import jax.numpy as np
from jax.numpy import interp
from jax_cosmo.background import *
from jax_cosmo.scipy.ode import odeint
def E(cosmo, a):
r"""Scale factor dependent factor E(a) in the Hubble
parameter.
Parameters
----------
a : array_like
Scale factor
Returns
-------
E : ndarray, or float if input scalar
Square of the scaling of the Hubble constant as a function of
scale factor
Notes
-----
The Hubble parameter at scale factor `a` is given by
:math:`H^2(a) = E^2(a) H_o^2` where :math:`E^2` is obtained through
Friedman's Equation (see :cite:`2005:Percival`) :
.. math::
E^2(a) = \Omega_m a^{-3} + \Omega_k a^{-2} + \Omega_{de} a^{f(a)}
where :math:`f(a)` is the Dark Energy evolution parameter computed
by :py:meth:`.f_de`.
"""
return np.power(Esqr(cosmo, a), 0.5)
def df_de(cosmo, a, epsilon=1e-5):
r"""Derivative of the evolution parameter for the Dark Energy density
f(a) with respect to the scale factor.
Parameters
----------
cosmo: Cosmology
Cosmological parameters structure
a : array_like
Scale factor
epsilon: float value
Small number to make sure we are not dividing by 0 and avoid a singularity
Returns
-------
df(a)/da : ndarray, or float if input scalar
Derivative of the evolution parameter for the Dark Energy density
with respect to the scale factor.
Notes
-----
The expression for :math:`\frac{df(a)}{da}` is:
.. math::
\frac{df}{da}(a) = =\frac{3w_a \left( \ln(a-\epsilon)-
\frac{a-1}{a-\epsilon}\right)}{\ln^2(a-\epsilon)}
"""
return (3 * cosmo.wa * (np.log(a - epsilon) - (a - 1) / (a - epsilon)) /
np.power(np.log(a - epsilon), 2))
def dEa(cosmo, a):
r"""Derivative of the scale factor dependent factor E(a) in the Hubble
parameter.
Parameters
----------
a : array_like
Scale factor
Returns
-------
dE(a)/da : ndarray, or float if input scalar
Derivative of the scale factor dependent factor in the Hubble
parameter with respect to the scale factor.
Notes
-----
The expression for :math:`\frac{dE}{da}` is:
.. math::
\frac{dE(a)}{da}=\frac{-3a^{-4}\Omega_{0m}
-2a^{-3}\Omega_{0k}
+f'_{de}\Omega_{0de}a^{f_{de}(a)}}{2E(a)}
Notes
-----
The Hubble parameter at scale factor `a` is given by
:math:`H^2(a) = E^2(a) H_o^2` where :math:`E^2` is obtained through
Friedman's Equation (see :cite:`2005:Percival`) :
.. math::
E^2(a) = \Omega_m a^{-3} + \Omega_k a^{-2} + \Omega_{de} a^{f(a)}
where :math:`f(a)` is the Dark Energy evolution parameter computed
by :py:meth:`.f_de`.
"""
return (0.5 *
(-3 * cosmo.Omega_m * np.power(a, -4) -
2 * cosmo.Omega_k * np.power(a, -3) +
df_de(cosmo, a) * cosmo.Omega_de * np.power(a, f_de(cosmo, a))) /
np.power(Esqr(cosmo, a), 0.5))
def growth_factor(cosmo, a):
"""Compute linear growth factor D(a) at a given scale factor,
normalized such that D(a=1) = 1.
Parameters
----------
cosmo: `Cosmology`
Cosmology object
a: array_like
Scale factor
Returns
-------
D: ndarray, or float if input scalar
Growth factor computed at requested scale factor
Notes
-----
The growth computation will depend on the cosmology parametrization, for
instance if the $\gamma$ parameter is defined, the growth will be computed
assuming the $f = \Omega^\gamma$ growth rate, otherwise the usual ODE for
growth will be solved.
"""
if cosmo._flags["gamma_growth"]:
return _growth_factor_gamma(cosmo, a)
else:
return _growth_factor_ODE(cosmo, a)
def growth_factor_second(cosmo, a):
"""Compute second order growth factor D2(a) at a given scale factor,
normalized such that D(a=1) = 1.
Parameters
----------
cosmo: `Cosmology`
Cosmology object
a: array_like
Scale factor
Returns
-------
D2: ndarray, or float if input scalar
Growth factor computed at requested scale factor
Notes
-----
The growth computation will depend on the cosmology parametrization,
as for the linear growth. Currently the second order growth
factor is not implemented with $\gamma$ parameter.
"""
if cosmo._flags["gamma_growth"]:
raise NotImplementedError(
"Gamma growth rate is not implemented for second order growth!")
return None
else:
return _growth_factor_second_ODE(cosmo, a)
def growth_rate(cosmo, a):
"""Compute growth rate dD/dlna at a given scale factor.
Parameters
----------
cosmo: `Cosmology`
Cosmology object
a: array_like
Scale factor
Returns
-------
f: ndarray, or float if input scalar
Growth rate computed at requested scale factor
Notes
-----
The growth computation will depend on the cosmology parametrization, for
instance if the $\gamma$ parameter is defined, the growth will be computed
assuming the $f = \Omega^\gamma$ growth rate, otherwise the usual ODE for
growth will be solved.
The LCDM approximation to the growth rate :math:`f_{\gamma}(a)` is given by:
.. math::
f_{\gamma}(a) = \Omega_m^{\gamma} (a)
with :math: `\gamma` in LCDM, given approximately by:
.. math::
\gamma = 0.55
see :cite:`2019:Euclid Preparation VII, eqn.32`
"""
if cosmo._flags["gamma_growth"]:
return _growth_rate_gamma(cosmo, a)
else:
return _growth_rate_ODE(cosmo, a)
def growth_rate_second(cosmo, a):
"""Compute second order growth rate dD2/dlna at a given scale factor.
Parameters
----------
cosmo: `Cosmology`
Cosmology object
a: array_like
Scale factor
Returns
-------
f2: ndarray, or float if input scalar
Second order growth rate computed at requested scale factor
Notes
-----
The growth computation will depend on the cosmology parametrization,
as for the linear growth rate. Currently the second order growth
rate is not implemented with $\gamma$ parameter.
"""
if cosmo._flags["gamma_growth"]:
raise NotImplementedError(
"Gamma growth factor is not implemented for second order growth!")
return None
else:
return _growth_rate_second_ODE(cosmo, a)
def _growth_factor_ODE(cosmo, a, log10_amin=-3, steps=128, eps=1e-4):
"""Compute linear growth factor D(a) at a given scale factor,
normalised such that D(a=1) = 1.
Parameters
----------
a: array_like
Scale factor
amin: float
Mininum scale factor, default 1e-3
Returns
-------
D: ndarray, or float if input scalar
Growth factor computed at requested scale factor
"""
# Check if growth has already been computed
if not "background.growth_factor" in cosmo._workspace.keys():
# Compute tabulated array
atab = np.logspace(log10_amin, 0.0, steps)
def D_derivs(y, x):
q = (2.0 - 0.5 *
(Omega_m_a(cosmo, x) +
(1.0 + 3.0 * w(cosmo, x)) * Omega_de_a(cosmo, x))) / x
r = 1.5 * Omega_m_a(cosmo, x) / x / x
g1, g2 = y[0]
f1, f2 = y[1]
dy1da = [f1, -q * f1 + r * g1]
dy2da = [f2, -q * f2 + r * g2 - r * g1**2]
return np.array([[dy1da[0], dy2da[0]], [dy1da[1], dy2da[1]]])
y0 = np.array([[atab[0], -3.0 / 7 * atab[0]**2],
[1.0, -6.0 / 7 * atab[0]]])
y = odeint(D_derivs, y0, atab)
# compute second order derivatives growth
dyda2 = D_derivs(np.transpose(y, (1, 2, 0)), atab)
dyda2 = np.transpose(dyda2, (2, 0, 1))
# Normalize results
y1 = y[:, 0, 0]
gtab = y1 / y1[-1]
y2 = y[:, 0, 1]
g2tab = y2 / y2[-1]
# To transform from dD/da to dlnD/dlna: dlnD/dlna = a / D dD/da
ftab = y[:, 1, 0] / y1[-1] * atab / gtab
f2tab = y[:, 1, 1] / y2[-1] * atab / g2tab
# Similarly for second order derivatives
# Note: these factors are not accessible as parent functions yet
# since it is unclear what to refer to them with.
htab = dyda2[:, 1, 0] / y1[-1] * atab / gtab
h2tab = dyda2[:, 1, 1] / y2[-1] * atab / g2tab
cache = {
"a": atab,
"g": gtab,
"f": ftab,
"h": htab,
"g2": g2tab,
"f2": f2tab,
"h2": h2tab,
}
cosmo._workspace["background.growth_factor"] = cache
else:
cache = cosmo._workspace["background.growth_factor"]
return np.clip(interp(a, cache["a"], cache["g"]), 0.0, 1.0)
def _growth_rate_ODE(cosmo, a):
"""Compute growth rate dD/dlna at a given scale factor by solving the linear
growth ODE.
Parameters
----------
cosmo: `Cosmology`
Cosmology object
a: array_like
Scale factor
Returns
-------
f: ndarray, or float if input scalar
Growth rate computed at requested scale factor
"""
# Check if growth has already been computed, if not, compute it
if not "background.growth_factor" in cosmo._workspace.keys():
_growth_factor_ODE(cosmo, np.atleast_1d(1.0))
cache = cosmo._workspace["background.growth_factor"]
return interp(a, cache["a"], cache["f"])
def _growth_factor_second_ODE(cosmo, a):
"""Compute second order growth factor D2(a) at a given scale factor,
normalised such that D(a=1) = 1.
Parameters
----------
a: array_like
Scale factor
amin: float
Mininum scale factor, default 1e-3
Returns
-------
D2: ndarray, or float if input scalar
Second order growth factor computed at requested scale factor
"""
# Check if growth has already been computed, if not, compute it
if not "background.growth_factor" in cosmo._workspace.keys():
_growth_factor_ODE(cosmo, np.atleast_1d(1.0))
cache = cosmo._workspace["background.growth_factor"]
return interp(a, cache["a"], cache["g2"])
def _growth_rate_ODE(cosmo, a):
"""Compute growth rate dD/dlna at a given scale factor by solving the linear
growth ODE.
Parameters
----------
cosmo: `Cosmology`
Cosmology object
a: array_like
Scale factor
Returns
-------
f: ndarray, or float if input scalar
Second order growth rate computed at requested scale factor
"""
# Check if growth has already been computed, if not, compute it
if not "background.growth_factor" in cosmo._workspace.keys():
_growth_factor_ODE(cosmo, np.atleast_1d(1.0))
cache = cosmo._workspace["background.growth_factor"]
return interp(a, cache["a"], cache["f"])
def _growth_rate_second_ODE(cosmo, a):
"""Compute second order growth rate dD2/dlna at a given scale factor by solving the linear
growth ODE.
Parameters
----------
cosmo: `Cosmology`
Cosmology object
a: array_like
Scale factor
Returns
-------
f2: ndarray, or float if input scalar
Second order growth rate computed at requested scale factor
"""
# Check if growth has already been computed, if not, compute it
if not "background.growth_factor" in cosmo._workspace.keys():
_growth_factor_ODE(cosmo, np.atleast_1d(1.0))
cache = cosmo._workspace["background.growth_factor"]
return interp(a, cache["a"], cache["f2"])
def _growth_factor_gamma(cosmo, a, log10_amin=-3, steps=128):
r"""Computes growth factor by integrating the growth rate provided by the
\gamma parametrization. Normalized such that D( a=1) =1
Parameters
----------
a: array_like
Scale factor
amin: float
Mininum scale factor, default 1e-3
Returns
-------
D: ndarray, or float if input scalar
Growth factor computed at requested scale factor
"""
# Check if growth has already been computed, if not, compute it
if not "background.growth_factor" in cosmo._workspace.keys():
# Compute tabulated array
atab = np.logspace(log10_amin, 0.0, steps)
def integrand(y, loga):
xa = np.exp(loga)
return _growth_rate_gamma(cosmo, xa)
gtab = np.exp(odeint(integrand, np.log(atab[0]), np.log(atab)))
gtab = gtab / gtab[-1] # Normalize to a=1.
cache = {"a": atab, "g": gtab}
cosmo._workspace["background.growth_factor"] = cache
else:
cache = cosmo._workspace["background.growth_factor"]
return np.clip(interp(a, cache["a"], cache["g"]), 0.0, 1.0)
def _growth_rate_gamma(cosmo, a):
r"""Growth rate approximation at scale factor `a`.
Parameters
----------
cosmo: `Cosmology`
Cosmology object
a : array_like
Scale factor
Returns
-------
f_gamma : ndarray, or float if input scalar
Growth rate approximation at the requested scale factor
Notes
-----
The LCDM approximation to the growth rate :math:`f_{\gamma}(a)` is given by:
.. math::
f_{\gamma}(a) = \Omega_m^{\gamma} (a)
with :math: `\gamma` in LCDM, given approximately by:
.. math::
\gamma = 0.55
see :cite:`2019:Euclid Preparation VII, eqn.32`
"""
return Omega_m_a(cosmo, a)**cosmo.gamma
def Gf(cosmo, a):
r"""
FastPM growth factor function
Parameters
----------
cosmo: dict
Cosmology dictionary.
a : array_like
Scale factor.
Returns
-------
Scalar float Tensor : FastPM growth factor function.
Notes
-----
The expression for :math:`Gf(a)` is:
.. math::
Gf(a)=D'_{1norm}*a**3*E(a)
"""
f1 = growth_rate(cosmo, a)
g1 = growth_factor(cosmo, a)
D1f = f1 * g1 / a
return D1f * np.power(a, 3) * np.power(Esqr(cosmo, a), 0.5)
def Gf2(cosmo, a):
r""" FastPM second order growth factor function
Parameters
----------
cosmo: dict
Cosmology dictionary.
a : array_like
Scale factor.
Returns
-------
Scalar float Tensor : FastPM second order growth factor function.
Notes
-----
The expression for :math:`Gf_2(a)` is:
.. math::
Gf_2(a)=D'_{2norm}*a**3*E(a)
"""
f2 = growth_rate_second(cosmo, a)
g2 = growth_factor_second(cosmo, a)
D2f = f2 * g2 / a
return D2f * np.power(a, 3) * np.power(Esqr(cosmo, a), 0.5)
def dGfa(cosmo, a):
r""" Derivative of Gf against a
Parameters
----------
cosmo: dict
Cosmology dictionary.
a : array_like
Scale factor.
Returns
-------
Scalar float Tensor : the derivative of Gf against a.
Notes
-----
The expression for :math:`gf(a)` is:
.. math::
gf(a)=\frac{dGF}{da}= D^{''}_1 * a ** 3 *E(a) +D'_{1norm}*a ** 3 * E'(a)
+ 3 * a ** 2 * E(a)*D'_{1norm}
"""
f1 = growth_rate(cosmo, a)
g1 = growth_factor(cosmo, a)
D1f = f1 * g1 / a
cache = cosmo._workspace['background.growth_factor']
f1p = cache['h'] / cache['a'] * cache['g']
f1p = interp(np.log(a), np.log(cache['a']), f1p)
Ea = E(cosmo, a)
return (f1p * a**3 * Ea + D1f * a**3 * dEa(cosmo, a) + 3 * a**2 * Ea * D1f)
def dGf2a(cosmo, a):
r""" Derivative of Gf2 against a
Parameters
----------
cosmo: dict
Cosmology dictionary.
a : array_like
Scale factor.
Returns
-------
Scalar float Tensor : the derivative of Gf2 against a.
Notes
-----
The expression for :math:`gf2(a)` is:
.. math::
gf_2(a)=\frac{dGF_2}{da}= D^{''}_2 * a ** 3 *E(a) +D'_{2norm}*a ** 3 * E'(a)
+ 3 * a ** 2 * E(a)*D'_{2norm}
"""
f2 = growth_rate_second(cosmo, a)
g2 = growth_factor_second(cosmo, a)
D2f = f2 * g2 / a
cache = cosmo._workspace['background.growth_factor']
f2p = cache['h2'] / cache['a'] * cache['g2']
f2p = interp(np.log(a), np.log(cache['a']), f2p)
E_a = E(cosmo, a)
return (f2p * a**3 * E_a + D2f * a**3 * dEa(cosmo, a) +
3 * a**2 * E_a * D2f)