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Photon (Non)Conservation in the Reduced Speed of Light Approximation and How to (Almost) Fix It

This paper demonstrates that while photon non-conservation errors in the Reduced Speed of Light approximation can be exactly quantified and corrected for simple numerical schemes, such corrections fail to maintain accuracy in semi-realistic cosmic reionization simulations, resulting in significant deviations from full speed-of-light reference models.

Original authors: Nickolay Y. Gnedin

Published 2026-06-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nickolay Y. Gnedin

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 Big Picture: A Cosmic Traffic Jam

Imagine trying to simulate how light travels through the universe to ionize (strip electrons from) gas clouds. In reality, light moves incredibly fast—the speed of light ( $c$ ). However, computers are slow. If a computer tries to calculate light moving at its true speed across a digital grid, it has to take tiny, tiny steps (like a snail crossing a highway) to avoid making mistakes. This would take millions of years to run a single simulation.

To fix this, scientists use a trick called the "Reduced Speed of Light" (RSL) approximation. They tell the computer, "Let's pretend light moves 10 or 100 times slower than it actually does." This makes the simulation run much faster, like a car driving on a local road instead of a highway.

The Problem: The Missing Photons

The paper argues that while this trick saves time, it breaks the rules of physics in a specific way: it loses photons.

Think of it like a water hose filling a bucket.

Real Life: The hose sprays water at full pressure. The bucket fills up exactly as much as the water source provides.
The RSL Trick: You turn the nozzle down so the water trickles out slowly. The computer simulates this slow trickle.
The Glitch: Because the water is trickling so slowly in the simulation, the computer thinks the bucket is filling up slower than it should. By the time the simulation "catches up" to the real timeline, the bucket is actually empty compared to reality. The photons (water) haven't disappeared; the computer just forgot to count them because they were moving too slowly in the math.

The author, Nickolay Gnedin, shows that in two major recent universe simulations (called CROC and THESAN), this "missing water" is likely why they got different results. Even though they used similar physics, the way they handled the "slow light" trick caused their universes to look different.

The First Fix: Counting the Missing Drops

The author first asks: "Can we at least count how many photons we lost?"

He says yes. By looking closely at the math the computer uses, he can calculate exactly how many photons were "dropped" by the slow-speed approximation.

The Analogy: Imagine you are tracking a delivery truck that is driving 10 mph instead of 60 mph. You know exactly how many packages it should have delivered by noon. You can calculate that it is currently 50 packages short.
The Result: In simple, controlled tests (a box with random light sources), the author successfully counted these missing photons. When he added the count back in, the simulation was accurate to within 1%.

The Second Fix: Putting the Drops Back

Counting the missing photons is easy; putting them back where they belong is the hard part.

The author tries to fix the simulation by adding the missing photons back in as a "background" glow.

The Analogy: Imagine you realize your slow truck is 50 packages short. You decide to just drop 50 packages randomly on the ground to make up the difference.
The Problem: In the real world, those packages should have been delivered to specific houses. In the simulation, because the "light" is moving slowly, the missing photons are actually far away from the source. But the computer doesn't know exactly where they are. So, the author has to guess, treating them like a fog or a background glow that fills the empty spaces.

The Results: Good News and Bad News

The author tests this "fix" in two scenarios:

Simple Tests: In a uniform box of gas, the fix works beautifully. The simulation becomes almost perfect (sub-percent error).
Realistic Simulations: When he tries this on a more realistic model of the early universe (cosmic reionization), the fix fails to save the day.
- The "fog" of missing photons doesn't land in the right spots.
- The simulation still shows the universe becoming ionized (clearing up) too quickly.
- When comparing specific points in the simulation to a "perfect" (but very expensive) full-speed-of-light model, the errors are huge—sometimes off by 10% to 100% in empty spaces.

The Conclusion

The paper concludes with a sobering message:

We know the "Reduced Speed of Light" trick causes photons to go missing.
We can count exactly how many are missing.
But, we currently have no good way to put them back in the right places during a complex simulation.

The author isn't offering a final solution. Instead, he is raising a red flag to the scientific community. He is saying, "We think we know how to speed up these simulations, but we are losing accuracy in a way we can't fully fix yet. We need to work together to find a better way, or our models of the early universe might be fundamentally flawed."

Note on the "Acknowledgments": The author humorously admits in the final section that he used an AI (ChatGPT) to help overcome writer's block and write the first two paragraphs of the introduction, highlighting that even the author of this technical paper relies on modern tools to get started.

Technical Summary: Photon (Non)Conservation in the Reduced Speed of Light Approximation

Problem Statement
The "Reduced Speed of Light" (RSL) approximation is a standard technique in cosmological simulations used to accelerate radiative transfer (RT) calculations by replacing the true speed of light ( $c$ ) with a reduced value ( $\hat{c} \ll c$ ). While necessary for computational feasibility, this approximation introduces a fundamental flaw: it fails to conserve photons when the radiation field evolves rapidly in time.

This non-conservation manifests differently depending on the specific formulation of the RSL approximation used in the literature:

Flavor A: Sets the time derivative coefficient to $\hat{c}$ but keeps the source term coefficient at $c$ . This preserves the correct steady-state solution but leads to a severe undercount of photons in optically thin, rapidly evolving regimes.
Flavor B: Sets both coefficients to $\hat{c}$ . This appears to conserve photon numbers in the optically thin limit but distorts the steady-state solution, effectively treating a "photon" as only a fraction ( $\hat{c}/c$ ) of a real photon, thereby undercounting ionization rates.

The author notes that these conservation errors likely explain discrepancies between recent "second-generation" reionization simulations (e.g., CROC vs. THESAN), where radiation sources are similar, but the resulting radiation fields differ significantly in their power spectra and timing.

Methodology
The paper utilizes the ALTAIR simulation framework (currently in development) to investigate these issues. The study employs a semi-implicit numerical scheme based on the Variable Eddington Tensor (VET) class of models, specifically testing three variants:

Explicit scheme (CFL = 0.25).
Half-step semi-implicit scheme (CFL = 0.5).
Full-step semi-implicit scheme (CFL = 0.75).

The methodology proceeds in two stages:

Quantification: The author derives an exact analytical expression for the number of "missing" photons ( $\Lambda_{RSL}$ ) based on the discretized numerical scheme. This allows for the precise calculation of the photon deficit without running a full-speed-of-light simulation.
Restoration: The paper proposes a method to restore these missing photons by treating them as a spatially uniform "background" component ( $N_{BG}$ ). This approach, inspired by the CROC simulations, solves an ordinary differential equation (ODE) for the mean background density and couples it to the local RT equations via an ansatz ( $N_{BG}(\vec{x}, t) = \bar{N}_{BG}(t)f_0(\vec{x}, t)$ ). Two schemes are tested for this restoration: a "4-equation scheme" and a computationally cheaper "1-equation scheme."

Key Results

Simple Tests (Uniform Density): In tests with a uniform density box and randomly distributed sources, the missing photons can be counted exactly. When the calculated deficit is added back to the simulation (either as a global correction or via the background ansatz), photon number conservation is restored to sub-percent accuracy. This holds true for both constant and variable reduced speed of light implementations.
Semi-Realistic Simulations (Cosmic Reionization): When applied to semi-realistic reionization simulations (20 $h^{-1}$ $h^{- 1}$ cMpc box, 256 $^3$ $^{3}$ particles), the results are less favorable.
- While the global ionization history (volume- and mass-weighted neutral fractions) is reproduced reasonably well, the overlap of ionized regions occurs faster in the RSL runs compared to the full-speed-of-light reference.
- Point-wise errors remain significant. In voids during the overlap epoch, errors reach and exceed an order of magnitude. Even at later times, errors in low-radiation regions (where the background component dominates) remain around 0.1 dex (25%).
- The "1-equation scheme" performs noticeably worse than the "4-equation scheme" in these realistic scenarios.

Significance and Claims
The paper does not claim to have solved the RSL photon conservation problem entirely. Instead, its primary significance lies in:

Explicit Demonstration: Providing a rigorous derivation showing exactly how and why photon non-conservation occurs in RSL approximations and quantifying the missing photons for specific numerical schemes.
Partial Mitigation: Demonstrating that while one can count missing photons exactly, adding them back into a simulation is a much harder task. The proposed "background" ansatz improves global statistics but fails to correct local point-wise errors in realistic, inhomogeneous environments.
Community Call to Action: The author concludes that the lack of photon conservation is a likely culprit for discrepancies between major reionization simulations. The paper aims to underscore the problem and encourage the community to develop more robust solutions, noting that a true "golden solution" may require larger, full-speed-of-light simulations that are currently computationally prohibitive.

The work builds on prior findings by Cain (2024) and Ocvirk et al. (2019) but extends them by providing a practical numerical framework for counting and attempting to restore the missing photons, ultimately highlighting the limitations of current ad-hoc fixes.