On the (un)testability of the general free scalar–tensor gravity for the Solar System tests

Abstract

Recently, Zhang et al. [Eur. Phys. J. C 84, 381, https://doi.org/10.1140/epjc/s10052-024-12723-8 (2024)] constrained the Yukawa correction in the general free scalar–tensor gravity by borrowing the measured values of the parametrized post-Newtonian (PPN) parameters γ-1=(2.1±2.3)×10-5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma -1=(2.1\\pm 2.3)\ imes 10^{-5}$$\\end{document} and β-1=(-4.1±7.8)×10-5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta -1=(-\\,4.1\\pm 7.8)\ imes 10^{-5}$$\\end{document} in the Solar System tests. They firstly defined two PPN parameters γ(r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma (r)$$\\end{document} and β(r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta (r)$$\\end{document} in the gravity, which depend on the radial distance r and include the Yukawa-type parameters. Then, by comparing between the experiments’ results of two PPN parameters and their γ(r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma (r)$$\\end{document} and β(r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta (r)$$\\end{document}, the authors claimed that they gave the tightest bound on the gravity by the Cassini tracking experiment. However, we find their approach is not rigorous. In this paper, corresponding astronomical experiments have been physically modelled by considering the lightlike and the timelike geodesics in the general free scalar–tensor gravity. Contrary to the wrong results in Zhang et al.’s work, it is shown that the Cassini tracking experiment is insensitive to the general free scaler–tensor gravity. Furthermore, we also find that the time delay and the light deflection are all independent of the gravity. Due to an additional Yukawa-type advance in the periastron shift, we derive very much improved bounds on the Yukawa-type parameters of this gravity.