Geometric Cosmology models: statistical analysis with observational data

This work investigates alternative geometric cosmological models featuring an infinite tower of higher-order curvature invariants and employs cosmic chronometers, Type Ia supernovae, and globular cluster age data to demonstrate that while certain variants are ruled out, specific cases of the GILA model successfully account for current observational constraints.

The Gist

Imagine the universe as a giant, expanding balloon. For decades, scientists have used a standardized "instruction manual" called ΛCDM (Lambda-CDM) to explain how this balloon is inflated. This manual relies on two main ingredients: General Relativity (Einstein's theory of gravity) and a mysterious, invisible push called Dark Energy (represented by the Greek letter Lambda, Λ), which causes the universe to expand ever faster.

However, this manual has some cracks. The mathematics do not quite make sense when scientists try to calculate how much "Dark Energy" should exist, and different methods for measuring the universe's expansion rate yield contradictory answers (a problem known as the "Hubble Tension"). For this reason, scientists are searching for new instruction manuals that do not need to invent a mysterious "Dark Energy" fluid.

This work examines a new, highly mathematical manual called Geometric Cosmology (GC).

The New Idea: Gravity as a Multi-Layered Cake

Instead of adding a new ingredient (Dark Energy) to the universe, this theory suggests that gravity itself is more complex than Einstein thought.

Imagine Einstein's gravity as a simple, flat pancake. The new theory proposes that gravity is actually a multi-layered cake with an infinite number of layers. Each layer represents a more complex mathematical "curvature" of space.

• The Standard Model: Only the bottom pancake (Einstein's gravity) + a magical push (Dark Energy).
• The New Model: A tall tower cake where the shape of the layers themselves generates the "push" needed to expand the universe. No magical push required; geometry does the work.

The Three Flavors of the New Cake

The authors tested three specific ways to stack this infinite cake:

1. GILA (Geometric Inflation and Late Acceleration): This is the most complex stack. It features layers designed to explain both the very early, rapid expansion of the universe (inflation) and the current acceleration.
2. GR-Deformation: This is a "slightly bent" pancake. It stays very close to Einstein's original theory but includes a tiny adjustment to the gravitational constant.
3. Non-GR Contribution: This removes the original pancake entirely. It is a stack consisting only of the complex, higher-order layers, without standard Einsteinian gravity at the base.

The Test: The "Age of the Universe" Challenge

To check whether these new cake recipes work, the authors compared them with real data:

• Supernovae: Distant exploding stars that serve as "standard candles" for measuring distances.
• Cosmic Chronometers: Ancient galaxies that act like stopwatches for measuring the expansion rate.
• The "Grandparents" Constraint: This is the unique twist of the work. The authors examined globular clusters (dense groups of very old stars). These are the "grandparents" of the universe.

The Logic: If your new instruction manual says the universe is only 10 billion years old, but we have "grandparents" (stars) that are 12.7 billion years old, your manual is wrong. The universe must be older than its oldest stars.

The Results: Who Passed the Test?

The authors had to create a special computer program to test these models because the mathematics were so "stiff" (like trying to balance a tower of Jenga blocks on a wobbly table) that standard testing methods failed.

Here is what they found:

• The "GR-Deformation" and "Non-GR" Models: These did not pass the test. When the authors crunched the numbers, these models predicted a universe that was too young. They could not allow enough time for the formation of the oldest stars (the grandparents). Consequently, these models were ruled out.
• The "GILA" Model: Some versions of this complex cake stack passed the test. They predicted a universe old enough to host the oldest stars and fit the data from supernovae and galaxies.
  • However, there is a catch. While these GILA models work, they are not better than the old standard manual (ΛCDM). When the authors compared the mathematics, the data still slightly favored the standard model with Dark Energy.

The Big Takeaway

This work did not find a new "winner" that could replace the standard model of the universe. Instead, it accomplished something equally important:

1. It set a new rule: It proved that any new theory of gravity must pass the "oldest star" test. If a theory says the universe is younger than its oldest stars, it is dead.
2. It built a new tool: Since the mathematics were so difficult, the authors developed a new statistical method (a new way to probe the data) to handle these complex theories.
3. It narrowed the field: It showed that while the idea of "Geometric Cosmology" is interesting, the specific versions they tested (except for some GILA variations) do not fit the data better than our current best estimate.

In short: The universe is a complex place. Although we found a few new recipes that might work, the old favorite (ΛCDM) is still the most popular choice in the kitchen, and every new recipe must prove it can cook a universe old enough to host its oldest stars.

Technical Summary

Technical Conclusion: Geometric Cosmology Models with Observational Data

Problem Statement
While the standard $\Lambda$CDM cosmological model successfully describes most observational data, it faces significant theoretical challenges, most notably the discrepancy between the observed cosmological constant value and the theoretical prediction of vacuum energy (which differs by 60–120 orders of magnitude). Furthermore, observable tensions persist, such as the "Hubble tension" between measurements of the Hubble constant ($H_0$) in the early universe (Planck) and in the late universe (Supernovae/Cepheids), as well as recent indications from DESI that dark energy may not behave like a simple cosmological constant. These problems motivate the exploration of alternative frameworks, particularly modified gravity theories that do not require a cosmological constant or dark energy fluid to explain late-time acceleration.

This work investigates a specific class of Geometric Cosmology (GC) models. These models arise from an extension of the Einstein-Hilbert action that includes an infinite tower of higher-order curvature invariants. The theory is constructed to satisfy "Einstein conditions," ensuring second-order field equations and the absence of ghost modes. The study focuses on the evolution of the universe in late phases within this framework, assuming a characteristic function $F(H)$ in the modified Friedmann equations that takes an exponential form.

Methodology
The authors analyze three specific families of solutions derived from the GC framework:

1. GILA (Geometric Inflation and Late Acceleration): Here, the linear Ricci term is absent ($\alpha_1 = 0$).
2. GR Deformation: A slight deformation of General Relativity, where $-1 < \alpha_1 < 0$.
3. Non-GR Contribution: A case where the linear Ricci term is completely removed ($\alpha_1 = -1$).

The study confronts these models with three distinct observational datasets:

• Pantheon Plus + SH0ES (PPS): A compilation of 1701 Type Ia supernovae (SNIa) used to constrain distance moduli.
• Cosmic Chronometers (CC): Differential age measurements of passively evolving galaxies used to directly estimate the Hubble parameter $H(z)$.
• Globular Cluster Ages (GC): A constraint on the lower limit of the age of the universe (AoU). The authors adopt a conservative lower bound of 12.7 billion years (based on the age of the oldest clusters of approximately 12.2 billion years plus a formation onset of 0.5 billion years).

Statistical Approach and Novelty
A key methodological contribution of this paper is the development of a dedicated statistical analysis technique. Standard Markov Chain Monte Carlo (MCMC) methods proved unfeasible due to:

1. The stiffness of the Friedmann equations in certain regions of the parameter space.
2. The non-trivial geometry of the prior region imposed by the globular cluster age constraint (a hard lower bound rather than a Gaussian prior).

To address this, the authors implemented a Monte Carlo sampling methodology (distinct from MCMC) comprising three stages:

1. Consistency Check: Verifying whether the confidence regions from CC and SNIa data overlap for each model family.
2. Application of Constraints: Constructing a grid in the parameter space, calculating the total $\chi^2$, and discarding points where the predicted universe age falls below 12.7 billion years.
3. Posterior Sampling: Sampling the surviving region of the parameter space to generate marginalized posteriors and confidence contours.

Key Results
The statistical analysis yields the following conclusions regarding the viability of the proposed models:

• Excluded Models:

  • The entire GR Deformation family is excluded. While some parameter sets fit the SNIa and CC data, they predict a universe age below the 12.7 billion-year threshold required by globular cluster data.
  • The Non-GR Contribution family is largely excluded. Models with exponents $s=\{2,3,4\}$ fail the consistency test between CC and SNIa data. Models with $s=\{5,6,7\}$ pass the consistency test but are subsequently excluded by the globular cluster age constraint.

• Viable but Disfavored Models:

  • Specific subfamilies of the GILA model (particularly those with exponent pairs $(r,s) = (3,4), (3,5), (3,6)$) are capable of simultaneously explaining SNIa, CC, and globular cluster age data.
  • For these viable GILA models, the authors derive meaningful constraints on the free parameters ($H_0$, $\beta$, and $M_{abs}$). Notably, the data constrain the parameter $\beta$ and limit high values of $H_0$, while $H_0$ remains largely unconstrained beyond the prior limits.
  • Despite their viability, Model Comparison Criteria (AIC/BIC) show a strong statistical preference for the standard $\Lambda$CDM model over the investigated GILA models. The $\Delta$AIC and $\Delta$BIC values indicate that $\Lambda$CDM provides a significantly better fit to the data given the number of parameters.

Significance and Claims
The paper claims that its primary significance lies not in establishing a new preferred cosmological model, but in establishing a robust methodology for testing alternative geometric cosmologies. Specifically:

• It demonstrates the critical importance of including globular cluster age constraints, a factor often overlooked in the literature that effectively excludes entire classes of otherwise viable modified gravity models.
• It provides the first systematic statistical test of this specific class of models with an infinite tower of curvature invariants against current high-precision data.
• It establishes a clear framework (the Monte Carlo sampling method) for handling stiff equations and hard priors, which can be applied to future studies with different characteristic functions ($F(H)$) or alternative gravity theories.

The authors conclude that while the specific exponential ansatz function for $F(H)$ investigated here does not offer a preferred alternative to $\Lambda$CDM, the work successfully excludes specific functional forms and paves the way for future research exploring alternative choices of the characteristic function within the framework of Geometric Cosmology.