Thurston geometries and parameter constraints from SNIa data

Technical Summary: Thurston Geometries and Parameter Constraints from SNIa Data

Problem Statement
The standard cosmological model ($\Lambda$CDM) relies on the Cosmological Principle, assuming a homogeneous and isotropic universe described by the Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. However, recent observations—including anomalies in the Cosmic Microwave Background (CMB) power spectrum, alignments of optical and radio polarization vectors, and large-scale velocity flows exceeding $\Lambda$CDM predictions—suggest potential violations of large-scale statistical isotropy. This paper investigates whether extending the standard model to include homogeneous but anisotropic spacetime geometries, specifically the topologically unique Thurston geometries, can better accommodate these observational hints and constrain potential cosmic anisotropies.

Methodology
The authors analyze three specific Thurston geometries that allow for anisotropy while preserving homogeneity: $R \times H^2/S^2$, Nil, and Solv. Unlike standard FLRW models which use a single scale factor, these models introduce anisotropy via distinct scale factors ($a(t)$ and $b(t)$) in different spatial directions.

1. Theoretical Framework: The authors derive the metric tensors for these geometries and establish the Einstein field equations for a perfect matter fluid source. They define key anisotropic parameters, including shear ($\sigma$), eccentricity ($e^2$), and a preferred axis ($\hat{\lambda}$).
2. Redshift and Distance Relations: Analytical expressions for redshift ($z$) and direction-dependent luminosity distance ($d_L$) are derived for each geometry. These expressions depend on the angle $\alpha$ between the isotropic plane and the line of sight to the supernova.
3. Evolution Equations: A set of coupled differential equations governing the evolution of cosmological parameters (Hubble parameter $h$, matter density $\Omega_m$, curvature density $\Omega_\kappa$, shear, and eccentricity) is formulated with respect to a dimensionless time variable $\tau = \ln(A)$, where $A$ is the mean scale factor.
4. Data Analysis: The study utilizes the Pantheon+ & SH0ES compilation of 1701 Type Ia Supernova (SNIa) light curves. A Bayesian likelihood analysis is performed using the Polychord sampler within the Cobaya framework. The $\chi^2$ function is minimized against the full covariance matrix of the Pantheon+ data to constrain model parameters.
5. Comparison: The results of the anisotropic Thurston models are compared against the standard flat $\Lambda$CDM model.

Key Contributions

• Derivation of Anisotropic Evolution Equations: The paper provides a complete set of evolution equations for $R \times H^2/S^2$, Nil, and Solv geometries, explicitly linking the scale factors to observable quantities like luminosity distance and redshift.
• Parameter Constraints: It presents 2$\sigma$ constraints on fundamental parameters ($H_0$, $\Omega_m$, $\Omega_\Lambda$) and anisotropic parameters (shear $\sigma$, eccentricity $e^2$, curvature $\Omega_\kappa$, and preferred axis) for these specific geometries.
• Curvature Radius Estimation: The study calculates the physical curvature radii ($L_0$) for each geometry and compares them to the Hubble radius ($R_0$), testing the hypothesis that these geometries operate on scales larger than the observable universe.

Results

• Model Preference: The Bayesian evidence ($\log Z$) and minimum $\chi^2$ values indicate that the flat $\Lambda$CDM model remains the most favored description of the data. Among the Thurston geometries, $R \times H^2/S^2$ is the most strongly favored, followed by Nil and Solv, though the differences in fit quality are described as negligible.
• Anisotropy Indicators: The $R \times H^2/S^2$ geometry exhibits the highest mean values for shear ($\approx 0.036$) and eccentricity ($\approx 0.107$), suggesting the largest physical deviation from isotropy among the tested models.
• Hubble Tension: The Nil and Solv geometries constrain a slightly lower mean Hubble parameter ($H_0 \approx 0.72$) compared to $\Lambda$CDM ($H_0 \approx 0.73$). The authors suggest this could imply a slower expansion rate in these models, potentially alleviating tension between CMB-derived and local $H_0$ measurements.
• Curvature and Preferred Axis: All anisotropic models constrain small but non-zero curvature density parameters ($\Omega_\kappa$). The data favors a positive $\Omega_\kappa$ for $R \times H^2/S^2$, which implies a negative curvature parameter $\kappa$ (favoring $R \times H^2$ over $R \times S^2$). Furthermore, the constrained preferred axes for all Thurston geometries show mutual agreement up to 1$\sigma$ and align with directions sourced by soft X-ray effects identified in previous literature.
• Length Scales: The calculated curvature radii ($L_0$) for the anisotropic geometries are significantly larger than the Hubble radius ($R_0$), with ratios $L_0/R_0$ ranging from $\approx 1.8$ to $\approx 7.6$. This supports the claim that the size of individual Thurston geometries exceeds the current Hubble scale.

Significance and Claims
The paper concludes that while the Pantheon+ & SH0ES data currently favors the standard flat $\Lambda$CDM model, the Thurston geometries provide a viable framework for exploring large-scale isotropy violations. The authors note "mild evidence" of large-scale isotropy violation, particularly in the non-zero constraints on shear and eccentricity at the 2$\sigma$ level.

The significance of this work lies in its attempt to distinguish between different Thurston geometries using purely isotropic sources (SNIa) and its derivation of specific constraints on cosmic shear, eccentricity, and preferred axes. The authors modestly state that more data is required to establish stringent constraints capable of definitively deciding in favor of Thurston models over the standard model. They suggest that future analyses utilizing improved datasets (such as DES, DESI, and JWST) are necessary to further investigate the potential of these anisotropic spacetimes to explain observed cosmic skewness.