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Linear perturbation theory and structure formation in a Brans-Dicke theory of gravity without dark matter

This paper investigates a Brans-Dicke gravity model without dark matter that successfully explains galactic dynamics and cosmic acceleration but fails to account for early structure formation due to a suppressed gravitational field at high redshifts and predicts a discrepancy between the gravitational effects on massive particles and photons.

Original authors: Lorenzo Gervani, Antonaldo Diaferio, Francesco Pace, Andrea Pierfrancesco Sanna

Published 2026-02-02

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CC BY 4.0

Original authors: Lorenzo Gervani, Antonaldo Diaferio, Francesco Pace, Andrea Pierfrancesco Sanna

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

The Big Idea: A Universe Without "Invisible Stuff"

Imagine you are trying to understand why a spinning merry-go-round doesn't fly apart. In our standard understanding of the universe (the $\Lambda$ CDM model), we say there must be a huge amount of invisible, heavy "glue" called Dark Matter holding it together, plus a mysterious "push" called Dark Energy making the universe expand faster.

This paper asks a different question: What if there is no invisible glue at all?

The authors propose a theory where the universe contains only the stuff we can see (stars, gas, planets—called "baryonic matter"). To explain why galaxies spin fast and why the universe is expanding, they suggest that gravity itself works differently than Einstein described. They use a specific version of "Modified Gravity" called Brans–Dicke theory.

The Magic Ingredient: The "Gravity Lens"

In this theory, there is a special, invisible field (a scalar field, let's call it $\phi$ ) that fills the entire universe. Think of this field like a variable-density glass or a lens that sits between matter and gravity.

The "Refracted Gravity" Effect:
In normal physics, gravity is like light traveling through a vacuum. In this theory, gravity travels through this "glass" field.
- In empty space: The glass is thin, and gravity acts normally.
- Inside a galaxy: The density of stars is high. The theory says the "glass" gets thicker or changes its properties here. This change acts like a gravitational boost. It makes gravity stronger than it "should" be for the amount of visible matter present.
- The Result: This extra boost explains why stars on the edge of galaxies spin so fast without needing invisible Dark Matter to hold them. It's like the gravity is being "refracted" or amplified by the field, similar to how a lens focuses light.
The "Cosmic Engine":
This same field ( $\phi$ ) also acts as the engine for the universe's expansion. Instead of needing "Dark Energy" to push the universe apart, the field naturally drives the universe to expand faster over time. So, one single invisible field replaces both Dark Matter and Dark Energy.

The Experiment: Testing the Theory

The authors built a mathematical model to see if this idea holds up when we look at the history of the universe. They ran two main tests:

Test 1: The Background Story (The Hubble Parameter)

They asked: "If we run the universe forward in time with this new gravity, does it look like the real universe we observe?"

The Setup: They used data from "cosmic chronometers" (essentially measuring the age of galaxies at different distances) to see how fast the universe is expanding at different times.
The Finding: They found that the theory can match the observed expansion rate of the universe, but only if the "background value" of their invisible field is quite large.
The Catch: This creates a puzzle. For the theory to work on small scales (like inside a galaxy) and explain the missing mass, the field needs to be small. But to make the universe expand correctly, the field needs to be large. It's like trying to tune a radio to hear a faint station (galaxy dynamics) while simultaneously turning the volume up so loud it drowns out everything else (cosmic expansion).

Test 2: The Growth of Structure (The "Baby" Universe)

This is the most critical part of the paper. They asked: "If we start with a tiny, smooth universe and let gravity pull matter together, do galaxies form at the right time?"

The Problem: In their model, the "boost" to gravity is controlled by a factor that changes over time. The authors found that in the early universe (when galaxies were just starting to form), this boost was suppressed.
The Analogy: Imagine trying to build a sandcastle. In the real universe, the waves (gravity) are strong enough to pile up the sand quickly. In this theory, the waves are too weak at the beginning. The sand just sits there flat.
The Consequence: Because the gravitational boost is too weak early on, structures (galaxies) don't form until much later in the universe's history (after redshift $z < 1$ ).
The Conflict: We know from telescopes that fully formed galaxies existed very early in the universe (at high redshifts). This theory predicts they wouldn't exist yet. Therefore, the theory fails to explain how the universe looks today.

A Quirky Side Effect: The "Double Boost"

The paper also notes a strange quirk in how this field interacts with light versus matter.

Matter (Stars/Planets): The field gives them a "double boost" of gravity.
Light (Photons): The field only gives them a "single boost."
Why it matters: This difference could change how we see gravitational lensing (where massive objects bend light). If we look at a galaxy cluster, the way it bends light might look different than the way it pulls on stars. This is a specific prediction that future telescopes could check to prove or disprove the theory.

The Conclusion

The authors conclude that while their theory is elegant—replacing two mysterious invisible components (Dark Matter and Dark Energy) with one single field—it doesn't work with the current observations of how galaxies formed.

The theory predicts that the universe was "too quiet" in its early days, delaying the formation of galaxies until it was too late to match what we actually see. The authors suggest that maybe if they tweaked the mathematical rules of the field (changing the shape of the "glass"), they could fix this, but with the current rules, the theory is in trouble.

In short: It's a clever idea that gravity might be "refracted" to explain the universe without dark matter, but the math says it would have taken too long for galaxies to form, which contradicts what we see in the sky.

Technical Summary: Linear Perturbation Theory and Structure Formation in a Brans–Dicke Theory without Dark Matter

Problem Statement
The standard $\Lambda$ CDM cosmological model, while successful on large scales, faces persistent tensions regarding the Hubble constant ( $H_0$ ), the lithium abundance problem, the nature of dark energy, and small-scale galactic anomalies (e.g., core-cusp, dwarf galaxy problems). Furthermore, the direct detection of Weakly Interactive Massive Particles (WIMPs) remains elusive. Modified gravity theories offer an alternative by replacing dark matter and dark energy with modifications to the gravitational sector. Specifically, this paper investigates a Brans–Dicke scalar-tensor theory of gravity where a single scalar field $\phi$ accounts for both the accelerated expansion of the universe and the dynamics of galaxies without invoking dark matter. The theory is constructed to reduce to "Refracted Gravity" (RG) in the weak-field limit, a non-relativistic theory that reproduces galactic dynamics via a gravitational permittivity function. The central problem addressed is whether this specific Brans–Dicke model, defined by $W(\phi) = -1$ and $V(\phi) = -\Xi\phi$ , can successfully describe the formation of large-scale cosmic structure via linear perturbation theory, consistent with observational data.

Methodology
The authors develop a comprehensive linear perturbation theory for the specified Brans–Dicke model within a flat, matter-dominated, homogeneous, and isotropic universe.

Background Evolution: The authors derive modified Friedmann equations and the Klein-Gordon equation for the scalar field. They obtain an analytic solution for a radiation-free, matter-dominated universe and validate it against numerical solutions that include radiation. They constrain the model parameters ( $\phi_0$ and $\Omega_{\Xi0}$ ) by fitting the predicted Hubble parameter evolution $H(z)$ against cosmic chronometer data.
Perturbation Analysis: The study applies linear perturbation theory to the perturbed Einstein field equations, the Klein-Gordon equation, and the continuity/Euler equations in the Newtonian gauge. The analysis separates scalar and tensor perturbations.
Key Derivations:
- Sound Speed: The authors calculate the sound speed of the scalar field ( $c_{s,\delta\phi}$ ) to determine if the field clusters.
- Growth Factor: They derive the differential equation governing the growth factor of matter density perturbations.
- Gravitational Potentials: The Laplacians of the two gravitational potentials ( $\Psi$ and $\Phi$ ) are computed to assess the theory's predictions for gravitational lensing.
- Quasi-Static Approximation: A quasi-static limit is used to further isolate the role of the specific degrees of freedom in the evolution of density perturbations.

Key Contributions and Results

Background Consistency: The model successfully reproduces the observed redshift evolution of the Hubble-Lemaître parameter $H(z)$ . Numerical fitting indicates that values of the present-day scalar field $\phi_0 > 1$ are favored, with the parameter space saturating the upper bound derived from the condition that $H(a)$ remains real. However, this creates a tension with the weak-field limit: if $\phi_0 > 1$ and $\phi$ increases with density, the scalar field would be too large on galactic scales to provide the necessary "gravitational boost" (via $\phi = 2\epsilon$ ) to explain missing mass without dark matter.
Scalar Field Clustering: The analysis confirms that the sound speed of the scalar field is luminal ( $c_{s,\delta\phi} = 1$ ). Consequently, the scalar field does not cluster; it propagates perturbations at the speed of light, suppressing any clumping. This distinguishes the model from Unified Dark Matter (UDM) models where the scalar field itself forms the structure. In this theory, structure formation is driven by the gravitational boost from the background scalar field rather than the clustering of the field itself.
Structure Formation Delay: The most significant negative result concerns the growth of structure. The specific form of the potential $V(\phi) = -\Xi\phi$ leads to a growth factor equation where the source term coefficient is proportional to $H^{-1}(z)$ . This causes the gravitational field to be strongly suppressed at early times (high redshift). As a result, the model predicts that significant structure formation is delayed until redshift $z < 1$ .
Observational Disagreement: This delay in structure formation is in direct disagreement with observations of formed galaxies at much higher redshifts ( $z \gg 1$ ).
Gravitational Lensing: The specific choices of $W(\phi)$ and the linear $V(\phi)$ imply that the scalar field generates twice the gravitational boost on massive particles compared to photons. This asymmetry suggests potentially observable consequences for gravitational lensing phenomena, distinct from General Relativity predictions.

Significance and Claims
The paper concludes that while the proposed Brans–Dicke theory successfully unifies the description of cosmic acceleration and galactic dynamics in the background and weak-field limits, it fails to account for the observed timeline of large-scale structure formation. The specific functional forms chosen for the kinetic term ( $W(\phi) = -1$ ) and the potential ( $V(\phi) = -\Xi\phi$ ), which are necessary to recover Refracted Gravity in the weak-field limit, lead to a suppression of gravitational growth at early times that contradicts observational evidence.

The authors modestly state that it remains to be investigated whether alternative choices for $W(\phi)$ and $V(\phi)$ —which could still reduce to Refracted Gravity in the weak-field limit—might alleviate these problems. The study serves as a critical test of the viability of this specific class of scalar-tensor theories, highlighting a tension between recovering galactic dynamics and reproducing the correct history of cosmic structure growth.