Testing the cosmological Euler equation: viscosity, equivalence principle, and gravity beyond general relativity

This paper extends the testing of the cosmological Euler equation by introducing a generalized observable for equivalence principle violations and a model-independent parameter for dark matter viscosity, demonstrating that future Stage-IV surveys like SKA2 can constrain the latter to a precision of approximately $10^{-7}$.

The Gist

Imagine the universe as a giant, expanding ocean. In this ocean, there are invisible islands of "dark matter" that hold everything together. For decades, scientists have assumed these islands are like perfect, frictionless ice: they slide around without any resistance, and they all fall at the exact same speed when pulled by gravity. This assumption is the "standard model" of cosmology.

But what if that ice isn't perfect? What if it's actually honey? Or maybe molasses? What if, instead of falling identically, some particles fall slightly faster than others because they are being pushed by a hidden "fifth force"?

This paper is a detective story. The authors are asking: "Can we test if our 'perfect ice' assumption is wrong, even if the universe is actually made of sticky honey, without getting confused?"

Here is the breakdown of their investigation using simple analogies:

1. The Three Suspects

The authors are looking for three things that might be messing with our view of the universe:

• Viscosity (The Honey): Dark matter might be "sticky." Just like honey resists being stirred, sticky dark matter would resist moving, slowing down the formation of galaxy clusters.
• Equivalence Principle Violation (The Uneven Slide): Einstein's rule says everything falls at the same speed in gravity. But what if dark matter is like a heavy bowling ball and normal matter is like a feather, and they fall at different speeds?
• Modified Gravity (The Hidden Hand): Maybe gravity itself works differently on huge scales, acting like a hidden hand pushing or pulling things in ways we don't expect.

2. The Problem: The "Mix-Up"

The tricky part is that these three suspects look very similar.

• If dark matter is sticky (viscous), it slows down galaxy growth.
• If gravity is modified, it can also slow down galaxy growth.
• If the Equivalence Principle is broken, it can also change how galaxies move.

It's like trying to figure out why a car is moving slowly. Is it because the engine is weak (modified gravity)? Is it because the brakes are stuck (viscosity)? Or is the driver pressing the gas pedal differently (Equivalence Principle violation)? Usually, you can't tell them apart just by looking at the speed.

3. The Solution: The "Two-Tracer" Trick

The authors propose a clever way to separate these suspects. Instead of looking at just one type of galaxy, they suggest looking at two different populations (like "Bright" galaxies and "Faint" galaxies) at the same time.

Think of it like a race between a Ferrari and a Tractor.

• If the road is just bumpy (standard gravity), both vehicles slow down predictably.
• If the road is covered in honey (viscosity), the Ferrari might get stuck more than the Tractor, or vice versa, depending on their shape.
• If there is a hidden wind (modified gravity), it might push the Ferrari harder than the Tractor.

By comparing how the "Ferrari" and "Tractor" move relative to each other, and by looking at the universe from a very specific, relativistic angle (using the Doppler effect and gravitational redshift, which are like the "sound" and "color" shifts of light), the authors found a way to isolate the "honey."

4. The New Clue: $C_{vis,0}$

The paper introduces a new "smoking gun" number called $C_{vis,0}$.

• Think of this as a "Stickiness Meter" for the universe today.
• The authors developed a way to measure this stickiness directly from the data of future galaxy surveys, without needing to guess what the universe looked like at the beginning of time.
• They found that even a tiny amount of stickiness (like a drop of honey in a swimming pool) leaves a huge fingerprint on the way galaxies cluster, especially on small scales.

5. The Future: The Super-Surveys

To catch these subtle effects, we need super-powerful eyes. The paper forecasts what will happen when three massive new telescopes come online:

1. DESI (The Desert Surveyor)
2. Euclid (The European Space Agency's eye)
3. SKA2 (The Square Kilometre Array, a radio telescope the size of a continent)

The Verdict:
The authors ran the numbers (forecasts) and found that SKA2 will be the ultimate detective. It will be able to measure the "Stickiness Meter" ($C_{vis,0}$) with incredible precision—down to a level of 0.0000001.

Why Does This Matter?

• If they find stickiness: We know dark matter isn't just cold, boring particles. It has a complex, fluid nature. This changes our understanding of how galaxies form.
• If they find no stickiness: We can confidently say that any weird behavior we see in the universe is due to Modified Gravity or Equivalence Principle violations, not sticky dark matter. This would be a massive breakthrough for physics, potentially proving Einstein wrong on cosmic scales.

In a Nutshell

This paper is a blueprint for how to use the next generation of giant telescopes to taste the "texture" of the invisible universe. They've figured out how to tell the difference between a universe made of sticky honey and a universe where gravity has a secret sauce, ensuring that we don't get fooled by the mix-up. It's a robust, model-independent way to test the fundamental laws of physics using the largest structures in existence.

Technical Summary

Here is a detailed technical summary of the paper "Testing the cosmological Euler equation: viscosity, equivalence principle, and gravity beyond general relativity" by Zheng, Schneider, and Amendola.

1. Problem Statement

The cosmological Euler equation governs the evolution of velocity perturbations in large-scale structure (LSS) and encodes the Weak Equivalence Principle (EP), which states that all test particles fall identically in a gravitational field. Standard cosmological analyses typically assume Dark Matter (DM) behaves as a "cold," pressureless, perfect fluid. However, the microscopic nature of DM is unknown, and it may possess dissipative properties (viscosity).

The paper addresses three interconnected challenges:

1. Viscosity: How does DM viscosity (both bulk $\zeta$ and shear $\eta_s$) modify the Euler equation and structure growth?
2. Degeneracy: Does DM viscosity mimic or obscure signatures of EP violations and Modified Gravity (MG)?
3. Observability: Can upcoming Stage-IV galaxy surveys (DESI, Euclid, SKA2) distinguish between viscosity, EP violation, and MG in a model-independent way?

Previous studies often relied on strong assumptions about structure growth parametrizations or ignored relativistic effects. This work aims to lift the perfect-fluid assumption and provide a robust, model-independent framework to test these effects.

2. Methodology

Theoretical Framework

• Viscous Fluid Dynamics: The authors treat DM as a pressureless fluid with constant bulk and shear viscosities within Eckart's theory of relativistic viscous fluids. They derive the perturbed continuity and Euler equations in the Newtonian gauge.
• Modified Gravity & EP Violation: They generalize the Euler equation to include phenomenological free functions:
  • $\Theta$ and $\Gamma$: Friction and fifth-force terms acting on DM (encoding EP violation).
  • $\mu_G$ and $\eta$: Effective gravitational coupling and gravitational slip (encoding MG).
• Relativistic Galaxy Counts: Instead of relying solely on Redshift-Space Distortions (RSD), the authors utilize relativistic galaxy number counts. They include corrections up to order $\lambda = H/k$ (where $H$ is the Hubble parameter and $k$ is the wavenumber), such as gravitational redshift, Doppler effects, and lensing magnification.
• Two-Tracer Approach: The analysis employs two distinct galaxy populations (Bright and Faint) to break degeneracies between bias and cosmological parameters.

Key Derivations

• Generalized EP Parameter ($\tilde{E}_P$): The authors extend the observable $\mathcal{E}_P$ (defined in previous work [1]) to a viscous generalization $\tilde{E}_P$. They demonstrate that in the limit of small viscosity, $\tilde{E}_P$ reduces to the standard EP test, provided viscosity terms are negligible.
• Viscosity Parameter ($C_{vis}$): A new model-independent observable is identified:
  $$C_{vis} = \frac{1}{3}\frac{Z + E}{1 - Z}$$
  where $Z$ and $E$ are dimensionless parameters related to bulk and shear viscosity, respectively. $C_{vis}$ characterizes the present-day viscosity ($C_{vis,0}$).
• Scale Dependence: Crucially, viscosity introduces a term proportional to $\lambda^{-2}$ (or $k^2$) in the Euler equation. This leads to a scale-dependent suppression of structure growth, which is distinct from the scale-independent effects of standard MG parameters in the sub-horizon limit.

Forecast Analysis

• Fisher Matrix Formalism: The authors construct a Fisher matrix to forecast constraints on parameters using data from DESI, Euclid, and SKA Phase 2 (SKA2).
• Parameter Space: The analysis varies 10 free parameters per redshift bin, including growth rates ($\tilde{\beta}$), bias parameters, Alcock-Paczynski parameters, and the shot noise amplitude.
• Model Independence: The spectral shape of the power spectrum is left free to vary in $k$-bins, avoiding assumptions about the initial power spectrum or specific growth parametrizations.

3. Key Contributions

1. Generalization of the EP Test: The paper proves that the observable $\tilde{E}_P$ remains a valid "smoking gun" for EP violation even in the presence of DM viscosity, provided the viscosity is small ($C_{vis,0} \lesssim 10^{-6}$). If EP violation occurs at a level comparable to viscosity ($O(10^{-6})$), the two effects become degenerate.
2. Identification of $C_{vis,0}$: The authors identify $C_{vis,0}$ as a measurable, model-independent quantity that encodes the present-day viscosity of dark matter. They show that while bulk and shear viscosities are degenerate individually, their combination $C_{vis}$ can be tightly constrained.
3. Relativistic Effects as a Probe: The study highlights that relativistic corrections (specifically the Doppler term $V' \cdot \hat{n}$) are essential for measuring viscosity. The viscosity signal appears at order $\lambda^{-2}$ in the power spectrum, making it detectable even for very small viscosity coefficients.
4. Two-Tracer Advantage: The analysis confirms that cross-correlating two galaxy populations with different biases is necessary to disentangle viscosity from other cosmological parameters and to measure the scale-dependent imprint of viscosity.

4. Results

• Constraints on Viscosity ($C_{vis,0}$):
  • SKA2: Provides the tightest constraints, with a $1\sigma$uncertainty of **$1.08 \times 10^{-7}$** on$C_{vis,0}$.
  • Euclid: Achieves constraints at the level of $3.32 \times 10^{-7}$.
  • DESI: Yields constraints of approximately $1.44 \times 10^{-6}$.
  • These constraints are significantly tighter than previous limits and are achieved with minimal theoretical assumptions.
• Bias Ratio Sensitivity: The constraining power improves with larger differences in galaxy bias between the two tracers. For SKA2, an asymmetric bias ratio ($b_F/b_B = 0.2$) improves the constraint on $C_{vis,0}$ to $7.46 \times 10^{-8}$.
• EP Violation Constraints: Constraints on the EP violation parameter $\tilde{E}_P$ are weaker than in previous viscous-free studies. This is because the analysis treats magnification bias parameters ($\alpha_B, \alpha_F$) as free variables to remain conservative, leading to degeneracies with $\tilde{E}_P$. Future improvements in measuring magnification bias are required to tighten these bounds.
• Degeneracy: The study confirms that in the general case, viscosity parameters ($Z, E$) are degenerate with EP violation parameters ($\Theta, \Gamma$). However, the scale-dependent nature of viscosity allows for its separation in the small-viscosity limit.

5. Significance

• Robustness of Gravity Tests: The paper establishes that the Weak Equivalence Principle can still be tested using large-scale structure data even if Dark Matter is not a perfect fluid. The "null test" for EP violation remains valid unless the violation is extremely subtle (at the $10^{-6}$ level).
• New Window on Dark Matter: It provides a pathway to directly measure the dissipative properties of Dark Matter (viscosity) using cosmological surveys, moving beyond the standard cold dark matter paradigm without assuming specific microscopic models.
• Survey Optimization: The results guide the design and analysis of future Stage-IV surveys (DESI, Euclid, SKA), emphasizing the critical role of relativistic corrections, multi-tracer strategies, and precise bias modeling.
• Model Independence: By avoiding assumptions about the background expansion or initial power spectrum shape, the proposed method offers a more robust test of fundamental physics compared to previous approaches that relied on specific parametrizations of structure growth.

In conclusion, this work demonstrates that upcoming galaxy surveys will be capable of placing stringent, model-independent constraints on dark matter viscosity, thereby opening a new avenue to test the fundamental nature of gravity and dark matter on cosmological scales.