Scalar Spectral Index Research Articles

Primordial power spectra of cosmological fluctuations with generalized uncertainty principle and maximum length quantum mechanics

The existence of the cosmological particle horizon as the maximum measurable length $l_{max}$ in the universe leads to a generalization of the quantum uncertainty principle (GUP) to the form $\Delta x \Delta p \geq \frac{\hbar}{2}\frac{1}{1-\alpha\Delta x^2} $, where $\alpha\equiv l_{max}^{-2}$. The effects of this GUP on simple quantum mechanical systems has been shown recently by one of the authors\cite{Perivolaropoulos:2017rgq} to be extremely small (beyond current measurements) due to the extremely large scale of the current particle horizon. This is not the case in the Early Universe during the quantum generation of the inflationary primordial fluctuation spectrum. We estimate the effects of such GUP on the primordial fluctuation spectrum and on the corresponding spectral index. We generalize the field commutation (GFC) relation to $[\varphi(k),\pi_{\varphi}(k')]$=$i\delta(k-k')\frac{1}{1-\mu\varphi^2(k)}$, where $\mu\sim \alpha^2\equiv l_{max}^{-4}$ is a GFC parameter, $\varphi$ denotes a scalar field and $\pi_{\varphi}$ denotes its canonical conjugate momentum. We obtain the predicted primordial perturbation spectrum as $P_S(k)=P_S^{(0)}(k)\left(1+\frac{\bar{\mu}}{k}\right)$ where $\bar{\mu}\equiv\mu V_* \simeq \sqrt{\alpha}= l_{max}^{-1}$ (here $V_*\simeq l_{max}^3$ is the volume corresponding to $l_{max}$) and $P_S^{(0)}(k)$ is the standard primordial spectrum obtained in the context of the Heisenberg uncertainty principle ($\mu=0$). We show that the predicted scalar spectral index is $n_s=1-\lambda-\frac{\bar{\mu}}{k}$ where $\lambda$ is a slow-roll parameter. Using observational constraints on the scale dependence of the spectral index $n_s$ we show that the $2\sigma$ range of $\alpha$ corresponds to $l_{max}\gtrsim 10^{26} m $ which is of the same order as the current particle horizon.

Constraining chameleon field driven warm inflation with Planck 2018 data

We investigate warm inflationary scenario in which the accelerated expansion of the early Universe is driven by chameleon-like scalar fields. Due to the non-minimal coupling between the scalar field and the matter sector, the energy-momentum tensor of each fluid component is not conserved anymore, and the generalized balance equation is obtained. The new source term in the energy equation can be used to model warm inflation. On the other hand, if the coupling function varies slowly, the model reduces to the standard model used for the description of cold inflation. To test the validity of the warm chameleon inflation model, the results for warm inflationary scenarios are compared with the observational Planck2018 Cosmic Microwave Background data. In this regard, the perturbation parameters such as the amplitude of scalar perturbations, the scalar spectral index and the tensor-to-scalar ratio are derived at the horizon crossing in two approximations, corresponding to the weak and strong dissipative regimes. As a general result it turns out that the theoretical predictions of the chameleon warm inflationary scenario are consistent with the Planck 2018 observations.

Inflation driven by a holographic energy density

In this letter we study a model of inflation in which the inflationary regimen comes from a type of holographic energy density. In particular, we consider the Granda-Oliveros proposal for the holographic energy density, which contains two free dimensionless parameters, α and β. This holographic energy density is associated to the so-called Granda-Oliveros infrared cutoff (G-O cutoff). Additionally, since in the inflationary regimen the energy scales are very high, it is necessary to modify the G-O cutoff taking into account a correction due to the ultraviolet cutoff. In this way, we obtain an algebraic equation which implicitly includes the Hubble parameter (as a function of the e-folding number, N) and from this, we calculate the Hubble slow-roll parameters and the values of the inflationary observables: the scalar spectral index of the curvature perturbations and its running, the tensor spectral index and the tensor-to-scalar ratio. Finally, since the values for these inflationary observables are known (Planck 2018 observations), we present constraints on the parameters α and β to make this a viable model.

Chaotic inflation from the MSSM along flat D -term trajectory

Within the MSSM we propose the chaotic inflationary scenario in which the inflaton field is a combination of sleptons and the Higgs field states evolving along the $D$-term flat direction. In the inflation and postinflation reheating processes, a decisive role is played by the MSSM Yukawa superpotential. The vacuum energy during the inflationary era is mainly from the muonic Yukawa coupling, while the inflaton decay and subsequent reheating process dominantly proceeds due to the strange quark Yukawa term. Because of these, the presented scenario is predictive, and the results obtained agree well with cosmological observations. In particular, the scalar spectral index and the tensor-to-scalar ratio are, respectively, ${n}_{s}\ensuremath{\simeq}0.966$ and $r=0.00117$. The reheating temperature is found to be ${T}_{r}\ensuremath{\simeq}7.2\ifmmode\times\else\texttimes\fi{}{10}^{7}\text{ }\text{ }\mathrm{GeV}$.

Chaplygin gas inspired non-canonical scalar field warm inflation

We consider a warm inflation universe model in the presence of generalized Chaplygin gas and modified Chaplygin gas by taking Dirac-Born-Infeld non-canonical scalar field. For developing various inflationary parameters, we assume quadratic potential and quartic potential and Anti-de-Sitter warp factor for the Dirac-Born-Infeld inflation. We develop the number of e-folds, scalar as well as tensor power spectrum, scalar spectral index and tensor-scalar ratio in the terms of ϕ under slow-roll approximation by assuming the constant sound speed. We establish the graphical behavior of r−N,ns−N and α−N for both Chaplygin gas models. The results of r−N,ns−N and α−N are 0.972−0.02+0.02,−0.0005−0.001+0.001,0.0004−0.0004+0.0004 (for generalized Chaplygin gas with quadratic potential) and 0.97−0.05+0.05,0.0045−0.0055+0.0055,0.0045−0.0045+0.0045 (generalized Chaplygin gas with quartic potential), respectively. Also, the results of inflationary parameters are 0.97−0.02+0.02,0.0015−0.0025+0.0025,0.125−0.125+0.125 (for modified Chaplygin gas with quadratic potential) and 0.94−0.05+0.05,0.01−0.015+0.015,0.0185−0.0185+0.0185 (modified Chaplygin gas with quartic potential), respectively. It is found that these results are well matched with 68% and 95% CL constraints of the Planck 2018 TT,TE,EE+lowP+lensing data.

Reheating constraints on a two-field inflationary model

Once inflation comes to an end, the Universe enters a short stage named reheating. Making coherent oscillations about the minimum of its potential, the inflaton field transforms into the particles that occupy the Universe at later times. Yet, there is no direct cosmological observables corresponding to the reheating epoch. However, one can achieve some indirect limits, simply by taking into account cosmological evolution for observable CMB scales between the Hubble crossing time during inflation and the present time, for example. Moreover, since the reheating period takes place before the radiation dominated era, it can be modeled by an effective equation-of-state parameter, wreh. There is a limited range of reasonable values for wreh which can be regarded as the reheating period. In this paper, considering a two-field model, we investigate the reheating epoch so as to impose some constraints on the parameter space of the model. To do so, we are first required to obtain the reheating e-folds number as well as its temperature. These parameters can be written in terms of the scalar spectral index which let us perform numerical analysis in this model. Therefore, we can set some constraints on the reheating parameters which correspond to observationally viable values of the scalar spectral index.

Intermediate inflation with non-canonical scalar field in the low anisotropy Universe

The behavior of a non-canonical scalar field within an anisotropic Bianchi type I, spatially homogeneous Universe in the framework of the intermediate inflation will be studied. It will be examined on the condition that both the anisotropy and non-canonical sources come together if there is any improvement in compatibility with the observational data originated from Planck 2015. Based on this investigation, it can be observed that automatically a steep potential which can manage inflation in a better way will be obtained. Additionally, as a common procedure for an inflationary study, we shall try to calculate the related inflationary observables such as the amplitude of the scalar perturbations, scalar and tensor spectral indices, tensor-to-scalar ratio, the running spectral index and the number of e-folds. As an exciting part of our results, we will find that our model has a good consistency compared to data resulting from CMB and different Planck results. To justify our claims, the well-known canonical inflationary scenario in an anisotropic Bianchi type I Universe will also be evaluated.

Constant-roll approach to noncanonical inflation

The scenario of constant-roll inflation is studied where the inflaton is a scalar field with modified kinetic term, known as non-canonical scalar field. This modification leads to some changes in the slow-roll parameters, and also by taking the second slow-roll parameter as a constant, the differential equation for the Hubble parameter is altered as well. Assuming $\eta=\beta$ and reconsidering the perturbation equations makes it clear that there should be some modified terms in the scalar spectral index and amplitude of scalar perturbations. After finding the exact solution, the main perturbation parameters are obtained at the horizon crossing time. Then by plotting the $r-n_s$ diagram it is shown that for some specific values of $\beta$ one could find a good agreement with observational data. In addition, considering the attractor behavior of the model leads to this result that this feature could be appropriately satisfied.

Nonminimal inflation in supersymmetric GUTs with U(1)R × Zn symmetry

A supersymmetric hybrid inflation framework is employed to realize a class of nonminimal inflation models with [Formula: see text] global symmetry. This framework naturally incorporates models based on grand unified theories by avoiding the most commonly faced monopole problem. The predictions of inflationary observables, the scalar spectral index [Formula: see text]–0.966 and the tensor to scalar ratio [Formula: see text]–0.0045, are in perfect agreement with the Planck 2018 data. For sub-Planckian values of the field the [Formula: see text] symmetry is only allowed for [Formula: see text].

Slow-roll inflation with exponential potential in scalar-tensor models

A study of the slow-roll inflation for an exponential potential in the frame of the scalar-tensor theory is performed, where non-minimal kinetic coupling to curvature and non-minimal coupling of the scalar field to the Gauss-Bonnet invariant are considered. Different models were considered with couplings given by exponential functions of the scalar field, that lead to graceful exit from inflation and give values of the scalar spectral index and the tensor-to-scalar ratio in the region bounded by the current observational data. Special cases were found, where the coupling functions are inverse of the potential, that lead to inflation with constant slow-roll parameters, and it was possible to reconstruct the model parameters for given ns and r. In first-order approximation the standard consistency relation maintains its validity in the model with non-minimal coupling, but it modifies in presence of Gauss–Bonnet coupling. The obtained Hubble parameter during inflation, Hsim 10^{-5} M_p and the energy scale of inflation V^{1/4}sim 10^{-3} M_p, are consistent with the upper bounds set by latest observations.

Inflation from fermions with curvature-dependent mass

A model of inflation realization driven by fermions with curvature-dependent mass is studied. Such a term is derived from the Covariant Canonical Gauge Theory of gravity (CCGG) incorporating Dirac fermions. We obtain an initial de Sitter phase followed by a successful exit, and moreover, we acquire the subsequent thermal history, with an effective matter era, followed finally by a dark-energy epoch. This behavior is a result of the effective `weakening' of gravity at early times, due to the decreased curvature-dependent fermion mass. Through the perturbation level, we obtain the scalar spectral index and the tensor-to-scalar ratio, which are in agreement with the Planck observations. The efficiency of inflation from fermions with curvature-dependent mass, at both the background and perturbation level, reveals the capabilities of the scenario and makes it a good candidate for the description of nature.

Holographic inflation

We apply the holographic principle at the early universe, obtaining an inflation realization of holographic origin. Such a consideration has equal footing with its well-studied late-time application, and moreover the decrease of the horizons at early times naturally increases holographic energy density at inflationary scales. Taking as Infrared cutoff the particle or future event horizons, and adding a simple correction due to the Ultraviolet cutoff, whose role is non-negligible at the high energy scales of inflation, we result in a holographic inflation scenario that is very efficient in incorporating inflationary requirements and predictions. We first extract analytically the solution of the Hubble function in an implicit form, which gives a scale factor evolution of the desired e-foldings. Furthermore, we analytically calculate the Hubble slow-roll parameters and then the inflation-related observables, such as the scalar spectral index and its running, the tensor-to-scalar ratio, and the tensor spectral index. Confronting the predictions with Planck 2018 observations we show that the agreement is perfect and in particular deep inside the 1σ region.

Inflation model selection revisited

We update the constraints on the power spectra of primordial curvature perturbation and tensor perturbation including Planck data 2015 (P15) and recently released BICEP2/Keck data (BK15), Baryon Acoustic Oscillation data and the Type Ia supernovae data. We find that the upper limits of tensor-to-scalar ratio are $0.061$, $0.064$ and $0.072$ at $95\%$ confidence level (CL) in the $\Lambda$CDM+$r$, $\Lambda$CDM+$r$+$\alpha_s$ and $\Lambda$CDM+$r$+$\alpha_s$+$\beta_s$ models respectively, where $\alpha_s$ and $\beta_s$ are the running of scalar spectral index and running of running. The inflation model with a concave potential is favored at more than $95\%$ CL. In addition, parametrizing the slow-roll parameter $\epsilon\sim 1/N^p$, where $N$ is the e-folding number before the end of inflation and taken in the range of $[50,60]$ and $[14,75]$ respectively, we conclude that the inflation model with a monomial potential $V(\phi)\sim \phi^n$ is disfavored at more than $95\%$ CL, and both the Starobinsky inflation model and brane inflation model are still consistent with the data.

Inflation and leptogenesis in a U(1) -enhanced supersymmetric model

Motivated by the flavored Peccei-Quinn symmetry for unifying flavor physics and string theory, we investigate a supersymmetric extension of standard model (SM) for an explanation of inflation and leptogenesis by introducing $U(1)$ symmetries such that the $U(1)$-$[gravity]^2$ anomaly-free condition together with the SM flavor structure demands additional sterile neutrinos as well as no axionic domain-wall problem. Such additional neutrinos may play a crucial role as a bridge between leptogenesis and new neutrino oscillations along with high energy cosmic events. In a realistic moduli stabilization, we show that the moduli backreaction effect on the inflationary potential leads to the energy scale of inflation with the inflaton mass in a way that the power spectrum of the curvature perturbation and the scalar spectral index are to be well fitted with the latest Planck observation. We suggest that a new leptogenesis scenario could naturally be implemented via Affleck-Dine mechanism. So we show that the resultant baryon asymmetry, constrained by the sum of active neutrino masses and new high energy neutrino oscillations, crucially depends on the reheating temperature $T_{\rm reh}$. And we show that the model has a preference on $T_{\rm reh}\sim10^3$ TeV, which is compatible with the required $T_{\rm reh}$ to explain the baryon asymmetry of the Universe.

Reheating constraints on the inflaton and dark matter: Swampland conjecture

In this paper, we studied the phenomenological consequences of recently proposed swampland conjecture on the inflationary models though constraints on reheating. If dark matter is assumed to be produced during reheating, the conjecture will provide further constraints on the dark matter parameter space through its current relic abundance. As has been pointed out already and also analyzed in our present paper any successful inflationary scenario is in clear tension with the aforementioned conjecture in its current form. However considering the swampland parameters to be free and constrained by the inflationary observables, we studied in detail its consequence on the reheating and dark matter phenomenology. We point out the connection between swampland conjecture and the scalar spectral index $n_s$ by PLANCK within current $2 \sigma$ range, and associated constraints imposed on the reheating temperature and the dark matter annihilation cross-section.

Observational constraints on two-field warm inflation

We study the two-field warm inflation models with a double quadratic potential and a linear temperature dependent dissipative coefficient. We derived the evolution equation of all kinds of perturbations without assuming slow-roll approximation, and obtained the curvature power spectrum at the end of inflation with a fully numerical method. Then we compute the scalar spectral index $n_s$, tensor-to-scalar ratio $r$ for several representative potentials, and compare our results with observational data. At last, we use Planck data to constrain the parameters in our models. This work is a natural extension of single-field warm inflation, and the aim of this work is to present some features of multi-field warm inflation using a simple two-field model.

U(1)R inspired inflation model in no-scale supergravity

We consider a cosmological inflation scenario based on a no-scale supergravity sector with $U(1)_R$ symmetry. It is shown that a tree level $U(1)_R$ symmetric superpotential alone does not lead to a slowly rolling scalar potential. A deformation of this tree level superpotential by including an explicit $R$ symmetry breaking term beyond the renomalizable level is proposed. The resulting potential is found to be similar (but not exactly the same) to the one in Starobinsky inflation model. We emphasize that for successful inflation, with the scalar spectral index $n_s \sim 0.96$ and the tensor-to-scalar ratio $r < 0.08$, a correlation between the mass parameters in the superpotential and the vacuum expectation value of the modulus field $T$ in the K\"ahler potential must be adopted.

Quasimatter bounce equivalent to Starobinsky inflation

In this paper we construct a bounce model that mimics the Starobinsky inflationary model. Our construction relies on Wands' duality, which shows that the Mukhanov-Sasaki equation has a symmetry transformation by changing appropriately its time-dependent mass term. One of the advantages of this constructive method is that one can control every contribution to the primordial power spectrum and check how far we can emulate a given primordial model with a different scenario. In particular, we show that mapping the Starobinsky inflation into a quasi-matter bounce gives the correct relation between the scalar spectral index $n_s-1$ and the tensor-to-scalar ratio $r$.

Reconstructing braneworld inflation

The reconstruction of a braneworld inflationary universe considering the parametrization (or attractor) of the scalar spectral index ${n}_{s}(N)$ in terms of the number of $e$-folding $N$ is developed. We also study the possibility that the reconstruction for the scenario of braneworld inflation can be realized in terms of the tensor-to-scalar ratio $r(N)$. For both reconstruction methodologies, we consider a general formalism in order to obtain the effective potential as a function of the cosmological parameters ${n}_{s}(N)$ or $r(N)$. For both reconstruction methods, we consider the specific examples for large $N$ in the framework of the slow-roll approximation as the attractor ${n}_{s}\ensuremath{-}1\ensuremath{\propto}{N}^{\ensuremath{-}1}$ for the scalar spectral index and the attractor $r\ensuremath{\propto}{N}^{\ensuremath{-}2}$ for the tensor-to-scalar ratio. In this context and depending on the attractors used, we find different expressions for the effective potential $V(\ensuremath{\phi})$, as also the constraints on the parameters present in the reconstruction.

Power law, simple symmetry breaking and Hilltop potentials inspired warm inflationary dynamics

We study the warm inflationary scenario in the context of the modified Chaplygin gas by assuming the generalized form of the dissipative coefficient Γ=Γ(T,ϕ) and standard scalar field model during weak and strong dissipative regimes. We use various types of well-known potentials such as power law potential, simple symmetry breaking potential and Hilltop model in order to explore the inflationary parameters like scalar spectral index (ns) and tensor-to-scalar ratio (r) etc under the slow-roll approximation. We discuss, in detail, the dependence of r on the spectral index ns and observed the conditions in both dissipative regimes (weak as well as strong). Our results are well corroborated with the recent observational data like WMAP(7 + 9) and Planck data.