content_image/397416.jpg content_image/397421.jpg content_image/397426.jpg content_image/397427.jpg content_image/397425.jpg content_image/397419.jpg content_image/397422.jpg content_image/397423.jpg content_image/397424.jpg content_image/397418.jpg content_image/397420.jpg content_image/397417.jpg
Finally, for the equation of motion of the dark matter particle, consider Eq. (10). Using Eqs. (3, 4), this can be reduced to 

Obviously the left hand side is the conventional geodesic equation and the right hand side is the new fifth force due to the coupling to the scalar field. Note that be- cause g ab − u $^{a}$ub = h ab is the projection tensor that projects any 4-tensor into the 3-space perpendicular to u $^{a}$, so ( g ab − u $^{a}$u$^{b}$) ∇ a = ˆ ∇ b is the spatial derivative in the 3-space of the observer and perpendicular to u $^{a}$; con- sequently the fifth force C$_{ϕ}$ ( ϕ ) C ( ϕ ) ˆ ∇ $_{a}$ϕ = ˆ ∇ a log C ( ϕ ) has no component parallel to u a (the time component), indicat- ing that the energy density of CDM will be conserved and only the particle trajectories are modified, as mentioned in [23]. Remember that u a in Eq. (22) is the 4-velocity of individual particles, but from Eq. (15) we see that δϕ is computed in the fundamental observer’s frame (where density perturbation is calculated), so if we also want to work on Eq. (22) in the fundamental observer’s frame (so that we can use the δϕ from Eq. (15) directly), then we must rewrite Eq. (22) by substituting 

<img src='content_image/397416.jpg'>

up to first order in perturbations, in which ˜ velocity of the fundamental observer and v is the peculiar velocity of the particle. Then the first term in the above expression is the gradient of δϕ observed by the funda- mental observer (rather than an observer comoving with the particle) and the second term is a velocity dependent acceleration [21]. In [22] it is claimed that the second term is of big importance; in our simulations, however, this term will be neglected (from here on) because it de- pends on ˙ ϕ , which is very small due to the chameleon nature of the model. We have checked in a linear pertur- bation computation that removing this term only changes the matter power spectrum by less than 0.0001%. 

Now in the non-relativistic limit the spatial compo- nents of Eq. (22) can be written as 

where t is the physical time coordinate. If we instead use the comoving coordinate x , then this becomes 

where we have used Eq. (19). The canonical momentum 

<img src='content_image/397421.jpg'>

<img src='content_image/397426.jpg'>

<img src='content_image/397427.jpg'>

conjugate to x is p = a $^{2}$˙ x so we have now 

in which Eq. (26) is for CDM particle and Eq. (27) is for baryons. Note that according to Eq. (26) the quantity a log[ C ( ϕ )] acts as a new piece of potential: the potential for the fifth force. This is an important observation and we will come back to it later when we calculate the escape velocity of CDM particles within a virialized halo. 

Eqs. (15, 21, 25, 26, 27) will be used in the code to eval- uate the forces on the dark matter particles and evolve their positions and momenta in time. 

In our numerical simulation we use a modified version of MLAPM ([32], see III A), and we will have to change our above equations in accordance with the internal units used in that code. Here we briefly summarize the main features. 

MLAPM code uses the following internal units (with subscript $_{c}$): 

in which B is the present size of the simulation box and H$_{0}$ is the present Hubble constant, and ρ , with subscript, can represent the density for either CDM ( ρ$_{c,}$$_{CDM}$ ) or baryons ( ρ$_{c,}$$_{B}$ ). Using these newly-defined quantities, it is easy to check that Eqs. (25, 26, 21, 15) could be rewritten as 

and 

<img src='content_image/397425.jpg'>

<img src='content_image/397419.jpg'>

<img src='content_image/397422.jpg'>

<img src='content_image/397423.jpg'>

<img src='content_image/397424.jpg'>

<img src='content_image/397418.jpg'>

<img src='content_image/397420.jpg'>

## C. Internal Units 

<img src='content_image/397417.jpg'>

