Bimetric MOND as a framework for variable-$G$ theories -- local systems and cosmology

This paper explores extending Bimetric MOND (BIMOND) to create variable-$G$ theories that naturally satisfy local constraints on the constancy of Newton's constant while allowing for a different, enhanced effective gravitational constant ($G_e$) in cosmological settings to potentially explain dark matter effects without new degrees of freedom.

The Gist

Here is an explanation of Mordehai Milgrom's paper, "Bimetric MOND as a framework for variable-G theories," translated into everyday language with creative analogies.

The Big Picture: A Cosmic "Volume Knob"

Imagine the universe is a giant concert hall. For decades, physicists have been trying to explain why the music (the movement of stars and galaxies) sounds different than the sheet music (Newton's laws and Einstein's General Relativity) predicts.

The standard solution (the "Dark Matter" theory) says: "The sheet music is right, but there's an invisible choir of ghosts (Dark Matter) singing along that we can't see."

Milgrom's paper proposes a different idea: The sheet music itself changes depending on how loud the music is.

He suggests that the "volume knob" of gravity (a value called $G$) isn't actually a fixed constant. Instead, it's a variable volume knob that changes based on the situation. This theory is built on a framework called BIMOND (Bimetric MOND), which is like having two sets of sheet music playing at the same time.

The Core Concept: Two Worlds, One Rulebook

To understand this, imagine the universe has two layers:

1. Our World: Where we live, made of normal matter (stars, planets, us).
2. The Twin World: A parallel layer that might have its own "twin matter," but we can't see it directly.

In this theory, gravity isn't just one thing; it's an interaction between these two layers. The "distance" or "difference" between these two layers determines how strong gravity feels.

• The Analogy: Think of two dancers (the two layers of the universe).
  • When they move in perfect sync (high acceleration, like near the Sun), they are so close together that they act like a single, standard dancer. Gravity behaves exactly as Einstein predicted.
  • When they drift apart (low acceleration, like at the edge of a galaxy), the tension between them changes the rules. Gravity gets a "boost" without needing invisible ghosts.

The Problem: The "Volume Knob" Can't Be Broken

There is a major catch. If gravity's strength ($G$) changes, we should see weird things happening everywhere.

• If $G$ changed, the Earth might drift away from the Sun.
• Stars might burn out at the wrong speed.
• Pulsars (cosmic lighthouses) might tick at the wrong rhythm.

Scientists have tested this with extreme precision, and so far, $G$ seems perfectly constant in our solar system and in high-speed, high-gravity environments.

Milgrom's Solution:
He proposes a "Smart Volume Knob" (Variable-$G$ theory). This knob is designed to be context-aware:

1. In High-Pressure Zones (Solar System, Black Holes): The knob locks itself to the standard setting. It says, "Everything is fine, gravity is normal." This satisfies all the strict rules we have for our solar system.
2. In Low-Pressure Zones (The Edge of Galaxies, The Early Universe): The knob slowly turns up the volume. It says, "Here, gravity needs to be stronger to explain what we see."

The "Cosmic Coincidence"

Milgrom points out a funny coincidence. The acceleration where gravity starts acting weird (in galaxies) is almost exactly the same as the acceleration caused by the expansion of the universe.

• Analogy: It's like finding out that the speed limit on your local street is exactly the same number as the speed of light. It suggests that local physics and the big picture of the universe are connected.

His theory tries to use this connection to explain Dark Energy (the force pushing the universe apart) and Dark Matter (the invisible glue holding galaxies together) without needing new particles. Instead, the "glue" is just gravity behaving differently when things are moving slowly.

How It Works in Practice

Milgrom introduces a mathematical "switch" (called $M_G$) that controls this volume knob.

• In the Solar System: The switch is off. Gravity is standard. No weirdness.
• In the Early Universe (Big Bang): The switch is off. This is crucial because if gravity were different back then, the formation of elements (like Helium) would have been totally different, and we wouldn't be here.
• In the Modern Universe (Matter-Dominated Era): The switch slowly turns on. As the universe expands and things get "slower" and more spread out, the effective gravity ($G_e$) becomes stronger (about $2\pi$ times stronger).

Why is this cool?
In the standard model, to explain why the universe expands the way it does, we need a huge amount of invisible "Dark Matter."
In Milgrom's model, we don't need that invisible stuff. We just need gravity to get a little stronger naturally as the universe evolves. It's like realizing you don't need a heavier backpack to walk up a hill; you just need to realize the hill is steeper than you thought.

The "Twin" Twist

The theory uses two "metrics" (mathematical maps of space and time).

• Imagine you are walking on a trampoline (our space).
• There is a second trampoline right underneath it (the twin space).
• If the trampolines are perfectly aligned, you walk normally.
• If they wiggle differently, the interaction between them creates a "phantom" force that looks like extra gravity.

Milgrom suggests that in the very beginning, the two trampolines were perfectly aligned (symmetric), so gravity was normal. But as the universe grew, random wiggles (fluctuations) made them drift apart, turning on the "variable gravity" effect.

The Bottom Line

This paper doesn't claim to have solved everything. Milgrom admits it's a "framework" or a "playground" for ideas, not a finished product. He hasn't built a perfect model that fits every single observation yet.

However, the main takeaway is powerful:
It shows that we can build a theory where gravity changes its strength only in the places where we need it to (the deep cosmos) while staying perfectly normal in the places where we've tested it (our solar system). It offers a way to explain the universe's expansion and galaxy behavior without needing to invent a new, invisible substance, but rather by tweaking the rules of gravity itself.

In short: Gravity isn't a rigid law written in stone; it's a flexible rule that adapts to the environment, and this paper shows us how to write the instructions for that flexibility.

Technical Summary

Here is a detailed technical summary of the paper "Bimetric MOND as a framework for variable-G theories: Local systems and cosmology" by Mordehai Milgrom.

1. Problem Statement

The paper addresses two major challenges in modern cosmology and gravitational physics:

1. The Nature of Dark Matter and Dark Energy: Standard $\Lambda$CDM cosmology relies on non-baryonic dark matter and dark energy to explain galactic rotation curves and the expansion history of the universe. Modified Newtonian Dynamics (MOND) successfully explains galactic dynamics without dark matter but requires a relativistic extension to address cosmology.
2. The Constraints on Variable-$G$ Theories: Variable-$G$ theories (VGTs) propose that Newton's gravitational constant, $G$, is not constant but depends on spacetime or dynamical fields. However, such theories face stringent constraints from high-acceleration, subcosmological systems (e.g., the Solar System, pulsars, stellar evolution, gravitational wave emission), which require $G$ to be effectively constant.
3. The Coincidence Problem: The MOND acceleration constant $a_0$ is numerically close to cosmological scales ($cH_0$ and $c\sqrt{\Lambda/3}$). This suggests a deep link between local galactic dynamics and cosmology, yet standard theories struggle to unify them without fine-tuning.

The central question is: Can a relativistic MOND theory be constructed that allows for a variable effective gravitational constant ($G_e$) in cosmology (to mimic dark matter effects) while strictly recovering standard General Relativity (GR) with a constant $G$ in all high-acceleration, subcosmological regimes?

2. Methodology

Milgrom utilizes Bimetric MOND (BIMOND) as the foundational framework. BIMOND is a relativistic theory involving two metrics:

• $g_{\mu\nu}$: The metric to which standard matter couples minimally.
• $\hat{g}_{\mu\nu}$: A "twin" metric, potentially coupled to "twin matter."

The interaction between these metrics is governed by a scalar function $\tilde{M}$ constructed from the "relative-acceleration tensor" $C^\alpha_{\beta\gamma} = \Gamma^\alpha_{\beta\gamma} - \hat{\Gamma}^\alpha_{\beta\gamma}$ (the difference of the Levi-Civita connections).

The Proposed Extension (VGMOND):
Milgrom generalizes BIMOND into a class of Variable-G MOND (VGMOND) theories. The core modification involves the gravitational action:
$$I_B^G = -\frac{1}{16\pi G} \int d^4x \, (1 + M_G) \left[ |g|^{1/2}R + |\hat{g}|^{1/2}\hat{R} + (|g|^{1/2} + |\hat{g}|^{1/2})\ell_M^{-2}\tilde{M} \right]$$
Where:

• $M_G$ is a dimensionless function of the same scalar variables used in $\tilde{M}$.
• The effective gravitational constant is defined as $G_e \equiv G / (1 + M_G)$.
• The theory is analyzed using two formulations:
  1. Metric Formulation: Connections are derived from the metrics (Levi-Civita).
  2. Einstein-Palatini Formulation: Connections are treated as independent degrees of freedom.

The methodology involves analyzing the behavior of $M_G$ in different limits (high vs. low acceleration, non-relativistic vs. relativistic) to ensure compliance with observational constraints.

3. Key Contributions

A. Phenomenon-Dependent $G$

The paper proposes a mechanism where the variability of $G$ is phenomenon-dependent rather than strictly spacetime-dependent.

• High-Acceleration Regime: In systems with accelerations $a \gg a_0$ (Solar System, pulsars, black holes), $M_G \to 0$, ensuring $G_e \approx G$. This automatically satisfies all tight observational constraints on the inconstancy of $G$ derived from these systems.
• Cosmological Regime: In the low-acceleration, large-scale universe, $M_G$ can become non-zero (and potentially negative), leading to $G_e \neq G$. This allows the theory to mimic the effects of dark matter on the expansion history without invoking actual dark matter particles.

B. Resolution of the "Coincidence" via Metric Symmetry

Milgrom leverages the property that in BIMOND, if the two metrics are identical ($g_{\mu\nu} = \hat{g}_{\mu\nu}$), the relative-acceleration scalars vanish.

• Early Universe: The theory posits that the universe began with exact metric symmetry (e.g., during inflation or the Big Bang). In this state, scalars vanish, $M_G = 0$, and $G_e = G$. This satisfies Big Bang Nucleosynthesis (BBN) constraints, which require $G$ to be standard at early times.
• Late Universe: As random fluctuations break the symmetry between the two sectors, the scalars become non-zero, $M_G$ evolves, and $G_e$ deviates from $G$, potentially driving the accelerated expansion or mimicking dark matter density ($\rho_{eff} \approx 2\pi \rho_b$).

C. Scalar Variable Classification and Lensing Constraints

The paper rigorously classifies the scalar variables used in the interaction terms:

• Type 1 Scalars: Contain terms like $(\nabla \phi^*)^2$. These dominate in high-acceleration systems.
• Type 2 Scalars: Contain mixed terms involving metric perturbations $h^*_{ij}$ but no $(\nabla \phi^*)^2$ terms. These vanish in the non-relativistic (NR) limit for standard galactic systems but can be significant in cosmology due to time-dependence.
• Lensing Constraint: The paper refines previous constraints, showing that "correct" gravitational lensing (where light and matter follow the same potential) requires specific combinations of these scalars. It demonstrates that Type 2 scalars can be used in $M_G$ without violating lensing constraints in local systems because they vanish in the NR limit.

D. Avoidance of Ostrogradsky Instabilities

The modified action introduces higher-order derivatives (third derivatives in the field equations). While this typically leads to Ostrogradsky instabilities (ghosts), Milgrom argues that because the higher-derivative terms appear linearly in the Lagrangian, the theory is degenerate. Furthermore, the proposed non-analytic behavior of $M_G$ (e.g., exponential suppression) near the GR limit may evade standard ghost theorems that rely on analytic expansions.

4. Results and Examples

• Construction of Viable $M_G$: Milgrom constructs a specific functional form for $M_G$ using Type 1 ($Y$) and Type 2 ($X$) scalars:
  $$M_G = -D \zeta(X) \lambda(Y)$$
  Where $\zeta(X) \to 0$ as $X \to 0$ (low acceleration/cosmology) and $\lambda(Y) \to 0$ as $Y \to \infty$ (high acceleration).

  • Result: This ensures $M_G \approx 0$ for all high-acceleration systems (satisfying Solar System/Pulsar constraints) and for static low-acceleration galactic systems (preserving standard BIMOND success).
  • Cosmological Effect: $M_G$ becomes non-zero only in the cosmological context where time-dependent fluctuations break metric symmetry, allowing $G_e$ to vary and mimic dark matter effects ($G_e \approx 2\pi G$).

• Cosmological Implications:

  • The theory predicts that $G_e$ was standard during BBN ($M_G \ll 1$) but evolves to $G_e > G$ (or $G_e < G$ depending on the sign of $M_G$) during the matter-dominated era.
  • This variation can account for the observed expansion history without dark matter, effectively replacing the need for $\rho_{DM} \approx 5\rho_b$ with a dynamical modification of gravity.

• Twin Matter: The paper clarifies that "twin matter" in BIMOND does not act as dark matter in local galaxies. It is a separate sector that interacts only gravitationally. The "dark matter" effect in cosmology arises from the interaction between the metrics (the $M_G$ term), not from the presence of twin matter itself.

5. Significance

1. Unification of Local and Cosmological Physics: The framework provides a theoretical mechanism to reconcile the success of MOND in galactic dynamics with the need for "dark" components in cosmology, all within a single relativistic theory.
2. Natural Variable-$G$ Mechanism: It offers a solution to the "Variable-$G" problem: how to have a varying$G$in cosmology without violating local constraints. The separation is achieved naturally via the acceleration scales ($a_0$) and the specific behavior of the scalar fields.
3. Alternative to Dark Matter: It demonstrates that the expansion history of the universe could be driven by a variable effective gravitational constant rather than a mysterious dark matter fluid, potentially resolving the coincidence problem ($\rho_{DM} \approx 5\rho_b$) by relating the two via a dynamical ratio.
4. Heuristic Framework: While the paper admits that a fully consistent model matching all cosmological data (CMB, structure formation) is not yet presented, it establishes a robust framework (VGMOND) for exploring these possibilities. It highlights the importance of non-analytic behaviors and degenerate higher-derivative theories in avoiding instabilities.

Conclusion:
Milgrom successfully demonstrates that BIMOND can be extended to a class of Variable-$G$ theories (VGMOND) that are consistent with all local high-acceleration constraints while offering a dynamical explanation for cosmological phenomena usually attributed to dark matter. The key innovation is the use of metric-symmetry breaking and specific scalar variable dependencies to isolate cosmological $G$-variability from local physics.