Hubble tension in lepton asymmetric cosmology with an extra radiation

We study the fit of cosmological models with two additional free parameters $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and ${\xi }_e$ in addition to the parameters of $\mathrm{\Lambda }$CDM. We introduce extra radiation components such as hot axions or sterile neutrinos in addition to the energy density of neutrinos with large neutrino degeneracy. Then, a larger $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is allowed without spoiling Big Bang Nucleosynthesis (BBN), as positive neutrino degeneracy ${\xi }_e$ could improve BBN fit. By analysing the data from Planck, baryon acoustic oscillation (BAO), BBN and type-Ia supernovae (SNeIa), it can be seen that the Hubble tension can be ameliorated for ${\xi }_e\simeq 0.04$ and $0.3 \lesssim \Delta N_$\mathrm{e}\mathrm{f}\mathrm{f}$ \lesssim 0.6$.