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where we have used the concentrationc=rvir/rsand the dimensionless mass profile m(x) =
Z x
0
ydm(x0)x02dx0 = log(1 +x)− x
1 +x. (2.0.4)
The virial radius is defined as the radius at which the dark matter halo has a characteristic average density equal to∆vir(z)times the critical density,
ρc(z) = 3H2(z)
8πG . (2.0.5)
The Hubble parameter relative to its present value is given by H(z)
H0 =£
Ωm(1 +z)3+ Ωde(1 +z)3(1+wde)¤1/2
, (2.0.6)
whereΩmandΩderepresent the density of the matter and dark energy components, respectively, relative to the critical density today. Using numerical simulations, Bryan and Norman (1998) found forΛCDM cosmologies that the parametric form
∆vir(z) = 18π2+ 82[Ω(z)−1]−39[Ω(z)−1]2 , (2.0.7) where
Ω(z)≡Ωm(1 +z)3
· H0 H(z)
¸2
, (2.0.8)
provides a good fit over a wide range ofΛCDM cosmological models. Once the characteristic average density is chosen, the virial mass and radius become uniquely related.
The concentration parameter is defined asc=rvir/rsand is derived by fitting the dark matter mass distribution (e.g. Mvir, M200, M500, M1500, M2500) as a function of radii (e.g. rvir, r200, r500, r1500, r2500). Simulated dark matter halos exhibit scatter about the above concentration scaling relation. To account for this we allow the concentration to be a free parameter and fix it by fitting to the thermal SZ profile. Another effect that will cause scatter in our estimate of the concentration is the asymmetry of the dark matter distribution, which our model does not capture.
Once the dark matter density is specified we can obtain the gravitational potential of the halo as
φ(x) =−GM(r)/r=φ0f(x), (2.0.9)
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withf(x)andφ0determined by
f(x) =m(x)/x, φ0 =−4πG r2sρs. (2.0.10) The velocity dispersion of the dark matter is defined as
σ2(x) = σ20s(x), σ02 = 4πGρsrs2 (2.0.11) where
s(x) = x(1 +x)2
· σc−
Z c
x
u−(1 +u) log(1 +u) u3(1 +u)3 du
¸
(2.0.12) withσcis proportional to the surface pressure. Forσcwe adopt the expression
σc= π2
2 −log(c) 2 − 1
2c− 1
2(1 +c)2 − 3 1 +c +
µ1 2 + 1
2c2 − 2 c − 1
1 +c
¶
log(1 +c) +3
2log2(1 +c) + 3 Z 0
−c
log(1−x)dlog(x) (2.0.13) derived by Łokas and Mamon (2001) by solving the Jeans equation for a constant velocity anisotropy and isotropic orbits.
The total gravitational energy of the dark matter halo is given by Wdm(c) = 1
2W0dm Z c
0
f(x)ydm(x)x2dx, (2.0.14) with
W0dm= 4πφ0ρsrs3, (2.0.15) and the total kinetic energy is given by
Tdm(c) = T0dm Z c
0
s(x)ydm(x)x2dx, (2.0.16) where
T0dm = 6πσ20ρsrs3. (2.0.17)
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Stellar Mass, Initial Gas Energy and Pressure
A fraction of the original baryons,f∗would have formed stars leaving a fractionfgas= 1−f∗ of gas in the halo. We place the stars in the central region of the halo out to some radiusxs,which is obtained by integrating the mass profile of the stars (assumed proportional to the dark matter profile) out toxsto give the total stellar mass, that is by solving
4πr3sρsm(xs)fb =M∗ (2.0.18)
The exact prescription for M∗ follows that given in the appendix of Bode et al. (2009) and includes mass lost from winds and supernovae. The stellar mass exhibits a very weak dependence on redshift, particularly forz <1. Note that removing stars also increases the average gas energy.
The resulting gas energy, found by multiplying the energy in dark matter by the baryon fraction, is thus obtained by integrating the gravitational and kinetic energies fromxs toc,such that
Wgas(c) = 1 2W0gas
Z c
xs
f(x)ydm(x)x2dx, W0gas =fbW0dm (2.0.19) and
Tgas(c) =T0gas Z c
xs
s(x)ydm(x)x2dx, T0gas=fbT0dm (2.0.20) Applying the virial theorem: Wdm(c) + 2Tdm(c)−4πr3virPs = 0to the halo allows us to find the initial surface pressure of the halo, and consequently the initial gas surface pressure given by Psgas=fbPs.
Gas Equilibrium Distribution
We now present the equilibrium distributions after the gas has settled into the dark matter poten- tial. Assuming that the gas is a polytrope
ρgas(x) =ρ0ygas(x), Pgas(x) = P0ygasγ (x) (2.0.21) in hydrostatic equilibrium with the dark matter potential allows us to solve for the form of the gas profile
ygas(x) = (1−β[φ0−φ(x)])1/(γ−1), (2.0.22)
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with
β =−φ0
γ−1 γ
1 1 +δrel
ρ0 P0 ≡β0
ρ0
P0 (2.0.23)
whereδrel is the fractional nonthermal contribution to the pressure from shocks and a magnetic field, for example.
The normalisation of the gas density and gas pressure is obtained by requiring the con- servation of energy and constancy of the surface pressure. We model changes to the gas en- ergy in terms of three contributions: expansion or contraction of the gas, dynamical transfer of energy from the dark matter, and feedback from collapsed objects such as supernovae and AGN. The change in energy of the gas due to a change in volume, reaching a final radius, rf = cfrs, is given by ∆EP = 4π3 (c3 − c3f)Psgasr3s while the dynamical energy transferred from the dark matter during virialisation is assumed to be proportional to the initial dark matter energy, ∆ED = ²D|Wdm(c) +Tdm(c)|,with ²D measuring the fraction of energy transferred.
The change in energy from AGN and supernovae feedback is assumed to be proportional to the mass in formed stars, ∆EF = ²FMFc2 which is derived from the stellar mass by taking gas recycling into account – for the redshift range that we are interested in, 0 < z < 1, the mass in formed stars is roughly fifty percent larger than the stellar mass. We follow the prescription forMF given in Bode et al. (2009), which makes use of the ‘fossil’ model by Nagamine et al.
(2006). In this particular model, different stellar populations are considered as well as stellar mass loss due to supernovae and winds.
Taking into account energy conservation; i.e. Einitial+ ∆EP + ∆ED+ ∆EF =Ef inal, and matching the surface and exterior pressures of the gas gives two equations which can be solved to obtain the unknown parameters,cf andβ. These are, respectively,
2Wgas(c) + Tgas(c) + 4π(c3−c3f)rs3Psgas+²D|Wdm(c) +Tdm(c)|+²FMFc2 =φ0MgasIWgas(cf, β) IMgas(cf, β) +3
2(1 + 2δrel) µβ
β0
¶ Mgas
ITgas(cf, β)
IMgas(cf, β) (2.0.24)
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and
yγgas(cf;β) = 1 +f∗ 1 +δrel
σcσ02 m(c)
µβ β0
¶
IMgas(cf, β). (2.0.25) Here we have defined
IWgas(cf, β) = Z cf
0
f(x)ygas(x;β)x2dx (2.0.26) ITgas(cf, β) =
Z cf
0
yγgas(x;β)x2dx (2.0.27)
IMgas(cf, β) = Z cf
0
ygas(x;β)x2dx (2.0.28)
and the gas mass
Mgas=fgasfbMvir = 4πrs3ρ0IMgas(cf, β). (2.0.29) The above equation, describing the conservation of gas mass, allows us to obtain ρ0 oncecf is known, which in turn allows us to solve forP0 fromβ.
Physical Gas Model Parameters
In our study we consider two different gas models which are characterised by various parameters.
The first model is of an adiabatic type, where there is no feedback, star formation or dynamical energy exchange. This model is analogous to the ‘zero model’ in Bode et al. (2009). The second model we study is the standard model, which incorporates all the forementioned effects. For such a model we choose fiducial values for the feedback parameter, ²F = 4×10−6 and dynamical energy exchange parameter, ²D = 0.05. This model is similar to the one of the same name in Bode et al. (2009). The exact prescription for the baryon and stellar fractions encompassed within each of the models is presented in §4.5.3. All other parameters such as the concentration and dark matter profile index,α, are treated identically for each of the models.