Model-independent reconstruction of f(T) and f(R) gravity

In this paper, we revisit the model-independent reconstruction of f ( T ) and f ( R ) gravity at the cosmological background and linear matter density perturbation levels respectively via Gaussian process by using the currently available cosmic observations, which consist Pantheon+ SNe Ia samples, observed Hubble parameter H ( z ) and the redshift space distortion fσ ₈(z) f σ 8 ( z ) data points. For the f ( T ) gravity, we find the reconstructed form of f ( T ) from the background and perturbation levels cannot match each other well in 1 σ σ regions, it might imply the tension between the background and perturbation evolutions. For the f ( R ) gravity, due to the existence of an extra degree for freedom Ḟ F ˙ , the form of f ( R ) can be reconstructed by the addition of the growth rate function. The results show that the reconstructed form of f ( R ) might be viable in the redshift range of $$z<0.75$$ z < 0.75 based on the currently available cosmic observations, for filling the requirement of f,R>0 f , R > 0 and f,RR>0 f , R R > 0 to avoid a tachyonic instability and be ghosts free in theory.