Can neutrino-assisted early dark energy models ameliorate the $H_0$ tension in a natural way?

The idea of neutrino-assisted early dark energy ($\nu$EDE), where a coupling between neutrinos and the scalar field that models early dark energy (EDE) is considered, was introduced with the aim of reducing some of the fine-tuning and coincidence problems that appear in usual EDE models. In order to be relevant in ameliorating the $H_0$ tension, the contribution of EDE to the total energy density ($f_\text{EDE}$) should be around 10\% near the redshift of matter-radiation equality. We verify under which conditions $\nu$EDE models can fulfill these requirements for a model with a quartic self-coupling of the EDE field and an exponential coupling to neutrinos. We find that in the situation where the EDE field is frozen initially, the contribution to $f_\text{EDE}$ can be significant but it is not sensitive to the neutrino-EDE coupling and does not address the EDE coincidence problem. On the other hand, if the EDE field starts already dynamical at the minimum of the effective potential, it tracks this time-dependent minimum that presents a feature triggered by the neutrino transition from relativistic to nonrelativistic particles. This feature generates $f_\text{EDE}$ in a natural way at around this transition epoch, that roughly coincides with the matter-radiation equality redshift. For the set of parameters that we considered we did not find values that satisfy the requirements on the background cosmological evolution to mitigate the Hubble tension in a natural way in this particular $\nu$EDE model.