and copious production of PBHs of arbitrary mass. We quantify the variability of the shape of the…
If the expansion of the early Universe was not close to de Sitter, the statistical imprints of the primordial density perturbation on the cosmic microwave background can be quite different from those derived in slow-roll inflation. In this paper we study the inflationary signatures of all single-field models which are free of ghost-like instabilities. We allow for a rapid change of the Hubble parameter and the speed of sound of scalar fluctuations, in a way that is compatible with a nearly scale-invariant spectrum of perturbations, as supported by current cosmological observations. Our results…
We study the evolution of spatially flat Friedmann-Lemaître-Robertson-Walker universe for chaotic and Starobinsky potentials in the framework of modified loop quantum cosmologies. These models result in a non-singular bounce as in loop quantum cosmology, but with far more complex modified Friedmann dynamics with higher order than quadratic terms in energy density. For the kinetic energy dominated bounce, we obtain analytical solutions using different approximations and compare with numerical evolution for various physical variables. The relative error turns out to be less than $0.3\%$ in the b…
The current epoch of accelerated cosmic expansion is postulated to be driven by dark energy, which in the standard model takes the form of a cosmological constant with equation of state parameter $w=-1$. We propose an innovative perspective over the nature of dark energy by drawing a parallel with inflation, which we assume to be driven by a single scalar field, the inflaton. The inflaton was not a cosmological constant, as indicated by the fact that inflation ended and by the Planck satellite's constraint of $n_s\neq 1$ at $8σ$ confidence. Therefore, it is interesting to verify whether its eq…
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 study the implications of the recently proposed Trans-Planckian Censorship Conjecture (TCC) for early universe cosmology and in particular inflationary cosmology. The TCC leads to the conclusion that if we want inflationary cosmology to provide a successful scenario for cosmological structure formation, the energy scale of inflation has to be lower than $10^9$ GeV. Demanding the correct amplitude of the cosmological perturbations then forces the generalized slow-roll parameter $ε$ of the model to be very small ($<10^{-31}$). This leads to the prediction of a negligible amplitude of primordi…
We carry out a numerical calculation of the bispectrum in generalised trajectories of canonical, single--field inflation. The trajectories are generated in the Hamilton-Jacobi (HJ) formalism based on Hubble Slow Roll (HSR) parameters. The calculation allows generally shape and scale dependent bispectra, or dimensionless $f_{NL}$, in the out-of-slow-roll regime. The distributions of $f_{NL}$ for various shapes and HSR proposals are shown as an example of how this procedure can be used within the context of Monte Carlo exploration of inflationary trajectories. We also show how allowing out-of-sl…
In the context of scalar-tensor theories of gravity we compute the third-order corrected spectral indices in the slow-roll approximation. The calculation is carried out by employing the Green's function method for scalar and tensor perturbations in both the Einstein and Jordan frames. Then, using the interrelations between the Hubble slow-roll parameters in the two frames we find that the frames are equivalent up to third order. Since the Hubble slow-roll parameters are related to the potential slow-roll parameters, we express the observables in terms of the latter which are manifestly invaria…
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…
The bispectrum of single-field inflationary trajectories in which the speed of sound of the inflationary trajectories $c_s$ is constant but not equal to the speed of light $c=1$ is explored. The trajectories are generated as random realisations of the Hubble Slow-Roll (HSR) hierarchy and the bispectra are calculated using numerical techniques that extends previous work. This method allows for out-of-slow-roll models with non-trivial time dependence and arbitrarily low $c_s$. The ensembles obtained using this method yield distributions for the shape and scale-dependence of the bispectrum and th…
We investigate the influence on the inflationary predictions from the reheating processes characterized by the $e$-folding number $N_{\mathrm{reh}}$ and the effective equation-of-state parameter $w_{\mathrm{reh}}$ during the reheating phase. For the first time, reheating processes can be constrained in the $N_{\mathrm{reh}}\!-\!w_{\mathrm{reh}}$ plane from Planck 2015. We find that for Higgs inflation with a nonminimal coupling to gravity, the predictions are insensitive to the reheating phase for current CMB measurements. We also find that the spontaneously broken SUSY inflation and axion mon…
We consider the familiar problem of a minimally coupled scalar field with quadratic potential in flat Friedmann-Lemaître-Robertson-Walker cosmology to illustrate a number of techniques and tools, which can be applied to a wide range of scalar field potentials and problems in e.g. modified gravity. We present a global and regular dynamical systems description that yields a global understanding of the solution space, including asymptotic features. We introduce dynamical systems techniques such as center manifold expansions and use Padé approximants to obtain improved approximations for the `attr…
Slow roll solutions to inflationary potentials have been widely believed to be the only universal attractor. Over the last few years there has been growing interest in a new class of inflationary models known as Constant-Roll Inflation. Constant roll solutions are a generalization of "ultra-slow roll" dynamics, where the first slow roll parameter is small, but the second slow roll parameter $η$ is larger than unity. In Ultra-slow Roll Inflation, the large-$η$ solution is a dynamical transient, relaxing exponentially to the attractor de Sitter solution. In the constant roll generalization, rece…
Recently, the nonperturbative quantization scheme of loop quantum gravity has been extended to the Brans-Dicke theory and the corresponding loop quantum Brans-Dicke cosmology has been derived, which provides an essential platform to explore inflationary models in this framework. In this paper, we consider two inflation models, the Starobinsky and $α$-attractor inflation whose cosmological predictions are in excellent agreement with Planck data, and study systematically their pre-inflationary dynamics as well as the slow-roll inflation. We show that for both models, the background evolution of …
In this work, we studied the slow-roll approximation of cosmic inflation within the context of $f(R,T)$ gravity, where $R$ is the scalar curvature, and $T$ is the trace of the energy-momentum tensor. By choosing a minimal coupling between matter and gravity, we obtained the modified slow-roll parameters, the scalar spectral index ($n_s$), the tensor spectral index ($n_{\textrm{T}}$), and the tensor-to-scalar ratio ($r$). We computed these quantities for a general power-law potential, Natural & Quartic Hilltop inflation, and the Starobinsky model, plotting the trajectories on the $(n_s,r)$ plan…
From the assumption that the slow roll parameter $ε$ has a Lorentzian form as a function of the e-folds number $N$, a successful model of a quintessential inflation is obtained, as succinctly studied in \cite{Benisty:2020xqm}. The form corresponds to the vacuum energy both in the inflationary and in the dark energy epochs and satisfies the condition to climb from small values of $ε$ to $1$ at the end of the inflationary epoch. We find the corresponding scalar Quintessential Inflationary potential with two flat regions. Moreover, a reheating mechanism is suggested with numerical estimation for …
We study the inflationary period driven by a fermionic field which is non-minimally coupled to gravity in the context of the constant-roll approach. We consider the model for a specific form of coupling and perform the corresponding inflationary analysis. By comparing the result with the Planck observations coming from CMB anisotropies, we find the observational constraints on the parameters space of the model and also the predictions the model. We find that the values of $r$ and $n_{s}$ for $-1.5<β\leq-0.9$ are in good agreement with the observations when $|ξ|=0.1$ and $N=60$.
In this work we revisit Wald's cosmic no-hair theorem in the context of accelerating Bianchi cosmologies for a generic cosmic fluid with non-vanishing anisotropic stress tensor and when the fluid energy momentum tensor is of the form of a cosmological constant term plus a piece which does not respect strong or dominant energy conditions. Such a fluid is the one appearing in inflationary models. We show that for such a system anisotropy may grow, in contrast to the cosmic no-hair conjecture. In particular, for a generic inflationary model we show that there is an upper bound on the growth of an…
In previous work, two of us have proposed a model of inflationary magnetogenesis based on a rolling auxiliary field able both to account for the magnetic fields inferred by the (non) observation of gamma-rays from blazars, and to start the galactic dynamo, without incurring in any strong coupling or strong backreaction regime. Here we evaluate the correction to the scalar spectrum and bispectrum with respect to single-field slow-roll inflation generated in that scenario. The strongest constraints on the model originate from the non-observation of a scalar bispectrum. Nevertheless, even when th…
In this paper, we study a non-canonical extension of a supergravity-motivated model acting as a vivid counterexample to the cosmic no-hair conjecture due to its unusual coupling between scalar and electromagnetic fields. In particular, a canonical scalar field is replaced by the string-inspired Dirac-Born-Infeld one in this extension. As a result, exact anisotropic inflationary solutions for this Dirac-Born-Infeld model are figured out under a constant-roll condition. Furthermore, numerical calculations are performed to verify that these anisotropic constant-roll solutions are indeed attractiv…
We compute the non-Gaussianities that arise in single field, slow roll inflationary models arising from arbitrary homogeneous initial states, as well as subleading contributions to the power spectrum. Non Bunch-Davies vacuum initial states can arise if the transition to the single field, slow roll inflation phase occurs only shortly before observable modes left the horizon. They can also arise from new physics at high energies that has been integrated out. Our general result for the bispectrum exhibits several features that were previously seen in special cases.
We summarize our work on the generation of primordial black holes in a type of two-field inflationary models. The key ingredient is a sharp turn of the background trajectory in field space. We show that certain classes of solutions to the equations of motion exhibit precisely this kind of behavior. Among them we find solutions, which describe a transition between an ultra-slow roll and a slow roll phases of inflation.
We investigate constraints on power spectra of the primordial curvature and tensor perturbations with priors based on single-field slow-roll inflation models. We stochastically draw the Hubble slow-roll parameters and generate the primordial power spectra using the inflationary flow equations. Using data from recent observations of CMB and several measurements of geometrical distances in the late Universe, Bayesian parameter estimation and model selection are performed for models that have separate priors on the slow-roll parameters. The same analysis is also performed adopting the standard pa…
In this work we address the reheating issue in the context of $F(R)$ gravity, for theories that the inflationary era does not obey the slow-roll condition but the constant-roll condition is assumed. As it is known, the reheating era takes place after the end of the inflationary era, so we investigate the implications of a constant-roll inflation era on the reheating era. We quantify our considerations by calculating the reheating temperature for the constant-roll $R^2$ model and we compare to the standard reheating temperature in the context of $F(R)$ gravity. As we demonstrate, the new reheat…
The previously introduced class of two-parametric phenomenological inflationary models in General Relativity in which the slow-roll assumption is replaced by the more general, constant-roll condition is generalized to the case of $f(R)$ gravity. A simple constant-roll condition is defined in the original Jordan frame, and exact expressions for a scalaron potential in the Einstein frame, for a function $f(R)$ (in the parametric form) and for inflationary dynamics are obtained. The region of the model parameters permitted by the latest observational constraints on the scalar spectral index and t…
We present a simple toy model corresponding to a network of frustrated topological defects of domain walls or cosmic strings that exist previous to the standard slow-roll inflationary era of the universe. Such a network (i) can produce a slower inflationary era than that of the standard scenario if it corresponds to a network of frustrated domain walls or (ii) can induce a vanishing universal acceleration; i.e., the universe would expand at a constant speed, if it corresponds to a network of frustrated cosmic strings red. Those features are phenomenologically modeled by a Chaplygin gas that ca…
In this work we generalize the constant-roll condition for minimally coupled canonical scalar field inflation. Particularly, we shall assume that the scalar field satisfies the condition $\ddotϕ=α(ϕ) V'(ϕ)$, and we derive the field equations under this assumption. We call the framework extended constant-roll framework. Accordingly we calculate the inflationary indices and the corresponding observational indices of inflation. In order to demonstrate the inflationary viability, we choose three potentials that are problematic in the context of slow-roll dynamics, namely chaotic, linear power-law …
We consider a simple model of modified gravity interacting with a single scalar field $φ$ with weakly coupled exponential potential within the framework of non-Riemannian spacetime volume-form formalism. The specific form of the action is fixed by the requirement of invariance under global Weyl-scale symmetry. Upon passing to the physical Einstein frame we show how the non-Riemannian volume elements create a second canonical scalar field $u$ and dynamically generate a non-trivial two-scalar-field potential $U_{\rm eff}(u,φ)$ with two remarkable features: (i) it possesses a large flat region fo…
Using a principal component (PC) basis that accommodates order unity features in the slow roll parameters as fine as 1/10 of a decade across more than 2 decades of the inflationary expansion, we test slow roll and single field inflation with the WMAP7 data. Detection of any non-zero component would represent a violation of ordinary slow roll and indicate a feature in the inflaton potential or sound speed. Although one component shows a deviation at the 98% CL, it cannot be considered statistically significant given the 20 components tested. The maximum likelihood PC parameters only improves 2D…
A Class of Exact Single-Field Inflationary Solutions beyond Slow Roll
Chongqing University of Science and Technology（重庆科技大学）；Zhengzhou University（郑州大学）；Chongqing University of Posts & Telecommunications（重庆邮电大学）；Chongqing University（重庆大学）
