# An Accurate Fitting Function For Scale-dependent Growth Rate in Hu-Sawicki $f(R)$ Gravity

## TL;DR

This paper introduces a highly accurate, simple fitting formula for the scale-dependent growth rate in Hu-Sawicki $f(R)$ gravity, enabling precise predictions for cosmological observations across various scales and redshifts.

## Contribution

The paper provides a new fitting function for the growth rate in Hu-Sawicki $f(R)$ gravity with sub-percent accuracy, validated over a wide range of scales, redshifts, and model parameters.

## Key findings

- Achieves sub-percent maximum error in fitting the growth rate.
- Valid across scales $10^{-4} \,\leq k \leq 5$ Mpc$^{-1}$ and redshifts $0 \leq z \leq 3$.
- Useful for cosmological probes like redshift space distortions and galaxy velocities.

## Abstract

We present a simple fitting formula for the scale-dependent growth rate of Hu-Sawicki model in $f(R)$ modified gravity. We compare the accuracy of the fitting function against numerical results and report achieving a sub-percent maximum error for all different studied cases. The validity of the fitting function is tested against a wide range of scales, $10^{-4} \le k \le 5 \ [\mathrm{Mpc}^{-1}]$ , for various redshift values in the range of $z \in [0.0,\ 3.0]$ and different $|f_R^0|$ values. This formula is useful in producing predictions for cosmological probes that are sensitive to growth rate such as redshift space distortions, galaxy-galaxy correlation function, peculiar velocity statistics and pairwise velocities of galaxy clusters.