Number Of E-folds Research Articles

The open effective field theory of inflation

Abstract In our quest to understand the generation of cosmological perturbations, we face two serious obstacles: we do not have direct information about the environment experienced by primordial perturbations during inflation, and our observables are practically limited to correlators of massless fields, heavier fields and derivatives decaying exponentially in the number of e-foldings. The flexible and general framework of open systems has been developed precisely to face similar challenges. Building on previous work, we develop a Schwinger-Keldysh path integral description for an open effective field theory of inflation, describing the possibly dissipative and non-unitary evolution of the Goldstone boson of time translations interacting with an unspecified environment, under the key assumption of locality in space and time. Working in the decoupling limit, we study the linear and interacting theory in de Sitter and derive predictions for the power spectrum and bispectrum that depend on a finite number of effective couplings organised in a derivative expansion. The smoking gun of interactions with the environment is an enhanced but finite bispectrum close to the folded kinematical limit. We demonstrate the generality of our approach by matching our open effective theory to an explicit model. Our construction provides a standard model to simultaneously study phenomenological predictions as well as quantum information aspects of the inflationary dynamics.

Reheating Constraints and the H0 Tension in Quintessential Inflation

In this work, we focus on two important aspects of modern cosmology: reheating and Hubble constant tension within the framework of a unified cosmic theory, namely the quintessential inflation connecting the early inflationary era and late-time cosmic acceleration. In the context of reheating, we use instant preheating and gravitational reheating, two viable reheating mechanisms when the evolution of the universe is not affected by an oscillating regime. After obtaining the reheating temperature, we analyze the number of e-folds and establish its relationship with the reheating temperature. This allows us to connect, for different quintessential inflation models (in particular for models coming from super-symmetric theories such as α-attractors), the reheating temperature with the spectral index of scalar perturbations, thereby enabling us to constrain its values. In the second part of this article, we explore various alternatives to address the H0 tension. From our perspective, this tension suggests that the simple Λ-Cold Dark Matter model, used as the baseline by the Planck team, needs to be refined in order to reconcile its results with the late-time measurements of the Hubble constant. Initially, we establish that quintessential inflation alone cannot mitigate the Hubble tension by solely deviating from the concordance model at low redshifts. The introduction of a phantom fluid, capable of increasing the Hubble rate at the present time, becomes a crucial element in alleviating the Hubble tension, resulting in a deviation from the Λ-Cold Dark Matter model only at low redshifts. On a different note, by utilizing quintessential inflation as a source of early dark energy, thereby diminishing the physical size of the sound horizon close to the baryon–photon decoupling redshift, we observe a reduction in the Hubble tension. This alternative avenue, which has the same effect of a cosmological constant changing its scale close to the recombination, sheds light on the nuanced interplay between the quintessential inflation and the Hubble tension, offering a distinct perspective on addressing this cosmological challenge.

Non-extensive entropy and power-law inflation: Implications for observations

This study explores the interaction between non-extensive entropic FLRW cosmology and the power-law inflationary model, with a focus on the overlap between the scalar spectral index “[Formula: see text]” and the tensor-to-scalar ratio “r”. Based on a conjecture that non-extensive entropy alters the energy–momentum content of the cosmic fluid, the analysis examines how these overlaps shift with different model parameters and compares the findings to those from Bekenstein–Hawking (BH) entropic cosmology. The study highlights the impact of Tsallis, Rényi, and Sharma–Mittal entropies, uncovering a significant correlation between “[Formula: see text]” and “r” that suggests a deeper connection in power-law inflationary dynamics. The results demonstrate that non-extensive entropies not only enable viable inflation with a graceful exit but also address limitations inherent in the standard BH entropic framework, emphasizing the importance of precise parameter estimation. Specifically, Tsallis entropy allows for power-law inflation with [Formula: see text] to [Formula: see text] in alignment with Planck 2018 data. Moreover, the [Formula: see text] parameter in Rényi and Sharma–Mittal entropy models must be extremely small ([Formula: see text] in Planck mass units) to achieve successful power-law inflation with an e-folding number around 55–65, suggesting a unified thermodynamic perspective in cosmological studies.

Study of cosmic evolution admitting thermodynamic analysis

This article examines the cosmic evolution in the framework of symmetric teleparallel theory, characterized by the function of non-metricity scalar (Q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mathcal {Q})$$\\end{document}. We use the e-folding number and reconstruction method with a suitable parametrization of the scale factor to obtain the functional form of symmetric teleparallel theory. Using this reconstructed model, we examine the behavior of different cosmographic parameters to demonstrate the bouncing scenarios of the cosmos by considering the contraction and expansion phases of cosmos before and after the bouncing point, respectively. It is found that the null energy condition is violated which shows that the singularity issue can be resolved in this extended theoretical framework. Moreover, we observe that the acceleration occurs near the bouncing point and the reconstructed model aligns with the current cosmic expansion. Finally, we check the validity of second law of thermodynamics in the bouncing framework of our model.

Analyzing inflationary parameters and swampland conjectures of modified cosmology according to various entropies

This paper investigates the inflationary phase of the FRW universe within the context of generalized Chaplygin gas. We explore modified cosmological scenarios incorporating two entropy corrections, namely Tsallis and Barrow. These proposed entropies lead to modifications in the Friedmann equations. By incorporating the generalized Chaplygin gas into these equations, we determine various parameters including the slow roll parameters, number of e-folds, curvature perturbation, tensor-to-scalar ratio, and spectral index. We observe variations in the values of the tensor-to-scalar ratio r and the scalar spectral index ns with respect to the Tsallis and Barrow parameters. Additionally, we plot the r−ns plane. For the Tsallis entropy, we obtain r≤0.012, r≤0.027, and r≤0.008, with ns lying between (0.961,0.964), (0.957,0.966), and (0.956,0.964), respectively. In Barrow cosmology, we find r≤0.0037, r≤0.12, and r≤0.010, while ns falls within (0.92,0.97), (0.855,0.995), and (0.960,0.966) for the chaotic, SFI, and exponential potentials, respectively. These graphical analyses demonstrate the compatibility of our models with the latest Planck data. Moreover, by considering these entropy corrections, we determine the bounds of the Swampland de Sitter conjecture for generalized Chaplygin gas. In all cases, this bound remains less than unity, satisfying the required condition for the Swampland de Sitter conjecture.

Analysis of initial singularity admitting viable bounce models

This article examines the cosmic evolution in the framework of symmetric teleparallel theory, characterized by the function of non-metricity scalar (Q). We use the e-folding number and reconstruction method with a suitable parametrization of the scale factor to obtain the functional form of symmetric teleparallel theory. Using this reconstructed model, we examine the behavior of different cosmographic parameters to demonstrate the bouncing scenarios of the cosmos by considering the contraction and expansion phases of cosmos before and after the bouncing point, respectively. It is found that the null energy condition is violated which shows that the singularity issue can be resolved in this extended theoretical framework. Moreover, we observe that the acceleration occurs near the bouncing point and the reconstructed model aligns with the current cosmic expansion. Finally, we check the validity of second law of thermodynamics in the bouncing framework of our model.

A thermodynamic model of inflation without inflaton field

In the framework of geometrothermodynamics, we explore how to construct an inflationary cosmic fluid without requiring inflaton fields. To do so, we employ standard radiation and matter as barotropic background, speculating on the existence of a further real fluid, interpreted as a possible matter with pressure constituent. We thus illustrate its main characteristics, showing how it may face the flatness and horizon problems, yielding an equivalent number of e-foldings, compatible with theoretical expectations. Moreover, we calculate the associated equilibrium and kinetic temperatures, and compute out-of-equilibrium properties, in terms of particle production and effective reheating mechanism. Accordingly, we revise how inflation ends and how to fix the free parameters of our model from observations. Last but not least, we propose how to obtain a bare cosmological constant that could relate the late-time acceleration to inflation, aiming to unify the two approaches. Finally, we discuss the corresponding theoretical implications of our treatment in contrast to the standard approach reliant on scalar field potentials, emphasizing the main advantages and disadvantages.

Loop correction and resummation of vertex functions for a self interacting scalar field in the de Sitter spacetime

We consider a massless and minimally coupled self interacting quantum scalar field theory in the inflationary de Sitter background of dimension four. The self interaction potential is taken to be either quartic, λϕ4/4!, or quartic plus cubic, λϕ4/4!+βϕ3/3! (λ>0). We compute the four and three point vertex functions up to two loop. The purely local or partly local part of these renormalised loop corrected vertex functions grow unboundedly after sufficient number of de Sitter e-foldings, due to the appearances of secular logarithms. We focus on the purely local part of the vertex functions and attempt a resummation of them in terms of the dynamically generated mass of the scalar field at late times. Such local logarithms have sub-leading powers compared to the non-local leading ones which can be resummed via the stochastic formalism. The variation of these vertex functions are investigated with respect to the tree level couplings numerically. Since neither the secular effect, nor the dynamical generation of field mass is possible in the Minkowski spacetime, the above phenomenon has no flat spacetime analogue. We have also compared our result with the ones that could be found via the recently proposed renormalisation group techniques. All these results suggest that at late times the value of the non-perturbative vertex function should be less than the tree level coupling.

Impact of dark sector preheating on CMB observables

The prediction of a nearly scale-invariant spectrum of curvature and tensor fluctuations is among the main features of cosmic inflation. The current measurements of the primordial fluctuations in the cosmic microwave background (CMB) provide tight constraints on the amplitude of the scalar and tensor spectra, and the scalar tilt. However, the precise connection between these observables and a given inflationary model, depends on the expansion history between the end of inflation and the beginning of the radiation dominated era, which corresponds to the reheating epoch. This mapping between horizon exit and reentry of fluctuations, parametrized by the number of e-folds N*, can therefore be affected by the presence of a transient epoch of non-perturbative particle production during reheating (preheating). Using a combination of perturbative and lattice computations, we quantify the impact of preheating in a non-equilibrated dark matter sector on the CMB observables, under the assumption of a simultaneous perturbative decay of the inflaton into Standard Model particles. Combined with structure formation constraints, this allows us to impose stringent bounds on the post-inflationary reheating temperature.

Loop blow-up inflation

We present a new model of string inflation driven by a blow-up Kähler modulus of type IIb compactifications with a potential generated by string loops. Slow-roll is naturally realized thanks to the fact that the blow-up mode is a leading-order flat direction lifted by string loops which are unavoidable and generate a plateau at large field values. We check that throughout the whole inflationary dynamics the effective field theory is under control. We perform a phenomenological analysis determining the exact number of efoldings by studying the post-inflationary evolution. We determine the values of the microscopic parameters which lead to agreement with CMB data, together with the prediction of a tensor-to-scalar ratio of order r ∼ 10−5.

Warm inflation triggered by entropies of some recent dark energy models within gravity

This manuscript aims to study cosmic warm inflation (WI) in the framework of - gravity, where Q represents the nonmetricity (NM) scalar. To accomplish this task, we introduce the Tsallis, Renyi, and Barrow holographic dark energy (HDE) entropies into the standard Friedmann equations. Utilizing the slow-roll (SR) approximation, we find exact analytic solutions for the inflaton field, the effective potential necessary to produce inflation, and the scale factor for both low- and high-dissipative regimes. We calculate key parameters, including SR parameters, the number of e-folds, the scalar spectral index and its running, and finally tensor-to-scalar ratio to assess the accuracy of the chosen DE models in light of the published observational data. The allowed ranges of the involved free parameters are found from the limits on inflationary observables imposed by the Planck data. It is concluded that the obtained results are consistent with proposed theoretical predictions up to the confidence level.

Introduction to the Number of e-Folds in Slow-Roll Inflation

In this review, a pedagogical introduction to the concepts of slow-roll inflationary universe and number of e-folds is provided. In particular, the difference between the basic notion of e-folds (Ne) the total number of e-folds (NT) and the number of e-folds before the end of inflation (N) is outlined. The proper application of the number of e-folds before the end of inflation is discussed both as a time-like variable for the scalar field evolution and as a key parameter for computing inflationary predictions.

Post-inflationary leptogenesis and dark matter production: metric versus Palatini formalism

We investigate production of non-thermal dark matter particles and heavy sterile neutrinos from inflaton during the reheating era, which is preceded by a slow-roll inflationary epoch with a quartic potential and non-minimal coupling (ξ) between inflaton and gravity. We compare our analysis between metric and Palatini formalism. For the latter, the tensor-to-scalar ratio, r, decreases with ξ. We find that for ξ = 0.5 and number of e-folds ~ 60, r can be as small as ~ O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document} (10−3) which may be validated at future reaches of upcoming CMB observation such as CMB-S4 etc. We identify the permissible range of Yukawa coupling yχ between inflaton and fermionic DM χ, to be O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document} (10−3.5) ≳ yχ ≳ O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document} (10−20) for metric formalism and O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document} (10−4) ≳ yχ ≳ O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document} (10−11) for Palatini formalism which is consistent with current PLANCK data and also within the reach of future CMB experiments. For the scenario of leptogenesis via the decay of sterile neutrinos produced from inflaton decay, we also investigate the parameter space involving heavy neutrino mass MN1 and Yukawa coupling yN1 of sterile neutrino with inflaton, which are consistent with current CMB data and successful generation of the observed baryon asymmetry of the universe via leptogenesis. In contrast to metric formalism, in the case of Palatini formalism, for successful leptogenesis to occur, we find that yN1 has a very narrow allowable range and is severely constrained from the consistency with CMB predictions.

Observational analysis on warm tachyon scalar field inflation via Woods–Saxon, valley hybrid, exponential, quartic, quasi exponential potentials

In this paper, we study the tachyon scalar field inflationary scenario considering FRW universe in context of modified Chaplygin gas model. For this purpose, we consider some well-known potential models such as Woods–Saxon potential , Valley hybrid potential, exponential potential, quartic potential, and Quasi exponential potential. We develop the relations for the number of e-folds N, slow roll parameters ϵ,η, scalar spectral index ns, scalar to tensor ratio r, and running scalar spectral index αs. To understand how the inflationary parameters behave, the planes of ns−r and ns−αs are plotted. Through these plotted planes, we check the consistency of these inflationary parameters for assumed potentials in light of the Planck 2018 observational data and compare our outcomes with these observation by incorporating observational contours. We found that the obtained results are compatible and consistent with the recent observational data.

Dynamics of Cosmological Scalar Fields Revisited

This paper reviews the dynamics of a single isotropic and homogeneous scalar field φ(t) in the context of cosmological models. A non-standard approach to the solution of the Einstein–Klein–Gordon equations is described which uses the scalar field as the evolution parameter for cosmic dynamics. General conclusions about the qualitative behaviour of the solutions can be drawn, and examples of how to obtain explicit solutions for some cosmological models of interest are given. For arbitrary potentials, analytical results can be obtained from the slow-roll approximation by using a series expansion for the Hubble parameter H[φ], from which a quantitative estimate for the number of e-folds of expansion is obtained. This approach is illustrated with the examples of quadratic potentials and hilltop models, with special consideration of Higgs-type potentials. The GUT-scale is shown to come out of such a model quite naturally. Finally, it is discussed how to find scalar potentials giving rise to a predetermined scalar-field behaviour and the associated evolution of the scale factor.

Stringy spacetime uncertainty principle and a modified trans-Planckian censorship criterion

We study the implications of the stringy spacetime uncertainty relation (STUR) for inflationary cosmology. By demanding that no fluctuation modes that exit the Hubble radius are affected by the nonlocality resulting from the STUR, we find an upper bound on the number of e-foldings of inflation. The bound is a factor of 2 weaker than what results from the trans-Planckian censorship criterion (TCC). By demanding that the inflationary phase is simultaneously consistent with STUR and sufficiently long for inflation to provide a causal explanation of structure on the scale of the current Hubble radius, we find an upper bound on the energy scale of inflation. The bound is less restrictive than what follows from the TCC, but it remains in conflict with canonical single-field inflation models. Published by the American Physical Society 2024

Bayesian analysis of a generalized Starobinsky model with reheating constraints

We study a generalization of the Starobinsky model adding a term of the form R 2p to the Einstien-Hilbert action. we take the power p as a parameter of the model and explore the constraints from CMB plus BAO data through a Bayesian analysis, thus exploring a range of values for the exponent parameter. We incorporate a reheating phase to the model through the background matter content (equation of state) and the duration of this period (number of e-foldings of reheating). We find that incorporating information from reheating imposes constraints on cosmological quantities, more stringent than the case of no reheating when tested with the Planck+BAO data. The inferred value of the exponent parameter is statistically consitent with p = 1, favoring the original Starobinsky potential. Moreover, we report tighter constraints on p and the number of e-folds in comparison with previous works. The obtained values for other inflationary observational parameters, such as the scalar spectral index ns and the scalar amplitude of perturbations As , are consistent with prior measurements. Finally we present the alternative use of consistency relations in order to simplify the parameter space and test the generalized Starobinsky potential even more efficiently.

Solution for cosmological observables in the Starobinsky model of inflation

This paper focuses on the Starobinsky model of inflation in the Einstein frame. We derive solutions for various cosmological observables, such as the scalar spectral index ns\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_s$$\\end{document}, the tensor-to-scalar ratio r and their runnings, as well as the number of e-folds of inflation, reheating, and radiation, with minimal assumptions. The impact of reheating on inflation is explored by constraining the equation of state parameter ωre\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _{re}$$\\end{document} at the end of reheating. An equation linking inflation with reheating is established, which is solved for the spectral index ns\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_s$$\\end{document}. Using consistency relations of the model, we determine the other observables while the number of e-folds during inflation Nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_k$$\\end{document}, and the number of e-folds during reheating Nre\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_{re}$$\\end{document} are determined by their respective formulas involving ns\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_s$$\\end{document}. We find remarkable agreement between the Starobinsky model and current measurements of the power spectrum of primordial curvature perturbations and the present bounds on the spectrum of primordial gravitational waves.

An approximate application of quantum gravity to the rotation problem

Arbitrary initial conditions allow solutions of Einstein’s field equations for General Relativity to have arbitrarily large relative rotation of matter and inertial frames. The ‘Rotation Problem’ is to explain why the measured relative rotation rate is so small. As it turns out, nearly any reasonable theory of quantum gravity can solve the rotation problem by phase interference. Even as early as about a quarter of a second after the initial simgularity, quantum cosmology would limit the cosmologies that contribute significantly to a path integral calculation to have relative rms rotation rates less than about 10−51 radians per year. Those calculations are based on using 50 e-foldings during inflation. For 55 or 60 e-foldings, the cosmologies contributing significantly to the path integral would have even smaller relative rotation rates. In addition, although inflation dominates the calculation, even if there had been no inflation, the cosmologies contributing significantly to the path integral would have relative rotation rates less than about 10−32 radians per year at about a quarter of a second after the initial singularity. These calculations are insensitive to the details of the theory of quantum gravity because the main factor depends only on the size of the visible Universe, the Planck time, the free-space speed of light, the Hubble parameter, and the number of e-foldings during inflation. These calculations use the Einstein–Hilbert action in quantum gravity, including large-scale relative rotation of inertial frames and the matter distribution, in which each ‘path’ is a cosmology with a different rms relative rotation rate. The calculations include inflation for 50, 55, and 60 e-foldings, and for values of the dependence of relative rotation rate on cosmological scale factor a as a −m for various values of m. The calculation shows that the action is an extremum at zero rms relative rotation rate.

Constraints on real scalar inflation from preheating using LATTICEEASY* *Wei Cheng was Supported by the Natural Science Foundation of Chongqing, China (CSTB2022NSCQ-MSX0432), the Science and Technology Research Project of Chongqing Education Commission (KJQN202200621), and the Chongqing Human Resources and Social Security Administration Program (D63012022005). Ruiyu Zhou was Supported by the Chongqing Natural Science Foundation (CSTB2022NSCQ-MSX0534). This work was supported in

In this paper, we undertake a detailed study of real scalar inflation using LATTICEEASY simulations to investigate preheating phenomena. Generally, the scalar inflation potential with non-minimal coupling can be approximated using a quartic potential. We observe that the evolutionary behavior of this potential remains unaffected by the coupling coefficient. Furthermore, the theoretical predictions for the scalar spectral index ( ) and tensor-to-scalar power ratio (r) are independent of this coefficient. Consequently, the coefficients of this model are not constrained by Planck observations. Fortunately, the properties of preheating after inflation provide a viable approach to examining these coefficients. Through LATTICEEASY simulations, we trace the evolution of particle number density, scale factor, and energy density during the preheating process. Subsequently, we derive the parameters, such as the energy ratio (γ) and the e-folding number of preheating ( ), which facilitate further predictions of and r. We successfully validate real scalar inflation model using preheating in LATTICEEASY simulations based on the analytical relationship between preheating and inflation models.