Exponential vs Gaussian: A Guide to Not Being Fooled By Dense Gas

When The Universe Lies To Your Instruments

In January 2026, a team of astronomers published a paper that fundamentally changed our understanding of mysterious objects discovered by the James Webb Space Telescope.

They did not discover new physics.
They did not propose exotic matter.
They did not invoke phenomena beyond our current theories.

They simply looked more carefully at the shape of a spectral line.

And realised everyone had been measuring the wrong thing.

The Problem With Assumptions

When astronomers observe broad emission lines in spectra, the standard interpretation is straightforward: broad lines mean fast-moving gas. The broader the line, the faster the motion. The faster the motion, the more massive the central object holding that gas in orbit.

This works remarkably well in most circumstances. It has been tested across decades of observations. It is how we measure black hole masses throughout the universe.

But it relies on a critical assumption: that line broadening comes primarily from velocity—from Doppler shifts as gas orbits at thousands of kilometres per second.

In the little red dots discovered by JWST at cosmic dawn, that assumption was wrong.

Exponential vs Gaussian: A Difference That Matters

A Gaussian line profile looks like a bell curve. It is what you expect from random velocities—gas moving in all directions with some average speed. Plot it on normal axes, and you see a smooth peak that falls off symmetrically on both sides.

An exponential line profile is different. It falls off much more gradually in the wings. On normal axes, it looks somewhat similar to a Gaussian—similar enough that you might not immediately notice the difference, especially in noisy data.

But plot it on semi-logarithmic axes, and the distinction becomes unmistakable.

A Gaussian curves.
An exponential forms a straight line.

This matters because exponential profiles indicate a completely different physical process: electron scattering through dense ionised gas.

What Electron Scattering Does To Light

Imagine a photon trying to escape from near a black hole, but between the source and your telescope sits an extraordinarily dense cloud of ionised gas—so dense that every photon bounces off electrons thousands of times before finally escaping.

Each bounce changes the photon’s direction slightly. Each scatter adds a small random shift to the photon’s wavelength. By the time the light reaches your detector, it has taken a chaotic, pinball-like path through the gas cloud.

The result is line broadening—but not from velocity.

From scattering.

The photons were never moving at thousands of kilometres per second. They were just bouncing around inside a very dense medium before finally reaching us.

The Measurement That Changed Everything

Vasily Rusakov, Darach Watson, and their colleagues examined the highest-quality JWST spectra of little red dots and did something surprisingly simple: they plotted the emission line profiles on semi-logarithmic scales.

The lines formed straight lines over several orders of magnitude.

Textbook electron scattering.

This meant the broad emission lines that everyone had been interpreting as high-velocity gas orbiting billion-solar-mass black holes were actually narrow emission lines from much smaller black holes, broadened by scattering through Compton-thick ionised gas cocoons.

The black hole mass estimates dropped by a factor of one hundred.

Not because the black holes changed.

Because the measurements had been fooled by dense gas all along.

Why This Matters Beyond Astronomy

This discovery is not just about correcting black hole masses in the early universe. It is about epistemic humility—the recognition that our instruments measure what they measure, not necessarily what we think they measure.

Broad spectral lines usually mean velocity. Usually. But in extreme environments with dense ionised gas, they can mean electron scattering instead. The difference is subtle in appearance but fundamental in interpretation.

For years, astronomers looked at these objects and saw impossible black holes that violated formation theory. The exotic explanations proliferated: maybe black holes formed differently in the early universe, maybe they grew faster than theory allowed, maybe new physics was required.

The actual explanation was simpler and more humbling: we were being fooled by a very effective disguise made of ionised gas.

The Shape Of A Line Contains A Story

Spectroscopy is forensic work. Every photon carries information about the environment it passed through. The width of a line, the shape of its wings, the presence or absence of asymmetry—all of these contain clues about physical conditions billions of light-years away.

But interpreting those clues requires careful attention to which physical processes produce which signatures.

Gaussian profiles: velocity dispersion, thermal broadening, turbulence.
Exponential profiles: electron scattering through optically-thick gas.
Lorentzian profiles: natural line broadening, collisional broadening, certain types of turbulence.

Each has a characteristic shape. Each tells a different story about what happened to the light before it reached us.

What Was Hiding In The Early Universe

The little red dots are now understood to be young supermassive black holes—not billions of solar masses, but hundreds of thousands to millions—wrapped in extraordinarily dense cocoons of ionised gas.

These cocoons are only light-days across but contain enough material to scatter every photon multiple times before it escapes. The gas is so dense that electron column densities reach 10²⁴ particles per square centimetre. The black holes are accreting near the Eddington limit, radiating tremendous luminosities whilst simultaneously being obscured by the very material that fuels them.

This represents an entirely new phase of black hole evolution that had been hiding for 12 billion years behind what amounted to a very effective disguise.

And we only found it by looking more carefully at the shape of a spectral line and asking: is this really telling us what we think it’s telling us?

The Management Lesson

The Square-Haired Boss would like to remind employees that sometimes the most important question is not “what are we measuring?” but “what process produced the measurement we’re seeing?”

Assumptions work beautifully until they don’t. Standard interpretations serve us well in most cases but can fail catastrophically in extreme environments. And occasionally, revolutionary discoveries emerge not from dramatic new observations but from asking whether everyone’s been interpreting familiar data correctly.

Before proposing exotic new physics, consider: are we being fooled by dense gas?

It is a more common problem than anyone expected.