Big Bang Nucleosynthesis with $f(R)$ gravity scalarons and astrophysical consequences

$f(R)$ gravity is one of the serious alternatives of general relativity having a large range of astronomical consequences. In this work, we study Big Bang Nucleosynthesis (BBN) in $f(R)$ gravity theory. We consider modification to gravity due to the existence of primordial black holes in the radiation era which introduce additional degrees of freedom known as scalarons. We calculate the light element abundances by using the BBN code PArthENoPE. It is found that for a range of scalaron mass $(2.2-3.5) \times 10^4$ eV, the abundance of lithium is lowered by $3-4$ times the value predicted by general relativistic BBN which is a level desired to address the cosmological lithium problem. For the above scalaron mass range the helium abundance is within the observed bound. However, the deuterium abundance is found to be increased by $3-6$ times the observed primordial abundance. It calls for a high efficiency of stellar formation and evolution processes for destruction of primordial deuterium which is suggested as possible in scalaron gravity. A novel relation between scalaron mass and black hole mass has been used to show that the above scalaron mass range corresponds to primordial black holes of sub-planetary mass ($\sim 10^{19}$ g) serving as one of the potential candidates of non-baryonic dark matter. We infer Big Bang equivalence of power law $f(R)$ gravity with primordial black holes that are detectable with upcoming gravitational wave detectors.