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 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 + 0.091(r/r_c)^2)^{-8} where :math:`r_c` is the core radius, corresponding to the radius where the density drops by half its central value it has, as parameters: :param kappa_0: central convergence :param theta_c: core radius (in arcseconds) """ _s = 0.000001 # numerical limit for minimal radius param_names = ['kappa_0', 'theta_c', 'center_x', 'center_y'] # rule of thumb: m_phys = 10^-15 m_noCosmo, M_phys = 10^21 M_noCosmo lower_limit_default = {'kappa_0': 0, 'theta_c': 0, 'center_x': -100, 'center_y': -100} upper_limit_default = {'kappa_0': 1., 'theta_c': 100, 'center_x': 100, 'center_y': 100}
[docs] def rhotilde(self, kappa_0, theta_c): """ Computes the central density in angular units :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :return: central density in 1/arcsec """ num_factor = 2048 * np.sqrt(0.091) /(429 * np.pi) return kappa_0 * num_factor / theta_c
def _lensing_integral(self, x): """ The analitic result of the integral entering the computation of the lensing potential, that is ..math:: \int dy/y (1 - (1 + y^2)^{-13/2}) :param x: evaluation point of the integral :return: result of the antiderivative in x """ denominator = 3465*(x**2 +1)**(5.5) numerator = 3465*x**10 + 18480*x**8 + 39963*x**6 + 44154*x**4 + 25399*x**2 + 6508 return np.log(np.sqrt(x**2 +1) + 1) - numerator/denominator
[docs] def function(self, x, y, kappa_0, theta_c, center_x=0, center_y=0): """ :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 center_x: center of halo (in angular units) :param center_y: center of halo (in angular units) :return: lensing potential (in arcsec^2) """ x_ = x - center_x y_ = y - center_y r = np.sqrt(x_** 2 + y_** 2) r = np.maximum(r, self._s) Integral_factor = self._lensing_integral(np.sqrt(0.091) * r / theta_c) prefactor = 2/ 13 * kappa_0 * theta_c**2 / 0.091 return prefactor * Integral_factor
[docs] def alpha_radial(self, r, kappa_0, theta_c): """ returns the radial part of the deflection angle :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param r: radius where the deflection angle is computed :return: radial deflection angle """ prefactor = 2/ 13 * kappa_0 * theta_c**2 / 0.091 denominator_factor = (1 + 0.091 * r**2/theta_c**2)**(6.5) return prefactor/r * (1 - 1/denominator_factor)
[docs] def derivatives(self, x, y, kappa_0, theta_c, center_x=0, center_y=0): """ 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 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) * x_ / R f_y = self.alpha_radial(R, kappa_0, theta_c) * y_ / R return f_x, f_y
[docs] def hessian(self, x, y, kappa_0, theta_c, center_x=0, center_y=0): """ :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 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) prefactor = 2/ 13 * kappa_0 * theta_c**2 / 0.091 # denominator factor denominator = 1 + 0.091 * R**2/theta_c**2 factor1 = 13 * 0.091 * denominator**(-7.5) / (theta_c**2 * R**2) factor2 = 1/R**4 * (1 - denominator**(-6.5)) 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): """ 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) :return: rho(R) density in angular units """ rhotilde = self.rhotilde(kappa_0, theta_c) return rhotilde/(1 + 0.091* (R/theta_c)**2)**8
[docs] def density_lens(self, r, kappa_0, theta_c): """ 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) :return: density rho(r) """ return self.density(r, kappa_0, theta_c)
[docs] def kappa_r(self, R, kappa_0, theta_c): """ 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 :return: convergence at r """ return kappa_0 * (1 + 0.091 * (R/theta_c)**2)**(-15/2)
[docs] def density_2d(self, x, y, kappa_0, theta_c, center_x=0, center_y=0): """ 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 :type R: float/numpy array :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :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)
def _mass_integral(self, x): """ Returns the analitic result of the integral appearing in mass expression """ numerator = x * (3465 * x**12 + 23100 * x**10 + 65373 * x**8 + 101376*x**6 + 92323*x**4 + 48580 * x**2 - 3465) denominator = 215040 * (x**2 + 1)**7 result = 33 * np.arctan(x) / 2048 + numerator/denominator return result
[docs] def mass_3d(self, R, kappa_0, theta_c): """ 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) :return: mass of soliton in angular units """ rhotilde = self.rhotilde(kappa_0, theta_c) prefactor = 4. * np.pi * rhotilde * theta_c**3 / (0.091)**(1.5) m_3d = prefactor * (self._mass_integral(R/theta_c * np.sqrt(0.091)) - self._mass_integral(0) ) return m_3d
[docs] def mass_3d_lens(self, r, kappa_0, theta_c): """ mass enclosed a 3d sphere or radius r :param R: :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param rho0: central density in angular units :return: mass """ m_3d = self.mass_3d(r, kappa_0, theta_c) return m_3d
[docs] def mass_2d(self, R, kappa_0, theta_c): """ 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) :return: mass enclosed in 2d sphere """ m_2d = np.pi * R * self.alpha_radial(R, kappa_0, theta_c) return m_2d