Constraining Scale-Dependent Growth in f(R) Gravity with Future 21 cm Surveys

Recent observations, particularly from DESI, have provided intriguing hints of dynamical behaviour in late-time dark energy. Modified gravity theories offer a compelling framework for interpreting such phenomena, with $f(R)$ gravity emerging as one of the most extensively studied examples. A central challenge in these models, however, lies in determining the precise functional form of $f(R)$. Nevertheless, several viable models have been proposed that successfully reproduce the standard $\Lambda$CDM cosmology at high red shifts while generating late-time cosmic acceleration without an explicit dark energy component. Within this framework, the evolution of the linear matter density contrast becomes scale dependent, leading to a growth index that varies with both scale and redshift. In this work, we explore the capability of forthcoming 21 cm observations to constrain the growth index, as well as the combined neutral hydrogen (HI) bias and growth-rate parameter. Our results indicate that future 21 cm surveys can provide meaningful, though moderate, support for these modified gravity scenarios.