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A Baseline model for Modified Newtonian Mechanics I: The Early Universe

This paper proposes a new single-metric cosmological model that smoothly interpolates between the Schwarzschild and FLRW metrics within a Modified Newtonian Dynamics framework, introducing a density-dependent acceleration scale to explain early massive galaxy formation without invoking Cold Dark Matter.

Original authors: James C. C. Wong, Ray J. Rivers

Published 2026-03-26

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CC BY 4.0

Original authors: James C. C. Wong, Ray J. Rivers

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, expanding balloon. For decades, scientists have been trying to figure out how galaxies form on the surface of this balloon.

The standard story (called ΛCDM) says that to make galaxies stick together, you need a lot of invisible "glue" called Dark Matter. Without this glue, the visible stars and gas shouldn't be able to clump together fast enough to form the massive galaxies we see today.

However, a new telescope (the James Webb Space Telescope) has started taking pictures of the very early universe. It's seeing huge, mature galaxies that formed way too quickly for the standard story to explain. It's like walking into a bakery and finding a fully baked, decorated cake on the counter just one minute after the oven was turned on. The standard recipe says that should take hours.

This paper proposes a new recipe. It suggests we don't need the invisible glue (Dark Matter) to explain these early galaxies. Instead, we need to change the rules of how gravity works when the universe is young and expanding.

Here is the breakdown of their idea using simple analogies:

1. The Problem: Two Different Rulebooks

Currently, physicists use two different "rulebooks" for gravity:

The Local Rulebook (Schwarzschild): This works great for things close to a heavy object, like a planet orbiting a star. It's the gravity we know and love.
The Global Rulebook (FLRW): This works for the whole universe. It describes how space itself is stretching out like the expanding balloon.

The problem is that these two rulebooks don't play nice together. Usually, scientists try to patch them together with a "vacuole" (a sharp boundary where one rule stops and the other starts). The authors say, "That's too clunky. Let's blend them smoothly."

2. The Solution: A Smooth Blend (VMOND)

The authors propose a new "metric" (a mathematical map of space and time) that acts like a smooth gradient rather than a hard wall.

Near a galaxy: Gravity acts like the Local Rulebook.
Far away: Gravity acts like the Global Rulebook.
In between: There is a "sweet spot" where the two mix.

They call this VMOND (Vacuum Modified Newtonian Dynamics). It's a cousin to an older theory called MOND, but with a twist.

3. The Twist: Gravity Gets Stronger in the Past

In the old theories, the "switch" that turns on extra gravity (when things get too slow or far apart) was a fixed setting, like a thermostat set to a specific temperature.

In this new model, the "thermostat" changes depending on how fast the universe is expanding.

Today: The universe is expanding slowly. The extra gravity is weak, so we don't notice it much in our solar system (which is why the Oort cloud, a distant ring of icy objects, behaves normally).
The Early Universe: The universe was expanding much faster. In this new model, the "thermostat" turns up the heat. Gravity becomes super-strong at galactic scales.

The Analogy: Imagine you are trying to build a sandcastle.

Standard Model: You have a weak wind (expansion) blowing your sand away. You need a heavy bucket (Dark Matter) to hold the sand in place so you can build a castle.
This Paper's Model: In the early universe, the wind was actually helping you. The rules of physics changed so that the sand naturally clumped together much faster, even without the heavy bucket.

4. Why This Matters: The "Early Cake" Problem

Because this new gravity is so strong in the early, fast-expanding universe, it allows clumps of gas to collapse into galaxies much faster than the standard model predicts.

The Result: This explains why the James Webb Space Telescope sees massive, mature galaxies at the "dawn of time." They didn't need millions of years to form; the new gravity rules let them bake in a fraction of the time.

5. Does it break anything?

The authors checked their math against things we know well:

Our Solar System: The effect is so tiny here that it doesn't mess up the orbits of planets or the distant Oort cloud. It passes the "local test."
Galaxy Clusters: It fits the data for how big galaxy clusters can get without flying apart.

The Bottom Line

This paper suggests that we might not need to hunt for invisible Dark Matter particles to explain the early universe. Instead, the universe itself might have had a different "personality" when it was young. Gravity was stronger, expansion was faster, and galaxies were able to form quickly and easily, just like the new telescope is showing us.

It's a "baseline" idea—a new starting point for physicists to test. It doesn't claim to solve everything (like the very largest scales of the universe), but it offers a fresh, elegant way to explain the mystery of those ancient, massive galaxies.

1. The Problem

The paper addresses two major crises in modern cosmology and astrophysics:

The "Early Universe" Crisis: Recent data from the James Webb Space Telescope (JWST) reveals the existence of massive, morphologically mature galaxies at extremely high redshifts ( $z \sim 14$ ). The standard $\Lambda$ CDM (Lambda Cold Dark Matter) model predicts that structure formation via hierarchical merging of small halos is too slow to produce such massive structures within the first few hundred million years of the universe.
The Mass Discrepancy Crisis: Neither standard $\Lambda$ CDM (which posits invisible dark matter) nor canonical Modified Newtonian Dynamics (MOND) fully resolves all observational tensions. $\Lambda$ CDM faces issues at small scales (cusp-core problem, missing satellites) and large scales (Hubble tension, $\sigma_8$ tension). Canonical MOND, while successful at galactic scales, fails at larger scales (clusters) and in high-acceleration regimes (like the Solar System's Oort cloud), and lacks a robust relativistic framework compatible with gravitational wave constraints (GW170817).

The authors argue that the current models struggle to reconcile the rapid formation of early structures without invoking new particle physics or ad-hoc modifications that violate other observational constraints.

2. Methodology

The authors propose a new geometric framework called VMOND (Velocity-based Modified Newtonian Dynamics). Instead of adding new matter components to the Stress-Energy Tensor or modifying the laws of gravity phenomenologically, they construct a single metric that interpolates smoothly between two asymptotic regimes of General Relativity (GR):

Small Scales: The Schwarzschild-Lemaître (SL) metric (describing a central mass in a static or locally bound system).
Large Scales: The Friedmann-Lemaître-Robertson-Walker (FLRW) metric (describing an expanding homogeneous universe).

Key Technical Steps:

Metric Construction: They utilize the Lemaître-Tolman (LT) metric class. They define the radial coordinate $r$ as a function of comoving distance $\varrho$ and time $\tau$ that interpolates between the SL solution ( $r \propto (\varrho - c\tau)^{2/3}$ ) and the FLRW solution ( $r = a(\tau)\varrho$ ).
Geodesic Equation: The core innovation is the definition of the particle's free-fall velocity. The authors postulate that the total radial velocity $\dot{r}$ is the algebraic sum of the Hubble flow velocity ($Hr$) and the Newtonian free-fall velocity ( $-\sqrt{2GM/r}$ ):
$\frac{\dot{r}}{r} = H - \sqrt{\frac{2GM}{r^3}}$
This contrasts with standard perturbation theory where accelerations are added, not velocities.
Parameter Selection: They select the interpolation parameter $\eta = 3/2$ . While Baker (2013) rejected this value due to concerns about isotropy, the authors argue that in an adiabatic, pressureless approximation, isotropy is naturally maintained, and this choice is required to recover the correct Newtonian potential.
Stress-Energy Tensor: The model strictly adheres to the standard Stress-Energy Tensor containing only baryonic matter and a cosmological constant. No "dark matter" particles are added.

3. Key Contributions

The VMOND Metric: A new exact solution that bridges the gap between local gravitational collapse and cosmic expansion without discrete boundaries (unlike the Einstein-Straus vacuole model).
Variable Acceleration Scale ( $a_0$ ): Unlike canonical MOND where the acceleration scale $a_0$ is a universal constant, in VMOND, the transition scale depends on the background matter density and the Hubble parameter:
$a_0 \sim \frac{1}{2}H^2 r$
This implies that at high redshifts (where $H(z)$ is large), the "non-Newtonian" regime extends to much larger distances, providing stronger effective gravity.
Effective "Phantom" Density: The interpolation generates an effective non-Newtonian acceleration term that can be mathematically interpreted as an additional density $\rho_{MH} = \sqrt{\rho_M \rho_H}$ . Crucially, this is a purely dynamical gravitational effect, not a physical fluid, and it exerts zero pressure.
Modified Perturbation Theory: The authors derive a new overdensity evolution equation. In the standard Newtonian limit, overdensity $\delta$ grows as $a(t)$ (linear). In VMOND, the presence of the extra potential term leads to a growth rate of:
$\delta \propto a(t)^2$
This represents a significantly faster growth of structure in the early universe.

4. Results

Solar System Consistency: The model predicts a turnaround radius ( $r_{TA}$ ) for the Solar System of $\sim 2.44 \times 10^7$ AU. This is two orders of magnitude larger than the Oort cloud ( $\sim 10^5$ AU), ensuring that the Solar System remains entirely within the Newtonian regime, resolving the conflict canonical MOND has with Oort cloud observations.
Galactic Cluster Turnaround: The model relaxes the theoretical upper bound on the turnaround radius for galactic clusters derived in $\Lambda$ CDM, bringing it into better agreement with observational data.
High-Redshift Structure Formation:
- Using the modified growth rate ( $\delta \propto a^2$ ), the authors calculate the turnaround redshift ( $z_{ta}$ ) for initial density perturbations.
- For a plausible initial overdensity ( $\delta_{int} \approx 2.8 \times 10^{-3}$ ), the model predicts a turnaround redshift of $z_{ta} \approx 29$ .
- The subsequent free-fall time allows these structures to virialize and form massive galaxies by $z \approx 15$ .
- This timeline is consistent with JWST observations of massive galaxies at $z > 14$ , which $\Lambda$ CDM cannot explain without invoking non-standard physics or extreme bias.
Low-Redshift Behavior: At low redshift ( $z \to 0$ ), the Hubble parameter $H$ decreases, causing the non-Newtonian effects to diminish, effectively recovering standard $\Lambda$ CDM behavior on large scales where dark matter is typically invoked for CMB acoustic peaks.

5. Significance

A New Baseline: The paper proposes VMOND not necessarily as a final replacement for all dark matter theories, but as a "baseline metric" that must be accounted for. It suggests that the "missing mass" problem at galactic scales might be a geometric consequence of interpolating between local and global metrics, rather than a need for new particles.
Resolution of the JWST Tension: It offers a mechanism to explain the early appearance of massive galaxies without requiring exotic dark matter properties or violating the speed of light (Lorentz invariance).
Unification of Scales: It provides a continuous framework that respects General Relativity, avoids the artificial boundaries of vacuole models, and naturally transitions from Newtonian dynamics to cosmological expansion.
Implications for Dark Matter: The authors suggest that if this metric is correct, the role of non-baryonic dark matter might be restricted to the very early universe (CMB scales) or large-scale structure, while galactic dynamics are dominated by this modified metric. It decouples the need for dark matter halos from the explanation of rotation curves and early galaxy formation.

In conclusion, Wong and Rivers present a geometric solution that modifies the effective gravitational force in the early universe through a smooth metric interpolation, offering a potential solution to the "too big to fail" and "too early to form" problems currently plaguing the standard cosmological model.