In Milgrom's modified Newtonian dynamics (MOND) framework, the dynamical mass of a galaxy is fully determined by its baryonic matter distribution. We fit the distribution of cold and hot gas halos, focusing on hot gas, around SDSS central galaxies using weak lensing signals from the DECaLS survey in MOND. The central galaxies are classified into two samples, the total galaxies and star-forming galaxies. We find that hot gas halo densities nearly follow Plummer's profile for both samples across all mass bins. The rotation curves of the galaxy samples are also demonstrated. The efficiency of converting gas into stars, $M_*/(M_* + M_{\rm g})$, is between 0.3 and 0.8 in all mass bins of the star-forming sample, which is higher than in the total sample. We also calculate gas mass using the satellite kinematics method in MOND. A constant, mildly radial anisotropy or isotropy in satellite motion leads to good agreement between the satellite kinematics and weak lensing methods. Combining both methods, we observe a baryonic mass to line-of-sight velocity dispersion of satellites ($M_{\rm b}$-$\sigma_{\rm s}$) relation. In addition, we examine more sophisticated models in MOND, including external field effects and Osipkov-Merritt anisotropy profiles, and find them unnecessary. A simple, mildly radial constant anisotropic model or an isotropic model, even when isolated, already aligns with observations. The strong concordance between the two methods suggests that weak lensing signals reliably measure the dynamical mass of central galaxies and can constrain the distribution of missing baryons in galaxy clusters.
We present constraints on the $f(R)$ gravity model using a sample of 1,005 galaxy clusters in the redshift range $0.25 - 1.78$ that have been selected through the thermal Sunyaev-Zel'dovich effect (tSZE) from South Pole Telescope (SPT) data and subjected to optical and near-infrared confirmation with the Multi-component Matched Filter (MCMF) algorithm. We employ weak gravitational lensing mass calibration from the Dark Energy Survey (DES) Year 3 data for 688 clusters at $z < 0.95$ and from the Hubble Space Telescope (HST) for 39 clusters with $0.6 < z < 1.7$. Our cluster sample is a powerful probe of $f(R)$ gravity, because this model predicts a scale-dependent enhancement in the growth of structure, which impacts the halo mass function (HMF) at cluster mass scales. To account for these modified gravity effects on the HMF, our analysis employs a semi-analytical approach calibrated with numerical simulations. Combining calibrated cluster counts with primary cosmic microwave background (CMB) temperature and polarization anisotropy measurements from the Planck2018 release, we derive robust constraints on the $f(R)$ parameter $f_{R0}$. Our results, $\log_{10} |f_{R0}| < -5.32$ at the 95 % credible level, are the tightest current constraints on $f(R)$ gravity from cosmological scales. This upper limit rules out $f(R)$-like deviations from general relativity that result in more than a $\sim$20 % enhancement of the cluster population on mass scales $M_\mathrm{200c}>3\times10^{14}M_\odot$.