A minimal fractional deformation of Newtonian gravity

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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 gravity not as a rigid, unchangeable law written in stone, but as a piece of elastic fabric. For centuries, we've thought this fabric was perfectly smooth and uniform (that's Newton's view). Then, Einstein came along and showed us the fabric could stretch and warp in complex ways (General Relativity).

But what if the fabric itself has a slightly different texture? What if it has a tiny, almost invisible "friction" or "memory" built into it?

This paper proposes exactly that. The author, S. M. M. Rasouli, suggests a "Fractional Deformation" of gravity. Think of this as adding a single, tiny knob (called $\alpha$ ) to the universe's gravity settings.

If you turn the knob to 1: The universe behaves exactly like Newton described it. No surprises.
If you turn the knob just a tiny bit away from 1 (like 1.000002): The universe starts behaving in a slightly new way, but in a way that actually solves some of the biggest headaches in modern physics.

Here is the breakdown of what this paper does, using simple analogies:

1. The Big Promise: One Theory to Rule Them All

Usually, physicists have to use different rulebooks for different sizes:

Small scale (Solar System): We use Newton or Einstein to explain why Mercury orbits the way it does.
Huge scale (The Universe): We use "Dark Energy" and "Dark Matter" to explain why the universe is expanding and accelerating.

This paper argues: "Wait, we don't need two different rulebooks."
By just tweaking that one tiny knob ( $\alpha$ ), the same math that explains a planet orbiting a star also explains the entire history of the universe—from the Big Bang to today's expansion. It's like finding a single key that opens both your front door and your car.

2. The "Cosmic Friction" (The Fractional Part)

The paper introduces a concept called a "fractional kernel." Imagine you are running through a pool.

Newtonian Gravity: You are running on dry land. You move smoothly, and your path is predictable.
Fractional Gravity: You are running through water. There is a tiny bit of drag or "friction" that depends on how long you've been running.

This "friction" isn't strong enough to stop you, but it subtly changes your path over time. In the universe, this subtle change is enough to:

Create an early period of rapid expansion (Inflation) without needing exotic new particles.
Cause the universe to speed up today (Dark Energy) without needing a mysterious "Cosmological Constant."

3. Testing the Theory: The Solar System Check

A new theory is useless if it breaks the things we already know work. The author tested this "Fractional Gravity" against two famous experiments:

The Mercury Test (The Wobbly Orbit): Mercury's orbit wobbles slightly more than Newton predicted. Einstein explained this with curved space. This paper shows that the "Fractional Friction" (the $\alpha$ knob) creates the exact same wobble.
The Light Test (The Bending Beam): When light from a distant star passes near the Sun, it bends. Einstein predicted exactly how much. This paper shows that the Fractional model predicts the same amount of bending.

The Catch: For this to work, the knob $\alpha$ must be set extremely close to 1. The difference is so small it's like the difference between a grain of sand and a mountain. But that tiny difference is enough to change the physics.

4. Solving the "Cosmic Mysteries"

The paper suggests this simple tweak solves three massive problems that have baffled scientists for decades:

The "Why is the Universe accelerating?" Problem: Instead of inventing a mysterious "Dark Energy" that pushes the universe apart, the fractional "friction" naturally causes the expansion to speed up over time. It's like a car that slowly accelerates on its own because of how the engine is built, not because someone is pressing the gas.
The "Huge Numbers" Problem (Hierarchy): Why is the expansion rate during the Big Bang ( $10^{37}$ ) so much bigger than it is today ( $10^{-18}$ )? It's like asking why a whisper can turn into a scream. In this model, the tiny knob $\alpha$ changes its function over time, naturally creating this massive difference without needing to fine-tune the universe.
The "Hubble Tension" Problem: Scientists are arguing about the exact speed of the universe's expansion. Some measurements say it's fast; others say it's slow. This paper suggests that because the "friction" changes slightly over time, the expansion rate might actually be different depending on when you measure it. This tiny variation could explain why scientists are getting different answers.

The Bottom Line

This paper is a bold suggestion: Maybe gravity isn't as simple as Newton thought, but it doesn't need to be as complicated as Einstein's full theory requires for the whole universe.

By adding a tiny, mathematical "texture" to gravity (the fractional parameter), we might be able to explain the motion of planets, the bending of light, and the birth and death of the universe all with one single, elegant equation. It's a "minimalist" approach that tries to fix the universe's biggest puzzles by turning a single, tiny dial.

1. Problem Statement

Modern theoretical physics faces significant challenges in unifying gravitational dynamics across vastly different scales. While General Relativity (GR) accurately describes Solar System tests and large-scale cosmology, explaining the full cosmic evolution (inflation, dark energy, accelerated expansion) typically requires introducing exotic components like dark energy, scalar fields, or a cosmological constant ( $\Lambda$ ). Furthermore, fundamental puzzles remain unresolved, including:

The Cosmological Constant Problem (the discrepancy between the observed vacuum energy and theoretical predictions).
The Hierarchy Problem between the inflationary Hubble scale and the present Hubble scale.
The Hubble Tension (the discrepancy between early-universe and late-universe measurements of the Hubble constant).

The paper asks whether these phenomena can arise not from new degrees of freedom, but from a minimal modification of the effective structure of classical Newtonian gravity itself.

2. Methodology

The author employs a Fractional Newtonian Framework (FNF) characterized by a single deformation parameter, $\alpha$ .

Fractional Action: The theory is built on a fractional extension of the Newtonian gravitational action:
$S_\alpha = \frac{1}{\Gamma(\alpha)} \int_{t'}^{\bar{t}} \xi(\bar{t}) L(r, \dot{r}; \alpha) d\bar{t}$
where $\xi(\bar{t}) = \left(\frac{\bar{t}-t'}{t_*}\right)^{\alpha-1}$ is a time-dependent kernel. In the limit $\alpha \to 1$ , the theory recovers standard Newtonian mechanics.
Unified Potential: The model utilizes a single effective potential, $V_{\text{eff}}(r; \alpha)$ , to describe both planetary motion and photon trajectories. This potential is decomposed into a Newtonian term and fractional corrections:
$V_{\text{eff}}(r; \alpha) = -\frac{m\mu}{r} - \frac{m\mu A(\alpha)}{r^3} - \frac{m\mu B(\alpha)}{r} e^{-r/\lambda}$
Here, the $r^{-3}$ term represents a power-law correction, and the Yukawa term ( $e^{-r/\lambda}/r$ ) represents finite-range modifications.
Weak-Field Regime: The analysis assumes a weak deformation regime where $|\alpha - 1| \equiv \epsilon \ll 1$ .
Key Tests: The paper applies this framework to two classical weak-field tests:
1. Perihelion Precession of Mercury (PPM): Derived via a fractional generalization of Binet's equation.
2. Gravitational Deflection of Light (GDL): Derived using Fermat's principle with the spacetime refractive index derived from the effective potential.

3. Key Contributions

Unified Description: The paper demonstrates that a single parameter $\alpha$ $α$ and a single unified potential can simultaneously describe:
- Cosmological evolution (from pre-inflation to the current accelerated expansion).
- Solar System dynamics (planetary orbits and light deflection).
Derivation of Constraints: It derives specific observational bounds on the deformation parameter $\alpha$ from Solar System data, showing consistency with previous cosmological constraints.
Resolution of Cosmological Puzzles: It proposes a mechanism within this minimal framework to address the hierarchy of scales and the Hubble tension without introducing a fundamental cosmological constant.

4. Key Results

A. Perihelion Precession of Mercury (PPM)

The fractional Binet equation yields an orbital solution with a shifted angular period. The perihelion shift per orbit is:
$\Delta \phi_{\text{frac}} \approx 6\pi |\alpha - 1| \ell^2 \left( \frac{GM}{h^2} \right)^2$
where $\ell$ is a characteristic macroscopic length scale (chosen as the geometric mean of the orbital radius and the solar radius, $\ell = \sqrt{a R_\odot}$ ).
Constraint: Comparing the fractional prediction to General Relativity and observational data yields:
$|\alpha - 1| \simeq \frac{h^2}{\ell^2} \simeq \frac{GM_\odot}{R_\odot} \approx 2 \times 10^{-6}$
This confirms that the deformation is extremely small ( $\epsilon \ll 1$ ), consistent with the model's assumptions.

B. Gravitational Deflection of Light (GDL)

The $r^{-3}$ correction contributes negligibly to light deflection (suppressed by $\sim 10^{-3}$ relative to the Newtonian term).
The Yukawa term becomes relevant for light grazing the solar limb ( $b \approx R_\odot$ ) if the interaction scale $\lambda \sim R_\odot$ .
Constraint: To match the GR prediction for light deflection ( $\delta_{GR} = 4GM/b$ ), the coefficient $B(\alpha)$ must be of order unity. Using a minimal parametrization $B(\alpha) \propto (\alpha-1)$ , the analysis again yields:
$|\alpha - 1| \simeq 2 \times 10^{-6}$
This result independently confirms the constraint derived from the Mercury perihelion shift, validating the unified potential approach.

C. Cosmological Implications

Cosmological Constant: The observed accelerated expansion is interpreted as an effective cosmological constant $\Lambda_{\text{eff}} \sim 3\kappa^2(\alpha)$ , which dynamically emerges from the fractional structure. The smallness of $\Lambda$ is naturally linked to the small deviation $|\alpha - 1| \ll 1$ .
Hierarchy of Scales: The ratio between the inflationary Hubble scale ( $H_{\text{inf}}$ ) and the current Hubble scale ( $H_0$ ) is determined by different functional dependencies on $\alpha$ . Small variations in $\alpha$ can naturally generate the massive hierarchy ( $H_{\text{inf}}/H_0 \sim 10^{55}$ ) without fine-tuning.
Hubble Tension: The model predicts a residual time-dependent correction to the Hubble parameter, $\Delta H \equiv (\alpha - 1)/(2t)$ . For $|\alpha - 1| \approx 0.15$ , this correction is $\sim 5 \text{ km s}^{-1} \text{ Mpc}^{-1}$ , which is of the same order as the current observational discrepancy between early and late-time Hubble constant measurements.

5. Significance

This paper provides a compelling argument for a minimalist approach to modified gravity. By deforming Newtonian gravity with a single fractional parameter $\alpha$ , the framework achieves a unified phenomenological description that:

Recovers Standard Physics: It reduces exactly to Newtonian gravity and GR limits when $\alpha \to 1$ .
Passes Local Tests: It is consistent with high-precision Solar System tests (Mercury's orbit and light deflection) under the strict bound $|\alpha - 1| \approx 2 \times 10^{-6}$ .
Explains Cosmic Evolution: It reproduces the $\Lambda$ CDM history (inflation, radiation/matter eras, acceleration) without exotic matter.
Offers New Perspectives: It suggests that the cosmological constant, the hierarchy of scales, and the Hubble tension may be artifacts of the fractional nature of spacetime dynamics rather than requiring new fundamental fields or constants.

The work suggests that the "dark sector" of cosmology might be an emergent phenomenon arising from the fractional structure of gravity itself.