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Big Bang Nucleosynthesis (BBN) is a strong probe for constraining new physicsincluding gravitation. $f(R)$ gravity theory is an interesting alternative togeneral relativity which introduces additional degrees of freedom known asscalarons. In this work we demonstrate the existence of black hole solutions in$f(R)$ gravity and develop a relation between scalaron mass and black holemass. We have used observed bound on the freezeout temperature to constrainscalaron mass range by modifying the cosmic expansion rate at the BBN epoch.The mass range of primordial black holes (PBHs) which are astrophysical darkmatter candidates is deduced. The range of scalaron mass which does not spoilthe BBN era is found to be $10^{-16}-10^4 \text{ eV}$ for both relativistic andnon-relativistic scalarons. The window $10^{-16}-10^{-14}$ eV of scalaron massobtained from solar system constraint on PPN parameter is compatible with theBBN bound derived in this work. The PBH mass range is obtained as$10^6-10^{-14}\text{ }M_{\odot}$. Scalarons constrained by BBN are alsoeligible to accommodate axion like dark matter particles. The problem ofultra-light PBHs ($M \le 10^{-24} \text{ }M_\odot$) not constrained by thepresent study of BBN is still open. Estimation of deuterium (D) fraction andrelative D+$^3$He abundance in the $f(R)$ gravity scenario shows that the BBNhistory mimics that of general relativity. While the PBH mass range is eligiblefor non-baryonic dark matter, the BBN bounded scalarons provide with anindependent strong field test of $f(R)$ gravity. The PBH mass range obtained inthe study is discussed in relation to future astronomical measurements.

Constraining primordial black hole masses through $f(R)$ gravity scalarons in Big Bang Nucleosynthesis