Testing the cosmic distance-duality relation with localized fast radio bursts: a cosmological model-independent study

1. Problem Statement

The Etherington Cosmic Distance-Duality Relation (CDDR) is a fundamental identity in metric theories of gravity, defined as $\eta(z) \equiv D_L(z) / [(1+z)^2 D_A(z)] = 1$. This relation holds provided that photons travel along null geodesics and their number is conserved. Testing this relation is crucial because any statistically significant deviation ($\eta \neq 1$) could indicate:

• Violations of photon conservation (e.g., cosmic opacity or photon-axion mixing).
• Exotic physics beyond standard General Relativity.
• Systematic errors in distance measurements.

Historically, CDDR tests have paired Type Ia Supernovae (SNe Ia, providing luminosity distance $D_L$) with other tracers of angular diameter distance ($D_A$), such as galaxy clusters or Baryon Acoustic Oscillations (BAO). However, these methods often rely on specific cosmological models or suffer from different systematic uncertainties.

The Gap: Fast Radio Bursts (FRBs) offer a unique, independent probe. Their observed Dispersion Measure (DM) encodes the integrated electron column density along the line of sight, which is sensitive to the expansion history $H(z)$ and geometry, but does not rely on photon flux. This paper aims to test the CDDR by comparing SNe Ia $D_L$ with FRB-reconstructed $D_A$ in a cosmological model-independent framework.

2. Methodology

The authors employ a data-driven, machine-learning approach to reconstruct distances without assuming a parametric form for the expansion history $H(z)$.

A. Data Sources

• FRBs: A cleaned sample of 122 localized FRBs with measured redshifts and Dispersion Measures (DMs).
  • Preprocessing: Contributions from the Milky Way ($DM_{MW}$) and Galactic Halo ($DM_{halo} = 50 \text{ pc cm}^{-3}$) are subtracted.
  • Quality Control: Events with $DM_{obs} - DM_{MW} < 80 \text{ pc cm}^{-3}$ are removed to avoid unphysical negative extragalactic DMs. Outliers (e.g., FRB 210117, FRB 190520) and extremely local events (e.g., FRB 200110E) are excluded.
• SNe Ia: The Pantheon+ compilation, utilizing corrected apparent magnitudes and the full statistical + systematic covariance matrix. The SH0ES calibrator subset is removed to avoid external distance anchors.

B. Reconstruction of Angular Diameter Distance ($D_A$) from FRBs

The core innovation is the reconstruction of $D_A(z)$ directly from the DM-redshift relation using Artificial Neural Networks (ANNs):

1. Extragalactic DM ($DMEG$): The mean extragalactic DM is reconstructed as a smooth function of redshift, $\langle DMEG(z) \rangle$, using an ensemble of feed-forward multilayer perceptrons (MLPs).
2. Anchoring: To ensure physical consistency, the reconstruction is anchored such that the cosmological IGM contribution vanishes at $z=0$ ($\langle DM_{IGM}(0) \rangle = 0$). The intercept $DMEG(0)$ is interpreted as the mean host/near-source contribution ($DM_{host}$).
3. Deriving $H(z)$ and $D_A(z)$:
   • The derivative of the mean DM curve, $d\langle DM_{IGM} \rangle / dz$, is computed numerically.
   • Using the relation $\dot{D}_C(z) = \frac{1}{A F(z)} \frac{d\langle DM_{IGM} \rangle}{dz}$ (where $A$ and $F(z)$ are known physical constants and ionization factors), the comoving distance derivative is recovered.
   • $D_C(z)$ is obtained by integration, and $D_A(z)$ is derived via $D_A = D_C / (1+z)$.
4. Uncertainty Propagation: Bootstrap resampling is used to generate an ensemble of reconstructions. This captures the non-diagonal covariance (correlated uncertainties) across redshift, which is critical for the subsequent likelihood analysis.

C. Likelihood Frameworks

Two complementary methods are implemented to compare $D_L$ (from SNe) and $D_A$ (from FRBs) via the parameter $\eta(z)$:

• Method A (Direct): Uses individual Pantheon+ SNe. The FRB $D_A$ is interpolated to SN redshifts. The total covariance combines the SN sub-matrix and the FRB reconstruction covariance.
• Method B (ANN/Reconstructed): Reconstructs a smooth SN Hubble diagram on the FRB redshift grid using ANNs trained on Monte Carlo realizations of the Pantheon+ data. This allows for a direct comparison on a common grid, propagating the full Pantheon+ covariance into a non-diagonal prediction matrix.

D. Parametrization

Deviations from CDDR are modeled using three one-parameter ansätze for $\eta(z)$, all satisfying $\eta(0)=1$:

1. Linear: $\eta(z) = 1 + \epsilon z$
2. Linear Fractional: $\eta(z) = 1 + \epsilon \frac{z}{1+z}$
3. Power Law: $\eta(z) = (1+z)^\beta$

3. Key Contributions

1. Model-Independence: The study avoids assuming a specific cosmological model (e.g., $\Lambda$CDM) for $H(z)$. The expansion history is inferred purely from the data-driven reconstruction of the FRB DM-redshift relation.
2. Novel Covariance Handling: The authors explicitly account for the correlated uncertainties induced by global reconstructions (both for FRBs and SNe in Method B). By retaining the full non-diagonal covariance matrices, they avoid artificially tightening constraints that would occur if redshift points were treated as independent.
3. Data-Driven Host Constraint: By anchoring the reconstruction at $z=0$, the study derives a robust, data-driven constraint on the average host contribution: $DM_{host} = 128.8 \pm 34.1 \text{ pc cm}^{-3}$ (at 3$\sigma$ confidence).
4. Dual-Likelihood Validation: The use of two distinct statistical approaches (direct vs. reconstructed) serves as a robust internal cross-check, ensuring results are not artifacts of specific data compression or interpolation schemes.

4. Results

• CDDR Consistency: Both likelihood implementations yield posteriors consistent with the standard CDDR ($\eta(z) = 1$). There is no statistically significant evidence for deviations from the distance-duality relation at current precision.
• Parameter Constraints:
  • For the linear model, $\epsilon = -0.0221 \pm 0.0482$ (Method A).
  • For the power-law model, $\beta = -0.0355 \pm 0.0837$ (Method A).
  • All parameters are consistent with zero (no deviation) within $1\sigma$.
• Redshift Dependence: The reconstructed $\eta(z)$ bands remain close to unity across the entire redshift range probed ($z \approx 0$ to $2.148$).
  • At the highest redshift ($z \approx 2.15$), the constraints allow for deviations of at most $\sim 10\%$ (e.g., $\eta(z_{max}) \approx 0.95 \pm 0.10$ for the linear model), but the posterior comfortably encompasses $\eta=1$.
• Degeneracy: A clear anti-correlation is observed between the deviation parameter ($\theta$) and the SN absolute magnitude ($M_B$). This is expected as a redshift-dependent tilt in $\eta(z)$ can mimic a shift in the SN zero-point calibration. However, the data breaks this degeneracy sufficiently to confirm $\eta \approx 1$.
• Method Comparison: Method B (ANN) yields slightly wider credible intervals than Method A due to the additional reconstruction variance, but the results are fully compatible, confirming the robustness of the conclusions.

5. Significance

• Validation of Fundamental Physics: The study provides a stringent, model-independent confirmation of the Etherington relation, reinforcing the validity of standard metric gravity and photon conservation up to $z \approx 2.15$.
• FRBs as Cosmological Probes: It demonstrates the maturity of FRBs as a tool for cosmology. By reconstructing $D_A$ without relying on standard candles, FRBs provide a completely independent systematic check on the cosmic distance ladder.
• Methodological Blueprint: The paper establishes a rigorous framework for combining heterogeneous datasets (SNe and FRBs) with correlated uncertainties. This "covariance-aware" approach is essential for future high-precision tests as FRB samples grow.
• Future Prospects: The study highlights that current constraints are limited by the sparsity of FRBs at high redshift ($z > 1$). As the localized FRB sample expands, this model-independent pipeline will enable progressively sharper tests of cosmic opacity and new physics.