class NFW_ELLIPSE(LensProfileBase):
"""
this class contains functions concerning the NFW profile with an ellipticity defined in the potential
parameterization of alpha_Rs and Rs is the same as for the spherical NFW profile
from Glose & Kneib: https://cds.cern.ch/record/529584/files/0112138.pdf
relation are: R_200 = c * Rs
"""
profile_name = 'NFW_ELLIPSE'
param_names = ['Rs', 'alpha_Rs', 'e1', 'e2', 'center_x', 'center_y']
lower_limit_default = {
'Rs': 0,
'alpha_Rs': 0,
'e1': -0.5,
'e2': -0.5,
'center_x': -100,
'center_y': -100
}
upper_limit_default = {
'Rs': 100,
'alpha_Rs': 10,
'e1': 0.5,
'e2': 0.5,
'center_x': 100,
'center_y': 100
}
def __init__(self, interpol=False, num_interp_X=1000, max_interp_X=10):
"""
:param interpol: bool, if True, interpolates the functions F(), g() and h()
:param num_interp_X: int (only considered if interpol=True), number of interpolation elements in units of r/r_s
:param max_interp_X: float (only considered if interpol=True), maximum r/r_s value to be interpolated
(returning zeros outside)
"""
self.nfw = NFW(interpol=interpol,
num_interp_X=num_interp_X,
max_interp_X=max_interp_X)
self._diff = 0.0000000001
super(NFW_ELLIPSE, self).__init__()
def function(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""
returns elliptically distorted NFW lensing potential
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
:param e1: eccentricity component in x-direction
:param e2: eccentricity component in y-direction
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: lensing potential
"""
x_, y_ = param_util.transform_e1e2_square_average(
x, y, e1, e2, center_x, center_y)
R_ = np.sqrt(x_**2 + y_**2)
rho0_input = self.nfw.alpha2rho0(alpha_Rs=alpha_Rs, Rs=Rs)
if Rs < 0.0000001:
Rs = 0.0000001
f_ = self.nfw.nfwPot(R_, Rs, rho0_input)
return f_
def derivatives(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""
returns df/dx and df/dy of the function, calculated as an elliptically distorted deflection angle of the
spherical NFW profile
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
:param e1: eccentricity component in x-direction
:param e2: eccentricity component in y-direction
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: deflection in x-direction, deflection in y-direction
"""
x_, y_ = param_util.transform_e1e2_square_average(
x, y, e1, e2, center_x, center_y)
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = abs(1 - q)
R_ = np.sqrt(x_**2 + y_**2)
rho0_input = self.nfw.alpha2rho0(alpha_Rs=alpha_Rs, Rs=Rs)
if Rs < 0.0000001:
Rs = 0.0000001
f_x_prim, f_y_prim = self.nfw.nfwAlpha(R_, Rs, rho0_input, x_, y_)
f_x_prim *= np.sqrt(1 - e)
f_y_prim *= np.sqrt(1 + e)
f_x = cos_phi * f_x_prim - sin_phi * f_y_prim
f_y = sin_phi * f_x_prim + cos_phi * f_y_prim
return f_x, f_y
def hessian(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""
returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy
the calculation is performed as a numerical differential from the deflection field. Analytical relations are possible
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
:param e1: eccentricity component in x-direction
:param e2: eccentricity component in y-direction
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
"""
alpha_ra, alpha_dec = self.derivatives(x, y, Rs, alpha_Rs, e1, e2,
center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, Rs, alpha_Rs,
e1, e2, center_x,
center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, Rs, alpha_Rs,
e1, e2, center_x,
center_y)
f_xx = (alpha_ra_dx - alpha_ra) / diff
f_xy = (alpha_ra_dy - alpha_ra) / diff
f_yx = (alpha_dec_dx - alpha_dec) / diff
f_yy = (alpha_dec_dy - alpha_dec) / diff
return f_xx, f_xy, f_yx, f_yy
def mass_3d_lens(self, R, Rs, alpha_Rs, e1=1, e2=0):
"""
:param R: radius (in angular units)
:param Rs:
:param alpha_Rs:
:param e1:
:param e2:
:return:
"""
return self.nfw.mass_3d_lens(R, Rs, alpha_Rs)
def density_lens(self, r, Rs, alpha_Rs, e1=1, e2=0):
"""
computes the density at 3d radius r given lens model parameterization.
The integral in the LOS projection of this quantity results in the convergence quantity.
:param r: 3d radios
:param Rs: turn-over radius of NFW profile
:param alpha_Rs: deflection at Rs
:return: density rho(r)
"""
return self.nfw.density_lens(r, Rs, alpha_Rs)
class NFW_ELLIPSE(object):
"""
this class contains functions concerning the NFW profile
relation are: R_200 = c * Rs
"""
def __init__(self):
self.nfw = NFW()
self._diff = 0.000001
def function(self, x, y, Rs, theta_Rs, q, phi_G, center_x=0, center_y=0):
"""
returns double integral of NFW profile
"""
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = min(abs(1. - q), 0.99)
xt1 = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
xt2 = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
R_ = np.sqrt(xt1**2 + xt2**2)
rho0_input = self.nfw._alpha2rho0(theta_Rs=theta_Rs, Rs=Rs)
if Rs < 0.0001:
Rs = 0.0001
f_ = self.nfw.nfwPot(R_, Rs, rho0_input)
return f_
def derivatives(self,
x,
y,
Rs,
theta_Rs,
q,
phi_G,
center_x=0,
center_y=0):
"""
returns df/dx and df/dy of the function (integral of NFW)
"""
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = min(abs(1. - q), 0.99)
xt1 = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
xt2 = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
R_ = np.sqrt(xt1**2 + xt2**2)
rho0_input = self.nfw._alpha2rho0(theta_Rs=theta_Rs, Rs=Rs)
if Rs < 0.0001:
Rs = 0.0001
f_x_prim, f_y_prim = self.nfw.nfwAlpha(R_, Rs, rho0_input, xt1, xt2)
f_x_prim *= np.sqrt(1 - e)
f_y_prim *= np.sqrt(1 + e)
f_x = cos_phi * f_x_prim - sin_phi * f_y_prim
f_y = sin_phi * f_x_prim + cos_phi * f_y_prim
return f_x, f_y
def hessian(self, x, y, Rs, theta_Rs, q, phi_G, center_x=0, center_y=0):
"""
returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy
"""
alpha_ra, alpha_dec = self.derivatives(x, y, Rs, theta_Rs, q, phi_G,
center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, Rs, theta_Rs,
q, phi_G, center_x,
center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, Rs, theta_Rs,
q, phi_G, center_x,
center_y)
f_xx = (alpha_ra_dx - alpha_ra) / diff
f_xy = (alpha_ra_dy - alpha_ra) / diff
f_yx = (alpha_dec_dx - alpha_dec) / diff
f_yy = (alpha_dec_dy - alpha_dec) / diff
return f_xx, f_yy, f_xy
def mass_3d_lens(self, R, Rs, theta_Rs, q=1, phi_G=0):
"""
:param R:
:param Rs:
:param theta_Rs:
:param q:
:param phi_G:
:return:
"""
return self.nfw.mass_3d(R, Rs, theta_Rs)
class NFW_ELLIPSE(object):
"""
this class contains functions concerning the NFW profile
relation are: R_200 = c * Rs
"""
param_names = ['Rs', 'alpha_Rs', 'e1', 'e2', 'center_x', 'center_y']
lower_limit_default = {
'Rs': 0,
'alpha_Rs': 0,
'e1': -0.5,
'e2': -0.5,
'center_x': -100,
'center_y': -100
}
upper_limit_default = {
'Rs': 100,
'alpha_Rs': 10,
'e1': 0.5,
'e2': 0.5,
'center_x': 100,
'center_y': 100
}
def __init__(self, interpol=False, num_interp_X=1000, max_interp_X=10):
self.nfw = NFW(interpol=interpol,
num_interp_X=num_interp_X,
max_interp_X=max_interp_X)
self._diff = 0.0000000001
def function(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""
returns double integral of NFW profile
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = min(abs(1. - q), 0.99)
xt1 = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
xt2 = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
R_ = np.sqrt(xt1**2 + xt2**2)
rho0_input = self.nfw._alpha2rho0(alpha_Rs=alpha_Rs, Rs=Rs)
if Rs < 0.0000001:
Rs = 0.0000001
f_ = self.nfw.nfwPot(R_, Rs, rho0_input)
return f_
def derivatives(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""
returns df/dx and df/dy of the function (integral of NFW)
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = min(abs(1. - q), 0.99)
xt1 = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
xt2 = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
R_ = np.sqrt(xt1**2 + xt2**2)
rho0_input = self.nfw._alpha2rho0(alpha_Rs=alpha_Rs, Rs=Rs)
if Rs < 0.0000001:
Rs = 0.0000001
f_x_prim, f_y_prim = self.nfw.nfwAlpha(R_, Rs, rho0_input, xt1, xt2)
f_x_prim *= np.sqrt(1 - e)
f_y_prim *= np.sqrt(1 + e)
f_x = cos_phi * f_x_prim - sin_phi * f_y_prim
f_y = sin_phi * f_x_prim + cos_phi * f_y_prim
return f_x, f_y
def hessian(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""
returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy
"""
alpha_ra, alpha_dec = self.derivatives(x, y, Rs, alpha_Rs, e1, e2,
center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, Rs, alpha_Rs,
e1, e2, center_x,
center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, Rs, alpha_Rs,
e1, e2, center_x,
center_y)
f_xx = (alpha_ra_dx - alpha_ra) / diff
f_xy = (alpha_ra_dy - alpha_ra) / diff
f_yx = (alpha_dec_dx - alpha_dec) / diff
f_yy = (alpha_dec_dy - alpha_dec) / diff
return f_xx, f_yy, f_xy
def mass_3d_lens(self, R, Rs, alpha_Rs, e1=1, e2=0):
"""
:param R:
:param Rs:
:param alpha_Rs:
:param q:
:param phi_G:
:return:
"""
return self.nfw.mass_3d(R, Rs, alpha_Rs)
class TestTNFW(object):
def setup(self):
self.nfw = NFW()
self.tnfw = TNFW()
def test_deflection(self):
Rs = 0.2
theta_Rs = 0.1
r_trunc = 1000000000000 * Rs
x = np.linspace(0.0 * Rs, 5 * Rs, 1000)
y = np.linspace(0., 1, 1000)
xdef_t, ydef_t = self.tnfw.derivatives(x, y, Rs, theta_Rs, r_trunc)
xdef, ydef = self.nfw.derivatives(x, y, Rs, theta_Rs)
np.testing.assert_almost_equal(xdef_t, xdef, 5)
np.testing.assert_almost_equal(ydef_t, ydef, 5)
def test_potential(self):
Rs = 0.2
theta_Rs = 0.1
r_trunc = 1000000000000 * Rs
x = np.linspace(0.1 * Rs, 5 * Rs, 1000)
y = np.linspace(0.2, 1, 1000)
pot_t = self.tnfw.nfwPot((x ** 2 + y ** 2) ** .5, Rs, theta_Rs, r_trunc)
pot = self.nfw.nfwPot((x ** 2 + y ** 2) ** .5, Rs, theta_Rs)
np.testing.assert_almost_equal(pot, pot_t, 4)
def test_gamma(self):
Rs = 0.2
theta_Rs = 0.1
r_trunc = 1000000000000 * Rs
x = np.linspace(0.1 * Rs, 5 * Rs, 1000)
y = np.linspace(0.2, 1, 1000)
g1t, g2t = self.tnfw.nfwGamma((x ** 2 + y ** 2) ** .5, Rs, theta_Rs, r_trunc, x, y)
g1, g2 = self.nfw.nfwGamma((x ** 2 + y ** 2) ** .5, Rs, theta_Rs, x, y)
np.testing.assert_almost_equal(g1t, g1, 5)
np.testing.assert_almost_equal(g2t, g2, 5)
def test_hessian(self):
Rs = 0.2
theta_Rs = 0.1
r_trunc = 1000000000000 * Rs
x = np.linspace(0.1 * Rs, 5 * Rs, 100)
y = np.linspace(0.2, 1, 100)
xxt, yyt, xyt = self.tnfw.hessian(x, y, Rs, theta_Rs, r_trunc)
xx, yy, xy = self.nfw.hessian(x, y, Rs, theta_Rs)
np.testing.assert_almost_equal(xy, xyt, 4)
np.testing.assert_almost_equal(yy, yyt, 4)
np.testing.assert_almost_equal(xy, xyt, 4)
Rs = 0.2
r_trunc = 5
xxt, yyt, xyt = self.tnfw.hessian(Rs, 0, Rs, theta_Rs, r_trunc)
xxt_delta, yyt_delta, xyt_delta = self.tnfw.hessian(Rs+0.000001, 0, Rs, theta_Rs, r_trunc)
npt.assert_almost_equal(xxt, xxt_delta, decimal=6)
def test_density_2d(self):
Rs = 0.2
theta_Rs = 0.1
r_trunc = 1000000000000 * Rs
x = np.linspace(0.1 * Rs, 3 * Rs, 1000)
y = np.linspace(0.2, 0.5, 1000)
kappa_t = self.tnfw.density_2d(x, y, Rs, theta_Rs, r_trunc)
kappa = self.nfw.density_2d(x, y, Rs, theta_Rs)
np.testing.assert_almost_equal(kappa, kappa_t, 5)
def test_numerical_derivatives(self):
Rs = 0.2
theta_Rs = 0.1
r_trunc = 1.5 * Rs
diff = 1e-9
x0, y0 = 0.1, 0.1
x_def_t, y_def_t = self.tnfw.derivatives(x0,y0,Rs,theta_Rs,r_trunc)
x_def_t_deltax, _ = self.tnfw.derivatives(x0+diff, y0, Rs, theta_Rs,r_trunc)
x_def_t_deltay, y_def_t_deltay = self.tnfw.derivatives(x0, y0 + diff, Rs, theta_Rs,r_trunc)
actual = self.tnfw.hessian(x0,y0,Rs,theta_Rs,r_trunc)
f_xx_approx = (x_def_t_deltax - x_def_t) * diff ** -1
f_yy_approx = (y_def_t_deltay - y_def_t) * diff ** -1
f_xy_approx = (x_def_t_deltay - y_def_t) * diff ** -1
numerical = [f_xx_approx,f_yy_approx,f_xy_approx]
for (approx,true) in zip(numerical,actual):
npt.assert_almost_equal(approx,true)