def _flat_age(self, z):
r"""Age of the universe in Gyr at redshift ``z``.
For :math:`\Omega_{rad} = 0` (:math:`T_{CMB} = 0`; massless neutrinos)
the age can be directly calculated as an elliptic integral [1]_.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
t : `~astropy.units.Quantity` ['time']
The age of the universe in Gyr at each input redshift.
References
----------
.. [1] Thomas, R., & Kantowski, R. (2000). Age-redshift relation for
standard cosmology. PRD, 62(10), 103507.
"""
# Use np.sqrt, np.arcsinh instead of math.sqrt, math.asinh
# to handle properly the complex numbers for 1 - Om0 < 0
prefactor = (2. / 3) * self._hubble_time / np.emath.sqrt(1 - self._Om0)
arg = np.arcsinh(
np.emath.sqrt((1 / self._Om0 - 1 + 0j) / (aszarr(z) + 1.0)**3))
return (prefactor * arg).real
def de_density_scale(self, z):
r"""Evaluates the redshift dependence of the dark energy density.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
I : ndarray or float
The scaling of the energy density of dark energy with redshift.
Returns `float` if the input is scalar.
Notes
-----
The scaling factor, I, is defined by :math:`\rho(z) = \rho_0 I`,
and in this case is given by
.. math::
I = \left(1 + z\right)^{3 \left(1 + w_0 - w_z\right)}
\exp \left(-3 w_z z\right)
"""
z = aszarr(z)
zp1 = z + 1.0 # (converts z [unit] -> z [dimensionless])
return zp1**(3. * (1. + self._w0 - self._wz)) * exp(-3. * self._wz * z)
def _dS_lookback_time(self, z):
r"""Lookback time in Gyr to redshift ``z``.
The lookback time is the difference between the age of the Universe now
and the age at redshift ``z``.
For :math:`\Omega_{rad} = 0` (:math:`T_{CMB} = 0`; massless neutrinos)
the age can be directly calculated.
.. math::
a = exp(H * t) \ \text{where t=0 at z=0}
t = (1/H) (ln 1 - ln a) = (1/H) (0 - ln (1/(1+z))) = (1/H) ln(1+z)
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
t : `~astropy.units.Quantity` ['time']
Lookback time in Gyr to each input redshift.
"""
return self._hubble_time * log(aszarr(z) + 1.0)
def inv_efunc(self, z):
r"""Function used to calculate :math:`\frac{1}{H_z}`.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
E : ndarray or float
The inverse redshift scaling of the Hubble constant.
Returns `float` if the input is scalar.
Defined such that :math:`H_z = H_0 / E`.
"""
Or = self._Ogamma0 + (self._Onu0 if not self._massivenu else
self._Ogamma0 * self.nu_relative_density(z))
zp1 = aszarr(z) + 1.0 # (converts z [unit] -> z [dimensionless])
return (zp1**3 * (Or * zp1 + self._Om0) + self._Ode0)**(-0.5)
def efunc(self, z):
"""Function used to calculate H(z), the Hubble parameter.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
E : ndarray or float
The redshift scaling of the Hubble constant.
Returns `float` if the input is scalar.
Defined such that :math:`H(z) = H_0 E(z)`.
"""
Or = self._Ogamma0 + (self._Onu0 if not self._massivenu else
self._Ogamma0 * self.nu_relative_density(z))
zp1 = aszarr(z) + 1.0 # (converts z [unit] -> z [dimensionless])
return sqrt(zp1**3 * (Or * zp1 + self._Om0) +
self._Ode0 * zp1**(3.0 * (1 + self._w0)))
def de_density_scale(self, z):
r"""Evaluates the redshift dependence of the dark energy density.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
I : ndarray or float
The scaling of the energy density of dark energy with redshift.
Returns `float` if the input is scalar.
Notes
-----
The scaling factor, I, is defined by :math:`\rho(z) = \rho_0 I`,
and in this case is given by
:math:`I = \left(1 + z\right)^{3\left(1 + w_0\right)}`
"""
return (aszarr(z) + 1.0)**(3.0 * (1. + self._w0))
def de_density_scale(self, z):
r"""Evaluates the redshift dependence of the dark energy density.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
I : ndarray or float
The scaling of the energy density of dark energy with redshift.
Returns `float` if the input is scalar.
Notes
-----
The scaling factor, I, is defined by :math:`\rho(z) = \rho_0 I`,
and in this case is given by :math:`I = 1`.
"""
z = aszarr(z)
return np.ones(z.shape) if hasattr(z, "shape") else 1.0
def w(self, z):
r"""Returns dark energy equation of state at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
w : ndarray or float
The dark energy equation of state.
Returns `float` if the input is scalar.
Notes
-----
The dark energy equation of state is defined as
:math:`w(z) = P(z)/\rho(z)`, where :math:`P(z)` is the pressure at
redshift z and :math:`\rho(z)` is the density at redshift z, both in
units where c=1. Here this is given by :math:`w(z) = w_0 + w_z z`.
"""
return self._w0 + self._wz * aszarr(z)
def _EdS_age(self, z):
r"""Age of the universe in Gyr at redshift ``z``.
For :math:`\Omega_{rad} = 0` (:math:`T_{CMB} = 0`; massless neutrinos)
the age can be directly calculated as an elliptic integral [1]_.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
t : `~astropy.units.Quantity` ['time']
The age of the universe in Gyr at each input redshift.
References
----------
.. [1] Thomas, R., & Kantowski, R. (2000). Age-redshift relation for
standard cosmology. PRD, 62(10), 103507.
"""
return (2. / 3) * self._hubble_time * (aszarr(z) + 1.0)**(-1.5)
def w(self, z):
r"""Returns dark energy equation of state at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
w : ndarray or float
The dark energy equation of state
Returns `float` if the input is scalar.
Notes
-----
The dark energy equation of state is defined as
:math:`w(z) = P(z)/\rho(z)`, where :math:`P(z)` is the pressure at
redshift z and :math:`\rho(z)` is the density at redshift z, both in
units where c=1. Here this is :math:`w(z) = w_0`.
"""
z = aszarr(z)
return self._w0 * (np.ones(z.shape) if hasattr(z, "shape") else 1.0)
def efunc(self, z):
"""Function used to calculate H(z), the Hubble parameter.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
E : ndarray or float
The redshift scaling of the Hubble constant.
Returns `float` if the input is scalar.
Defined such that :math:`H(z) = H_0 E(z)`.
"""
# We override this because it takes a particularly simple
# form for a cosmological constant
Or = self._Ogamma0 + (self._Onu0 if not self._massivenu else
self._Ogamma0 * self.nu_relative_density(z))
zp1 = aszarr(z) + 1.0 # (converts z [unit] -> z [dimensionless])
return np.sqrt(zp1**2 * ((Or * zp1 + self._Om0) * zp1 + self._Ok0) +
self._Ode0)
def w(self, z):
r"""Returns dark energy equation of state at redshift ``z``.
Parameters
----------
z : Quantity-like ['redshift'], array-like, or `~numbers.Number`
Input redshift.
Returns
-------
w : ndarray or float
The dark energy equation of state
Returns `float` if the input is scalar.
Notes
-----
The dark energy equation of state is defined as
:math:`w(z) = P(z)/\rho(z)`, where :math:`P(z)` is the pressure at
redshift z and :math:`\rho(z)` is the density at redshift z, both in
units where c=1. Here this is :math:`w(z) = w_p + w_a (a_p - a)` where
:math:`a = 1/1+z` and :math:`a_p = 1 / 1 + z_p`.
"""
apiv = 1.0 / (1.0 + self._zp.value)
return self._wp + self._wa * (apiv - 1.0 / (aszarr(z) + 1.0))
def test_invalid(self, z, exc):
"""Test :func:`astropy.cosmology.utils.aszarr`."""
with pytest.raises(exc):
aszarr(z)
def test_valid(self, z, expect):
"""Test :func:`astropy.cosmology.utils.aszarr`."""
got = aszarr(z)
assert np.array_equal(got, expect)
