Matter density perturbations in Quadratic Gravity

We investigate linear matter-density perturbations in full quadratic gravity. Working in the longitudinal gauge and adopting the sub-horizon and quasi-static approximations, we obtain an algebraic expression for the effective gravitational coupling in the theory and the equation for evolution of matter density perturbations. We identify the positive-real poles of this coupling and map their dependence on the quadratic couplings. In the formal infrared continuation, the reduced coupling recovers its general-relativistic value, whereas at large physical wavenumber it is suppressed when both quadratic couplings are nonzero; the single-coupling limits reproduce distinct behaviors. Numerical integrations of the growing mode are performed up to the first pole and show that pole-free regions of the reduced system can support regular matter-growth solutions, while poles obstruct a continuous late-time evolution. Our results show that quadratic gravity cannot generally reproduce the standard cosmological evolution of perturbations and in the regions where it can, it closely matches the behavior of f(R) theories.