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The idea of neutrino-assisted early dark energy ($\nu$EDE), where a couplingbetween neutrinos and the scalar field that models early dark energy (EDE) isconsidered, was introduced with the aim of reducing some of the fine-tuning andcoincidence problems that appear in usual EDE models. In order to be relevantin ameliorating the $H_0$ tension, the contribution of EDE to the total energydensity ($f_\text{EDE}$) should be around 10\% near the redshift ofmatter-radiation equality. We verify under which conditions $\nu$EDE models canfulfill these requirements for a model with a quartic self-coupling of the EDEfield and an exponential coupling to neutrinos. We find that in the situationwhere the EDE field is frozen initially, the contribution to $f_\text{EDE}$ canbe significant but it is not sensitive to the neutrino-EDE coupling and doesnot address the EDE coincidence problem. On the other hand, if the EDE fieldstarts already dynamical at the minimum of the effective potential, it tracksthis time-dependent minimum that presents a feature triggered by the neutrinotransition from relativistic to nonrelativistic particles. This featuregenerates $f_\text{EDE}$ in a natural way at around this transition epoch, thatroughly coincides with the matter-radiation equality redshift. For the set ofparameters that we considered we did not find values that satisfy therequirements on the background cosmological evolution to mitigate the Hubbletension in a natural way in this particular $\nu$EDE model.

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