MODERN COSMOLOGY

Dark matter halos 455

Table 15.3.Comparison of the NFW and Mooreet alprofiles.

Parameter NFW Mooreet al

Density ρ=x( 1 ρ+^0 x) 2 ρ=x 1. (^5) ( 1 ρ+^0 x) 1. 5
x=r/rs ρ∝x−^3 forx 1 ρ∝x−^3 forx 1
ρ∝x−^1 forx 1 ρ∝x−^1.^5 forx 1
ρ/ρ 0 = 1 / 4 .00 atx= 1 ρ/ρ 0 = 1 / 2 .00 atx= 1
ρ/ρ 0 = 1 / 21 .3atx= 2. 15 ρ/ρ 0 = 1 / 3 .35 atx= 1. 25
Mass
M= 4 πρ 0 rs^3 f(x) f(x)=ln( 1 +x)− 1 +xx f(x)=^23 ln( 1 +x^3 /^2 )
M=Mvirf(x)/f(C)
Mvir=^43 πρcr
0 δTHr^3 vir
Concentration CNFW= 1. 72 CMoore CMoore=CNFW/ 1. 72
C=rvir/rs (for the sameMvirandrmax)
C 1 / 5 ≈ 0. 86 f(CCNFWNFW)+ 0. 1363 C 1 / 5 = CMoore
[( 1 +C^3 Moore/^2 )^1 /^5 − 1 ]^2 /^3
(error<3% forCNFW= 5 −30)C 0. 0 ≈ CMoore
[C^3 Moore/^10 − 1 ]^2 /^3
Cγ=− 2 =CNFW Cγ=− 2 = 23 /^2 CMoore
Cγ=− 2 =≈ 2. 83 CMoore
Circular velocity
v^2 c=GMrvirvirCxff((Cx)) xmax≈ 2. 15 xmax≈ 1. 25
v^2 c=vmax^2 xmaxx f(fx(maxx)) v^2 max≈ 0. 216 v^2 virfC(C) v^2 max≈ 0. 466 vvir^2 fC(C)
v^2 vir=GMrvirvir
(say,r>rs) as closely as possible, we may choose to change the ratio of the
characteristic radiirs,NFW/rs,Moorein such a way that both profiles reach the
maximum circular velocityvcat the same physical radiusrmax. In this case, the
formal concentration of the Mooreet alprofile is 1.72 times smaller than that of
the NFW profile. Indeed, with this normalization the profiles look very similar
in the outer parts as one finds in figure 15.14. Table 15.3 also gives two other
‘concentrations’. The concentrationC 1 / 5 is defined as the ratio of virial radius
to the radius, which encompasses one-fifth of the virial mass (Avila-Reeseet al
1999). For halos withCNFW≈ 5 .5 this one-fifth mass concentration is equal to
CNFW. One can also define the concentration as the ratio of the virial radius to
the radius at which the logarithmic slope of the density profile is equal to−2.
This scale corresponds torsfor the NFW profile and≈ 0. 35 rsfor the Mooreet al
profile.