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Can nonlocal gravity really explain dark energy?

This paper demonstrates that nonlocal modifications to General Relativity, specifically an Infinite Derivative Gravity model of the Ricci scalar without a cosmological constant or scalar fields, can dynamically explain the Universe's accelerated expansion and offers a technique to derive verifiable predictions such as a decreasing effective gravitational constant.

Original authors: Salvatore Capozziello, Anupam Mazumdar, Giuseppe Meluccio

Published 2026-08-04

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Salvatore Capozziello, Anupam Mazumdar, Giuseppe Meluccio

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, cosmic movie. For decades, physicists have been the directors, using a script called General Relativity to explain how gravity works. This script is a masterpiece; it perfectly predicts how planets orbit stars and how light bends around black holes. But when the directors tried to film the entire history of the universe, the script started to glitch. In the very beginning (the "ultraviolet" or high-energy regime), the math breaks down and explodes. But the bigger mystery is happening right now, in the "infrared" or low-energy regime. We know the universe isn't just coasting along; it's speeding up, expanding faster and faster. To explain this, the standard script adds a mysterious character called "Dark Energy," which acts like a cosmic anti-gravity force. However, this character is a bit of a placeholder. We don't know what it is, and when we try to calculate its value using quantum physics, the numbers are off by a staggering 120 zeros. It's like trying to balance a checkbook and finding you're off by the weight of a galaxy.

So, scientists are asking: Is Dark Energy a real substance, or is our script for gravity just missing a few pages? This is where the work of Salvatore Capozziello, Anupam Mazumdar, and Giuseppe Meluccio comes in. They are exploring a radical idea: what if gravity isn't just about what happens here and now, but is also influenced by what happened there and then? In physics, we usually think of gravity as a local interaction—like a magnet pulling a paperclip right next to it. But "nonlocal" gravity suggests that the gravitational pull at one point in space is also affected by the history of the universe at other times. Think of it like a conversation where your current sentence is shaped not just by what your friend is saying right now, but by everything they said yesterday, last week, and last year. The authors propose that these "echoes" from the past could be the real reason the universe is accelerating, potentially doing away with the need for a mysterious Dark Energy substance entirely.

The Paper's Story: Gravity with a Memory

In their paper, titled "Can nonlocal gravity really explain dark energy?", the authors investigate a specific version of this idea called Infinite Derivative Gravity. Instead of adding a new ingredient like Dark Energy to the cosmic soup, they tweak the recipe for gravity itself. They suggest that at very low energies (the scale of the whole universe), gravity has a "memory." It doesn't just react to the matter right in front of it; it integrates information from the entire history of the universe's expansion.

To test this, the team built a mathematical model of the universe that includes these nonlocal effects. They didn't just guess; they derived complex equations that describe how the universe should evolve if gravity really does have this long-term memory. The math was incredibly tricky, involving infinite sums and integrals that are usually a nightmare to solve. To get around this, the authors developed a clever technique. They essentially "localized" the problem, turning those infinite, history-hugging equations into a simpler form that could be solved, much like turning a complex, multi-layered cake recipe into a manageable set of instructions.

What did they find?
The results are fascinating. When they ran their model, they discovered that the nonlocal gravitational effects naturally kick in at the right time in the universe's history. For billions of years, the universe behaved exactly as standard gravity predicts, with matter slowing down the expansion. But then, as the universe grew older and cooler, the nonlocal "memory" effects became dominant. Suddenly, the expansion started to accelerate, just like we observe today.

Crucially, this acceleration happened without introducing a cosmological constant (the standard "Dark Energy" placeholder) or any new scalar fields. The acceleration emerged purely from the way gravity interacts with itself over cosmic time. The authors suggest that this mechanism could be the true driver of the universe's current speed-up.

A Twist in the Gravity:
One of the most playful and surprising predictions of their model is that the strength of gravity itself might be changing. In our current era, the model suggests that the "effective" gravitational constant—the number that tells us how strong gravity is—has decreased. Specifically, the authors calculate that gravity is currently about 82% as strong as it was in the past (a reduction from $G$ to $G/1.22$ ). You can think of this as the universe's "glue" getting slightly weaker, allowing the cosmic fabric to stretch out faster. This isn't a bug; it's a feature. The weakening of gravity acts as a repulsive force, mimicking the effects of Dark Energy.

How sure are they?
The authors are careful not to claim they have solved the universe's biggest mystery. They describe their findings as a "toy model" that shows it is possible for nonlocal gravity to explain dark energy. They point out that their solution relies on certain approximations and that the mathematical coefficients in their infinite series are still somewhat of a mystery. However, they argue that the model is consistent with observations and offers a compelling alternative to the standard view. They also note that this approach connects nicely to another famous theory called Starobinsky gravity, which is used to explain the rapid expansion of the very early universe (inflation). This hints that nonlocal gravity might be the key to unifying the story of the Big Bang and the story of today's accelerating universe.

What's next?
The paper suggests that we might be able to test these ideas. If gravity really is getting weaker, we might see subtle changes in how planets orbit or how light travels over vast distances. Future missions like the Euclid telescope or data from the Lunar Laser Ranging experiment could potentially spot these tiny deviations. The authors conclude that while we can't say for sure yet, the idea that gravity has a memory is a powerful, efficient mechanism that could finally explain why the universe is speeding up, all without needing to invent a mysterious new substance. It's a reminder that sometimes, the answer to the universe's biggest problems isn't a new ingredient, but a better understanding of the old ones.

Technical Summary: Nonlocal Gravity as a Dynamical Mechanism for Dark Energy

Problem Statement
General Relativity (GR), while successful across various energy scales, faces significant challenges in the infrared (IR) regime, specifically regarding the accelerated expansion of the late Universe. The standard explanation involves a cosmological constant ( $\Lambda$ ) or unknown scalar fields, yet these introduce severe theoretical issues, most notably the vacuum energy discrepancy of $\sim 120$ orders of magnitude. The authors investigate whether nonlocal modifications to GR, specifically within the framework of Infinite Derivative Gravity (IDG), can dynamically address the dark energy problem without introducing $\Lambda$ or new scalar fields. The goal is to determine if IR-scale nonlocal gravitational effects can naturally generate accelerated expansion.

Methodology
The authors analyze a nonlocal gravitational action based on IDG, restricted to terms involving the Ricci scalar $R$ and its nonlocal corrections. The action is defined as:
$S = \int d^4x \sqrt{-g} \frac{1}{2} \left[ M_P^2 R + R \bar{F}_1(\Box) R \right]$
where $\bar{F}_1(\Box) = \sum_{n=1}^{\infty} f_{1-n} \Box^{-n}$ represents an infinite series of inverse d'Alembertian operators, introducing nonlocality. $M_P$ is the reduced Planck mass, and $M$ is an IR mass scale.

To solve the resulting complex integro-differential cosmological equations, the authors employ a "formal localisation" technique. They impose an ansatz where the nonlocal operator acting on the Ricci scalar is approximated by a linear relation: $\Box^{-1}R = r_1 R + r_2$ . This allows the nonlocal terms to be recast into a local form parameterized by $r_1$ and $r_2$ .

The study focuses on a homogeneous, isotropic, and flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe. The authors propose an approximate solution for the scale factor $a(t)$ in the late Universe ( $t > t_*$ , where $t_*$ marks the transition from matter domination to dark energy domination) of the form $a(t) \propto e^{\frac{\lambda}{2}t^2}$ . This ansatz is designed to match the standard matter-dominated era ( $a \propto t^{2/3}$ ) at the transition point $t_*$ and reproduce accelerated expansion thereafter.

Key Contributions and Results

Derivation of Localized Equations: The authors successfully derived the equations of motion for the nonlocal IDG model and applied the localization ansatz to simplify the cosmological equations into fourth-order nonlinear differential equations for the scale factor, avoiding explicit integral operators.
Approximate IR Solution: By adopting the exponential scale factor ansatz, the authors demonstrated that the nonlocal corrections become significant only in the IR limit (low energies), leaving the early Universe and matter-dominated era consistent with the Standard Model of Cosmology. The parameter $\lambda$ was fixed by matching the scale factor at the transition redshift $z_* \approx 0.295$ .
Connection to Starobinsky Gravity: Substituting the approximate solution back into the action reveals that the IDG model effectively reduces to a correction of the Starobinsky gravity model ( $R + \alpha R^2$ ). The nonlocal series sums to a dimensionless coefficient $\bar{F}_1(\bar{\lambda})$ , which acts as a correction to the gravitational constant and the coefficient of the $R^2$ term.
Effective Gravitational Constant: A central result is the calculation of the effective gravitational constant in the current cosmological era. The analysis yields a negative value for the nonlocal coefficient $\bar{F}_1(\bar{\lambda})$ , implying a reduction in the effective gravitational strength:
$G_{\text{eff}} \approx \frac{G}{1.22} \approx 0.82 G$
This weakening of gravity is proposed as the dynamical mechanism driving the observed accelerated expansion, effectively mimicking dark energy.
Mass Scale Consistency: The IR mass scale $M$ required for consistency is found to be of the order of the Hubble constant ( $M \sim H_0 \sim 10^{-42}$ GeV), linking the nonlocal effects directly to the current cosmic expansion rate.

Significance and Claims
The paper claims that nonlocal gravity corrections emerging at specific IR scales offer an efficient mechanism to address the dark energy problem without resorting to a cosmological constant or ad-hoc scalar fields. The model suggests that the accelerated expansion is a manifestation of nonlocal gravitational effects that remain "hidden" until the Universe cools sufficiently for these IR effects to dominate.

The authors highlight that their approach recovers the Starobinsky model improved with nonlocal corrections, potentially unifying inflationary regimes (UV) with the dark energy era (IR). They note that the model predicts a testable decrease in the effective gravitational constant, which could be probed by future cosmological missions (e.g., JWST, Euclid, DESI) or Lunar Laser Ranging experiments. However, the authors remain cautious, acknowledging that the infinite series of coefficients requires specific convergence properties and that the presence of ghosts (due to the negative sign of the $R^2$ coefficient) needs further demonstration in curved spacetime. The study concludes that while the mathematical complexity is high, the formalism allows for predictive power without truncating the infinite derivative terms, offering a viable alternative to standard dark energy paradigms.

The Paper's Story: Gravity with a Memory

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