Comment on Cosmological constraints on unimodular gravity models with diffusion (arXiv:2211.07424): thermodynamic inadmissibility of the H0 tension resolution mechanism

This paper demonstrates that diffusion-based Unimodular Gravity models proposed to resolve the $H_0$ tension are thermodynamically inadmissible because they violate the second law of thermodynamics by requiring a negative energy transfer rate ($\dot{Q} < 0$) that contradicts the necessary condition for alleviating the tension.

The Gist

The Big Problem: The "Hubble Tension"

Imagine the universe is a giant balloon being blown up. Astronomers are trying to measure exactly how fast it's expanding (this speed is called the Hubble Constant, or $H_0$).

There is a major disagreement in the scientific community:

• Team A (The Past): Looks at the "baby photos" of the universe (the Cosmic Microwave Background) and calculates the speed. They get one number.
• Team B (The Present): Looks at "current photos" (nearby stars and supernovae) and measures the speed directly. They get a faster number.

These two numbers don't match. This is the Hubble Tension. It's like trying to measure your height as a baby and as an adult, but the math says you should be the same height, yet you are clearly taller now. Something is missing from our physics.

The Proposed Solution: The "Leaky Balloon"

To fix this, some physicists proposed a theory called Unimodular Gravity. They suggested that energy isn't perfectly conserved in the universe.

Think of the universe as a leaky water balloon:

• The Water: Represents Matter (stars, gas, dark matter).
• The Air Pressure: Represents Dark Energy (the force pushing the balloon to expand).

The proposed solution (by researchers in papers [10] and [11]) was that the "water" (matter) is slowly leaking into the "air pressure" (dark energy). As the water leaks out, the pressure builds up, making the balloon expand faster. This extra push would explain why the universe is expanding faster today than our old models predicted, potentially fixing the Hubble Tension.

The New Paper's Discovery: The "Thermodynamic Police"

The author of this new paper, Mauricio Cataldo, says: "Hold on a minute. While this idea is clever, it breaks the most fundamental rule of physics: The Second Law of Thermodynamics."

To understand this, imagine the universe as a bank account:

• The Second Law of Thermodynamics is like a strict bank rule: You can't just magically create wealth, and you can't move money in a way that makes the total "disorder" (entropy) of the system decrease.
• In our universe, for things to get more "disordered" (which they naturally do), energy usually needs to flow into the matter (the water), not out of it.

Cataldo's Argument:

1. The Rule: If you have a fluid (like the cold dark matter in the universe) and you want the universe to obey the laws of thermodynamics, energy must flow from the Dark Energy into the Matter.
   • Analogy: Imagine a hot cup of coffee (Dark Energy) cooling down and warming up the cold room (Matter). The heat flows from hot to cold. The "disorder" increases.
2. The Proposal's Mistake: The "Leaky Balloon" models proposed by the other researchers do the exact opposite. They have energy flowing from Matter into Dark Energy.
   • Analogy: This would be like the cold room suddenly sucking heat out of the coffee, making the coffee colder and the room hotter, all while the total disorder decreases. It's like a broken refrigerator that gets colder the more you open the door.
3. The Result: Cataldo proves mathematically that if you force energy to flow from Matter to Dark Energy (to fix the Hubble Tension), you violate the Second Law of Thermodynamics. The universe would essentially be running a "perpetual motion machine" that is impossible.

The "No-Go" Theorem

The paper concludes with a "No-Go Theorem." Think of this as a traffic sign that says "DO NOT ENTER."

• The Sign: You cannot have a model where energy flows from Matter to Dark Energy (to speed up expansion) AND still obey the laws of thermodynamics.
• The Implication: The specific "leaky balloon" ideas proposed in papers [10] and [11] are thermodynamically inadmissible. They are mathematically consistent with gravity equations, but they are physically impossible because they break the rules of heat and energy.

Summary in a Nutshell

• The Goal: Fix the disagreement about how fast the universe is expanding.
• The Idea: Let matter leak energy into dark energy to speed things up.
• The Problem: This direction of energy flow is "backwards" according to the laws of thermodynamics. It's like trying to un-mix a dropped egg.
• The Verdict: This specific way of fixing the problem is impossible. The universe cannot work that way.

What does this mean for the future?
It doesn't mean the Hubble Tension is unsolvable. It just means we can't solve it using this specific type of energy leak. Scientists will need to find a different mechanism—one that doesn't break the rules of thermodynamics—to explain why the universe is expanding faster than we thought.

Technical Summary

1. Problem Statement

The paper addresses the persistent $H_0$ tension, the significant discrepancy between the Hubble constant value inferred from Cosmic Microwave Background (CMB) observations (assuming the standard $\Lambda$CDM model) and the value measured directly from local distance indicators.

Recent proposals within the framework of Unimodular Gravity (UG) have attempted to resolve this tension by introducing a diffusion mechanism. Specifically, works by Perez, Sudarsky, and Wilson-Ewing [10] and Landau et al. [11] suggest that energy diffuses from the matter sector (cold dark matter) into an effective cosmological term ($\Lambda_{\text{eff}}$). This energy transfer increases the late-time expansion rate, potentially reconciling the conflicting $H_0$ values.

The Core Issue: While these models are observationally motivated, the authors argue that the specific mechanism required to alleviate the $H_0$ tension (energy flowing from matter to dark energy) has not been tested against the Second Law of Thermodynamics. The paper investigates whether these diffusion models are thermodynamically admissible.

2. Methodology

The author employs a rigorous thermodynamic analysis within the Unimodular Gravity framework for a spatially flat FLRW universe containing pressureless matter (Cold Dark Matter, CDM) and an effective cosmological term.

• Theoretical Framework:

  • The effective cosmological term is defined as $\Lambda_{\text{eff}}(t) = \Lambda_0 + Q(t)$, where $\Lambda_0$ is a constant and $Q(t)$ is a diffusion function representing the violation of energy-momentum conservation.
  • The continuity equation for pressureless matter ($p_m=0$) in the presence of diffusion is:
    $$\dot{\rho}_m + 3H\rho_m = -\dot{Q}$$
  • Here, $\dot{Q} > 0$ implies energy flow from matter to $\Lambda_{\text{eff}}$, while $\dot{Q} < 0$ implies the reverse.

• Thermodynamic Derivation:

  • Starting from the Gibbs equation for a pressureless fluid: $T dS = dU + p_m dV = d(\rho_m V)$.
  • By taking the time derivative and substituting the continuity equation, the author derives the rate of entropy change:
    $$T \dot{S} = -a^3 \dot{Q}$$
  • Since temperature $T > 0$ and volume $a^3 > 0$, the Second Law of Thermodynamics ($\dot{S} > 0$) imposes a strict condition on the diffusion function:
    $$\dot{Q} < 0$$
  • In terms of redshift $z$, this requires $dQ/dz > 0$ (meaning $Q$ must decrease as the universe expands).

• Model Testing:

  • The derived condition ($\dot{Q} < 0$) is applied to the specific functional forms of $Q(t)$ proposed in Refs. [10] and [11].
  • The author also establishes a general "no-go" theorem for the entire class of $\Lambda$CDM+diffusion models with pressureless matter.

3. Key Contributions

1. Derivation of a General Thermodynamic Admissibility Condition: The paper proves that for any $\Lambda$CDM+diffusion model in UG with pressureless matter, the Second Law of Thermodynamics strictly requires $\dot{Q} < 0$. This means energy must flow from the effective cosmological term into the matter sector to increase entropy.
2. Identification of Structural Incompatibility: The author demonstrates that the mechanism proposed to solve the $H_0$ tension requires $\dot{Q} > 0$ (energy flowing from matter to $\Lambda_{\text{eff}}$ to increase the expansion rate). This creates a fundamental, structural contradiction: the condition required to solve the $H_0$ tension is the exact opposite of the condition required for thermodynamic admissibility.
3. No-Go Theorem: The paper formalizes a theorem stating that no choice of the diffusion function $Q(t)$ can simultaneously satisfy the Second Law of Thermodynamics and produce a growing effective cosmological term ($\dot{\Lambda}_{\text{eff}} > 0$) within this specific framework.

4. Results

The analysis explicitly evaluates the two primary proposals:

• Perez, Sudarsky, and Wilson-Ewing [10]:

  • Model A (Sudden Transfer): Requires a jump in $\Lambda_{\text{eff}}$ at a specific redshift, implying $\dot{\Lambda}_{\text{eff}} > 0$ and $\dot{Q} > 0$. This violates the Second Law ($T\dot{S} < 0$).
  • Model B (Anomalous Decay): Requires a continuous growth of $\Lambda_{\text{eff}}$ ($\dot{Q} > 0$). This also violates the Second Law.
  • Note: The original authors explicitly identified $\dot{\Lambda}_{\text{eff}} > 0$ as necessary for their solution but did not address the thermodynamic consequence.

• Landau et al. [11]:

  • This model uses a smoothed step function for energy transfer. To alleviate the $H_0$ tension, the model requires a positive energy transfer amplitude ($\Delta \rho > 0$), which corresponds to $Q$ increasing with time ($\dot{Q} > 0$).
  • The statistical analysis in Ref. [11] shows that the region where $H_0$ tension is alleviated ($\Delta \rho_\Lambda > 0$) is precisely the region that is thermodynamically inadmissible. The region consistent with thermodynamics ($\Delta \rho_\Lambda < 0$) fails to resolve the tension.

5. Significance and Conclusion

• Structural Obstruction: The paper concludes that the specific class of Unimodular Gravity models relying on diffusion from matter to dark energy to solve the $H_0$ tension is thermodynamically inadmissible. This is not a flaw in specific parameter choices but a fundamental structural incompatibility within the $\Lambda$CDM+diffusion framework with pressureless matter.
• Implications for Future Research:
  • The results do not rule out Unimodular Gravity as a whole.
  • They rule out the specific mechanism of matter-to-dark-energy diffusion for pressureless fluids.
  • Future viable models must either:
    1. Propose a different physical mechanism for $H_0$ resolution that does not rely on $\dot{\Lambda}_{\text{eff}} > 0$.
    2. Consider matter components with non-zero pressure (equation of state $w \neq 0$), which would alter the thermodynamic derivation.
• Constraint on Model Building: Thermodynamic consistency must be treated as a necessary constraint alongside observational viability when constructing diffusion-based cosmological models.

In summary, Cataldo demonstrates that the "fix" for the $H_0$ tension proposed in recent literature violates the Second Law of Thermodynamics, rendering these specific models physically invalid.