  In this paper, we investigate the momentum coupling between early dark energy
(EDE) and cold dark matter to alleviate cosmological tensions. EDE has
exhibited promising efficacy in addressing the Hubble tension, but it
exacerbates the large-scale structure tension. We consider the interaction
between EDE and cold dark matter, introducing a pure momentum exchange between
them to alleviate the large-scale structure tension introduced by the EDE
model. We find that this coupling model is consistent with the EDE model,
yielding a higher value for $H_0$, which can resolve the Hubble tension.
Additionally, the new model exhibits a suppressive effect on structure growth,
contributing to the alleviation of the large-scale structure tension. By
utilizing the Markov Chain Monte Carlo method and incorporating various
cosmological data, the coupling model constrains the best-fit values for $H_0$
to be $72.23$ km/s/Mpc and for $S_8$ to be 0.8192. Compared to the $\Lambda$CDM
model, the new models have not fully resolved the large-scale structure
tension. However, in contrast to the best-fit value of 0.8316 for $S_8$
obtained from the EDE model, the new model alleviates the negative impact of
the EDE model.
