[Paper Review] Inflation from a Non-Local Theory of Gravity

Philip Stephens|ArXiv.org|Aug 19, 2009

Cosmology and Gravitation Theories3 references3 citations

TL;DR

This paper investigates inflation in a non-local gravity theory with an infinite series of higher-derivative curvature terms $ f(R, \square R) = R + c_0 R^2 + \sum_{i=1}^\infty c_i R \square^i R $, showing it yields dynamics identical to $ R + R^2 $ gravity and chaotic inflation with a $ \phi^4 $ potential. The model reproduces COBE-normalized scalar perturbations and predicts a scalar-to-tensor ratio of $ r \approx 0.24 $, slightly above WMAP constraints but potentially reconcilable with $ N_{\text{remaining}} = 70 $ e-folds.

ABSTRACT

This paper studies the inflationary dynamics of a Non-Local Theory of gravity. This theory, based around derivatives of the Ricci Scalar in the Einstein-Hilbert action, was previously found to be Ghost free, and to give rise to a bouncing cosmology. Perturbation spectra are calculated and found to be similar to those for chaotic inflation.

Motivation & Objective

• To investigate inflationary dynamics in a non-local gravity theory with arbitrarily high-derivative curvature terms.
• To determine whether such a theory can generate viable inflation without introducing a fundamental scalar field.
• To evaluate the model's consistency with cosmological data, particularly COBE normalization and WMAP constraints on spectral index and tensor-to-scalar ratio.
• To compare the results directly with $ R+R^2 $ gravity and chaotic inflation without relying on conformal transformations.
• To assess the phenomenological viability of the model in light of current observational bounds.

Proposed method

• The paper uses direct variational methods on the action $ \mathcal{L} = f(R, \square R)\sqrt{-g} $, avoiding conformal transformations.
• It derives the trace of the field equations for the non-local Lagrangian $ f(R, \square R) = R + c_0 R^2 + \sum_{i=1}^\infty c_i R \square^i R $, reducing to a differential equation in the Ricci scalar $ R $.
• Under homogeneous and isotropic (FRW) conditions, the d’Alembertian reduces to time derivatives, allowing exponential solutions for $ R \propto e^{\beta t} $.
• The model is analyzed for inflationary solutions by solving the resulting ODEs for $ R $, leading to exponential growth in the scale factor.
• Perturbation spectra are computed using standard inflationary formalism, with $ \delta_H^2(k) $ and spectral index $ n $ evaluated at $ N_{\text{remaining}} = 50 $ e-folds.
• The COBE normalization and tensor-to-scalar ratio $ r $ are computed and compared to WMAP data, with sensitivity to the choice of $ N_{\text{remaining}} $.

Experimental results

Research questions

• RQ1 Can a non-local gravity theory with an infinite series of $ R \square^i R $ terms generate viable inflation without a fundamental scalar field?
• RQ2 Is the inflationary dynamics of this non-local theory equivalent to $ R + R^2 $ gravity or chaotic $ \phi^4 $ inflation?
• RQ3 What are the predictions for the scalar spectral index $ n $, its running, and the tensor-to-scalar ratio $ r $, and how do they compare to WMAP and COBE data?
• RQ4 Does the model remain ghost-free and asymptotically free, as previously claimed, and does this affect inflationary dynamics?
• RQ5 Can the model be reconciled with observational constraints if the number of e-folds before horizon exit is increased beyond 50?

Key findings

• The non-local gravity model $ f(R, \square R) = R + c_0 R^2 + \sum_{i=1}^\infty c_i R \square^i R $ yields inflationary dynamics identical to $ R + R^2 $ gravity.
• The model is dynamically equivalent to chaotic inflation with a $ \phi^4 $ potential, with the scalar field identified via direct field equations rather than conformal transformation.
• The scalar spectral index is $ n = 0.941 $, and its running is $ dn/d\ln K = 5.77 \times 10^{-4} $, consistent with WMAP data.
• The COBE normalization is satisfied with $ M = 6.16 \times 10^{-6} M_{\text{Pl}} \approx 1.5 \times 10^{13} \, \text{GeV}/c^2 $, matching known results from conformal methods.
• The predicted tensor-to-scalar ratio is $ r = 0.24 $, slightly above the WMAP 2-\sigma upper limit of $ r < 0.2 $, but potentially reconcilable with $ N_{\text{remaining}} = 70 $ e-folds.
• The model avoids the need for a fundamental scalar field, with inflation driven purely by non-local gravity terms, and energy density stored in $ R $ decaying at the end of inflation.