Number Of E-folds Research Articles

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.

Study of warm inflation using irreversible thermodynamical description within Rastall gravity

This manuscript aims to study warm inflation via “irreversible thermodynamics of open systems” with matter creation/decay within Rastall theory of gravity. Interacting scalar field and radiation are assumed to be the components of cosmological fluid in spatially flat FRW universe model. Considering early universe as an open system, and implementing the thermodynamics along with the dynamical equations in Rastall gravity on the interacting cosmological fluid leads to modify the standard formalism of warm inflationary model, including the creation(decay) pressure, which is considered as part of the energy–momentum tensor explicitly. The analysis is done using constant as well as Higgs potential. Under slow-roll approximation and introducing some dimensionless variables, numerical solutions of the thermodynamical equations (scale factor, number of scalar field particles, energy densities of scalar field and radiation and temperature) are obtained and represented graphically. For appropriate set of initial conditions, these results depict the phase transition from acceleration to deceleration phase. Using obtained solutions, we calculate inflationary observable like “slow-roll parameters, number of e-folds, scalar/tensor power spectra, scalar/tensor spectral indices and tensor-to-scalar ratio”. By constraining the free model parameters, the theoretical predictions of the under lying model are compared with the Planck-2018 data.

Quintessential constant-roll inflation

We investigate a single field model in the context of the constant-roll inflation in which inflaton moves down to the minimum point of the potential with a constant rate of rolling. We use a quintessential inflationary model obtained by a Lorentzian function which is dependent on the number of e-folds. We present the inflationary analysis for the model and find the observational constraints on the parameters space using the observations of CMB anisotropies i.e. the Planck and Keck/array datasets. We find the observationally acceptable values of the Width of the Lorentzian function as 0.3<Γ≤0.5 at the 68% CL and Γ≤0.3 at the 95% CL when ξ=120, |β|=0.02 and N=60. Also, we acquire the observationally favoured values of the amplitude of the Lorentzian function as 400<ξ≤600 at the 68% CL and ξ≤400 at the 95% CL when Γ=0.1, |β|=0.02 and N=60. Moreover, we study the model from the Weak Gravity Conjecture approach using the swampland criteria.

Characterizing the postinflationary reheating history: Single daughter field with quadratic-quadratic interaction

We study the evolution of the energy distribution and equation of state of the Universe from the end of inflation until the onset of either radiation domination (RD) or a transient period of matter domination (MD). We use both analytical techniques and lattice simulations. We consider two-field models where the inflaton $\Phi$ has a monomial potential after inflation $V(\Phi) \propto |\Phi - v|^p$ ($p\geq2$), and is coupled to a daughter field $X$ through a quadratic-quadratic interaction $g^2\Phi^2 X^2$. We consider two situations, depending on whether the potential has a minimum at $i)$ $v = 0$, or $ii)$ $v > 0$. In the scenario $i)$, the final energy transferred to $X$ is independent of $g^2$ and entirely determined by $p$: it is negligible for $p < 4$, and of order $\sim 50\%$ for $p \geq 4$. The system goes to MD at late times for $p = 2$, while it goes to RD for $p > 2$. In the later case, we can calculate exactly the number of e-folds until RD as a function of $g^2$, and hence predict accurately inflationary observables like the scalar tilt $n_s$ and the tensor-to-scalar ratio $r$. In the scenario $ii)$, the energy is always transferred completely to $X$ for $p>2$, as long as its effective mass $m_X^2 = g^2(\Phi-v)^2$ is not negligible. For $p=2$, the final ratio between the energy densities of $X$ and $\Phi$ depends strongly on $g^2$. For all $p \ge 2$, the system always goes to MD at late times.

An exact solution approach to warm inflation using Tsallis and Barrow holographic dark energy entropy within Rastall gravity.

This paper studies the phenomena of warm inflation in a modified cosmological scenario within Rastall gravity. In this context, we modify the standard Friedmann equations using recently proposed Tsallis and Barrow holographic dark energy entropies, alternatively. For both entropies, the exact solutions including the inflaton field, potential required to produce inflation and the scale factor are obtained under slow-roll approximation for low- and high-dissipative regimes. We obtain exponentially growing scale factors for both dark energy models. To confront the theoretical predictions of these models with recent observational data, we calculate the slow-roll parameters, number of e-folds, scalar spectral index, running of scalar spectral index and the tensor-to-scalar ratio for both extremes of dissipation. The Planck bounds on inflationary trajectories are being used to constraint the free parameters. The results are interesting and physically viable up to 2sigma level.

Sliding naturalness: cosmological selection of the weak scale

We present a cosmological solution to the electroweak hierarchy problem. After discussing general features of cosmological approaches to naturalness, we extend the Standard Model with two light scalars very weakly coupled to the Higgs and present the mechanism, which we recently introduced in a companion paper to explain jointly the electroweak hierarchy and the strong-CP problem. In this work we show that this solution can be decoupled from the strong-CP problem and discuss its possible implementations and phenomenology. The mechanism works with any standard inflationary sector, it does not require weak-scale inflation or a large number of e-folds, and does not introduce ambiguities related to eternal inflation. The cutoff of the theory can be as large as the Planck scale, both for the Cosmological Constant and for the Higgs sector. Reproducing the observed dark matter relic density fixes the couplings of the two new scalars to the Standard Model, offering a target to future axion or fifth force searches. Depending on the specific interaction of the scalars with the Standard Model, the mechanism either yields rich phenomenology at colliders or provides a novel joint solution to the strong-CP problem. We highlight what predictions are common to most realizations of cosmological selection of the weak scale and will allow to test this general framework in the near future.

Warm Higgs\u2013Starobinsky inflation

In this work, we investigate the Higgs–Starobinsky (HS) model in the context of warm inflation scenario. The dissipative parameter as a linear form of temperature of warm inflation is considered with strong and weak regimes. We study the HS model in the Einstein frame using the slow-roll inflation framework. The inflationary observables are computed and then compared with the Plank 2018 data. With the sizeable number of e-folds and proper choices of parameters, we discover that the predictions of warm HS model present in this work are in very good agreement with the latest Planck 2018 results. More importantly, the parameters of the HS model are also constrained by using the data in order to make warm HS inflation successful.

Swampland conjecture and inflation model from brane perspective

Over the past few decades, inflation models have been studied by researchers from different perspectives and conditions in order to introduce a model for the expanding universe. In this paper, we introduce a modified f(R) gravitational model as (R + γ R p ) in order to examine a new condition for inflation models. Given that our studies are related to a modified f(R) gravitational model on the brane, therefore we will encounter modified cosmological parameters. So, we first introduce these modified cosmological parameters such as spectral index, a number of e-folds and etc. Then, we apply these conditions to our modified f(R) gravitational model in order to adapt to the swampland criteria. Finally, we determine the range of each of these parameters by plotting some figures and with respect to observable data such as Planck 2018.

Reheating constraints on k -inflation

In this work we revisit constraints on K-inflation with DBI kinetic term and power-law kinetic term from reheating. For DBI kinetic term we choose monomial potentials, $V\propto \phi^n$ with $n=2/3\,,1\,,\,2$ and $4$, and natural inflaton potential, and for power-law kinetic term we choose quadratic, quartic and exponential potentials. The phase of reheating can be parameterized in terms of reheating temperature $T_{re}$, number of e-folds during reheating $N_{re}$ and effective equation of state during reheating $w_{re}$. These parameters can be related to the spectral index $n_s$ and other inflationary parameters depending on the choice of inflaton kinetic term and potential. By demanding that $w_{re}$ should have a finite range and $T_{re}$ should be above electroweak scale, one can obtain the bounds on $n_s$ that can provide bounds on tensor-to-scalar ratio $r$. We find, for K-inflation with DBI kinetic term and quadratic and quartic potentials, that the upper bound on $r$ for physically plausible value of $0\le w_{re} \le 0.25$ is slightly larger than the Planck-2018 and BICEP2/Keck array bound, and for $n=2/3$ and $1$, the reheating equation of state should be less than $0$ to satisfy Planck-2018 joint constraints on $n_s$ and $r$. However, natural inflation with DBI kinetic term is compatible with Planck-2018 bounds on $r$ and joint constraints on $n_s$ and $r$ for physically plausible range $0 \le w_{re}\le 0.25$. The quadratic and quartic potential with power-law kinetic term are also compatible with Planck-2018 joint constraints on $n_s$ and $r$ for $0\le w_{re} \le 1$. However, for exponential potential with power-law kinetic term, the equation of state during reheating $w_{re}$ should be greater than $1$ for $r-n_s$ predictions to lie within $68\%$C.L. of joint constraints on $n_s$ and $r$ from Planck-2018 observations.

Quantum aspects of chaos and complexity from bouncing cosmology: A study with two-mode single field squeezed state formalism

Circuit Complexity, a well known computational technique has recently become the backbone of the physics community to probe the chaotic behaviour and random quantum fluctuations of quantum fields. This paper is devoted to the study of out-of-equilibrium aspects and quantum chaos appearing in the universe from the paradigm of two well known bouncing cosmological solutions viz. Cosine hyperbolic and Exponential models of scale factors. Besides circuit complexity, we use the Out-of-Time Ordered correlation (OTOC) functions for probing the random behaviour of the universe both at early and the late times. In particular, we use the techniques of well known two-mode squeezed state formalism in cosmological perturbation theory as a key ingredient for the purpose of our computation. To give an appropriate theoretical interpretation that is consistent with the observational perspective we use the scale factor and the number of e-foldings as a dynamical variable instead of conformal time for this computation. From this study, we found that the period of post bounce is the most interesting one. Though it may not be immediately visible but an exponential rise can be seen in the complexity once the post bounce feature is extrapolated to the present time scales. We also find within the very small acceptable error range a universal connecting relation between Complexity computed from two different kinds of cost functionals-linearly weighted and geodesic weighted with the OTOC. Furthermore, from the complexity computation obtained from both the cosmological models under consideration and also using the well known Maldacena (M) Shenker (S) Stanford (S) bound on quantum Lyapunov exponent, \lambda\leq 2\pi/\betaλ≤2π/β for the saturation of chaos, we estimate the lower bound on the equilibrium temperature of our universe at the late time scale. Finally, we provide a rough estimation of the scrambling time scale in terms of the conformal time.

Chaotic inflation and reheating in generalized scalar-tensor gravity

In the present work, we study slow-roll inflation in scalar-tensor gravity theories in the presence of both the non-minimal coupling between the scalar field and curvature, and the Galileon self-interaction of the scalar field. Furthermore, we give predictions for the duration of reheating as well as for the reheating temperature after inflation. After working out the expressions for the power spectra of scalar and tensor perturbations in the case of a general non-minimal coupling function that depends solely on the scalar field and a general scalar potential, we focus on the special cases of the power-law coupling function and chaotic quadratic inflation. Thus, under the slow-roll approximation we confront the predictions of the model with the current PLANCK constraints on the spectral index n s and the tensor-to-scalar ratio r using the n s-r plane. We found that the combination of the non-minimal coupling and Galileon self-interaction effects allows us to obtain better results for r than in the case in which each effect is considered separately. Particularly, we obtained that the predictions of the model are in agreement with the current observational bounds on n s and r within the 95 % C.L region and also slightly inside the 68 % C.L region. Also, we investigate the oscillatory regime after the end of inflation by solving the full background equations, and then we determine the upper bound for the Galileon and non-minimal coupling parameters under the condition that the scalar field oscillates coherently during reheating. Finally, after approximating reheating by a constant equation of state, we derive the relations between the reheating duration, the temperature at the end of reheating, its equation of state, and the number of e-folds of inflation and then we relate them all to the inflationary observables.

On the α-attractor T-models

We carry out a fully analytical study of the phenomenology of α-attractor T-models defined by the potential V = V0tanhp(λϕ/M_pl). We obtain expressions for the number of e-folds during inflation Nke in terms of the scalar spectral index ns and independently in terms of the tensor-to-scalar ratio r. From these expressions we obtain exact solutions for both ns and r in terms of Nke along with their expansions for large Nke, in full agreement with known expressions. Eliminating the parameter λ from the model in terms of ns and r we can obtain exact solutions for r in terms of ns and Nke which allows us to reproduce, in particular, numerical solutions presented by the Planck Collaboration for the monomial potentials. We explicitly show how these solutions are contained in the solutions for the α-attractors and are also the end points of these. Finally, by also eliminating the global scale V0 in terms of the observables ns and r we show how in the appropriate limit the α-attractor potential exactly reduces to the monomials potential. We also briefly show that for α-attractor E-models, which generalize the Starobinsky potential in the Einstein frame, a similar transition occurs.

Multi-field inflation from single-field models

Quantization implies independent degrees of freedom that do not appear in the classical theory, given by fluctuations, correlations, and higher moments of a state. A systematic derivation of the resulting dynamical systems is presented here in a cosmological application for near-Gaussian states of a single-field inflation model. As a consequence, single-field Higgs inflation is made viable observationally by becoming a multi-field model with a specific potential for a fluctuation field interacting with the inflaton expectation value. Crucially, non-adiabatic methods of semiclassical quantum dynamics reveal important phases that can set suitable initial conditions for slow-roll inflation (in combination with the uncertainty relation), and then end inflation after the observationally preferred number of e-folds. New parameters in the interaction potential are derived from properties of the underlying background state, demonstrating how background non-Gaussianity can affect observational features of inflation or, conversely, how observations may be used to understand the quantum state of the inflaton.

Reconstructing inflation in scalar-torsion f(T,phi ) gravity

It is investigated the reconstruction during the slow-roll inflation in the most general class of scalar-torsion theories whose Lagrangian density is an arbitrary function f(T,phi ) of the torsion scalar T of teleparallel gravity and the inflaton phi . For the class of theories with Lagrangian density f(T,phi )=-M_{pl}^{2} T/2 - G(T) F(phi ) - V(phi ), with G(T)sim T^{s+1} and the power s as constant, we consider a reconstruction scheme for determining both the non-minimal coupling function F(phi ) and the scalar potential V(phi ) through the parametrization (or attractor) of the scalar spectral index n_{s}(N) and the tensor-to-scalar ratio r(N) as functions of the number of e-folds N. As specific examples, we analyze the attractors n_{s}-1 propto 1/N and rpropto 1/N, as well as the case rpropto 1/N (N+gamma ) with gamma a dimensionless constant. In this sense and depending on the attractors considered, we obtain different expressions for the function F(phi ) and the potential V(phi ), as also the constraints on the parameters present in our model and its reconstruction.

Gauge-field-induced torsion and cosmic inflation

In this paper, inflation in the framework of Einstein–Cartan theory is revisited. Einstein–Cartan theory is a natural extension of the General Relativity with nonvanishing torsion. The connection on Riemann–Cartan space–time is only compatible with the cosmological principal for a particular form of torsion. We also show this form to be compatible with gauge invariance principle for non-Abelian and Abelian gauge fields under a certain deviced coupling procedure. We adopt an Abelian gauge field in the form of “cosmic triad”. The dynamical field equations are obtained and shown to sustain cosmic inflation with a large number of e-folds. We emphasize that at the end of inflation, torsion vanishes and the theory of Einstein–Cartan reduces to the General Relativity with the usual FRW geometry.

Inflation in mimetic [formula omitted] gravity

In this paper, we employ mimetic f(R,T) gravity coupled with Lagrange multiplier and mimetic potential to yield viable inflationary cosmological solutions consistent with latest Planck and BICEP2/Keck Array data. We present here three viable inflationary solutions of the Hubble parameter (H) represented by H(N)=(AexpβN+BαN)γ, H(N)=(AαN+BlogN)γ, and H(N)=(AeβN+BlogN)γ, where A, β, B, α, γ are free parameters, and N represents the number of e-foldings. We carry out the analysis with the simplest minimal f(R,T) function of the form f(R,T)=R+χT, where χ is the model parameter. We report that for the chosen f(R,T) gravity model, viable cosmologies are obtained compatible with observations by conveniently setting the Lagrange multiplier and the mimetic potential.

Noncommutative effective loop quantum cosmology: Inclusion of a potential term

We construct and study a simple noncommutative scheme (theta-deformation) for the effective Loop Quantum Cosmology of the flat Friedmann-Lema\^itre-Robertson-Walker model in the presence of a homogeneous scalar field $\phi$ with a potential $\mathcal{V}(\phi)=\frac{1}{2}m^2\phi^2$. We first conduct a simple analysis from the corresponding Hamilton equations of motion considering a generic term $\mathcal V(\phi)$. It is observed that the characteristic Big Bounce of Loop Quantum Cosmology is preserved under such noncommutative extension. When specializing to the quadratic case, numerical solutions to the corresponding Hamilton equations exhibiting an early inflationary epoch with a sufficiently large number of e-foldings are found. It is concluded that, in this noncommutative setup, solutions exist which are in the overall compatible with the early universe predicted by standard (effective) Loop Quantum Cosmology (i.e. a bouncing and inflationary early universe). The issue of the genericness of a sufficiently long inflationary period on the space of solutions in this noncommutative construct remains to be addressed.

On Viability of Inflation in Nonminimal Kinetic Coupling Theory

We consider initial conditions for inflation in non-minimal kinetic coupling theory. If inflation is driven solely by kinetic term with no potential, the resulting number of e-folds depends only upon initial velocity of the scalar field $\dot \phi$. We write down the expression for the number of e-folds explicitly, and show that for physically reasonable values of the coupling constant, we can get 60 e-folds only for exponentially big initial $\dot \phi$. When the scalar field potential is taken into account, on the other hand, we need not very large $\dot \phi$ to start with for 60 e-folds, on the other hand, some initial condition lead to physically inadmissible eternal inflation. We show numerically that in the measure used in the present paper only smaller part of initial conditions lead to eternal inflation for reasonable values of the coupling constant.