Constraints on nDGP gravity from SPT galaxy clusters with DES and HST weak-lensing mass calibration and from Planck PR4 CMB anisotropies

TL;DR

This work tests the normal-branch DGP gravity by leveraging the abundance of massive galaxy clusters detected by SPT and calibrated with weak-lensing mass measurements from DES and HST, complemented by Planck PR4 CMB data. The authors model the nDGP modification to structure formation with a calibrated halo mass function that accounts for the Vainshtein screening and solve the linear growth with a time-dependent factor involving $\beta(a)$, embedding these in a Bayesian cluster-population framework. They validate the nDGP halo mass function against BRIDGE $N$-body simulations and perform a joint Planck PR4 plus cluster analysis to break degeneracies; the resulting constraint is $1/ sqrt{H_0 r_c} < 1.41$ (95% c.l.) when neutrinos are included, improving the Planck-only bound and approaching the strongest limits in the literature. The work demonstrates the efficacy of combining cluster abundances with CMB data to probe gravity on cosmological scales and foreshadows substantial gains from upcoming surveys and deeper WL data.

Abstract

We present constraints on the normal branch of the Dvali-Gabadadze-Porrati (nDGP) braneworld gravity model from the abundance of massive galaxy clusters. On scales below the nDGP crossover scale $r_{\rm c}$, the nDGP model features an effective gravity-like fifth force that alters the growth of structure, leading to an enhancement of the halo mass function (HMF) on cluster scales. The enhanced cluster abundance allows for constraints on the nDGP model using cluster samples. We employ the SPT cluster sample, selected through the thermal Sunyaev-Zel'dovich effect (tSZE) with the South Pole Telescope (SPT) and with mass calibration using weak-lensing data from the Dark Energy Survey (DES) and the Hubble Space Telescope (HST). The cluster sample contains 1,005 clusters with redshifts $0.25 < z < 1.78$, which are confirmed with the Multi-Component Matched Filter (MCMF) algorithm using optical and near-infrared data. Weak-lensing data from DES and HST enable a robust mass measurement of the cluster sample. We use DES Year 3 data for 688 clusters with redshifts $z < 0.95$, and HST data for 39 clusters with redshifts $ 0.6 < z <1.7$. We account for the enhancement in the HMF through a semi-analytic correction factor to the $νΛ$CDM HMF derived from the spherical collapse model in the nDGP model. We then further calibrate this model using $N$-body simulations. In addition, for the first time, we analyze the primary cosmic microwave background (CMB) temperature and polarization anisotropy measurements from Planck PR4 within the nDGP model. We obtain a competitive constraint from the joint analysis of the SPT cluster abundance with the Planck PR4 data, and report an upper bound of $1/\sqrt{H_0r_{\rm c}}< 1.41$ at $95\%$ when assuming a cosmology with massive neutrinos.

Figures (6)

• Figure 1: Enhancement of the nDGP HMF with respect to the corresponding $\Lambda$CDM cosmology from Eq. \ref{['eq:nDGP_enhancement']} in colored lines for different strengths of the nDGP model at mean cluster sample redshift $z = 0.6$. The enhancement depends on the strength of the nDGP model and grows exponentially with mass. The black dashed line shows the $\Lambda$CDM limit with no enhancement, i. e. $\mathcal{R} = 1$.
• Figure 2: Comparison between the HMF enhancement predicted by the BRIDGE simulations, $\mathcal{R}_{\rm BRIDGE}$, and the semi-analytical model, $\mathcal{R}_{\rm SAM}$, across various redshifts. The data points are color-coded by $\log(H_0r_{rm c})$ values. Gray dashed lines indicate $\pm10\%$ deviation as a visual reference. The top four panels present the uncorrected comparison based on Eq. \ref{['eq:nDGP_enhancement']}, while the bottom four panels show the results after applying the correction factor from Eq. \ref{['eq:nDGP_enhancement_correction']}, resulting in improved agreement. Error bars reflect the jackknife covariance estimated from the BRIDGE simulation.
• Figure 3: Posterior distribution on $\Omega_{\rm m}$, $\log_{10}A_s$, $\sigma_8$ and ${1/\sqrt{H_0r_{\rm c}}}$ ($68\,\%$ and $95\,\%$ credible regions) for the nDGP SPT-clusters$\times$WL analysis in red and for reference the $\nu\Lambda$CDM analysis from Bocquet24II in blue. The cluster dataset alone cannot meaningfully constrain the nDGP parameter ${1/\sqrt{H_0r_{\rm c}}}$. Compared to the $\nu\Lambda$CDM analysis, the constraint on $\Omega_{\rm m}$ remains the same while $\sigma_8$ (which is computed using the nDGP linear growth equation) is shifted to higher values due to the enhanced growth in nDGP.
• Figure 4: Posterior distribution on $\Omega_{\rm m}$, $A_s$, $\sigma_8$ and ${1/\sqrt{H_0r_{\rm c}}}$ ($68\,\%$ and $95\,\%$ credible regions) for the nDGP SPT-cluster$\times$WL analysis in red, Planck PR4 in green and the combination in purple. The joint analysis places a competitive constraint on the nDGP parameter ${1/\sqrt{H_0r_{\rm c}}} < 1.41$ at $95\,\%$ credibility.
• Figure 5: Left: 1D posterior distribution for the nDGP parameter ${1/\sqrt{H_0r_{\rm c}}}$ for the different Planck datasets with and without varying the lensing parameter $A_{\rm lens}$ in solid and dashed lines, respectively. Planck PR3 indicated a $3.7\,\sigma$ detection on the nDGP model. This detection goes away if $A_{\rm lens}$ is varied as well as with the Planck PR4 analysis pipeline. Right: Posterior distribution for ${1/\sqrt{H_0r_{\rm c}}}$ and $A_{\rm lens}$ for Planck PR3 and Planck PR4. Both datasets show a negative correlation between the two parameters, as both have a smoothing effect on the high $l$ CMB power spectrum. For reference, the black dashed lines show the fiducial value of $A_{\rm lens}$ and the black dotted line shows the recovered value of the Planck PR3 analysis.