Constraining f (R ) gravity using future galaxy cluster abundance and weak-lensing mass calibration datasets

We present forecasts for constraints on the Hu and Sawicki f (R ) modified gravity model using realistic mock data representative of future cluster and weak lensing surveys. We create mock thermal Sunyaev-Zel'dovich effect selected cluster samples for SPT-3G and CMB-S4 and the corresponding weak gravitational lensing data from next-generation weak-lensing (ngWL) surveys like Euclid and Rubin. We employ a state-of-the-art Bayesian likelihood approach that includes all observational effects and systematic uncertainties to obtain constraints on the f (R ) gravity parameter log₁₀|f_{R 0}|. In this analysis we vary the cosmological parameters [Ω_m,Ω_νh²,h ,A_s,n_s,log₁₀|f_{R 0}|], which allows us to account for possible degeneracies between cosmological parameters and f (R ) modified gravity. The analysis accounts for f (R ) gravity via its effect on the halo mass function which is enhanced on cluster mass scales compared to the expectations within general relativity (GR). Assuming a fiducial GR model, the upcoming cluster dataset SPT -3 G ×ngWL is expected to obtain an upper limit of log₁₀|f_{R 0}|<-5.95 at 95% credibility, which significantly improves upon the current best bounds. The CMB -S 4 ×ngWL dataset is expected to improve this even further to log₁₀|f_{R 0}|<-6.23 . Furthermore, f (R ) gravity models with log₁₀|f_{R 0}|≥-6 , which have larger numbers of clusters, would be distinguishable from GR with both datasets. We also report degeneracies between log₁₀|f_{R 0}| and Ω_m as well as σ₈ for log₁₀|f_{R 0}|>-6 and log₁₀|f_{R 0}|>-5 respectively. Our forecasts indicate that future cluster abundance studies of f (R ) gravity will enable substantially improved constraints that are competitive with other cosmological probes.