Model independent test of the FLRW metric and the curvature in light of DESI DR2

Technical Summary: Model Independent Test of the FLRW Metric and Curvature in Light of DESI DR2

Problem and Motivation
The standard concordance model of cosmology, $\Lambda$CDM, relies on the fundamental assumption that the Universe is homogeneous and isotropic, encoded in the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. Despite the model's success, tensions persist, particularly regarding the Hubble constant ($H_0$) and recent hints from the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) suggesting a dynamical dark energy equation of state. While parametric fits of alternative models are common, model-independent reconstructions of the expansion history offer a direct way to test the consistency of observational data with the underlying assumptions of FLRW and spatial flatness without assuming a specific Dark Energy (DE) model. This work aims to perform such a test using the latest DESI DR2 Baryon Acoustic Oscillation (BAO) data combined with Type Ia Supernova (SNIa) compilations.

Methodology
The authors employ a model-independent "litmus test" known as the $\mathcal{O}_k$ diagnostic, originally proposed by [9, 10]. This diagnostic tests the consistency of the FLRW metric and the spatial curvature parameter $\Omega_{k,0}$.

1. Data Sources:

   • BAO: Transverse ($d_M/r_d$) and radial ($d_H/r_d$) modes from DESI DR2.
   • SNIa: Two compilations are used: Pantheon+ (1550 SNIa, $0.001 < z < 2.26$) and DES Dovekie (1820 SNIa, $0.025 < z < 1.144$, a re-analysis of DES Y5).
   • Note: While SNIa standardization assumes $\Lambda$CDM, the constraints on light-curve hyper-parameters are established to be independent of the cosmological model [22].

2. Iterative Smoothing:

   • Since BAO data are sparse and available only at specific redshifts, the authors use an iterative smoothing algorithm [13, 23, 24] to reconstruct the distance modulus $\mu(z)$, the dimensionless comoving distance $D(z)$, and its derivative $D'(z)$ at the BAO redshifts.
   • The algorithm uses a Gaussian kernel with a smoothing scale $\Delta = 0.3$.
   • Reconstructions are normalized to remove dependence on the absolute magnitude $M_B$ and $H_0$, ensuring the results are independent of these specific parameter choices.

3. The $\mathcal{O}_k$ Diagnostic:

   • The diagnostic is defined as $\mathcal{O}_k(z) = \Theta^2(z) - 1$, where $\Theta(z) = h(z)D'(z)$. In a valid FLRW universe, $\mathcal{O}_k(z)$ should be constant and equal to $\Omega_{k,0}$.
   • The authors combine the reconstructed $D(z)$ and $D'(z)$ from SNIa with the BAO measurements to compute $\mathcal{O}_k$ at BAO redshifts.

4. Selection Criteria and Consistency Tests:

   • $\chi^2$ Selection: Reconstructions are filtered based on $\Delta\chi^2 < 0$, meaning only those providing a better fit to the data than the best-fit flat $\Lambda$CDM model are retained. Two criteria are tested: fit to SNIa alone ($\Delta\chi^2_{\text{SNIa}}$) and fit to SNIa plus the transverse BAO mode ($\Delta\chi^2_{\text{SNIa}+d_M}$).
   • P-value Test: To ensure the reconstructed $\mathcal{O}_k$ is consistent with a constant (FLRW metric), a p-value test is applied. Reconstructions with $p < 0.05$ (rejecting the hypothesis that $\mathcal{O}_k$ is constant) are discarded. Only reconstructions passing this test are used to estimate $\Omega_{k,0}$.

Key Results

• Pantheon+ & DESI DR2:

  • When using the $\Delta\chi^2_{\text{SNIa}} < 0$ criterion, the majority of reconstructions deviate significantly from a constant $\mathcal{O}_k$ at high redshift ($z > 1$), indicating inconsistency with the FLRW metric. Only $\sim 1\%$ of reconstructions pass the p-value test.
  • These few FLRW-consistent reconstructions show a preference for positive curvature ($\Omega_{k,0} > 0$), with a median value of $\Omega^{\text{med}}_{k,0} = 0.064^{+0.048}_{-0.099} \pm 0.038$.
  • However, when applying the stricter $\Delta\chi^2_{\text{SNIa}+d_M} < 0$ criterion (incorporating transverse BAO), the tension is resolved. The reconstructions inconsistent with FLRW are rejected, and the remaining set is fully consistent with the FLRW metric. The median curvature shifts slightly to $\Omega^{\text{med}}_{k,0} = 0.045^{+0.045}_{-0.081} \pm 0.038$.
  • Restricting the analysis to $z < 1.13$ (removing high-z Pantheon+ and high-z DESI points) yields $\Omega^{\text{med}}_{k,0} = 0.102^{+0.056}_{-0.129} \pm 0.064$, suggesting the tension is driven by sparse, high-redshift data.

• DES Dovekie & DESI DR2:

  • This combination shows excellent agreement. 100% of reconstructions pass the p-value test under both selection criteria.
  • The median curvature is negative: $\Omega^{\text{med}}_{k,0} = -0.102^{+0.100}_{-0.005} \pm 0.043$.
  • All reconstructions remain consistent with flatness ($\Omega_{k,0}=0$) within $3\sigma$.

• Robustness Checks:

  • Smoothing Scale: Varying $\Delta$ between 0.2 and 0.4 shows that smaller scales reinforce dataset-specific preferences (positive $\Omega_k$ for Pantheon+, negative for DES Dovekie), while larger scales ($\Delta=0.4$) increase consistency with flatness across datasets.
  • DES Y5 vs. DES Dovekie: The updated DES Dovekie data show slightly better agreement with flatness than the original DES Y5 data, reducing the maximum deviation from flatness from $-3.28\sigma$ to $-2.51\sigma$.

Significance and Claims
The paper claims to provide a robust, model-independent test of the FLRW metric and spatial curvature using the latest DESI DR2 data. The primary significance lies in demonstrating that:

1. Data Consistency: The apparent tension between Pantheon+ SNIa and DESI DR2 BAO at high redshifts manifests as a violation of the FLRW metric in model-independent reconstructions. This tension is resolved when requiring consistency with the transverse BAO mode, suggesting the high-redshift Pantheon+ data may be unreliable for such reconstructions due to sparsity and systematics (e.g., Malmquist bias).
2. Curvature Constraints: The results are consistent with a flat universe within $3\sigma$ across all data combinations and selection criteria. While slight preferences for positive (Pantheon+) or negative (DES Dovekie) curvature appear, these are not statistically significant.
3. Methodological Validation: The study validates the use of the $\mathcal{O}_k$ diagnostic combined with iterative smoothing as a powerful tool to test the fundamental assumptions of cosmology without relying on specific Dark Energy models.

The authors conclude that their results are consistent with Planck 2018 and previous model-independent tests, reinforcing the FLRW metric and flatness as valid descriptions of the Universe within current observational limits, while highlighting the need for better high-redshift SNIa observations or alternative probes (e.g., standard sirens) to extend these tests further.