Strongest constraint in $f(R) = R+ \alpha R^2$ gravity: stellar stability

In the metric approach of $f(R)$ theories of gravity, the fourth-order field equations are often recast as effective Einstein equations in the presence of standard matter and a curvature fluid (which gathers all the extra terms), always in the Jordan frame. In this picture, we investigate the strong gravity regime of the $f(R) = R+ \alpha R^2$ model. In particular, we focus on the stability of a compact star composed by a mixture of ordinary matter -- described by a polytropic equation of state -- and an effective curvature fluid in an otherwise standard Einstein gravity, so that we are able to apply the usual equations that govern the radial adiabatic oscillations of relativistic stars. Our new restriction on the free parameter is $\alpha \lesssim 2.4 \times 10^8\ \text{cm}^2$ in order to guarantee stellar stability, about $100$ times more restrictive than previous results (based on mass-radius relations alone) in the literature.