On the Energy Distribution of the Galactic Center Excess' Sources

By employing a neural network simulation-based inference approach to jointly analyze spatial and spectral data, this study demonstrates that the Galactic Center Excess is consistent with a diffuse dark matter signal or an exceptionally large population of dim point sources, challenging earlier conclusions that favored a smaller number of brighter sources.

The Gist

Imagine the center of our galaxy, the Milky Way, is like a bustling city at night. Astronomers have been staring at this "city" with a powerful telescope called Fermi, looking for a specific kind of glow. They found a mysterious, bright haze of gamma-ray light right in the middle. This is called the Galactic Center Excess (GCE).

For years, scientists have argued about what causes this glow. There are two main suspects:

1. Dark Matter: The invisible stuff that makes up most of the universe. If it exists, it might be colliding with itself and creating this glow. This would look like a smooth, even fog.
2. Millisecond Pulsars: These are tiny, spinning dead stars (like lighthouses) that are too small to see individually but are so numerous that, together, they create a glow. This would look like a fog made of billions of tiny, distinct dots.

The Old Way of Looking

Previously, scientists tried to solve this mystery by looking at the shape of the glow on a map. They asked, "Is this smooth fog, or is it a collection of dots?"

• The Problem: Their tools were like trying to identify a crowd of people in a foggy room by only looking at the shadows on the wall. They had to ignore the color of the light (the energy) because their math was too complicated to handle both shape and color at the same time.
• The Old Conclusion: Because they ignored the color, earlier studies suggested the glow was made of thousands of distinct, relatively bright "lighthouses" (pulsars).

The New Approach: A Smart AI Detective

In this paper, the authors introduce a new tool: a Neural Network (a type of artificial intelligence). Think of this AI as a detective who can look at the city map and the color of every single light simultaneously.

Instead of just looking at the shape, the AI analyzes the energy spectrum (the "color" or "temperature" of the light). The authors trained this AI on millions of simulated galaxies to learn the difference between a smooth fog and a crowd of tiny dots, paying close attention to how their colors differ.

The Big Discovery

When the AI analyzed the real data from the Fermi telescope, it found something surprising:

1. The "Lighthouses" are incredibly dim: When the AI included the energy information, it realized that if the glow is made of individual stars (pulsars), those stars must be extremely faint.
2. Too many to count: To create the glow we see with such faint stars, you would need tens of thousands of them (around 35,000 to 200,000). This is far more than the few hundred suggested by the old methods.
3. The "Fog" is the winner: Because the stars are so incredibly faint, they blur together so perfectly that they look exactly like a smooth fog. In fact, the AI found that the data is almost indistinguishable from a smooth fog (Poisson emission).

The Analogy: The Rain vs. The Sprinkler

Imagine you are looking at a wet sidewalk.

• The Old View: You thought the wetness came from a few powerful sprinklers (bright pulsars) spraying water.
• The New View: The AI looked at the size of the water droplets (energy). It realized that if it was sprinklers, they would have to be so weak and so numerous that they are basically just a fine mist.
• The Conclusion: At that level of faintness, you can't tell the difference between a million tiny sprinklers and a single, smooth cloud of mist. The data fits the "smooth cloud" (Dark Matter) just as well as, or better than, the "million sprinklers."

What This Means

The paper doesn't say, "We found Dark Matter." Instead, it says:

• The evidence that the glow is made of distinct, bright stars is much weaker than we thought.
• If the glow is made of stars, there are so many of them that they act exactly like a smooth fog.
• This makes the Dark Matter explanation (the smooth fog) a very strong contender again, because the "star" explanation requires an absurdly large number of incredibly faint stars to work.

The authors also warn that their results depend heavily on how well we understand the background "noise" of the galaxy. If our map of the background is slightly wrong, the number of required stars could change. But the key takeaway is that adding energy information to the analysis changes the story completely, pushing the "star" theory toward a level where it looks just like the "Dark Matter" theory.

Technical Summary

Technical Summary: On the Energy Distribution of the Galactic Center Excess' Sources

Problem Statement
The Galactic Center Excess (GCE), an observed surplus of gamma-ray emission from the inner Milky Way, remains a subject of intense debate regarding its origin. While some analyses suggest it could be evidence for annihilating dark matter (DM), others argue it arises from a population of unresolved astrophysical point sources, such as millisecond pulsars. Previous attempts to distinguish between these scenarios using Non-Poissonian Template Fitting (NPTF) and similar likelihood-based methods have faced significant limitations. These methods typically rely on two major approximations to ensure computational tractability: (1) neglecting pixel-to-pixel correlations in the photon count map, and (2) discarding spectral (energy) information. Furthermore, earlier machine learning approaches, specifically those utilizing Convolutional Neural Networks (CNNs), successfully addressed the first approximation but retained the second, analyzing data in a single energy bin. This lack of spectral information limits the ability to disentangle the GCE from astrophysical backgrounds, as the GCE possesses a distinct energy spectrum compared to dominant diffuse backgrounds.

Methodology
The authors present a simulation-based inference framework that extends previous CNN methodologies to jointly analyze spatial and spectral data. The core of their approach involves two independent neural networks trained on simulated Fermi-LAT maps:

1. Data Generation: The authors generate one million simulated Fermi maps using a region of interest within 25° of the Galactic Center (excluding the plane and bright sources). Crucially, the data is divided into ten logarithmically spaced energy bins (2–20 GeV). The simulations incorporate energy-dependent templates for diffuse emission (π⁰ + bremsstrahlung, inverse Compton), isotropic emission, Fermi bubbles, and point-source components (GCE and Galactic disk). The point-source components are modeled using a generalized NFW profile for the GCE and a double-exponential for the disk.
2. CNN Architecture: The framework utilizes the DeepSphere library, which applies graph-based convolutions on HEALPix tessellations.
   • Input: Photon-count maps with shape $n_{pix} \times n_{bins}$, where energy bins act as channels analogous to color channels in standard CNNs.
   • Network 1 (Flux Fraction): Estimates the fractional contribution of each template to the total flux in each energy bin. It is trained using a negative log-likelihood loss with a Gaussian distribution to estimate uncertainties.
   • Network 2 (Source-Count Distribution): Infers the underlying brightness distribution (Source-Count Distribution, SCD, or $dN/dF$) for the point-source components. This network employs a simulation-based inference approach rooted in quantile regression. It predicts binned histograms of the cumulative flux distribution using the "Earth mover's pinball loss," allowing for the reconstruction of the full posterior distribution across flux bins.
3. Training Strategy: The networks are trained on mock data where priors are chosen to ensure the inclusion of maps where point sources are so dim they are indistinguishable from Poisson emission (the DM hypothesis). This addresses the physical degeneracy where an infinite number of infinitely dim sources is mathematically equivalent to Poisson emission.

Key Contributions

• First Energy-Dependent Analysis: This work provides the first simultaneous inference of the GCE spectrum and its Source-Count Distribution (SCD) using a machine learning approach that explicitly incorporates energy information.
• Removal of Spectral Approximation: By treating energy bins as input channels and simulating energy-dependent instrument responses (specifically the Point Spread Function), the method moves beyond the single-bin approximation that characterized previous likelihood-free analyses.
• Robustness Testing: The authors perform extensive systematic studies, varying diffuse background models (comparing Model O and Model A), GCE spatial morphologies (NFW, Einasto, Burkert), and CNN training priors (modality, brightness, spectral jitter) to assess the stability of their inferences.

Results

• Dimmer Source Population: The inclusion of energy information drives the inferred SCD to be significantly dimmer than results from previous likelihood-based studies (e.g., NPTF 2016) and even previous energy-independent CNN analyses. The median prediction for the GCE SCD peaks at fluxes where sources produce, on average, less than one photon.
• Consistency with Poisson Emission: Quantitatively, for the best-fit background model (Model O), the excess is essentially consistent with Poisson emission. At 95% confidence, the network can only exclude 3% of the emission as being inconsistent with Poisson statistics. This suggests that if the GCE is composed of point sources, there must be an exceptionally large number of them (median prediction $\mathcal{O}(10^5)$, or >35,000 at 90% confidence), rather than the hundreds preferred by earlier analyses.
• Spectral Recovery: The CNN recovers a spectrum for the GCE consistent with earlier studies, but with significantly reduced bin-to-bin variability compared to Poisson likelihood fits, reflecting the correlation across energy bins learned during training.
• Systematic Sensitivity: The results are robust to variations in the disk template and GCE morphology (within reasonable bounds). However, the results are sensitive to the choice of the diffuse background model. When trained on "Model A" (a model that fits the data worse than Model O), the inferred SCD becomes brighter (closer to the 1-photon threshold), though it remains dimmer than earlier likelihood results. This highlights that the background diffuse model remains the primary systematic uncertainty.

Significance and Claims
The paper claims that the addition of energy information acts as a powerful lever arm, shifting the interpretation of the GCE away from a population of resolvable point sources toward a scenario consistent with either a truly diffuse origin (such as dark matter annihilation) or a population of sources so numerous and dim that they are statistically indistinguishable from Poisson noise.

The authors emphasize that while their method undermines one of the key pieces of evidence for a point-source origin (the brightness of the sources), the conclusion remains subject to the caveat of background systematics. They state that their findings do not definitively prove the DM origin but rather demonstrate that the point-source hypothesis requires a much larger, dimmer population than previously thought to be viable. The work serves as a benchmark for future machine learning applications in gamma-ray astronomy, suggesting that further improvements in handling background systematics and expanding the energy range are necessary to definitively resolve the nature of the GCE.