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PAC in DESI. I. Galaxy Stellar Mass Function into the $10^{6}{\rm M}_{\odot}$ Frontier

This paper presents an improved Photometric Objects Around Cosmic Webs (PAC) method applied to DESI and DECaLS data to derive the galaxy stellar mass function down to $10^{5.3}{\rm M}_{\odot}$ , revealing that red galaxies dominate the low-mass population below $10^{7.6}{\rm M}_{\odot}$ and establishing a pathway for future surveys to probe the entire local galaxy population.

Original authors: Kun Xu (U Penn, Durham, SJTU), Y. P. Jing, Shaun Cole, Carlos S. Frenk, Sownak Bose, Willem Elbers, Wenting Wang, Yirong Wang, Samuel Moore, J. Aguilar, S. Ahlen, D. Bianchi, D. Brooks, T. Claybaugh

Published 2026-04-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kun Xu (U Penn, Durham, SJTU), Y. P. Jing, Shaun Cole, Carlos S. Frenk, Sownak Bose, Willem Elbers, Wenting Wang, Yirong Wang, Samuel Moore, J. Aguilar, S. Ahlen, D. Bianchi, D. Brooks, T. Claybaugh, A. de la Macorra, Arjun Dey, J. E. Forero-Romero, E. Gaztañaga, S. Gontcho A Gontcho, G. Gutierrez, K. Honscheid, M. Ishak, T. Kisner, S. E. Koposov, M. Landriau, L. Le Guillou, R. Miquel, J. Moustakas, C. Poppett, F. Prada, I. Pérez-Ràfols, G. Rossi, E. Sanchez, D. Sprayberry, G. Tarlé, B. A. Weaver, H. Zou

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: Counting the Invisible

Imagine you are trying to count every single person in a massive, dark city. You have two tools:

A High-Resolution Map (Spectroscopy): This gives you the exact address and ID of the people you can see clearly (bright stars/galaxies), but it's expensive and slow, so you can only map the wealthy, bright districts. You miss the poor, dim neighborhoods.
A Deep, Grainy Satellite Photo (Photometry): This covers the whole city, including the dark, poor slums. You can see something is there, but the photo is blurry. You can't tell if a blurry blob is a person, a dog, or a pile of trash, and you don't know exactly where they live in 3D space.

The Problem: For decades, astronomers have tried to count the "poor" (low-mass) galaxies using only the High-Resolution Map. But because the map isn't deep enough, they keep missing the smallest, dimmest galaxies. This leads to a bad census.

The Solution (The PAC Method): This paper introduces a clever trick called PAC (Photometric Objects Around Cosmic Webs). Think of it as using the High-Resolution Map to "tag" the blurry blobs in the Satellite Photo. By seeing how the blurry blobs cluster around the known, mapped galaxies, we can statistically figure out the properties (like mass and color) of the blurry ones without needing to take a perfect photo of every single one.

What They Did: The "DESI" Upgrade

The authors used a new, super-powerful telescope survey called DESI (Dark Energy Spectroscopic Instrument).

The New Map: DESI is like a super-fast scanner that mapped 40 million galaxies (40 times more than previous surveys).
The New Photo: They used the DECaLS survey, which is a very deep, wide-angle photo of the sky.

They combined these two to create a census of galaxies that goes much deeper into the "dark" universe than ever before. They managed to count galaxies as small as 1 million times the mass of our Sun (10⁶ solar masses). Previous methods could only reliably count down to about 100 million solar masses.

The "Aha!" Moment: The Red Dwarf Surprise

When they finally counted these tiny galaxies, they found something shocking.

The Old View: We thought small galaxies were mostly "blue" (young, star-forming, like a bustling construction site).
The New View: Below a certain size, red galaxies (old, dead, like abandoned ghost towns) actually outnumber the blue ones!

The Analogy: Imagine you are counting cars in a city. You expect to see mostly shiny, new sports cars (blue) in the suburbs. But when you look at the tiny, rusted-out cars in the alleyways, you find there are actually more of those old, red sedans than sports cars. The paper found that the "ghost town" galaxies are the most common type of tiny galaxy in the universe.

Why This Matters: The "Dark Matter" Detective

Why do we care about counting tiny, dead galaxies?

Dark Matter: Dark matter is the invisible glue holding galaxies together. Some theories say dark matter is "warm" (moving fast), which would prevent tiny galaxies from forming. Others say it's "cold" (moving slow), allowing thousands of tiny galaxies to form.
- The Analogy: If you find a pile of tiny pebbles (tiny galaxies) in a river, it tells you the water (dark matter) was calm enough to let them settle. If the water was raging, the pebbles would wash away. By counting these tiny galaxies, the authors are testing if our theories about dark matter are correct.
The "Local Void" Problem: Previous surveys were often done in our "neighborhood" (the Local Group), which happens to be a bit empty (a void). It's like trying to count all the trees in a forest by only standing in a clearing; you'd think trees are rare. This paper used a method that averages out these empty spots, giving a true "cosmic average."

The "De-blending" Challenge

There was one tricky part. In the deep photos, some tiny galaxies looked like they were made of multiple pieces, or parts of a big galaxy looked like small galaxies (like seeing a spiral arm and thinking it's a whole new galaxy).

The Fix: The authors had to "mask" (cover up) the areas around big, bright galaxies to make sure they weren't accidentally counting broken pieces of a big galaxy as new, tiny ones. Even after this correction, they confirmed that tiny galaxies are incredibly numerous.

The Bottom Line

This paper is a major leap forward. By combining a new, massive map (DESI) with a deep photo (DECaLS) using a smart statistical trick (PAC), the authors have:

Pushed the boundary: They can now reliably count galaxies 100 times smaller than before.
Found a surprise: Tiny galaxies are mostly red and old, not blue and young.
Set the stage: This method opens the door for future telescopes (like the Roman Space Telescope or LSST) to map the entire population of galaxies in our universe, helping us solve the mysteries of dark matter and how the universe began.

In short: They built a better net to catch the smallest, most elusive fish in the cosmic ocean, and they found that the ocean is teeming with a type of fish we didn't expect to be so common.

1. Problem Statement

The Galaxy Stellar Mass Function (GSMF) is a fundamental observable for understanding galaxy formation, star formation history, and quenching mechanisms. However, measuring the GSMF at the low-mass end ( $M_* < 10^9 M_\odot$ ) using traditional spectroscopic surveys (e.g., SDSS, GAMA) is severely limited by:

Flux Limits: Spectroscopic surveys are typically magnitude-limited, missing faint, low-mass galaxies.
Local Volume Bias: Measurements relying on the "Local Volume" often suffer from cosmic variance, specifically underestimating densities due to the Local Void (LV).
Selection Effects: Spectroscopic samples often have complex selection functions that bias the sample toward specific galaxy types (e.g., missing red, low-mass satellites).

While photometric surveys (e.g., DECaLS) are much deeper, they lack precise redshifts, making it difficult to isolate physical properties like stellar mass without relying on noisy photometric redshifts ( $z_{photo}$ ).

2. Methodology: The Improved PAC Method

The authors apply and significantly enhance the Photometric Objects Around Cosmic webs (PAC) method. This technique cross-correlates a deep photometric sample (pop2) with a spectroscopic tracer sample (pop1) to statistically recover the physical properties of the photometric objects without needing individual redshifts.

Key Improvements over Previous Iterations (Paper I & III):

Elimination of Redshift Bins: The previous method required dividing pop1 into narrow redshift bins, assuming pop2 objects shared the bin's mean redshift. This introduced errors in physical property calculations (luminosity distance, rest-frame colors) at low redshifts. The new method calculates physical properties for the entire photometric sample across the full redshift range ( $0 < z < 1$ ) using interpolation on a set of 44 reference redshifts, removing the need for binning.
New Estimator: A new weighted estimator for the projected cross-correlation function ( $\bar{n}_2 w_p$ ) is introduced. It incorporates weights ( $\bar{S}_2/r_1^2$ ) directly into the pair counting (Landy-Szalay estimator), accounting for the variation in comoving distance and surface density for individual sources rather than bin averages.
Color Cuts: Conservative color cuts in $g-r$ vs. $r-z$ space are applied to the photometric sample to exclude foreground and background galaxies, reducing noise and improving the signal-to-noise ratio of the correlation.
Stellar Mass Limit Determination: A novel method is developed to determine the completeness limit of DECaLS at different redshifts by comparing angular surface densities in deep regions vs. the full sample, rather than relying on external catalogs like GAMA which may be incomplete at low masses.
Bias Correction (Mass & Color): To derive the GSMF, the authors assume galaxy bias is determined by both stellar mass and color. They separate the analysis into blue and red populations. For red galaxies, where the spectroscopic sample (DESI BGS) is incomplete at low masses, they model the correlation function ( $w_p$ ) using a power-law assumption constrained by high-mass data to extrapolate to lower masses.

Data Used:

Spectroscopic Tracers: DESI Year 1 (Y1) Bright Galaxy Survey (BGS) Bright sample ( $z < 0.6$ , $r < 19.5$ ).
Photometric Source: DECam Legacy Survey (DECaLS) Data Release 9 ( $r \sim 23.5$ ).
Cosmology: $\Omega_m = 0.268, \Omega_\Lambda = 0.732, H_0 = 71$ km/s/Mpc.

3. Key Contributions

Methodological Rigor: The paper presents a mathematically rigorous, bin-free version of the PAC method, validated to be accurate to within 3% even at very low redshifts ( $z < 0.1$ ) and small scales ( $r_p \sim 1.25 h^{-1}$ Mpc).
Extension to the $10^6 M_\odot$ Frontier: The study successfully measures the GSMF down to stellar masses of $10^{5.3} M_\odot$ for blue galaxies and $10^{6.3} M_\odot$ for red galaxies, a significant leap from previous limits of $\sim 10^8 M_\odot$ .
Fragmentation Correction: The authors identify and quantify "fake" sources in deep photometric data (e.g., fragmented spiral arms or star-forming regions misidentified as galaxies). They implement a masking strategy around massive galaxies to correct the GSMF, finding that while the fake fraction ( $F_{frag}$ ) is low ( $<5\%$ ) for $M_* > 10^{5.0} M_\odot$ , it rises to $\sim 40\%$ at $M_* \sim 10^{4.6} M_\odot$ .
Red Galaxy Dominance: The study reveals a sharp upturn in the red GSMF at low masses, indicating that red galaxies dominate the population below $10^{7.6} M_\odot$ , a result not captured by previous local volume surveys.

4. Key Results

GSMF Shapes:
- Blue Galaxies: Well-described by a single Schechter function with a low-mass slope $\alpha_{blue} = -1.54 \pm 0.02$ .
- Red Galaxies: Well-described by a double Schechter function with a steep low-mass slope $\alpha_{red} = -2.50 \pm 0.08$ .
- Total Population: Best fit by a triple Schechter function.
Low-Mass Behavior: The red GSMF shows a steep rise below $10^{8.6} M_\odot$ , overtaking the blue population at $M_* < 10^{7.6} M_\odot$ . This suggests dwarf galaxies are predominantly red, challenging current models of "environmental quenching" which predict a less steep upturn.
Comparison with Previous Studies:
- The results agree with GAMA and SDSS at high masses ( $M_* > 10^{10} M_\odot$ ) but diverge significantly at low masses.
- The discrepancy with local volume surveys (GAMA, Local Group) is attributed to Local Void bias (under-densities in the local universe) and the inability of shallow spectroscopic surveys to detect faint red dwarfs.
- After applying a Local Void correction (based on ELUCID simulations), the DESI BGS $V_{max}$ results align with the PAC results down to $10^{7.6} M_\odot$ .
Validation: The PAC-derived GSMF at $z < 0.2$ agrees well with high-redshift measurements from COSMOS-Web ( $0.2 < z_p < 0.8$ ) down to $10^{7.6} M_\odot$ , confirming the robustness of the method and the existence of a high number density of dwarf galaxies.

5. Significance and Future Prospects

Dark Matter Constraints: The number density of galaxies at $M_* \sim 10^6 M_\odot$ is highly sensitive to the nature of dark matter (e.g., Warm Dark Matter models predict a cutoff). The steep slope observed here places strong constraints on dark matter particle mass.
Reionization Physics: The abundance of low-mass galaxies is a key tracer for the epoch of reionization, as these galaxies are the primary sources of ionizing photons.
Dwarf Galaxy Formation: The finding that low-mass dwarfs are predominantly red challenges existing semi-analytic models and suggests a need to re-evaluate quenching mechanisms in low-mass halos.
Stage-IV Surveys: This work demonstrates the power of the PAC method for Stage-IV surveys (LSST, Euclid, Roman). These future surveys, which will be 2–3 magnitudes deeper than DECaLS, will allow the PAC method to probe the entire galaxy population in the local universe, potentially resolving the "missing satellites" problem and refining our understanding of structure formation.

In summary, this paper establishes a new, rigorous standard for measuring the low-mass end of the galaxy stellar mass function, revealing a previously hidden population of red dwarf galaxies and providing critical data for cosmological models.