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Inflationary cosmology in modified gravity theories

Kazuharu Bamba, Sergei D. Odintsov

arXiv:1503.00442v1hep-thastro-ph.COgr-qchep-ph

TL;DR

The paper reviews modified-gravity inflation to generalize Starobinsky’s R^2 model and covers gravity modifications, quantum-anomaly effects, and R^2 inflation in loop quantum cosmology. It uses conformal-frame reformulations and related reconstruction approaches, reporting Planck-compatible scalar spectral indices and tensor-to-scalar ratios while also discussing F(R) bounce cosmology.

  • Problem

    The paper addresses how inflationary models in modified gravity can generalize Starobinsky inflation and reproduce Planck-compatible observables.

  • Method

    The paper reviews inflation from gravity modifications, the quantum anomaly, and R^2 gravity in loop quantum cosmology, together with F(R) reconstruction and bounce cosmology.

  • Results

    The reviewed inflationary models yield scalar spectral indices and tensor-to-scalar ratios compatible with Planck results.

  • Takeaways & Limitations

    Modified-gravity inflation provides models consistent with the stated Planck observables, while F(R) gravity also supports discussion of cosmological bounce scenarios.

Abstract

from arXiv · show

We review inflationary cosmology in modified gravity such as $R^2$ gravity with its extensions in order to generalize the Starobinsky inflation model. In particular, we explore inflation realized by three kinds of effects: modification of gravity, the quantum anomaly, and the $R^2$ term in loop quantum cosmology. It is explicitly demonstrated that in these inflationary models, the spectral index of scalar modes of the density perturbations and the tensor-to-scalar ratio can be consistent with the Planck results. Bounce cosmology in $F(R)$ gravity is also explained.

I. INTRODUCTION

The paper reviews inflationary cosmology in modified gravity, extending Starobinsky’s R^2 model and considering quantum-anomaly and loop-quantum-cosmology effects. It aims to explain inflationary models compatible with Planck results and related F(R) bounce cosmology.

  • Observational context: Trace-anomaly-driven inflation with an R^2 term is supported by WMAP and Planck data.The R^2 or higher-derivative □R contribution is treated as an effective gravitational action and can produce sufficiently long inflation with a graceful exit.
  • Scope: The review generalizes Starobinsky inflation by studying R^2 gravity and further F(R) extensions.R^2 gravity combines the Einstein-Hilbert term with an R^2 term and can be viewed as a form of F(R) gravity.
  • Inflationary mechanisms: It examines inflation driven by gravity modifications, the quantum trace anomaly, and the R^2 term in loop quantum cosmology.The LQC treatment incorporates quantum effects into the inflationary framework.
  • Additional topic: The review also presents recent consequences of reconstructing F(R) gravity for cosmological bounce scenarios.Its organization includes a dedicated discussion of reconstructing F(R) gravity to realize a bounce in LQC.

B. Slow-roll inflation

The paper formulates F(R) inflation through an Einstein-frame scalar field and analyzes its slow-roll dynamics. It then describes reconstructing F(R) models from Einstein- or Jordan-frame descriptions to extend Starobinsky inflation.

  • Slow-roll conditions: Slow-roll inflation assumes potential domination and negligible inflaton acceleration, requiring the slow-roll parameters to remain much smaller than unity.The framework uses 3H^2/κ^2 ≈ V(ϕ), |ϕ̈| ≪ |3Hϕ̇|, and ε, η ≪ 1.
  • Inflationary observables: The number of e-folds is defined by N_e ≡ ln(a_f/a_i) and relates the inflationary scale-factor growth to the scalar-field evolution.The initial and final values correspond to the beginning and end of inflation.
  • Inflationary observables: The scalar spectral index and tensor-to-scalar ratio are expressed through the slow-roll parameters as n_s−1 = −6ε+2η and r = 16ε.These relations connect the slow-roll dynamics to the observables used in Planck comparisons.
  • Reconstruction: The review reconstructs F(R) models either from an Einstein-frame potential or directly in the Jordan frame to generalize Starobinsky inflation.The stated purpose is to study extended forms of the Starobinsky potential and reconstruct the corresponding Jordan-frame action.

1. Extension of the Starobinsky inflation model

The extended Starobinsky model introduces modified higher-curvature structure and analyzes its inflationary dynamics. For 60 e-folds, the model yields scalar and tensor observables compatible with Planck constraints.

  • Model construction: The reconstructed model reduces to Starobinsky inflation when c1 = c3, while c1 > c3 gives an extension with a cosmological constant.The construction reproduces the Einstein-Hilbert term through the parameter choice −c2/(2c1) = 1.
  • Model assumptions: The model assumes c3 = 0 and introduces a positive parameter γ1 with γ1/M_Pl ≪ 1 so higher-curvature corrections can appear.The inflaton rolls from a large negative amplitude toward a minimum at V(ϕ = 0) = −γ1/(4κ^2).
  • Inflationary dynamics: Exponential inflation is realized with a(t) = a_i exp(H_inf t), and the analysis derives the corresponding inflaton solution and slow-roll parameters.The inflationary solution is obtained from the gravitational field equations and is accompanied by an approximately specified initial field value.
  • Observational compatibility: For N_e = 60, the model gives n_s = 0.967 and r = 3.00 × 10^-3, consistent with the stated Planck constraints.The comparison uses Planck’s n_s = 0.9603±0.0073 and r < 0.11 bounds.
  • Context: The discussion situates the model among studies of scalar-field, perfect-fluid, and F(R)-gravity descriptions and quantum corrections to inflation.It also notes prior BICEP2-related observations and discussions of foreground subtraction.

2. Power-law corrections to general relativity

The paper studies power-law corrections to general relativity and reconstructs F(R) gravity directly in the Jordan frame. The resulting examples produce inflationary observables compatible with Planck analysis, while frame equivalence remains debated.

  • Power-law model: A generic power-law correction is added to the Einstein-Hilbert term, with β > 0, constants R_c and Λ_p, and q > 1 with q ≠ 2.The model is analyzed as an extended gravitational form for inflation.
  • Power-law results: For q = 1.99, the model yields n_s = 0.962 and r = 1.08 × 10^-3, compatible with Planck analysis.The paper presents this as an inflationary result of the power-law-corrected model.
  • Frame dependence: The paper notes that Jordan- and Einstein-frame cosmology may differ and treats them as different cosmological theories for discussion.It reports debates over conformal-frame equivalence and acknowledges that results may differ without the scalar-tensor transformation.
  • Jordan-frame reconstruction: Jordan-frame reconstruction uses the e-fold variable and Hubble function to express R and derive a second-order differential equation for F(R).The procedure solves for N̄(R), then formulates the Friedmann equation in terms of F_R and F_RR.
  • Jordan-frame results: For two parameter choices, the reconstructed model gives (n_s, r) = (0.963, 6.89 × 10^-2) and (0.965, 5.84 × 10^-2), respectively.The paper states that both values of n_s and r are indicated by the Planck analysis.

III. TRACE-ANOMALY DRIVEN INFLATION IN MODIFIED GRAVITY

The section reviews how the quantum trace anomaly arises in curved-space quantum field theory and contributes to modified-gravity inflation. It specifies anomaly coefficients for scalar, fermion, and vector fields while neglecting graviton and higher-derivative conformal-scalar contributions.

  • Quantum anomaly: The quantum anomaly appears through renormalization and is formulated using the trace of the energy-momentum tensor in four-dimensional spacetime.The formulation uses curvature invariants including the Weyl-tensor square, Gauss-Bonnet invariant, and covariant d’Alembertian.
  • Quantum anomaly: The anomaly coefficients α1, α2, and α3 depend on the numbers of real scalar, Dirac fermion, and vector fields.The review defines these coefficients explicitly and neglects graviton and higher-derivative conformal-scalar contributions.
  • Quantum anomaly: The qualitative consequences do not depend on the signs of α1, α2, and α3, although α1 and α2 are set positive in the analysis.For N = 4 SU(N) super Yang-Mills theory, the coefficients are related to the field multiplicities through a large- N parameter.
  • Quantum anomaly: The R2 term acts as a higher-curvature correction to Einstein gravity or contributes to the energy-momentum tensor as matter.This interpretation is given for Yang-Mills theory in curved spacetime.

B. F(R) gravity with the quantum anomaly

This section combines the quantum-anomaly action with R2 gravity and an additional function f(R). It derives the corresponding gravitational equations and neglects the radiation integration constant during inflation near the Planck scale.

  • F(R) gravity with the quantum anomaly: The modified action contains the quantum-anomaly action, the R2 term, and an additional function f(R).For original Starobinsky inflation, the additional function satisfies f(R) = 0.
  • F(R) gravity with the quantum anomaly: The trace equation and FLRW field equations are expressed through f(R), its derivative fR(R), and the effective energy density and pressure.The effective quantities obey the conservation equation ρ̇eff + 3H(ρeff + Peff) = 0.
  • F(R) gravity with the quantum anomaly: Quantum-anomaly contributions enter the effective energy density and pressure through higher-derivative terms involving H and its time derivatives.The derivation uses conservation of the effective energy density and pressure.
  • F(R) gravity with the quantum anomaly: The radiation integration constant is set to zero because radiation can be neglected relative to the quantum anomaly and modified-gravity deviation during inflation near the Planck scale.The constant corresponds to the energy density of radiation in the quantum state.

C. de Sitter solutions by the trace anomaly

The section studies de Sitter solutions generated by trace-anomaly effects in exponential F(R) gravity and analyzes their stability and inflationary observables. The model can produce an unstable inflationary solution with a graceful exit and Planck-consistent perturbation parameters.

  • C. de Sitter solutions by the trace anomaly: For R/Rc ≪ 1, f(R) approaches zero, whereas for R/Rc ≫ 1 the Λc term acts as a cosmological constant during the early universe.Thus the model approaches R2 gravity with the quantum anomaly at late times and includes a cosmological-constant-like contribution during inflation.
  • C. de Sitter solutions by the trace anomaly: Expanding the exponential correction makes the trace-anomaly-driven de Sitter solution unstable, which can lead to a graceful exit from inflation.The instability condition is derived for the de Sitter solution with positive α2 and α4.
  • C. de Sitter solutions by the trace anomaly: The de Sitter solution is required to be unstable if it is to describe inflation, because inflation must end.Perturbations are introduced as H = Hde Sitter + δH(t) and analyzed through their characteristic exponents.
  • C. de Sitter solutions by the trace anomaly: For u = 3, Λcα2κ2 = 0.125, and Ne = 76, the model gives ns = 0.960 and r = 1.20 × 10^-3.These quantities are evaluated for trace-anomaly-driven inflation in exponential gravity during slow roll.
  • C. de Sitter solutions by the trace anomaly: The exponential-gravity trace-anomaly model can explain the inflationary observables considered in the section.The de Sitter solution used for this conclusion is the inflationary solution derived in the high-curvature limit.

IV. R2 GRAVITY IN LOOP QUANTUM COSMOLOGY (LQC)

The section reviews R2 gravity with holonomy corrections in loop quantum cosmology and examines its curvature and Hubble-rate behavior. Under a stated parameter condition, curvature and |H| remain bounded, preventing singularities in this setting.

  • R2 gravity in LQC: The analysis formulates F(R) gravity in the Einstein frame within loop quantum cosmology using Hamiltonian dynamics and holonomy corrections.The construction assumes a spatially flat FLRW background and uses the LQC Poisson-bracket relation.
  • R2 gravity in LQC: The LQC Hamiltonian constraint yields a Friedmann equation with holonomy corrections and a critical energy density ρcritical.The review uses a Hilbert space of almost periodic functions and a Hamiltonian with general holonomy corrections.
  • R2 gravity in LQC: R2 gravity can develop curvature singularities in the early universe, whereas its LQC formulation can avoid them.The section explicitly analyzes the R2 action in LQC to establish this difference.
  • R2 gravity in LQC: For αSκ2 < 1/(18ρcritical), the absolute value of R is bounded.The Einstein-frame potential is used in deriving the curvature bound.
  • R2 gravity in LQC: The bounded curvature implies bounded |H|, and no singularity appears in R2 gravity for LQC.The Hubble-rate relation connects the Einstein-frame and Jordan-frame descriptions.

C. Loop quantum R2 gravity in the Einstein frame

The Einstein-frame analysis of loop-quantum-corrected R^2 gravity finds trajectories that contract, bounce at critical density, and re-expand around a critical point. The resulting inflationary observables can agree with Planck data and recover Starobinsky inflation when holonomy corrections vanish.

  • R^2 gravity is analyzed in the Einstein frame, where the equations are simpler than in the Jordan frame.
  • The dynamics are symmetric between contraction and expansion under (Ĥ, t̂) → (−Ĥ, −t̂).A contraction trajectory maps to a corresponding expansion trajectory.
  • All trajectories begin and end at the critical point (Ψ̂, dΨ̂/dt̂) = (1, 0), corresponding to both the universe’s beginning and end.The critical point has ρ̂ = 0.
  • Holonomy corrections produce a bounce at ρ̂ = ρ̂critical as contraction changes into expansion.Trajectories approach the critical-density line, bounce, and then return toward the critical point.
  • For N̂e = 68.0 and 8αSκ^2ρ̂critical = 8.50, the model gives n̂s = 0.967 and r̂ = 2.55 × 10^-3, compatible with Planck data.In the limit ρ̂critical → ∞, the observables become those of the original Starobinsky model.

D. Loop quantum R2 gravity in the Jordan frame

The Jordan-frame analysis shows that holonomy corrections enable a cosmological bounce, while their absence produces an early-universe singularity. The framework also reconstructs F(R) gravity models capable of realizing matter-bounce cosmology.

  • A Jordan-frame bounce occurs when an Einstein-frame trajectory intersects the curve on which H becomes zero.The negative-sign case applies for Ψ̂ > 1, while the positive-sign case applies for 0 < Ψ̂ < 1.
  • The relevant Jordan-frame curve is an ellipse, parabola, or hyperbola according to the sign of B+ for 0 < Ψ̂ < 1.The curve is an ellipse for B+ > 0, a parabola for B+ = 0, and a hyperbola for B+ < 0.
  • In the Jordan frame, the universe begins and ends at (H, Ḣ) = (0, 0).
  • The Jordan-frame bounce is generated by holonomy corrections; without them, a singularity appears at the universe’s early stage.
  • The paper reconstructs an F(R) theory in which a matter bounce can occur within loop quantum cosmology.The construction uses a matter density ρ = ρ̄m / [(3/4)t^2 + 1] and solves for the corresponding F(R).

VI. CONCLUSIONS

The paper reviews modified-gravity inflation, including extensions of Starobinsky R^2 gravity, quantum-anomaly inflation, and R^2 inflation with loop-quantum-cosmology corrections. It also surveys bounce cosmology in F(R) gravity and reports Planck-compatible perturbation observables in the three inflationary models.

  • Inflationary models: The review studies inflation from gravity-modification terms, quantum anomalies, and R^2 gravity with loop-quantum-cosmology holonomy corrections.These approaches extend or generalize the Starobinsky inflation framework.
  • Modified gravity: The authors transform F(R) models from the Jordan frame to the Einstein frame and reconstruct inflationary potentials with slow-roll dynamics.The Einstein-frame description is presented as general gravity plus a scalar-field theory.
  • Modified gravity: The reviewed F(R) models include extensions of R^2 gravity and general relativity with power-law correction terms.These models are considered as extended versions of the Starobinsky inflation model.
  • Bounce cosmology: Holonomy corrections allow bounces in the Jordan frame and remove cosmic singularities that appear in ordinary R^2 gravity.The review analyzes R^2 gravity for loop quantum cosmology in both Einstein and Jordan frames.
  • Observational consistency: The three inflationary models yield a scalar spectral index and tensor-to-scalar ratio compatible with Planck data.The conclusion reports this compatibility across the modified-gravity, quantum-anomaly, and loop-quantum-cosmology settings.
  • Bounce cosmology: The review also discusses reconstructed F(R) models realizing matter-bounce, super-bounce, and ekpyrotic scenarios in loop quantum cosmology.The super-bounce is described as exhibiting two-times bounce behaviors.
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