[Paper Review] Reconstructing the f(R) gravity from the holographic principle

L. N. Granda|ArXiv.org|Dec 9, 2008

Cosmology and Gravitation Theories2 references3 citations

TL;DR

This paper proposes a holographic f(R) gravity model by matching the geometrical effective energy density in f(R) gravity to the holographic energy density, using a novel infrared cut-off dependent on Hubble parameter and its derivative. The reconstruction yields a power-law f(R) Lagrangian compatible with supernova data, valid at low redshift, with distinct behaviors for quintessence (ω > -1) and phantom (ω < -1) dark energy phases.

ABSTRACT

An holographic f(R) gravity model of dark energy is proposed. The correspondence between the f(R) geometrical effective energy density with the holographic density, allows the reconstruction of the f(R) gravity in flat FRW background in the Einstein frame. The proposed infrared cut-off for the holographic energy density depends on two parameters which are fit using the luminosity versus redshift data, allowing a suitable reconstruction in two representative cases of the EoS parameter: for $ω&gt;-1$ and $ω

Motivation & Objective

• To construct a phenomenological f(R) gravity model that effectively describes holographic dark energy in the late-time universe.
• To use the holographic principle with a specific infrared cut-off (αH² + βḢ) to constrain the form of f(R) gravity.
• To fit the model parameters α and β using luminosity-distance versus redshift data from type Ia supernovae.
• To reconstruct f(R) in the Einstein frame for two representative cases: ω > -1 (quintessence) and ω < -1 (phantom).
• To ensure consistency with local gravity tests via boundary conditions on f′(R) and f′′(R) at z = 0.

Proposed method

• Adopts a holographic energy density ρΛ = 3Mp²(αH² + βḢ), where α and β are free parameters constrained by observational data.
• Assumes dark energy dominance in the Friedmann equation, leading to H² = αH² + βḢ, which yields a power-law solution H ∝ 1/t.
• Derives the luminosity distance dL(z) from the redshift-dependent Hubble parameter H(z) = H₀(1+z)^{(α−1)/β}.
• Reconstructs f(R) by expressing the Hubble parameter in terms of the Ricci scalar R, using the relation R ∝ H² + Ḣ.
• Solves the differential equation for f(R) in terms of redshift, transforming to R-dependent form via z → R using the power-law behavior.
• Applies consistency conditions f′(R) = 1 and f′′(R) = 0 at z = 0 to determine integration constants Φ⁺₀ and Φ⁻₀.

Experimental results

Research questions

• RQ1 Can the holographic energy density with the proposed IR cut-off αH² + βḢ be consistently matched to an f(R) gravity model?
• RQ2 How does the reconstructed f(R) Lagrangian behave for different dark energy equation-of-state parameters (ω > -1 and ω < -1)?
• RQ3 What is the functional form of f(R) that reproduces the luminosity distance data from type Ia supernovae under this holographic framework?
• RQ4 How do the parameters α and β, derived from observational data, influence the curvature power-law structure of f(R)?
• RQ5 To what extent does the reconstructed f(R) model remain consistent with local gravity constraints at z = 0?

Key findings

• The reconstructed f(R) model takes the form of a power-law Lagrangian with three terms: one proportional to R^{3p/2}, and two with exponents (p+3±√((p−1)²+8p))/4.
• For α > 1 and β > 0, the model describes a quintessence-like phase (ω > -1); for α < 1 and β > 0, it describes a phantom phase (ω < -1).
• The model is valid at low redshift, where the power-law approximation H ∝ 1/t matches observational data, as confirmed by luminosity distance plots.
• The term with the lowest power dominates in the late-time evolution, e.g., for α = 1.1 and β = 0.5, R^{2−√7/4} dominates.
• The integration constants Φ⁺₀ and Φ⁻₀ are determined by imposing f′(R) = 1 and f′′(R) = 0 at z = 0, ensuring consistency with Newtonian gravity.
• The model is approximate and phenomenological, valid within the adopted low-redshift approximation, and does not fully recover non-linear f(R) theories.