class LensCosmo(object):
"""
class to manage the physical units and distances present in a single plane lens with fixed input cosmology
"""
def __init__(self, z_lens, z_source, cosmo=None):
"""
:param z_lens: redshift of lens
:param z_source: redshift of source
:param cosmo: astropy.cosmology instance
"""
self.z_lens = z_lens
self.z_source = z_source
self.background = Background(cosmo=cosmo)
self.nfw_param = NFWParam()
def a_z(self, z):
"""
convert redshift into scale factor
:param z: redshift
:return: scale factor
"""
return 1. / (1. + z)
@property
def h(self):
return self.background.cosmo.H(0).value / 100.
@property
def D_d(self):
"""
:return: angular diameter distance to the deflector [Mpc]
"""
return self.background.D_xy(0, self.z_lens)
@property
def D_s(self):
"""
:return: angular diameter distance to the source [Mpc]
"""
return self.background.D_xy(0, self.z_source)
@property
def D_ds(self):
"""
:return: angular diameter distance from deflector to source [Mpc]
"""
return self.background.D_xy(self.z_lens, self.z_source)
@property
def D_dt(self):
"""
:return: time delay distance [Mpc]
"""
return (1 + self.z_lens) * self.D_d * self.D_s / self.D_ds
@property
def epsilon_crit(self):
"""
returns the critical projected mass density in units of M_sun/Mpc^2 (physical units)
:return: critical projected mass density
"""
if not hasattr(self, '_Epsilon_Crit'):
const_SI = const.c**2 / (4 * np.pi * const.G
) #c^2/(4*pi*G) in units of [kg/m]
conversion = const.Mpc / const.M_sun # converts [kg/m] to [M_sun/Mpc]
factor = const_SI * conversion #c^2/(4*pi*G) in units of [M_sun/Mpc]
self._Epsilon_Crit = self.D_s / (
self.D_d * self.D_ds) * factor #[M_sun/Mpc^2]
return self._Epsilon_Crit
def phys2arcsec_lens(self, phys):
"""
convert physical Mpc into arc seconds
:param phys: physical distance [Mpc]
:return: angular diameter [arcsec]
"""
return phys / self.D_d / const.arcsec
def arcsec2phys_lens(self, arcsec):
"""
convert angular to physical quantities for lens plane
:param arcsec: angular size at lens plane [arcsec]
:return: physical size at lens plane [Mpc]
"""
return arcsec * const.arcsec * self.D_d
def arcsec2phys_source(self, arcsec):
"""
convert angular to physical quantities for source plane
:param arcsec: angular size at source plane [arcsec]
:return: physical size at source plane [Mpc]
"""
return arcsec * const.arcsec * self.D_s
def kappa2proj_mass(self, kappa):
"""
convert convergence to projected mass M_sun/Mpc^2
:param kappa: lensing convergence
:return: projected mass [M_sun/Mpc^2]
"""
return kappa * self.epsilon_crit
def mass_in_theta_E(self, theta_E):
"""
mass within Einstein radius (area * epsilon crit) [M_sun]
:param theta_E: Einstein radius [arcsec]
:return: mass within Einstein radius [M_sun]
"""
mass = self.arcsec2phys_lens(theta_E)**2 * np.pi * self.epsilon_crit
return mass
def mass_in_coin(self, theta_E):
"""
:param theta_E: Einstein radius [arcsec]
:return: mass in coin calculated in mean density of the universe
"""
chi_L = self.background.T_xy(0, self.z_lens)
chi_S = self.background.T_xy(0, self.z_source)
return 1. / 3 * np.pi * (
chi_L * theta_E * const.arcsec
)**2 * chi_S * self.background.rho_crit #[M_sun/Mpc**3]
def time_delay_units(self, fermat_pot, kappa_ext=0):
"""
:param fermat_pot: in units of arcsec^2 (e.g. Fermat potential)
:param kappa_ext: unit-less
:return: time delay in days
"""
D_dt = self.D_dt / (1. - kappa_ext) * const.Mpc # eqn 7 in Suyu et al.
return D_dt / const.c * fermat_pot / const.day_s * const.arcsec**2 # * self.arcsec2phys_lens(1.)**2
def time_delay2fermat_pot(self, dt):
"""
:param dt: time delay in units of days
:return: Fermat potential in units arcsec**2 for a given cosmology
"""
D_dt = self.D_dt * const.Mpc
return dt * const.c * const.day_s / D_dt / const.arcsec**2
def nfw_angle2physical(self, Rs_angle, alpha_Rs):
"""
converts the angular parameters into the physical ones for an NFW profile
:param alpha_Rs: observed bending angle at the scale radius in units of arcsec
:param Rs: scale radius in units of arcsec
:return: M200, r200, Rs_physical, c
"""
Rs = Rs_angle * const.arcsec * self.D_d
theta_scaled = alpha_Rs * self.epsilon_crit * self.D_d * const.arcsec
rho0 = theta_scaled / (4 * Rs**2 * (1 + np.log(1. / 2.)))
rho0_com = rho0 / self.h**2 * self.a_z(self.z_lens)**3
c = self.nfw_param.c_rho0(rho0_com)
r200 = c * Rs
M200 = self.nfw_param.M_r200(
r200 * self.h / self.a_z(self.z_lens)) / self.h
return rho0, Rs, c, r200, M200
def nfw_physical2angle(self, M, c):
"""
converts the physical mass and concentration parameter of an NFW profile into the lensing quantities
:param M: mass enclosed 200 rho_crit in units of M_sun
:param c: NFW concentration parameter (r200/r_s)
:return: alpha_Rs (observed bending angle at the scale radius, Rs_angle (angle at scale radius) (in units of arcsec)
"""
rho0, Rs, r200 = self.nfwParam_physical(M, c)
Rs_angle = Rs / self.D_d / const.arcsec # Rs in arcsec
alpha_Rs = rho0 * (4 * Rs**2 * (1 + np.log(1. / 2.)))
return Rs_angle, alpha_Rs / self.epsilon_crit / self.D_d / const.arcsec
def nfwParam_physical(self, M, c):
"""
returns the NFW parameters in physical units
:param M: physical mass in M_sun
:param c: concentration
:return:
"""
r200 = self.nfw_param.r200_M(M * self.h) / self.h * self.a_z(
self.z_lens) # physical radius r200
rho0 = self.nfw_param.rho0_c(c) * self.h**2 / self.a_z(
self.z_lens)**3 # physical density in M_sun/Mpc**3
Rs = r200 / c
return rho0, Rs, r200
def sis_theta_E2sigma_v(self, theta_E):
"""
converts the lensing Einstein radius into a physical velocity dispersion
:param theta_E: Einstein radius (in arcsec)
:return: velocity dispersion in units (km/s)
"""
v_sigma_c2 = theta_E * const.arcsec / (4 *
np.pi) * self.D_s / self.D_ds
return np.sqrt(v_sigma_c2) * const.c / 1000
def sis_sigma_v2theta_E(self, v_sigma):
"""
converts the velocity dispersion into an Einstein radius for a SIS profile
:param v_sigma: velocity dispersion (km/s)
:return: theta_E (arcsec)
"""
theta_E = 4 * np.pi * (
v_sigma * 1000. / const.c)**2 * self.D_ds / self.D_s / const.arcsec
return theta_E
class LensCosmo(object):
"""
class to manage the physical units and distances present in a single plane lens with fixed input cosmology
"""
def __init__(self, z_lens, z_source, cosmo=None):
"""
:param z_lens: redshift of lens
:param z_source: redshift of source
:param cosmo: astropy.cosmology instance
"""
self.z_lens = z_lens
self.z_source = z_source
self.background = Background(cosmo=cosmo)
self.nfw_param = NFWParam(cosmo=cosmo)
def a_z(self, z):
"""
convert redshift into scale factor
:param z: redshift
:return: scale factor
"""
return 1. / (1. + z)
@property
def h(self):
return self.background.cosmo.H(0).value / 100.
@property
def dd(self):
"""
:return: angular diameter distance to the deflector [Mpc]
"""
return self.background.d_xy(0, self.z_lens)
@property
def ds(self):
"""
:return: angular diameter distance to the source [Mpc]
"""
return self.background.d_xy(0, self.z_source)
@property
def dds(self):
"""
:return: angular diameter distance from deflector to source [Mpc]
"""
return self.background.d_xy(self.z_lens, self.z_source)
@property
def ddt(self):
"""
:return: time delay distance [Mpc]
"""
return (1 + self.z_lens) * self.dd * self.ds / self.dds
@property
def sigma_crit(self):
"""
returns the critical projected lensing mass density in units of M_sun/Mpc^2
:return: critical projected lensing mass density
"""
if not hasattr(self, '_sigma_crit_mpc'):
const_SI = const.c**2 / (4 * np.pi * const.G
) # c^2/(4*pi*G) in units of [kg/m]
conversion = const.Mpc / const.M_sun # converts [kg/m] to [M_sun/Mpc]
factor = const_SI * conversion # c^2/(4*pi*G) in units of [M_sun/Mpc]
self._sigma_crit_mpc = self.ds / (
self.dd * self.dds) * factor # [M_sun/Mpc^2]
return self._sigma_crit_mpc
@property
def sigma_crit_angle(self):
"""
returns the critical surface density in units of M_sun/arcsec^2 (in physical solar mass units)
when provided a physical mass per physical Mpc^2
:return: critical projected mass density
"""
if not hasattr(self, '_sigma_crit_arcsec'):
const_SI = const.c**2 / (4 * np.pi * const.G
) # c^2/(4*pi*G) in units of [kg/m]
conversion = const.Mpc / const.M_sun # converts [kg/m] to [M_sun/Mpc]
factor = const_SI * conversion # c^2/(4*pi*G) in units of [M_sun/Mpc]
self._sigma_crit_arcsec = self.ds / (
self.dd * self.dds) * factor * (
self.dd * const.arcsec)**2 # [M_sun/arcsec^2]
return self._sigma_crit_arcsec
def phys2arcsec_lens(self, phys):
"""
convert physical Mpc into arc seconds
:param phys: physical distance [Mpc]
:return: angular diameter [arcsec]
"""
return phys / self.dd / const.arcsec
def arcsec2phys_lens(self, arcsec):
"""
convert angular to physical quantities for lens plane
:param arcsec: angular size at lens plane [arcsec]
:return: physical size at lens plane [Mpc]
"""
return arcsec * const.arcsec * self.dd
def arcsec2phys_source(self, arcsec):
"""
convert angular to physical quantities for source plane
:param arcsec: angular size at source plane [arcsec]
:return: physical size at source plane [Mpc]
"""
return arcsec * const.arcsec * self.ds
def kappa2proj_mass(self, kappa):
"""
convert convergence to projected mass M_sun/Mpc^2
:param kappa: lensing convergence
:return: projected mass [M_sun/Mpc^2]
"""
return kappa * self.sigma_crit
def mass_in_theta_E(self, theta_E):
"""
mass within Einstein radius (area * epsilon crit) [M_sun]
:param theta_E: Einstein radius [arcsec]
:return: mass within Einstein radius [M_sun]
"""
mass = self.arcsec2phys_lens(theta_E)**2 * np.pi * self.sigma_crit
return mass
def mass_in_coin(self, theta_E):
"""
:param theta_E: Einstein radius [arcsec]
:return: mass in coin calculated in mean density of the universe
"""
chi_L = self.background.T_xy(0, self.z_lens)
chi_S = self.background.T_xy(0, self.z_source)
return 1. / 3 * np.pi * (
chi_L * theta_E * const.arcsec
)**2 * chi_S * self.background.rho_crit #[M_sun/Mpc**3]
def time_delay_units(self, fermat_pot, kappa_ext=0):
"""
:param fermat_pot: in units of arcsec^2 (e.g. Fermat potential)
:param kappa_ext: unit-less external shear not accounted for in the Fermat potential
:return: time delay in days
"""
D_dt = self.ddt * (1. - kappa_ext) * const.Mpc # eqn 7 in Suyu et al.
return D_dt / const.c * fermat_pot / const.day_s * const.arcsec**2 # * self.arcsec2phys_lens(1.)**2
def time_delay2fermat_pot(self, dt):
"""
:param dt: time delay in units of days
:return: Fermat potential in units arcsec**2 for a given cosmology
"""
D_dt = self.ddt * const.Mpc
return dt * const.c * const.day_s / D_dt / const.arcsec**2
def nfw_angle2physical(self, Rs_angle, alpha_Rs):
"""
converts the angular parameters into the physical ones for an NFW profile
:param alpha_Rs: observed bending angle at the scale radius in units of arcsec
:param Rs: scale radius in units of arcsec
:return: rho0, Rs, c, r200, M200
"""
Rs = Rs_angle * const.arcsec * self.dd
theta_scaled = alpha_Rs * self.sigma_crit * self.dd * const.arcsec
rho0 = theta_scaled / (4 * Rs**2 * (1 + np.log(1. / 2.)))
rho0_com = rho0 / self.h**2
c = self.nfw_param.c_rho0(rho0_com, self.z_lens)
r200 = c * Rs
M200 = self.nfw_param.M_r200(r200 * self.h, self.z_lens) / self.h
return rho0, Rs, c, r200, M200
def nfw_physical2angle(self, M, c):
"""
converts the physical mass and concentration parameter of an NFW profile into the lensing quantities
:param M: mass enclosed 200 rho_crit in units of M_sun (physical units, meaning no little h)
:param c: NFW concentration parameter (r200/r_s)
:return: Rs_angle (angle at scale radius) (in units of arcsec), alpha_Rs (observed bending angle at the scale radius
"""
rho0, Rs, r200 = self.nfwParam_physical(M, c)
Rs_angle = Rs / self.dd / const.arcsec # Rs in arcsec
alpha_Rs = rho0 * (4 * Rs**2 * (1 + np.log(1. / 2.)))
return Rs_angle, alpha_Rs / self.sigma_crit / self.dd / const.arcsec
def nfwParam_physical(self, M, c):
"""
returns the NFW parameters in physical units
:param M: physical mass in M_sun
:param c: concentration
:return: rho0, Rs, r200
"""
r200 = self.nfw_param.r200_M(
M * self.h, self.z_lens) / self.h # physical radius r200
rho0 = self.nfw_param.rho0_c(
c, self.z_lens) * self.h**2 # physical density in M_sun/Mpc**3
Rs = r200 / c
return rho0, Rs, r200
def nfw_M_theta_vir(self, M):
"""
returns virial radius in angular units of arc seconds on the sky
:param M: physical mass in M_sun
:return: angle (in arc seconds) of the virial radius
"""
r200 = self.nfw_param.r200_M(
M * self.h, self.z_lens) / self.h # physical radius r200
theta_r200 = r200 / self.dd / const.arcsec
return theta_r200
def sis_theta_E2sigma_v(self, theta_E):
"""
converts the lensing Einstein radius into a physical velocity dispersion
:param theta_E: Einstein radius (in arcsec)
:return: velocity dispersion in units (km/s)
"""
v_sigma_c2 = theta_E * const.arcsec / (4 * np.pi) * self.ds / self.dds
return np.sqrt(v_sigma_c2) * const.c / 1000
def sis_sigma_v2theta_E(self, v_sigma):
"""
converts the velocity dispersion into an Einstein radius for a SIS profile
:param v_sigma: velocity dispersion (km/s)
:return: theta_E (arcsec)
"""
theta_E = 4 * np.pi * (v_sigma * 1000. /
const.c)**2 * self.dds / self.ds / const.arcsec
return theta_E
def uldm_angular2phys(self, kappa_0, theta_c):
"""
converts the anguar parameters entering the LensModel Uldm() (Ultra Light
Dark Matter) class in physical masses, i.e. the total soliton mass and the
mass of the particle
:param kappa_0: central convergence of profile
:param theta_c: core radius (in arcseconds)
:return: m_eV_log10, M_sol_log10, the log10 of the masses, m in eV and M in M_sun
"""
D_Lens = self.dd * 10**6 # in parsec
Sigma_c = self.sigma_crit * 10**(-12) # in M_sun / parsec^2
r_c = theta_c * const.arcsec * D_Lens
rho0 = 2048 * np.sqrt(0.091) * kappa_0 * Sigma_c / (429 * np.pi * r_c)
m_log10 = -22 + 0.5 * np.log10(190 / rho0 * (r_c / 100)**(-4))
M_log10 = 9 + np.log10(160 * 1.4 / r_c) - 2 * (m_log10 + 22)
return m_log10, M_log10
def uldm_mphys2angular(self, m_log10, M_log10):
"""
converts physical ULDM mass in the ones, in angular units, that enter
the LensModel Uldm() class
:param m_log10: exponent of ULDM mass in eV
:param M_log10: exponent of soliton mass in M_sun
:return: kappa_0, theta_c, the central convergence and core radius (in arcseconds)
"""
D_Lens = self.dd * 10**6 # in parsec
Sigma_c = self.sigma_crit * 10**(-12) # in M_sun/parsec^2
m22 = 10**(m_log10 + 22)
M9 = 10**(M_log10 - 9)
r_c = 160 * 1.4 * m22**(-2) * M9**(-1) # core radius in parsec
rho0 = 190 * m22**(-2) * (r_c / 100)**(
-4) # central density in M_sun/parsec^3
kappa_0 = 429 * np.pi * rho0 * r_c / (2048 * np.sqrt(0.091) * Sigma_c)
theta_c = r_c / D_Lens / const.arcsec
return kappa_0, theta_c
