Stability of cosmological singularity-free solutions in quadratic gravity

We introduce a large family of homogeneous and isotropic cosmological solutions in quadratic gravity which are singularity-free at early and late times. This kind of smooth solutions only emerge beyond the nonstandard de Sitter branch $3\ensuremath{\alpha}<\ensuremath{\beta}$, $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ being the couplings of the ${R}^{2}$ and $\ensuremath{-}{R}_{\ensuremath{\mu}\ensuremath{\nu}}^{2}$ terms. We have analyzed the stability of these singularity-free solutions by computing the second-order variation of the action. We show that the expected instability is similar to that of the standard $\mathrm{\ensuremath{\Lambda}}$ Cold Dark Matter ($\mathrm{\ensuremath{\Lambda}}$ CDM) scenario. Thus, they provide cosmological background scenarios as legitimate as that of the Standard Model. The complete analysis shows that a singular big bang scenario can emerge from a singularity-free cosmological scenario when the parameters of the theory are slightly modified, which provides a bridge between the two very different cosmological scenarios with similar instabilities.