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

Abstract 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 $$f\sigma _8(z)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>8</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> 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 $${\dot{F}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>F</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:math> , 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 $$z&lt;0.75$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0.75</mml:mn> </mml:mrow> </mml:math> based on the currently available cosmic observations, for filling the requirement of $$f_{,R}&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$f_{,RR}&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mi>R</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> to avoid a tachyonic instability and be ghosts free in theory.