Number Of E-folds Research Articles

Primordial black holes from Higgs inflation with a Gauss-Bonnet coupling

Primordial black holes (PBHs) can be the source for all or a part of today's dark matter density. Inflation provides a mechanism for generating the seeds of PBHs in the presence of a temporal period where the velocity of an inflaton field $\phi$ rapidly decreases toward 0. We compute the primordial power spectra of curvature perturbations generated during Gauss-Bonnet (GB) corrected Higgs inflation in which the inflaton field has not only a nonminimal coupling to gravity but also a GB coupling. For a scalar-GB coupling exhibiting a rapid change during inflation, we show that curvature perturbations are sufficiently enhanced by the appearance of an effective potential $V_{\rm eff}(\phi)$ containing the structures of plateau-type, bump-type, and their intermediate type. We find that there are parameter spaces in which PBHs can constitute all dark matter for these three types of $V_{\rm eff}(\phi)$. In particular, models with bump- and intermediate-types give rise to the primordial scalar and tensor power spectra consistent with the recent Planck data on scales relevant to the observations of cosmic microwave background. This property is attributed to the fact that the number of e-foldings $\Delta N_c$ acquired around the bump region of $V_{\rm eff}(\phi)$ can be as small as a few, in contrast to the plateau-type where $\Delta N_c$ typically exceeds the order of 10.

Model independent bounds for the number of e-folds during the evolution of the universe

We present a simple procedure to obtain universal bounds for quantities of cosmological interest, such as the number of e-folds during inflation, reheating, and radiation, as well as the reheating temperature. The main assumption is to represent each of the various epochs of evolution of the universe as being due to a single substance changing instantaneously into the next, describing a new era of evolution of the universe. This assumption, commonly used to obtain solutions of the Friedmann equations for simple cosmological models, is implemented here to find model-independent bounds on cosmological quantities of interest. In particular, we find that the bound Nk ≈ 56 for -1/3 < ω re < 1/3 is very robust as an upper bound on the number of e-folds during inflation and also as a lower bound when ω re > 1/3, where ω re is the effective equation of state parameter during reheating. These are model-independent results that any single-field model of inflation should satisfy. As an example we illustrate with the basic α attractor model the usual model dependent approach, and the one presented here, and show how they complement each other.

On Warm Natural Inflation and Planck 2018 Constraints

We investigate natural inflation with non-minimal coupling to gravity, characterized either by a quadratic or a periodic term, within the warm inflation paradigm during the slow-roll stage, in both strong and weak dissipation limits; and show that, in the case of a T-linearly dependent dissipative term, it can accommodate the spectral index ns and tensor-to-scalar ratio r observables given by Planck 2018 constraints, albeit with a too-small value of the e-folding number to solve the horizon problem, providing, thus, only a partial solution to natural inflation issues, assuming a T-cubically dependent dissipative term can provide a solution to this e-folding number issue.

Primordial black holes from stochastic tunnelling

If the inflaton gets trapped in a local minimum of its potential shortly before the end of inflation, it escapes by building up quantum fluctuations in a process known as stochastic tunnelling. In this work we study cosmological fluctuations produced in such a scenario, and how likely they are to form Primordial Black Holes (PBHs). This is done by using the stochastic-δ N formalism, which allows us to reconstruct the highly non-Gaussian tails of the distribution function of the number of e-folds spent in the false-vacuum state. We explore two different toy models, both analytically and numerically, in order to identify which properties do or do not depend on the details of the false-vacuum profile. We find that when the potential barrier is small enough compared to its width, ΔV/V < Δϕ 2/M Pl 2, the potential can be approximated as being flat between its two local extrema, so results previously obtained in a “flat quantum well” apply.Otherwise, when Δ V/V < V/M Pl 4, the PBH abundance depends exponentially on the height of the potential barrier, and when Δ V/V > V/M Pl it depends super-exponentially (i.e. as the exponential of an exponential) on the barrier height. In that later case PBHs are massively produced.This allows us to quantify how much flat inflection points need to be fine-tuned.In a deep false vacuum, we also find that slow-roll violations are typically encountered unless the potential is close to linear. This motivates further investigations to generalise our approach to non–slow-roll setups.

Kinetically stabilized inflation

In this work, we propose a string-inspired two fields inflation model to address the fine-tuning problem that the standard inflation model suffers. The fast-rolling tachyon mathcal{T} originated from the D-brane and anti-D-brane pair annihilation locks the inflaton φ slowly rolling on a Higgs-like potential Vleft(varphi right)=-{m}_{varphi}^2{varphi}^2+lambda {varphi}^4 and drives a kinetically stabilized (KS) inflation. Our numerical simulation confirms such a solution is a dynamic attractor. In particular, for λ < 0.8 × 10−3, the e-folding number contributed by the KS inflation phase can be larger than 62 to solve the horizon and flatness problems of Big Bang theory. Notably, this KS inflation generates a nearly scale-invariant primordial curvature perturbations spectrum consistent with current cosmic microwave background (CMB) observations. It predicts a low tensor-to-scalar ratio, which the current primordial gravitational wave background (the B-modes in CMB) searches favor.

Universal accelerating cosmologies from 10d supergravity

We study 4d Friedmann-Lemaître-Robertson-Walker cosmologies obtained from time-dependent compactifications of Type IIA 10d supergravity on various classes of 6d manifolds (Calabi-Yau, Einstein, Einstein-Kähler). The cosmologies we present are universal in that they do not depend on the detailed features of the compactification manifold, but only on the properties which are common to all the manifolds belonging to that class. Once the equations of motion are rewritten as an appropriate dynamical system, the existence of solutions featuring a phase of accelerated expansion is made manifest. The fixed points of this dynamical system, as well as the trajectories on the boundary of the phase space, correspond to analytic solutions which we determine explicitly. Furthermore, some of the resulting cosmologies exhibit eternal or semi-eternal acceleration, whereas others allow for a parametric control on the number of e-foldings. At future infinity, one can achieve both large volume and weak string coupling. Moreover, we find several smooth accelerating cosmologies without Big Bang singularities: the universe is contracting in the cosmological past (T < 0), expanding in the future (T > 0), while in the vicinity of T = 0 it becomes de Sitter in hyperbolic slicing. We also obtain several cosmologies featuring an infinite number of cycles of alternating periods of accelerated and decelerated expansions.

The paradigm of quantum cosmology through Dunkl fractional Laplacian operators and fractal dimensions

The semi-classical regime of fractional quantum cosmology has been discussed based on the Wheeler-DeWitt equation and ADM-Hamiltonian formalism. Fractionality has been introduced through two different approaches: the first one is based on fractional Dunkl Laplacian operator and the second one based on fractal calculus. Our models generalize the recent approaches of fractional quantum cosmology introduced in literature. In the first approach, it was revealed that power-law inflation and large number of e-folds may be obtained for various values of the fractional parameter. However, the second approach is plausible only if the cosmological constant decays with the scale factor of the universe. We have confronted our model with observations based on the Bayesian hierarchical model that describes the full distance ladder from nearby geometric anchors through Cepheids to Hubble-Flow SNe we have estimated the numerical values of the fractional order parameter. In both scenarios, the number of e-folds is too large.Pacs numbers: 05.45.Df, 45.10.Hj, 98.80.-k

Islands and the de Sitter entropy bound

The de Sitter (dS) entropy bound gives the maximal number of e-folds that non-eternal inflation can last before violating the thermodynamical interpretation of dS space. This semiclassical argument is the analogue, for dS space, of the Black-Hole information paradox. We use techniques developed to address the latter, namely the island formula, to calculate semiclassically the fine-grained entropy as seen by a Minkowskian observer after inflation and find that this follows a Page-like curve, never exceeding the thermodynamic dS entropy. This calculation, performed for a CFT in 2D gravity, suggests that the semiclassical expectation should be modified in such a way that the entropy bound might actually not be present.

Finite inflation in curved space

We investigate the effects of non-zero spatial curvature on cosmic inflation in the light of cosmic microwave background (CMB) anisotropy measurements from the Planck 2018 legacy release and from the 2015 observing season of BICEP2 and the Keck Array. Even a small percentage of non-zero curvature today would significantly limit the total number of e-folds of the scale factor during inflation, rendering just-enough inflation scenarios with a kinetically dominated or fast-roll stage prior to slow-roll inflation more likely. Finite inflation leads to oscillations and a cutoff towards large scales in the primordial power spectrum and curvature pushes them into the CMB observable window. Using nested sampling, we carry out Bayesian parameter estimations and model comparisons taking into account constraints from reheating and horizon considerations. We confirm the preference of CMB data for closed universes with Bayesian odds of over $100:1$ and with a posterior on the curvature density parameter of $\Omega_{K,0}=-0.051\pm0.017$ for a curvature extension of LCDM and $\Omega_{K,0}=-0.031\pm0.014$ for Starobinsky inflation. Model comparisons of various inflation models give similar results as for flat universes with the Starobinsky model outperforming most other models.

Special cases of the multi-measure model – Understanding the prolonged inflation

The multi-measure model (MMM), in which one modifies the action to include both the Riemannian measure and a non-Riemannian one, has proven to be able to produce viable Universe evolution scenarios. In this article we consider two special cases of the multi-measure model, in which we first decouple the two kinetic terms in the Lagrangian entirely and later remove the dark charge of the model. We show numerically that those special cases still possess the needed evolutionary stages of the Universe and furthermore, for them one can obtain a sufficient number of e-folds of the early inflation. In the first case, the inflaton still moves backwards on the effective potential during inflation, while in the second, it does not, meaning that this behavior comes from the dark charge. We connect the model with hyperinflationary models and investigate how the different epochs are born from the interplay between the two scalar fields. We demonstrate that there is a dynamically induced slow-roll epoch, which is prolonged by the complicated movement of the two scalars in the field space. Finally, we show that while the adiabatic speed of sound can become imaginary, the phase speed of sound remains real.

Primordial black holes and gravitational waves in teleparallel Gravity

In this paper, we consider the possible effect of the teleparallel gravity on the production of the primordial black holes (PBH) and on the gravitational waves (GWs). We investigate the relationship between the slow roll, the e-folds number and the teleparallel parameters. We show that in the case of the teleparallel parameter delta =3, the e-folds number reaches the values 50 and 60 in consistency with the contour plot of the (r,n_{s}) plane obtained by Planck data at 1sigma and 2sigma C.L. Furthermore, we use the fraction of the energy density and the variance of the density perturbations approach to study the abundance of the production PBH. We find that the PBH overproduction can be satisfied for specific values of parameters of the non-adiabatic curvature power spectrum at some narrow parametric resonance. Moreover, to explain the GWs expected by observations, the duration of preheating should be bounded by values less or equal to 2. This bound is in agreement with values of the tensor-to-scalar ratio and the spectral index constrained by Planck data.

Holographic β Function in de Sitter Space

The scale invariance of the universe is slightly broken by slow roll parameters. It is likely the slow roll is dual to the random walk. We investigate the distribution function of the conformal zeromode. We identify de Sitter entropy $S_{dS}$ with the distribution entropy of the conformal zeromode $\rho(\omega)$. We have collected convincing support on our postulate. The semiclassical evidence is that the both are given by the gravitational coupling $1/g=\log N/2$ where $g=G_NH^2/\pi$ and $N$ is the e-folding number. We show the renormalized distribution function obeys gravitational Fokker-Planck equation (GFP) and Langevin equations . Under the Gaussian approximation, they boil down to a simple first order partial differential equation. The identical equation is derived by the thermodynamic arguments in the inflationary space-time. GFP determines the evolution of de Sitter entropy of the universe. It coincides with $\beta$ function of $g$. We find two types of the solutions of GFP:(1) UV complete spacetime and (2) inflationary spacetime with power potentials. The maximum entropy principle favors the scenario: (a) born small $\epsilon$ and (b) grow large by inflation. We like to convey the emerging notion of de Sitter duality. The inflationary universe: (bulk/geometrical) is dual to the stochastic space-time on the boundary (cosmological horizon ) as the both are the solutions of GFP.

Generalized Uncertainty Principle with Maximal Momentum: QFT, Quantum Black Holes, and Some Cosmological Implications

This paper is a continuation of the earlier works written by the author and devoted to studies of the applicability boundary of a quantum field theory (QFT). Based on the results from black hole thermodynamics at all energy scales, this work demonstrates that, provided the Generalized Uncertainty Principle with maximal momentum at Planck scale and without a minimal length for the well-known quantum field theory, there exists a natural ultraviolet applicable boundary(cut-off) distant from the Planck scales. Some important cosmological implications from the obtained results are considered. Specifically, the number of e-folds in the inflation pattern is estimated considering the quantum-gravity corrections. Besides, particular cosmological parameters are evaluated in this case in the explicit form.

Helical magnetogenesis with reheating phase from higher curvature coupling and baryogenesis

We investigate the generation of helical magnetic fields and address the baryon asymmetry of the universe from an inflationary magnetogenesis scenario, in which the conformal and parity symmetries of the electromagnetic field are broken through its coupling to the Ricci scalar and to the Gauss–Bonnet invariant via the dual field tensor, so that the generated magnetic field can have a helical nature. Depending on the reheating mechanism, we consider two different cases - (1) instantaneous reheating scenario, in which case the reheating e-fold number is zero, and (2) Kamionkowski reheating scenario which is parametrized by a non-zero e-fold number, a reheating equation of state parameter and a given reheating temperature. It is demonstrated that for both the reheating mechanisms, the generated magnetic fields can be compatible with the observations for suitable range of the model parameter present in the non-minimal coupling of the electromagnetic field. Actually the present magnetogenesis model does not produce sufficient hierarchy between the electric and magnetic fields at the end of inflation, and thus the electric field is not able to sufficiently induce (or enhance) the magnetic field during the Kamionkowski reheating stage. This in turn makes both the reheating cases almost similar from the perspective of magnetic field’s evolution. Furthermore we find that the magnetic fields at the galactic scale with strength ∼ 1 0 − 13 G can lead to the resultant value of the ratio of the baryonic number density to the entropy density as large as ∼ 1 0 − 10 , which is consistent with the observational data.

Non-minimal coupled warm inflation with quantum-corrected self-interacting inflaton potential

In this work, we investigate the non-minimally coupled scenario in the context of warm inflation with quantum-corrected self-interacting potential. We transform the potential in the Jordan frame to the Einstein frame and consider a dissipation parameter of the form Gamma = C_{T}T with C_T being a coupling parameter. We focus on the strong regime of which the interaction between inflaton and radiation fluid has been taken into account. We compute inflationary observables and constrain the parameters of our model using current Planck 2018 data. With the sizeable number of e-folds and proper choices of parameters, we discover that allowed values of C_T lie in the range 0.014lesssim C_{T}lesssim 0.020 in which the predictions are in good agreement with the latest Planck 2018 results at the 2,sigma confident level.

Early and late universe holographic cosmology from a new generalized entropy

We propose a new four-parameter entropy function that generalizes the Tsallis, Rényi, Barrow, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies for suitable limits of the parameters. Consequently, we address the early and late universe cosmology corresponding to the proposed four-parameter entropy function. As a result, it turns out that the entropic cosmology from the generalized entropy function can unify the early inflation to the late dark energy era of the universe. In such a unified scenario, we find that – (1) the inflation era is described by a quasi de-Sitter evolution of the Hubble parameter, which has an exit at around 58 e-folding number, (2) the inflationary observable quantities like the spectral index for primordial scalar perturbation and the tensor-to-scalar ratio are simultaneously compatible with the recent Planck data, and (3) regarding the late time cosmology, the dark energy EoS parameter is found to be consistent with the Planck result for the same values of the entropy parameters that lead to the viable inflation during the early universe. Furthermore, we show that the entropic cosmology from the proposed entropy function is equivalent to holographic cosmology, where the respective holographic cut-offs are determined in terms of either particle horizon and its derivative or future horizon and its derivative.

On the number of e-folds in the Jordan and Einstein frames

We investigate how inflationary predictions are affected by the difference in the number of e-folds between the Jordan and Einstein frames. We study several test models in relation to a Jordan frame defined by the common Higgs-inflation-like non-minimal coupling to gravity and consider two different formulations of gravity: metric and Palatini. We find that the difference is quite contained in case of a metric Jordan frame while can be quite remarkable in case of a Palatini Jordan frame. We also discuss a way to overcome the discrepancy by using a frame invariant physical distance and a consequently frame invariant number of e-folds.

A Natural Inflation inspired model

We propose a modification of the Natural Inflation (NI) potential in such a way that the spontaneous symmetry breaking scale $f$ can take values less than one (in Planck units). The proposed potential seems simple enough, however, its consequences are difficult to calculate analytically. Therefore, we illustrate the feasibility of the model by considering some numerical examples that easily satisfy the conditions imposed on the observables $n_s$ and $r$ by the most recent observations, while at the same time maintaining the number of e-folds during the inflationary epoch within the expected range.

Dynamical torsion suppression in Brans–Dicke inflation and Lorentz violation

Paul and SenGupta (Eur Phys J C 79:591, 2019) have recently shown that the reason why torsion effects are not present in today s universe is that they are suppressed in the bulk of a 5D braneworld. In this case only curvature effects remain. In this paper our goals are to investigate how this idea applies in the 4D Brans–Dicke inflation in two cases: First in the spin-torsion energy density scaling as sim {a^{-6}}, whereas matter density scales at a^{-3}. Here a(t) is the cosmic evolution parameter in the cosmic time t. In this case when a is given by the de Sitter expansion spin-torsion density it is highly suppressed. the second is the inflation in the framework of Einstein–Cartan–Brans–Dicke (ECBD) gravity used here, to build an effective inflaton potential and computing the number of its e-folds. Spin-torsion density contributes to a decrease on the number of e-folds when the ratio of spin-torsion density to matter density is not suppressed as in Early Universe. At present universe it is easy to show that by the end of inflation, spin-torsion density is suppressed with respect to matter-curvature density with a ratio of a^{-3} or t^{-3} by the end of inflation. When torsion possesses dynamical degrees of freedom, contortion is also shown to be suppressed by the end of BD inflation. The Kalb–Ramond fields in the 5D braneworld are equivalent to torsion. At present universe we find Ksim {10^{-39}~hbox {GeV}} for contortion. Such an estimate is much more stringent than the value found by Russell et al. (Phys Rev Lett 100, 2008) from torsion Lorentz violation (LV) with dual masers experiment.

A simple parametrisation for coupled dark energy

As an alternative to the popular parametrisations of the dark energy equation of state, we construct a quintessence model where the scalar field has a linear dependence on the number of e-folds. Constraints on more complex models are typically limited by the degeneracies that increase with the number of parameters. The proposed parametrisation conveniently constrains the evolution of the dark energy equation of state as it allows for a wide variety of time evolutions. We also consider a non-minimal coupling to cold dark matter. We fit the model with Planck and KiDS observational data. The CMB favours a non-vanishing coupling with energy transfer from dark energy to dark matter. Conversely, gravitational weak lensing measurements slightly favour energy transfer from dark matter to dark energy, with a substantial departure of the dark energy equation of state from -1.