⚛️ general relativity

Joint Curvature and Growth Rate measurements with Supernova Peculiar Velocities and the CMB

By combining Type Ia supernova peculiar velocity measurements with CMB data, this study demonstrates that these datasets are highly complementary for simultaneously constraining matter clustering ( $\sigma_8$ ), curvature ( $\Omega_k$ ), and the growth index ( $\gamma$ ), revealing hints of positive curvature and showing that including SH0ES $H_0$ data recasts the Hubble tension as a preference for negative curvature and suppressed structure growth.

Original authors: Camilo Crisman, Miguel Quartin, João Rebouças

Published 2026-07-08

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CC BY 4.0

Original authors: Camilo Crisman, Miguel Quartin, João Rebouças

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 the universe as a giant, expanding ocean. For decades, astronomers have been trying to map this ocean by measuring how far away "islands" (supernovae) are and how fast the water is moving. This paper is about a new, clever way to read the map by looking not just at where the islands are, but at how they are drifting.

Here is a breakdown of the paper's findings using simple analogies:

1. The Problem: The "Drifting" Islands

Usually, when astronomers look at Type Ia supernovae (exploding stars), they use them as "standard candles" to measure distance. It's like seeing a lighthouse: if you know how bright it should be, you can tell how far away it is based on how dim it looks.

However, these stars aren't just sitting still in the water. They are being pushed and pulled by the gravity of massive clusters of matter (like invisible underwater mountains). This causes them to have a "peculiar velocity"—a drift speed that isn't just from the expansion of the universe, but from local gravitational tugs.

The Analogy: Imagine trying to measure the speed of a river by watching leaves float by. If you only look at the leaves, you might think the river is moving at a steady pace. But if you notice that some leaves are getting caught in whirlpools or pulled toward rocks, you realize the water's flow is more complex. This paper uses those "whirlpools" (the drift) to learn about the shape of the riverbed (the universe).

2. The New Tool: Listening to the Drift

The authors combined two massive lists of supernovae (Pantheon+ and DES-Y5) with data from the Cosmic Microwave Background (CMB)—which is like a baby picture of the universe taken 13.8 billion years ago.

They treated the "drift" of the supernovae as a new type of data. Instead of ignoring the drift as noise, they used it to measure how fast cosmic structures (like galaxy clusters) are growing.

The Analogy: Think of the CMB data as a blueprint of a house, and the supernova drift as the sound of the house settling. By listening to the settling sounds and looking at the blueprint together, you can figure out if the house is built on solid ground or if the foundation is shifting.

3. The Big Discovery: The Universe Might Be Curved

When they combined the "drift" data with the "baby picture" (CMB), they found something surprising.

The Result: The data hints that the universe might have positive curvature.
The Analogy: Imagine drawing a triangle on a flat piece of paper; the angles add up to 180 degrees. If you draw a triangle on a basketball (a sphere), the angles add up to more than 180 degrees. The authors' data suggests our universe is more like the basketball than the flat paper.
The Confidence: They aren't 100% sure yet. It's like a detective saying, "The evidence points to a curved universe with about 95% confidence, but we need more clues to be absolutely certain."

4. Solving a Mystery: The "Growth Index"

Cosmologists have a parameter called $\gamma$ (gamma) that describes how fast matter clumps together over time. General Relativity (Einstein's theory of gravity) predicts a specific value for this.

The Finding: When the authors combined their new drift data with the CMB, the value for $\gamma$ matched Einstein's prediction perfectly.
The Analogy: It's like testing a recipe. Einstein said, "Add exactly 2 cups of flour." Previous measurements were a bit fuzzy, but this new method (combining the drift with the baby picture) confirmed that the universe is indeed using exactly 2 cups of flour.

5. The "Hubble Tension" Puzzle

There is a famous disagreement in physics called the "Hubble Tension." One group of scientists measures the universe's expansion rate using the CMB (the baby picture) and gets one number. Another group measures it using nearby supernovae and gets a faster number. They don't match.

The Paper's Take: When the authors added their new "drift" data to the mix, they found that the tension didn't disappear, but it changed shape.
The Analogy: Imagine two people arguing about how fast a car is going. One says 60 mph, the other says 70 mph. The authors said, "If we assume the car is driving on a curved road (curvature) and the engine is running slightly differently (growth rate), the argument changes."
The Catch: If they force the data to agree with the "faster" measurement (using a specific local dataset called SH0ES), the math breaks down and suggests the universe is curved in a weird way and gravity is behaving strangely. The authors warn that this might just be a mathematical trick to make the numbers match, rather than a real physical discovery.

Summary

This paper is like upgrading a GPS system. Instead of just looking at the map (CMB) or just looking at the speedometer (supernovae), they combined the map with the "wind resistance" (peculiar velocities).

What they learned: The universe is likely flat or very slightly curved (like a sphere), and gravity is working exactly as Einstein predicted.
The limitation: The current data is a bit "fuzzy" (like a low-resolution photo), so they can't be 100% sure about the curvature yet. But by combining these two different ways of looking at the universe, they got a much clearer picture than before.

The authors conclude that this method is a powerful new tool that helps break down the "blind spots" in our understanding of the cosmos, even before the next generation of telescopes comes online.

Technical Summary: Joint Curvature and Growth Rate Measurements with Supernova Peculiar Velocities and the CMB

Problem Statement
Current cosmological analyses rely heavily on Type Ia supernovae (SN) as distance indicators to constrain background expansion parameters. However, SN magnitudes also contain correlated uncertainties arising from peculiar velocities (PV) induced by the large-scale structure of the Universe. While traditionally treated as noise, these PV correlations encode information about the growth of cosmic structures. The paper addresses the potential of utilizing SN peculiar velocities, in combination with Cosmic Microwave Background (CMB) data, to simultaneously constrain the amplitude of matter clustering ( $\sigma_8$ ), the growth index ( $\gamma$ ), and spatial curvature ( $\Omega_k$ ). This approach aims to test General Relativity (GR) and explore extensions to the standard $\Lambda$ CDM model, particularly in the context of current tensions such as the $H_0$ discrepancy.

Methodology
The authors analyze two major SN catalogues: Pantheon+ (approx. 1550 SNe, $0.001 < z < 2.3$ ) and the Dark Energy Survey 5-Year (DES-Y5) sample (approx. 1800 SNe, $0.025 < z < 1.13$ ). A subset of the newer DES-Dovekie catalogue is also used for comparison.

Peculiar Velocity Modeling: The analysis focuses on low-redshift subsamples ( $z < 0.1$ for Pantheon+, $z < 0.2$ for DES) where PV correlations are significant. The authors employ a Gaussian likelihood where the covariance matrix $C$ is decomposed into:
- $C_{PV}$ : Linear perturbation theory correlations derived from the velocity-velocity correlation function, dependent on the growth factor $D(a)$ and the growth index $\gamma$ .
- $C_{nonlin}$ : A diagonal matrix modeling non-linear velocity dispersions via a nuisance parameter $\sigma_v$ .
- $C_{cat}$ : Standard catalogue uncertainties (photometry, calibration, etc.).
  The authors ensure uniform treatment by removing fiducial non-linear terms from the Pantheon+ and DES-Dovekie catalogues and replacing them with a coherent term computed via their pipeline.
CMB Data: The study incorporates Planck PR4 temperature and polarization anisotropies (via HiLLiPoP and LoLLiPoP likelihoods) and CMB lensing data (Planck PR3). The analysis considers the lensing amplitude parameter $A_L$ both fixed to unity and as a free parameter.
Parameter Space: The authors perform Markov Chain Monte Carlo (MCMC) analyses sampling over standard cosmological parameters and extended models including:
- Free spatial curvature ( $\Omega_k$ ).
- Free growth index ( $\gamma$ ), where $f(a) \simeq \Omega_m(a)^\gamma$ .
- Free lensing amplitude ( $A_L$ ).
- Nuisance parameters for SN systematics and CMB systematics.
- A Gaussian prior on $\omega_b$ from Big Bang Nucleosynthesis (BBN) is used when CMB data is not included.

Key Contributions

Complementarity of Probes: The paper demonstrates that SN peculiar velocities and CMB data possess orthogonal degeneracy directions in the $\sigma_8$ - $\gamma$ - $\Omega_k$ parameter space. This complementarity allows for the simultaneous measurement of these parameters, which is difficult with either probe alone.
Joint Constraints: By combining SN PV with Planck CMB data, the authors achieve meaningful constraints on $\Omega_k$ , $\sigma_8$ , and $\gamma$ without relying on external distance ladder calibrations (like SH0ES) for the primary analysis.
Hubble Tension Analysis: The study investigates how the inclusion of free $\Omega_k$ , $\gamma$ , and $A_L$ affects the $H_0$ tension between CMB and local measurements (SH0ES).

Results

$\sigma_8$ Constraints: Using SN data alone in a flat $\Lambda$ CDM model, the authors find $\sigma_8 = 0.73 \pm 0.22$ (Pantheon+) and $\sigma_8 = 0.70^{+0.31}_{-0.38}$ (DES-Y5). These values are consistent with CMB and galaxy survey measurements, showing improved accuracy over previous analyses using older catalogues like JLA.
Curvature and Growth Index: When combining CMB with SN PV and allowing $\Omega_k$ $Ω_{k}$ and $\gamma$ $γ$ to vary:
- Curvature: The combined data hints at positive spatial curvature (negative $\Omega_k$ ), with results of $\Omega_k = -0.011 \pm 0.006$ (Pantheon+) and $\Omega_k = -0.013^{+0.005}_{-0.006}$ (DES-Y5). These exclude flatness at the $2.0\sigma$ and $2.6\sigma$ levels, respectively.
- Growth Index: The measured growth indices are consistent with GR predictions ( $\gamma \approx 0.55$ ): $\gamma = 0.519^{+0.061}_{-0.099}$ (Pantheon+) and $\gamma = 0.500^{+0.054}_{-0.098}$ (DES-Y5).
- Precision: The joint analysis achieves $\sim 15\%$ precision on $\gamma$ and $\sim 1\%$ on $\sigma_8$ , comparable to state-of-the-art galaxy surveys.
Lensing Amplitude ( $A_L$ ): Including $A_L$ as a free parameter significantly degrades the precision of $\gamma$ due to strong degeneracy but does not significantly shift the posterior means for $\sigma_8$ , $\Omega_k$ , or $H_0$ . The lensing amplitude remains consistent with the standard prediction ( $A_L \approx 1$ ).
Hubble Tension:
- Without SH0ES, the joint CMB+SN posterior yields $H_0 \approx 63.2 \pm 2.0$ km/s/Mpc.
- When the SH0ES prior ( $H_0 = 73.30 \pm 1.04$ ) is naively included, the tension between the datasets is not resolved by the extra degrees of freedom. Instead, the tension is "recast": the model shifts to favor a significantly negative curvature ( $\Omega_k \approx 0.009$ ) and a higher growth index ( $\gamma \approx 0.69$ ), excluding the flat GR case ( $\Omega_k=0, \gamma=0.55$ ) at $>4.4\sigma$ . This suggests the tension persists as a conflict between the datasets rather than being alleviated by modified gravity or curvature.

Significance and Claims
The paper claims that peculiar velocities from Type Ia supernovae provide a powerful, independent probe of large-scale structure that transcends the traditional role of SN as mere distance indicators. The authors assert that:

Robustness: The results from Pantheon+ and DES-Y5 are compatible with each other and show better agreement with CMB results than previous compilations (e.g., JLA), lending robustness to the analysis.
Model Testing: The combination of SN PV and CMB allows for advanced testing of models with extra degrees of freedom (curvature, modified growth) with high precision, even without next-generation low-redshift SN surveys.
Tension Manifestation: The analysis indicates that the $H_0$ tension cannot be resolved simply by introducing curvature or modified growth parameters; rather, forcing a reconciliation leads to unphysical or highly significant deviations from the standard flat $\Lambda$ CDM model, suggesting the tension is a fundamental discrepancy between the probes.

The authors conclude that while current catalogues are limited in size, the methodology is established and future precision will increase with next-generation surveys (e.g., LSST, ZTF). They plan to extend this work by combining these probes with galaxy surveys in future studies.