Cosmological test of a length-preserving biconnection gravity

Technical Summary: Cosmological Test of a Length-Preserving Biconnection Gravity

Problem Statement and Motivation
The paper addresses the theoretical shortcomings of the standard $\Lambda$CDM model, specifically the fine-tuning and coincidence problems associated with the cosmological constant. While modified gravity theories based on non-Riemannian geometries (such as $f(R)$, Weyl, and Finsler geometries) have been proposed to explain cosmic acceleration, many face issues such as the non-conservation of vector lengths during autoparallel transport in standard Weyl geometry. The authors investigate a specific extension of General Relativity (GR) known as biconnection gravity, constructed from the Schrödinger connection and its dual. This framework aims to preserve vector lengths (length-preserving) while introducing non-Riemannian geometric degrees of freedom that could naturally encode an effective dark energy sector without invoking exotic matter fields.

Methodology
The study employs a metric formalism where the gravitational field equations retain the structural form of Einstein's equations, but the Ricci tensor and scalar are replaced by "mutual" curvature quantities derived from the Schrödinger connection ($\Gamma$) and its dual ($\tilde{\Gamma}$).

1. Theoretical Framework: The authors define the mutual tensor $K^\lambda_{\sigma\mu}$ as the difference between the two connections. The symmetric combination of the connections recovers the Levi-Civita connection (ensuring GR is recovered at the background level), while their difference encodes non-Riemannian degrees of freedom.
2. Cosmological Derivation: Generalized Friedmann equations are derived for a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe. The non-Riemannian contributions appear as effective energy density ($\rho_{\text{eff}}$) and pressure ($p_{\text{eff}}$), effectively acting as a dark energy component.
3. Parametrization: To close the system of equations, the effective dark energy equation of state (EoS), $\omega_{\text{de}}(z)$, is parametrized using five distinct models:
   • B$\Lambda$CDM (constant $\omega = -1$)
   • $\omega$CDM (constant free parameter $\omega$)
   • Chevallier-Polarski-Linder (CPL)
   • Barboza-Alcaniz
   • Logarithmic EoS
4. Observational Constraints: The model parameters are constrained using a Markov Chain Monte Carlo (MCMC) analysis against a combined dataset comprising:
   • DESI DR2 (Dark Energy Spectroscopic Instrument) Baryon Acoustic Oscillation (BAO) data.
   • Pantheon+ Type Ia Supernovae compilation.
   • Cosmic Chronometer (CC) $H(z)$ measurements.
5. Model Comparison: Statistical performance is evaluated using the Akaike Information Criterion (AICc), Bayesian Information Criterion (BIC), and Deviance Information Criterion (DIC) relative to the standard $\Lambda$CDM model.
6. Dynamical Analysis: The authors utilize cosmographic parameters (deceleration $q$, jerk $j$, snap $s$), Statefinder diagnostics ($\{j, s\}$ and $\{q, j\}$ planes), and the $Om(z)$ diagnostic to characterize the expansion history and distinguish between dark energy models.

Key Results

• Statistical Fit: All five biconnection parameterizations provide an excellent fit to the observational data, with reduced chi-square values ($\chi^2_{\text{d.o.f}}$) close to unity.
• Hubble Parameter: The models yield a current Hubble rate $H_0 \approx 70.7$ km s$^{-1}$ Mpc$^{-1}$, which is consistent across all biconnection parametrizations and statistically distinct from the SH0ES value ($73.2 \pm 1.3$), though the tension remains at the $\sim 1.4\text{--}1.6\sigma$ level.
• Model Selection:
  • According to AICc and DIC, the B$\Lambda$CDM model is the most preferred, followed by $\omega$CDM, CPL, Logarithmic, and Barboza-Alcaniz. The $\Lambda$CDM model is the least favored among the tested set.
  • Under the BIC, which penalizes complexity more heavily, B$\Lambda$CDM remains the top model, but $\Lambda$CDM moves to second place, surpassing the more complex dynamical parametrizations.
• Cosmographic Evolution:
  • All models predict a transition from deceleration to acceleration at a redshift $z_t \approx 0.7$, consistent with $\Lambda$CDM.
  • The jerk parameter $j_0$ for the biconnection models is generally lower than the $\Lambda$CDM value ($j=1$), suggesting a phantom-like trend in the current epoch for most models.
  • The Statefinder analysis indicates that the biconnection models evolve from a quintessence-like regime ($s > 0, j < 1$) in the past toward a phantom-like regime ($s < 0, j > 1$) in the future. Most models cross the $\Lambda$CDM fixed point ($s=0, j=1$) in the recent past ($z \sim 0.03\text{--}0.08$), except for $\omega$CDM which crosses in the future.
• $Om(z)$ Diagnostic: The $Om(z)$ diagnostic confirms that the geometric sector generates a dynamical dark energy component. Most parametrizations exhibit a negative slope (quintessence-like) in the past, transitioning to a positive slope (phantom-like) in the recent past/future, whereas $\omega$CDM remains in the phantom regime in the future.

Significance and Claims
The paper claims that length-preserving biconnection gravity offers a geometrically motivated extension to General Relativity where cosmic acceleration arises from non-Riemannian degrees of freedom rather than a fundamental cosmological constant.

• Competitiveness: The framework is shown to be statistically competitive with $\Lambda$CDM, with the B$\Lambda$CDM variant emerging as the most supported model across information criteria.
• Dynamical Nature: Unlike the static $\Lambda$CDM, the biconnection framework naturally produces a dynamical dark energy component that evolves from a quintessence-like to a phantom-like regime, crossing the phantom divide.
• Background Consistency: The authors emphasize that at the background level, the symmetric part of the construction recovers GR, ensuring the model remains consistent with standard cosmological observations while offering a distinct geometric origin for acceleration.
• Limitations: The authors modestly note that the analysis is restricted to the background level. They state that a full perturbative treatment and confrontation with structure formation data (e.g., redshift-space distortions, CMB) are necessary for a more stringent test of the theory.