The idea of neutrino-assisted early dark energy (νEDE), 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 H0​ tension, the contribution of EDE to the total energy
density (fEDE​) should be around 10\% near the redshift of
matter-radiation equality. We verify under which conditions νEDE 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 fEDE​ 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 fEDE​ 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 νEDE model.Comment: 6 pages, 4 figures. New version with more detailed analysi