Dynamics of clusters of galaxies with extended $f(\chi)$ gravity

In this article, we present the results of a fourth order perturbation analysis of the metric theory of gravity $f(\chi) = \chi^{3/2}$, with $\chi$ a suitable dimensionless Ricci scalar. Such model corresponds to a specific $f(R)$ metric theory of gravity, where the mass of the system is included into the gravitational field's action. In previous works we have shown that, up to the second order in perturbations, this theory reproduces flat rotation curves of galaxies and the details of the gravitational lensing in individual, groups and clusters of galaxies. Here, leaving fixed the results from our previous works, we show that the theory reproduces the dynamical masses of 12 Chandra X-ray galaxy clusters, without the need of dark matter, through the metric coefficients up to the fourth order of approximation. In this sense, we calculate the first relativistic correction of the $f(\chi)$ metric theory and apply it to fit the dynamical masses of the clusters of galaxies.