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 fσ8(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\sigma _8(z)$$\\end{document} 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 1σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document} 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 F˙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\dot{F}}$$\\end{document}, 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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z<0.75$$\\end{document} based on the currently available cosmic observations, for filling the requirement of f,R>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_{,R}>0$$\\end{document} and f,RR>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_{,RR}>0$$\\end{document} to avoid a tachyonic instability and be ghosts free in theory.