Analytical constraints on gravitational models with a quadratic Weyl tensor

We set analytical constraints on the parameter space of models of gravity containing a term quadratic in Weyl curvature $-\alpha C^2$. In this class of models, there are four propagating tensorial degrees of freedom, two vector degrees of freedom, and two scalar degrees of freedom, $\delta_m$ and $\Phi$, corresponding to gauge invariant perturbations in the matter density and the gravitational Bardeen potential, respectively. We consider the era of matter domination, and requiring that growth of perturbations are recovered in the scalar sector and classical instabilities are eliminated in the vector and tensor sectors, we obtain bounds on the free coupling parameter to the quadratic Weyl curvature term, $10^{14}H^2_0 \lesssim \alpha^{-1} \ll M^2_{\text{cutoff}}$, where $M_{\text{cutoff}}$ is the cutoff scale of the low energy effective field theory, and $\alpha^{-1}$ is proportional to the masses of the additional propagating degrees of freedom.