6.2 The Superfluid Density Model
6.2.3 Matching the two phases
After the superfluid and normal phases of the dark matter density were outlined, some way of transitioning between the two had to be devised. In reality, there would be a statistical probability that a dark matter particle would undergo a phase transition to superfluid phase, with the probability increasing in high density or low temperature environments. This would result in the aforementioned mixing phase. However, as this mixing phase was not included in the model for simplification purposes, a different approach had to be taken.
As an approximation, the superfluid core radius was chosen to be the radius at which both the dark matter density and pressure were continuous between the two phases, i.e,ρs(Rc) =
ρn(Rc) =ρcandPs(Rc) =Pn(Rc).
From the EoS, the pressure of the superfluid phase was determined. The normal phase pressure was determined by assuming hydrostatic equilibrium. The parameter which needed to be calculated was thereforeRc. The procedure, developed in Hodson et al. (2016), is il-
lustrated below, using the isothermal halo as an example, though the same procedure can be applied to the NFW-like halo.
Determining the enclosed mass
The first step in doing this was to derive an expression for the total mass of a given system, which was done by integrating the density components. The expression for enclosed mass,
Chapter 6. Superfluid Dark Matter in Galaxy Clusters
outside the core radius was derived to be,
Mgrav(r) =Mc+Mn(r) +Mb(r) =4π Z Rc 0 ρs(r0)r02dr0+4π Z r Rc ρn(r0)r02dr0+Mb(r) =8πp2K Z Rc 0 Æ −Φ(r)r02dr0+4πR2cρc[r−Rc] +Mb(r), (6.11)
where Mcis the mass of the superfluid core (enclosed mass atr =Rc),Mn(r)is the enclosed
mass of the normal phase component, which is only present at radii r >Rc hence the limits on the integral, and Mb(r)is the enclosed baryon mass. The above expression is not valid for r<Rcand is only valid for an isothermal normal component. This expression is sufficient for determining the pressure of the normal phase component at the boundaryr=Rc.
Solving the Poisson Equation
The next step was to determine the superfluid density. As mentioned, this was done by solving the Possion equation inside the core and calculating the density via Eqn 6.7. For this, a central value for the potential was assumed. It should be noted that this result would change with the inclusion of the phonon force, which would add some energy density into the system inside the core.
Calculating the Core Radius
To determine the core radius, the imposed constraint of continuity of pressure was assumed. This led to the equation,
ρ3 c 12K2 = Z ∞ Rc ρn(r)GNMgrav(r) r2 dr=R 2 cρc Z ∞ Rc GNMgrav(r) r4 dr, (6.12)
where the left hand side of the equation is the pressure of the superfluid phase dark matter at the core radius, derived from the EoS, and the right hand side is the pressure of the normal phase component at this radius. Mg r av(r)is the total enclosed mass including both baryons
6.2. The Superfluid Density Model
rameter,ρc. In order to findRc,ρchad to be written in terms of the core radius,ρc(Rc). This
involved enforcing the second constraint, being the continuity of density. From the previous section, the Possion equation has been solved inside the superfluid core, for a given central potential value. Therefore,
ρc=ρs(Rc) =2K
Æ
−2Φ(Rc). (6.13) Putting all the above ingredients together, Eqn 6.12 was able to be solved numerically to find
Rc. This then gave the radius at which the dark matter changed from being described as a superfluid and became normal phase dark matter. It must be stressed that this is not physical as the superfluid core should continue until the density of superfluid particles drops to zero. However, due to the two phase approximation made (no mixing phase) the pressure balancing approach for determining the core radius allowed a first test to be conducted. An interesting follow up study might involve rigorously modelling a system under the superfluid paradigm and comparing it to the pressure matching model described above.
Determining the virial radius
In the previous sections, an algorithm for determining the mass profile of an object, in the con- text of superfluid dark matter was prescribed. This was done by solving a system of equations, all beginning with choosing a central boundary potential for the system. The next issue which required addressing was defining the extent of the dark matter halo as they are not assumed to extend indefinitely.
As the entire point of the superfluid dark matter paradigm is to mimic the properties of
ΛCDM on the largest of scales, it seemed reasonable to adopt the conventional way of deter- mining the radius of the dark matter halo radius, defining the edge of the halo as the virial radius,Rv such that,
3M(Rv) 4πR3 v =200ρcrit(zv). (6.14) where ρcrit(zv) = 3H 2(zv) 8πGN . (6.15)
andzv is the redshift of the object. Equation 6.14 was then numerically solved forRv where that mass was derived from Eqn 6.11.
Chapter 6. Superfluid Dark Matter in Galaxy Clusters
Making Virial Mass the Free Parameter
Summarising the previous sections, the process for determining the dark matter density profile was,
• Choose central potential,Φ0
• Solve Poisson equation within superfluid core,
• Determine pressure within and outwith the superfluid core
• Determine radius at which outside and inside pressure are equal
• Find density at this radius
• Determine virial mass by determining the virial radius
AssumingΛm3 is a global constant, the only free parameter, once the baryon distribution was defined, was the central potential2. InΛCDM, when haloes are described, the common free parameters are the total mass and the concentration. Therefore, making the total mass the free parameter would not only be more in line with the standard model, it would also be more useful to make predictions.
This link was achieved using interpolation functions. The above procedure was conducted for several values for the central potential, say {Φ01,Φ02...Φ0n}. From this an array of core radii was constructed, {Rc01,Rc02...Rc0n}, along with core densities{ρc01,ρc02...ρc0n}, virial
radii{Rvir01,Rvir02...Rvir0n} and virial masses{Mvir01,Mvir02...Mvir0n}. It was then possible to make plots ofΦ0vs Mvir,Rcvs Mvir,ρcvs Mvir, andRvirvs Mvir. Then, interpolation procedures
were implemented to make continuous functions of virial mass as functions of core radius, core density, central potential and virial radius. This allowed all quantities to be calculated from the choice of virial mass.
Physical vs Model Core radius