⚛️ general relativity

Beyond dust in local cosmography: perturbative Vlasov predictions

This paper uses linear Vlasov perturbation theory to demonstrate that while velocity dispersion in structure formation generates an intrinsic dipole in the local distance-redshift relation through pressure gradients, this effect is too small to explain the observed low-redshift dipole, which is instead dominated by the relative motion between the matter and CMB frames.

Original authors: Chris Clarkson, Roy Maartens

Published 2026-10-20

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CC BY 4.0

Original authors: Chris Clarkson, Roy Maartens

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

In the vast, expanding universe, astronomers often rely on a simplified picture of how matter moves. They imagine the cosmos as a smooth, flowing fluid where every piece of dust or dark matter at a given location shares the same speed and direction. This "dust" model works well when the universe is calm, but as gravity pulls matter together to form galaxies and clusters, the flow becomes chaotic. Streams of particles cross paths, and at any single point in space, matter can be moving in many different directions at once. This jumbling creates a kind of internal pressure, not from collisions like gas molecules, but from the sheer randomness of the motion. While individual particles follow the straightest possible paths through space, the average flow of the crowd can be pushed and pulled by the gradients of this pressure. Understanding this subtle push is crucial for measuring the universe's expansion, because if we ignore it, our maps of cosmic distances could be slightly off.

A team of researchers has now calculated exactly how much this internal pressure changes the local flow of the universe. They used a sophisticated mathematical framework, known as Vlasov perturbation theory, to track how these random motions evolve from the smooth early universe into the clumpy structure we see today. By building on a standard model of cosmology and using estimates of how dark matter halos generate random speeds, they simulated the behavior of matter on scales relevant to nearby galaxies. Their work reveals that while this pressure does indeed create a new force that accelerates the average flow of matter, the effect is surprisingly small. The most significant change they found is a tiny adjustment to the "shear," which describes how the expansion of space stretches and distorts the flow in different directions. On the scales they studied, this correction amounts to less than one and a half percent of the usual stretching effect.

The researchers were particularly interested in whether this pressure could explain a mysterious dipole, or a directional bias, observed in the motion of nearby galaxies. When astronomers look at the universe, they see a slight preference for galaxies moving in one direction, which has been interpreted as a sign that our local patch of the universe is moving relative to the cosmic background. Some had wondered if the internal pressure of the matter itself could be the source of this bias. The study shows that it cannot. The intrinsic dipole generated by the pressure of the random motions is more than a thousand times smaller than the dipole actually measured. The observed bias is almost entirely due to the motion of our local group of galaxies through the universe, not the internal jiggling of the matter itself.

This finding clarifies the limits of our current models. The researchers confirmed that while pressure does alter the growth of cosmic structures and introduces a tiny, constant acceleration to the local flow, it is far too weak to account for the large-scale motions we observe. The sensitivity of previous measurements to very nearby galaxies, which caused some confusion in earlier studies, is likely a limitation of trying to describe a chaotic, multi-stream flow as a single smooth fluid, rather than a failure of the underlying physics. The study also highlights that the pressure's effect on the expansion rate is most noticeable in the next layer of detail, where it slightly modifies how the expansion changes with distance. However, even these corrections remain small compared to the dominant effects of the universe's expansion and the motion of the observer.

Ultimately, the paper provides a rigorous check on the "dust" approximation that underpins much of modern cosmology. It confirms that for the purpose of mapping the local universe, treating matter as a simple fluid is a very good approximation, with errors that are negligible for current observations. The internal pressure generated by the chaotic motion of dark matter is real and calculable, but it is a whisper in a storm. It does not drive the large-scale flows that astronomers measure, nor does it significantly distort our view of the universe's expansion. The work serves as a precise calibration, ensuring that when we look out at the cosmos, we are interpreting the signals of motion and expansion with the correct understanding of the invisible forces at play.

Technical Summary: Beyond Dust in Local Cosmography

Problem Statement
Standard cosmographic analyses rely on the "dust" approximation, which assumes matter flows as a pressureless fluid with a unique velocity at every spacetime event. However, as cosmic structure forms, particle streams cross, creating regions where multiple velocities coexist at the same position. This generates velocity dispersion, which manifests as isotropic and anisotropic pressure. While individual dark matter particles follow geodesics, the gradients of these pressures accelerate the mean matter flow. This paper investigates the magnitude of this acceleration and its impact on local cosmography, specifically the distance–redshift relation, which is traditionally expressed in terms of local expansion, shear, and 4-acceleration. Previous work (Paper I) established the theoretical framework for this acceleration; this study quantifies its effects using linear Vlasov perturbation theory (VPT).

Methodology
The authors employ linear Vlasov perturbation theory about a flat $\Lambda$ CDM Friedmann-Lemaître-Robertson-Walker (FLRW) background to model collisionless matter.

Theoretical Framework: The study utilizes velocity moments and cumulants to describe the phase-space distribution. The hierarchy is truncated at the second cumulant ( $c_{\text{max}}=2$ ) for acceleration spectra and spatial windows, and at the third cumulant ( $c_{\text{max}}=3$ ) for local Hubble and deceleration multipoles to capture pressure time derivatives.
Input Physics: The mean velocity dispersion history, $\epsilon(a)$ , is not derived from first principles but prescribed based on halo estimates from reference [21], which combines Navarro-Frenk-White density profiles and halo mass functions. This history accounts for random motions generated by nonlinear structure formation.
Calculation: The authors evolve linear fluctuations of density, velocity, and pressure about this mean dispersion. They calculate the resulting corrections to the kinematic quantities (expansion $\Theta$ , shear $\sigma_{ab}$ , and acceleration $A_a$ ) and project these onto the local distance–redshift relation.
Observables: The analysis focuses on local cosmographic coefficients (Hubble dipole, shear, deceleration multipoles) and the intrinsic dipole in the distance–redshift relation. These are evaluated using Gaussian and top-hat spatial windows ( $R \approx 13.4$ and $30 \, h^{-1}\text{Mpc}$ ) and volume-weighted radial integrals up to $z_{\text{max}} = 0.01$ .

Key Contributions and Results

Shear Correction: At zeroth order in redshift, the largest correction to the dust model is to the Hubble shear. The inclusion of velocity dispersion alters the shear root-mean-square (rms) by 0.3% to 1.3% depending on the smoothing scale, relative to the dust shear rms.
Intrinsic Acceleration Dipole: The pressure gradients generate an intrinsic dipole in the local expansion ( $H_a = -A_a$ ). The calculated sky rms of this intrinsic dipole is approximately $8.4 \times 10^{-7}$ (at $30 \, h^{-1}\text{Mpc}$ ) and $4.0 \times 10^{-6}$ (at $13.4 \, h^{-1}\text{Mpc}$ ).
Comparison with Observations: The calculated intrinsic dipole is three to four orders of magnitude smaller than the observed low-redshift dipole in the Cosmicflows-4 (CF4) catalogue. The observed dipole is dominated by the Doppler effect arising from the motion of the local matter frame relative to the Cosmic Microwave Background (CMB).
Redshift-Dependent Effects: At the next order in redshift ( $O(z)$ ), corrections to the deceleration parameter arise from changes in the growth of structure (altering expansion and shear gradients) and explicit acceleration terms. The growth-induced dipole correction can exceed the shear correction at finer scales, reaching an rms of $3.6 \times 10^{-4}$ at $13.4 \, h^{-1}\text{Mpc}$ .
Truncation Sensitivity: Including the third cumulant (which describes velocity distribution asymmetry) changes the observer amplitudes by less than 1%, confirming that the second-order truncation is sufficient for these linear calculations.
Nonlinear Modulation: The authors note that while the mean dispersion is fixed, nonlinear density modes can modulate the local production of dispersion. Sensitivity tests suggest this modulation could change the acceleration dipole by order unity, though this requires nonlinear pressure-density correlations not captured in the linear VPT framework.

Significance and Claims
The paper claims that while velocity dispersion introduces physically real corrections to local cosmography, these effects are too small to explain the systematic shifts observed in CF4 fits when extending to very low redshifts ( $z < 0.01$ ).

Limit of Smooth-Flow Description: The deterioration of the dust-based fit at low redshift is interpreted primarily as a limit of the "smooth-flow" cosmographic description itself, rather than a failure of the $\Lambda$ CDM model or a missing large-scale acceleration.
Dipole Origin: The intrinsic dipole generated by pressure gradients is negligible compared to the Doppler dipole caused by observer motion relative to the CMB.
Cosmological Principle: The study emphasizes that the assumed FLRW background prevents an independent test of the cosmological principle within this framework. However, it suggests that measuring the intrinsic dipole (via the acceleration term $H_1$ ) in future fits could empirically test the geodesic-flow assumption used in arguments for the cosmological principle.
Future Requirements: The authors conclude that connecting local Taylor coefficients to catalogue fits requires finite-distance light-cone calculations, as the local gradient expansion lacks a small parameter ( $k\chi \sim 1$ ) at representative depths, even for dust.

In summary, the paper provides a rigorous perturbative calculation showing that while pressure effects from velocity dispersion are non-zero and modify shear and growth rates, they do not generate a significant intrinsic dipole capable of mimicking or explaining the observed low-redshift kinematic anomalies.

Technical Summary: Beyond Dust in Local Cosmography

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