content_image/314627.jpg content_image/314628.jpg content_image/314629.jpg content_image/314630.jpg content_image/314631.jpg content_image/314632.jpg
where ρ$_{d}$ is back ground DE energy density. Converting the time derivative into the derivative with respect to the scale factor a , we can write ¨ R = (˙ a ) 2 d $^{2}$R da 2 + ¨ a dR da and change the Raychaudhuri’s equation into 

<img src='content_image/314627.jpg'>

where ζ = ρ cluster m /ρ$_{m}$ . Therefore, we have 

<img src='content_image/314628.jpg'>

where the prime denotes the derivative with respective to the scale factor a of the universe. 

In order to solve these equations, we set the initial conditions R ∼ a and R ′ = 1 at the starting point z = 3200 which approximately corresponds to the epoch of matter-radiation equality. At the starting point the cluster is comoving with the background expansion. In company with the spherical model, we investigate the linear perturbation in the subhorizon approximation satisfying [19] 

<img src='content_image/314629.jpg'>

and combine with (27,28) by starting from the initial condition ζ$_{i}$ = δ$_{mi}$ + 1 in Eqs. (28) till the point when the spherical model collapses R ( a$_{coll}$ ) ≈ 0 to obtain the critical over density δ$_{c}$ above which objects collapse. This linear extrapolated density threshold is useful in the following computation on the halo abundances. 

If the DE distribution is not homogeneous, the energy content inside the spherical region consists of multi-fluids. When DE does not fully trace DM, the four velocity of DE and DM are different u a ( d ) ̸ = u a ( m ) and 

<img src='content_image/314630.jpg'>

where γ = (1 − v 2 $_{d}$) − 1 / 2 is Lorentz-boost factor and v a d is the relative velocity of DE fluid observed by the observer rested on the DM frame. If DE fully follows DM, v a d = 0. Now we consider the non-comoving perfect fluids [20], 

<img src='content_image/314631.jpg'>

where h ab = g ab + u $^{a}$ub is the projection operator [20]. Inserting (30) into the second equation of (31) and denoting u a ( m ) by u $^{a}$, the energy momentum tensor for DE reads, 

<img src='content_image/314632.jpg'>

