arXiv subjects

Gabriel German

Publications and source records attributed to Gabriel German.

At least 19 recordsLinked to original sources

arXiv2025

Viability of generalized $\alpha$-inflation from Planck, ACT, and DESI Data

We study inflationary constraints from reheating on two classes of single-field inflationary models: a generalized $\alpha$-attractor with potential $V(\phi) = V_0 \!\left(1 - \sech^p\!\left[\phi/(\sqrt{6\alpha}\,M_{Pl})\right]\right)$ and the $\alpha$-Starobinsky model with potential $V(\phi) = V_0 \!\left(1 - e^{- \sqrt{2/(3\alpha)}\, \phi / M_{Pl}} \right)^2$. Using a semi-analytical relation that connects inflationary dynamics to reheating, we solve for the horizon-crossing field value and calculate the scalar spectral index $n_s$, the tensor-to-scalar ratio $r$, and the reheating temperature $T_{re}$. We consider three perturbative decay channels of the inflaton: gravitational, Yukawa (fermionic), and scalar. The parameter space spanned by $(\alpha,p,\omega_{re})$ for the first model and $(\alpha,\omega_{re})$ for the second is explored and compared with recent measurements from Planck and ACT DR6, as well as BAO from DESI DR2. For clarity, we emphasize that ACT DR6 alone is fully compatible with Planck regarding $n_s$. The upward shift toward $n_s \simeq 0.975$ appears only when DESI DR2 BAO are combined with CMB datasets (Planck+DESI, ACT+DESI, SPT-3G+DESI), and is likewise seen in the P--ACT--LB2+DESI combination. CMB-only combinations (e.g., Planck+ACT or Planck+ACT+SPT-3G) primarily refine constraints and do not by themselves raise $n_s$. Our comparisons and model assessments are therefore made with this distinction in mind: CMB-only constraints versus CMB+DESI combinations. We conclude that both models remain consistent with current CMB-only data in restricted regions of parameter space, and that residual tension with the higher $n_s$ favored by CMB+DESI persists.

astro-ph.CO↗

arXiv2025

Perturbative analysis of the reheating dynamics of $\alpha$-attractors

We study the reheating phase following inflation in the context of single-field models, focusing on the perturbative decay of the inflaton into lighter particles. A general analytical framework is presented to compute the reheating temperature $T_{re}$ and related quantities by combining cosmological observations with model-dependent parameters. We derive expressions for $T_{re}$ for three types of interactions: gravitational, scalar, and Yukawa-type fermionic couplings, and apply these results to the class of $\alpha$-attractor inflationary models, which exhibit attractor behavior in the $(n_s, r)$ plane. The main goal of this work is to investigate how key cosmological quantities such as $T_{re}$, $N_{re}$, and $m_\phi$ among others, evolve with the scalar spectral index $n_s$ and the Yukawa coupling constant $y$, within a consistent analytical framework. Although the formulas used are approximate, they are sufficient to capture the qualitative behavior of the relevant quantities across a wide range of parameter values. Here, we are not interested in precise numerical approximations or data analysis, but rather in understanding the general trends and dependence of cosmological quantities of interest. In particular, tendencies observed in the figures, such as the sensitivity of $T_{re}$ to the coupling strength and the equation-of-state parameter $\omega_{re}$, reflect physical features that are not strongly affected by the approximations involved.

astro-ph.CO↗

arXiv2024

Reheating Dynamics in Inflationary Cosmology: Insights from $\alpha$-Attractor and $\alpha$-Starobinsky Models

Reheating in inflationary cosmology is essential for understanding the early universe, influencing particle production, thermalization, and the primordial power spectrum. Crucial quantities defined during the reheating epoc, such as the equation of state parameter $\omega_{re}$, reheating temperature $T_{re}$, and the number of $e$-folds $N_{re}$, affect inflationary observables like the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$. We analyze two classes of inflationary models: generalized $\alpha$-attractor models and the $\alpha$-Starobinsky generalization. These models, motivated by supergravity and string theory, exhibit attractor behavior, ensuring strong predictions and have been studied extensively before. A salient novelty of this study, compared to previous works, is the inclusion of an analytical expression for the reheating temperature, $T_{\text{re}}$, which makes it a dynamical quantity. This is crucial for determining all the cosmological quantities analyzed in this work. Our results show a universal scaling behavior for a tightly bounded $T_{re}$ in both models. We believe this is the first time that $T_{re}$ is so closely determined. This work complements previous Bayesian and numerical studies by providing detailed numerical and analytical insights into the evolution of cosmological observables and reheating parameters, offering also constraints on inflationary models based on observational data.

astro-ph.CO↗

arXiv2024

Primordial black hole formation during slow-reheating: A review

In this paper we review the possible mechanisms for the production of primordial black holes (PBHs) during a slow-reheating period {in which the energy transfer of the inflaton field to standard model particles becomes effective at slow temperatures}, offering a comprehensive examination of the theoretical foundations and conditions required for each of formation channel. In particular, we focus on post-inflationary scenarios where there are no self-resonances and the reheating epoch can be described {by the inflaton evolving in} a quadratic-like potential. In the hydrodynamical interpretation of this field during the slow-reheating epoch, the gravitational collapse of primordial fluctuations is subject to conditions on their sphericity, limits on their spin, as well as a maximum velocity dispersion. We show how to account for all conditions and show that PBHs form with different masses depending on the collapse mechanism. Finally we show, through an example, how PBH production serves to probe both the physics after primordial inflation, as well as the primordial powerspectrum at the smallest scales.

astro-ph.CO↗

arXiv2023

Inflationary models constrained by reheating

The study of reheating in inflationary models is crucial for understanding the early universe and gaining insights into inflationary dynamics and parameters. The reheating temperature $T_{re}$ and the duration of the reheating phase, quantified by the number of $e$-folds $N_{re}$, have significant implications for particle production, thermalization, and the primordial power spectrum. The duration of reheating affects particle abundance, including dark matter, and shapes the primordial power spectrum and cosmic microwave background anisotropies. By combining cosmological observations and theoretical considerations, we can constrain both $T_{re}$ and $N_{re}$, which in turn constrain the spectral index $n_s$, tensor-to-scalar ratio $r$, and inflation model parameters. Utilizing consistency relations among observables, such as $n_s$ and $r$, provides additional constraints on inflationary models and determines bounds for other observables like the running of the scalar spectral index. These bounds are valuable for assessing the viability of models and can serve to specify priors in Bayesian analyses of specific models. As an example of how to proceed, we study in detail a particular case of a generalized $\alpha$-attractor model that accurately reproduces observed quantities. We present equations for the conditions of instantaneous reheating, establish consistency relations, and explore the generalized $\alpha$-attractor model using cosmological data.

astro-ph.CO↗

arXiv2023

Solution for cosmological observables in the Starobinsky model of inflation

This paper focuses on the Starobinsky model of inflation. We derive solutions for various cosmological observables, such as the scalar spectral index $n_s$, 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. We establish an equation that connects inflation and reheating, which can be solved for the spectral index $n_s$. Using consistency relations of the model, we determine the other observables, the number of $e$-folds during inflation $N_k$, and the number of $e$-folds during reheating $N_{re}$. The impact of reheating on inflation is explored by constraining the equation of state parameter $\omega_{re}$ at the end of reheating. 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.

astro-ph.CO↗

arXiv2023

Constraining inflationary potentials with inflaton PBHs

If, after primordial inflation, the universe undergoes a relatively long reheating period, it could present a phase of matter domination supported by the oscillating inflaton field. During this epoch, small perturbations from the inflaton that reenter the cosmological horizon could virialize to form \textit{inflaton} structures. If the primordial overdensities are large enough, their associated inflaton structures could collapse to form primordial black holes (PBHs) [L.E.Padilla, J.C.Hidalgo and K.A.Malik, Phys.Rev.D, vol.106, p.023519, Jul 2022; hereinafter P1]. For this to happen at a considerable rate, the primordial power spectrum should be enhanced at small scales, a feature typically induced in single-field inflation through an ultra-slow roll phase (produced by a nearly-inflection point in the inflationary potential). In this article we consider two specific inflationary potentials that present this nearly-inflection point and we look at the PBH formation rate through the mechanism proposed in P1. We report on constraints to these two specific models from the bounds to PBH abundances. This serves as an illustration of the usefulness of the PBH formation mechanism proposed in P1.

astro-ph.CO↗

arXiv2022

Bayesian analysis for a class of $\alpha$-attractor inflationary models

We perform a Bayesian study of a generalization of the basic $\alpha$-attractor T model given by the potential $V(\phi)=V_0\left[1-\text{sech}^{p}\left(\phi/\sqrt{6\alpha}M_{pl}\right)\right]$ where $\phi$ is the inflaton field and the parameter $\alpha$ corresponds to the inverse curvature of the scalar manifold in the conformal or superconformal realizations of the attractor models. Such generalization is characterized by the power $p$ which includes the basic or base model for $p=2$. Once the priors for the parameters of the $\alpha$-attractor potential are set by numerical exploration, we perform the corresponding statistical analysis for the cases $p=1\, , 2\, , 3\, ,4$, and derive posteriors. Considering the original $\alpha$-attractor potential as the base model, we calculate the evidence for our generalization, and conclude that the $p=4$ model is preferred by the CMB data. We also present constraints for the parameter $\alpha$. Interestingly, all the cases studied prefer a specific value for the tensor-to-scalar ratio given by $r\simeq 0.0025$.

astro-ph.CO↗

arXiv2022

Production of PBHs from inflaton structure

At times prior to Big Bang Nucleosynthesis, the universe could show a primordial structure formation period if dominated by a fast oscillating inflaton field during reheating. In this context, we have postulated a new mechanism of primordial black hole formation [L. E. Padilla, J. C. Hidalgo, and K. A. Malik, Phys. Rev. D, vol. 106, p. 023519, Jul 2022], that draws the analogy between an extended reheating era and the scalar field dark matter model, stipulating the gravitational collapse of inflaton halos and inflaton stars. In this paper we look at the requirements for the realization of this new mechanism. We show that a generic primordial power spectrum with a peak at small scales is most suitable for the production of a considerable number of PBHs. When such requirement is met, and if reheating lasts long enough, large populations of PBHs with $M_{\rm PBH}\sim 1~\mathrm{gram}$ may be produced. We find in particular, that the mass fraction of PBHs is orders of magnitude larger than that obtained when PBHs form via direct collapse in a universe dominated by radiation or pressure-less dust. Looking at observable implications of our findings, we explore the possibility that the PBHs component may dominate the energy density of the universe at some point after the end of reheating.

astro-ph.CO↗

arXiv2022

A comparison between the Jordan and Einstein Frames in Brans-Dicke theories with torsion

In recent years, gravitational models motivated by quantum corrections to gravity which introduce higher order terms like $R^{2}$ or terms in which the Riemann tensor is not symmetric have been studied by several authors in the form of a general Brans-Dicke type model containing the Ricci scalar, the Holst term and the Nieh-Yan invariant. In this paper we focus on the less explored Jordan frame of such theories and in the comparison between both this frame and the Einstein one. Furthermore, we discuss the role of the transformation of the torsion under conformal transformations and show that the transformation proposed in this paper (extended conformal transformation) contains a special case of the projective transformation of the connection used in some of the papers that motivated this work. We discuss the role and advantages of the extended conformal transformation and show that this new approach can have interesting consequences by working with different variables such as the metric and torsion. Moreover, we study the stability of the system via a dynamical analysis in the Jordan frame, this in order to analyze whether or not we have the fixed points that can be later identified as the inflationary attractor and the unstable fixed point where inflation could take place. Finally we study the scale invariant case of the general model in the Jordan frame. We find out that both the scalar spectral index and the tensor-to-scalar ratio are in agreement with the latest Planck results.

gr-qc↗

arXiv2021

New generalization of the simplest $\alpha$-attractor $T$ model

The simplest $\alpha$-attractor $T$ model is given by the potential $V=V_0\tanh^2(\lambda\phi/M_{pl})$. However its generalization to the class of models of the type $V = V_0 \tanh^p (\lambda \phi/M_{pl})$ is difficult to interpret as a model of inflation for most values of $p$. Keeping the basic model, we propose a new generalization, where the final potential is of the form $V = V_0 (1-\sech^p (\lambda \phi/M_{pl}))$, which does not present any of the problems that plague the original generalization, allowing a successful interpretation as a model of inflation for any value of $p$ and, at the same time, providing the potential with a region where reheating can occur for any $p$ (including odd and fractional values) without difficulty. In the cases $p = 1, 2, 4$ we obtain the solutions $r(n_s, N_{ke})$ where $r$ is the tensor-to-scalar ratio, $n_s$ the spectral index and $N_{ke}$ the number of $e$-folds during inflation. We also show how these solutions connect to the $\phi^2$ monomial.

astro-ph.CO↗

arXiv2021

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.

astro-ph.CO↗

arXiv2021

On the $\alpha$-attractor T-models

We carry out a fully analytical study of the phenomenology of $\alpha$-attractor T-models defined by the potential $V = V_0\tanh^ p\left(\lambda \phi/M_{pl}\right)$. We obtain expressions for the number of e-folds during inflation $N_{ke}$ in terms of the scalar spectral index $n_s$ and independently in terms of the tensor-to-scalar ratio $r$. From these expressions we obtain exact solutions for both $n_s$ and $r$ in terms of $N_{ke}$ along with their expansions for large $N_{ke}$, in full agreement with known expressions. Eliminating the parameter $\lambda$ from the model in terms of $n_s$ and $r$ we can obtain exact solutions for $r$ in terms of $n_s$ and $N_{ke}$ 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 $\alpha$-attractors and are also the end points of these. Finally, by also eliminating the global scale $V_0$ in terms of the observables $n_s$ and $r$ we show how in the appropriate limit the $\alpha$-attractor potential exactly reduces to the monomials potential. We also briefly show that for $\alpha$-attractor E-models, which generalize the Starobinsky potential in the Einstein frame, a similar transition occurs.

astro-ph.CO↗

arXiv2020

Quartic hilltop inflation revisited

We implement a procedure by which the parameters present in the potential of Quartic Hilltop Inflation (QHI) are eliminated in favor of the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$. By doing this it is posible to obtain in a straightforward and simple way the equations of a previous analysis where an analytical treatment of QHI in the large field limit is given. This procedure also allows a more precise discussion of general properties of the model. Also, using a constraint from the reheating epoch it is possible to find bounds for the parameters of the model as well as for quantities of interest such as the running of the scalar index, the reheating temperature and the inflationary scale. Since the bounds found come from expressions given exclusively in terms of $n_s$ and $r$ they will continue to narrow as the measurements of the observables $n_s$ and $r$ become more sensitive.

astro-ph.CO↗

arXiv2020

Constraining $\alpha$-attractor models from reheating

We eliminate the parameters originally present in models of inflation of the $\alpha$-attractor type in favor of the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$. We then write expressions for the number of $e$-folds during reheating $ N_ {re} $. By imposing reasonable conditions on $N_{re}$ we can restrict $n_s$ and $r$ and in turn, we use these constraints in order to find bounds for cosmological quantities of interest such as the number of $e$-folds during inflation and the radiation dominated eras, as well as for the reheating temperature and the running index. The minimum condition that $N_ {re}$ must satisfy is $N_ {re}\geq 0$ which we use to constrain the cosmological quantities mentioned above. In particular, we find that the tensor-to-scalar ratio $r$ (and as a consequence the energy scale of inflation) is bounded from below. We provide figures illustrating the behavior of these quantities as functions of $r$ for several values of $n_s$ and tables containing the bounds so obtained.

astro-ph.CO↗

arXiv2020

Constraints for the running index independent of the parameters of the model

By writing the running of the scalar spectral index completely in terms of the scalar index $n_s$ and the tensor-to-scalar ratio $r$ we are able to impose constraints to models of inflation which are independent of the parameters of the model in question. We write analytical expressions for the running index of Natural Inflation, two models of the type Mutated Hilltop Inflation and the Starobinsky model. The resulting formulae for the running depend exclusively on $n_s$ and/or $r$ and will keep tightening the running index further as additional conditions and observations constrain the scalar and the tensor-to-scalar indices.

astro-ph.CO↗

arXiv2020

Evolution of the universe during the inflationary epoch

We often find in the literature solutions to the Friedmann and fluid equations for simple cosmological models during the matter, radiation or cosmological constant dominated epochs. However no solutions appear for the inflationary era dominated by the potential energy of a scalar field due, perhaps, to the fact that we do not have as yet a strongly favored model of inflation; there are, of course, very well motivated models which fit the data. The purpose of this article is to study with some detail the evolution of the Universe during inflation in the slow-roll approximation. Taking the Starobinsky model as an example, we display exact solutions for the time evolution of the scalar field $\phi(t)$, scale factor $a(t)$, Hubble function $H(t)$, equation of state parameter $\omega(t)$ and acceleration of the scale factor $\ddot{a}(t)$ among other quantities of interest.

gr-qc↗

arXiv2020

Measuring the expansion of the universe

We draw a figure from where it is possible to measure the number of e-folds of expansion of the universe with a ruler. We find model independent bounds for the number of e-folds during inflation, reheating and radiation. We also give a lower bound to the size of the universe at the beginning of observable inflation. Finally, we show that consistency with a relevant diagram requires the existence of a new form of energy to drive the present expansion of the universe.

astro-ph.CO↗