Model-independent test of the cosmic distance duality relation with recent observational data

Using two model-independent methods and recent observational data from Type Ia supernovae, baryon acoustic oscillations, cosmic chronometers, and gamma-ray bursts, this study confirms the validity of the cosmic distance duality relation within a 1σ confidence level, finding no evidence for its violation.

The Gist

Imagine the universe as a giant, expanding balloon. For decades, cosmologists have been trying to measure how fast this balloon is inflating and how big it is at different stages of its life. To do this, they use two different "rulers" to measure distance:

1. The "Brightness" Ruler (Luminosity Distance): If you know how bright a lightbulb should be, you can tell how far away it is by how dim it looks. In space, we use exploding stars (Supernovae) as these "standard lightbulbs."
2. The "Size" Ruler (Angular Diameter Distance): If you know how big a tree should be, you can tell how far away it is by how small it looks. In space, we use the "frozen sound waves" from the early universe (Baryon Acoustic Oscillations) as our standard-sized trees.

The Cosmic Rulebook
There is a fundamental rule in physics called the Cosmic Distance Duality Relation (CDDR). It's like a universal law of optics that says: "If you measure the distance using the Brightness Ruler, and then measure it using the Size Ruler, the two numbers must match perfectly, adjusted for the fact that the universe is expanding."

If these two rulers ever disagree, it would be a massive earthquake in physics. It would mean:

• Gravity works differently than Einstein thought.
• Light is disappearing or being created out of nowhere (like ghosts eating photons).
• We are missing a whole layer of reality.

What This Paper Did
The author, Xing Wu, decided to put this rulebook to the ultimate test using the latest, most accurate data available. Think of this as a "stress test" for the universe's rulebook.

The paper used two different detective methods to see if the two rulers agreed:

Method 1: The "Smooth Curve" Detective

Imagine you are trying to guess the shape of a hidden curve by looking at a few dots.

• The Strategy: The author used a mathematical "smooth curve" (called the PAge parametrization) to describe how the universe has expanded over time. This curve is flexible enough to fit almost any theory of the universe.
• The Data: They plugged in data from:
  • Supernovae (SN): The "Brightness Ruler."
  • BAO: The "Size Ruler."
  • Cosmic Chronometers: A way to measure the age of the universe directly (like counting tree rings).
  • Gamma-Ray Bursts (GRB): These are incredibly bright explosions from the very early universe. The author hoped these would act as "long-distance runners" to test the rule at extreme distances.
• The Result: The two rulers agreed perfectly. The "Brightness" and "Size" measurements matched the rulebook within the margin of error.
• The Twist: The "long-distance runners" (Gamma-Ray Bursts) were a bit too noisy. It's like trying to hear a whisper from a mile away in a windstorm; the signal was too fuzzy to add much value compared to the clear data from closer distances.

Method 2: The "Shape-Shifter" Detective

This method was even more flexible. Instead of assuming a specific shape for the curve, the author used a technique called Gaussian Process (GP).

• The Strategy: Imagine you have a flexible rubber sheet. You pin down the "Brightness" data points on the sheet, and the rubber naturally stretches to connect them without assuming any specific formula. Then, you check if the "Size" data points fit onto that same rubber sheet.
• The Result: Even without forcing the data into a specific shape, the rubber sheet held up perfectly. The two rulers still agreed.

The "Hubble Tension" Plot Twist

There is a famous mystery in cosmology called the Hubble Tension. It's like two groups of scientists measuring the speed of the expanding balloon:

• Group A (Early Universe): Uses the Cosmic Microwave Background (the "baby photos" of the universe) and says the balloon is expanding at speed X.
• Group B (Late Universe): Uses nearby stars and says it's expanding at speed Y (which is faster).

Usually, if you force Group A's speed and Group B's speed to be true at the same time, the math breaks.

• The Paper's Finding: When the author forced these two conflicting speeds to be true simultaneously, the "Brightness" and "Size" rulers did start to disagree. They looked like the rulebook was broken!
• The Conclusion: But the author realized this wasn't a broken rulebook. It was just a sign that the two groups (Early vs. Late) are currently disagreeing with each other. The "violation" of the rule was just a symptom of the Hubble Tension, not a new law of physics.

The Bottom Line

The Cosmic Distance Duality Relation is still standing.

• The universe's "Brightness Ruler" and "Size Ruler" are in perfect agreement.
• The latest data from the Pantheon+ and DES Dovekie supernova surveys (the two most recent and best datasets) tell the same story.
• There is no evidence that light is vanishing, gravity is broken, or that we need exotic new physics to explain how distances work.

In simple terms: The universe is behaving exactly as the standard rules of physics predict. The only thing "broken" is our ability to agree on exactly how fast the universe is expanding right now, but that's a different puzzle entirely!

Technical Summary

1. Problem Statement

The Cosmic Distance Duality Relation (CDDR), defined as $\eta(z) \equiv d_L / [(1+z)^2 d_A] = 1$, is a fundamental pillar of modern cosmology. It relies on three key assumptions:

1. Gravity is described by a metric theory.
2. Photons follow null geodesics.
3. Photon number is conserved (no cosmic opacity).

Violations of the CDDR ($\eta \neq 1$) could signal new physics, such as photon coupling to exotic particles, non-standard gravity, or cosmic opacity. While numerous studies have tested the CDDR, most rely on low-redshift data ($z \lesssim 2.3$) or specific cosmological models (e.g., $\Lambda$CDM). There is a need for model-independent tests using the latest high-precision data (PantheonPlus, DES Dovekie, DESI DR2) to constrain potential violations across a broader redshift range without assuming a specific late-time cosmological model.

2. Methodology

The author employs two distinct, model-independent methods to test the CDDR using a combination of Type Ia Supernovae (SN), Baryon Acoustic Oscillations (BAO), Cosmic Chronometers (CC), and Gamma-Ray Bursts (GRB).

Method I: Parametric Approach with PAge Parametrization

• Framework: Uses the PAge parametrization, a model-independent description of the expansion history characterized by the cosmic age. It approximates the Hubble parameter $H(z)$ and distances using only three parameters ($p_{age}, \eta, H_0$), providing a robust approximation for a wide class of cosmological models up to high redshifts ($z \sim 10^4$).
• CDDR Violation Models: Tests four parametric forms of the violation $\eta(z)$:
  • $P1: \eta(z) = 1 + \eta_0 z$
  • $P2: \eta(z) = 1 + \eta_0 \frac{z}{1+z}$
  • $P3: \eta(z) = 1 + \eta_0 \ln(1+z)$
  • $P4: \eta(z) = (1+z)^{\eta_0}$
• Data Combination: Jointly constrains cosmological parameters and $\eta_0$ using:
  • SN: PantheonPlus and the latest DES Dovekie samples.
  • BAO: DESI DR2 data.
  • CC: 33 Cosmic Chronometer data points for calibration.
  • GRB: A118 sample (118 long GRBs) to extend redshift coverage to $z \sim 8$.
• Calibration Strategies: Compares three calibration approaches to break degeneracies between absolute magnitude ($M_B$) and sound horizon ($r_d$):
  1. Distance Ladder: SH0ES prior on $M_B$.
  2. Inverse Distance Ladder: Planck 2018 prior on $r_d$.
  3. Model-Independent: Using CC data to provide a length scale.

Method II: Non-Parametric Approach with Gaussian Processes (GP)

• Framework: Avoids assuming a specific functional form for $\eta(z)$.
• Procedure:
  1. Reconstructs the luminosity distance $d_L(z)$ directly from SN data using Gaussian Process (GP) regression.
  2. Constructs the observed violation parameter $\eta_{obs}$ at the specific redshifts of BAO data points: $\eta_{obs}(z) = \frac{\tilde{d}_L(z) \zeta}{\tilde{d}_M(z)(1+z)}$, where $\zeta$ is a dimensionless parameter combining $M_B$ and $r_d$.
  3. Constrains the parametric forms of $\eta(z)$ using these $\eta_{obs}$ values.
• Data: Uses PantheonPlus and DES Dovekie combined with transverse BAO data. Note: GRB data are excluded here because no $d_A$ data exists at comparable high redshifts.

3. Key Contributions

1. Integration of Latest Data: First comprehensive CDDR test utilizing the DES Dovekie SN sample (a reanalysis of DES5yr with improved systematics) alongside PantheonPlus and DESI DR2 BAO data.
2. Dual-Method Validation: Provides a robust cross-check by comparing a parametric method (PAge) with a non-parametric method (GP reconstruction).
3. GRB Constraint Analysis: Rigorously evaluates the utility of GRB data (A118 and A123) for CDDR tests, demonstrating their limited constraining power due to large intrinsic scatter in the Amati relation.
4. Calibration Sensitivity: Systematically investigates how different calibration choices (SH0ES vs. Planck vs. CC) affect the CDDR test results and the Hubble tension.
5. Resolution of Tensions: Analyzes the "aB tension" (discrepancy in the intercept parameter $a_B$) between PantheonPlus and previous DES samples, showing that the new DES Dovekie sample is more compatible with PantheonPlus.

4. Key Results

• Validity of CDDR: Both methods confirm that the CDDR holds within $1\sigma$ for all tested parametrizations ($P1-P4$). No significant evidence for violation ($\eta_0 \neq 0$) is found.
• Impact of GRB Data:
  • GRB data (A118/A123) extend the redshift range to $z \sim 8$ but provide weak constraints compared to SN+BAO+CC.
  • The large intrinsic scatter ($\sigma_{int} \sim 0.4-0.5$) in the Amati relation dominates the error budget, making GRB data less effective for tightening CDDR constraints in the current analysis.
• Calibration Dependence:
  • When using standard priors (SH0ES $M_B$ or Planck $r_d$) or CC data individually, results are consistent with $\eta_0 = 0$.
  • Crucial Finding: Simultaneously imposing the SH0ES $M_B$ prior and the Planck $r_d$ prior leads to a spurious violation of the CDDR at the $3\sigma - 4\sigma$ level. This suggests that apparent CDDR violations in such scenarios are artifacts of the Hubble tension (inconsistency between early- and late-universe measurements) rather than new physics.
• SN Dataset Comparison:
  • PantheonPlus and DES Dovekie yield consistent results for CDDR tests.
  • The new DES Dovekie sample resolves previous tensions with PantheonPlus regarding the $a_B$ parameter, showing improved compatibility compared to the older DES5yr sample.
• Method II Consistency: The non-parametric GP reconstruction yields results consistent with Method I, reinforcing the conclusion that no CDDR violation exists within current observational uncertainties.

5. Significance and Conclusion

This paper establishes a robust, model-independent baseline for the Cosmic Distance Duality Relation using the most recent cosmological datasets. The primary significance lies in:

• Ruling out Model Bias: By using PAge and GP, the study avoids biases introduced by assuming a specific cosmological model (like $\Lambda$CDM).
• Clarifying the Hubble Tension: It demonstrates that forcing consistency between early-universe (CMB) and late-universe (SN/Cepheid) distance scales can artificially induce apparent CDDR violations. This highlights the importance of calibration choices in interpreting fundamental relations.
• Future Outlook: While current data supports the CDDR, the paper emphasizes that future high-precision data from facilities like the Roman Space Telescope, Rubin Observatory, Euclid, and CSST, along with improved GRB physics understanding, will be essential for tighter, non-parametric tests at higher redshifts.

In summary, the CDDR remains valid within $1\sigma$ across all tested scenarios, and any apparent deviations are likely attributable to calibration tensions rather than a breakdown of fundamental physics.