Starobinsky Inflation Research Articles

Scalaron-Higgs inflation

In scalaron-Higgs inflation the Standard Model Higgs boson is non-minimally coupled to gravity and the Einstein-Hilbert action is supplemented by the quadratic scalar curvature invariant. For the quartic Higgs self-coupling λ fixed at the electroweak scale, we find that the resulting inflationary two-field model effectively reduces to a single field model with the same predictions as in Higgs inflation or Starobinsky inflation, including the limit of a vanishing non-minimal coupling. For the same model, but with the scalar field a priori not identified with the Standard Model Higgs boson, we study the inflationary consequences of an extremely small λ. Depending on the initial conditions for the inflationary background trajectories, we find that the two-field dynamics either again reduces to an effective single-field model with a larger tensor-to-scalar ratio than predicted in Higgs inflation and Starobinsky inflation, or involves the full two-field dynamics and leads to oscillatory features in the inflationary power spectrum. Finally, we investigate under which conditions the inflationary scenario with extremely small λ can be realized dynamically by the Standard Model renormalization group flow and discuss how the scalaron-Higgs model can provide a natural way to stabilize the electroweak vacuum.

Fuzzy Euclidean wormholes in the inflationary universe

In this paper, we investigate complex-valued Euclidean wormholes in the Starobinsky inflation. Due to the properties of the concave inflaton potential, the classicality condition at both ends of the wormhole can be satisfied, as long as the initial condition of the inflaton field is such that it is located sufficiently close to the hilltop. We compare the probabilities of classicalized wormholes with the Hartle-Hawking compact instantons and conclude that the Euclidean wormholes are probabilistically preferred than compact instantons, if the inflation lasts more than 50 e-foldings. Our result assumes that the Euclidean path integral is the correct effective description of quantum gravity. This opens a new window for various future investigations that can be either confirmed or refuted by future experiments.

On the equivalence of Starobinsky and Higgs inflationary models in gravity and supergravity

Starobinsky inflation and Higgs inflation in gravity, and their equivalence based on the common inflationary potential are extended to supergravity in the proper framework, where the Starobinsky and Higgs descriptions of inflation arise in two different gauges of the same supergravity model.

Late-time acceleration and inflation in a Poincaré gauge cosmological model

A self-consistent model which can unify Starobinsky inflation and the ΛCDM model in the framework of Poincaré gauge cosmology (PGC) is studied in this work, without extra inflaton and “dark energy”. We start from the general nine-parameter PGC Lagrangian and get two Friedmann-like analytical solutions with the certain ghost- and tachyon-free conditions on the Lagrangian parameters. The scalar torsion h-determined solution is consistent with Starobinsky cosmology in the early-time and the pseudo-scalar torsion f-determined solution contains naturally a constant “dark energy” density, 3(α−1)2/8B0, which covers the ΛCDM model in the late-time. According to the latest observations, we estimate the magnitudes of parameters: α−1 ≃ 8.07 × 10−56, B0 ≃ 5.76 × 10−28 GeV−2 in the natural units.

Inflation model selection revisited

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

Note on stability in conformally connected frames

Dynamics in conformally connected frames are often referred to as 'equivalent'. In the context of cosmology, while the perturbations remain invariant for a single scalar field model, the background equations differ and therefore the dynamics. However, since the background dynamics are not the same, it is not clear whether the attractor nature of the solutions remains the same in all conformally connected frames, i.e., a stable solution in one frame implies an equivalent stable solution in another frame. To answer the question, in this work, we first consider power law cosmology in the Brans-Dicke theory as well as in the conformal Einstein frame. We show that, in this case, the attractor behavior is indeed equivalent under conformal transformation, i.e., an attractor solution in one frame implies an attractor solution in another frame. However, the decay rates of the deviations from the fixed points are different in the two frames. We are able to relate the behavior and find that the difference is due to the difference in e-fold 'clocks' in different frames, i.e., $\Delta N$ in different frames differ from one another. We show that the behavior is indeed true for any model in conformally connected frames and obtain the general 'equivalence' relation. In the context of inflation, we consider two models: Starobinsky and chaotic inflation, and explicitly point out the differences in these two frames. We show that the duration of inflation in any model in the Jordon frame is always higher than the Einstein frame.

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

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

Quasimatter bounce equivalent to Starobinsky inflation

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

Cosmological Probes of Supersymmetric Field Theory Models at Superhigh Energy Scales

The lack of positive results in searches for supersymmetric (SUSY) particles at the Large Hadron Collider (LHC) and in direct searches for Weakly Interacting Massive Particles (WIMPs) in the underground experiments may hint to a super-high energy scale of SUSY phenomena beyond the reach of direct experimental probes. At such scales the supergravity models based on Starobinsky inflation can provide the mechanisms for both inflation and superheavy dark matter. However, it makes the indirect methods the only way of testing the SUSY models, so that cosmological probes acquire the special role in this context. Such probes can rely on the nontrivial effects of SUSY physics in the early Universe, which are all model-dependent and thus can provide discrimination of the models and their parameters. The nonstandard cosmological features like Primordial Black Holes (PBHs) or antimatter domains in a baryon-asymmetric universe are discussed as possible probes for high energy scale SUSY physics.

Towards distinguishing variants of non-minimal inflation

We study models of inflation where the scalar field ϕ that drives inflation is coupled non-minimally to gravity via ξ ϕ2 R, or where the gravity sector is enlarged by an R2 term. We consider the original Higgs inflation, Starobinsky inflation, and two different versions of a scenario where the inflaton is a scalar field other than the Higgs, and discuss if they can be distinguished from each other by measuring the tensor-to-scalar ratio and runnings of the spectral index of primordial curvature perturbations, on top of the amplitude and spectral index of the perturbations. We consider both metric and Palatini theories of gravity, showing how detailed studies of non-minimally coupled models can help to identify the inflaton field and how they may provide for a way to also distinguish between different theories of gravity in the present context.

Inflation in loop quantum cosmology

We develop a consistent analytic approach to determine the conditions under which slow roll inflation can arise when the inflaton is the same scalar field that is responsible for the bounce in Loop Quantum Cosmology (LQC). We find that the requirement that the energy density of the field is fixed at the bounce having to match a critical density has important consequences for its future evolution. For a generic potential with a minimum, we find different scenarios depending on the initial velocity of the field and whether it begins life in a kinetic or potential energy dominated regime. For chaotic potentials that start in a kinetic dominated regime, we find an initial phase of superinflation independent of the shape of the potential followed by a damping phase which slows the inflaton down, forcing it to turnaround and naturally enter a phase of slow-roll inflation. If we begin in a potential energy dominated regime, then the field undergoes a period where the corrections present in LQC damp its evolution once again forcing it to turnaround and enter a phase of slow roll inflation. On the other hand we show for the Starobinsky potential that inflation never occurs when we begin in a potential dominated regime. In fact traditional Starobinsky inflation has to start in a kinetic energy dominated regime, with corresponding tighter constraints on the initial value of the field for successful inflation than in the conventional case. Comparing our analytic results to published numerical ones, we find remarkable agreement especially when we consider the different epochs that are involved. In particular the values of key observables obtained from the two approaches are in excellent agreement, opening up the possibility of obtaining analytic results for the evolution of the density perturbations in these models.

Polonyi–Starobinsky supergravity with inflaton in a massive vector multiplet with DBI and FI terms

We propose the Starobinsky-type inflationary model in the matter-coupled N = 1 four-dimensional supergravity with the massive vector multiplet that has inflaton (scalaron) and goldstino amongst its field components, whose action includes the Dirac–Born–Infeld-type kinetic term and the generalized (new) Fayet–Iliopoulos-type term, without gauging the R-symmetry. The N = 1 chiral matter (‘hidden sector’) is described by the modified Polonyi model needed for spontaneous supersymmetry breaking after inflation. We compute the bosonic action and the scalar potential of the model, and show that it can accommodate the positive (observed) cosmological constant (as the dark energy) and the spontaneous supersymmetry breaking at high scale after the Starobinsky inflation.

Pre-inflationary dynamics of Starobinsky inflation and its generization in loop quantum Brans-Dicke cosmology

Recently, the nonperturbative quantization scheme of loop quantum gravity has been extended to the Brans-Dicke theory and the corresponding loop quantum Brans-Dicke cosmology has been derived, which provides an essential platform to explore inflationary models in this framework. In this paper, we consider two inflation models, the Starobinsky and α-attractor inflation whose cosmological predictions are in excellent agreement with Planck data, and study systematically their pre-inflationary dynamics as well as the slow-roll inflation. We show that for both models, the background evolution of a flat Friedmann-Lemaître-Robertson-Walker universe in general can be divided into three different phases: the pre-inflationary quantum phase, quantum-to-classical transition, and the slow-roll inflation. The pre-inflationary dynamics are dominated by the quantum geometry effects of loop quantum Brans-Dicke cosmology and the corresponding Universe could be either initially expanding or contracting, depending on the initial velocity of inflaton field. It is shown that the detailed evolution of pre-inflationary quantum phase also depend on specific inflation models. After the pre-inflationary quantum phase, the universe gradually evolves into the slow-roll inflation with some of initial conditions for Starobinsky and α-attractor potentials. In addition, to be consistent with observational data, we also find the restricted parameter space of initial conditions that could produce at least 60 e-folds during the slow-roll inflation.

Logarithmic-corrected R2 gravity inflation in the presence of Kalb-Ramond fields

In this paper we shall study the inflationary aspects of a logarithmic corrected R2 Starobinsky inflation model, in the presence of a Kalb-Ramond field in the gravitational action of F(R) gravity. Our main interest is to pin down the effect of this rank two antisymmetric tensor field on the inflationary phenomenology of the F(R) gravity theory at hand. The effects of the Kalb-Ramond field are expected to be strong during the inflationary era, however as the Universe expands, the energy density of the Kalb-Ramond field scales as ∼ a−6 so dark matter and radiation dominate over the Kalb-Ramond field effects. In general, antisymmetric fields constitute the field content of superstring theories, and thus their effect at the low-energy limit of the theory is expected to be significant. As we will show, for a flat Friedmann-Robertson-Walker metric, the Kalb-Ramond field actually reduces to a scalar field, so it is feasible to calculate the observational indices of inflation. We shall calculate the spectral index and the tensor-to-scalar ratio for the model at hand, by assuming two conditions for the resulting Kalb-Ramond scalar field, the slow-roll and the constant-roll condition. As we shall demonstrate, in both the slow-roll and constant-roll cases, compatibility with the latest observational data can be achieved. Also the effect of the Kalb-Ramond field on the inflationary phenomenology is to increase the amount of the predicted primordial gravitational radiation, in comparison to the corresponding f(R) gravities, however the results are still compatible with the observational data.

Electroweak relaxation of cosmological hierarchy

A simple model for the late-time cosmic acceleration problem is presented in the Starobinsky inflation with a negative bare cosmological constant as well as a nonminimal coupling to the Higgs boson. After electroweak symmetry breaking, the Starobinsky inflaton has been frozen until very recently, becoming a thawing quintessence, and a comparable magnitude to the observed dark energy density can be achieved without fine-tuning. Our proposal essentially reduces the cosmological constant problem into the electroweak hierarchy problem, and its late-time behaviour is also consistent with the recently proposed swampland criteria.

Nonlinear anisotropy growth in Bianchi-I spacetime in metric f(R) cosmology

The present work is related to anisotropic cosmological evolution in metric $f(R)$ theory of gravity. The initial part of the paper develops the general cosmological dynamics of homogeneous anisotropic Bianchi-I spacetime in $f(R)$ cosmology. The anisotropic spacetime is pervaded by a barotropic fluid which has isotropic pressure. The paper predicts nonlinear growth of anisotropy in such spacetimes. In the later part of the paper we display the predictive power of the nonlinear differential equation responsible for the cosmological anisotropy growth in various relevant cases. We present the exact solutions of anisotropy growth in Starobinsky inflation driven by quadratic gravity and exponential gravity theory. Semi-analytical results are presented for the contraction phase in quadratic gravity bounce. The various examples of anisotropy growth in Bianchi-I model universe shows the complex nature of the problem at hand.

Inflation as a spontaneous symmetry breaking of Weyl symmetry

In this paper we study a novel realization of inflation, based on Weyl invariant gravity with torsion. We show that requiring the classical action for the scalar field to be Weyl invariant introduces a dilaton which induces a non trivial modification of the field space geometry of the scalar sector, which allows for inflationary phase that begins at the conformal point of the inflaton ψ, i.e. ⟨ψ⟩=0. Since the model is Weyl invariant, the inflaton condensation models a process of spontaneous Weyl symmetry breaking. For a wide range of parameters the spectral observables of the model are in good agreement with the CMB measurements, such that the scalar spectral index and the tensor-to-scalar ratio approximately agree with those of Starobinsky's inflation, i.e. ns≃ 0.96–0.97 and r ≈ 3× 10−3. The simplest version of our model contains two scalar degrees of freedom, one of them being an exactly flat direction. If that degree is excited early on in inflation and if inflation lasts for about 60 e-foldings, we find that the Universe undergoes a short period of kination that predates inflation. Such a period strongly suppresses the amplitude of large scale CMB temperature fluctuations providing thus an elegant explanation for the lack of power in the lowest CMB multipoles.

Black hole remnants may exist if Starobinsky inflation occurred

Zero temperature black hole solutions to the semiclassical backreaction equations are investigated. Evidence is provided that certain components of the stress-energy tensors for free quantum fields at the horizon only depend on the local geometry near the horizon. This allows the semiclassical backreaction equations to be solved near the horizon. It is found that macroscopic uncharged zero temperature black hole solutions to the equations may exist if the coefficient of one of the higher derivative terms in the gravitational Lagrangian is large enough and of the right sign for Starobinsky inflation to have occurred in the early Universe.

Consistent early and late time cosmology from the RG flow of gravity

We investigate the compatibility of cosmological constraints on inflation and the cosmological constant with the asymptotic safety scenario of quantum gravity. The effective action is taken to be of f(R) form, truncated to second order. The flow generated by the Functional Renormalisation Group Equation is analysed and it is found to allow for trajectories that are compatible with the observational constraints on the parameters of the action, both at early and late cosmological times. In particular, the gravitational effective dynamics generated in the trans-Planckian regime flows to Starobinsky inflation at early times and to standard Einstein Gravity with a cosmological constant at late times. Moreover, the cosmological constant acquires an energy dependence at 10−2 eV, increasing from its current value of 10−66 eV2 on Hubble scale to a value of 1030 eV2 at inflation scale.

Two-loop corrections to Starobinsky-Higgs inflation

Higgs inflation and $R^2$-inflation (Starobinsky model) are two limits of the same quantum model, hereafter called Starobinsky-Higgs. We analyse the two-loop action of the Higgs-like scalar $\phi$ in the presence of: 1) non-minimal coupling ($\xi$) and 2) quadratic curvature terms. The latter are generated at the quantum level with $\phi$-dependent couplings ($\tilde\alpha$) even if their tree-level couplings ($\alpha$) are tuned to zero. Therefore, the potential always depends on both Higgs field $\phi$ and scalaron $\rho$, hence multi-field inflation is a quantum consequence. The effects of the quantum (one- and two-loop) corrections on the potential $\hat W(\phi,\rho)$ and on the spectral index are discussed, showing that the Starobinsky-Higgs model is in general stable in their presence. Two special cases are also considered: first, for a large $\xi$ in the quantum action one can integrate $\phi$ and generate a "refined" Starobinsky model which contains additional terms $\xi^2 R^2\ln^p (\xi \vert R\vert/\mu^2)$, $p=1,2$ ($\mu$ is the subtraction scale). These generate corrections linear in the scalaron to the "usual" Starobinsky potential and a "running" scalaron mass. Second, for a small fixed Higgs field $\phi^2 \ll M_p^2/\xi$ and a vanishing classical coefficient of the $R^2$-term, we show that the "usual" Starobinsky inflation is generated by the quantum corrections alone, for a suitable non-minimal coupling ($\xi$).