Can Distance Duality Violation Save Late-time Solutions to the Hubble Tension?

Technical Summary: Can Distance Duality Violation Save Late-time Solutions to the Hubble Tension?

Problem Statement
The paper addresses the persistent $>5\sigma$ discrepancy between the local determination of the Hubble constant ($H_0 \approx 73$ km s$^{-1}$ Mpc$^{-1}$) from Cepheid-calibrated Type Ia supernovae (SNIa) and the value inferred from Planck CMB observations within the $\Lambda$CDM framework ($H_0 \approx 67$ km s$^{-1}$ Mpc$^{-1}$). While numerous late-time modifications to the expansion history have been proposed to resolve this "Hubble tension," they have met with limited success. The authors frame the tension as a calibration inconsistency in the $r_d$–$M_B$ plane (sound horizon vs. supernova absolute magnitude), noting that if the sound horizon ($r_d$) and supernova calibration ($M_B$) are held fixed, any purely late-time resolution necessitates a violation of the Cosmic Distance Duality Relation (CDDR). The central question is whether such a violation is observationally viable.

Methodology
The authors employ a model-independent analysis that treats the CDDR as a fundamental anchor to re-evaluate the tension. Their methodology proceeds in three stages:

1. Recasting the Tension in the $r_d$–$M_B$ Plane:
   Using the CDDR definition $\eta(z) \equiv d_L(z) / [(1+z)^2 d_A(z)]$, where $\eta=1$ implies validity, the authors combine uncalibrated SNIa data (Pantheon+) and BAO data (DESI DR2). They derive a constraint on the combination $A = M_B + 5 \log_{10}(r_d / \text{Mpc} \cdot \eta)$. By fixing $\eta=1$, they determine the allowed region in the $r_d$–$M_B$ plane. They compare this against the fiducial calibration derived from Planck ($r_d \approx 147$ Mpc) and SH0ES ($M_B \approx -19.25$), demonstrating that the tension manifests as a $7.0\sigma$ deviation from the CDDR-consistent band.

2. Decoupling Late-Time Expansion from $H_0$:
   The authors show that for fixed $r_d$ and standard distance duality, the combination of BAO, SNIa, and the CDDR determines $H_0$ independently of the late-time expansion history. They derive the relation:
   $$H_0 \propto \frac{1}{\eta} (r_d) 10^{(A-M)/5}$$
   This demonstrates that reconciling the tension with fixed calibrations requires a shift in $\eta$ (violation of duality) rather than a modification of the expansion history $H(z)$. To resolve the tension, an $\mathcal{O}(8-10\%)$ violation of the CDDR ($\eta \approx 0.92$) is required.

3. Testing Viability of CDDR Violation Mechanisms:
   The authors decompose potential CDDR violations into two distinct physical mechanisms and constrain them separately using current data:

   • Etherington Reciprocity Violation ($\alpha$): This modifies angular-diameter distances ($d_A$). The authors develop a new geometric probe using transverse and radial BAO measurements combined with cosmic-chronometer (CC) data. By assuming reciprocity violations affect only the angular sector, they reconstruct the reciprocity parameter $\alpha(z)$ without relying on luminosity-distance measurements.
   • Photon Number Non-Conservation ($\beta$): This modifies only the luminosity distance ($d_L$). They utilize constraints from the thermal Sunyaev–Zel'dovich (tSZ) effect in galaxy clusters, which constrain the evolution of the CMB temperature with redshift, to limit deviations from standard photon conservation.

Key Results

• Geometric Constraint on Reciprocity: The combined BAO and CC analysis yields a tight constraint on the reciprocity parameter at low redshifts: $\bar{\alpha} = 0.99 \pm 0.008$ (around $z \sim 1$). This effectively rules out the large distortions in angular diameter distances required to resolve the tension.
• Constraint on Photon Number: Constraints from tSZ measurements limit the photon non-conservation parameter to the percent level ($\beta(1) = 1.011 \pm 0.013$), far below the $\sim 8-10\%$ deviation needed.
• Joint Exclusion: When plotting constraints in the $\alpha$–$\beta$ plane (where $\eta = \alpha\beta$), the region required to alleviate the Hubble tension (e.g., $\eta \approx 0.92$) is strongly disfavoured by the intersection of independent observational limits.
• Independence from Expansion History: The analysis confirms that fixing $r_d$ and $M_B$ while imposing the CDDR yields $H_0 = 66.6 \pm 0.7$ km s$^{-1}$ Mpc$^{-1}$, regardless of the late-time expansion model. Consequently, late-time modifications (e.g., dynamical dark energy, interacting dark energy, modified gravity) cannot resolve the tension without violating the CDDR.

Significance and Claims
The paper concludes that for fixed sound-horizon and supernova calibrations, no modification confined to the late-time expansion history—even one violating distance duality—can resolve the Hubble tension.

The authors argue that the required level of CDDR violation is strongly excluded by current data, specifically:

1. Geometric constraints from BAO and cosmic chronometers rule out reciprocity violations of the necessary magnitude.
2. tSZ constraints rule out photon non-conservation of the necessary magnitude.
3. Direct model-independent tests of the CDDR also show no significant deviation.

Therefore, the paper posits that the Hubble tension cannot be solved by late-time physics alone. The resolution must instead lie in early-Universe physics (modifying $r_d$) or unresolved local systematics (modifying the $M_B$ calibration or the distance ladder), rather than in exotic late-time propagation effects or expansion history modifications. The work provides a rigorous framework for testing non-standard physics (non-metric gravity, photon decay) against the specific requirements of the Hubble tension, placing stringent limits on their viability.