Addressing the $H_0$ tension through matter with pressure and no early dark energy

This paper proposes and validates a new $\Lambda_{\omega_s}$CDM cosmological model featuring a subdominant "matter with pressure" fluid that resolves the Hubble tension by modifying the sound horizon and expansion rate without relying on early dark energy, showing a statistical preference over the standard $\Lambda$CDM model.

The Gist

The Big Problem: The Universe is Growing Too Fast (or Too Slow?)

Imagine you are trying to measure how fast a car is driving.

• Method A (The Local View): You stand on the side of the road with a radar gun. You measure the car right here, right now. You get a speed of 73 mph.
• Method B (The Historical View): You look at the car's dashboard from 13 billion years ago (when it was brand new) and try to calculate how fast it should be going today based on its engine specs and fuel consumption. You calculate a speed of 67 mph.

In cosmology, this is the Hubble Tension.

• Method A uses nearby stars and supernovae (the "local" universe).
• Method B uses the Cosmic Microwave Background (CMB), which is the "baby picture" of the universe from 380,000 years after the Big Bang.

The numbers don't match. The universe seems to be expanding faster today than our best theories predict it should be. This is a huge headache for physicists because it suggests our "rulebook" for the universe (called the $\Lambda$CDM model) is missing a page.

The Usual Suspects: Early Dark Energy

For a while, the most popular idea to fix this was Early Dark Energy (EDE).

• The Analogy: Imagine the universe is a balloon. Standard physics says the air inside (matter and radiation) pushes it out, but a mysterious "dark energy" is pulling it apart even faster. EDE suggests that early on, there was a temporary burst of extra "push" (like a sudden injection of helium) that made the balloon grow faster for a short time, then vanished.
• The Problem: While this fixes the math, it feels a bit like cheating. It requires a very specific, "fine-tuned" mechanism that turns on and off at the exact right moment. It's like a magic trick that only works if you pull the rabbit out of the hat at exactly 3:00 PM.

The New Idea: "Squeezed" Matter

The authors of this paper propose a different solution. Instead of adding a temporary burst of energy, they suggest we missed a type of matter that has pressure.

• The Analogy: Think of the early universe as a crowded dance floor.
  • Normal Matter (Dust): These are people standing still or walking slowly. They take up space but don't push back hard.
  • Radiation (Light/Photons): These are people running around wildly, bouncing off walls. They push hard.
  • The New Fluid (Pressure Matter): The authors suggest there is a third group of people. They aren't running as fast as the radiation, but they aren't standing still like the dust. They are jostling and shoving. They have "pressure."

In physics terms, this is a fluid where the particles push against each other (positive pressure), but not as violently as light does.

How This Fixes the Speedometer

Why does adding "shoving people" fix the speed difference?

1. The Sound Horizon: In the early universe, sound waves traveled through the plasma (the hot soup of particles). The distance these waves traveled before the universe cooled down is called the "sound horizon." This distance is the "ruler" we use to measure the universe's expansion.
2. The Squeeze: If you add this new "shoving" matter, it changes the sound speed. It's like adding a thick syrup to the dance floor. The waves move differently.
3. The Result: This new fluid makes the "ruler" (the sound horizon) slightly shorter than we thought.
4. The Fix: If the ruler is shorter, but the angle we see it at (from the CMB) stays the same, the math forces us to conclude that the universe must be expanding faster to match our local measurements.

It's like realizing your tape measure was actually a rubber band that had stretched out. Once you realize the ruler was wrong, the speed of the car suddenly makes sense without needing a magic helium injection.

Why This is Better Than the Old Idea

The authors compare their idea to the "Early Dark Energy" idea in a table (Table I in the paper). Here is the simple version:

Feature	Early Dark Energy (The Old Idea)	Matter with Pressure (The New Idea)
Behavior	It's a "sprinter." It shows up for a split second, does its job, and then disappears.	It's a "marathon runner." It's always there, but it's quiet and doesn't take over.
Complexity	Needs a very specific, complicated machine to turn it on and off.	It's simple. It just follows the standard rules of physics (like a gas).
Origin	Usually requires mysterious "scalar fields" (invisible energy fields).	Could be real particles, like heavy photons or "Proca fields" (massive light particles).

The Evidence: Did They Win?

The authors ran massive computer simulations (using a tool called CLASS, which is like a super-accurate universe simulator) to see if their idea fits the data.

• The Result: Yes! Their model fits the data better than the standard model.
• The Numbers: They found that this new fluid exists, but it's very small. It's about 45% as dense as light (photons) in the early universe. It's a "subdominant" player—it doesn't take over the party, but it changes the music just enough to fix the rhythm.
• The Equation: They found the "pressure" of this fluid is slightly less than that of light. It's not "stiff" (hard as a rock), but it's not "soft" (like dust) either. It's somewhere in between.

What Could This "Shoving Matter" Actually Be?

The authors speculate on what this fluid might be made of in the real world:

1. Massive Photons: What if light particles (photons) actually have a tiny bit of mass? This would make them act like this "shoving" fluid.
2. Dark Photons: A cousin of the photon that lives in a "dark sector" and interacts weakly with us.
3. Scalar Fields: A type of energy field that behaves like a fluid.

The Bottom Line

The universe is expanding faster than we thought. Instead of inventing a magical, temporary energy burst to explain it, these scientists suggest we simply missed a type of matter that pushes back.

It's a simpler, more elegant solution. It's like realizing the car isn't speeding up because of a hidden turbo; it's just that we were measuring the speed with a slightly stretched-out tape measure. Once we account for the "pressure" of this new fluid, the tape measure shrinks back to size, and the speed of the universe finally makes sense.

Technical Summary

1. Problem Statement: The Hubble Tension

The paper addresses the persistent Hubble tension, a significant discrepancy between local measurements of the Hubble constant ($H_0$) and values inferred from the Cosmic Microwave Background (CMB) under the standard $\Lambda$CDM model.

• Local Measurements: The SH0ES collaboration reports $H_0 \approx 73.04 \pm 1.04$ km/s/Mpc using Cepheid-calibrated distance ladders.
• CMB Inference: Planck data within the $\Lambda$CDM framework yields $H_0 \approx 67.36 \pm 0.54$ km/s/Mpc.
• Significance: This $\sim 4.1\sigma$ discrepancy suggests potential new physics beyond the standard model.
• Current Approaches: Solutions are generally categorized into late-time (modifying expansion history after recombination) or early-time (modifying pre-recombination physics). Late-time solutions are disfavored due to tight constraints on $H(z)$. Early-time solutions often rely on Early Dark Energy (EDE) models involving scalar fields that temporarily increase the expansion rate before recombination to reduce the sound horizon ($r_s$).

2. Methodology and Proposed Model

The authors propose a novel early-time solution that avoids scalar fields and the fine-tuning issues associated with EDE. Instead, they introduce a barotropic fluid with positive pressure (matter with pressure).

• The $\Lambda\omega_s$CDM Paradigm:

  • The model extends $\Lambda$CDM by adding a fluid component $s$ with an equation of state (EoS) $P_s = \omega_s \rho_s$, where $\omega_s > 0$ (satisfying the Zel'dovich limit).
  • Unlike radiation ($\omega = 1/3$) or stiff matter ($\omega = 1$), the authors explore the regime $0 < \omega_s < 1/3$.
  • Key Constraint: The fluid must remain subdominant to radiation and dust throughout the universe's history to avoid altering CMB anisotropies significantly.

• Physical Mechanism:

  • The presence of this fluid modifies the adiabatic sound speed ($c_s$) of the photon-baryon plasma before recombination.
  • The modified sound speed is given by:
    $$c_s^2 = c^2 \frac{1 + 3\omega_s W}{3(1 + R + W)}$$
    where $W$ is a dimensionless ratio involving the new fluid density and photon density, and $R$ is the baryon-photon ratio.
  • By altering $c_s$ and the expansion rate $H(z)$, the model reduces the comoving sound horizon ($r_s$).
  • Since the angular scale of the sound horizon ($\theta_* = r_s / D_A$) is tightly constrained by CMB observations, a reduction in $r_s$ necessitates a reduction in the angular diameter distance $D_A$, which effectively increases the inferred value of $H_0$.

• Numerical Analysis:

  • The authors modified the CLASS (Cosmic Linear Anisotropy Solving System) Boltzmann code to include this extra fluid component.
  • They performed a Monte Carlo Markov Chain (MCMC) analysis using three datasets:
    1. Pantheon+ (1701 Type Ia Supernovae) + SH0ES (Cepheid anchors).
    2. DESI-BAO (DR2 Baryon Acoustic Oscillations).
    3. CMB Shift Parameters ($R$ and $l_A$) from Planck.
  • Priors were set to ensure the fluid density $\Omega_s(z_*)$ remains smaller than radiation $\Omega_r(z_*)$ and matter $\Omega_m(z_*)$ at recombination.

3. Key Contributions and Results

A. Statistical Preference

The $\Lambda\omega_s$CDM model is statistically preferred over the standard $\Lambda$CDM model:

• Log-Likelihood: The new model achieves a significantly higher log-likelihood ($\ln \mathcal{L} = -654.86$) compared to $\Lambda$CDM ($-671.60$).
• Information Criteria: The model yields lower Akaike (AIC) and Bayesian (BIC) Information Criteria values, with $\Delta$AIC $\approx 29$ and $\Delta$BIC $\approx 18$, indicating strong evidence in favor of the new paradigm.

B. Best-Fit Parameters

The MCMC analysis yielded the following best-fit values:

• Hubble Constant: $H_0 = 74.30^{+0.88}_{-0.82}$ km/s/Mpc (resolving the tension with local measurements).
• Equation of State: $\omega_s = 0.293^{+0.003}_{-0.009}$. This value is slightly less than that of radiation ($\omega_\gamma = 1/3$), satisfying the condition $\omega_s \lesssim \omega_\gamma$.
• Relative Density: The ratio of the new fluid's density to photon density is $\Omega_s / \Omega_\gamma = 0.45$.
• Subdominance: The fluid density is confirmed to be subdominant to radiation and matter at recombination ($z_* \approx 1090$) and at later epochs, ensuring minimal disruption to the CMB power spectrum.

C. Physical Interpretation

The authors propose physical origins for this "matter with pressure":

1. Proca-type Vector Fields: Massive vector fields (dark photons) where the mass term alters the EoS from $1/3$ to slightly lower values. The derived mass for such a field is $m_D \approx 0.064$ eV.
2. Thermal Scalar Fields: Generalized K-essence models or thermalized scalar fields behaving as quasi-quintessence.
3. Exclusion: The model rules out pure stiff matter ($\omega = 1$) and pure dark energy models that vanish after recombination.

4. Significance and Implications

• Alternative to EDE: This work provides a robust alternative to Early Dark Energy. Unlike EDE models, which require specific dynamical mechanisms to "turn off" after recombination (leading to fine-tuning issues), the proposed fluid has a constant EoS and is always present but naturally subdominant. It does not require a phase transition or a specific potential shape.
• Thermodynamic Consistency: The fluid satisfies the Zel'dovich limit ($\omega > 0$), implying positive pressure and attractive gravity, distinguishing it from dark energy.
• Minimal Extension: The model achieves a resolution to the Hubble tension with a minimal extension to the standard cosmological model, adding only one new fluid component with a constant EoS.
• Future Directions: The authors suggest that this fluid could be associated with dark radiation, axions, or Proca fields. Future work will focus on the impact of this fluid on structure formation and refining bounds on its fundamental nature.

Conclusion

The paper successfully demonstrates that introducing a subdominant, barotropic fluid with a positive equation of state ($\omega_s \approx 0.29$) can resolve the Hubble tension by modifying the early-universe sound horizon. The $\Lambda\omega_s$CDM model is statistically favored over $\Lambda\bar{C}$DM and offers a physically motivated alternative to scalar-field-based Early Dark Energy, avoiding fine-tuning while remaining consistent with CMB, BAO, and Supernova data.