Testing generalized logotropic models with cosmic growth

We check the dynamical and observational features of four typologies of logotropic dark energy models, leading to a thermodynamic cosmic speed up fueled by a single fluid that unifies dark energy and dark matter. We first present two principal Anton-Schmidt fluids where the Gr\"uneisen parameter ₆ is free to vary and then fixed to the special value ₆=56. We also investigate the pure logotropic model, corresponding to ₆=-16. Finally, we propose a new logotropic paradigm that works as a generalized logotropic fluid, in which we split the role of dark matter and baryons. We demonstrate that the logotropic paradigms may present drawbacks in perturbations, showing a negative adiabatic sound speed which make perturbations unstable. We thus underline which model is favored over the rest. The Anton-Schmidt model with ₆=56 is ruled out while the generalized logotropic fluid seems to be the most suitable one, albeit weakly disfavored than the model. To fix numerical constraints, we combine low- and higher-redshift domains through experimental fits based on Monte Carlo Markov Chain procedures, taking into account the most recent Pantheon supernovae Ia catalog, Hubble measurements, and ₈ data points based on the linear growth function for the large scale structures. We also consider two model selection criteria to infer the statistical significance of the four models under examination. We conclude there is a statistical advantage to handle the Anton-Schmidt fluid with the Gr\"uneisen parameter free to vary and/or fixed to ₆=-16. The generalized logotropic fluid indicates suitable results, more statistically favored than the other models until the sound speed is positive, becoming unstable in perturbations elsewhere. We emphasize that the paradigm works statistically better than any kind of logotropic and generalized logotropic models, while the Chevallier-Polarski-Linder parametrization is statistically comparable with logotropic scenarios. Finally, we propose that generalizing the Gr\"uneisen parameter by including the effects of temperature would guarantee the sound speed to be positive definite at all redshifts.