Redshift evolution of the Hubble constant: Constraints and new insights from an interacting dark energy model

This paper proposes a modified interacting dark energy model that successfully describes a redshift-dependent evolution of the Hubble constant, supported by comprehensive observational data ranging from late-Universe probes to CMB constraints, thereby offering a viable theoretical mechanism to alleviate the Hubble tension.

The Gist

The Big Mystery: The Universe's Speedometer is Broken

Imagine the universe is a giant car speeding down a highway. Astronomers have a speedometer called the Hubble Constant ($H_0$) that tells us how fast the universe is expanding right now.

But here's the problem: The speedometer is giving two different readings.

1. The "Local" Reading: If we look at nearby stars and supernovae (like checking the speedometer while driving through town), the universe is zooming along at about 73 units.
2. The "Ancient" Reading: If we look at the "baby picture" of the universe (the Cosmic Microwave Background, or CMB, from 13.8 billion years ago) and calculate what the speed should be today, it says the universe is only going 67 units.

This difference is called the "Hubble Tension." It's like if your GPS said you were driving 60 mph, but your speedometer said 80 mph. Something is wrong with the map, the car, or the physics.

The New Idea: The Engine is Changing Gears

This paper proposes a solution. The authors suggest that the universe isn't just a car with a broken speedometer; rather, the engine itself is changing how it works as the car ages.

They introduce a concept called Interacting Dark Energy (IDE).

• Dark Matter is the invisible "cargo" in the back of the car.
• Dark Energy is the mysterious "fuel" pushing the car forward.

In the standard model, these two never talk to each other. But in this new model, Dark Energy and Dark Matter are having a conversation. They are exchanging energy.

The Analogy: The Leaky Fuel Tank

Imagine Dark Energy is a fuel tank, and Dark Matter is a heavy cargo box.

• In the old model, the fuel tank stays full, and the cargo stays heavy. The car's speed is predictable.
• In this new model, there is a leak (or a transfer pipe) between the fuel tank and the cargo box.
  • As the universe gets older (time passes), Dark Energy slowly leaks energy into Dark Matter.
  • This changes the "weight" of the cargo and the "power" of the fuel.
  • The Result: The car's speed doesn't stay constant. It changes depending on when you measure it.

What the Paper Found

The authors built a mathematical model of this "leaky" universe and tested it against the latest data from powerful telescopes (like DESI and Planck). Here is what they discovered:

1. The Speed Depends on Time (Redshift)
They found that the "speed" of the universe ($H_0$) isn't a single number. It's a curve.

• In the past (High Redshift): The universe was expanding slower.
• Now (Low Redshift): The universe is expanding faster.
• The Analogy: Imagine a runner who starts a race slowly but gets a second wind halfway through. If you measure their speed at the start, you get one number. If you measure it at the finish line, you get a higher number. The paper suggests the universe is doing exactly this.

2. The "Magic Number" ($\alpha$)
They calculated a specific number, called $\alpha$, which measures how fast this "speed change" is happening.

• Using data from the nearby universe (supernovae, galaxies), they found $\alpha$ is small but positive (about 0.01). This confirms the "speeding up" trend.
• The Twist: When they added data from the very early universe (the CMB), the number $\alpha$ dropped to almost zero (about 0.00005).

3. Why the Drop? The "Baby" Universe was Quiet
Why did the number change? The authors explain that in the very early universe, the "leak" between Dark Energy and Dark Matter was blocked.

• The Metaphor: Think of the early universe as a crowded room where everyone is holding hands tightly (strong coupling between matter and light). In this crowded room, the Dark Energy and Dark Matter couldn't exchange energy.
• As the universe expanded and cooled, the crowd dispersed, the "hands" let go, and the energy exchange (the leak) started.
• This explains why the early universe looks like the standard model (slow speed), but the modern universe looks different (fast speed).

The Conclusion: A Smooth Bridge

This model is exciting because it acts as a bridge.

• It explains why local measurements see a fast universe (because the energy exchange is active now).
• It explains why ancient measurements see a slow universe (because the exchange was frozen back then).
• It resolves the "Hubble Tension" not by saying the measurements are wrong, but by saying the universe's expansion rate is evolving.

In short: The universe isn't breaking; it's just growing up. As it ages, the interaction between its invisible components changes, causing the expansion rate to shift. This new "Interacting Dark Energy" model provides a consistent story that fits all the data, from the baby universe to the adult universe today.

Technical Summary

1. Problem Statement

The paper addresses two critical challenges in modern cosmology:

• The Hubble Tension: A persistent $\sim5\sigma$ discrepancy between the local measurement of the Hubble constant ($H_0 \approx 73.04$ km s$^{-1}$ Mpc$^{-1}$ from SH0ES) and the early Universe inference ($H_0 \approx 67.4$ km s$^{-1}$ Mpc$^{-1}$ from Planck CMB under $\Lambda$CDM).
• Redshift-Dependent $H_0$: Recent observational analyses suggest that the effective Hubble constant inferred from data is not constant but decreases with increasing redshift, following a phenomenological power-law relation: $H_0(z) = H_0(1+z)^{-\alpha}$, where $\alpha \sim 10^{-2}$.

The authors aim to determine if this observed redshift evolution can be explained by a fundamental physical mechanism rather than being merely a phenomenological fit or a systematic error. They propose that Interacting Dark Energy (IDE) provides a self-consistent theoretical origin for this behavior.

2. Methodology

Theoretical Framework

The authors construct a modified IDE model within a flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe containing Dark Matter (DM), Phantom Dark Energy (DE), and Radiation.

• Interaction: Energy is exchanged between DM and DE via a coupling term $Q = \delta H \rho_m$, where $\delta$ is a dimensionless coupling parameter.
• Scaling Solution: They adopt a specific scaling relation where the ratio of DE to DM densities evolves as $r = \rho_e/\rho_m \propto a^\xi$. To ensure theoretical consistency (avoiding divergences and solving the coincidence problem), they fix $\xi = 3$.
• Derivation of $H_0(z)$: By solving the continuity equations with $\xi=3$, the authors derive the modified Friedmann equation. In the late Universe (neglecting radiation), they demonstrate that the expansion rate effectively behaves as:
  $$H(z) \simeq H_0 (1+z)^{-\alpha} \sqrt{\Omega_{m,0}(1+z)^3 + (1-\Omega_{m,0})}$$
  where $\alpha = \delta/2$. This establishes a direct physical link between the interaction strength $\delta$ and the phenomenological evolution parameter $\alpha$.

Data and Analysis

The study utilizes a comprehensive dataset to constrain the model parameters ($\alpha, \Omega_{m,0}, H_0, \Omega_b h^2$):

1. Baryon Acoustic Oscillations (BAO):
   • DESI DR2: 25 data points spanning $0.1 < z < 4.2$ (including BGS, LRG, ELG, QSO, and Ly$\alpha$ forest).
   • SDSS: Pre-DESI measurements (6dF, BOSS, WiggleZ).
2. Cosmic Chronometers (CC): 32 model-independent $H(z)$ measurements ($0.07 < z < 1.965$).
3. Type Ia Supernovae (SNIa): The Pantheon sample (1048 SNIa, $0.01 \le z \le 2.26$).
4. Cosmic Microwave Background (CMB): Planck 2018 distance priors (shift parameter $R$, acoustic scale $l_A$, and baryon density $\omega_b$).

The analysis employs Markov Chain Monte Carlo (MCMC) simulations (using the emcee package) to maximize the likelihood function and derive posterior distributions. Model selection is evaluated using the Akaike Information Criterion (AIC).

3. Key Contributions

• Physical Mechanism for $H_0(z)$ Evolution: Unlike previous studies that treat $H_0(z) \propto (1+z)^{-\alpha}$ as a purely phenomenological ansatz, this paper derives it from first principles within an IDE framework. It proves that dark-sector energy exchange naturally induces a redshift-dependent effective Hubble constant.
• Unification of Late and Early Universe Constraints: The model successfully reconciles the high $H_0$ values preferred by late-Universe data with the low $H_0$ required by CMB data by allowing the interaction strength to vary effectively with redshift (suppressed at high $z$, active at low $z$).
• Robustness Check: The authors perform a sensitivity analysis on the scaling parameter $\xi$ (varying between 2.90 and 3.10) to ensure the results are not artifacts of the specific choice $\xi=3$.

4. Key Results

• Late-Universe Constraints (DESI + SDSS + CC + SNIa):

  • The interaction parameter is constrained to $\alpha = 0.0107^{+0.0032}_{-0.011}$.
  • The best-fit value is on the order of $10^{-2}$, confirming a statistically significant decreasing trend of $H_0$ with redshift.
  • The IDE model is preferred over the standard $\Lambda$CDM model with $\Delta \text{AIC} = -2.55$ (substantial evidence).
  • The derived $H_0$ is $69.85 \pm 0.15$ km s$^{-1}$ Mpc$^{-1}$.

• Joint Constraints (Including CMB):

  • Incorporating Planck CMB data tightens the constraint on $\alpha$ to the order of $10^{-5}$ ($\alpha \approx 5.6 \times 10^{-5}$).
  • Physical Interpretation: This drastic tightening implies that dark-sector interactions are strongly suppressed at high redshifts ($z \sim 1100$). This is consistent with the strong baryon-photon coupling in the early Universe, which would inhibit such interactions.
  • The joint fit yields $H_0 = 69.67 \pm 0.13$ km s$^{-1}$ Mpc$^{-1}$, acting as a "bridge" between local and early Universe measurements.
  • The model preference strengthens with CMB data ($\Delta \text{AIC} = -3.95$).

• Consistency: The extrapolation of the late-Universe trend ($\alpha \sim 10^{-2}$) to the last scattering surface ($z=1100$) yields an $H_0(z)$ value consistent with the CMB-inferred $H_0$, demonstrating internal consistency across cosmic epochs.

5. Significance

• Resolution of Hubble Tension: The paper suggests that the Hubble tension may not require new physics in the early Universe (like early dark energy) but rather a dynamical evolution in the dark sector that becomes relevant at late times. The IDE model provides a mechanism where $H_0$ evolves naturally, bridging the gap between local and CMB measurements.
• Theoretical Validation: It transforms the parameter $\alpha$ from a fitting coefficient into a physical quantity derived from the dark-sector coupling strength ($\delta$), providing a dynamical origin for the observed redshift evolution.
• Future Directions: The results underscore the necessity of combining probes across different redshift regimes. The authors suggest future work should incorporate additional data sources like the Tip of the Red Giant Branch (TRGB), Fast Radio Bursts (FRBs), and Gamma-Ray Bursts (GRBs) to further refine these constraints.

In conclusion, the study presents a theoretically consistent and observationally viable IDE model that explains the redshift evolution of the Hubble constant, offering a promising pathway to alleviating the Hubble tension without violating early Universe constraints.