⚛️ general relativity

Consistent cosmological structure formation on all scales in relativistic extensions of MOND

This paper proposes and applies a method to derive consistent equations for cosmological structure formation across all scales in relativistic MOND theories, specifically generalized Einstein-Aether models, demonstrating that a single free function governs the background, linear, and non-linear regimes while clarifying that galactic-scale MOND behavior does not necessarily imply cosmological-scale modifications.

Original authors: Daniel B Thomas, Ali Mozaffari, Tom Zlosnik

Published 2026-07-31

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CC BY 4.0

Original authors: Daniel B Thomas, Ali Mozaffari, Tom Zlosnik

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

Imagine the universe as a giant, invisible ocean. For decades, the best map we have for this ocean is called the "Standard Model" (or $\Lambda$ CDM). It says the ocean is mostly made of two mysterious ingredients: "Dark Energy," which pushes the water apart, and "Cold Dark Matter," a ghostly substance that doesn't shine but acts like invisible glue, holding galaxies together. This map works incredibly well for most of the ocean, but it stumbles a bit when we look at the tiny, swirling eddies inside the galaxies.

Some scientists think the map is wrong, not because the ingredients are missing, but because the rules of how the water moves (gravity) are different than we thought. This idea is called MOND (Modified Newtonian Dynamics). It suggests that when gravity gets very weak—like on the edges of galaxies—it doesn't fade away as fast as our current rules say it should. The problem is, MOND was originally just a "rule of thumb" for galaxies. To make it a real, complete theory of the universe, it needs to be wrapped in a "relativistic extension," a fancy suit of armor that lets it work not just for galaxies, but for the whole expanding cosmos, from the biggest waves to the smallest ripples.

The big question has always been: If you fix the rules for galaxies, do those same rules automatically fix the rules for the whole universe? Or do you end up with a theory that works for a galaxy but breaks the universe? Until now, scientists have been trying to patch these theories together piece by piece, like building a car by welding a bicycle engine to a jet turbine. They often used different rules for the big picture and the small picture, which is risky. If the rules don't match, the whole theory might collapse.

This paper, written by D. B. Thomas, A. Mozaffari, and T. Zlosnik, acts like a master architect who finally draws up a single, consistent blueprint for the whole car. They focus on a specific type of "relativistic extension" called Generalised Einstein-Aether (GEA) theories. Their goal was to derive a single set of equations that can describe how matter clumps together to form galaxies and clusters, whether those clumps are tiny, huge, or somewhere in between. They wanted to see if the "MOND magic" that fixes galaxy rotation curves could also survive the journey to the cosmic scale without breaking the laws of the universe.

Here is what they found: You cannot simply copy-paste the galaxy rules to the universe. They discovered that for these theories to work consistently, the "free function" (the mathematical knob you turn to get MOND behavior) must be the same everywhere. When they turned this knob to get the MOND behavior needed for galaxies, they found a surprising result: it usually doesn't create MOND behavior for the whole universe.

In fact, for the theory to remain consistent, the universe's expansion (how fast the cosmos is growing) and the basic rules for how gravity works on large scales must look exactly like the standard "Dark Matter" rules. The "MOND magic" only appears in specific, narrow circumstances. Specifically, they found that for MOND to show up on cosmic scales, a specific parameter in the theory (called $\alpha$ ) must be exactly zero. If $\alpha$ is zero, the universe expands exactly as it does in the standard model, and the "Newtonian" part of gravity (the normal kind) remains unchanged on large scales. The "MOND" part only kicks in when gravity gets very weak, just like in galaxies.

This means that if you want a theory that fixes galaxy rotation without breaking the universe, you can't just have "MOND everywhere." You have to have a theory that looks exactly like the standard model on the big scales and only changes its behavior on the small, weak-gravity scales. The authors showed that existing computer simulations of the universe that tried to force MOND rules onto the whole cosmos were likely inconsistent because they didn't use this single, unified set of equations.

The paper doesn't claim to have solved the mystery of dark matter or proved that MOND is the right answer. Instead, it provides the necessary toolkit to test these ideas fairly. It shows that if a theory has MOND behavior in galaxies, it doesn't guarantee it will have it in the cosmos. In many cases, the theory forces the universe to behave normally, leaving the "MOND" effects only for the galaxies. This is a crucial check: it tells us that we can't just assume a theory works everywhere because it works in one place. To truly test these ideas, we need to run new, consistent computer simulations using the equations the authors derived, ensuring that the rules for the galaxy and the rules for the universe are stitched together perfectly. Until we do that, we can't say for sure if these modified gravity theories can replace the need for dark matter.

Technical Summary: Consistent Cosmological Structure Formation on All Scales in Relativistic Extensions of MOND

Problem Statement
The paper addresses a critical gap in testing Modified Newtonian Dynamics (MOND) against the standard $\Lambda$ CDM paradigm. While MOND successfully describes galactic dynamics, it is a phenomenological description that requires embedding within a relativistic theory of gravity (a "relativistic extension") to be a complete cosmological model. A significant theoretical challenge arises because different regimes of cosmological structure formation—linear perturbation theory (large scales), non-linear structure formation (small scales), and weak-field galactic dynamics—are typically studied in isolation within these relativistic extensions.

In General Relativity (GR), the equations governing these regimes are remarkably similar, allowing for a seamless transition from linear to non-linear scales in N-body simulations. However, in relativistic extensions of MOND (such as Generalised Einstein-Aether theories), the equations in these different regimes often differ significantly. Previous attempts to run cosmological N-body simulations with MONDian behavior have often involved "extrapolating" galactic-scale modifications (like a modified Poisson equation) into cosmological regimes without deriving the consistent equations that govern the full range of scales. This disjointed approach raises questions about whether a single, coherent cosmology can exist where MONDian behavior arises simultaneously on galactic and cosmological scales, and whether existing simulations accurately reflect specific relativistic theories.

Methodology
The authors propose a method to derive a single, consistent set of equations governing cosmological structure formation on all scales (from super-horizon to non-linear scales) for a specific relativistic theory. They apply the Post-Friedmann formalism, a weak-field, post-Newtonian-like expansion in powers of $1/c$ adapted for a Friedmann-Lemaître-Robertson-Walker (FLRW) background.

The core of their methodology involves:

Expansion: Expanding the metric, matter fields, and the specific vector field of the theory (Generalised Einstein-Aether or GEA) in powers of $1/c$ .
Limit Derivation: Deriving the equations for two distinct limits:
- The Linear Perturbation limit (valid on large scales).
- The Newtonian limit (valid on small, non-linear scales).
Re-summed Potentials: Converting the metric potentials into "re-summed" potentials ( $\psi_P, \phi_P, \vec{\omega}$ ) which allow for a unified description across scales.
Stitching: Combining the two limits into a single set of "all-scales" equations. This is achieved by identifying terms that are present in both limits and removing terms that only exist in a hypothetical "intermediate regime" (a regime where neither linear theory nor the Newtonian limit applies). The authors argue that for a consistent cosmology to be testable, such an intermediate regime should not exist.
Case Study: They apply this method to Generalised Einstein-Aether (GEA) theories, which feature an additional timelike vector field $A_\mu$ and a free function $F(K)$ , where $K$ is a scalar constructed from the derivatives of $A_\mu$ .

Key Contributions and Derivations
The paper derives the complete set of equations for GEA theories that describe the evolution of cosmological inhomogeneities on all scales. Key technical results include:

Unified Equations: Equations (3.22)–(3.27) provide the background evolution, linear perturbations, and non-linear structure formation for GEA theories using a single free function $F(K)$ . This ensures that the choice of $F(K)$ is consistent across the background, large scales, and small scales.
Vector Potential Source: The authors derive a specific source equation for the vector potential $\vec{\omega}$ (Equation 3.27). This equation allows for a quantitative test of the validity of the Newtonian limit assumption in N-body simulations; if the vector potential is small, the Newtonian approximation holds.
Consistent Expansion of $M^2$ and $F(K)$ : The authors carefully analyze the power counting of the mass scale $M$ and the function $F(K)$ in the $1/c$ expansion. They propose a specific scaling ( $M^2 \to M_*^2 c^2$ ) that ensures the homogeneous and inhomogeneous parts of the equations behave consistently, particularly when $\bar{K} \neq 0$ .

Results and Phenomenology
The application of these equations to GEA theories yields several significant findings regarding the emergence of MONDian behavior:

Decoupling of Scales: MONDian behavior on galactic scales does not necessarily imply MONDian behavior on cosmological scales. The conditions required for MOND to manifest in the cosmological Poisson equation are distinct from those in the galactic weak-field limit.
The $\alpha = 0$ Condition: The authors find that for cosmological MONDian behavior to arise in GEA theories, the model parameter $\alpha$ $α$ (a combination of the theory's constants) must be zero.
- If $\alpha \neq 0$ : The leading-order cosmological equations do not exhibit MONDian behavior, even if the same $F(K)$ is used. The cosmological evolution follows standard Friedmann equations (potentially with a modified Newton's constant), and the MONDian terms appear only at higher orders that do not dominate the dynamics.
- If $\alpha = 0$ : The leading-order terms in the Poisson equation cancel out the standard GR terms, leaving a dominant term that reproduces the deep MOND Poisson equation. In this specific case, cosmological MONDian behavior is possible.
Constraints on $\alpha = 0$ Models: When $\alpha = 0$ $α = 0$ and cosmological MOND arises:
- The Friedmann equations governing the background expansion remain unmodified from their GR form.
- The Newtonian branch of the cosmological Poisson equation (where $|\nabla \Phi| \gg a_0$ ) also reverts to the standard GR form.
- This implies that for these theories to exhibit MOND on cosmological scales, they must essentially look like GR in the linear regime and on large scales, with MONDian effects only appearing in the non-linear, deep-MOND regime.
Relation to Existing Simulations: The paper clarifies the relationship between existing cosmological MOND N-body simulations and relativistic extensions.
- Simulations that simply impose a modified Poisson equation on cosmological scales do not necessarily correspond to a consistent GEA cosmology (specifically for $\alpha \neq 0$ ).
- However, for the specific case of $\alpha = 0$ , existing MOND N-body codes could represent structure formation in a consistent GEA cosmology, provided the initial conditions and background expansion are chosen to match the GR-like constraints derived in the paper.

Significance and Claims
The paper claims to provide the first "complete and coherent" cosmology for a relativistic extension of MOND (specifically GEA) that covers all necessary scales and regimes with a single set of equations.

Methodological Advance: The primary contribution is the method itself, which allows for the derivation of consistent all-scales equations for any relativistic extension of MOND. This enables the running of cosmological N-body simulations that are theoretically consistent with the underlying relativistic theory, rather than relying on ad-hoc modifications.
Theoretical Constraints: The work demonstrates that achieving MONDian behavior on cosmological scales places severe constraints on the theory. Specifically, it suggests that theories capable of cosmological MOND must essentially reduce to GR in the linear regime and for the background expansion. This limits the ability of such theories to replace Cold Dark Matter (CDM) entirely without introducing additional matter species.
Validation: The derived equations include a built-in test (via the vector potential $\vec{\omega}$ ) to verify the validity of the Newtonian limit in simulations, a crucial step for ensuring the reliability of numerical results in modified gravity.

The authors conclude that while their method allows for consistent testing of these theories, the specific results for GEA suggest that cosmological MOND is a highly constrained possibility, and that many relativistic extensions may only exhibit MONDian behavior on galactic scales while behaving like GR on cosmological scales. They note that this framework can be applied to other theories, such as the recently proposed AeST (Aether Scalar Tensor) theory, to determine their cosmological viability.

Technical Summary: Consistent Cosmological Structure Formation on All Scales in Relativistic Extensions of MOND

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