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)