content_image/408316.jpg content_image/408314.jpg content_image/408310.jpg content_image/408303.jpg content_image/408313.jpg content_image/408304.jpg content_image/408305.jpg content_image/408311.jpg content_image/408317.jpg content_image/408315.jpg content_image/408312.jpg
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 

<img src='content_image/408316.jpg'>

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 

it is straightforward to show that the energy momentum tensor for the scalar field is given by 

<img src='content_image/408314.jpg'>

So the total energy momentum tensor is 

<img src='content_image/408310.jpg'>

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 

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 

<img src='content_image/408303.jpg'>

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 

where □ = ∇ $^{a}$∇ $_{a}$. Using Eq. (4) it can be rewritten as 

Eqs. (8, 9, 10, 11) summarize all the physics that will be used in our analysis. 

<img src='content_image/408313.jpg'>

<img src='content_image/408304.jpg'>

<img src='content_image/408305.jpg'>

<img src='content_image/408311.jpg'>

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 

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. 

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; 

<img src='content_image/408317.jpg'>

<img src='content_image/408315.jpg'>

<img src='content_image/408312.jpg'>

## B. Nonrelativistic Limit 

