def scale_radius(conc, z_halo,Om=0.3, Ol=0.7, Or=0.0,
A_200=5.71, B_200=-0.084, C_200=-0.47, h_scale=0.7):
'''
purpose: compute the scale radius given the concentration parameter
default parameters based on full profiles of Duffy et al. 2008
input:
conc = concentration parameter at r200
z_halo = redshift of that halo
Om, Ol, Or = cosmological paramters
A200 = duffy et al parameter for concentration radius relationship
B200 = "
C200 = "
h_scale = reduced hubble parameter
output:
scale radius in Mpc
'''
#unit conversion values
minMpc = 3.08568025*10**22 #m in a Megaparsec
kginMsun = 1.988e30 #kg
rho_cr = cosmo.rhoCrit(z_halo,h_scale,Om,Ol,Or) #in kg/m**3
#the h_scale is multiplied because the pivotal mass
#m_pivotal = 2e12 is in unit of M_sun h_scale^(-1)
m_200 = profiles.nfwM200(conc,z_halo, A_200,B_200,C_200,h_scale)*\
kginMsun
r_200 = (m_200/(4*np.pi/3*200*rho_cr))**(1/3.) #in m
r_s = r_200 / conc / minMpc
return r_s
def nfwparam_extended(M_200, z, h_scale=0.7, Om=0.3, Ol=0.7, Or=0.0):
'''
This is the same as nfwparam except that it offers extended output.
Inputs:
M_200 = [array of floats; units=e14 M_sun]
Outputs:
del_c, r_s = characteristic overdensity of the CDM halo, scale radius of
the halo (Mpc)
r_200 (Mpc)
c = concentration
rho_s (M_sun/Mpc^3)
Assumes Duffy et al. 2008 M_200 vs. c relationship.
'''
# calculate the concentration parameter based on Duffy et al. 2008
# for full samples profile
A200 = 5.71
B200 = -0.084
C200 = -0.47
rho_cr = cosmo.rhoCrit(z, h_scale, Om, Ol, Or) / kginMsun * minMpc ** 3
# calculate the r_200 radius
r_200 = (M_200 * 1e14 * 3 / (4 * numpy.pi * 200 * rho_cr)) ** (1 / 3.)
c = A200 / (1 + z) ** numpy.abs(C200) * (M_200 * h_scale / 2e-2) ** (B200)
#c = 5.71/(1+z)**0.47*(M_200*h_scale/2e12)**(-0.084)
del_c = 200 / 3. * c ** 3 / (numpy.log(1 + c) - c / (1 + c))
r_s = r_200 / c
rho_s = del_c * rho_cr
return del_c, r_s, r_200, c, rho_s
def nfwparam(M_200, z, A200=5.71, B200=-0.084, C200=-0.47,
h_scale=0.7, Om=0.3, Ol=0.7, Or=0.0, debug=False):
'''
Inputs:
M_200 = [array of floats; units=1e14 M_sun]
Outputs:
del_c, r_s = characteristic overdensity of the CDM halo, scale radius of
the halo (Mpc)
Assumes Duffy et al. 2008 M_200 vs. c relationship.
'''
assert numpy.sum(M_200 < 5e2) / M_200.size, "M_200 has to be in units of \
1e14 M_sun check your input M_200"
# calculate the concentration parameter based on Duffy et al. 2008
# for full samples profile
rho_cr = cosmo.rhoCrit(z, h_scale, Om, Ol, Or) / kginMsun * minMpc ** 3
# calculate the r_200 radius
r_200 = (M_200 * 1e14 * 3 / (4 * numpy.pi * 200 * rho_cr)) ** (1 / 3.)
# the h_scale is multiplied because the scaling relationship uses
# 2e-2 h_scale^{-1} using 1e14 Msun as unit
c = A200 * ((1 + z) ** C200) * (M_200 * h_scale / 2e-2) ** (B200)
del_c = 200 / 3. * c ** 3 / (numpy.log(1 + c) - c / (1 + c))
r_s = r_200 / c
if debug is True:
print "c = {0}".format(c)
print "r_200 = {0}".format(r_200)
return del_c, r_s
def nfw_den(del_c, r_s, r, z, h=0.7, Om=0.3, Ol=0.7, Or=0):
'''
NFW density at radius r [kg/m^3]. Note that this function accepts a single number
or list for r.
del_c = characteristic overdensity of the CDM halo
r_s = scale radius of the halo
r = radius of interest [same units as r_s]
z = halo redshift
'''
rho_crit = cosmo.rhoCrit(z, h, Om, Ol, Or)
return rho_crit * del_c / (r / r_s * (1. + r / r_s) ** 2)
def NFWprop(M_200,z,c):
'''
Determines the NFW halo related properties. Added this for the case of user
specified concentration.
Input:
M_200 = [float; units:M_sun] mass of the halo. Assumes M_200 with
respect to the critical density at the halo redshift.
z = [float; unitless] redshift of the halo.
c = [float; unitless] concentration of the NFW halo.
'''
# CONSTANTS
rho_cr = cosmo.rhoCrit(z)/kginMsun*minMpc**3
#calculate the r_200 radius
r_200 = (M_200*3/(4*numpy.pi*200*rho_cr))**(1/3.)
del_c = 200/3.*c**3/(numpy.log(1+c)-c/(1+c))
r_s = r_200/c
rho_s = del_c*rho_cr
return del_c, r_s, r_200, c, rho_s
def nfw_Sigmabar(del_c,r_s,r,z,h=0.7,Om=0.3,Ol=0.7,Or=0):
'''
NFW average surface mass density within radius r [kg/m^2]. Note that this
function accepts a single number or list for r.
del_c = characteristic overdensity of the CDM halo
r_s = scale radius of the halo [Mpc]
r = radius of interest [Mpc]
z = halo redshift
compare expressions in Umetsu 2010 table 1 and equations in discussion on
p.21 to get this expression
'''
rho_crit = cosmo.rhoCrit(z,h,Om,Ol,Or)
#convert r to an array so that the function will work for ints and arrays
r = r*numpy.ones(numpy.shape(r))
x = r/r_s
if numpy.sum(x==0)!=0 or r_s==0:
print 'profiles.nfw_Sigmabar: r or r_s = 0 leads to infinity, exiting'
sys.exit()
mask_lt = x<1
mask_eq = x==1
mask_gt = x>1
g = numpy.zeros(numpy.shape(x))
#if x<1
if numpy.sum(mask_lt) !=0:
g[mask_lt] = numpy.log(x[mask_lt]/2.)+\
2/numpy.sqrt(1-x[mask_lt]**2)*\
numpy.arctanh(numpy.sqrt((1-x[mask_lt])/(1+x[mask_lt])))
#elif x==1
if numpy.sum(mask_eq) !=0:
g[mask_eq] = numpy.log(x[mask_eq]/2.) + 1
#elif x > 1:
if numpy.sum(mask_gt) !=0:
g[mask_gt] = numpy.log(x[mask_gt]/2.)+\
2/numpy.sqrt(x[mask_gt]**2-1)*\
numpy.arctan(numpy.sqrt((x[mask_gt]-1)/(x[mask_gt]+1.)))
#if only single value of r was input then make the g array into a float
if numpy.size(g) == 1:
g = g[0]
return 4*del_c*rho_crit*r_s*g/x**2*minMpc
def nfw_Sigma(del_c,r_s,r,z,h=0.7,Om=0.3,Ol=0.7,Or=0):
'''
NFW surface mass density at radius theta [kg/m^2]. Note that this function
accepts a single number or list for r.
del_c = characteristic overdensity of the CDM halo
r_s = scale radius of the halo (Mpc)
r = radius of interest (Mpc)
z = halo redshift
compare expressions in Umetsu 2010 table 1 and equations in discussion on
p.21 to get this expression
'''
rho_crit = cosmo.rhoCrit(z,h,Om,Ol,Or)
#convert r to an array so that the function will work for ints and arrays
r = r*numpy.ones(numpy.shape(r))
x = r/r_s
if numpy.sum(r_s==0)!=0:
print 'profiles.nfw_Sigma: r_s = 0 leads to infinity, exiting'
sys.exit()
mask_lt = x<1
mask_eq = x==1
mask_gt = x>1
f = numpy.zeros(numpy.shape(x))
#if x<1
if numpy.sum(mask_lt) !=0:
f[mask_lt] = 1/(1.-x[mask_lt]**2)*\
(-1+2/numpy.sqrt(1-x[mask_lt]**2)*\
numpy.arctanh(numpy.sqrt((1-x[mask_lt])/(1.+x[mask_lt]))))
#elif x==1
if numpy.sum(mask_eq) !=0:
f[mask_eq] = 1/3.
#elif x > 1
if numpy.sum(mask_gt) !=0:
f[mask_gt] = 1/(x[mask_gt]**2-1.)*\
(1-2/numpy.sqrt(x[mask_gt]**2-1)*\
numpy.arctan(numpy.sqrt((x[mask_gt]-1)/(x[mask_gt]+1.))))
#if only single value of r was input then make the f array into a float
if numpy.size(f) == 1:
f = f[0]
return 2*del_c*rho_crit*r_s*f*minMpc
def nfwparam(M_200,z,h_scale=0.7,Om=0.3,Ol=0.7,Or=0.0):
'''
Inputs:
M_200 = [array of floats; units=1e14 M_sun]
Outputs:
del_c, r_s = characteristic overdensity of the CDM halo, scale radius of
the halo (Mpc)
Assumes Duffy et al. 2008 M_200 vs. c relationship.
'''
#calculate the concentration parameter based on Duffy et al. 2008
#for full samples profile
A200 = 5.71
B200 = -0.084
C200 = -0.47
rho_cr = cosmo.rhoCrit(z,h_scale,Om,Ol,Or)/kginMsun*minMpc**3
#calculate the r_200 radius
r_200 = (M_200*1e14*3/(4*numpy.pi*200*rho_cr))**(1/3.)
#the h_scale is multiplied because the scaling relationship uses
# 2e-2 h_scale^{-1} using 1e14 Msun as unit
c = A200/(1+z)**numpy.abs(C200)*(M_200*h_scale/2e-2)**(B200)
del_c = 200/3.*c**3/(numpy.log(1+c)-c/(1+c))
r_s = r_200/c
return del_c, r_s
def ext_1D_reduced_shear(theta, conc, r_s):
'''
function for calculating the tangential shear given by a NFW profile
for fitting the concentration parameter and the scale radius r_s
This is written by adopting the expression in Umetsu
theta = range of radius of the halo that we 're considering,
unit in arcsec
conc = concentration parameter at the given radius
r_s = scale radius of the NFW profile (Mpc)
z_halo = halo redshift
### WARNING
### cosmological parameters are written within the function
h_scale = hubble scale H = h*100 km/s / Mpc
Om = matter energy density
Ol = dark energy density
Or radiation energy density
halo coord 1
halo coord 2
z_halo = redshift of the lens
z_source = redshift of the source galaxies
beta = ratio of D_LS and D_S see James Jee 's paper for exact def
'''
##latest parameters for El Gordo
z_halo = 0.87
z_source = 1.318
beta = 0.258
#old parameters
#z_halo =0.89
#z_source = 1.22
#beta = 0.216
#Cosmological parameters
#print "profile_1D.tan_1D_reduced_shear: this func uses its own "
#print "cosmological parameters"
Om = 0.3
Ol = 0.7
Or = 0.0
c = 3e5 #speed of light units km/s
G = 6.673*10**(-11) # m^3/kg/s^2
h_scale=0.7
kminMpc = 3.08568025*10**19 # km in a Megaparsec
minMpc = 3.08568025*10**22 #m in a Megaparsec
kginMsun = 1.988e30 #kg
#angular diameter distance to the lens
dl = cosmo.Da(z_halo,h_scale,Om,Ol) #in Mpc
del_c = 200/3.*conc**3/(np.log(1+conc)-conc/(1+conc)) #unitless
nfwSigmacr = c**2/(4*np.pi*G*dl*beta)*1000/kminMpc #in units of kg/m^2
#print 'nfwSigmacr in units of M_sun /pc^2 is ', nfwSigmacr/kginMsun*(3.086*10**16)**2
# in units of kg/m^3
rho_cr = cosmo.rhoCrit(z_halo,h_scale,Om,Ol,Or)#/kginMsun*minMpc**3
#if scale radius is not given then infer from Duffy et al.
#assuming full halo
#if r_s == np.nan:
#r_s = scale_radius(conc,z_halo,Om,Ol,Or)
kappa_s = 2*del_c*rho_cr*(r_s*minMpc)/nfwSigmacr
#just do a simple conversion from arcsec to physical distance using
#angular diameter distance
#make sure that the units is in arcsec
theta_s = r_s /dl*180.0/np.pi*60.*60.
x = theta / theta_s
#write an array with finer bins than x
fx = x #np.arange(np.min(x),np.max(x),(x[1]-x[0])/10.)
#code the expression for kappa as a function of x
#print kappa_s
func_kappa0 = lambda fx: kappa_s/(1-fx**2.)*\
(-1.+2./np.sqrt(1.-fx**2.)*\
np.arctanh(np.sqrt(1.-fx)/np.sqrt(1.+fx)))
func_kappa1 = lambda fx: kappa_s*1./3.
func_kappa2 = lambda fx: kappa_s/(fx**2.-1)*\
(1.-2./np.sqrt(fx**2.-1.)*\
np.arctan(np.sqrt(fx-1.)/np.sqrt(fx+1.)))
kappa = np.piecewise(fx,[fx<1.0, fx==1.0, fx>1.0],
[func_kappa0,func_kappa1,func_kappa2])
g_theta = np.piecewise(x, [x<1.0, x==1.0, x>1.0],
[lambda x: np.log(x/2.)+2./np.sqrt(1-x**2)*\
np.arctanh(np.sqrt(1-x)/np.sqrt(1+x)),
lambda x: np.log(x/2.)+1.,
lambda x: np.log(x/2.)+2./np.sqrt(x**2.-1.)*\
np.arctan(np.sqrt(x-1.)/np.sqrt(x+1.))])
kappa_bar = 2.*kappa_s/x**2.*g_theta
azim_kappa = kappa #no need to azimuthally smooth function since
#the function did n't contain azimuthal angular info to begin with
#, azim_bins = azimuthal_avg_shear_in_bins(kappa, fx, np.min(x), np.max(x)+x[1]-x[0], x[1]-x[0])
#print 'bins are ', x
#print 'azim_bins are ', azim_bins
#print 'size of kappa_bar is ', kappa_bar.size
#print 'size of azim_kappa is ', azim_kappa.size
#print 'kappa_bar is ', kappa_bar
#print 'azim_kappa is ', azim_kappa
red_shear= (kappa_bar-azim_kappa)/(1-azim_kappa)
return red_shear
