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jcalc.py
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jcalc.py
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#!/usr/bin/env python
"""
@file jcalc.py
@brief Python modules that are used to compute the line-of-sight
integral over a spherically symmetric DM distribution.
@author Matthew Wood <[email protected]>
@author Alex Drlica-Wagner <[email protected]>
"""
__author__ = "Matthew Wood"
__date__ = "12/01/2011"
import copy
import numpy as np
from scipy.integrate import quad
from scipy.interpolate import bisplrep
from scipy.interpolate import bisplev
from scipy.interpolate import interp1d, UnivariateSpline
import scipy.special as spfn
import scipy.optimize as opt
class LoSFn(object):
"""Integrand function for LoS parameter (J). The parameter alpha
introduces a change of coordinates x' = x^(1/alpha). The change
of variables means that we need make the substitution:
dx = alpha * (x')^(alpha-1) dx'
A value of alpha > 1 weights the points at which we sample the
integrand closer to x = 0 (distance of closest approach).
Parameters
----------
d: Distance to halo center.
xi: Offset angle in radians.
dp: Density profile.
alpha: Rescaling exponent for line-of-sight coordinate.
"""
def __init__(self,d,xi,dp,alpha=4.0):
self._d = d
self._d2 = d*d
self._xi = xi
self._sinxi = np.sin(xi)
self._sinxi2 = np.power(self._sinxi,2)
self._dp = dp
self._alpha = alpha
def __call__(self,xp):
#xp = np.asarray(xp)
#if xp.ndim == 0: xp = np.array([xp])
x = np.power(xp,self._alpha)
r = np.sqrt(x*x+self._d2*self._sinxi2)
rho2 = np.power(self._dp.rho(r),2)
return rho2*self._alpha*np.power(xp,self._alpha-1.0)
class LoSFnDecay(LoSFn):
def __init__(self,d,xi,dp,alpha=1.0):
super(LoSFnDecay,self).__init__(d,xi,dp,alpha)
def __call__(self,xp):
#xp = np.asarray(xp)
#if xp.ndim == 0: xp = np.array([xp])
x = np.power(xp,self._alpha)
r = np.sqrt(x*x+self._d2*self._sinxi2)
rho = self._dp.rho(r)
return rho*self._alpha*np.power(xp,self._alpha-1.0)
class LoSIntegralFn(object):
"""Object that computes integral over DM density squared along a
line-of-sight offset by an angle psi from the center of the DM
halo. We introduce a change of coordinates so that the integrand
is more densely sampled near the distance of closest of approach
to the halo center.
Parameters
----------
dist: Distance to halo center.
dp: Density profile.
alpha: Parameter determining the integration variable: x' = x^(1/alpha)
rmax: Radius from center of halo at which LoS integral is truncated.
"""
def __init__(self, dp, dist, rmax=None, alpha=3.0,ann=True):
if rmax is None: rmax = np.inf
self._dp = dp
self._dist = dist
self._rmax = rmax
self._alpha = alpha
self._ann = ann
def __call__(self,psi,dhalo=None):
"""Evaluate the LoS integral at the offset angle psi for a halo
located at the distance dhalo.
Parameters
----------
psi : array_like
Array of offset angles (in radians)
dhalo : array_like
Array of halo distances.
"""
if dhalo is None: dhalo = np.array(self._dist,ndmin=1)
else: dhalo = np.array(dhalo,ndmin=1)
psi = np.array(psi,ndmin=1)
if dhalo.shape != psi.shape:
dhalo = dhalo*np.ones(shape=psi.shape)
v = np.zeros(shape=psi.shape)
for i, t in np.ndenumerate(psi):
s0 = 0
s1 = 0
if self._ann:
losfn = LoSFn(dhalo[i],t,self._dp,self._alpha)
else:
losfn = LoSFnDecay(dhalo[i],t,self._dp,self._alpha)
# Closest approach to halo center
rmin = dhalo[i]*np.sin(psi[i])
# If observer inside the halo...
if self._rmax > dhalo[i]:
if psi[i] < np.pi/2.:
x0 = np.power(dhalo[i]*np.cos(psi[i]),1./self._alpha)
s0 = 2*quad(losfn,0.0,x0)[0]
x1 = np.power(np.sqrt(self._rmax**2 -
rmin**2),1./self._alpha)
s1 = quad(losfn,x0,x1)[0]
else:
x0 = np.power(np.abs(dhalo[i]*np.cos(psi[i])),
1./self._alpha)
x1 = np.power(np.sqrt(self._rmax**2 -
rmin**2),1./self._alpha)
s1 = quad(losfn,x0,x1)[0]
# If observer outside the halo...
elif self._rmax > rmin:
x0 = np.power(np.sqrt(self._rmax**2 -
rmin**2),1./self._alpha)
s0 = 2*quad(losfn,0.0,x0)[0]
v[i] = s0+s1
return v
class LoSIntegralFnFast(LoSIntegralFn):
"""Vectorized version of LoSIntegralFn that performs midpoint
integration with a fixed number of steps.
Parameters
----------
dist: Distance to halo center.
dp: Density profile.
alpha: Parameter determining the integration variable: x' = x^(1/alpha)
rmax: Radius from center of halo at which LoS integral is truncated.
nstep: Number of integration steps. Increase this parameter to
improve the accuracy of the LoS integral.
"""
def __init__(self, dp, dist, rmax=None, alpha=3.0,ann=True,nstep=400):
super(LoSIntegralFnFast,self).__init__(dp,dist,rmax,alpha,ann)
self._nstep = nstep
xedge = np.linspace(0,1.0,self._nstep+1)
self._x = 0.5*(xedge[1:] + xedge[:-1])
def __call__(self,psi,dhalo=None):
"""Evaluate the LoS integral at the offset angle psi for a halo
located at the distance dhalo.
Parameters
----------
psi : array_like
Array of offset angles (in radians)
dhalo : array_like
Array of halo distances.
"""
if dhalo is None: dhalo = np.array(self._dist,ndmin=1)
else: dhalo = np.array(dhalo,ndmin=1)
psi = np.array(psi,ndmin=1)
# if dhalo.shape != psi.shape:
# d = np.zeros(shape=psi.shape)
# d[:] = dhalo
# dhalo = d
# elif dhalo.ndim == 0: dhalo = np.array([dhalo])
# if psi.ndim == 0: psi = np.array([psi])
v = np.zeros(shape=psi.shape)
if self._ann: losfn = LoSFn(dhalo,psi,self._dp,self._alpha)
else: losfn = LoSFnDecay(dhalo,psi,self._dp,self._alpha)
# Closest approach to halo center
rmin = dhalo*np.sin(psi)
msk0 = self._rmax > dhalo
msk1 = self._rmax > rmin
# Distance between observer and point of closest approach
xlim0 = np.power(np.abs(dhalo*np.cos(psi)),1./self._alpha)
# Distance from point of closest approach to maximum
# integration radius
xlim1 = np.zeros(shape=psi.shape)
xlim1[msk1] = np.power(np.sqrt(self._rmax**2 - rmin[msk1]**2),
1./self._alpha)
# If observer inside the halo...
if np.any(msk0):
msk01 = msk0 & (psi < np.pi/2.)
msk02 = msk0 & ~(psi < np.pi/2.)
if np.any(msk01):
dx0 = xlim0/float(self._nstep)
dx1 = (xlim1-xlim0)/float(self._nstep)
x0 = np.outer(self._x,xlim0)
x1 = xlim0 + np.outer(self._x,xlim1-xlim0)
s0 = 2*np.sum(losfn(x0)*dx0,axis=0)
s1 = np.sum(losfn(x1)*dx1,axis=0)
v[msk01] = s0[msk01]+s1[msk01]
if np.any(msk02):
dx1 = (xlim1-xlim0)/float(self._nstep)
x1 = xlim0 + np.outer(self._x,xlim1-xlim0)
s0 = np.sum(losfn(x1)*dx1,axis=0)
v[msk02] = s0[msk02]
# If observer outside the halo...
if np.any(~msk0 & msk1):
dx0 = xlim1/float(self._nstep)
x0 = np.outer(self._x,xlim1)
s0 = 2*np.sum(losfn(x0)*dx0,axis=0)
v[~msk0 & msk1] = s0[~msk0 & msk1]
return v
class LoSIntegralSplineFn(object):
def __init__(self,dp=None,nx=40,ny=20):
self.dp = copy.copy(dp)
if self.dp is not None:
nx = 40
ny = 20
dhalo, psi = np.mgrid[1:2:ny*1j,0.001:2.0:nx*1j]
dhalo = np.power(10,dhalo)
psi = np.radians(psi)
f = LoSIntegralFn(self.dp)
self.z = f(dhalo,psi)
self.init_spline(dhalo,psi,self.z)
def init_spline(self,dhalo,psi,z):
"""Compute knots and coefficients of an interpolating spline
given a grid of points in halo distance (dhalo) and offset
angle (psi) at which the LoS integral has been computed.
"""
kx = 2
ky = 2
self._psi_min = psi.min()
self._tck = bisplrep(dhalo,psi,np.log10(z),s=0.0,kx=kx,ky=ky,
nxest=int(kx+np.sqrt(len(z.flat))),
nyest=int(ky+np.sqrt(len(z.flat))))
def __call__(self,dhalo,psi,rho=1,rs=1):
"""Compute the LoS integral using a 2D spline table.
Returns
-------
vals: LoS amplitude per steradian.
"""
dhalo = np.asarray(dhalo)
psi = np.asarray(psi)
if dhalo.ndim == 0: dhalo = np.array([dhalo])
if psi.ndim == 0: psi = np.array([psi])
if psi.ndim == 2 and dhalo.ndim == 2:
v = np.power(10,bisplev(dhalo[:,0],psi[0,:],self._tck))
else:
v = np.power(10,bisplev(dhalo,psi,self._tck))
v *= rho*rho*rs
return v
def SolidAngleIntegral(psi,pdf,angle):
""" Compute the solid-angle integrated j-value
within a given radius
Parameters
----------
psi : array_like
Array of offset angles (in radians)
pdf : array_like
Array of j-values at angle psi
angle : array_like
Maximum integration angle (in degrees)
"""
angle = np.asarray(angle)
if angle.ndim == 0: angle = np.array([angle])
scale=max(pdf)
norm_pdf = pdf/scale
bad = np.where(norm_pdf <= 0)[0]
idx = bad.min() if bad.size else len(pdf)
log_spline = UnivariateSpline(psi[:idx],np.log10(norm_pdf[:idx]),k=1,s=0)
spline = lambda r: 10**(log_spline(r))
integrand = lambda r: spline(r)*2*np.pi*np.sin(r)
integral = []
for a in angle:
integral.append(quad(integrand, 0, np.radians(a),full_output=True)[0])
integral = np.asarray(integral)
return integral*scale
class JProfile(object):
def __init__(self,losfn):
self._log_psi = np.linspace(np.log10(np.radians(0.001)),
np.log10(np.radians(90.)),1000)
self._psi = np.power(10,self._log_psi)
self._jpsi = losfn(self._psi)
self._jspline = UnivariateSpline(self._psi,self._jpsi,s=0,k=2)
@staticmethod
def create(dp,dist,rmax):
losfn = LoSIntegralFn(dp,dist,rmax=rmax)
return JProfile(losfn)
def __call__(self,psi):
return self._jspline(psi)
def integrate(self,psimax):
"""
Calculate the integrated J-factor out to a given angle.
Default integrated J-factor is returned in units of (Msun^2/kpc^5)
psimax : Maximum integration angle (degrees)
"""
xedge = np.linspace(0.0,np.radians(psimax),1001)
x = 0.5*(xedge[1:] + xedge[:-1])
domega = 2.0*np.pi*(-np.cos(xedge[1:])+np.cos(xedge[:-1]))
return np.sum(self._jspline(x)*domega)
def cumsum(self,psi):
x = 0.5*(psi[1:]+psi[:-1])
dcos = -np.cos(psi[1:])+np.cos(psi[:-1])
return np.cumsum(self._jspline(x)*dcos)
class DensityProfile(object):
""" DM density profile that truncates at a maximum DM density.
rho(r) = rho(r) for rho(r) < rhomax AND r > rmin
= rhomax for rho(r) >= rhomax
= rho(rmin) for r <= rmin
Parameters
----------
rhos : Density normalization parameter.
rmin : Inner radius interior to which the density will be fixed to
a constant value. (rhomax = rho(rmin)).
rhomax : Maximum DM density. If rhomax and rmin are both defined
the maximum DM density will be the lesser of rhomax and rho(rmin).
"""
def __init__(self,rhos,rmin=None,rhomax=None):
self._name = 'profile'
self._rmin=rmin
self._rhomax=rhomax
self._rhos = rhos
def setMassConcentration(self,mvir,c):
rhoc = 9.9E-30*Units.g_cm3
rvir = np.power(mvir*3.0/(177.7*4*np.pi*rhoc*0.27),1./3.)
rs = rvir/c
self._rs = rs
mrvir = self.mass(rvir)
self._rhos = self._rhos*mvir/mrvir
def rho(self,r):
r = np.asarray(r)
if r.ndim == 0: r = r.reshape((1))
if self._rhomax is None and self._rmin is None:
return self._rho(r)
elif self._rhomax is None:
rho = self._rho(r)
rho[r<self._rmin] = self._rho(self._rmin)
return rho
elif self._rmin is None:
rho = self._rho(r)
rho[rho>self._rhomax] = self._rhomax
return rho
else:
rho = self._rho(r)
rhomax = min(self._rho(self._rmin),self._rhomax)
rho[rho>rhomax] = rhomax
return rho
# return np.where(rho>self._rhomax,[self._rhomax],rho)
def set_rho(self,rho,r):
"""Fix the density normalization at a given radius."""
rhor = self._rho(r)
self._rhos = rho*self._rhos/rhor
def name(self):
return self._name
@staticmethod
def create(opts):
"""Method for instantiating a density profile object given the
profile name and a dictionary."""
o = {}
o.update(opts)
name = opts['type'].lower()
def extract(keys,d):
od = {}
for k in keys:
if k in d: od[k] = d[k]
return od
if o['rhos'] is None: o['rhos'] = 1.0
if name == 'nfw':
dp = NFWProfile(**extract(['rhos','rs','rmin'],o))
elif name == 'gnfw':
dp = GNFWProfile(**extract(['rhos','rs','rmin','gamma'],o))
elif name == 'isothermal':
dp = IsothermalProfile(**extract(['rhos','rs','rmin'],o))
elif name == 'einasto':
dp = EinastoProfile(**extract(['rhos','rs','rmin','alpha'],o))
elif name == 'burkert':
dp = BurkertProfile(**extract(['rhos','rs','rmin'],o))
else:
print 'No such halo type: ', name
sys.exit(1)
if 'rhor' in o:
dp.set_rho(o['rhor'][0]*Units.gev_cm3,
o['rhor'][1]*Units.kpc)
elif 'jval' in o:
print o
dp.set_jval(o['jval']*Units.gev2_cm5,
o['rs'],
o['dist'])
return dp
class BurkertProfile(DensityProfile):
""" Burkert (1995)
rho(r) = rhos/( (1+r/rs)(1+(r/rs)**2) )
"""
def __init__(self,rhos=1,rs=1,rmin=None,rhomax=None):
self._rs = rs
super(BurkertProfile,self).__init__(rhos,rmin,rhomax)
self._name = 'burkert'
def _rho(self,r):
x = r/self._rs
return self._rhos*np.power(1+x,-1)*np.power(1+x*x,-1)
def mass(self,r):
x = r/self._rs
return np.pi*self._rhos*(np.log(x**2+1)+2*np.log(x+1)-2*np.arctan(x))
class IsothermalProfile(DensityProfile):
""" Isothermal Profile
rho(r) = rhos/(1+(r/rs))**2
"""
def __init__(self,rhos=1,rs=1,rmin=None,rhomax=None):
self._rs = rs
super(IsothermalProfile,self).__init__(rhos,rmin,rhomax)
self._name = 'isothermal'
def _rho(self,r):
x = r/self._rs
return self._rhos*np.power(1+x,-2)
def mass(self,r):
raise Exception("IsothermalProfile mass not implemented")
class NFWProfile(DensityProfile):
""" Navarro, Frenk, and White (1996)
rho(r) = rhos/( (r/rs)(1+r/rs)**2)
"""
def __init__(self,rhos=1,rs=1,rmin=None,rhomax=None):
self._rs = rs
super(NFWProfile,self).__init__(rhos,rmin,rhomax)
self._name = 'nfw'
def set(self,rhos,rs):
self._rs = rs
self._rhos = rhos
def set_jval(self,jval,rs,dist):
print 'jval ', jval
print 'rs ', rs
print 'dist ', dist
rhos = np.sqrt(3./(4.*np.pi)*jval*dist**2/rs**3)
self._rs = rs
self._rhos = rhos
print 'rs ', self._rs/Units.kpc
print 'rhos ', np.log10(self._rhos/Units.msun_kpc3)
def mass(self,r):
x = r/self._rs
return 4*np.pi*self._rhos*np.power(self._rs,3)*(np.log(1+x)-x/(1+x))
def jval(self,r=None,rhos=None,rs=None):
if rhos is None: rhos = self._rhos
if rs is None: rs = self._rs
if r is not None:
x = r/rs
return (4*np.pi/3.)*rhos**2*rs**3*(1.-np.power(1.+x,-3))
else:
return (4*np.pi/3.)*rhos**2*rs**3
#(4*M_PI/3.)*std::pow(a(0),2)*std::pow(a(1),3)*(1.-std::pow(1+x,-3));
def _rho(self,r):
x = r/self._rs
return self._rhos*np.power(x,-1)*np.power(1+x,-2)
class EinastoProfile(DensityProfile):
""" Einasto profile
rho(r) = rhos*exp(-2*((r/rs)**alpha-1)/alpha)
"""
def __init__(self,rhos=1,rs=1,alpha=0.17,rmin=None,rhomax=None):
self._rs = rs
self._alpha = alpha
super(EinastoProfile,self).__init__(rhos,rmin,rhomax)
self._name = 'einasto'
def set(self,rhos,rs):
self._rs = rs
self._rhos = rhos
def mass(self,r):
x = r/self._rs
gamma = spfn.gamma(3./self._alpha)
return 4*np.pi*self._rhos*np.power(self._rs,3)/self._alpha* \
np.exp(2./self._alpha)* \
np.power(2./self._alpha,-3./self._alpha)* \
gamma*spfn.gammainc(3./self._alpha,
(2./self._alpha)*np.power(x,self._alpha))
def _rho(self,r):
x = r/self._rs
return self._rhos*np.exp(-2./self._alpha*(np.power(x,self._alpha)-1))
class GNFWProfile(DensityProfile):
""" Generalized NFW Profile
rho(r) = rhos/( (r/rs)^g(1+r/rs)**(3-g))
"""
def __init__(self,rhos=1,rs=1,gamma=1.0,rmin=None,rhomax=None):
self._rs = rs
self._gamma = gamma
super(GNFWProfile,self).__init__(rhos,rmin,rhomax)
self._name = 'nfw'
def set(self,rhos,rs):
self._rs = rs
self._rhos = rhos
def mass(self,r):
# x = r/self._rs
# return 4*np.pi*self._rhos*np.power(self._rs,3)*(np.log(1+x)-x/(1+x))
return 0
def _rho(self,r):
x = r/self._rs
return self._rhos*np.power(x,-self._gamma)* \
np.power(1+x,-(3-self._gamma))
class GeneralNFWProfile(DensityProfile):
""" Strigari et al. (2007)
rho(r) = rhos/( (r/rs)**a (1+(r/rs)**b )**(c-a)/b
Default: NFW profile
"""
def __init__(self,rhos=1,rs=1,a=1,b=1,c=3,rmin=None,rhomax=None):
self._rs = rs
self._a = a
self._b = b
self._c = c
super(GeneralNFWProfile,self).__init__(rhos,rmin,rhomax)
self._name = 'general_nfw'
def _rho(self,r):
x = r/self._rs
return self._rhos/(x**self._a*(1+x**self._b)**((self._c-self._a)/self._b))
class UniformProfile(object):
""" Uniform spherical profile
rho(r) = rhos for r < rs
rho(r) = 0 otherwise
"""
def __init__(self,rhos=1,rs=1):
self._name = 'uniform'
self._rhos = rhos
self._rs = rs
def _rho(self,r):
return np.where(r<rs,rhos,0)
class Units(object):
pc = 3.08568e18 # pc to cm
kpc = pc*1e3 # kpc to cm
msun = 1.98892e33 # solar mass to g
gev = 1.78266e-24 # gev to g
g = 1.0
m2 = 1E4
hr = 3600.
deg2 = np.power(np.pi/180.,2)
msun_pc3 = msun*np.power(pc,-3)
msun_kpc3 = msun*np.power(kpc,-3)
msun2_pc5 = np.power(msun,2)*np.power(pc,-5)
msun2_kpc5 = np.power(msun,2)*np.power(kpc,-5)
gev2_cm5 = np.power(gev,2)
gev_cm3 = np.power(gev,1)
gev_cm2 = np.power(gev,1)
g_cm3 = 1.0
cm3_s = 1.0
if __name__ == '__main__':
print "Line-of-sight Integral Package..."
import matplotlib.pyplot as plt
psi = np.linspace(0.01,0.1,500)
dp = NFWProfile(1,1)
fn0 = LoSIntegralFnFast(dp,100,10)
fn1 = LoSIntegralFn(dp,100,10)
dhalo = np.linspace(100,100,500)
v0 = fn0(dhalo,psi)
v1 = fn1(dhalo,psi)
delta = (v1-v0)/v0
print delta
plt.hist(delta,range=[min(delta),max(delta)],bins=100)
plt.show()