⚛️ general relativity

How greedy is the Universe? An entropy ledger and a causal envelope for early black hole growth

This paper refutes the hypothesis that the Universe maximizes entropy production by demonstrating that while supermassive black holes dominate total entropy, their actual growth rates remain orders of magnitude below the theoretical causal-hydrodynamic limits, indicating that cosmic structure formation is not driven by a maximum-entropy-production principle.

Original authors: Fernando Izaurieta, Cristian Quinzacara, Omar Valdivia

Published 2026-08-04

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Fernando Izaurieta, Cristian Quinzacara, Omar Valdivia

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

The Universe's Great Ledger: A Story of Greedy Algorithms and Cold Giants

Imagine the Universe as a giant, cosmic bank account. In this account, there is a special currency called entropy. You can think of entropy as a measure of "messiness" or "disorder." The Second Law of Thermodynamics, a fundamental rule of physics, says that this messiness can never go down; it can only go up. Every time a star burns, a cup of coffee cools, or a gas cloud collapses, the Universe gets a little bit messier, and the total entropy goes up.

For a long time, scientists have wondered: What is the Universe's favorite way to get messy? Is it the burning of stars? The explosion of supernovas? Or is there a more efficient, perhaps even "greedy," way to maximize this messiness? One popular idea suggested that the Universe might be trying to get as messy as possible as fast as possible, perhaps by turning everything into black holes, which are the messiest objects in existence. This paper asks a simple but profound question: Is the Universe actually trying to maximize its entropy, or is it just taking the path of least resistance, even if that path isn't the most efficient?

The Paper's Big Discovery: The Universe is a "Greedy" Algorithm, Not a Maximizer

This paper, written by Fernando Izaurieta, Cristian Quinzacara, and Omar Valdivia, acts like a cosmic accountant. They set out to calculate exactly how much entropy the Universe has produced over its 13.8-billion-year history and compared that to the maximum amount of entropy it could have produced if it had taken the most efficient route possible.

The result? The Universe is not a master planner trying to break the entropy record. Instead, it behaves like a greedy algorithm. In computer science, a greedy algorithm is a program that makes the best choice it can see right now, without looking ahead to the bigger picture. It optimizes step-by-step, missing out on the "global maximum" (the best possible outcome) because it gets stuck on local, easier options.

Here is what the authors found, broken down into three main discoveries:

1. The Black Hole Dominance (The "Cold" Giants)

First, the authors confirmed that almost all the entropy in the Universe comes from black holes, specifically the supermassive ones sitting at the centers of galaxies.

The Analogy: Imagine two ways to make a mess. One is to burn a log (like a star), which creates smoke and heat. The other is to swallow the log whole into a giant, invisible trash compactor (a black hole).
The Finding: The paper shows that the "trash compactor" method is vastly more efficient at creating entropy. The entropy produced by black hole horizons is 10^16 to 10^17 times larger (that's 10 quadrillion to 100 quadrillion times) than the entropy produced by all the starlight, dust, and radiation in the Universe combined.
Why? It's all about temperature. Entropy is like "energy divided by temperature." Black holes are incredibly cold (near absolute zero), so when they swallow energy, they generate a massive amount of entropy. Stars are hot, so they generate much less. The Universe is dominated by these cold giants, not the hot stars.

2. The "Late Peak" Surprise

If the Universe were trying to maximize entropy as fast as possible, you might expect it to have done so right at the beginning, when everything was chaotic and energetic.

The Finding: The authors found that the Universe's entropy production actually peaked much later, around 10 to 12 billion years ago (at a redshift of z ≈ 1–2), and has been slowing down since.
The Analogy: Think of it like compound interest. You don't make the most money on the day you open a bank account; you make the most money when your account has had time to grow large, and then you add more to it. Black holes are the same. The entropy production rate depends on the mass of the black hole. Since the biggest black holes took billions of years to assemble, the "entropy peak" happened only after the Universe had built up a massive "capital" of black holes. The Universe didn't rush; it waited for the giants to grow.

3. The "Envelope" and the Efficiency Gap

The authors calculated a theoretical "ceiling" or envelope: the fastest possible rate at which black holes could have grown if the Universe had taken the most efficient, "greedy" path allowed by the laws of physics. This path involves gas collapsing directly into black holes without breaking apart to form stars first.

The Reality Check: They compared the actual history of the Universe to this theoretical "perfect" path. The result is shocking: The Universe is operating at only 0.0001% to 0.001% (10^-7 to 10^-4) of its maximum potential efficiency.
The "Greedy" Mistake: Why is it so inefficient? Because the Universe is "greedy" in the wrong way. When a cloud of gas collapses, the local physics usually causes it to break apart and form stars (which is easy and happens often). However, to reach the "global maximum" of entropy, the gas would need to avoid forming stars and collapse directly into a black hole. This requires suppressing the local tendency to fragment. The Universe almost always chooses the local, easy option (stars), missing out on the massive entropy gain of direct black hole formation.
The "Little Red Dots": The paper notes that the few objects that do seem to follow this efficient path (growing fast, hidden in gas, and looking red) match the description of mysterious "little red dots" recently spotted by the James Webb Space Telescope (JWST). These might be the rare exceptions where the Universe actually tried to be "greedy" in the right way.

What the Paper Rules Out

The authors explicitly argue against the idea that the Universe is driven by a "Maximum Entropy Production Principle" (MEPP)—the idea that nature has a goal to create as much entropy as possible as fast as possible.

They show that if the Universe were following this principle, it would have collapsed into black holes immediately, creating a huge amount of entropy right at the start.
Instead, the Universe took a slow, step-by-step path, forming stars and waiting for black holes to grow.
They also rule out the idea that "dissipative adaptation" (a theory suggesting systems evolve to dissipate energy faster) explains this, because the conditions required for that theory don't apply to the formation of galaxies.

The Bottom Line

The Universe is not a strategic planner trying to break the entropy record. It is a greedy algorithm that takes the easiest path available at every step. It chooses to form stars because that's what gas clouds do naturally, even though this choice leaves 16 orders of magnitude of potential entropy unrealized. The Universe produces entropy overwhelmingly through black holes, but it does so with a tiny fraction of the efficiency it could have achieved.

In short: The Universe is messy, but it's not maximally messy. It's just messy enough, and it got there by taking the scenic route, not the express lane.

Technical Summary: "How greedy is the Universe? An entropy ledger and a causal envelope for early black hole growth"

Problem Statement
The observable Universe's entropy budget is overwhelmingly dominated by the horizons of supermassive black holes (SMBHs), exceeding the entropy of photons and neutrinos by approximately 15 orders of magnitude. This disparity has led to the suggestion that gravitational collapse into black holes is the Universe's preferred channel for entropy production, potentially driven by a Maximum Entropy Production Principle (MEPP). The paper investigates whether the Universe actually maximizes entropy production or if this dominance is merely a consequence of the second law of thermodynamics acting on a specific dynamical history. The central question is whether the Universe's structure formation behaves as a "greedy" algorithm that optimizes globally for entropy, or if it follows local dynamical constraints that result in a sub-optimal global entropy yield.

Methodology
The authors construct a quantitative framework consisting of three primary components:

The Entropy Ledger: A calculation of the cosmic entropy production rate, $dS/dt(z)$, resolved by channel.
- Black Hole Channel: Normalized via the Soltan argument to match the local SMBH mass density ( $\rho_{BH,0}$ ). The horizon entropy production rate is calculated using the Bekenstein–Hawking formula, $S \propto M^2$ , weighted by the accretion rate and the mass of the accretor ( $\dot{S}_{BH} \propto M\dot{M}$ ).
- Radiative Channels: Calculated using the Madau–Dickinson star formation history, accounting for starlight reprocessed by dust and AGN light.
- Parameters: The analysis uses Planck 2018 cosmological parameters and includes a "Little Red Dot" (LRD) bracket to account for potential JWST-era revisions suggesting higher accretion densities at $z \gtrsim 5$ .
The Causal-Hydrodynamic Envelope: A theoretical upper bound on entropy production defined by causality and gas dynamics, independent of thermodynamic selection.
- This envelope assumes every atomic-cooling halo channels its gas into a single black hole at the maximum sustained rate allowed by gravity ( $\dot{M} \sim c_s^3/G$ ), without radiative throttling (Eddington limit) or fragmentation into stars.
- This trajectory represents "maximal greed," where local dissipation is suppressed to allow monolithic collapse.
Efficiency Metric ( $F(z)$ ): A dimensionless ratio defined as the realized entropy production divided by the envelope entropy production. This quantifies how closely the Universe approaches its theoretical maximum.

Key Results

Dominance of the Horizon Channel: After the formation of the first black hole seeds, the horizon entropy production rate exceeds all radiative channels (starlight, dust, neutrinos) by a factor of $\sim 10^{16}–10^{17}$ . This confirms that the entropy budget is dominated by black holes, but the rate of production is the critical variable.
Temporal Structure of Production: The total entropy production rate peaks at redshift $z \approx 1–2$ (cosmic noon) and has declined since. This late peak is structurally determined by the weighting of entropy production by the mass of the accretor ( $\dot{S} \propto M\dot{M}$ ). Because the most massive black holes are assembled last via hierarchical merging, the peak in entropy production occurs after the peak in accretion rate. A Universe maximizing entropy would have front-loaded production (peaking at cosmic dawn), which is not observed.
Low Efficiency ( $F(z)$ ): The realized entropy production remains far below the causal-hydrodynamic envelope. The efficiency $F(z)$ ranges between $10^{-7}$ and $10^{-4}$ for $4 \leq z \leq 12$ . Even under the most generous "LRD bracket" assumptions (enhancing early accretion by a factor of 30), the Universe operates four to seven orders of magnitude below its maximal potential.
Phenomenological Consistency: The envelope trajectory (monolithic collapse in atomic-cooling halos with suppressed cooling) predicts an object that is obscured, radiatively inefficient, X-ray weak, and red. This closely matches the phenomenology of JWST "little red dots," suggesting these objects may be the rare instances where the Universe approaches the envelope, though they still fall short of it.

No-Go Results for Entropic Teleology
The paper presents three quantitative refutations of the hypothesis that structure formation is driven by a principle of maximum entropy production:

Boltzmann Weighting: If channel selection were weighted by entropy gain ( $P \propto e^{\Delta S}$ ), the Universe would have collapsed entirely into black holes immediately. The fact that gas fragments into stars in nearly every case (forgoing a factor of $\sim 10^{16}$ in entropy per episode) refutes this.
Dissipative Adaptation: Theorems regarding dissipative adaptation (e.g., England's work) rely on conditions (heat baths, local detailed balance) that do not apply to the macrostates of protogalactic clouds. There is no mechanism for self-gravitating systems to "choose" the global entropy maximum over local fragmentation.
Strong MEPP: A strong Maximum Entropy Production Principle fails because the dynamics of structure formation (local field theories) do not possess the capability to compare local optima against a global maximum. The Universe behaves like a greedy algorithm: it optimizes locally (fragmenting into stars) without access to the global maximum (monolithic collapse).

Significance and Conclusions
The paper concludes that while the Universe produces entropy overwhelmingly through black holes, it does so with extremely low efficiency relative to the limits imposed by causality and hydrodynamics. The "greed" of the Universe is local, not global.

Thermodynamic Role: The second law of thermodynamics acts as an auditor, recording the entropy of every process, but it does not act as a strategist determining the dynamical path. The observed history is a consequence of local physical processes (cooling, fragmentation, angular momentum transport) rather than a thermodynamic drive toward a global maximum.
The "Greedy Algorithm": Structure formation is characterized as a greedy algorithm that systematically forgoes the global entropy maximum (which requires suppressing local dissipation) in favor of the locally accessible channel (star formation).
Implications for JWST: The consistency between the envelope and "little red dots" suggests that cosmic dawn may have operated closer to the envelope (by a factor of $\sim 30$ ) than pre-JWST estimates, yet it remains orders of magnitude below the theoretical limit.

The paper asserts that the smallness of $F(z)$ and the late timing of the entropy peak are robust, order-of-magnitude results that rule out entropic teleologies as the primary driver of cosmic structure. The Universe begins far below its causally available entropy ceiling and approaches it only weakly, a fact that must be explained by dynamical selection effects rather than thermodynamic optimization.