Attractor inflation is a particularly robust framework for developing inflationary models that are insensitive to the details of the potential. Such models are most often considered in the metric formulation of gravity. However, non-minimal models may not necessarily maintain their attractor nature in the Palatini formalism where the connection is independent of the metric. In this work, we employ the $β$-function formalism to classify the strong coupling limit of inflationary models in both the metric and the Palatini approaches. Furthermore, we determine the range of values for the non-minimal coupling that lead to theories being observationally indistinguishable in metric and Palatini within current accuracy. Finally, we reconstruct the Jordan frame potential for $ξ$-attractors by imposing an explicit form for the $β$-function, demonstrating the effect that the choice of metric or Palatini has on the inflationary observables of the theory.
The inflationary dynamics and preheating in a model with a non-minimally coupled inflaton field in the metric and Palatini formalisms are studied in this paper. We find that in both formalisms, irrespective of the initial conditions, our Universe will evolve into a slow-roll inflationary era and then the scalar field rolls into an oscillating phase. The value of the scalar field at the end of the inflation in the Palatini formalism is always larger than that in the metric one, which becomes more and more obvious with the increase of the absolute value of the coupling parameter $|ξ|$. During th…
We present a simple generalisation of the $Λ$CDM model which on the one hand reaches very good agreement with the present day experimental data and provides an internal inflationary mechanism on the other hand. It is based on Palatini modified gravity with quadratic Starobinsky term and generalized Chaplygin gas as a matter source providing, besides a current accelerated expansion, the epoch of endogenous inflation driven by type III freeze singularity. It follows from our statistical analysis that astronomical data favors negative value of the parameter coupling quadratic term into Einstein-H…
With a scalar field non-minimally coupled to curvature, the underlying geometry and variational principle of gravity - metric or Palatini - becomes important and makes a difference, as the field dynamics and observational predictions generally depend on this choice. In the present paper we describe a classification principle which encompasses both metric and Palatini models of inflation, employing the fact that inflationary observables can be neatly expressed in terms of certain quantities which remain invariant under conformal transformations and scalar field redefinitions. This allows us to …
If the inflaton and the quintessence fields are identified, the background geometry evolves through a stiff epoch undershooting the expansion rate of a radiation-dominated plasma. For some classes of inflationary potentials this scenario is at odds with the current observational evidence since the corresponding tensor-to-scalar ratio is too large. Quintessential inflation is analyzed when the gravitational action is supplemented by a contribution quadratic in the Einstein-Hilbert term. In the Palatini formulation the addition such a term does not affect the scalar modes during the inflationary…
We present the impact of non-minimal coupling $ξϕ^2 R$ on the inflationary parameters by taking into account the models of single-field inflation with the inflaton that has a non-zero vacuum expectation value ($v$) after the period of inflation in Palatini gravity. We discuss the well-known symmetry breaking type potentials, namely the Higgs potential and Coleman-Weinberg potential. We show the inflationary predictions of these potentials, for both $ϕ>v$ and $ϕ<v$ inflation, the regions in the $v-ξ$ plane for which the values of $n_s$ and $r$ are in agreement with the recent measurements. We a…
We perform an analysis of models of chaotic inflation where the inflaton field $ϕ$ is coupled non-minimally to gravity via $ξϕ^n g^{μν}R_{μν}(Γ), n>0$. We focus on the Palatini theory of gravity, i.e. the case where the assumptions of the General Relativity are relaxed (that of the connection being the Levi-Civita one) and the gravitational degrees of freedom are encoded not only in the metric but also the connection $Γ$, which is treated as an independent variable. We show that in this case the famous attractor behaviour of simple non-minimally coupled models of inflation is lost. Therefore t…
When considered in the Palatini formalism, the Starobinsky model does not provide us with a mechanism for inflation due to the absence of a propagating scalar degree of freedom. By (non)--minimally coupling scalar fields to the Starobinsky model in the Palatini formalism we can in principle describe the inflationary epoch. In this article, we focus on the minimally coupled quartic and natural inflation models. Both theories are excluded in their simplest realization since they predict values for the inflationary observables that are outside the limits set by the Planck data. However, with the …
We consider $R^2$ inflation in the Palatini gravity assuming the existence of scalar fields, coupled to gravity in the most general manner. These theories, in the Einstein frame, and for one scalar field $h$, share common features with $K$ - inflation models. We apply this formalism for the study of popular inflationary models, whose potentials are monomials, $ V \sim h^{n} $, with $ n $ a positive even integer. We also study the Higgs model non-minimally coupled to gravity. Although these have been recently studied, in the framework of the Palatini approach, we show that the scalar power spec…
We study static neutron stars in the context of a class of non-minimally coupled inflationary potentials, the universal attractors. Universal attractors are known to generate a viable inflationary era, and they fall into the same category of inflationary phenomenology as the $R^2$ model and other well-known cosmological attractors. We present the essential features of universal attractors in both the Einstein and Jordan frame, and we extract the Tolman-Oppenheimer-Volkoff equations in the Einstein frame using the usual notation of theoretical astrophysics. We use a python 3 based double shooti…
The theory of General Relativity was established on a spacetime manifold equipped with a metric tensor, $(\mathcal{M}_4,\text{g})$, and the connection on $\mathcal{M}_4$ identified with the Levi-Civita one. Even though there are valid reasons to assume a torsionless manifold that preserves the metric, it was shown that dealing away with these assumptions the Levi-Civita condition can be reproduced at the level of equations of motion of GR. It was not long before the equivalence of General Relativity between the two descriptions, known as the Palatini or first-order formalism in which the conne…
Large-scale magnetogenesis is analyzed within the Palatini approach when the gravitational action is supplemented by a contribution that is nonlinear in the Einstein-Hilbert term. While the addition of the nonlinear terms does not affect the scalar modes of the geometry during the inflationary phase, the tensor-to-scalar ratio is nonetheless suppressed. In this context it is plausible to have a stiff phase following the standard inflationary stage provided the potential has a quintessential form. The large-scale magnetic fields can even be a fraction of the nG over typical length scales of the…
We give the procedure to reconstruct the extended inflationary potentials for attractors and use the $α$ attractor and the constant-roll model as examples to show how to reconstruct the class of extended inflationary potentials in the strong coupling limit. We also derive the strong coupling condition which the coupling constant $ξ$ satisfies. The class of extended inflationary potentials has the same attractor, and the reconstructed extended inflationary potentials are consistent with the observational constraints. The existence of the attractors from extended inflationary models further chal…
In this work, we show the effect of the non-minimal coupling $ξϕ^2 R$ on the inflationary parameters by considering the single-field inflation and present the inflationary predictions of the appealing potential for the particle physics viewpoint: Natural Inflation, an axion-like inflaton which has a cosine-type periodic potential and the inflaton naturally emerges as a pseudo-Nambu-Goldstone boson with a spontaneously broken global symmetry. We present the inflationary predictions for this potential, $n_s$, $r$, and $α=\mathrm{d}n_s/\mathrm{d}\ln k$. In addition, we assume standard thermal his…
We study how higher-order gravity affects Higgs inflation in the Palatini formulation. We first review the metric and Palatini formulations in comparative manner and discuss their differences. Next cosmic inflation driven by a scalar field and inflationary observables are discussed. After this we review the Higgs inflation and compute the inflationary observables both in the metric and Palatini formulations. We then consider adding higher-order terms of the curvature to the action. We derive the equations of motion for the most general action quadratic in the curvature that does not violate pa…
In this paper we study some classes of $α$-attractors models in the Jordan frame and we find the corresponding $F(R)$ gravity theory. We study analytically the problem at leading order and we investigate whether the attractor picture persists in the $F(R)$ gravity equivalent theory. As we show, if the slow-roll conditions are assumed in the Jordan frame, the spectral index of primordial curvature perturbations and the scalar-to-tensor ratio are identical to the corresponding observational indices of the $R^2$ model, a result which indicates that the attractor property is also found in the corr…
We show explicitly how the T model, E model, and Hilltop inflations are obtained from the general scalar-tensor theory of gravity with arbitrary conformal factors in the strong coupling limit. We argue that $ξ$ attractors can give any observables $n_s$ and $r$ by this method. The existence of attractors imposes a challenge to distinguish different models.
In the framework of classical scale invariance, we consider quadratic gravity in the Palatini formalism and investigate the inflationary predictions of the theory. Our model corresponds to a two-field scalar-tensor theory, that involves the Higgs field and an extra scalar field stemming from a gauge $U(1)_X$ extension of the Standard Model, which contains an extra gauge boson and three right-handed neutrinos. Both scalar fields couple nonminimally to gravity and induce the Planck scale dynamically, once they develop vacuum expectation values. By means of the Gildener-Weinberg approach, we desc…
We consider scalar field inflation in the Palatini formulation of general relativity. The covariant derivative of the metric is then non-zero. From the effective theory point of view it should couple to other fields. We write down the most general couplings between it and a scalar field that are quadratic in derivatives. We consider both the case when the torsion is determined by the field equations and the case when it is assumed to be zero a priori. We find the metric derivative terms can significantly modify inflationary predictions. We specialise to Higgs inflation and terms of only up to …
We show explicitly how the T-model, E-model, and Hilltop inflations are obtained from the inflation models with a non-canonical kinetic term and an arbitrary potential. By this method, any attractor of observables $n_s$ and $r$ is possible. The presence of attractors poses a challenge to differentiate inflation models.
A new setup of cosmic inflation with a periodic inflaton potential and conformal factor is discussed in the metric and Palatini formulations of gravity. As a concrete example, we focus on a natural-inflation-like inflaton potential, and show that the inflationary predictions fall into the allowed region of cosmic microwave background observations in both formulations.
We show explicitly how the E model attractor is obtained from the general scalar-tensor theory of gravity with arbitrary conformal factors in the strong coupling limit. By using conformal transformations, any attractor with the observables $n_s$ and $r$ can be obtained. The existence of attractors imposes a challenge to distinguish different models.
We construct models with the Gauss-Bonnet term multiplied to a function of the scalar field leading to inflationary scenario. The consideration is related with the slow-roll approximation. The cosmological attractor approach gives the spectral index of scalar perturbations which is in a good agreement with modern observation and allows variability for tensor-to-scalar ratio. We reconstruct models with variability of parameters which allow to reproduce cosmological attractor predictions for inflationary parameters in the leading order of $1/N$ approximation in the Einstein-Gauss-Bonnet gravity.
Marmara University（马尔马拉大学）；Ankara University（安卡拉大学）；Tallinn University of Technology（塔林理工大学）；National Institute of Chemical Physics and Biophysics（国家化学物理与生物物理学研究所）
We investigate the interplay between moduli dynamics and inflation, focusing on the KKLT-scenario and cosmological $α$-attractors. General couplings between these sectors can induce a significant backreaction and potentially destroy the inflationary regime; however, we demonstrate that this generically does not happen for $α$-attractors. Depending on the details of the superpotential, the volume modulus can either be stable during the entire inflationary trajectory, or become tachyonic at some point and act as a waterfall field, resulting in a sudden end of inflation. In the latter case there …
We investigate the observational viability of a class of $α$-attractors inflationary models in light of the most recent Cosmic Microwave Background (CMB) and Large-Scale Structure (LSS) data. By considering a double-well potential we perform a slow-roll analysis to study the behavior of this class of models, which is a continuous interpolation between the chaotic inflation for large values of $α$ and the universal attractor, i.e., $n_s=1- 2/N$ and $r=α12/N^2$ for small $α$, where $n_s$ is the scalar spectral index, $r$ is the tensor-to-scalar ratio, and $N$ is the e-fold number. In order to ex…
We study reheating of inflationary models with general non-minimal coupling $K(ϕ)R$ with $K(ϕ)\sim \sqrt{V(ϕ)}$ where $R$ is the Ricci scalar and $V$ is the inflaton potential. In particular, when we take the monomial potential $K(ϕ) \propto ϕ^m$ with $m \in \mathbb{Z}_+$, we provide general analytic expressions for cosmological observables. We consider a wide range of non-minimal coupling $ ξ\in [0, \infty) $ in metric and Palatini formalisms and derive the predictions for cosmological observables and the reheating temperature taking a general equation of state parameter $ w_\text{reh} $.
In this work we shall study the implications of a subclass of $E$-models cosmological attractors, namely of $a$-attractors, on hydrodynamically stable slowly rotating neutron stars. Specifically, we shall present the Jordan frame theory of the $a$-attractors, and by using a conformal transformation we shall derive the Einstein frame theory. We discuss the inflationary context of $a$-attractors in order to specify the allowed range of values for the free parameters of the model based on the latest cosmic-microwave-background-based Planck 2018 data. Accordingly, using the notation and physical u…
A reformulation of inflationary model analyses appeared recently, in which inflationary observables are determined by the structure of a pole in the inflaton kinetic term rather than the shape of the inflaton potential. We comprehensively study this framework with an arbitrary order of the pole taking into account possible additional poles in the kinetic term or in the potential. Depending on the setup, the canonical potential becomes the form of hilltop or plateau models, variants of natural inflation, power-law inflation, or monomial/polynomial chaotic inflation. We demonstrate attractor beh…
We present a class of inflationary potentials which are invariant under a special symmetry, which depends on the parameters of the models. As we show, in certain limiting cases, the inverse symmetric potentials are qualitatively similar to the $α$-attractors models, since the resulting observational indices are identical. However, there are some quantitative differences which we discuss in some detail. As we show, some inverse symmetric models always yield results compatible with observations, but this strongly depends on the asymptotic form of the potential at large $e$-folding numbers. In fa…
We study $α$-attractor models with both E-model and T-model potential in an extended Non-Minimal Derivative (NMD) inflation where a canonical scalar field and its derivatives are non-minimally coupled to gravity. We calculate the evolution of perturbations during this regime. Then by adopting inflation potentials of the model we show that in the large N and small $α$ limit, value of the scalar spectral index ns and tensor to scalar ratio r are universal. Next, we study reheating after inflation in this formalism. We obtain some constraints on the model's parameter space by adopting the results…
In this paper, we study a hybrid combination of Einstein-Hilbert action with curvature scalar $R$, and a function $f(\mathcal{R})$ in Palatini gravity within the context of inflationary scenario, from the Swampland conjecture point of view. This hybrid model has been paid attention in recent cosmological studies, and its applications have been widely studied in the literature. In this regard, using the Swampland conjecture (using ($ C_ {1} $) as the first component of dS swampland conjecture, which is obtained from the first derivative of the potential upon the potential and ($ C_ {2} $) as th…
A class of scalar-tensor theories (STT) including a non-metricity that unifies metric, Palatini and hybrid metric-Palatini gravitational actions with non-minimal interaction is proposed and investigated from the point of view of their consistency with generalized conformal transformations. It is shown that every such theory can be represented on-shell by a purely metric STT possessing the same solutions for a metric and a scalar field. A set of generalized invariants is also proposed. This extends the formalism previously introduced in \cite{kozak2019}. We then apply the formalism to Starobins…
This study focuses on the static neutron star perspective for two types of cosmological inflationary attractor theories, namely the induced inflationary attractors and the quadratic inflationary attractors. The two cosmological models can be discriminated cosmologically, since one of the two does not provide a viable inflationary phenomenology, thus in this paper we investigate the predictions of these theories for static neutron stars, mainly focusing on the mass and radii of neutron stars. We aim to demonstrate that although the models have different inflationary phenomenology, the neutron s…
We show that pure quadratic gravity with quantum loop corrections yields a viable inflationary scenario. We also show that a large family of models in the Jordan frame, with softly-broken scale invariance, corresponds to the same theory with linear inflaton potential in the Einstein frame. It follows that all these quasi scale-invariant models have the same relation between the tensor-to-scalar ratio and the scalar spectral index, which is also consistent with the current data. Thus, they form a family of attractors, which is sharply distinct from the recently discovered $α$-attractors of Kall…
We consider models of a scalar field coupled to quadratic $R\!+\!R^2$ gravity in the framework of the Palatini formulation. The resulting Einstein-frame generalized $k$-inflation effective theory is analyzed assuming that the constant-roll condition holds. We focus on a quartic self-interaction potential, a case of particular appeal modelling Higgs inflation, considering the cases of minimal and non-minimal coupling of the inflaton to gravity. For an appropriate range of the model parameters in the large field domain the obtained values for the inflationary observables are found in agreement w…
We study flat FLRW $α$-attractor $\mathrm{E}$- and $\mathrm{T}$-models by introducing a dynamical systems framework that yields regularized unconstrained field equations on two-dimensional compact state spaces. This results in both illustrative figures and a complete description of the entire solution spaces of these models, including asymptotics. In particular, it is shown that observational viability, which requires a sufficient number of $e$-folds, is associated with a solution given by a one-dimensional center manifold of a past asymptotic de Sitter state, where the center manifold structu…
Slow roll solutions to inflationary potentials have been widely believed to be the only universal attractor. Over the last few years there has been growing interest in a new class of inflationary models known as Constant-Roll Inflation. Constant roll solutions are a generalization of "ultra-slow roll" dynamics, where the first slow roll parameter is small, but the second slow roll parameter $η$ is larger than unity. In Ultra-slow Roll Inflation, the large-$η$ solution is a dynamical transient, relaxing exponentially to the attractor de Sitter solution. In the constant roll generalization, rece…
We investigate the effect of the quadratic correction $αR^2$ and non-minimal coupling $ξϕ^2 R$ on a quintessence model with an exponential potential $V(ϕ) = M^4\exp(-λϕ)$ in the Palatini formulation of gravity. We use dynamical system techniques to analyze the attractor structure of the model and uncover the possible trajectories of the system. We find that the quadratic correction cannot play a role in the late time dynamics, except for unacceptably large values of the parameter $α$; although it can play a role at early times. We find viable evolutions, from a matter-dominated phase to an acc…
We calculate the tree-level amplitudes for electrically neutral $2\to2$ scattering for the Standard Model Higgs doublet non-minimally coupled to the Ricci scalar. We consider both the metric and the Palatini formulation of gravity. We find the partial wave unitarity limit for a general background field value. Our results are in agreement with previous work. In the electroweak vacuum, tree-level unitarity is violated at $\sim M_\text{Planck}/ξ$ in the metric formulation, and at $\sim M_\text{Planck}/\sqrtξ$ in the Palatini formulation. In the inflationary large field background, the unitarity l…
We consider Einstein-Gauss-Bonnet (EGB) inflationary models using the effective potential approach. We present evolution equations in the slow-roll regime using the effective potential and the tensor-to-scalar ratio. The choice of the effective potential is related to an expression of the spectral index in terms of e-folding number $N_e$. The satisfaction of the slow-roll regime is mostly related to the form of the tensor-to-scalar ratio $r$. The case of $r\sim1/N^2_e$ leads to a generalization of $α$-attractors inflationary parameters to Einstein-Gauss-Bonnet gravity with exponential effectiv…
We study the process of the Primordial Black Holes (PBHs) production in the novel framework, namely $α$-attractor Galileon inflation (G-inflation) model. In our framework, we take the Galileon function as $G(ϕ)=G_{I}(ϕ)\left(1+G_{II}(ϕ)\right)$, where the part $G_{I}(ϕ)$ is motivated from the $α$-attractor inflationary scenario in its original non-canonical frame, and it ensures for the model to be consistent with the Planck 2018 observations at the CMB scales. The part $G_{II}(ϕ)$ is invoked to enhance the curvature perturbations at some smaller scales which in turn gives rise to PBHs formati…
The possibility to construct an inflationary scenario for renormalization-group improved potentials corresponding to the Higgs sector of finite gauge models is investigated. Taking into account quantum corrections to the renormalization-group potential which sums all leading logs of perturbation theory is essential for a successful realization of the inflationary scenario, with very reasonable parameter values. The inflationary models thus obtained are seen to be in good agreement with the most recent and accurate observational data. More specifically, the values of the relevant inflationary p…
Once quantum corrections are taken into account, the strong coupling limit of the $ξ$-attractor models (in metric gravity) might depart from the usual Starobinsky solution and move into linear inflation. Furthermore, it is well known that the metric and Palatini formulations of gravity lead to different inflationary predictions in presence of non-minimally couplings between gravity and the inflaton. In this letter we show that for a certain class of non-minimally coupled models, loop corrections will lead to a linear inflation attractor regardless of the adopted gravity formulation.
