Late-time and Big Bang nucleosynthesis constraints for generic modify gravity surveys

In this work, a new generic parameterisation for $f(R)$ theories is presented. Our proposal for a new equation of state can reproduce an $f(R)$-like evolution that describes late and early time universe within 1-$\sigma$ C.L when we use a combination of distance ladder measurements based on Cosmic Chronometers, Supernovae Ia, Baryon Acoustic Oscillation and finally, Cosmic Microwave Background and Lyman-$\alpha$ forest. Indeed, a family of $f(R)$ cosmological viable scenarios were extensively analysed in the light of late-time measurements, were an Eos reaches a precision better than $99.2\%$ over the numerical solutions for the field equations of this theory. Moreover, in this proposal we extended the study to find constraints at the very early time that can satisfy the Big Bang Nucleosynthesis data on helium fraction, $Y_{p}$. To perform this analysis, and with our generic $w_{f(R)}$ --which can be seemed it at the same level as other parameterisations into the pipeline and analysis of observational surveys-- we consider both background and linear perturbations evolution and constrain beyond the standard $\Lambda$CDM six cosmological parameters. While there are strong constraints at background on the free parameters of our $w_{f(R)}$, we found that $f(R)$ background viable models can set early constraints to the current Hubble constant $H_0$, which is in agreement with CMB data, but when late-time model-independent measurements are considered, $H_0$ is fully compatible with the $R^{H18}$ value. Finally, as an extension of these results, our proposal is capable to distinguish between $f(R)$ scenarios at both routes of the distance ladder showing a good approach to modify gravity at this level.