[Paper Review] Cosmology of non-local f(R) gravity

Ivan Dimitrijević, Бранко Драгович+3 more|arXiv (Cornell University)|Sep 14, 2015

Cosmology and Gravitation Theories22 references4 citations

TL;DR

This paper proposes a non-local f(R) gravity model with a generalized non-local term P(R)ℱ(□)Q(R), motivated by string and p-adic string theories. It derives the spectrum, establishes a ghost-free condition, and identifies algebraic criteria for classical stability of de Sitter solutions—showing that p+q=0 and p+q=2 always yield stable de Sitter phases, enabling potential unification of bouncing cosmologies and inflation.

ABSTRACT

We consider a modification of GR with a special type of a non-local f(R). The structure of the non-local operators is motivated by the string field theory and p-adic string theory. The spectrum is derived explicitly and the ghost-free condition for the model is formulated. We pay special attention to the classical stability of the de Sitter solution in our model and formulate the conditions on the model parameters to have a stable configuration. Relevance of unstable configurations for the description of the de Sitter phase during inflation is specifically discussed.

Motivation & Objective

• To develop a generalized non-local f(R) gravity model with P(R)ℱ(□)Q(R) structure, motivated by string and p-adic field theories.
• To derive the spectrum and formulate a ghost-free condition for the model, ensuring theoretical viability.
• To investigate classical stability of de Sitter solutions under general P(R) = R^p and Q(R) = R^q forms.
• To determine whether stable de Sitter configurations can be achieved, enabling a transition from bounce to inflation.
• To explore the cosmological relevance of unstable de Sitter phases for inflationary exit mechanisms.

Proposed method

• Formulate the action with a non-local operator ℱ(□) acting on curvature invariants P(R) and Q(R), using analytic functions of the d’Alembert operator.
• Derive the linearized equations of motion around de Sitter spacetime to analyze perturbations and stability.
• Apply the ghost-free condition requiring the operator ℒ(□) = M_P² + λR^{p+q-1}(p+q)(2−p−q)ℱ₀ to have at most one root and ω² > 0.
• Use the trace equation (61) and background constraint (64) to derive stability conditions in terms of model parameters.
• Reduce the stability condition to the inequality M_P² + λR^{p+q−1}(p+q)(2−p−q)f₀ > 0, which ensures real and positive ω².
• Analyze special cases (p+q=0, p+q=2, p=q=1/2) to identify stable configurations and their cosmological implications.

Experimental results

Research questions

• RQ1 Under what conditions is the de Sitter solution classically stable in the proposed non-local f(R) gravity model?
• RQ2 Can the ghost-free condition be explicitly formulated in terms of the model’s functional parameters?
• RQ3 Is it possible to unify a bouncing cosmology with inflation in a single non-local gravity model via stable de Sitter phases?
• RQ4 How do the parameters p and q in P(R)=R^p and Q(R)=R^q affect the stability of de Sitter solutions?
• RQ5 What is the role of the non-local scale ℳ in controlling the order of non-locality and cosmological behavior?

Key findings

• The ghost-free condition is satisfied if the operator ℒ(□) has at most one root, σ=0 or 1, and ω² > 0 in the linearized spectrum.
• The de Sitter phase is stable if the inequality M_P² + λR^{p+q−1}(p+q)(2−p−q)f₀ > 0 holds, ensuring real and positive ω².
• For p+q=0, the model corresponds to a generalized cosmological constant and always yields a stable de Sitter solution.
• For p+q=2, the model reduces to a generalized R² theory, which is a candidate for UV-complete quantum gravity and always yields stable de Sitter solutions.
• The case p=q=1/2 leads to a solvable system that reduces to a local gravity action, though solving the resulting equation (18) remains a complex open problem.
• Models with P=Q=R (p+q=2) produce eternally stable de Sitter phases, making them incompatible with inflationary exit unless modified, thus precluding direct use in bounce-inflation unification.