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Dark sector interactions in the $w \rightarrow -1$ limit: velocity locking in pure momentum exchange models

This paper reveals that in interacting dark energy and dark matter models with pure momentum exchange, a $w$ -dependent velocity-locking mechanism emerges as the dark energy equation of state approaches $-1$, causing the dark energy fluid to track dark matter velocities and shift power spectrum suppression to smaller scales, thereby explaining weaker observational constraints and demonstrating that neglecting dark energy perturbations leads to an overestimation of these constraints.

Original authors: Nathan Cruickshank, Robert Crittenden, Kazuya Koyama, Marco Bruni

Published 2026-07-08

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CC BY 4.0

Original authors: Nathan Cruickshank, Robert Crittenden, Kazuya Koyama, Marco Bruni

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Cosmic Tug-of-War

Imagine the universe is filled with two invisible fluids: Dark Matter (the stuff that holds galaxies together) and Dark Energy (the stuff pushing the universe apart). Usually, scientists think these two just ignore each other, except for gravity. But this paper explores a scenario where they actually "bump into" each other, like two crowds of people trying to walk through a hallway at the same time.

When they bump, they exchange momentum. Think of it like a game of tag: if the Dark Energy crowd is moving one way and the Dark Matter crowd is moving another, the collision slows the Dark Matter down. This slowing down changes how galaxies clump together, which astronomers can measure.

The Problem: The "Freeze" at the Edge

The paper focuses on a specific, tricky situation. Dark Energy has a property called an "equation of state" (let's call it $w$ ).

If $w$ is far from -1, the fluids behave normally.
If $w$ gets very close to -1, Dark Energy starts acting like a "vacuum" or a static background. It becomes very stiff and hard to move.

The authors wanted to know: What happens to the "bumping" (interaction) when Dark Energy gets this stiff?

The Discovery: The "Velcro" Effect

The paper reveals a surprising phenomenon the authors call "velocity locking."

Imagine Dark Matter and Dark Energy are two dancers.

Normal Mode ( $w$ is not -1): The dancers are wearing slippery shoes. They bump into each other, slow down a bit, but then they easily slide apart. The Dark Energy pushes back hard because it has high "pressure" (like a stiff spring). This keeps them from sticking together for long.
The "Freeze" Mode ( $w \to -1$ ): As Dark Energy gets closer to -1, it loses its ability to push back. It's like the dancer's shoes suddenly turn into Velcro.
- When they bump, the Dark Energy doesn't push back hard enough to break the connection.
- Instead, the Dark Energy gets "stuck" to the Dark Matter. They move together as a single unit.
- Because they are stuck together, the Dark Energy can't slow the Dark Matter down effectively. The "braking" effect disappears.

The Result: Hiding in the Small Details

Because of this "Velcro" effect, the interaction stops working on large scales (big clusters of galaxies) and only starts working on very tiny scales (small clumps).

The Shift: As Dark Energy gets closer to the "stiff" limit ( $w \to -1$ ), the point where the interaction stops working moves further and further to the right, into the realm of tiny, small-scale structures.
The Analogy: Imagine a filter that catches big rocks but lets sand through. As you change the settings, the filter starts catching only the tiniest grains of dust, letting everything else pass. The "cutoff point" shifts to smaller and smaller sizes.

This explains why previous studies found it hard to measure these interactions when $w$ is close to -1. The "signature" of the interaction (the slowing down of matter) gets pushed so far into the tiny scales that it becomes very hard to see with current telescopes.

The Mistake: Turning Off the Music

The paper also points out a common mistake scientists make. To make calculations easier, many researchers assume Dark Energy is "dead" or "frozen"—meaning they pretend it has no movement or internal changes ( $\delta_{DE} = 0$ and $\theta_{DE} = 0$ ).

Why this is wrong here: In the "Velcro" scenario, the movement of Dark Energy is exactly what keeps the lock from breaking.
The Consequence: If you pretend Dark Energy is frozen, you accidentally remove the "Velcro." You think the dancers are sliding apart when they are actually stuck together.
The Result: This leads scientists to believe they can measure the interaction strength very precisely, when in reality, the interaction is hiding in the small scales, making it much harder to detect. The paper argues that using this "frozen" assumption leads to overly optimistic (and incorrect) claims about how well we can constrain these models.

Summary

Interaction: Dark Matter and Dark Energy bump into each other, slowing down galaxy growth.
The Limit: When Dark Energy becomes very stiff ( $w \to -1$ ), it stops pushing back.
The Lock: Instead of pushing apart, the two fluids get "locked" together and move as one.
The Shift: This locking pushes the effect of the interaction to much smaller scales, making it harder to detect.
The Warning: If you ignore the movement of Dark Energy in your math, you miss this locking effect entirely and get the wrong answer about how strong the interaction is.

Technical Summary: Dark Sector Interactions in the $w \to -1$ Limit

Problem Statement
Interacting Dark Energy (DE) and Dark Matter (DM) models, specifically those involving pure momentum exchange, are proposed candidates to resolve cosmological tensions, particularly the $S_8$ tension regarding the amplitude of matter clustering. However, the theoretical behavior of these interactions in the limit where the DE equation of state approaches that of a cosmological constant ( $w \to -1$ ) remains poorly understood. In this limit, the standard fluid description breaks down because the DE fluid lacks a well-defined four-velocity, and the Euler equation becomes singular. Furthermore, current observational data (e.g., from DESI) suggest potential transitions between phantom ( $w < -1$ ) and non-phantom ( $w > -1$ ) regimes, necessitating a robust understanding of the $w \to -1$ boundary. Previous studies have noted that constraints on interaction strength weaken significantly as $w \to -1$ , often attributing this to parameter degeneracies, but the underlying physical mechanism has not been fully elucidated.

Methodology
The authors analyze the dynamics of pure momentum exchange models within the framework of linear perturbation theory in the conformal Newtonian gauge. They focus on two specific parametrisations of the interaction rate:

Dark Scattering (DS) Model: Analogous to Thomson scattering, where the interaction rate scales with the DE density ( $\Gamma \propto \rho_{DE}$ ).
Late-Time (LT) Interaction Model: Where the interaction rate strengthens at low redshifts, scaling inversely with DM density ( $\Gamma \propto 1/\rho_{DM}$ ).

The study utilizes the Einstein-Boltzmann solver CLASS to compute the linear matter power spectrum and growth rates ( $f\sigma_8$ ). The analysis compares models where DE perturbations are fully evolved ( $\delta_{DE} \neq 0, \theta_{DE} \neq 0$ ) against the common approximation where DE fluctuations are neglected ( $\delta_{DE} = \theta_{DE} = 0$ ). The authors systematically vary the equation of state $w$ from $-0.9$ to values extremely close to $-1 $(e.g.,$ -1 + 10^{-7}$) to isolate the effects of the $w \to -1$ limit on fluid dynamics and structure formation.

Key Contributions and Results

Discovery of a $w$ -Dependent Velocity-Locking Mechanism:
The paper demonstrates that a generic feature of pure momentum exchange models is a velocity-locking mechanism that becomes increasingly efficient as $w \to -1$ .
- Mechanism: As $w$ approaches $-1$, the DE density contrast ( $\delta_{DE}$ ) is heavily suppressed due to Hubble friction (since terms proportional to $1+w$ vanish in the continuity equation). Consequently, the DE pressure gradient, which normally resists the interaction drag on small scales, becomes negligible.
- Effect: The interaction drag force dominates over a much broader range of scales, forcing the DE velocity ( $\theta_{DE}$ ) to track the DM velocity ( $\theta_{DM}$ ). This "locking" prevents the suppression of DM perturbations until the scale is sufficiently small (high $k$ ) for the $k^2$ factor in the pressure term to overcome the drag.
- Scaling: The unlocking scale behaves as $k_{unlock} \propto 1/\sqrt{c_s^2(1+w)}$ . As $w \to -1$ , the onset of power spectrum suppression systematically shifts to smaller physical scales (higher $k$ ).
Impact on Observables:
- Power Spectrum: The shift in the suppression scale to higher $k$ means that for $w \approx -1$ , the suppression of the matter power spectrum occurs on smaller scales than for $w > -1$ .
- Growth Rate ( $f\sigma_8$ ): Because the suppression is confined to smaller scales, the overall impact on the integrated growth rate ( $f\sigma_8$ ) is reduced when $w \approx -1$ . This provides a physical explanation for the weaker observational constraints on interaction strength found in literature when $w$ is close to $-1$.
Failure of the $\delta_{DE} = \theta_{DE} = 0$ Approximation:
The authors rigorously demonstrate that the common approximation of neglecting DE perturbations fails catastrophically in the $w \to -1$ limit.
- By setting $\theta_{DE} = 0$ , the approximation artificially maximizes the velocity difference ( $\theta_{DM} - \theta_{DE}$ ), keeping the drag term fully active across all scales.
- This erroneously removes the velocity-locking mechanism, resulting in a scale-independent suppression that does not shift with $w$ .
- Consequently, using this approximation leads to a significant overestimation of the constraining power of cosmological data on the interaction strength, as it misses the physical mechanism that naturally weakens the interaction's signature at $w \approx -1$ .
Dependence on Sound Speed:
The study confirms that the unlocking scale is also dependent on the DE sound speed ( $c_s$ ). A lower sound speed further enhances the velocity-locking effect, shifting the suppression onset to even higher $k$ values.

Significance and Claims
The paper claims that the systematic shift in the power spectrum suppression scale is a generic feature of momentum-exchange models driven by the fluid dynamics of the DE component in the $w \to -1$ limit. The primary significance of this work is providing a physical explanation for the observed weakening of constraints on interaction strength as $w \to -1$ , moving beyond the attribution to mere parameter degeneracy.

The authors emphasize that accurate modeling of DE perturbations is crucial in this regime. They assert that neglecting DE velocity ( $\theta_{DE} = 0$ ) is not merely a minor simplification but a fundamental error in the $w \to -1$ limit, leading to incorrect predictions regarding the scale-dependence of the interaction and the resulting constraints on model parameters. This finding is particularly relevant for upcoming Stage IV surveys, which aim to probe the dark sector with high precision, especially given indications of potential phantom crossings in current data. The work also notes a phenomenological parallel with Type 3 scalar field momentum interaction models, suggesting shared observational signatures that warrant further investigation.