Bulk viscosity from early-time thermalization of cosmic fluids in light of DESI DR2 data

Using DESI DR2 BAO measurements, this paper constrains a model of interacting dark matter and radiation that induces bulk viscosity to resolve cosmic tensions, finding that the data disfavors such interactions prior to recombination and rules out the model as a solution to the Hubble tension.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Idea: A Cosmic Traffic Jam

Imagine the early universe as a giant, expanding dance floor. On this floor, there are two main groups of dancers:

1. Radiation (The Lightweights): These are photons (light particles) and hot plasma. They are moving super fast, bouncing off each other constantly.
2. Dark Matter (The Heavyweights): These are invisible, slow-moving particles that don't interact with light, but they have mass.

In the standard story of the universe (called the $\Lambda$CDM model), these two groups dance separately. They don't really talk to each other; they just move to the beat of the universe's expansion.

The New Hypothesis:
The authors of this paper asked: What if these two groups actually bumped into each other and tried to sync up their dance moves?

If the heavy dark matter and the light radiation interacted enough to reach a "thermal equilibrium" (basically, getting to the same temperature), it would create a weird friction. In physics, this is called Bulk Viscosity.

The Analogy: The Sticky Syrup Effect

Think of the universe's expansion like a car driving down a highway.

• Standard Model: The car is on a smooth road. It accelerates and slows down predictably based on how much fuel (matter) and air resistance (radiation) it has.
• The Viscous Model: Imagine the road suddenly gets coated in thick, sticky syrup right around the time the car is shifting gears (the moment when matter and radiation had equal energy).

This "syrup" is the bulk viscosity. It doesn't stop the car, but it adds a weird resistance that changes how fast the car goes. Specifically, this syrup gives the universe an extra little "push" during that specific era, making it expand slightly faster than expected.

Why Does This Matter? (The Hubble Tension)

Astronomers have a big problem right now called the Hubble Tension.

• Method A (The Baby Picture): When we look at the oldest light in the universe (the Cosmic Microwave Background), we calculate the universe's expansion rate to be about 67.
• Method B (The Adult Picture): When we look at nearby stars and galaxies today, we calculate the rate to be about 73.

These numbers don't match. Scientists are desperate for a theory that explains why.

The authors of this paper thought: "Maybe that 'sticky syrup' (viscosity) in the early universe gave the expansion an extra push, making the universe older or expanding faster, which could fix the mismatch between Method A and Method B."

The Test: The Cosmic Ruler

To test if this "sticky syrup" exists, the authors used data from DESI (Dark Energy Spectroscopic Instrument).

Think of the early universe as having a giant, frozen ripple in a pond. This ripple is called a Baryon Acoustic Oscillation (BAO). It's a standard "ruler" imprinted in the distribution of galaxies. We know exactly how big this ruler should be based on the physics of sound waves in the early plasma.

If the "sticky syrup" (viscosity) was real, it would have changed the speed of those sound waves. This would make the "ruler" look a different size to us today compared to what the standard model predicts.

The Result: The Syrup Wasn't There

The authors ran the numbers using the latest, most precise data from DESI (DR2). They looked for the signature of this viscosity.

The Verdict: Nope.

The data showed that the "ruler" is exactly the size the standard model predicts. There is no evidence of the "sticky syrup."

• The Constraint: They calculated that if this interaction did happen, it had to be incredibly weak—so weak that it's essentially zero.
• The Consequence: Because the viscosity is so small, it cannot provide the extra "push" needed to fix the Hubble Tension.

The Bottom Line

The paper concludes that Dark Matter and Radiation did not interact significantly in the early universe to create this viscosity.

• Good news: Our standard model of the universe is looking very solid.
• Bad news (for this theory): This specific idea cannot solve the mystery of why the universe's expansion rate is confusing us. We still need to find a different explanation for the Hubble Tension.

In a nutshell: The authors tried to find a "cosmic friction" that might explain a major astronomical puzzle. They used the most precise cosmic ruler available (DESI data) to look for it, but the ruler didn't bend. The friction isn't there, and the puzzle remains unsolved.

Technical Summary

Here is a detailed technical summary of the paper "Bulk viscosity from early-time thermalization of cosmic fluids in light of DESI DR2 data" by Hermano Velten and William Iania.

1. Problem Statement

The paper addresses the theoretical possibility that nonrelativistic dark matter (DM) and radiation interact in the early universe, reaching an approximate thermal equilibrium prior to recombination.

• The Mechanism: If these two adiabatic fluids (radiation and DM) interact weakly enough to be treated as a composite fluid but strongly enough to maintain thermal equilibrium, their differing cooling rates induce a transient bulk viscous pressure.
• The Motivation: Previous work (Ref. [1]) suggested that such a mechanism could alleviate the Hubble tension (the discrepancy between early-universe $H_0$ measurements from CMB and late-universe measurements) by increasing the expansion rate near the matter-radiation equality epoch ($z_{eq} \sim 3400$).
• The Conflict: While this mechanism boosts $H_0$, it also alters the sound speed of the baryon-photon fluid. The authors aim to test whether the magnitude of this interaction required to solve the Hubble tension is consistent with the latest high-precision Baryon Acoustic Oscillation (BAO) data from the DESI DR2 (Dark Energy Spectroscopic Instrument, Data Release 2).

2. Methodology

A. Theoretical Framework

The authors utilize the Eckart formalism for relativistic dissipative fluids to model the interaction between radiation ($r$) and cold dark matter ($m$).

• Thermalization Time Scale ($\tau$): The model introduces a free parameter $\tau_{eq}$, representing the characteristic time scale for the fluids to thermalize (reach a common temperature) around the epoch of matter-radiation equality.
• Bulk Viscous Pressure ($\Pi$): The interaction generates a bulk viscous pressure $\Pi = -3H\xi$, where $\xi$ is the bulk viscous coefficient.
  • The coefficient depends on the DM particle mass ($m_\chi$) and the thermalization time: $\xi \propto \tau_{eq} \cdot \tilde{\eta}(m_\chi)$.
  • The effect is transient, peaking around $z_{eq}$ and vanishing in the ultra-relativistic (early) and non-relativistic (late) limits.
• Modified Dynamics: The presence of $\Pi$ modifies the Friedmann continuity equation, leading to a dimensionless expansion rate $E(z)$ that deviates from the standard $\Lambda$CDM model.
• Sound Speed Correction: Crucially, the bulk viscosity introduces a non-adiabatic correction to the squared sound speed ($c_s^2$) of the baryon-photon fluid. The authors derive an analytical expression for this correction, noting that $\delta \rho_\Pi < 0$, which reduces the effective sound speed.

B. Observational Constraints

• Data Source: The study uses DESI DR2 BAO measurements, which provide cosmic distance ratios ($D_V/r_d$, $D_M/r_d$, $D_H/r_d$) across redshifts $z \in [0.295, 2.33]$. These are reported relative to a fiducial $\Lambda$CDM cosmology.
• Sound Horizon ($r_d$): The calculation of the sound horizon at the drag epoch ($z_d \approx 1060$) is sensitive to the integral of $c_s(z)/H(z)$. Since the model alters both $H(z)$ and $c_s(z)$, it directly impacts the predicted BAO observables.
• Statistical Analysis: A Bayesian analysis is performed on the logarithm of the free parameter, $\log_{10}(\tau_{eq})$.
  • Priors: A uniform prior is imposed with a theoretical upper bound derived from the requirement that the sound speed squared remains positive ($c_s^2 > 0$).
  • Likelihood: Constructed using the $\chi^2$ statistic comparing observed BAO distances with theoretical predictions derived from the modified expansion history and sound speed.

3. Key Contributions

1. Derivation of Non-Adiabatic Sound Speed: The paper provides a detailed derivation of how bulk viscosity modifies the sound speed of the baryon-photon fluid ($c_s^2$). This is a critical contribution because previous analyses often focused solely on the background expansion rate ($H_0$), neglecting the impact on the sound horizon which is the standard ruler for BAO.
2. Theoretical Upper Bound on Sound Speed: The authors establish a strict theoretical limit on the thermalization time $\tau_{eq} < 1.64 \times 10^{-8}$ s to ensure $c_s^2 > 0$ at all redshifts.
3. Mass Independence Robustness: The analysis demonstrates that the constraints on $\tau_{eq}$ are robust against variations in the dark matter particle mass ($m_\chi$) within the allowed range ($1-10$eV). The effective viscosity changes by less than an order of magnitude across this range, justifying the fixation of$m_\chi = 1$ eV for the statistical analysis.
4. Integration of DESI DR2 Data: This is one of the first applications of the new DESI DR2 dataset to constrain early-time interacting dark sector models, offering tighter constraints than previous DES Y3 data.

4. Results

• Constraint on Thermalization Time: The joint analysis of all DESI DR2 tracers yields a stringent 2$\sigma$ upper bound:
  $$\log_{10}(\tau_{eq} [\text{s}]) \lesssim -9.76$$
  This corresponds to $\tau_{eq} \lesssim 1.7 \times 10^{-10}$ s.
• Constraint on Bulk Viscosity: The corresponding dimensionless bulk viscous coefficient is constrained to:
  $$\frac{\tilde{\xi}}{E} \bigg|_{z_{eq}} \lesssim 5.94 \times 10^{-4} \quad (2\sigma)$$
• Comparison with Hubble Tension Solution:
  • Previous work suggested that $\tau_{eq} \sim 7 \times 10^{-8}$ s could significantly alleviate the Hubble tension (increasing $H_0$).
  • The new DESI DR2 bound ($\sim 10^{-10}$ s) is two orders of magnitude tighter than the value required to solve the tension.
  • At the constrained value ($\tau_{eq} \lesssim 10^{-9}$ s), the effect on the expansion rate is negligible, and the model fails to provide the necessary boost to $H_0$.

5. Significance and Conclusion

• Ruling Out the Model: The primary conclusion is that DESI DR2 data do not support the specific mechanism of early-time thermalization between dark matter and radiation proposed to solve the Hubble tension. The interaction time scale required to produce a measurable bulk viscosity is ruled out by the BAO data.
• Implications for Cosmic Tensions: This result suggests that this specific class of "bulk viscous" models cannot be the solution to the Hubble or $S_8$ tensions. The data favors the standard $\Lambda$CDM model over this interacting scenario.
• Methodological Note: The authors acknowledge that the Eckart formalism used has known issues with causality and stability. They suggest that future work should employ the Müller-Israel-Stewart (MIS) theory or other causal hydrodynamic frameworks to confirm these results, though they argue the physical conclusion (data disfavoring the interaction) likely holds regardless of the specific formalism.

In summary, the paper uses the precision of modern BAO data to place a "death knell" on a specific thermalization model, demonstrating that the required interaction strength is incompatible with the observed sound horizon and expansion history of the universe.