⚛️ general relativity

Cosmological Constant and Maximum of Entropy for de Sitter Space

This paper proposes that the inflationary cosmological constant of the early universe can be derived from the principle of maximum entropy, suggesting that the universe emerges from a minimum entropy state and evolves toward a 4-dimensional de Sitter space with a specific maximum entropy value.

Original authors: Igor Volovich

Published 2026-10-22

📖 4 min read🧠 Deep dive

CC0 1.0

Original authors: Igor Volovich

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 story of our universe often begins with a question of scale: how did everything we see, from the smallest atom to the most distant galaxy, come to be? For decades, physicists have relied on a concept called the cosmological constant to explain the expansion of the cosmos. Originally introduced to keep the universe static, this value now serves a different purpose, describing how the universe is currently speeding up its expansion due to a mysterious force known as dark energy. However, there is a second, older chapter in this story. In the very first fraction of a second after the universe began, it underwent a period of incredibly rapid growth called inflation. This early burst was also driven by a cosmological constant, but one that was vastly different from the one we measure today. Understanding why this early constant had the specific value it did remains one of the great puzzles in physics, as that value set the stage for everything that followed.

A new approach to solving this puzzle comes from Igor Volovich, a researcher at the Steklov Mathematical Institute in Russia. His work proposes a way to calculate the value of that early inflationary constant by looking at the universe through the lens of entropy, a measure of disorder or randomness. The core idea rests on a principle known as maximum entropy, which suggests that physical systems naturally evolve toward the state where they have the highest possible entropy. Volovich applies this principle to the very beginning of time, proposing that the universe emerged from a state of "nothing," which corresponds to a minimum of entropy. From that starting point, the universe would naturally settle into a configuration where its entropy is as high as it can possibly be.

To find this maximum, the researcher treated the number of dimensions in space and time not as a fixed fact, but as a variable that could change. While we experience the world as having three spatial dimensions and one time dimension, the mathematical tools used in this study allow for the exploration of universes with different numbers of dimensions. By calculating the entropy of a theoretical universe with a specific size and varying its dimensionality, the study searched for the point where entropy peaked. The results were striking: the researcher proposed that the radius of the early universe is such that the maximum entropy occurs precisely when the universe has four dimensions. This finding suggests that the universe is four-dimensional because the specific size of the early universe's horizon aligns the peak of the entropy function with that dimensionality.

Crucially, this maximum entropy state only occurs if the size of the universe at that moment matches a very specific value. By working backward from the condition that the maximum entropy must happen at four dimensions, the researcher calculated the exact size of the early universe's horizon. This size directly determines the value of the cosmological constant for that era. The calculation yields a specific number, approximately 9.087 when measured in the fundamental units of nature known as Planck units. This number is not a guess or a rough estimate; it is a precise result derived from the mathematical proposal that the universe maximizes its entropy under the assumption that spacetime dimension is a continuous parameter.

The study also explores what happens if the universe were to have fewer dimensions. It finds that for dimensions between zero and four, the entropy increases steadily as the number of dimensions grows, effectively treating the number of dimensions as a kind of time that moves the system toward its peak. At zero dimensions, the entropy is at its lowest, corresponding to the "nothing" from which the universe emerged. As the system evolves toward four dimensions, it climbs to the highest possible entropy state. This mathematical journey provides a potential explanation for why the early universe had the specific energy density it did, offering a new way to set the initial conditions for models of cosmic inflation.

While this work does not claim to have solved every mystery of the cosmos, it offers a compelling method for deriving a fundamental constant from first principles. Instead of accepting the value of the early cosmological constant as an arbitrary input, this approach suggests it is a necessary outcome of the universe seeking its most probable state under the proposed theoretical framework. The findings imply that the laws governing the birth of the universe are deeply connected to the geometry of space and the nature of entropy itself. By treating the dimension of space as a flexible parameter, the research bridges the gap between the abstract mathematics of quantum theory and the physical reality of our four-dimensional world, providing a concrete, calculated value for the force that drove the universe's first expansion.

Technical Summary: Cosmological Constant and Maximum of Entropy for de Sitter Space

Problem Statement
The paper addresses the theoretical challenge of determining the value of the inflationary cosmological constant ( $\Lambda$ ) in the early universe. While Einstein originally introduced a single cosmological constant, modern cosmology distinguishes between at least two: one governing the quasi-de Sitter inflationary phase of the early universe and another describing the current dark energy-driven acceleration. Existing approaches, such as those by Zeldovich linking vacuum fluctuations to $\Lambda$ , have not yielded a definitive first-principles calculation for the inflationary constant. The author posits that the ground state of the early universe, suggested by tunneling and no-boundary proposals to be de Sitter space, should be characterized by a specific radius (and thus a specific $\Lambda$ ) derived from fundamental thermodynamic principles.

Methodology
The author proposes a novel approach combining the principle of maximum entropy with the treatment of spacetime dimension ( $D$ ) as a continuous parameter. The methodology proceeds as follows:

Extension of the Maximum Entropy Principle: The standard principle, which selects the probability distribution with the highest entropy given current knowledge, is extended to include the dimensionality of spacetime as a variable. The observed dimension $D_0 = 4$ is treated not as a fixed input but as the result of an entropy maximization process.
Entropy Formulation: The entropy of a $D$ -dimensional de Sitter space with radius $\ell$ is defined using the Gibbons-Hawking temperature and the area of the cosmological horizon. The entropy function $S(\ell, D)$ is expressed as:
$S(\ell, D) = \frac{1}{4} \ell^{D-2} \Omega_{D-2} = \frac{1}{2} \ell^{D-2} \frac{\pi^{(D-1)/2}}{\Gamma((D-1)/2)}$
where $\Omega_{D-2}$ is the area of the unit sphere $S^{D-2}$ .
Analytical Continuation: The dimension $D$ is treated as a real variable rather than an integer, allowing for the use of calculus (differentiation with respect to $D$ ) and the inclusion of negative dimensions in the mathematical formalism.
Optimization Condition: The core hypothesis is that the universe emerges from a state of "nothing" (minimum entropy) and evolves to a state of maximum entropy. The specific radius $\ell_0$ (and corresponding $\Lambda$ ) is determined by finding the value of $\ell$ such that the entropy $S(\ell, D)$ reaches its global maximum exactly at the observed dimension $D = 4$ .

Key Results
By solving the extremum condition $\frac{\partial S(\ell, D)}{\partial D} = 0$ for $D=4$ , the author derives a specific value for the de Sitter radius $\ell_0$ :
$\ell_0^2 = \frac{1}{\pi} \exp\left\{\psi\left(\frac{3}{2}\right)\right\} \approx 0.3301$
where $\psi$ is the digamma function. Consequently, the inflationary cosmological constant in Planck units is computed as:
$\Lambda = \frac{3}{\ell_0^2} = 3\pi \exp\left\{-\psi\left(\frac{3}{2}\right)\right\} \approx 9.087$
In physical units, this corresponds to $\Lambda \approx 9.087 M_{pl}^2 \approx 4.3 \times 10^{-15} \text{ kg}^2$ .

The paper also notes the behavior of the entropy function:

For a fixed $\ell$ , $S(\ell, D)$ typically exhibits a single maximum for $D > 1$ .
At the specific value $\ell = \ell_0$ , the maximum of the entropy function occurs precisely at $D=4$ .
The entropy at $D=0$ for $\ell=\ell_0$ is negative ( $S \approx -0.482$ ), consistent with the concept of the universe emerging from a "nothing" state with minimal (potentially negative) entropy.

Significance and Claims
The paper claims to provide a theoretical computation of the inflationary cosmological constant based on the principle of maximum entropy and the analytical continuation of spacetime dimensions. The significance of this work lies in:

Deriving Initial Conditions: It offers a specific value for $\Lambda$ that can serve as an initial condition for inflationary models, specifically setting the initial Hubble parameter as $H(0) = 1/\ell_0$ .
Unifying Principles: It connects the observed dimensionality of spacetime ( $D=4$ ) with thermodynamic maximization, suggesting that the 4-dimensional nature of our universe is a consequence of the universe maximizing its entropy.
Mathematical Framework: It utilizes the concept of negative and continuous dimensions, a technique common in quantum field theory regularization and phase transition theories, to solve a cosmological problem.

The author maintains a modest tone, presenting this as a proposed approach to computing the constant rather than a definitive experimental proof, and notes that the discussion pertains to the post-Planck era creation of the universe.

Technical Summary: Cosmological Constant and Maximum of Entropy for de Sitter Space

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