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The Revised Evolutionary Volume Tolman Test: Cosmological Constraints from Galaxy Evolution

This paper proposes a revised "evolutionary volume Tolman test" that integrates modern measurements of galaxy evolution, such as star formation and merger histories, to constrain cosmological parameters and dark energy properties, demonstrating that this classic method can become competitive with supernova and CMB observations once astrophysical uncertainties are reduced to the 1–10% level by future missions like Euclid and LSST.

Original authors: Christopher J. Conselice, Edmund J. Copeland, Sergio Sevillano Muñoz

Published 2026-03-19

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CC BY 4.0

Original authors: Christopher J. Conselice, Edmund J. Copeland, Sergio Sevillano Muñoz

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

Imagine you are a detective trying to solve a mystery about the size and shape of a giant, expanding balloon. But there's a catch: you can't see the balloon itself. You can only see the stickers (galaxies) stuck to its surface.

For decades, scientists have tried to figure out how fast the balloon is stretching and what's inside it (a mysterious force called Dark Energy) by counting these stickers. But there was a huge problem: the stickers aren't static. They change color, they merge with other stickers to become bigger, and sometimes new ones appear. It was impossible to tell if the stickers were spreading out because the balloon was stretching, or just because the stickers themselves were changing.

This paper proposes a clever new way to solve this mystery. The authors, Christopher Conselice and his team, say: "Let's stop guessing and start tracking the stickers' life stories."

Here is the breakdown of their idea, using simple analogies:

1. The Old Problem: The "Moving Target"

In the 1930s, a scientist named Tolman suggested a test: "If we count how many galaxies we see at different distances, we can figure out the shape of the universe."

The Analogy: Imagine standing in a field and counting trees. If the field is flat, the number of trees grows in a predictable way as you look further out. If the field is curved, the count changes.
The Glitch: The problem is that trees grow, die, and merge. If you see a forest with fewer trees than expected, is it because the field is curved? Or is it because two trees grew together to become one giant tree? For a long time, scientists gave up on this method because they couldn't separate the "shape of the field" from the "growth of the trees."

2. The New Solution: The "Life Story" Tracker

The authors say, "We finally know enough about how trees grow to fix this!"
They have developed a new method called the Revised Evolutionary Volume Tolman Test. Instead of just counting the stickers, they are going to:

Know the history: They use data from deep space telescopes (like JWST) to know exactly how galaxies form stars and merge with each other over billions of years.
Do the math backwards: They take the galaxies we see today, subtract the effects of them merging and growing, and ask: "If these galaxies hadn't changed at all, how many should we have seen?"
Find the difference: If the number we should have seen doesn't match the number we actually see, that difference tells us about the shape of the universe and the nature of Dark Energy.

3. The Two Main "Sticker Changes"

The paper focuses on two main ways galaxies change, which act like "noise" in the data:

Star Formation (The "Growth" Effect): Galaxies make new stars, getting brighter and heavier. This is like a sticker getting a sticker-on-sticker, making it look bigger. If a galaxy grows heavy enough, it crosses a threshold and gets counted in a different category.
Mergers (The "Fusion" Effect): Two galaxies crash into each other and become one. This is like two stickers fusing into one giant sticker. This reduces the total count of individual stickers.

The authors created a complex mathematical model (a "recipe") that accounts for these growth and fusion events. By removing these effects from the data, they can isolate the pure signal of the universe's expansion.

4. The Goal: Catching "Dark Energy" in the Act

The biggest mystery in physics right now is Dark Energy—the invisible force pushing the universe apart.

The Standard Model (ΛCDM): Think of this as a car cruising at a steady speed.
Dynamical Dark Energy: Think of this as a car that is slowly speeding up or slowing down.

The authors show that if we can measure the galaxy counts with extreme precision (specifically, knowing the merger and growth rates to within 1–10%), we can tell the difference between a steady cruise and a speeding car.

The Catch: Right now, our "ruler" isn't precise enough. We don't know the galaxy growth rates perfectly yet.
The Future: The paper argues that upcoming massive surveys (like Euclid and LSST/Rubin) will map millions of galaxies. This will give us the data needed to measure those growth rates accurately. Once we do, this method could become just as powerful as the current top methods (like measuring the Cosmic Microwave Background or Supernovae).

Summary

Think of this paper as a new detective technique. Instead of just counting the suspects (galaxies) and getting confused because they keep changing clothes and merging into groups, the detective now has a complete file on every suspect's history.

By knowing exactly how they change, the detective can finally look at the crowd and say, "Ah! The reason the crowd is spaced out this way isn't because the people moved; it's because the room itself is expanding in a specific, mysterious way."

This method turns the chaotic history of galaxy evolution from a confusing obstacle into a powerful tool to measure the very fabric of our universe.

1. Problem Statement

The primary goal of modern cosmology is to constrain the nature of dark energy and the expansion history of the universe. While current methods rely heavily on the Cosmic Microwave Background (CMB), Type Ia Supernovae (SNe), and Baryon Acoustic Oscillations (BAO), there are emerging tensions (e.g., the Hubble Tension) and a need for independent probes.

Historically, the "Volume Test" or "Number Count Test" (proposed by Tolman in the 1930s) was abandoned as a viable cosmological tool. The fundamental issue was a degeneracy: observed changes in galaxy number counts over redshift could be caused by either:

Cosmological effects: Changes in the comoving volume of the universe due to expansion and curvature.
Astrophysical effects: The evolution of galaxies themselves (mergers, star formation, luminosity changes).

Because galaxy evolution was poorly understood and difficult to model, it was impossible to isolate the cosmological signal from the astrophysical noise. This paper aims to revive the Volume Test by leveraging modern, high-precision data on galaxy evolution to break this degeneracy.

2. Methodology

The authors propose a "Revised Evolutionary Volume Tolman Test." The core methodology involves calculating the expected number of galaxies in a comoving volume as a function of redshift, explicitly modeling the astrophysical processes that alter galaxy populations, and then comparing these predictions against different cosmological models.

A. Theoretical Framework

Stellar Mass Function (SMF): The starting point is the Schechter function, $\phi(M, z)$ , which describes the number density of galaxies as a function of stellar mass ( $M$ ) and redshift ( $z$ ).
Comoving Volume: The total number of galaxies $N_{tot}(z)$ is calculated by integrating the SMF over a comoving volume element $dV_c(z)$ . Crucially, $dV_c(z)$ depends on the Hubble parameter $H(z)$ , which varies based on the cosmological model (e.g., $\Lambda$ CDM vs. dynamical dark energy).
Cosmological Dependence: The authors introduce a correction for luminosity distances. Since mass measurements often assume a standard $\Lambda$ CDM cosmology, they adjust the minimum mass limit ( $M'_{min}$ ) for alternative cosmologies to ensure consistent mass thresholds across different expansion histories.

B. Modeling Astrophysical Evolution

To isolate the cosmological signal, the authors explicitly model how galaxy populations change due to astrophysics between redshift bins:

Star Formation (SFR):
- Modeled using a modified version of the Madau & Dickinson (2014) history, parameterized by Moster et al. (2013) to fit massive galaxies.
- Effect: Star formation increases the stellar mass of galaxies. This causes galaxies that were previously below the detection mass threshold ( $M_{min}$ ) to grow above it, effectively increasing the observed number count in a specific mass bin.
Galaxy Mergers:
- Major Mergers: Modeled using an empirical rate $R(z) \propto (1+z)^m e^{\gamma(1+z)}$ .
- Effect 1 (Decrease): Mergers reduce the total number of distinct galaxies (two become one).
- Effect 2 (Increase): Mergers between lower-mass galaxies (below $M_{min}$ ) can produce a single galaxy above $M_{min}$ , increasing the count in the observed bin.
- The authors use an iterative approach to calculate the net change in number density ( $\phi_T$ ) by accounting for both mass growth and merger fractions.

C. The "Clean" Evolution

By subtracting the calculated astrophysical evolution (star formation and mergers) from the observed number counts, the authors derive a "clean" evolution of the number density. Any residual deviation from the expected evolution is attributed to the cosmological volume expansion, allowing for the measurement of cosmological parameters.

3. Key Contributions

Revival of a Classic Test: The paper successfully updates the 1930s Tolman test by integrating modern, quantitative models of galaxy formation and evolution.
Decoupling Astrophysics and Cosmology: It provides a mathematical framework to separate the effects of galaxy evolution (mergers/SFR) from the expansion history of the universe.
Sensitivity Analysis: The authors introduce an error parameter $\epsilon$ to quantify how much the uncertainty in astrophysical parameters (merger rates, SFR) must be reduced to make this method competitive with current standards.
Future Projections: The study explicitly links the feasibility of this method to upcoming large-scale surveys (Euclid, LSST/Rubin), calculating the required survey areas to achieve necessary statistical precision.

4. Results

Differentiation of Dark Energy Models:
- The authors compared the standard $\Lambda$ CDM model ( $w_0 = -1, w_a = 0$ ) against dynamical dark energy models (using the CPL parameterization $w(z) = w_0 + w_a \frac{z}{1+z}$ ) based on recent DESI and Planck data.
- Figure 6 Results: The models predict different galaxy number counts per square degree. Dynamical dark energy models generally predict a lower count than $\Lambda$ CDM, with differences ranging from ~1% to ~5% depending on redshift and the specific $w_0, w_a$ values.
- Redshift Dependence: The divergence between models is most pronounced at $z \sim 2-4$ .
Impact of Matter Density ( $\Omega_m$ ):
- Variations in $\Omega_m$ (ranging from 0.29 to 0.36) induce significant changes in the predicted number counts (up to ~15% difference), demonstrating that this method is a powerful discriminator for matter density as well.
Current Limitations:
- With current observational uncertainties in merger rates and star formation histories (represented by $\epsilon = 1$ ), the astrophysical noise swamps the cosmological signal. The error bars are too large to distinguish between models.
Future Feasibility:
- To distinguish between models, the uncertainty in astrophysical parameters must be reduced by a factor of 10 to 100.
- Survey Requirements: Reducing Poisson noise by a factor of 100 requires a survey area of ~~525 deg². This is well within the scope of the Euclid (~~15,000 deg²) and LSST (~18,000 deg²) surveys.
- A 1000-fold reduction (required for extremely tight constraints) would theoretically require more than the full sky, but can be achieved by observing fainter galaxies (increasing $N$ ) and deeper integration.

5. Significance

Independent Cosmological Probe: This method offers a completely independent way to measure dark energy and the expansion history, distinct from SNe, CMB, or weak lensing. This is crucial for resolving current cosmological tensions.
Synergy with Next-Gen Surveys: The paper demonstrates that the upcoming era of "Big Data" astronomy (JWST, Euclid, LSST) will provide the necessary precision in galaxy evolution metrics to make this test viable.
Dual Utility: The method allows for the simultaneous measurement of cosmological parameters and the refinement of galaxy evolution models (merger rates, SFR histories). As cosmological constraints tighten, the method can be inverted to better understand galaxy physics, and vice versa.
Competitiveness: The authors conclude that once astrophysical uncertainties are reduced to the 1–10% level (achievable in the next decade), the Revised Evolutionary Volume Tolman Test will be competitive with the most precise current cosmological methods.