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Constraints on Rényi Entropy through Primordial Big-Bang Nucleosynthesis and Baryogenesis

This paper derives modified Friedmann equations based on Rényi entropy to constrain the Rényi parameter using Big-Bang Nucleosynthesis data, revealing that while the parameter ranges derived from Helium-4 and Deuterium overlap, their discrepancy with the Lithium-7 range suggests a potential mechanism for alleviating the long-standing Lithium problem in early-universe cosmology.

Original authors: Ahmad Sheykhi, Ava Shahbazi Sooraki

Published 2026-06-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ahmad Sheykhi, Ava Shahbazi Sooraki

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 used a standard rulebook (called the Big Bang theory) to predict how this balloon inflated and what ingredients were baked inside it during its first few minutes. This "baking" process is called Big-Bang Nucleosynthesis (BBN). It's responsible for creating the universe's first light elements: Hydrogen, Helium, and a tiny bit of Lithium.

Usually, the rulebook works perfectly. The predicted amount of Helium and Deuterium (a heavy form of Hydrogen) matches what we see in the sky. However, there's a stubborn glitch: the rulebook predicts three to four times more Lithium than we actually observe. This is known as the "Cosmological Lithium Problem."

In this paper, the authors ask: What if the rulebook isn't quite right? What if the "thermodynamics" (the physics of heat and energy) of the early universe works a little differently than we thought?

Here is a simple breakdown of their investigation:

1. The New Rulebook: Rényi Entropy

Standard physics uses a formula called "Boltzmann-Gibbs entropy" to measure disorder or information. The authors decided to swap this out for a more flexible formula called Rényi entropy.

The Analogy: Think of standard entropy like a strict accountant who adds up numbers linearly (1 + 1 = 2). Rényi entropy is like a more complex accountant who considers how the numbers interact with each other. It allows for "long-range" connections, which might happen if gravity and quantum mechanics are tangled together in the early universe.
The Result: By using this new formula, the authors derived new equations for how the universe expands. It's like changing the engine of a car; the car still drives, but the speed and fuel consumption change slightly.

2. The "MOND" Connection

Before diving into the Big Bang, the authors showed that this new entropy formula can also explain a weird phenomenon in galaxies called MOND (Modified Newtonian Dynamics).

The Analogy: In standard physics, stars at the edge of a galaxy should fly off because there isn't enough gravity holding them. MOND suggests the laws of gravity change at very low speeds. The authors showed that if you view gravity through the lens of Rényi entropy, you naturally get these modified laws without needing to invent "dark matter." It's like realizing the road itself is slightly curved, so the car doesn't need to go faster to stay on track.

3. Testing the New Theory: The Lithium Problem

The authors took their new "Rényi universe" and ran the numbers for the Big Bang baking process. They asked: Does this new engine fix the Lithium glitch?

They looked at three ingredients:

Helium-4: The standard recipe works well here.
Deuterium: The standard recipe works well here too.
Lithium-7: This is the problem child.

The Findings:

The Good News: The new Rényi rules can be tweaked (by adjusting a specific number called the "Rényi parameter," let's call it $\lambda$ ) to match the observed amounts of Helium and Deuterium perfectly.
The Bad News: The specific setting of $\lambda$ that fixes Helium and Deuterium is different from the setting needed to fix Lithium.
The "Almost" Solution: While the settings don't match perfectly, they are very close to each other. The Lithium range is just slightly off from the Helium/Deuterium range.
The Conclusion: The Rényi model doesn't completely solve the Lithium problem on its own with a single setting. However, the fact that the ranges are so close suggests that a slightly more complex version of this theory (perhaps one where the rules change as the universe cools) could solve the mystery. It opens a new door for physicists to walk through.

4. The Temperature Twist

The authors also looked at how time and temperature relate in this new universe.

The Analogy: Imagine the universe is a cooling cup of coffee. In the standard model, it cools at a predictable rate. In the Rényi model, the "cup" is slightly different.
The Finding: If you increase the Rényi parameter ( $\lambda$ ), the early universe stays hotter for longer. It's as if the universe expanded a bit slower, keeping the heat trapped inside for a longer time. This is a direct result of the new thermodynamic rules.

5. Creating Matter vs. Antimatter (Baryogenesis)

The universe is made of matter, but the Big Bang should have created equal amounts of matter and antimatter, which would have destroyed each other. Why is there anything left?

The Mechanism: The authors showed that the new Rényi rules naturally create a "wobble" in the universe's energy that pushes it out of perfect balance (thermal equilibrium).
The Result: This imbalance is exactly what is needed to explain why we have more matter than antimatter today. The new entropy rules provide a natural mechanism to satisfy the conditions required for the universe to exist as we know it.

Summary

This paper proposes that if we view the early universe through the lens of Rényi entropy (a more complex way of measuring disorder), we get a slightly different set of rules for how the universe expands and cools.

Does it fix everything? Not quite. It matches Helium and Deuterium perfectly but still struggles slightly with Lithium.
Is it useful? Yes. It brings the Lithium prediction much closer to reality than before, suggesting that the answer to the Lithium problem might lie in these subtle, non-standard thermodynamic rules.
Bonus: It also explains why the universe has more matter than antimatter and why galaxies spin the way they do, all from the same mathematical tweak.

In short, the authors didn't find the "magic bullet" to fix the Lithium problem instantly, but they found a very promising clue that suggests the universe's "thermostat" might be set to a slightly different, more complex setting than we previously thought.

Technical Summary: Constraints on Rényi Entropy through Primordial Big-Bang Nucleosynthesis and Baryogenesis

Problem Statement
The standard cosmological model, while successful in describing the evolution of the universe, faces persistent challenges, most notably the "Cosmological Lithium Problem," where the predicted abundance of primordial Lithium-7 ( $^7$ Li) exceeds observational data by a factor of 2–3. Additionally, the origin of the matter-antimatter asymmetry (baryogenesis) requires mechanisms that violate thermal equilibrium, a condition often difficult to satisfy in standard radiation-dominated eras. These discrepancies motivate the exploration of modified gravitational theories and nonextensive thermodynamics. Specifically, the authors investigate whether the Rényi entropy—a one-parameter generalization of Boltzmann-Gibbs entropy motivated by quantum gravitational effects and nonextensive statistical mechanics—can alter early-universe dynamics sufficiently to address these issues without violating constraints from other light elements (Deuterium and Helium-4) or baryogenesis data.

Methodology
The authors employ the thermodynamics-gravity correspondence conjecture, which posits that gravitational field equations can be derived from thermodynamic laws applied to the apparent horizon of the universe.

Derivation of Modified Dynamics: Starting with the Rényi entropy formula for a black hole horizon, $S_h = \frac{1}{\lambda} \ln(1 + \lambda S_{BH})$ , where $\lambda$ is the Rényi parameter, the authors apply the first law of thermodynamics ( $dE = T_h dS_h + W dV$ ) to the apparent horizon of a Friedmann-Robertson-Walker (FRW) spacetime. This yields modified Friedmann equations containing logarithmic correction terms dependent on $\lambda$ .
MOND Connection: The paper first demonstrates that the modified Newtonian dynamics (MOND) theory can be theoretically derived from Rényi-corrected entropy, establishing a link between nonextensive thermodynamics and galactic rotation curves.
BBN Analysis: The modified Friedmann equations are used to derive a temperature-dependent amplification factor, $Z(T)$ , which modifies the expansion rate $H(T)$ during the Big-Bang Nucleosynthesis (BBN) epoch. The authors calculate the resulting primordial abundances of Deuterium ( $^2$ H), Helium-4 ( $^4$ He), and Lithium-7 ( $^7$ Li) as functions of the Rényi parameter $\lambda$ and the baryon-to-photon ratio $\eta_{10}$ . These theoretical predictions are compared against precision observational data to constrain $\lambda$ .
Baryogenesis Analysis: The authors reformulate the energy density and pressure to include Rényi-induced fluctuations. They investigate whether these modifications can disrupt thermal equilibrium (satisfying the third Sakharov condition) to generate baryon asymmetry. By coupling the Ricci scalar derivative to the baryon current, they derive an expression for the baryon asymmetry parameter $\eta$ in terms of $\lambda$ and compare it with observational bounds.
Time-Temperature Relation: The relationship between cosmic time and temperature, $t(T)$ , is derived within the Rényi framework to analyze thermal evolution.

Key Contributions and Results

Modified Friedmann Equations: The study derives the first Friedmann equation in the form $H^2 - \alpha \ln(H^2) = \frac{8\pi G}{3}(\rho + \rho_\Lambda)$ (for a flat universe), where $\alpha = \lambda \pi / G$ . This introduces a logarithmic correction to the standard expansion rate.
Constraints on $\lambda$ from Light Elements:
- Helium-4 and Deuterium: The analysis reveals that the observational constraints on $^4$ He and $^2$ H allow for overlapping ranges of the Rényi parameter: $-3.18 \times 10^{-85} \lesssim \lambda \lesssim 1.19 \times 10^{-85}$ (from He) and $-1.054 \times 10^{-85} \lesssim \lambda \lesssim 7.96 \times 10^{-85}$ (from D).
- Lithium-7: The constraint derived from $^7$ Li is distinct and does not overlap with the He/D ranges: $-2.16 \times 10^{-84} \lesssim \lambda \lesssim -1.84 \times 10^{-84}$ .
- Lithium Problem: The authors conclude that a constant $\lambda$ cannot simultaneously resolve the Lithium discrepancy while remaining consistent with Deuterium and Helium observations. However, the proximity of the allowed ranges suggests that the Lithium problem might be alleviated in extended frameworks (e.g., temperature-dependent $\lambda$ ) where the expansion rate modification selectively affects the later $^7$ Li production epoch.
Baryogenesis Constraints: The study demonstrates that Rényi entropy corrections induce fluctuations in energy density and pressure that lead to a non-zero time derivative of the Ricci scalar ( $\dot{R} \neq 0$ ) during the radiation-dominated era. This naturally breaks thermal equilibrium, satisfying the third Sakharov condition. Comparing the theoretical baryon asymmetry with observational limits ( $\eta \lesssim 9.9 \times 10^{-11}$ ) yields a bound on the rescaled parameter $\tilde{\lambda} \lesssim 0.022$ . The authors note a partial overlap between the baryogenesis constraints and the BBN constraints.
Time-Temperature Evolution: The derived $t(T)$ relation shows that increasing the Rényi parameter $\lambda$ leads to higher temperatures in the early universe for a given cosmic time. This is attributed to nonextensive thermodynamic effects enhancing energy density and a modified expansion rate that prolongs high-temperature conditions.

Significance and Claims
The paper claims to provide the first systematic constraints on Rényi cosmology using primordial BBN and baryogenesis data. The authors assert that their analysis highlights the role of nonextensive thermodynamics in early-universe physics. While the constant- $\lambda$ model does not fully resolve the Lithium problem, the study suggests that the "small mismatch" between the parameter ranges for different elements opens a theoretical window for resolving the discrepancy within modified cosmological scenarios. Furthermore, the work establishes a theoretical link between Rényi entropy and the mechanism required for baryogenesis, showing that Rényi corrections can naturally fulfill the out-of-equilibrium condition necessary for generating matter-antimatter asymmetry. The results indicate that while Rényi corrections are phenomenologically mild (with $\lambda$ constrained to extremely small values, $\sim 10^{-85}$ , by BBN), they offer a viable framework for probing quantum gravitational effects in cosmology.