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.
