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Dekel_profile.py
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###################################################################
# ANALYTIC EXPRESSIONS FOR THE DEKEL+ PROFILE #
###################################################################
# This python program enables to compute the analytical #
# expressions of the Dekel+ dark matter density profile #
# Written by J. Freundlich, 22/04/2020 #
# #
# Based on: #
# - Zhao (1996), MNRAS, 278, 488 #
# - Dekel et al. (2017), MNRAS, 468, 1005 #
# - Freundlich et al. (2020), arXiv:2004.08395 #
# #
# params=(c,a,b,g,Mvir,Rvir) with b=2, g=3 #
# quantities in units of kpc, Msun, Gyr,... #
###################################################################
import numpy as np
from math import factorial
from scipy.special import betainc, gamma
from scipy.integrate import quad
G=4.499753324353496e-06 # [kpc^3 Msun^-1 Gyr^-2]
# DENSITY [Msun kpc^-3]
def rho_function(r, params, model='dekel'):
# DEKEL+ PROFILE
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
rs = float(Rvir)/c
x = r/rs
return (q_bar(c, a, b, g, Mvir, Rvir)*float(3.-a)/3.*(1.+float(3.-g)/(3.-a)*pow(x,1./b))/
np.array(pow(x,a)*pow((1.+pow(x,1./b)),b*(g-a)+1.)))
# NFW PROFILE
if model == 'nfw':
(c, Rvir, Mvir) = params
rs = float(Rvir)/c
x = r/rs
qb = Mvir*pow(c,3)/(4.*np.pi*pow(Rvir,3)*(np.log(1.+c)-c/(1.+c)))
return qb/(x*pow(1.+x,2))
# EINASTO PROFILE
if model == 'einasto':
(c, n, Rvir, Mvir) = params
rs = float(Rvir)/c
h = rs/float(pow(2*n,n))
raw_rho = np.exp(-pow((r/h),pow(n,-1)))
q = Mvir / quad( dM_einasto,1.e-8*Rvir,Rvir,args=(1.,h,n), epsabs=0., epsrel=1.e-5)[0]
return q*raw_rho
# DI CINTIO PROFILE
if model == 'dicintio':
(rho_s,r_s,a,b,g) = params
x=r/r_s
return rho_s/(pow(x,g)*pow(1+pow(x,a),(b-g)/a))
# GENERALIZED NFW PROFILE
if model == 'gnfw':
(c,a,Rvir,Mvir) = params
rs=float(Rvir)/c
x=r/rs
raw_rho=1./(pow(x,a)*pow(1.+x,3.-a))
rhoc = Mvir / quad( dM_gNFW,1.e-8*Rvir,Rvir,args=(1.,rs,a), epsabs=0., epsrel=1.e-5)[0]
return rhoc/(pow(x,a)*pow(1.+x,3.-a))
# MEAN DENSITY [Msun kpc^-3]
def brho_function(r, params, model='dekel'):
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
rs = float(Rvir)/c
x = r/rs
return q_bar(c, a, b, g, Mvir, Rvir)/np.array(pow(x,a)*pow((1+pow(x,1./b)),b*(g-a)))
if model == 'nfw':
return M_function(r, params, model)/(4.*np.pi/3.*pow(r,3))
if model == 'dicintio':
return M_function(r, params, model)/(4.*np.pi/3.*pow(r,3))
# ENCLOSED MASS [Msun]
def M_function(r, params, model='dekel'):
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
rs = float(Rvir)/c
x = r/rs
return Mvir*mu(c, a, b, g)*pow(x,3.-a)/np.array(pow(1.+pow(x,1./b),b*(g-a)))
if model == 'nfw':
(c, Rvir, Mvir) = params
rs = float(Rvir)/c
x = r/rs
return float(Mvir)*(np.log(1+pow(x,1.))-pow(x,1.)/(1+pow(x,1.)))/(np.log(1.+c)-c/(1.+c))
if model == 'dicintio':
(rho_s,r_s,a,b,g) = params
M_array=np.nan*np.ones_like(r)
for i in range(np.size(r)):
M_array[i]=quad(dM_dicintio,0,r[i],args=(rho_s,r_s,a,b,g))[0]
return M_array
if model == 'einasto':
(c, n, Rvir, Mvir) = params
rs = float(Rvir)/c
h = rs/float( pow(2*n,n) )
q = Mvir / quad( dM_einasto,1.e-8*Rvir,Rvir,args=(1.,h,n), epsabs=0., epsrel=1.e-5)[0]
M_array=np.nan*np.ones_like(r)
for i in range(np.size(r)):
M_array[i]=q * quad( dM_einasto,1.e-8*Rvir,r[i],args=(1.,h,n), epsabs=0., epsrel=1.e-5)[0]
return M_array
if model == 'gnfw':
(c,a,Rvir,Mvir) = params
rs=float(Rvir)/c
x=r/rs
rhoc = Mvir / quad( dM_gNFW,1.e-8*Rvir,Rvir,args=(1.,rs,a), epsabs=0., epsrel=1.e-5)[0]
M_array=np.nan*np.ones_like(r)
for i in range(np.size(r)):
M_array[i]= rhoc * quad( dM_gNFW,1.e-8*Rvir,r[i],args=(1.,rs,a), epsabs=0., epsrel=1.e-5)[0]
return M_array
# ORBITAL VELOCITY [kpc Gyr^-1]
def V_function(r, params, model='dekel'):
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
rs = float(Rvir)/c
x = r/rs
Vvir2 = G*Mvir/Rvir
return np.sqrt(Vvir2*mu(c, a, b, g)*c*pow(x,2.-a)/np.array(pow(1.+pow(x,1./b),b*(g-a))))
# LOGARITHMIC DENSITY SLOPE []
def s_function(r, params, model='dekel'):
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
rs = float(Rvir)/c
x = r/rs
return -(3.-g)/float(3.-a)/b*pow(x,1./b)/(1.+(3.-g)/float(3.-a)/b*pow(x,1./b)) + (a+(g+1./b)*pow(x,1./b))/np.array(1.+pow(x,1./b))
if model == 'dicintio':
(rho_s,r_s,a,b,g) = params
x = r/r_s
return g+(b-g)*x**a/(1.+pow(x,a))
if model == 'einasto':
(c, n, Rvir, Mvir) = params
rs = float(Rvir)/c
x=r/rs
return 2.*pow(x,1./n)
if model == 'gnfw':
(c,a,Rvir,Mvir) = params
rs=float(Rvir)/c
x=r/rs
return (a+3.*x)/(1+x)
# POTENTIAL [kpc^2 Gyr^-2]
def U_function(r, params, model='dekel'):
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
rs = float(Rvir)/c
x = r/rs
Vvir2 = G*Mvir/Rvir
if (b==1 and g==3):
return c*mu(c,a,b,g)/float(2.-a)*Vvir2*(pow(x/np.array(1.+x),2.-a)-pow(c/float(1.+c),2.-a))-Vvir2
if (b==2 and g==3):
chi = pow(x,0.5)/np.array(1.+pow(x,0.5))
chic = pow(c,0.5)/np.array(1.+pow(c,0.5))
return -Vvir2-2*c*mu(c,a,b,g)*Vvir2*((pow(chic,2*(2-a))-pow(chi,2*(2-a)))/float(2*(2-a))-(pow(chic,2*(2-a)+1)-pow(chi,2*(2-a)+1))/float(2*(2-a)+1))
else:
dU = lambda y: G*M_function(y,params,model)/array(pow(y,2))
return np.array([-Vvir2-quad(dU,rt,Rvir)[0] for rt in r])
# VELOCITY DISPERSION [kpc Gyr^-1] (ISOTROPIC)
# WARNING: THIS EXPRESSION IS FOR AN UNTRUNCATED HALO!
def sigmar_function(r,params, model='dekel'):
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
rs = float(Rvir)/c
x = r/rs
X = chi(x)
Xc= chi(c)
factor=2.*G*Mvir/Rvir*c*mu(c, a, b, g)*x**a*pow(1.+pow(x,0.5),2.*(3.5-a))
u = 4*(1.-a)
if u>0: # slightly more rapid
sigmar2=factor*(Binc(u,9,1)-Binc(u,9,X))
else: #(Binc(u,9+2*n,1)-Binc(u,9+2*n,X))
sigmar2=factor*Dbetainc(u,9,X)
return np.sqrt(sigmar2)
# VELOCITY DISPERSION [kpc Gyr^-1] (ISOTROPIC) AS A SUM
# (Zhao 1996, Eq. 19, Freundlich+2020a, Eq. B8)
# WARNING: THIS EXPRESSION IS FOR AN UNTRUNCATED HALO!
def sigmar_function_sum(r,params, model='dekel'):
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
rs = float(Rvir)/c
x = r/rs
X = chi(x)
Xc= chi(c)
Vvir2 = G*Mvir/Rvir
u = 4*(1.-a)
return np.sqrt(2.*Vvir2*c/pow(Xc,2.*(3.-a))*pow(x,3.5)/pow(X,2.*(3.5-a)) \
* ( (1.-X**u)/u - 8.*(1.-X**(u+1.))/(u+1.) \
+ 28.*(1.-X**(u+2.))/(u+2.) - 56.*(1.-X**(u+3.))/(u+3.) \
+ 70.*(1.-X**(u+4.))/(u+4.) - 56.*(1.-X**(u+5.))/(u+5.) \
+ 28.*(1.-X**(u+6.))/(u+6.) - 8.*(1.-X**(u+7.))/(u+7.) \
+ (1.-X**(u+8.))/(u+8.)))
# VELOCITY DISPERSION [kpc Gyr^-1] (ISOTROPIC) AS ANOTHER SUM
# (Freundlich+2020a, Eq. B10)
# WARNING: THIS EXPRESSION IS FOR AN UNTRUNCATED HALO!
def sigmar_function_sum2(r,params, model='dekel'):
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
rs = float(Rvir)/c
x = r/rs
X = chi(x)
Xc= chi(c)
Vvir2 = G*Mvir/Rvir
u = 4*(1.-a)
return np.sqrt(2.*Vvir2 *c/pow(Xc,2.*(3.-a)) * x**a*(1.+x**0.5)**(2.*(3.5-a)) \
* (B9(u,1)-B9(u,X)))
# DISTRIBUTION FUNCTION [in Virial units, i.e., Vvir^-3 Rvir^-3]
def distribution_function(E,xE,params, model='dekel'):
'''
xE=Psi^-1(Ei,params,model) with Psi=-U the relative potential
xE can notably be obtained numerically from Psi(xE)=E
'''
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
if np.size(E)==1:
fE= 2./(np.sqrt(8.)*np.pi)*quad(distribution_integrand,xE,c, args=(E,params,model))[0]-1/np.sqrt(2)/np.pi*dnudpsi(c,params)/dpsi(c,params)*np.sqrt(E+U_adim(c,params))
if np.size(E)>1:
fE=np.nan*np.ones_like(np.array(E))
for i,Ei in enumerate(E):
xmin_i=xE[i]
try:
fE[i]=2./(np.sqrt(8.)*np.pi)*quad(distribution_integrand,xmin_i,c, args=(Ei,params,model))[0]-1/np.sqrt(2)/np.pi*dnudpsi(c,params)/dpsi(c,params)*np.sqrt(Ei-psi(c,params))
except:
fE[i]=nan
return fE
###################################################################
# MASS-DEPENDENT PRESCRIPTIONS
# DI CINTIO PROFILE
def get_params_dicintio(rvir,mstar,mvir,fit_rvir='BN'):
logMM=np.log10(mstar/mvir)
# alpha, beta, gamma from Di Cintio + 14
a=2.94-np.log10((10**(logMM+2.33))**-1.08+(10**(logMM+2.33))**2.29)
b=4.23+1.34*logMM+0.26*logMM**2
g=-0.06+np.log10((10**(logMM+2.56))**-0.68+10**(logMM+2.56))
# DM concentration from Dutton & Maccio 14
if fit_rvir=='BN':
c_DM=10**(1.025-0.097*np.log10(mvir/1.e12*0.671))
elif fit_rvir=='R200':
c_DM=10**(0.905-0.101*np.log10(mvir/1.e12*0.671))
# SPH concentration from Di Cintio + 14
c_SPH=(1.0+0.00003* np.exp(3.4*(logMM+4.5)))*c_DM
r_2=rvir/c_SPH
r_s=r_2*((b-2.)/(2.-g))**(1./a)
rho_s=mvir/quad(dM_dicintio,0,rvir,args=(1.,r_s,a,b,g))[0]
return (rho_s,r_s,a,b,g)
# DEKEL+ PROFILE
def get_params_dekel(rvir,mstar,mvir,fit_func='new'):
logMM=np.log10(mstar/mvir)
if fit_func=='old':
popt_s1=[1.76438231e-01, 1.71787327e-02, 1.42328182e+02, 1.23488025e+00]
popt_c2=[1.86247747e+02, 1.37125654e+00, 0.00000000e+00, 1.41574162e-01]
c2_DMO=10**(1.025-0.097*np.log10(mvir*0.671/1e12))
s1_DMO=(1.+0.03*c2_DMO)/(1.+0.01*c2_DMO)
s1_ratio=s1_function1(mstar/mvir,*popt_s1)
s1=s1_ratio*s1_DMO
c2_ratio=exp1min_func(mstar/mvir,*popt_c2)
c2=c2_ratio*c2_DMO
a=(1.5*s1-2.*(3.5-s1)*np.sqrt(0.01)*np.sqrt(c2))/(1.5-(3.5-s1)*np.sqrt(0.01)*np.sqrt(c2))
c=((s1-2.)/((3.5-s1)*np.sqrt(0.01)-1.5/np.sqrt(c2)))**2
if fit_func=='new':
popt_s1=[1.30278262e-03,2.86267536e+00,3.20335633e-01]
popt_c2=[1.86247747e+02, 1.37125654e+00, 0.00000000e+00, 1.41574162e-01]
# Note: the popt_c2 parameters with function exp1min_func are equivalent to the parameters
# quoted in Table 1 of Freundlich+20b and their Eq. (49), with c1=popt_c2[0]=c'/x0^nu.
c2_DMO=10**(1.025-0.097*np.log10(mvir*0.671/1e12))
s1_DMO=(1+0.03*c2_DMO)/(1+0.01*c2_DMO)
s1_ratio=s1_zhao_dmo(mstar/mvir,*popt_s1)
s1=s1_ratio*s1_DMO
c2_ratio=exp1min_func(mstar/mvir,*popt_c2)
c2=c2_ratio*c2_DMO
a=(1.5*s1-2.*(3.5-s1)*np.sqrt(0.01)*np.sqrt(c2))/(1.5-(3.5-s1)*np.sqrt(0.01)*np.sqrt(c2))
c=((s1-2.)/((3.5-s1)*np.sqrt(0.01)-1.5/np.sqrt(c2)))**2
return (c, a, 2, 3, rvir, mvir)
def s1_zhao_dmo(x,x0,nu,spp):
return 1./(1.+pow(x/x0,nu))+spp*np.log10(1.+pow(x/x0,nu))
def s1_function1(x,x1,x2,nu1,nu2):#,eta):
return 1.+np.log10(pow(1.+x/x1,-nu1)+pow(x/x2,nu2))
def exp1min_func(x,c1,nu,delta,mu):
return 1.+c1*pow(x,nu)-pow(x,mu)-c1*pow(1.e-6,nu)+pow(1.e-6,mu)
###################################################################
# CONVERSION BETWEEN (a,c), (s1,c2), and (s1,cmax) for s1=s(r1)
# CONCENTRATION PARAMETER CORRESPONDING TO s(r2)=2 []
def c2_function(params, model='dekel'):
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
return c*(1.5/(2.-a))**2
if model == 'einasto':
(c, n, Rvir, Mvir) = params
return c
if model == 'enfw':
(c,a,Rvir,Mvir) = params
return c/(2.-a)
def params_from_s1cmax(s1,cmax,r1,rvir,mvir):
x12=np.sqrt(r1/rvir)
c12=np.sqrt(cmax)
a=(s1-2.*(3.5-s1)*x12*c12)/(1.-(3.5-s1)*x12*c12)
c=((s1-2.)/((3.5-s1)*x12*c12-1./c12))**2
return (c, a, 2, 3, rvir, mvir)
def params_from_s1c2(s1,c2,r1,rvir,mvir):
x12=np.sqrt(r1/rvir)
c12=np.sqrt(c2)
a=(1.5*s1-2.*(3.5-s1)*x12*c12)/(1.5-(3.5-s1)*x12*c12)
c=((s1-2.)/((3.5-s1)*x12-1.5/c12))**2.
return (c, a, 2, 3, rvir, mvir)
###################################################################
# LENSING PROPERTIES (NUMERICAL INTEGRATIONS)
# PROJECTED SURFACE DENSITY [Msun kpc^-2]
def surfdens(r, params,truncated=True):
(c, a, b, g, Rvir, Mvir) = params
rc = float(Rvir)/c
x = r/rc
rhoc=q_bar(c, a, b, g, Mvir, Rvir)*float(3.-a)/3.
if truncated:
xmax=c
else:
xmax=np.inf
if np.size(r)==1:
if x==c:
return 0.
else:
return 2.*rhoc*rc*quad(integrand_surfdens,x,xmax,args=(x,a))[0]
else:
Sigma_array=np.nan*np.ones_like(r)
for i in range(np.size(r)):
if x[i]==c:
Sigma_array[i]=0.
else:
Sigma_array[i]=2.*rhoc*rc*quad(integrand_surfdens,x[i],xmax,args=(x[i],a))[0]
return Sigma_array
def integrand_surfdens(x,X,a):
return pow(x,1.-a)*pow(1.+pow(x,0.5),2.*a-7)/np.sqrt(pow(x,2)-pow(X,2))
# PROJECTED CUMULATIVE MASS [Msun]
def cummass(r,params,truncated=True):
cummass=np.nan*np.ones_like(r)
for i in range(np.size(r)):
cummass[i]=2.*np.pi*quad(integrand_cummass,0.,r[i],args=(params,truncated))[0]
return cummass
def integrand_cummass(r,params,truncated=True):
return r*surfdens(r, params, truncated)
# CONVERGENCE [in virial units, i.e., Mvir/pi Rvir^2, Sigma_crit=1]
def convergence(r,params,truncated=True):
(c, a, b, g, Rvir, Mvir) = params
return cummass(r,params,truncated)/Mvir*pow(Rvir/r,2)
# SHEAR [in virial units, i.e., Mvir/pi Rvir^2, Sigma_crit=1]
def shear(r,params,truncated=True):
(c, a, b, g, Rvir, Mvir) = params
surfvir=Mvir/(np.pi*Rvir**2)
sigmab=cummass(r,params,truncated)/Mvir*pow(Rvir/r,2)
sigma=surfdens(r, params,truncated)/surfvir
return sigmab-sigma
###################################################################
# AUXILIARY FUNCTIONS
def q_bar(c, a, b, g, Mvir, Rvir):
brho_vir = Mvir/(4*np.pi/3.*pow(Rvir,3))
return brho_vir*pow(c,a)*pow((1.+pow(c,1./b)),b*(g-a))
def mu(c, a, b, g):
return pow(1.+pow(c,1./b),b*(g-a))/float(pow(c,3.-a))
def calc_total_mass(r, rho, Rmax):
return sum(rho[r<=Rmax]*4*np.pi*r[r<=Rmax]**2*np.concatenate(([r[0]], np.diff(r[r<=Rmax]))))
def dM_dicintio(r,rho_s,r_s,a,b,g):
return 4.*np.pi*rho_s*pow(r,2)/(pow(r/r_s,g)*pow(1.+pow(r/r_s,a),(b-g)/a))
def dM_gNFW(r,rho_s,r_s,g):
return 4.*np.pi*rho_s*pow(r,2)/( pow(r/r_s,g) * pow(1.+r/r_s,3-g) )
def dM_einasto(r,q,h,n):
return 4.*np.pi*pow(r,2) * q*np.exp( -pow( (r/h), 1./n ) )
def Binc(a,b,x):
return betainc(a,b,x)*gamma(a)*gamma(b)/gamma(a+b)
def Bintegrand(t,a,b):
return pow(t,a-1.)*pow(1-t,b-1.)
def Dbetainc(a,b,x1,x2=1):
if np.size(x1)==1:
return quad(Bintegrand,x1,x2,args=(a,b,))[0]
elif np.size(x1)>1:
Dbeta=np.nan*np.zeros_like(x1)
for i in range(np.size(x1)):
Dbeta[i]=quad(Bintegrand,x1[i],x2,args=(a,b,))[0]
return Dbeta
def chi(x):
u = pow(x,0.5)
return u/(1.+u)
def B9(a,x):
"""
Incomplete beta function B(a,b,x) with b=9
betainc(a,9,x)
"""
B=0.
for i in range(9):
B+=factorial(8)/factorial(i)*gamma(a)/gamma(a+9-i)*x**(a+8-i)*(1.-x)**i
return B
###################################################################
# AUXILIARY FUNCTIONS FOR THE DISTRIBUTION FUNCTION
def distribution_integrand(x,E,params,model='dekel'):
(c, a, b, g, Rvir, Mvir) = params
integrand= np.sqrt(E+U_adim(x, params, model='dekel'))*lambda_(x,params)
return np.nan_to_num(integrand)
def U_adim(x, params, model='dekel'): # U in [GMvir/Rvir]
if model == 'dekel':
(c, a, b, g, Rvir, Mvir) = params
if (b==1 and g==3):
return c*mu(c,a,b,g)/float(2.-a)*(pow(x/np.array(1.+x),2-a)-pow(c/float(1.+c),2.-a))-1.
if (b==2 and g==3):
chi = pow(x,0.5)/np.array(1.+pow(x,0.5))
chic = pow(c,0.5)/np.array(1.+pow(c,0.5))
return -1.-2*c*mu(c,a,b,g)*((pow(chic,2*(2-a))-pow(chi,2*(2-a)))/float(2*(2-a))-(pow(chic,2*(2-a)+1)-pow(chi,2*(2-a)+1))/float(2*(2-a)+1))
else:
dU = lambda y: M_function(y,params,model)/Mvir/array(pow(y,2))
return array([-1.-quad(dU,xt,c)[0] for xt in x])
def lambda_(x,params):
return (g_prim(x,params)*dpsi(x,params)-dnudpsi(x,params)*d2psi(x,params))/pow(dpsi(x,params),2)
def g_prim(x,params):
(c, a, b, g, Rvir, Mvir) = params
return (-3+a)*c**2*(48*a+3*(35+44*a)*x**0.5+4*x*(77+24*a)+245*x**1.5)/32./np.pi/(x+x**1.5)**4
def dnudpsi(x,params):
(c, a, b, g, Rvir, Mvir) = params
factor=(3.-a)*pow(c,2)/(4.*np.pi)
numerator=2.*a+(5.25+3.*a)*pow(x,0.5)+8.75*x
denominator=x**3*pow(1.+pow(x,0.5),3)
return factor*numerator/denominator
def d2psi(x,params):
(c, a, b, g, Rvir, Mvir) = params
return c*mu(c, a, b, g)*pow(1.+1./pow(x,0.5),2*a)*(-1+a+2*pow(x,0.5))/pow(1.+pow(x,0.5),7)
def dpsi(x,params):
(c, a, b, g, Rvir, Mvir) = params
return -c*mu(c, a, b, g)/pow(x,a-1)/pow(1.+pow(x,0.5),6-2*a)
###################################################################