We have carried out detailed $k$-mode analysis of single-inflaton slow-roll inflationary perturbations including quantum fluctuations by setting up non-linear coupled differential equations of inflaton field ($ϕ$), its perturbation ($δϕ$) and the metric perturbation (the Bardeen potential, $Φ_B$), and calculated the number of e-folds ($N$), scalar spectral index ($n_s$), tensor spectral index ($n_h$), scalar power spectrum ($Δ_s$), tensor power spectrum ($Δ_h$), tensor-to-scalar ratio ($r$) and the Hubble parameter ($H$) for different $k$ values at the horizon crossing. In these calculations we have employed the $E$-model $α$-attractor potentials which are found to display slow-roll behaviour. The values of $n_s$ and $r$ obtained by us are consistent with those given by the well-known universal $α$-attractor formulae. We got $n_s = 0.956908$ and $r= 0.005571 $ at $k = 10^6$ Planck unit for the value of the potential parameter $n = 1$. These can be compared with the Planck-2018 data viz., $n_s = 0.9649\pm 0.0042$ at $68\%$ CL, $r<0.064$ at $95\%$ CL and ACT-2020 data viz., $n_s = 0.9691\pm 0.0041$ at $68\%$ CL.
We show that the inflaton coupling to all other fields is exponentially suppressed during inflation in the cosmological $α$-attractor models. In the context of supergravity, this feature is a consequence of the underlying hyperbolic geometry of the moduli space which has a flat direction corresponding to the inflaton field. A combination of these factors protects the asymptotic flatness of the inflaton potential.
We construct the simplest inflationary $α$-attractor models in supergravity: it has only one scalar, the inflaton. There is no sinflaton since the inflaton belongs to an orthogonal nilpotent superfield where the sinflaton depends on fermion bilinears. When the local supersymmetry is gauge-fixed, these models have only one single real scalar (the inflaton), a graviton and a massive gravitino. The sinflaton, sgoldstino and inflatino are all absent from the physical spectrum in the unitary gauge. The orthogonality condition leads to the simplest Kähler potential for the inflaton, while preserving…
We study cosmological $α$-attractors in superconformal/supergravity models, where $α$ is related to the geometry of the moduli space. For $α=1$ attractors \cite{Kallosh:2013hoa} we present a generalization of the previously known manifestly superconformal higher curvature supergravity model \cite{Cecotti:1987sa}. The relevant standard 2-derivative supergravity with a minimum of two chiral multiplets is shown to be dual to a 4-derivative higher curvature supergravity, where in general one of the chiral superfields is traded for a curvature superfield. There is a degenerate case when both matter…
We study inflation in the $α-$attractor model under a non-slow-roll dynamics with an ansatz proposed by Gong \& Sasaki \cite{Gong:2015ypa} of assuming $N=N\left(ϕ\right)$. Under this approach, we construct a class of local shapes of inflaton potential that are different from the T-models. We find this type of inflationary scenario predicts an attractor at $n_{s}\sim0.967$ and $r\sim0.00055$. In our approach, the non-slow-roll inflaton dynamics are related to the $α-$parameter which is the curvature of Kähler geometry in the SUGRA embedding of this model.
In this short paper we outline a recipe for the reconstruction of $F(R)$ gravity starting from single field inflationary potentials in the Einstein frame. For simple potentials one can compute the explicit form of $F(R)$, whilst for more involved examples one gets a parametric form of $F(R)$. The $F(R)$ reconstruction algorithm is used to study various examples: power-law $ϕ^n$, exponential and $α$-attractors. In each case it is seen that for large $R$ (corresponding to large value of inflaton field), $F(R) \sim R^2$. For the case of $α$-attractors $F(R) \sim R^2$ for all values of inflaton fi…
Inflationary $α$-attractor models in supergravity, which provide excellent fits to the latest observational data, are based on the Poincare disk hyperbolic geometry. We refine these models by constructing Kahler potentials with built-in inflaton shift symmetry and by making a canonical choice of the goldstino Kahler potential. The refined models are stable with respect to all scalar fields at all $α$, no additional stabilization terms are required. The scalar potential V has a nearly Minkowski minimum at small values of the inflaton field $φ$, and an infinitely long dS valley of constant depth…
We revisit the notion of slow-roll in the context of general single-field inflation. As a generalization of slow-roll dynamics, we consider an inflaton $ϕ$ in an attractor phase where the time derivative of $ϕ$ is determined by a function of $ϕ$, $\dotϕ=\dotϕ(ϕ)$. In other words, we consider the case when the number of $e$-folds $N$ counted backward in time from the end of inflation is solely a function of $ϕ$, $N=N(ϕ)$. In this case, it is found that we need a new independent parameter to properly describe the dynamics of the inflaton field in general, in addition to the standard parameters c…
Recently, few cosmological models with additional non-Riemannian volume form(s) have been proposed. In this article we use Supernovae type Ia experimental data to test one of these models which provides a unified description of both dark energy via dynamically generated cosmological constant and dark matter as a "dust" fluid due to a hidden nonlinear Noether symmetry. It turns out that the model allows various scenarios of the future Universe evolution and in the same time perfectly fits contemporary observational data. Further, we investigate the influence of an additional inflaton field with…
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…
Cosmological alpha-attractors give a natural explanation for the spectral index n_s of inflation as measured by Planck while predicting a range for the tensor-to-scalar ratio r, consistent with all observations, to be measured more precisely in future detection of gravity waves. Their embedding into supergravity exploits the hyperbolic geometry of the Poincare disk or half-plane. These geometries are isometric under Mobius transformations, which include the shift symmetry of the inflaton field. We introduce a new Kahler potential frame that explicitly preserves this symmetry, enabling the infl…
The slow-roll approximation to inflation is ultimately justified by the presence of inflationary attractors for the orbits of the solutions of the dynamical equations in phase space. There are many indications that the inflaton field couples nonminimally to the spacetime curvature: the existence of attractor points for inflation with nonminimal coupling is demonstrated, subject to a condition on the inflaton potential and the value of the coupling constant.
Inflationary $α$-attractor models can be naturally implemented in supergravity with hyperbolic geometry. They have stable predictions for observables, such as $n_s=1-{2/ N_e} $, assuming that the potential in terms of the original geometric variables, as well as its derivatives, are not singular at the boundary of the hyperbolic disk, or half-plane. In these models, the potential in the canonically normalized inflaton field $φ$ has a plateau, which is approached exponentially fast at large $φ$. We call them exponential $α$-attractors. We present a closely related class of models, where the pot…
The inflaton is highly likely to settle in a squeezed vacuum state after inflation. The relic inflaton after inflation and reheating undergoes a damped oscillatory motion and contributes to the effective cosmological constant. We interpret the renormalized energy density from the squeezed vacuum state as an effective cosmological constant. Using the recent observational data on the cosmological constant, we find the constraint on the squeeze parameter of the inflaton in the early universe.
Recently, several broad classes of inflationary models have been discovered whose cosmological predictions are stable with respect to significant modifications of the inflaton potential. Some classes of models are based on a non-minimal coupling to gravity. These models, which we will call $ξ$-attractors, describe universal cosmological attractors (including Higgs inflation) and induced inflation models. Another class describes conformal attractors (including Starobinsky inflation and T-models) and their generalization to $α$-attractors. The aim of this paper is to elucidate the common denomin…
In this paper we consider the $α-$attractor model and study inflation under a non-slow-roll dynamics. More precisely, we follow the approach recently proposed by Gong and Sasaki \cite{Gong:2015ypa} by means of assuming $N=N\left(ϕ\right)$. Within this framework we obtain a family of functions describing the local shape of the potential during inflation. We study a specific model and find an inflationary scenario predicting an attractor at $n_{s}\approx0.967$ and $r\approx5.5\times10^{-4}$. We further show that considering a non-slow-roll dynamics, the $α-$attractor model can be broaden to a wi…
In a recent work, we demonstrated that a modified gravity model in which a scalar "darkon" field is coupled to both the standard Riemannian metric and to another non-Riemannian volume form is compatible with observational data from Supernovae Type Ia. Here, we investigate a more complicated model with an additional "inflaton" scalar field. We demonstrate numerically that the model can qualitatively reproduce the Universe inflation epoch, matter dominated epoch, and present accelerating expansion in a seamless way. We show that such solutions occur only when the model parameters are within a ve…
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…
The cosmological constant (term) is the simplest way, presently known, to illustrate the accelerating expansion of the universe. However, because of/despite its simple appearance, there is much confusion surrounding its essence. Theorists have been asking questions for years: Is there a mechanism to explain this term? Is it really a constant or a variable? Moreover, it seems that we have created a huge gulf separating the theories of inflation and accelerating expansion. Can we eliminate such an uncomfortable discontinuity? Abstract In this paper, we will journey to see the growth of the unive…
We present a cosmological model arising from a gravitational theory with an infinite tower of higher-order curvature invariants that can reproduce the entire evolution of the Universe: from inflation to late-time acceleration, without invoking an inflaton nor a cosmological constant. The theory is Einsteinian-like. The field equations for a Friedmann-Lemaître-Robertson-Walker metric are of second-order and can reproduce a late-time evolution that is consistent with the acceleration provided by the cosmological constant at low redshift. Our results force us to reinterpret the nature of dark ene…
The $N=1$ supergravity models of cosmological inflation with inflaton belonging to a massive vector multiplet and spontaneous SUSY breaking after inflation are reformulated as the supersymmetric $U(1)$ gauge theories of a massless vector superfield interacting with the Higgs and Polonyi chiral superfields, all coupled to supergravity. The $U(1)$ gauge sector is identified with the $U(1)$ gauge fields of the super-GUT coupled to supergravity, whose gauge group has a $U(1)$ factor. A positive cosmological constant (dark energy) is included. The scalar potential is calculated, and its de Sitter v…
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…
We construct new models of inflation and spontaneous supersymmetry breaking in de Sitter vacuum, with a single chiral superfield, where inflaton is the superpartner of the goldstino. Our approach is based on hyperbolic Kähler geometry, and a gauged (non-axionic) $U(1)_R$ symmetry rotating the chiral scalar field by a phase. The $U(1)_R$ gauge field combines with the angular component of the chiral scalar to form a massive vector, and single-field inflation is driven by the radial part of the scalar. We find that in a certain parameter range they can be approximated by simplest Starobinsky-like…
We construct $α$-attractor versions of hybrid inflation models. In these models, the potential of the inflaton field $φ$ is uplifted by the potential of the second field $χ$. This uplifting ends due to a tachyonic instability with respect to the field $χ$, which appears when $φ$ becomes smaller than some critical value $φ_{c}$. In the large $N$ limit, these models have the standard universal $α$-attractor predictions. In particular, $n_{s }= 1- {2 \over N}$ for the exponential attractors. However, in some special cases the large $N$ limit is reached only beyond the horizon, for $N \gtrsim 60$.…
We construct the gauge invariant free action for cosmological perturbations for the nonminimally coupled inflaton field in the Jordan frame. For this the phase space formalism is used, which keeps track of all the dynamical and constraint fields. We perform explicit conformal transformations to demonstrate the physical equivalence between the Jordan and Einstein frames at the level of quadratic perturbations. We show how to generalize the formalism to the case of a more complicated scalar sector with an internal symmetry, such as Higgs inflation. This work represents a first step in developing…
We introduce a novel non-minimal coupling between gravity and the inflaton sector. Remarkably, for large values of this coupling all models asymptote to a universal attractor. This behavior is independent of the original scalar potential and generalizes the attractor in the phi^4 theory with non-minimal coupling to gravity. The attractor is located in the `sweet spot' of Planck's recent results.
Sneutrino inflation employs the fermionic partners of the inflaton and stabilizer field as right-handed neutrinos to realize the seesaw mechanism for light neutrino masses. We show that one can improve the latest version of this scenario and its consistency with the Planck data by embedding it in the theory of cosmological $α$-attractors.
We study post-inflationary evolution in $α$-attractor T-models of inflation. We consider the dynamics of both scalar fields present in these models: the inflaton and the spectator, as a negative field-space curvature may lead to geometrical destabilization of the spectator. We perform state-of-the-art lattice simulations with a dedicated numerical code optimized for those models. We corroborate earlier findings that the perturbations of the spectator field are much more unstable than the perturbations of the inflaton field, so the dynamics of early stages of preheating is dominated by the evol…
In construction of an inflationary model, one usually assumes that the matter sector of the gravitational action is minimally coupled to the background. It means that the matter (inflaton) part of the action is coupled with the same metric of the gravitational part. We elaborate on this assumption and investigate some of the consequences. We assume that the gravitational and the inflaton sectors belong to different units (or conformal frames). We show that this coupling can convert a single-field inflationary model into a two-field one with a mixed kinetic term. The energy-momentum tensor of t…
In this paper, we apply reconstruction techniques to recover the potential parameters for a particular class of single-field models, the $α$-attractor (supergravity) models of inflation. This also allows to derive the inflaton vacuum expectation value at horizon crossing. We show how to use this value as one of the input variables to constrain the postaccelerated inflationary phase. We assume that the tensor-to-scalar ratio $r$ is of the order of $10^{-3}$ , a level reachable by the expected sensitivity of the next-generation CMB experiments.
The evolution of the quintessence field during a phase of chaotic inflation is studied. The inflaton and the quintesssence field are described in a supergravity framework where the coupling between the inflaton and quintessence is induced by non-renormalisable operators suppressed by the Planck mass. We show that the resulting quintessence potential during inflation possesses a time--dependent minimum playing the role of an attractor. The presence of this attractor forces the quintessence field to be small during inflation. These initial conditions are such that the quintessence field is on tr…
Recently, it has been shown that during preheating the entropy modes circulating in the loops, which correspond to the inflaton decay products, meaningfully modify the cosmological correlation functions at superhorizon scales. In this paper, we determine the significance of the same effect when reheating occurs in the perturbative regime. In a typical two scalar field model, the magnitude of the loop corrections are shown to depend on several parameters like the background inflaton amplitude in the beginning of reheating, the inflaton decay rate and the inflaton mass. Although the loop contrib…
A special class of conformal gravity theories is proposed to solve the long standing problem of the fine-tuned cosmological constant. In the proposed model time evolution of the inflaton field leaves behind a nearly vanishing, but finite value of dark energy density of order (a few meV)$^4$ to explain the late-time accelerating universe. A multiple scalar inflaton field is assumed to have a conformal coupling to the Ricci scalar curvature in the lagrantian, which results in, after a Weyl rescaling to the Einstein metric frame, modification of inflaton kinetic and potential terms along with its…
In this paper, the dynamical heteroclinic orbit and attractor have been employed to make the late-time behaviors of the model insensitive to the initial condition and thus alleviates the fine tuning problem in cosmological dynamical system of barotropic fluid and quintessence with a double exponential potential. The late-time asymptotic behavior of the double exponential model does not always correspond to the single case. The heteroclinic orbits are the non-stationary solutions and in general they will interpolate between the critical points. Although they can not be shown analytically, yet a…
I show that the problem of realizing inflation in theories with random potentials of a limited number of fields can be solved, and agreement with the observational data can be naturally achieved if at least one of these fields has a non-minimal kinetic term of the type used in the theory of cosmological $α$-attractors.
For evolution of flat universe, we classify late time and future attractors with scaling behavior of scalar field quintessence in the case of potential, which, at definite values of its parameters and initial data, corresponds to exact scaling in the presence of cosmological constant.
Recently, we have performed research on the subject of the cosmological constant problem. The scenario is based on two postulates for inflationary theory: one is that inflaton $ϕ$ can interact with radiation (relativistic particles); the other is that radiation will be created continuously during and after the epoch of inflation. According to these postulates and from a "macroscopic perspective", we discover that radiation can be viewed as a product of the interaction between $\dotϕ$ and some "effective kinetic frictional force" that exists in inflaton dynamics. Deducing and surmising from "ef…
We provide a unified description of cosmological $α$-attractors and late-time acceleration, in excellent agreement with the latest Planck data. Our construction involves two superfields playing distinctive roles: one is the dynamical field and its evolution determines inflation and dark energy, the other is nilpotent and responsible for a landscape of vacua and supersymmetry breaking. We prove that the attractor nature of the theory is enhanced when combining the two sectors: cosmological attractors are very stable with respect to any possible value of the cosmological constant and, interestin…
Results of a general study on the dynamics of cosmological scalar fields with arbitrary potentials are presented. Exact and approximate attractor solutions are found, with applications to quintessence, moduli stabilization and inflation.
We derive general analytic formulae for the power spectrum and spectral index of the curvature perturbation produced during inflation driven by a multi-component inflaton field, up to the second order in the slow-roll approximation. We do not assume any specific properties of the potential or the metric on the scalar field space, except for the slow-roll condition, Einstein gravity, and the absence of any permanent isocurvature modes.
We propose a new scenario for generating a relic density of non-relativistic dark matter in the context of heterotic string theory. Contrary to standard thermal freeze-out scenarios, dark-matter particles are abundantly produced while still relativistic, and then decouple from the thermal bath due to the sudden increase of their mass above the universe temperature. This mass variation is sourced by the condensation of an order-parameter modulus, which is triggered when the temperature T(t) drops below the supersymmetry breaking scale M(t), which are both time-dependent. A cosmological attracto…
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…
We propose a new class of inflationary models in which inflation takes place while the inflaton is climbing up a potential hill due to a coupling to gravity. We study their attractor behavior, and investigate its relation with known attractors. We also discuss a possible realization of this type of models with the natural inflation, and show that the inflationary predictions come well within the region consistent with the observation of the cosmic microwave background.
Adiabatic (curvature) perturbations are produced during a period of cosmological inflation that is driven by a single scalar field, the inflaton. On particle physics grounds -- though -- it is natural to expect that this scalar field is coupled to other scalar degrees of freedom. This gives rise to oscillations between the perturbation of the inflaton field and the perturbations of the other scalar degrees of freedom, similar to the phenomenon of neutrino oscillations. Since the degree of the mixing is governed by the squared mass matrix of the scalar fields, the oscillations can occur even if…
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 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 discuss implications of the latest BICEP/Keck data release for inflationary models, with special emphasis on the cosmological attractors which can describe all presently available inflation-related observational data. These models are compatible with any value of the tensor to scalar ratio $r$, all the way down to $r = 0$. Some of the string theory motivated models of this class predict $10^{-3} \leq r \leq 10^{-2}$. The upper part of this range can be explored by the ongoing BICEP/Keck observations.
We reconsider the fine-tuning problem of scalar-driven inflation arising from the need to couple the inflaton to ordinary matter in order to make reheating efficient. Quantum fluctuations of this matter induce Coleman-Weinberg corrections to the inflaton potential, depending (for de Sitter background) in a complex way on the ratio of the inflaton to the Hubble parameter. These corrections are not Planck-suppressed and cannot be completely subtracted because they are not even local for a general geometry. A previous study showed that it is not satisfactory to subtract a local function of just t…
Recent cosmological observations are in good agreement with the scalar spectral index $n_s$ with $n_s-1\sim -2/N$, where $N$ is the number of e-foldings. Quadratic chaotic model, Starobinsky model and Higgs inflation or $α$-attractors connecting them are typical examples predicting such a relation. We consider the problem in the opposite: given $n_s$ as a function of $N$, what is the inflaton potential $V(ϕ)$. We find that for $n_s-1=-2/N$, $V(ϕ)$ is either $\tanh^2(γϕ/2)$ ("T-model") or $ϕ^2$ (chaotic inflation) to the leading order in the slow-roll approximation. $γ$ is the ratio of $1/V$ at…
