Equivalence of scalar-tensor theories and scale-dependent gravity

This paper establishes a novel, bidirectional equivalence between scale-dependent gravity and specific scalar-tensor theories, demonstrating that the scale-setting relation can be promoted to a dynamical field and that these frameworks can be uniquely embedded into one another at both the action and field equation levels.

The Gist

Imagine the universe as a giant, complex video game. For decades, the "engine" running this game has been General Relativity (GR), a set of rules written by Einstein that explains how gravity works. It's a fantastic engine; it predicts planetary orbits and black holes perfectly. But lately, physicists have noticed some glitches. The game seems to have invisible "dark matter" holding galaxies together and a mysterious "dark energy" pushing the universe apart. The standard engine can't explain these glitches without adding weird, invisible cheat codes.

So, scientists are trying to rewrite the engine code. Two popular new versions have emerged: Scale-Dependent Gravity and Scalar-Tensor Theories.

This paper is like a master translator who just discovered that these two different versions of the engine are actually speaking the same language, just with different accents. Here is the breakdown of what they found, using some everyday analogies.

1. The Two Competing Theories

The Problem: Both theories try to fix the "glitches" in gravity, but they do it differently.

• Scalar-Tensor Theories (STT): Imagine the fabric of space-time is a trampoline. In standard gravity, the trampoline is just rubber. In STT, imagine there is a mysterious invisible fluid (a "scalar field") flowing through the rubber. This fluid changes how heavy or light the trampoline feels depending on where you are. If you are in a dense crowd (like a solar system), the fluid acts up and hides the extra forces, but in empty space, it might change how gravity works.
• Scale-Dependent Gravity (SD): Imagine the rules of the game change depending on the "zoom level." If you look at gravity from far away (cosmic scale), the rules are one way. If you zoom in close (quantum scale), the rules change. In this theory, the "strength" of gravity (the Newton constant) isn't a fixed number; it's a running variable that changes based on a "scale" (let's call it $k$). It's like the game engine saying, "At this zoom level, gravity is 10% stronger."

2. The Big Discovery: They Are Twins

The authors of this paper, Philipp Neckam and his team, proved that these two theories are mathematically equivalent.

Think of it like this:

• Theory A says: "Gravity changes because of a fluid flowing through space."
• Theory B says: "Gravity changes because the zoom level of the universe changes."

The paper shows that the fluid in Theory A is actually just the zoom level in Theory B wearing a disguise. You can translate the equations of one directly into the other.

• From SD to STT: If you have a model where gravity changes with the zoom level ($k$), you can mathematically turn that zoom level into a "fluid" (scalar field). The math works perfectly, and the fluid behaves exactly like the zoom level did.
• From STT to SD: Conversely, if you have a model with a fluid, you can treat that fluid as the "zoom level" of the universe.

The Catch: The translation isn't always one-to-one.

• Going from SD to STT is like translating a book into a different language: there is usually one perfect translation.
• Going from STT to SD is like translating a book into a different language, but you have a choice of dialects. You can pick different ways to define the "zoom level," leading to multiple possible SD versions of the same STT.

3. The "Scale" Becomes a Character

One of the most exciting parts of the paper is a shift in perspective. In traditional Scale-Dependent Gravity, the "zoom level" ($k$) was just a background setting, like the resolution on your TV screen. It wasn't a "thing" that moved or changed dynamically.

But when they translated it into Scalar-Tensor theory, they realized that to make the math work, the zoom level ($k$) has to be a character in the story. It needs its own "energy" and "movement."

• The Analogy: Imagine you are watching a movie. In the old view, the camera zooming in and out was just a camera trick. In this new view, the camera itself is a living actor that moves around, has its own momentum, and interacts with the scene. The paper shows that for the theories to be equivalent, the "zoom" must be treated as a real, dynamic object with its own rules of motion.

4. Why This Matters (The "Triangle" of Gravity)

Physicists already knew that:

1. Scale-Dependent Gravity is related to $f(R)$ Gravity (a theory where gravity depends on the curvature of space in a complex way).
2. Scalar-Tensor Theories are also related to $f(R)$ Gravity.

This paper connects the missing link: Scale-Dependent Gravity $\leftrightarrow$ Scalar-Tensor Theories.

Now, we have a complete "Equivalence Triangle." If you understand one corner of the triangle, you automatically understand the other two. This is huge because:

• Some problems are easier to solve in one theory than the other.
• Experimental data (like from atom interferometers or black hole observations) that constrains one theory can now be used to test the others.
• It suggests that the "fluid" and the "zoom level" are just two different ways of describing the same underlying quantum nature of gravity.

5. The Reality Check: Not All Models Are Created Equal

The authors didn't just do the math; they tested it with real-world examples.

• Good News: When they took popular "screened" theories (like Chameleons and Symmetrons, which hide their effects in dense areas) and translated them into Scale-Dependent Gravity, the results were smooth and healthy.
• Bad News: When they took some existing Scale-Dependent models (specifically those used for black holes and cosmology) and tried to translate them into Scalar-Tensor theories, things got messy. The resulting "fluid" models often had infinite energy or broke the laws of physics.

Why? Because many existing Scale-Dependent models assume the "zoom level" is static and doesn't follow its own rules of motion. But the translation requires the zoom level to be dynamic. If you force a static zoom level into a dynamic fluid, the fluid goes crazy.

The Bottom Line

This paper is a Rosetta Stone for modified gravity. It tells us that the "fluid" theories and the "zoom-level" theories are actually the same thing.

• For the layperson: It's like realizing that "temperature" and "average kinetic energy of molecules" are the same thing. You can describe a hot cup of coffee using either concept, and they will predict the exact same behavior.
• For the future: Now that we know they are equivalent, physicists can use the tools from one side to solve problems on the other. It also suggests that if we want to understand the quantum nature of gravity, we might need to treat the "scale" of the universe not just as a setting, but as a dynamic player in the cosmic game.

Technical Summary

1. Problem Statement

The paper addresses the lack of a comprehensive understanding of the interrelations between different modified theories of gravity. Specifically, while it is well-established that:

• Scalar-Tensor Theories (STTs) can be mapped to $f(R)$-gravity.
• Scale-Dependent (SD) Gravity (inspired by Asymptotic Safety) can be mapped to $f(R)$-gravity (under specific conditions).
• $f(R)$-gravity can be mapped to STTs.

There was no general, rigorous equivalence established directly between Scale-Dependent Gravity and Scalar-Tensor Theories. SD gravity typically treats the scale-setting parameter $k(x)$ as a non-dynamical background field (often with vanishing kinetic terms), whereas STTs treat the scalar field $\phi$ as a dynamical entity. The authors aim to bridge this gap, determining if and how SD gravity can be fully embedded into the framework of STTs, and vice versa, both at the level of actions and field equations.

2. Methodology

The authors employ a systematic comparison of the mathematical structures of the two theories in both the Jordan frame and the Einstein frame.

• Action Comparison: They start with the Jordan frame action for SD gravity (Eq. 7), which includes a running Newton constant $G(k)$, a running cosmological constant $\Lambda(k)$, and a newly introduced kinetic term for the scale parameter $k(x)$ with a prefactor $B(k)$. They compare this to the Jordan frame action of a single-field STT with a conformal coupling $A(\phi)$ and a canonical kinetic term.
• Field Redefinitions: They derive the necessary field redefinitions ($k \leftrightarrow \phi$) and functional mappings between the coupling functions ($G, \Lambda, B$) and the STT potentials ($V, A$) to make the actions identical.
• Field Equations Analysis: They derive the modified Einstein equations and the equations of motion (EOM) for the scalar field $\phi$ and the scale parameter $k$. They verify that the EOMs map onto each other consistently, paying special attention to the divergence of the energy-momentum tensor and the Bianchi identities.
• Consistency Checks: They perform a "triangle" consistency check by mapping $f(R)$-gravity $\to$ SD gravity $\to$ STTs and verifying that the original $f(R)$ theory is recovered.
• Application to Models: They apply the derived equivalence relations to specific physical models:
  • STTs $\to$ SD: Chameleon, Symmetron, and Environment-dependent Dilaton models.
  • SD $\to$ STTs: Einstein-Hilbert cosmology (using Null Energy Condition and Effective Lagrangian Fixed Point conditions) and Polytropic black hole solutions.

3. Key Contributions

A. Establishing the Equivalence

The paper proves that the set of "well-behaved" SD gravity theories can be uniquely embedded into STTs.

• Mapping SD to STT: Given a SD theory with functions $G(k)$, $\Lambda(k)$, and $B(k)$, one can uniquely define the STT conformal factor $A(\phi)$, potential $V(\phi)$, and kinetic structure. The scale parameter $k(x)$ is promoted to a dynamical scalar field $\phi$.
• Mapping STT to SD: Conversely, any STT can be written as an SD theory, but this mapping is not unique. There are multiple ways to define the scale-setting relation $k(\phi)$ and the kinetic prefactor $B(k)$. A convenient choice is $k(x) \equiv \phi(x)$.

B. Dynamical Nature of the Scale Parameter

A crucial theoretical advancement is the treatment of the scale-setting relation $k(x)$ as a dynamical field.

• In traditional SD gravity, $k(x)$ is often fixed by external conditions (e.g., curvature invariants) and lacks a kinetic term ($B(k)=0$).
• The authors show that for a full equivalence with STTs, $k(x)$ must possess a kinetic term. They derive the specific form of this kinetic term ($B(k)$) required to match the STT kinetic structure.
• They demonstrate that the Equation of Motion (EOM) for $k$ is intimately linked to the EOM for $\phi$. If the EOM for $k$ is not imposed in SD gravity, the equivalent STT scalar field will not satisfy its own EOM, rendering the equivalence inconsistent at the level of dynamics.

C. Consistency with $f(R)$-Gravity

The authors complete the "equivalence triangle" (SD $\leftrightarrow$ STT $\leftrightarrow$ $f(R)$). They show that:

• SD gravity with $B(k)=0$ maps to STTs with vanishing kinetic terms ($\kappa^2=0$), which corresponds to $f(R)$-gravity.
• The new equivalence relations are consistent with existing literature, confirming that the framework is robust and unifies these distinct approaches to modified gravity.

D. Einstein Frame Formulation

The paper provides the equivalence relations in the Einstein frame, where the equations of motion are significantly simpler (decoupled from the Ricci scalar). This facilitates practical applications and numerical solutions.

4. Results

• Uniqueness Conditions: For a SD theory to map to a valid STT, the running couplings must satisfy specific smoothness and positivity conditions:
  • $G(k) > 0$ (Gravity remains attractive).
  • $B(k) > -\frac{3}{4} \frac{1}{G(k)^3} \left(\frac{\partial G}{\partial k}\right)^2$ (Ensures the scalar field kinetic term has the correct sign to avoid ghosts).
• Physical Models (STT $\to$ SD): When applying the equivalence to screened scalar models (Chameleon, Symmetron, Dilaton), the resulting SD theories are well-behaved. They possess smooth running couplings and a canonical kinetic term for $k$. However, these resulting SD theories differ from standard literature models because they include the kinetic term for $k$.
• Physical Models (SD $\to$ STT): When applying the equivalence to standard SD models from the literature (which often use the Null Energy Condition or ELFP to fix $k$ without imposing its EOM), the resulting STTs are often ill-behaved.
  • The scalar potential $V(\phi)$ may become unbounded from below or singular.
  • The scalar field $\phi$ may diverge or fail to satisfy its EOM.
  • Conclusion: Many existing SD models are not physically equivalent to consistent STTs unless the EOM for $k$ is strictly imposed.
• Anomaly Cancellation: The paper shows that imposing the EOM for $k$ (or equivalently, the condition derived from the divergence of the energy-momentum tensor) is necessary to ensure the theory is anomaly-free (vanishing divergence of the effective energy-momentum tensor), consistent with the Bianchi identities.

5. Significance

• Unification of Frameworks: This work provides a rigorous mathematical bridge between Asymptotic Safety-inspired gravity (SD) and classical scalar-tensor extensions. It suggests that the "running" of couplings in quantum gravity can be effectively described by classical scalar fields.
• Reinterpretation of SD Gravity: It fundamentally changes the interpretation of SD gravity. Instead of viewing $k(x)$ as a mere bookkeeping parameter for renormalization group flow, it must be treated as a dynamical scalar field with its own kinetic energy and equation of motion to be consistent with standard field theory principles.
• Constraints on Models: The results imply that many existing SD gravity models (which do not impose the EOM for $k$) may not correspond to physically viable scalar-tensor theories. This provides a new consistency criterion for constructing SD models.
• Experimental Implications: Since screened scalar models (Chameleons, Symmetrons) map to specific SD theories with kinetic terms, the extensive experimental constraints on these scalar fields (from atom interferometry, solar system tests, etc.) can now be translated into constraints on the parameters of Scale-Dependent gravity.
• Future Directions: The paper opens avenues for applying STT techniques (e.g., inflationary dynamics, screening mechanisms) to SD gravity and vice versa. It also suggests a potential link to Higgs portal theories, further expanding the web of equivalences in modified gravity.

In summary, the paper establishes that Scale-Dependent gravity and Scalar-Tensor theories are not just related but are equivalent descriptions of the same physics, provided the scale-setting parameter is treated as a dynamical field satisfying its own equation of motion. This unification offers a powerful tool for exploring quantum gravity effects through the lens of classical field theory.