⚛️ high-energy theory

The Universe Originating from an Empty Planck-Size Torus

This paper proposes that a matter-free, three-torus universe driven by Casimir energy can evolve from a Planck-scale origin to a present size consistent with cosmic microwave background anomalies, provided specific parameters regarding particle content, inflation, and reheating are met.

Original authors: Bartosz Fornal

Published 2026-06-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bartosz Fornal

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 entire Universe not as an endless, flat sheet, but as a giant, invisible donut (or more accurately, a 3D donut shape called a "three-torus"). If you walked far enough in any direction in this Universe, you wouldn't hit a wall; you would simply loop back to where you started, just like in the video game Pac-Man.

This paper, written by physicist Bartosz Fornal, asks a big question: How did this donut-shaped Universe begin, and how big is it today?

Here is the story the paper tells, broken down into simple steps:

1. The Empty Start: A Universe with Nothing in it

Usually, we think the Big Bang started with a hot, dense soup of particles. But this paper imagines a different beginning.

The Scenario: At the very beginning (the "Planck time," which is the tiniest fraction of a second imaginable), the Universe was completely empty. No stars, no gas, no radiation. Just empty space shaped like a tiny donut.
The Engine: If there's nothing in it, what makes it expand? The paper suggests the engine is "Casimir Energy."
- The Analogy: Imagine two metal plates floating in a vacuum. Even in "empty" space, quantum physics says there are tiny, invisible waves buzzing around. If the plates are close together, some waves can't fit between them, creating a pressure difference that pushes the plates together. This is the Casimir effect.
- In this tiny, donut-shaped Universe, the "walls" of the donut trap these quantum waves. The pressure from these trapped waves acts like a fuel, pushing the Universe to expand, even though there is no matter inside it.

2. The Growth Spurt: From a Grain of Sand to a Room

The paper calculates how this empty donut grows using only that quantum pressure.

The Size: It starts at the size of a Planck length (about $10^{-35}$ meters—so small it's hard to even imagine).
The Expansion: Because the "fuel" (Casimir energy) behaves like radiation, the Universe expands steadily. The paper notes that the final size depends on a "tug-of-war" between two types of quantum particles: fermions (like electrons) and bosons (like photons).
- If the number of fermions and bosons is different (which it is in our known physics), the Universe grows to a specific size.
- The paper assumes a standard difference found in physics, which sets the starting size to be about two times the Planck length.

3. The Big Inflation: The Balloon Blows Up

Once the Universe gets big enough, a process called Inflation kicks in.

Think of this as a sudden, explosive blow to a balloon. The Universe expands exponentially fast, stretching from a microscopic speck to something the size of a room (about 35 cm) in a fraction of a second.
After inflation stops, the energy gets converted into the hot soup of particles we know as the Big Bang (a phase called "Reheating").

4. The Journey to Today: How Big is the Donut Now?

The paper tracks the Universe's growth through time:

Inflation: Expands it to the size of a room.
Reheating: The energy drops, and the Universe grows to the size of a large building (about 16 meters).
Radiation & Matter Eras: It keeps growing, eventually reaching the size of our solar system, then our galaxy, and finally the size of the observable Universe we see today.

The author calculates that if you start with that tiny, empty donut and use standard physics for inflation, the Universe today should be roughly 13 billion light-years across (or about 1.3 × 10²⁷ meters).

5. The "Goldilocks" Zone: Checking the Evidence

The paper doesn't just guess; it checks this idea against real data from the Cosmic Microwave Background (CMB), which is the "afterglow" of the Big Bang.

The Lower Limit (The "Too Small" Problem): If the Universe were too small, we would see repeating patterns in the sky (like seeing the same star in multiple directions). The Planck satellite hasn't seen these patterns yet, so the Universe must be at least a certain size.
The Upper Limit (The "Too Big" Problem): The CMB data shows a weird "anomaly" at the largest scales (low multipole moments). The temperature fluctuations are weaker than expected. The paper argues that if the Universe is a donut of a specific size, it naturally cuts off the very longest waves, explaining this weakness.
The Match: The paper finds a "sweet spot." If the Universe is a donut with a size between 0.8 and 1.5 times the current Hubble radius, it fits both the lack of repeating patterns and the strange weakness in the CMB data.

The Big Conclusion

The paper concludes that it is entirely possible for our Universe to have started as a tiny, empty, donut-shaped speck at the Planck time, powered solely by quantum vacuum energy.

The Prediction: This scenario predicts that the Universe today is just slightly larger than the "Hubble radius" (the size of the observable universe), which aligns perfectly with the strange "anomaly" seen in the CMB data.
The Takeaway: We might live in a finite, looping Universe that started with nothing but quantum pressure, and the specific size of that "donut" explains why the sky looks the way it does today.

In short: The Universe might be a cosmic donut that started empty, grew on quantum fumes, and is currently the "just right" size to explain the weird patterns we see in the oldest light in the sky.

Technical Summary: The Universe Originating from an Empty Planck-Size Torus

Problem Statement
The topology of the Universe remains an open question in modern cosmology. While observations constrain the geometry of spacetime, the global topology—specifically whether the Universe is compact and finite or infinite—is unknown. The Cosmic Microwave Background (CMB) offers the primary testing ground for these questions, yet searches for repeated patterns (e.g., circles in the sky) have yielded no strong conclusions. Furthermore, the observed suppression of power at low multipole moments in the CMB spectrum hints at a finite, compact topology but lacks a definitive theoretical origin. This paper investigates a specific scenario: a Universe with a three-torus ( $T^3$ ) topology that begins its evolution at the Planck time, devoid of matter or radiation, with its dynamics driven solely by Casimir energies.

Methodology
The authors model the spacetime metric of a three-torus using a volume modulus ( $b$ ) and five shape moduli ( $\Phi_T$ ). The evolution is analyzed through the following steps:

Casimir Energy Dynamics: In a compact $T^3$ topology, fields contribute to the energy density via Casimir energies. The authors utilize a regularized formula for these energies in the limit of massless fields ( $m_\alpha \to 0$ ). They demonstrate that the Casimir energy density scales as $\rho_{Cas} \propto 1/b^4$ , mimicking a radiation-like equation of state.
Moduli Stabilization: The equations of motion for the volume and shape moduli are derived from the action. The authors argue that Hubble friction damps the evolution of shape moduli. By selecting the global minimum of the Casimir energy in the shape moduli space (a specific symmetric configuration), the shape moduli remain constant, reducing the problem to the evolution of the volume modulus $b(t)$ .
Pre-Inflationary Evolution: Assuming the Universe starts with a size determined by the Planck length and the difference between fermionic ( $n_F$ ) and bosonic ( $n_B$ ) degrees of freedom, the expansion is driven purely by Casimir energy. The scale factor evolves as $b(t) \propto \sqrt{t}$ .
Inflation and Reheating: The model incorporates a standard inflationary epoch followed by reheating. The authors treat the number of e-folds ( $N$ ) and the energy density drop during reheating ( $D = \rho_{inf}/\rho_{reh}$ ) as free parameters. They derive a comprehensive relation linking the current size of the Universe, $b(T)$ , to these parameters and the initial degrees of freedom.
Constraints from CMB Data: The derived size $b(T)$ $b (T)$ is constrained against two observational limits:
- Lower Bound: The non-detection of matched circles in the CMB by the Planck satellite implies the Universe must be larger than a certain threshold ( $b(T) \gtrsim 0.8 \times 10^{27}$ m).
- Upper Bound/Hint: The observed suppression of the quadrupole moment (low- $\ell$ anomaly) suggests the Universe's size should be such that the infrared cutoff suppresses specific modes ( $0.6 \times 10^{27} \text{ m} \lesssim b(T) \lesssim 1.5 \times 10^{27}$ m).

Key Contributions and Results

Initial Conditions from Degrees of Freedom: The paper establishes a direct relationship between the size of the Universe at the Planck time and the net number of degrees of freedom ( $n_F - n_B$ ). For the Standard Model value ( $n_F - n_B = 62$ ), the initial size is approximately $2\pi \ell_P$ . For supersymmetric models where $n_F - n_B \approx 1$ , the initial size is closer to $2 \ell_P$ .
Evolutionary Trajectory: The authors trace the evolution from a Planck-size torus through the Casimir-dominated era, inflation, reheating, radiation, matter, and dark energy domination. They show that with natural parameters, a Planck-size origin can result in a Universe currently spanning several Hubble radii.
Parameter Constraints: By matching the derived size $b(T)$ $b (T)$ to the CMB constraints, the authors constrain the parameter space of inflation and reheating.
- For typical reheating energy drops ( $D \sim 10^4 - 10^8$ ), the required number of e-folds is tightly constrained to $N \sim 66.6 - 68.0$ .
- For instantaneous reheating ( $D=1$ ), the required e-folds are $N \in (68.2, 68.8)$ .
- A benchmark scenario with $N = 67.7$ and $D = 10^5$ yields a current universe size of $1.3 \times 10^{27}$ m, which satisfies both the lower bound from circle searches and the upper bound suggested by the low- $\ell$ anomaly.
Robustness: The authors note that the constraints on $N$ are relatively insensitive to the specific value of $n_F - n_B$ ; changing the initial degrees of freedom shifts the allowed region in the $(N, D)$ plane by a negligible amount ( $\Delta N \approx 1$ ).

Significance
The paper claims that a Universe with a three-torus topology, initially empty of matter and radiation, can naturally evolve from a Planck-scale size to its current dimensions solely through Casimir energy dynamics. This scenario provides a theoretical framework where the initial size is determined by the particle content of the theory (specifically the difference between fermionic and bosonic degrees of freedom).

The primary significance lies in the ability to reconcile the "low- $\ell$ anomaly" in the CMB (interpreted as a hint of finite topology) with the non-detection of repeated patterns. The authors demonstrate that there exists a specific, narrow range of inflationary parameters ( $N$ and $D$ ) consistent with both the Planck lower bound and the anomaly-derived upper bound. This suggests that if the Universe has a three-torus topology and began as an empty Planck-size torus, the parameters of inflation and reheating are highly constrained. The work also highlights that the specific metric choice (symmetric vs. rectangular) may influence the precise numerical factors but does not alter the fundamental viability of the scenario.