We consider conditions for existence and stability of a static cosmological solution in quadratic gravity. It appears that such a solution for a Universe filled by only one type of perfect fluid is possible in a wide range of the equation of state parameter w and for both positively and negatively spatially curved Universe. We show that the static solution for the negative curvature is always unstable if we require positive energy density of the matter content. On the other hand, a static solution with positive spatial curvature can be stable under certain restrictions. Stability of this solution with respect to isotropic perturbation requires that the coupling constant with the \(R^2\) therm in the Lagrangian of the theory is positive, and the equations of state parameter w is located in a rather narrow interval. Nevertheless, the stability condition does not require violation of the strong energy condition. Taking into account anisotropic perturbations leads to further restrictions on the values of coupling constants and the parameter w.

我们考虑二次引力中静态宇宙学解的存在和稳定性条件。看来，对于仅由一种完美流体填充的宇宙，在状态参数w的广泛范围内以及正空间弯曲和负空间弯曲的宇宙中，这样的解都是可能的。我们表明，如果我们需要物质含量的正能量密度，负曲率的静态解总是不稳定的。另一方面，具有正空间曲率的静态解在某些限制下是稳定的。该解相对于各向同性扰动的稳定性要求与理论拉格朗日量中\(R^2\) therm的耦合常数为正，状态参数方程w位于相当窄的区间内。然而，稳定条件不需要违反强能量条件。考虑各向异性扰动会进一步限制耦合常数和参数w 的值。