content_image/309607.jpg
<img src='content_image/309607.jpg'>

and inhomogeneous DE models without coupling to DM [27]. 

In our models, we find in Fig 5a that when the coupling is proportional to the energy density of DE( ξ$_{1}$ = 0 , ξ$_{2}$ ̸ = 0), in the normalization of local halos abundance, the departure from the cosmological constant model due to the interaction between dark sectors becomes smaller especially in the low mass bin when compared with the results by fixing σ$_{8}$ . The major impact due to the coupling emerges in the massive galaxy cluster as shown in Fig 5b. The signature of the DE perturbation is still not obvious which is similar to the result by fixing σ$_{8}$ . 

When the coupling is proportional to the energy density of DM, we see from Fig 6 that the departure from the 

