🔭 astrophysics

Constraints on the normal branch of DGP gravity from SPT galaxy clusters with DES and HST weak-lensing mass calibration and from Planck PR4 CMB anisotropies

This paper presents competitive constraints on the normal branch of Dvali-Gabadadze-Porrati (nDGP) modified gravity, deriving an upper bound of $1/\sqrt{(H_0r_{\rm c})}< 1.41$ at 95% confidence by combining the abundance of 1,005 SPT galaxy clusters with DES and HST weak-lensing mass calibration against Planck PR4 CMB anisotropy data.

Original authors: S. M. L. Vogt, S. Bocquet, C. T. Davies, J. J. Mohr, F. Schmidt, C. -Z. Ruan, B. Li, C. Hernändez-Aguayo, S. Grandis, L. E. Bleem, M. Klein, M. Aguena, S. Allam, F. Andrade-Oliveira, D. Bacon, D. Broo

Published 2026-06-15

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CC BY 4.0

Original authors: S. M. L. Vogt, S. Bocquet, C. T. Davies, J. J. Mohr, F. Schmidt, C. -Z. Ruan, B. Li, C. Hernändez-Aguayo, S. Grandis, L. E. Bleem, M. Klein, M. Aguena, S. Allam, F. Andrade-Oliveira, D. Bacon, D. Brooks, R. Camilleri, A. Carnero Rosell, J. Carretero, M. Costanzi, L. N. da Costa, M. E. da Silva Pereira, J. De Vicente, P. Doel, J. Garcïa-Bellido, P. Giles, D. Gruen, G. Gutierrez, S. R. Hinton, D. L. Hollowood, D. J. James, K. Kuehn, S. Lee, J. L. Marshall, J. Mena-Fernändez, F. Menanteau, R. Miquel, J. Myles, A. A. Plazas Malagön, A. Porredon, J. Prat, C. L. Reichardt, A. K. Romer, E. Sanchez, I. Sevilla-Noarbe, M. Smith, M. Soares-Santos, E. Suchyta, M. E. C. Swanson, C. To, V. Vikram, N. Weaverdyck

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 balloon. For decades, scientists have had a very successful "instruction manual" for how this balloon behaves, called the Standard Model (or ΛCDM). This manual says that gravity works exactly the way Einstein described it in his Theory of General Relativity. It also says that the balloon is being pushed apart by a mysterious force called "Dark Energy."

But what if the manual is missing a page? What if gravity works a little differently than Einstein thought, especially on the scale of huge clusters of galaxies?

This paper is like a team of cosmic detectives trying to find out if there is a "secret fifth force" of gravity hiding in the universe. They are testing a specific theory called nDGP (normal-branch Dvali-Gabadadze-Porrati).

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

1. The Suspect: A "Fifth Force"

In the nDGP theory, our universe is like a 4D sheet floating in a 5D room. On small scales (like our solar system), gravity acts normally. But on very large scales, gravity can "leak" into that 5th dimension.

The Analogy: Imagine a trampoline. If you put a heavy bowling ball on it, it curves the fabric. In our normal world, that curve is all there is. In this nDGP theory, the trampoline is actually sitting on a giant, invisible mattress underneath it. If you push hard enough, the trampoline dips into the mattress too, creating an extra "pull." This extra pull is the "fifth force."

2. The Crime Scene: Galaxy Clusters

To catch this fifth force, the scientists looked at galaxy clusters. These are massive groups of thousands of galaxies held together by gravity.

The Analogy: Think of galaxy clusters as "cosmic snowballs." In a normal universe (Einstein's rules), you can predict exactly how many snowballs of a certain size should exist based on how much snow (matter) is available.
The Twist: If that extra "fifth force" exists, it acts like a super-charged magnet. It pulls matter together more strongly than Einstein predicted. This means more massive snowballs (clusters) should form than the standard manual predicts.

3. The Evidence: Counting the Snowballs

The team used data from the South Pole Telescope (SPT) to find these clusters. They found 1,005 massive galaxy clusters.

The Problem: Just counting them isn't enough. You need to know their exact weight (mass) to see if the "fifth force" is making them heavier or more numerous than expected.
The Solution (The Scale): They used two other tools to weigh the clusters:
1. DES (Dark Energy Survey): A massive camera on the ground that looks at how light bends around the clusters (weak lensing).
2. HST (Hubble Space Telescope): A space telescope that took very sharp pictures of the most distant clusters.
- The Metaphor: Imagine trying to weigh a cloud. You can't put it on a scale. Instead, you look at how much it distorts the light of stars behind it. The more the light bends, the heavier the cloud. The team used this "light-bending" trick to weigh their 1,005 clusters.

4. The Comparison: The "Planck" Check

The team also looked at the Cosmic Microwave Background (CMB). This is the "afterglow" of the Big Bang, a baby picture of the universe.

The Analogy: If the galaxy clusters are the "adults" of the universe, the CMB is the "baby photo." The Planck satellite took the highest-resolution baby photo ever. The team used this photo to see what the universe looked like when it was very young, providing a baseline to compare against the "adult" clusters.

5. The Verdict: "Not Guilty" (But with a Caveat)

The scientists ran the numbers through a complex computer model that simulated how the universe would look if this "fifth force" existed.

The Result: They found that the number of galaxy clusters they observed matches the standard Einstein model perfectly.
The Conclusion: There is no strong evidence for this specific "fifth force." The universe seems to be following the standard rules of gravity.
The Limit: They couldn't prove the force doesn't exist at all, but they can say that if it does exist, it is very weak. They set a strict upper limit: the force must be weaker than a certain threshold (specifically, the parameter $1/\sqrt{H_0 r_c} < 1.41$ ).

Summary in One Sentence

The team weighed 1,005 giant galaxy clusters using ground and space telescopes and compared them to the universe's "baby photo" from the Planck satellite, finding that gravity behaves exactly as Einstein predicted, with no detectable "leakage" into a fifth dimension.

Technical Summary: Constraints on the Normal Branch of DGP Gravity from SPT Galaxy Clusters and Planck PR4

Problem and Motivation
The accelerated expansion of the Universe remains a fundamental open question in cosmology. While the standard $\Lambda$ CDM model explains this via a cosmological constant within General Relativity (GR), modified gravity theories offer alternative explanations. This paper focuses on the normal branch of the Dvali-Gabadadze-Porrati (nDGP) braneworld gravity model. In nDGP, gravity leaks into a fifth dimension on large scales, introducing a crossover scale $r_c$ . On scales smaller than $r_c$ , a Vainshtein screening mechanism restores GR locally, but a gravity-like fifth force persists, altering the growth of cosmic structure. Unlike the self-accelerating branch (which is unstable), the normal branch is theoretically consistent but requires a dark energy component to match the background expansion history of $\Lambda$ CDM. Consequently, the most stringent test of nDGP lies in the growth of structure, specifically the abundance of massive galaxy clusters, which is sensitive to the enhanced gravitational collapse predicted by the model.

Methodology
The authors perform a joint analysis combining galaxy cluster abundance data with Cosmic Microwave Background (CMB) anisotropy measurements to constrain the nDGP parameter $1/\sqrt{H_0 r_c}$ .

Cluster Sample and Mass Calibration:
- The study utilizes a sample of 1,005 galaxy clusters detected via the thermal Sunyaev-Zel'dovich effect (tSZE) by the South Pole Telescope (SPT) surveys (SPT-SZ, SPTpol ECS, SPTpol 500d).
- Clusters are confirmed using the Multi-Component Matched Filter (MCMF) algorithm with optical and near-infrared data from the Dark Energy Survey (DES) and the Wide-field Infrared Survey Explorer (WISE).
- To calibrate the relationship between observables (tSZE significance $\hat{\zeta}$ and optical richness $\hat{\lambda}$ ) and halo mass, the authors employ weak-lensing data. This includes DES Year 3 shape catalog data for 688 clusters ( $z < 0.95$ ) and targeted Hubble Space Telescope (HST) observations for 39 high-redshift clusters ( $0.6 < z < 1.7$ ).
- The analysis accounts for systematic uncertainties such as member contamination, miscentering, and shear calibration.
Theoretical Modeling (nDGP HMF):
- The authors model the Halo Mass Function (HMF) in nDGP by adapting the Sheth-Tormen formalism. They incorporate changes in spherical collapse dynamics caused by the Vainshtein screening mechanism, specifically calculating the critical collapse overdensity $\delta_c$ and virial overdensity $\Delta_{vir}$ within the nDGP framework.
- A semi-analytic enhancement factor $R$ is derived to scale the standard $\Lambda$ CDM HMF (specifically the Tinker HMF) to the nDGP prediction.
- To validate this semi-analytic approach, the authors compare it against the BRIDGE suite of N-body simulations. They identify a bias between the semi-analytic model and simulations for larger deviations from GR, modeling this bias as a linear function of the nDGP parameter and redshift. A correction factor is applied, reducing the variance between the model and simulations by 26%.
- The analysis assumes a cosmology with massive neutrinos ( $\nu\Lambda$ CDM) to account for their suppression of structure growth.
CMB Analysis:
- For the first time, the authors analyze primary CMB temperature and polarization anisotropy measurements from the Planck PR4 dataset within the nDGP framework.
- They utilize the HiLLiPoP and LoLLiPoP likelihoods, which feature improved systematic modeling that resolves the "CMB lensing anomaly" ( $A_{lens} \neq 1$ ) present in previous Planck releases. This anomaly was previously degenerate with modified gravity parameters.
- The nDGP linear power spectrum is computed using the MGCAMB code, incorporating the modified metric potentials.
Statistical Framework:
- A Bayesian analysis is performed using a cluster population model that links observables to the HMF. The likelihood function combines the cluster abundance likelihood with the Planck PR4 likelihood.
- Priors are applied to cosmological parameters ( $\Omega_m, \ln A_s, \Omega_\nu h^2$ ) and scaling relation parameters.

Key Contributions

First Planck PR4 nDGP Constraints: This work presents the first constraints on the nDGP model derived from Planck PR4 data, leveraging improved likelihoods that mitigate the $A_{lens}$ anomaly.
Improved HMF Calibration: The paper provides a refined semi-analytic model for the nDGP HMF, calibrated against BRIDGE N-body simulations with a specific bias correction for the nDGP parameter and redshift.
Joint Analysis: It performs a self-consistent joint analysis of SPT cluster abundance (with DES/HST weak-lensing mass calibration) and Planck PR4 CMB data, avoiding the reliance on external priors used in previous studies.

Results

Cluster Data Alone: The SPT cluster dataset alone cannot meaningfully constrain the nDGP parameter $1/\sqrt{H_0 r_c}$ due to strong degeneracies with standard cosmological parameters ( $\Omega_m$ and the primordial amplitude $A_s$ ). The cluster data alone yields constraints on $\Omega_m$ and $\sigma_8$ consistent with $\nu\Lambda$ CDM but with larger uncertainties in the nDGP scenario.
Planck PR4 Alone: Using Planck PR4 data alone, the authors derive an upper limit of $1/\sqrt{H_0 r_c} < 1.62$ at 95% credibility. This result is consistent with zero (GR) at the 95% level.
Joint Constraints: Combining SPT clusters and Planck PR4 data tightens the constraint to $1/\sqrt{H_0 r_c} < 1.41$ at 95% credibility. This represents a 15% improvement over the Planck-only constraint.
Comparison with Literature: When assuming massless neutrinos (to match conditions in other literature), the constraint tightens to $1/\sqrt{H_0 r_c} < 1.14$ . This is three times tighter than constraints derived from SDSS II DR7 monopole/quadrupole measurements and is comparable in tightness (within 17%) to the current tightest constraints from BOSS DR1 clustering wedge statistics combined with Planck 2015 priors.

Significance
The paper claims that the joint analysis provides competitive constraints on the nDGP modified gravity model, comparable to the most stringent existing limits derived from large-scale structure clustering statistics. The work demonstrates that galaxy cluster abundance, when combined with primary CMB data and robust weak-lensing mass calibration, serves as a powerful probe for testing deviations from General Relativity. The authors note that while current data does not show a statistically significant hint of nDGP, future surveys (SPT-3G, Simons Observatory, Euclid, LSST) will significantly improve sensitivity, potentially allowing for constraints on modified gravity without relying on external datasets. The resolution of the $A_{lens}$ anomaly in Planck PR4 is highlighted as crucial for obtaining reliable constraints on scale-independent modified gravity models.