In this letter we shall demonstrate that the viable $F(R)$ gravities can be classified mainly into two classes of inflationary attractors, either the $R^2$ attractors or the $α$-attractors. To show this, we shall derive the most general relation between the tensor-to-scalar ratio $r$ and the spectral index of primordial curvature perturbations $n_s$, namely the $r-n_s$ relation, by assuming that the slow-roll condition constrains the values of the slow-roll indices. As we show, the relation between the tensor-to-scalar ratio and the spectral index of the primordial curvature perturbations has the form $r\simeq \frac{48 (1-n_s)^2}{(4-x)^2}$, where the dimensionless parameter $x$ contains higher derivatives of the $F(R)$ gravity function with respect to the Ricci scalar, and it is a function of the $e$-foldings number $N$ and may also be a function of the free parameters of the various $F(R)$ gravity models. For $F(R)$ gravities which have a spectral index compatible with the observational data and also yield $x\ll 1$, these belong to the $R^2$-type of attractors, with $r\sim 3 (1-n_s)^2$, and these are viable theories. Moreover, in the case that $x$ takes larger values in specific ranges and is constant for a given $F(R)$ gravity, the resulting $r-n_s$ relation has the form $r\sim 3 α(1-n_s)^2$, where $α$ is a constant. Thus we conclude that the viable $F(R)$ gravities may be classified into two limiting types of $r-n_s$ relations, one identical to the $R^2$ model at leading order in $x$, and one similar to the $α$-attractors $r-n_s$ relation, for the $F(R)$ gravity models that yield $x$ constant. Finally, we also discuss the case that $x$ is not constant.
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 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.
Several recent misconceptions about the measure problem in inflation and the nature of inflationary attractors are addressed. We show that within the Hamiltonian system of flat Friedmann-Lemaître-Robertson-Walker cosmology coupled to a massive scalar field, the focussing of the Liouville measure on attractor solutions is brought about by a spread in a gauge degree of freedom - the spatial volume. Using this we show how the Liouville measure formulated on a surface of constant Hubble rate induces a probability distribution function on surfaces of other Hubble rates, and the attractor behaviour …
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…
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…
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 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 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…
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-minim…
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 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…
Recently a broad class of superconformal inflationary models was found leading to a universal observational prediction $n_s=1-2/N$ and $r=12/N^2$. Here we generalize this class of models by introducing a parameter $α$ inversely proportional to the curvature of the inflaton Kahler manifold. In the small curvature (large $α$) limit, the observational predictions of this class of models coincide with the predictions of generic chaotic inflation models. However, for sufficiently large curvature (small $α$), the predictions converge to the universal attractor regime with $n_s=1-2/N$ and $r=12α/N^2$…
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…
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.
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…
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.
Recently we identified a new class of (super)conformally invariant theories which allow inflation even if the scalar potential is very steep in terms of the original conformal variables. Observational predictions of a broad class of such theories are nearly model-independent. In this paper we consider generalized versions of these models where the inflaton has a non-minimal coupling to gravity with ξ<0 different from its conformal value ξ= -1/6. We show that these models exhibit attractor behavior. With even a slight increase of |ξ| from |ξ| = 0, predictions of these models for n_s and r rapid…
In a series of recent papers Kallosh, Linde, and collaborators have provided a unified description of single-field inflation with several types of potentials, ranging from power law to supergravity, in terms of just one parameter $α$. These so-called $α$-attractors predict a spectral index $n_{s}$ and a tensor-to-scalar ratio $r$, which are fully compatible with the latest Planck data. The only common feature of all $α$-attractors is a non-canonical kinetic term with a pole, and a potential analytic around the pole. In this paper, starting from the same Einstein frame with a non-canonical scal…
Qualitative dynamics of three different loop quantizations of spatially flat isotropic and homogeneous models is studied using effective spacetime description of the underlying quantum geometry. These include the standard loop quantum cosmology (LQC), its recently revived modification (referred to as mLQC-I), and another related modification of LQC (mLQC-II) whose dynamics is studied in detail for the first time. Various features of LQC, including quantum bounce and pre-inflationary dynamics, are found to be shared with the mLQC-I and mLQC-II models. We study universal properties of dynamics f…
In this paper we study the effects of spatial curvature of the metric on the phase space of vacuum $f(R)$ gravity. Particularly, we appropriately choose the variables of the dynamical system, in order for this to be autonomous, and we study the phase space of the resulting theory, focusing on de Sitter, matter and radiation domination fixed points. Our analysis indicates that the effect of spatial curvature on the phase space is radical, since it destabilizes all the stable de Sitter vacua of the flat spacetime vacuum $f(R)$ gravity phase space, making the phase space having non-trivial unstab…
We consider two classes of non-minimally coupled inflation models; those with a quadratic coupling of the inflaton to gravity, and the `Universal Attractor' models where the coupling is connected to the potential. We make a detailed analysis of the attractor structure in the latter case, identifying a shift of the attractor from the Starobinsky point and determining conditions for approach to the attractor. We then assess the viability of the models under different interpretations of the BICEP2 experiment's detection of B-mode polarisation, finding strong constraints on the non-minimal couplin…
We investigate the inflationary attractors in models of inflation inspired from general conformal transformation of general scalar-tensor theories to the Einstein frame. The coefficient of the conformal transformation in our study depends on both the scalar field and its kinetic term. Therefore the relevant scalar-tensor theories display the subset of the class I of the degenerate higher-order scalar-tensor theories in which both the scalar field and its kinetic term can non-minimally couple to gravity. We find that if the conformal coefficient $Ω$ takes a multiplicative form such that $Ω\equi…
We define a general class of superconformal inflationary attractor models studied in our previous papers [1,2], which include, among others, inflationary models of 'induced gravity' which were argued to retain perturbative unitarity up to the Planck scale [3]. The choice of the function $f(ϕ)$ in superconformal attractors for the models introduced in [3] is $f(ϕ)= ϕ^n - ξ^{-1}$. We present both superconformal and supergravity versions of these models, which were derived in [1,2] for arbitrary $f(ϕ)$, together with the universal conditions required for stabilization of the extra three moduli pr…
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 the slow-roll single field inflation in the context of the consistent $D\to4$ Einstein-Gauss-Bonnet gravity that was recently proposed in \cite{Aoki:2020lig}. In addition to the standard attractor regime, we find a new attractor regime which we call the Gauss-Bonnet attractor as the dominant contribution comes from the Gauss-Bonnet term. Around this attractor solution, we find power spectra and spectral tilts for the curvature perturbations and gravitational waves (GWs) and also a model-independent consistency relation among observable quantities. The Gauss-Bonnet term provides a nonl…
Among the remaining viable theories that can successfully describe the late-time era is the $k$-Essence theory and in this work we study in detail the phase space of $k$-Essence $f(R)$ gravity in vacuum. This theory can describe in a viable way the inflationary era too, so we shall study the phase space in detail, since this investigation may reveal general properties regarding the inflationary attractors. By appropriately choosing the dimensionless variables corresponding to the cosmological system, we shall construct an autonomous dynamical system, and we find the fixed points of the system.…
In this work, we revisit the dynamics of pre-inflationary universe with a family of $α-$attractor potentials, in the framework of loop quantum cosmology, in which the big bang singularity is generically resolved purely with quantum geometric effects, and replaced by a quantum bounce. At the bounce, the background evolution is divided into two distinct classes, the first is dominated by the kinetic energy of the inflaton field and the second by the potential energy. In both classes, we find the physically viable initial conditions numerically that provide not only the slow-roll inflation, but a…
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 discuss the introduction of a non minimal coupling between the inflaton and gravity in terms of the recently proposed $β$-function formalism for inflation\cite{Binetruy:2014zya}. Via a field redefinition we reduce to the case of minimally coupled theories. The universal attractor at strong coupling has a simple explanation in terms of the new field. Generalizations are discussed and the possibility of evading the universal attractor is shown.
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.
We show that if the $α$-attractor model is realized by the spontaneous breaking of the scale symmetry, then the stability and the dynamics of the vector field that gauges the scale symmetry can severely constrain the $α$-parameter as $5/6 < α< 1$ restricting the inflationary predictions in a very tiny region in the $n_s - r$ plane that are in great agreement with the latest Planck data. Although the different values of $α$ do not make a tangible difference for $n_s$ and $r$, they provide radically different scenarios for the post-inflationary dynamics which determines the standard BBN processe…
A generic outcome of theories with scalar-tensor coupling is the existence of inflationary attractors, either power-law or de Sitter. The fluctuations arising during this phase are Gaussian and their spectrum depends on the wavenumber $k$ according to the power-law $k^{1/(1-p)}$, where $p$ is the inflationary power-law exponent. We investigate to which extent these properties depend on the coupling function and on the potential. We find the class of models in which viable attractors exist. Within this class, we find that the cosmic expansion and the scaling of the fluctuation spectrum are inde…
We consider a broad class of inflationary models that arise naturally in supergravity. They are defined in terms of a parameter $α$ that determines the curvature and cutoff of these models. As a function of this parameter, we exhibit that the inflationary predictions generically interpolate between two attractor points. At small cutoff $α$, the resulting inflationary model is of plateau-type with $n_s = 1 - 2 / N$ and $r = 12 α/ N^2$. For $α= 1$, these predictions coincide with predictions of the Starobinsky model and Higgs inflation. In contrast, for large cutoff $α$, the theory asymptotes to…
We study the dynamics of the universe with a scalar field and an SU(2) non-Abelian Gauge (Yang-Mills) field. The scalar field has an exponential potential and the Yang-Mills field is coupled to the scalar field with an exponential function of the scalar field. We find that the magnetic component of the Yang-Mills field assists acceleration of the cosmic expansion and a power-law inflation becomes possible even if the scalar field potential is steep, which may be expected from some compactification of higher-dimensional unified theories of fundamental interactions. This power-law inflationary s…
Recent years have seen the introduction of various multi-field inflationary scenarios in which the curvature and geodesics of the scalar manifold play a crucial role. We outline a simple description that unifies these different proposals and discuss their stability criteria. We demonstrate how the underlying dynamics is governed by an effective potential, whose critical points and bifurcations determine the late-time behaviour of the system, thus unifying hyperinflation, angular, orbital and side-tracked inflation. Interestingly, we show that hyperinflation is a special case of side-tracked in…
Inflationary scenarios motivated by the Minimal Supersymmetric Standard Model (MSSM) where five scalar fields are non-minimally coupled to gravity are considered. The potential of the model and the function of non-minimal coupling are polynomials of two Higgs doublet convolutions. We show that the use of the strong coupling approximation allows to obtain inflationary parameters in the case when a combination of the four scalar fields plays a role of inflaton. Numerical calculations show that the cosmological evolution leads to inflationary scenarios fully compatible with observational data for…
The large class of inflationary models known as $α$- and $ξ$-attractors gives identical cosmological predictions at tree level (at leading order in inverse power of the number of efolds). Working with the renormalization group improved action, we show that these predictions are robust under quantum corrections. This means that for all the models considered the inflationary parameters $(n_s,r)$ are (nearly) independent on the Renormalization Group flow. The result follows once the field dependence of the renormalization scale, fixed by demanding the leading log correction to vanish, satisfies a…
We describe an induced inflation, which refers to a class of inflationary models with a generalized non-minimal coupling $ξg(ϕ) R$ and a specific scalar potential. The defining property of these models is that the scalar field takes a vev in the vacuum and thus induces an effective Planck mass. We study this model as a function of the coupling parameter $ξ$. At large $ξ$, the predictions of the theory are known to have an attractor behavior, converging to a universal result independent on the choice of the function $g(ϕ)$. We find that at small $ξ$, the theory approaches a second attractor. Th…
Inflation models make specific predictions for a tensor-scalar-scalar three-point correlation, or bispectrum, between one gravitational-wave (tensor) mode and two density-perturbation (scalar) modes. This tensor-scalar-scalar correlation leads to a local power quadrupole, an apparent departure from statistical isotropy in our Universe, as well as characteristic four-point correlations in the current mass distribution in the Universe. So far, the predictions for these observables have been worked out only for single-clock models in which certain consistency conditions between the tensor-scalar-…
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…
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…
The reconstruction of a k-essence inflationary universe, considering the unification between the swampland criteria and the attractor given by the scalar spectral index $n_S(N)$ together with the slow roll parameter $ε(N)$in terms of the number of $e$-folds $N$ is studied. In the context of a coupling of the form $L(ϕ)\,X$ in the k-essence model, we find the effective potential $V$ and the coupling parameter $L$ in terms of the scalar spectral index and the slow roll parameter under a general formalism. To apply the unification in our model, we consider some examples in order to rebuild the ef…
We consider a cosmological model consisting of two scalar fields defined in the hyperbolic plane known as hyperbolic inflation. For the background space, we consider a homogeneous and isotropic spacetime with nonzero curvature. We study the asymptotic behaviour of solutions and we search for attractors in the expanding regime. We prove that two hyperbolic inflationary stages are stable solutions that can solve the flatness problem and describe acceleration for both open and closed models, and additionally we obtain a Milne-like attractor solution for the open model. We also investigate the con…
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…
Since the standard inflationary paradigm is based on quantum field theory on classical space-times, it excludes the Planck era. Using techniques from loop quantum gravity, the paradigm is extended to a self-consistent theory from the Planck scale to the onset of slow roll inflation, covering some 11 orders of magnitude in energy density and curvature. This pre-inflationary dynamics also opens a small window for novel effects, e.g. a source for non-Gaussianities, which could extend the reach of cosmological observations to the deep Planck regime of the early universe.
We describe a broad class of multi-field inflationary models with spontaneously broken conformal invariance. It generalizes the recently discovered class of cosmological attractors with a single inflaton field. In the new multi-field theories, just as in the previously studied single-field models, the moduli space has a boundary (Kahler cone) in terms of the original homogeneous conformal variables. Upon spontaneous breaking of the conformal invariance and switching to the Einstein frame, this boundary moves to infinity in terms of the canonically normalized inflaton field. This results in the…
