Source code for lenstronomy.LensModel.Profiles.uldm

__author__ = 'lucateo'

# this file contains a class to compute the Ultra Light Dark Matter soliton profile
import numpy as np
import scipy.interpolate as interp
from scipy.special import gamma, hyp2f1
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
import lenstronomy.Util.constants as const
__all__ = ['Uldm']


[docs]class Uldm(LensProfileBase): """ This class contains functions concerning the ULDM soliton density profile, whose good approximation is (see for example https://arxiv.org/pdf/1406.6586.pdf ) .. math:: \\rho = \\rho_0 (1 + a(\\theta/\\theta_c)^2)^{-\\beta} where :math:`\\theta_c` is the core radius, corresponding to the radius where the density drops by half its central value, :math: `\\beta` is the slope (called just slope in the parameters of this model), :math: `\\rho_0 = \\kappa_0 \\Sigma_c/D_lens`, and :math: `a` is a parameter, dependent on :math: `\\beta`, chosen such that :math: `\\theta_c` indeed corresponds to the radius where the density drops by half (simple math gives :math: `a = 0.5^{-1/\\beta} - 1` ). For an ULDM soliton profile without contributions to background potential, it turns out that :math: `\\beta = 8, a = 0.091`. We allow :math: `\\beta` to be different from 8 to model solitons which feel the influence of background potential (see 2105.10873) The profile has, as parameters: :param kappa_0: central convergence :param theta_c: core radius (in arcseconds) :param slope: exponent entering the profile, default value is 8 """ _s = 0.000001 # numerical limit for minimal radius param_names = ['kappa_0', 'theta_c', 'slope', 'center_x', 'center_y'] lower_limit_default = {'kappa_0': 0, 'theta_c': 0, 'slope': 3.5, 'center_x': -100, 'center_y': -100} upper_limit_default = {'kappa_0': 1., 'theta_c': 100, 'slope': 10, 'center_x': 100, 'center_y': 100}
[docs] def rhotilde(self, kappa_0, theta_c, slope=8): """ Computes the central density in angular units :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: central density in 1/arcsec """ a_factor_sqrt = np.sqrt( (0.5)**(-1/slope) -1) num_factor = gamma(slope) / gamma(slope - 1/2) * a_factor_sqrt / np.sqrt(np.pi) return kappa_0 * num_factor / theta_c
[docs] def function(self, x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8): """ :param x: angular position (normally in units of arc seconds) :param y: angular position (normally in units of arc seconds) :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :param center_x: center of halo (in angular units) :param center_y: center of halo (in angular units) :return: lensing potential (in arcsec^2) """ from mpmath import hyp3f2 x_ = x - center_x y_ = y - center_y r = np.sqrt(x_** 2 + y_** 2) r = np.maximum(r, self._s) a_factor_sqrt = np.sqrt( (0.5)**(-1./slope) -1) if np.isscalar(r) == True: hypgeom = float(kappa_0 /2 * r**2 * hyp3f2(1, 1, slope - 0.5, 2, 2, -(a_factor_sqrt * r /theta_c )**2)) else: hypgeom = np.array([ kappa_0 /2. * r_i**2. * hyp3f2(1, 1, slope - 0.5, 2, 2, -(a_factor_sqrt * r_i / theta_c)**2.) for r_i in r], dtype=float) return hypgeom
[docs] def alpha_radial(self, r, kappa_0, theta_c, slope=8): """ returns the radial part of the deflection angle :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :param r: radius where the deflection angle is computed :return: radial deflection angle """ a_factor = (0.5)**(-1./slope) -1 prefactor = 2./(2*slope -3) * kappa_0 * theta_c**2 / a_factor denominator_factor = (1 + a_factor * r**2/theta_c**2)**(slope - 3./2) return prefactor/r * (1 - 1/denominator_factor)
[docs] def derivatives(self, x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8): """ returns df/dx and df/dy of the function (lensing potential), which are the deflection angles :param x: angular position (normally in units of arc seconds) :param y: angular position (normally in units of arc seconds) :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :param center_x: center of halo (in angular units) :param center_y: center of halo (in angular units) :return: deflection angle in x, deflection angle in y """ x_ = x - center_x y_ = y - center_y R = np.sqrt(x_**2 + y_**2) R = np.maximum(R,0.00000001) f_x = self.alpha_radial(R, kappa_0, theta_c, slope) * x_ / R f_y = self.alpha_radial(R, kappa_0, theta_c, slope) * y_ / R return f_x, f_y
[docs] def hessian(self, x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8): """ :param x: angular position (normally in units of arc seconds) :param y: angular position (normally in units of arc seconds) :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :param center_x: center of halo (in angular units) :param center_y: center of halo (in angular units) :return: Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2 """ x_ = x - center_x y_ = y - center_y R = np.sqrt(x_**2 + y_**2) R = np.maximum(R,0.00000001) a_factor = (0.5)**(-1./slope) -1 prefactor = 2./(2*slope -3) * kappa_0 * theta_c**2 / a_factor # denominator factor denominator = 1 + a_factor * R**2/theta_c**2 factor1 = (2*slope - 3) * a_factor * denominator**(1./2 - slope) / (theta_c**2 * R**2) factor2 = 1/R**4 * (1 - denominator**(3./2 - slope)) f_xx = prefactor * (factor1 * x_**2 + factor2 * (y_**2 - x_**2)) f_yy = prefactor * (factor1 * y_**2 + factor2 * (x_**2 - y_**2)) f_xy = prefactor * (factor1 * x_ * y_ - factor2 * 2*x_*y_) return f_xx, f_xy, f_xy, f_yy
[docs] def density(self, R, kappa_0, theta_c, slope=8): """ three dimensional ULDM profile in angular units (rho0_physical = rho0_angular Sigma_crit / D_lens) :param R: radius of interest :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: rho(R) density in angular units """ rhotilde = self.rhotilde(kappa_0, theta_c, slope) a_factor = (0.5)**(-1./slope) -1 return rhotilde/(1 + a_factor* (R/theta_c)**2)**slope
[docs] def density_lens(self, r, kappa_0, theta_c, slope=8): """ 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 radius :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: density rho(r) """ return self.density(r, kappa_0, theta_c, slope)
[docs] def kappa_r(self, R, kappa_0, theta_c, slope=8): """ convergence of the cored density profile. This routine is also for testing :param R: radius (angular scale) :param kappa_0: convergence in the core :param theta_c: core radius :param slope: exponent entering the profile :return: convergence at r """ a_factor = (0.5)**(-1./slope) -1 return kappa_0 * (1 + a_factor * (R/theta_c)**2)**(1./2 - slope)
[docs] def density_2d(self, x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8): """ projected two dimensional ULDM profile (convergence * Sigma_crit), but given our units convention for rho0, it is basically the convergence :param R: radius of interest :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: Epsilon(R) projected density at radius R """ x_ = x - center_x y_ = y - center_y R = np.sqrt(x_**2 + y_**2) return self.kappa_r(R, kappa_0, theta_c, slope)
def _mass_integral(self, x, slope=8): """ Returns the analitic result of the integral appearing in mass expression :param slope: exponent entering the profile :return: integral result """ hypF = np.real(hyp2f1(3./2, slope, 5./2, - x**2)) return 1./3 * x**3 * hypF
[docs] def mass_3d(self, R, kappa_0, theta_c, slope=8): """ mass enclosed a 3d sphere or radius r :param R: radius in arcseconds :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: mass of soliton in angular units """ rhotilde = self.rhotilde(kappa_0, theta_c, slope) a_factor = (0.5)**(-1./slope) -1 prefactor = 4. * np.pi * rhotilde * theta_c**3 / (a_factor)**(1.5) m_3d = prefactor * (self._mass_integral(R/theta_c * np.sqrt(a_factor), slope) - self._mass_integral(0, slope) ) return m_3d
[docs] def mass_3d_lens(self, r, kappa_0, theta_c, slope=8): """ mass enclosed a 3d sphere or radius r :param r: radius over which the mass is computed :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: mass enclosed in 3D ball """ m_3d = self.mass_3d(r, kappa_0, theta_c, slope) return m_3d
[docs] def mass_2d(self, R, kappa_0, theta_c, slope=8): """ mass enclosed a 2d sphere or radius r :param R: radius over which the mass is computed :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: mass enclosed in 2d sphere """ m_2d = np.pi * R * self.alpha_radial(R, kappa_0, theta_c, slope) return m_2d