⚛️ general relativity

Big-Bang Nucleosynthesis and WIMP Dark Matter Freeze-Out as Probes of Yukawa Cosmology

This paper investigates Yukawa cosmology by deriving modified Friedmann equations and applying them to Big-Bang Nucleosynthesis and WIMP freeze-out, revealing that while light element abundances and dark matter relic density place tight, complementary constraints on the Yukawa coupling $\alpha$ , the framework fails to resolve the Lithium Problem.

Original authors: Ava Shahbazi Sooraki, Ahmad Sheykhi

Published 2026-08-11

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Ava Shahbazi Sooraki, Ahmad Sheykhi

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, expanding balloon. For decades, scientists have been trying to figure out exactly how fast that balloon is inflating and what's inside it. The current "rulebook" for this cosmic balloon is called the Lambda-Cold Dark Matter (ΛCDM) model. It's a bit like a recipe that says the universe is made of normal stuff (like stars and you), invisible "cold dark matter" that holds galaxies together, and a mysterious "dark energy" pushing the balloon to expand faster. This recipe works great for most things, but it has a few stubborn glitches. One is a huge math problem about why the universe isn't expanding as fast as quantum physics predicts it should. Another is a weird mismatch with a specific ingredient: the universe seems to have way less Lithium-7 (a light element) than the recipe says it should have.

To fix these glitches, some scientists have started wondering if the rulebook for gravity itself needs a rewrite. Instead of gravity being a simple, unchanging force, what if it has a little "extra" push or pull that fades away over distance? This idea is called "Yukawa cosmology." Think of it like a magnet: normally, a magnet's pull is strong up close and weakens quickly. But in this new theory, there's an invisible layer of "magnetic dust" around the magnet that changes how the pull feels, depending on how far away you are. This paper asks a big question: If we tweak gravity with this "Yukawa dust," does it fix the universe's recipe? Specifically, does it help explain why the early universe cooked up the right amount of elements and why dark matter is hanging around in the exact amount we see today?

The Cosmic Kitchen: Testing a New Gravity Recipe

In this study, researchers Ava Shahbazi Sooraki and Ahmad Sheykhi from Shiraz University decided to test this "Yukawa dust" idea in the most extreme kitchen imaginable: the first few minutes after the Big Bang. They wanted to see if tweaking gravity could fix the universe's cooking without burning the whole meal.

The Setup: A New Way to Measure the Oven
First, the authors had to figure out how this new gravity would change the "oven temperature" of the early universe. In standard physics, the universe expands at a rate governed by Einstein's gravity. The authors used a clever trick involving thermodynamics (the study of heat and energy) to derive a new set of rules for how the universe expands if this "Yukawa dust" exists. They found that this new gravity introduces a tiny "correction factor" to the expansion rate.

Think of the universe's expansion like a car driving down a highway. In the standard model, the car cruises at a steady speed. In this new model, the car has a slightly different engine. If the "Yukawa coupling" (let's call it α, a number that measures how strong this extra gravity is) is positive, the car drives a bit slower. If α is negative, the car speeds up. This change in speed is crucial because it determines how fast the universe cools down, which in turn decides how many atoms get cooked up.

The First Test: Cooking the Elements (BBN)
The first thing they checked was Big-Bang Nucleosynthesis (BBN). This is the cosmic cooking phase that happened just minutes after the Big Bang, where protons and neutrons fused to create the first light elements: Helium-4, Deuterium (heavy hydrogen), and Lithium-7.

The authors compared their new "Yukawa expansion speed" against what we actually observe in the universe today:

Helium-4 and Deuterium: These two elements are like the "goldilocks" ingredients. They need the universe to expand at just the right speed to get the perfect amount. The researchers found that if the Yukawa parameter α is between -0.24 and 0.12, the universe cooks up the perfect amount of Helium and Deuterium, matching our observations perfectly. It's like finding the right dial setting on the oven.
Lithium-7: This is the troublemaker. In the standard recipe, the universe makes way too much Lithium-7 compared to what we see in old stars. The authors hoped that changing the expansion speed might fix this. They found that to get the Lithium amount right, α would need to be between -0.76 and -0.72.

The Verdict on Lithium: Here is the bad news for the Lithium problem. The "dial setting" needed to fix the Lithium (around -0.74) is completely different from the setting needed to fix the Helium and Deuterium (around 0). There is no single number for α that can satisfy all three ingredients at once. The authors conclude that while this Yukawa gravity idea is interesting, it cannot solve the Lithium Problem. The universe still has too much Lithium, and this specific tweak to gravity doesn't fix it.

The Second Test: The Dark Matter Freeze-Out
Next, the team looked at Dark Matter. They focused on a popular candidate called WIMPs (Weakly Interacting Massive Particles). Imagine WIMPs as invisible ghosts floating around in the early universe. As the universe expands and cools, these ghosts stop bumping into each other and "freeze out," leaving a leftover population that we see today as dark matter.

The amount of leftover ghosts depends heavily on how fast the universe was expanding when they froze. If the universe expands too fast, the ghosts freeze out too early, leaving too many. If it expands too slow, they freeze out too late, leaving too few.

The researchers calculated how the Yukawa gravity would change this freeze-out process. They found that for the leftover dark matter to match the precise amount we observe today (which is 0.120 ± 0.001 in specific units), the Yukawa parameter α must be very close to zero, specifically between -0.017 and 0.018.

The Grand Conclusion
When the authors put all the clues together, they found a sweet spot. The range of α that works for Dark Matter (-0.017 to 0.018) fits perfectly inside the range that works for Helium and Deuterium (-0.24 to 0.12). This means that if the Yukawa gravity parameter is very small (close to zero), the theory is consistent with both the cooking of elements and the amount of dark matter we see.

However, the story has a twist. The authors also looked at how this new gravity changes the relationship between time and temperature. They found that if α is positive, the universe expands more slowly, meaning it stays hotter for longer at any given moment in time. If α is negative, the universe expands faster and cools down quicker.

The Final Takeaway
This paper shows that "Yukawa cosmology" is a testable and viable idea for the early universe, provided the extra gravity effect is very weak. It successfully passes the tests for Helium, Deuterium, and Dark Matter. However, it explicitly fails to solve the mystery of the missing Lithium. The universe's Lithium problem remains unsolved by this particular theory, suggesting that if we want to fix that specific glitch, we'll need to look for a different kind of physics. The study confirms that by looking at the leftovers of the Big Bang and the ghosts of dark matter, we can put very strict limits on how much we can tweak the laws of gravity.

Technical Summary: Big-Bang Nucleosynthesis and WIMP Dark Matter Freeze-Out as Probes of Yukawa Cosmology

Problem Statement
The standard $\Lambda$ CDM cosmological model, while successful in describing large-scale structures and CMB anisotropies, faces profound theoretical challenges, most notably the cosmological constant problem. This discrepancy has motivated the exploration of modified gravity theories, including those introducing a Yukawa-type correction to the Newtonian potential. While such modifications are often tested on solar system and galactic scales, their implications for the very early universe remain under-explored. Specifically, it is unclear whether Yukawa cosmology can resolve the persistent "Lithium Problem"—the discrepancy between the predicted and observed primordial abundance of $^7$ Li—and how modified expansion rates affect the thermal freeze-out of Weakly Interacting Massive Particles (WIMPs).

Methodology
The authors derive the modified Friedmann equations for Yukawa cosmology by applying the first law of thermodynamics ($dE = TdS + WdV$) to the apparent horizon of a Friedmann-Robertson-Walker (FRW) universe.

Entropy Correction: Starting from Verlinde's entropic force scenario, the authors incorporate a Yukawa potential $\Phi(r) = -GM/r (1 + \alpha e^{-r/\lambda})$ to derive a modified entropy-area relation for the horizon. This yields an entropy expression containing an exponential correction term dependent on the coupling strength $\alpha$ and the characteristic range $\lambda$ .
Modified Friedmann Equations: Applying the thermodynamic first law to this modified entropy results in a modified Friedmann equation featuring a logarithmic correction term. In the early universe limit ( $R/\lambda \ll 1$ ), this leads to an effective gravitational constant $G_{\text{eff}} = G/(1+\alpha)$ and a modified Hubble parameter $H(T)$ .
Big-Bang Nucleosynthesis (BBN): The modified Hubble parameter is expressed as $H(T) = Z(T)H_{\text{GR}}(T)$ , where $Z(T)$ is an amplification factor dependent on $\alpha$ and $\lambda$ . The authors use numerical fits for primordial abundances of $^4$ He, Deuterium (D), and $^7$ Li as functions of $Z$ and the baryon-to-photon ratio $\eta_{10}$ . By comparing these theoretical predictions with observational data, they constrain $\alpha$ .
WIMP Freeze-Out: The modified expansion rate is incorporated into the Boltzmann equation governing WIMP number density. The authors derive an analytical expression for the relic abundance $\Omega_{\chi}h^2$ as a function of $\alpha$ , assuming a benchmark thermal WIMP with mass $m_\chi \sim 100$ GeV.
Time-Temperature Relation: The authors derive the modified relation between cosmic time and temperature in the radiation-dominated era to analyze thermal history deviations.

Key Contributions and Results

Derivation of Modified Equations: The paper provides a distinct derivation of modified Friedmann equations using thermodynamic principles on the apparent horizon, resulting in a specific logarithmic correction term that differs from previous entropic force approaches.
BBN Constraints:
- $^4$ He and Deuterium: The constraints from primordial $^4$ He and Deuterium abundances are mutually consistent, yielding a combined allowed range of $-0.24 \lesssim \alpha \lesssim 0.12$ .
- $^7$ Li (Lithium Problem): The constraint derived from $^7$ Li abundance is $-0.76 \lesssim \alpha \lesssim -0.72$ . The authors find that this range is completely disjoint from the ranges allowed by $^4$ He and Deuterium. Consequently, the study concludes that Yukawa cosmology cannot resolve the Lithium Problem within this framework, as no single value of $\alpha$ can simultaneously satisfy the abundances of all three elements.
WIMP Relic Abundance: The analysis of WIMP freeze-out yields an independent constraint on $\alpha$ based on the observed dark matter relic density ( $\Omega_{\text{CDM}}h^2 = 0.120 \pm 0.001$ ). The derived range is $-0.017 \lesssim \alpha \lesssim 0.018$ .
Combined Constraints: The WIMP constraint ( $-0.017 \lesssim \alpha \lesssim 0.018$ ) lies entirely within the BBN-allowed range for $^4$ He and Deuterium. This overlap suggests that Yukawa cosmology is a viable framework for early-universe physics, provided $\alpha$ is sufficiently small.
Time-Temperature Dynamics: The study demonstrates that the sign of $\alpha$ dictates the thermal history: positive $\alpha$ reduces the effective gravitational coupling, slowing the expansion and resulting in a hotter universe at a given cosmic time compared to standard cosmology. Conversely, negative $\alpha$ accelerates expansion, leading to a cooler universe.

Significance
The paper establishes BBN and dark matter relic abundance as complementary and powerful probes of modified gravity theories in the high-redshift, high-temperature universe. By confronting the modified expansion history with two independent cosmological observables, the authors confirm that Yukawa cosmology is a testable framework. While the model fails to resolve the Lithium Problem, its consistency with $^4$ He, Deuterium, and WIMP relic density constraints reinforces its viability for small coupling strengths. These findings extend existing bounds from late-time cosmological probes and local laboratory experiments, offering a more comprehensive picture of the constraints on Yukawa-type modifications to gravity.

The Cosmic Kitchen: Testing a New Gravity Recipe

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