it is straightforward to show that the energy momentum tensor for the scalar field is given by Eq. (3) is just the energy momentum tensor for a single dark matter particle. For a fluid with many particles the energy momentum tensor will be We will consider a special form for the scalar field po- tential, where µ and β are dimensionless constants while V$_{0}$ has mass dimension four. As has been discussed in [23], µ ≪ 1 to evade observational constraints and β can be set to − 1 without loss of generality, since we can always rescale ϕ as we wish. Meanwhile, the coupling between the scalar field and dark matter particle is chosen as where γ > 0 is yet another dimensionless constant char- acterizing the strength of the coupling. As discussed in [23], the two dimensionless parameters µ and γ have clear physical meanings: roughly speaking, µ controls the time when the scalar field becomes im- portant in cosmology while γ determines how important the scalar field would ultimately be. In fact, the potential given in Eq. (12) is partly motivated by the f ( R ) cosmol- ogy [11], in which the extra degree of freedom behaves as a coupled scalar field in the Einstein frame. As we can see from Eq. (12), the potential V → ∞ when ϕ → 0 while V → V$_{0}$ when ϕ → ∞ . In the latter case, however, C → ∞ , so that the effective total potential in which V is a volume microscopically large and macro- scopically small, and we have extended the 3-dimensional δ function to a 4-dimensional one by adding a time com- ponent. Here u a is the averaged 4-velocity of the collec- tion of particles inside this volume, and is not necessarily the same as the 4-velocity of the observer. Meanwhile, using So the total energy momentum tensor is where T CDM ab = ρ$_{CDM}$u$_{a}$u$_{b}$ , T S ab is the energy momentum tensor for all other matter species including baryons, and the Einstein equation is has a global minimum at some finite ϕ . If the total poten- tial V$_{eff}$ ( ϕ ) is steep enough around this minimum, then the scalar field becomes very heavy and thus follows its minimum dynamically, as is in the case of the chameleon cosmology (see e.g. [7]). If V$_{eff}$ is not steep enough at the minimum, however, the scalar field will experience a more complicated evolution. These two different cases can be obtained by choosing appropriate values of γ and µ : if γ is very large or µ is small then we run into the for- mer situation and if γ is small and µ is large we have the second. In reality, the situation can get even more com- plicated because when γ , which characterizes the cou- pling strength, increases, the CDM evolution could also get severely affected, which in turn has back-reactions on the scalar field itself. where G$_{ab}$ is the Einstein tensor. Note that due to the coupling between the scalar field ϕ and the dark mat- ter, the energy momentum tensors for either will not be conserved, and we have where throughout this paper we shall use a ϕ to denote the derivative with respect to ϕ . Finally, the scalar field equation of motion (EOM) from the given Lagrangian is ∇ $_{b}$T CDM ab = − C$_{ϕ}$ ( ϕ ) C ( ϕ ) ( g $^{ab}$L $_{CDM}$+ T CDM $^{ab}$) ∇ $_{b}$ϕ (10) □ ϕ + ∂V ( ϕ ) ∂ϕ = ∂C ( ϕ ) ∂ϕ L CDM □ ϕ + ∂V ( ϕ ) ∂ϕ + ρ$_{CDM}$ ∂C ( ϕ ) ∂ϕ = 0 . (11) where □ = ∇ $^{a}$∇ $_{a}$. Using Eq. (4) it can be rewritten as The N -body simulation only probes the motion of par- ticles at late times, and we are not interested in extreme conditions such as black hole formation/evolution, which mean that taking the non-relativistic limit of the above equations should be a sufficient approximation for our purpose. The existence of the scalar field and its (different) cou- plings to matter particles lead to the following changes to the ΛCDM model: Firstly, the energy momentum ten- sor has a new piece of contribution from the scalar field; Eqs. (8, 9, 10, 11) summarize all the physics that will be used in our analysis. T$_{ab}$ = ∇ $_{a}$ϕ ∇ $_{b}$ϕ − g$_{ab}$ [ 1 2 ∇ $_{c}$ϕ ∇ $^{c}$ϕ − V ( ϕ ) ] + C ( ϕ ) T CDM ab + T S ab (8) G$_{ab}$ = κT$_{ab}$ (9) V$_{eff}$ ( ϕ ) = V ( ϕ ) + ρ$_{CDM}$C ( ϕ ) (14) T ab = − 2 √ − g δ ( √ − g L ) δg$_{ab}$ (6) T ϕab = ∇ $^{a}$ϕ ∇ $^{b}$ϕ − g ab [ 1 2 ∇ $_{c}$ϕ ∇ $^{c}$ϕ − V ( ϕ ) ] . (7) C ( ϕ ) = exp( $_{γ}$√$_{κϕ}$$_{)}$$_{,}$ (13) T ab CDM = 1 V ∫ V d 4 $_{y}$√ − g m$_{0}$ √ − g δ ( y − x$_{0}$ ) ˙ x a $_{0}$˙ x b 0 = ρ$_{CDM}$u $^{a}$u$^{b}$, (5) V ( ϕ ) = V$_{0}$ [1 − exp ( $_{β}$√$_{κϕ}$$_{)]}$ $_{µ}$, (12) 3 B. Nonrelativistic Limit