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The $R+R^2$ model is successful in describing inflation, as it provides an excellent fit to the full set of available observational data. On the other hand, the same model is the simplest extension of general relativity which does not produce higher derivative ghosts and related instabilities. Long ago, it was proposed to treat all terms which cause higher derivative instabilities as small perturbations that could avoid the presence of ghosts in the spectrum. We put this proposal into practice and consider an explicit example of treating more complicated higher derivative terms as small perturbations over the $R+R^2$ model by introducing the $R\Box R$ term into the action. Within the described scheme, it is possible to obtain an upper bound on the coefficient of this non-scale-free sixth-derivative term by mapping the theory into a one-scalar field potential. It is shown that the result differs from treating this term on equal footing with other terms that requires mapping to a two-scalar field model, and in general leads to different observational consequences., Comment: 19 pages, 6 figures. Comment: One typo fixed

Slightly extending a right-handed neutrino version of the $3-3-1$ model, we show that it is not only possible to solve the strong CP problem but also to give the total dark matter abundance reported by the Planck collaboration. Specifically, we consider the possibility of introducing a $3-3-1$ scalar singlet to implement a gravity stable Peccei-Quinn mechanism in this model. Remarkably, for allowed regions of the parameter space, the arising axions with masses $m_a\approx$ meV can both make up the total dark matter relic density through non-thermal production mechanisms and be very close to the region to be explored by the IAXO helioscope., Comment: 21 pages, 4 figures