$O_k$ null test with multi-task Gaussian processes: cosmic curvature and data compatibility

1. Problem Statement

The paper addresses two critical challenges in modern cosmology:

• Model-Independent Verification: Determining whether the Universe is spatially flat ($\Omega_k = 0$) without relying on specific cosmological models (like $\Lambda$CDM) or assumptions about the nature of dark energy.
• Data Compatibility: Assessing whether different cosmological datasets (specifically Type Ia Supernovae and Baryon Acoustic Oscillations) are mutually consistent when combined.
• The "Zero-Point" Limitation: Traditional null tests for cosmic curvature (using the $\mathcal{O}_k$ diagnostic) often rely on absolute distance measurements from Type Ia Supernovae (SNe Ia). These require zero-point calibration, which introduces systematic uncertainties and limits the model-independence of the test.
• DESI Data Complexity: The Dark Energy Spectroscopic Instrument (DESI) provides BAO data in various forms (e.g., $F_{AP}$, $D_V/r_d$, $D_M/r_d$, $D_H/r_d$) with complex correlations. Standard single-task Gaussian Process (GP) methods fail to properly account for these correlations when reconstructing derived functions, potentially biasing results.

2. Methodology

The authors propose a novel framework combining Multi-Task Gaussian Processes (MGP) with specific distance ratio diagnostics tailored to DESI data.

• The $\mathcal{O}_k$ Null Test:
  The diagnostic is defined as $\mathcal{O}_k(z) = E(z)D'(z) - 1$. If $\mathcal{O}_k(z) = 0$, the universe is spatially flat.

  • Traditional Approach: Uses SNe Ia to reconstruct $D(z)$, suffering from zero-point issues.
  • Proposed Approach: Utilizes DESI BAO data which provides ratios independent of the sound horizon ($r_d$). The authors derive two specific forms:
    1. $\mathcal{O}_k(z) = F_{AP}(z) \frac{D'_M(z)}{D_M(z)} - 1$
    2. $\mathcal{O}_k(z) = F_{AP} \frac{D'_V}{D_V} + \frac{1}{3}F'_{AP} - \frac{1}{3}\frac{F_{AP}}{z} - 1$
  • These formulations eliminate dependence on $r_d$ and absolute distance scales, relying only on the Cosmological Principle.

• Multi-Task Gaussian Processes (MGP):
  Instead of reconstructing functions (like $F_{AP}$ and $D_V/r_d$) separately using Single-Task GP (SGP), the authors employ MGP.

  • Input: Joint datasets including correlations (e.g., $F_{AP}$ and $D_V/r_d$ with their covariance matrix).
  • Function: MGP simultaneously reconstructs the functions and their derivatives ($D'_M$, $D'_V$, $F'_{AP}$) while strictly preserving the statistical correlations between the input data points.
  • Advantage: This ensures that the derived null test statistics are not biased by ignoring the covariance structure inherent in DESI DR2 data.

• Data Sources:

  • DESI DR2: BAO measurements ($F_{AP}$, $D_V/r_d$, $D_M/r_d$, $D_H/r_d$).
  • SNe Ia: Pantheon Plus (PP), Union3 (U3), and DES Y5 (D5 and recalibrated D5A).

3. Key Contributions

1. Novel Null Test Formulation: Developed a curvature null test specifically designed for DESI BAO data that avoids absolute distance measurements and the SNe Ia zero-point problem.
2. MGP Implementation: Pioneered the use of Multi-Task Gaussian Processes to jointly reconstruct cosmological distance ratios and their derivatives from correlated DESI data, ensuring rigorous statistical handling of the covariance matrix.
3. Data Consistency Check: Demonstrated a method to test the compatibility between DESI BAO and SNe Ia datasets by comparing reconstructed distance ratios ($D_M/D'_M$) without assuming a specific cosmological model.
4. Identification of Low-Redshift Data Gaps: Identified that apparent deviations from flatness at low redshifts are likely artifacts of sparse data coverage rather than new physics.

4. Key Results

• Self-Consistency of DESI Data: The MGP reconstruction of $F_{AP}$ from joint data ($F_{AP}, D_V/r_d$) is consistent with reconstructions from individual datasets, confirming that DESI DR2 BAO data is internally self-consistent.
• Low-Redshift Curvature Anomaly ($z \lesssim 0.5$):
  • The $\mathcal{O}_k$ test indicates $\Omega_k \neq 0$ (non-zero curvature) at low redshifts ($z \lesssim 0.5$).
  • This deviation persists whether correlations are included (MGP) or ignored (SGP).
  • Crucial Finding: When the constraint $F'_{AP}(0) = 1$ (derived from a flat universe assumption) is imposed, the results become consistent with zero curvature.
  • Conclusion: The apparent non-zero curvature is likely an artifact caused by the lack of BAO data in the $z \lesssim 0.5$ range, rather than a genuine physical signal. This suggests the "phantom crossing" of dark energy favored by DESI might also be an artifact of the flat CPL parametrization.
• Data Compatibility:
  • Comparisons of $D_M/D'_M$ between DESI BAO and three different SNe Ia samples (PP, U3, D5) show broad compatibility.
  • The $\mathcal{O}_k$ test confirms that DESI BAO and SNe Ia data are mutually consistent, validating their joint use in cosmological inference.
• Sound Horizon Consistency: Under the assumption of a flat universe, the reconstructed sound horizon $r_d h$ is $100.61 \pm 1.62$ Mpc, which is consistent with the Planck 2018 value ($98.82 \pm 0.82$ Mpc).

5. Significance

• Robustness of Null Tests: The study validates the $\mathcal{O}_k$ diagnostic as a powerful, model-independent tool not only for testing spatial flatness but also for diagnosing data quality and compatibility.
• Methodological Advancement: The application of MGP to handle correlated BAO data sets a new standard for analyzing DESI and future large-scale structure surveys, preventing systematic errors from unaccounted correlations.
• Resolution of Tensions: The paper suggests that current tensions regarding non-zero curvature and dynamical dark energy (phantom crossing) at low redshifts may be driven by observational gaps (lack of low-$z$ BAO data) rather than new physics.
• Future Direction: The authors emphasize the urgent need for additional low-redshift BAO measurements to robustly test spatial flatness and distinguish between data artifacts and genuine cosmological evolution.