Cosmology / Research in progress

CosmoPaths

Exploring how modified gravity can connect radiation, matter and late-time cosmic acceleration.

Single crimson trajectory connecting three cosmological stages

Scientific Question

If a gravity model admits radiation, matter and accelerated expansion individually, can a physically admissible cosmic history connect them? The project investigates the relationship between local stability criteria and global cosmological evolution in metric f(R) gravity.

Current Scope

Research in progress, documented in two reports: an analytical and numerical study of exponential gravity, followed by a derivation and verification of the Amendola dynamical system. A compact formulation with matter and radiation is the next step.

The cosmological sequence

Connecting the eras of the Universe

The project asks whether the familiar local conditions for individual cosmological states are sufficient to permit a complete history within the physical region of the model.

Radiation → Matter → Late-time acceleration

This sequence is the target of the global analysis. Its existence has not yet been demonstrated by the present work.

First model

Exponential gravity

The first report studies a model in which the gravitational function contains an exponential correction to the Ricci curvature scalar R.

f (R) = R - β R_{s} (1 - e^{- R / R_{s}})

For β > 1 and R_s > 0, the report verifies the local matter criterion and the existence of a locally stable expanding de Sitter state. With u = R/R_s, the physical condition f_R > 0 requires u > ln β. These checks reproduce known local results and provide a baseline for the global investigation.

Illustrative numerical pilot

Following the expansion

A pilot with β = 2 follows the evolution of normalized densities, curvature and the deceleration parameter q. Negative q indicates accelerated expansion.

Four panels from the first report: radiation, matter and effective modified-gravity densities; curvature approaching de Sitter; deceleration across the illustrative history; and the numerically integrated transition to acceleration. Dashed curves show an early-time approximation and solid curves show numerical integration. — Figure from the report of 23 September 2026; original Spanish labels retained. Dashed curves: early high-curvature approximation. Solid curves: numerical integration beginning near u = 20. N = ln(a/a_ref); the reference scale factor is not calibrated to the observed present. This is an illustrative model calculation, without an observational fit.

The plotted history joins an early approximation to a late numerical integration. It is not a direct integration from radiation and does not establish global viability or robustness from the radiation era.

Dynamical systems

Local criteria and the global question

The functions m and r encode the model dependence in the conventional dynamical formulation. Subscripts R denote derivatives with respect to the Ricci scalar.

m = \frac{R f_{R R}}{f_{R}} r = - \frac{R f_{R}}{f}

The second report derives the four equations of the Amendola system, checks the Friedmann constraint and fluid conservation, and examines radiation, matter and de Sitter states. The stability calculation retains the dependence m(r).

It also distinguishes singular limits and cases where linearization cannot decide stability. Local fixed points and their eigenvalues do not, by themselves, demonstrate a physical trajectory connecting the cosmological eras.

Two Progress Reports

23 September 2026. Local criteria for exponential gravity and an illustrative cosmological evolution, with numerical consistency checks.

17 October 2026. Derivation and verification of the non-compact Amendola system, including local stability and the limitations of its coordinates.

Next Step

Construct and validate a compact dynamical system with matter and radiation, then examine its physical boundaries and possible connections between cosmological states. Whether additional global viability conditions are needed remains an open research question.

Research Context

Research carried out by Gerardo Mancebo under the supervision of Dr. Ricardo García-Salcedo. This page presents the scope and current progress of the ongoing cosmology project.

Foundational framework: Amendola et al. (2007). Exponential-gravity evolution: Bamba, Geng & Lee (2010).