forked from Aquila-Consortium/JaxPM_highres
adds growth functions from Chirag
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jaxpm/growth.py
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608
jaxpm/growth.py
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import jax.numpy as np
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from jax_cosmo.scipy.interpolate import interp
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from jax_cosmo.scipy.ode import odeint
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from jax_cosmo.background import *
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def E(cosmo, a):
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r"""Scale factor dependent factor E(a) in the Hubble
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parameter.
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Parameters
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----------
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a : array_like
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Scale factor
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Returns
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-------
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E : ndarray, or float if input scalar
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Square of the scaling of the Hubble constant as a function of
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scale factor
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Notes
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-----
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The Hubble parameter at scale factor `a` is given by
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:math:`H^2(a) = E^2(a) H_o^2` where :math:`E^2` is obtained through
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Friedman's Equation (see :cite:`2005:Percival`) :
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.. math::
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E^2(a) = \Omega_m a^{-3} + \Omega_k a^{-2} + \Omega_{de} a^{f(a)}
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where :math:`f(a)` is the Dark Energy evolution parameter computed
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by :py:meth:`.f_de`.
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"""
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return np.power(Esqr(cosmo, a), 0.5)
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def df_de(cosmo, a, epsilon=1e-5):
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r"""Derivative of the evolution parameter for the Dark Energy density
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f(a) with respect to the scale factor.
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Parameters
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----------
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cosmo: Cosmology
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Cosmological parameters structure
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a : array_like
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Scale factor
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epsilon: float value
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Small number to make sure we are not dividing by 0 and avoid a singularity
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Returns
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-------
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df(a)/da : ndarray, or float if input scalar
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Derivative of the evolution parameter for the Dark Energy density
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with respect to the scale factor.
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Notes
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-----
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The expression for :math:`\frac{df(a)}{da}` is:
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.. math::
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\frac{df}{da}(a) = =\frac{3w_a \left( \ln(a-\epsilon)-
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\frac{a-1}{a-\epsilon}\right)}{\ln^2(a-\epsilon)}
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"""
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return (
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3
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* cosmo.wa
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* (np.log(a - epsilon) - (a - 1) / (a - epsilon))
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/ np.power(np.log(a - epsilon), 2)
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)
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def dEa(cosmo, a):
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r"""Derivative of the scale factor dependent factor E(a) in the Hubble
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parameter.
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Parameters
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----------
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a : array_like
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Scale factor
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Returns
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-------
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dE(a)/da : ndarray, or float if input scalar
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Derivative of the scale factor dependent factor in the Hubble
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parameter with respect to the scale factor.
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Notes
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-----
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The expression for :math:`\frac{dE}{da}` is:
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.. math::
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\frac{dE(a)}{da}=\frac{-3a^{-4}\Omega_{0m}
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-2a^{-3}\Omega_{0k}
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+f'_{de}\Omega_{0de}a^{f_{de}(a)}}{2E(a)}
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Notes
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-----
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The Hubble parameter at scale factor `a` is given by
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:math:`H^2(a) = E^2(a) H_o^2` where :math:`E^2` is obtained through
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Friedman's Equation (see :cite:`2005:Percival`) :
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.. math::
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E^2(a) = \Omega_m a^{-3} + \Omega_k a^{-2} + \Omega_{de} a^{f(a)}
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where :math:`f(a)` is the Dark Energy evolution parameter computed
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by :py:meth:`.f_de`.
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"""
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return (
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0.5
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* (
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-3 * cosmo.Omega_m * np.power(a, -4)
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- 2 * cosmo.Omega_k * np.power(a, -3)
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+ df_de(cosmo, a) * cosmo.Omega_de * np.power(a, f_de(cosmo, a))
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)
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/ np.power(Esqr(cosmo, a), 0.5)
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)
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def growth_factor(cosmo, a):
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"""Compute linear growth factor D(a) at a given scale factor,
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normalized such that D(a=1) = 1.
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Parameters
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----------
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cosmo: `Cosmology`
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Cosmology object
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a: array_like
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Scale factor
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Returns
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-------
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D: ndarray, or float if input scalar
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Growth factor computed at requested scale factor
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Notes
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-----
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The growth computation will depend on the cosmology parametrization, for
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instance if the $\gamma$ parameter is defined, the growth will be computed
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assuming the $f = \Omega^\gamma$ growth rate, otherwise the usual ODE for
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growth will be solved.
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"""
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if cosmo._flags["gamma_growth"]:
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return _growth_factor_gamma(cosmo, a)
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else:
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return _growth_factor_ODE(cosmo, a)
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def growth_factor_second(cosmo, a):
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"""Compute second order growth factor D2(a) at a given scale factor,
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normalized such that D(a=1) = 1.
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Parameters
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----------
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cosmo: `Cosmology`
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Cosmology object
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a: array_like
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Scale factor
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Returns
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-------
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D2: ndarray, or float if input scalar
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Growth factor computed at requested scale factor
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Notes
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-----
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The growth computation will depend on the cosmology parametrization,
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as for the linear growth. Currently the second order growth
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factor is not implemented with $\gamma$ parameter.
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"""
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if cosmo._flags["gamma_growth"]:
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raise NotImplementedError(
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"Gamma growth rate is not implemented for second order growth!"
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)
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return None
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else:
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return _growth_factor_second_ODE(cosmo, a)
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def growth_rate(cosmo, a):
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"""Compute growth rate dD/dlna at a given scale factor.
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Parameters
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----------
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cosmo: `Cosmology`
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Cosmology object
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a: array_like
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Scale factor
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Returns
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-------
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f: ndarray, or float if input scalar
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Growth rate computed at requested scale factor
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Notes
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-----
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The growth computation will depend on the cosmology parametrization, for
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instance if the $\gamma$ parameter is defined, the growth will be computed
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assuming the $f = \Omega^\gamma$ growth rate, otherwise the usual ODE for
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growth will be solved.
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The LCDM approximation to the growth rate :math:`f_{\gamma}(a)` is given by:
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.. math::
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f_{\gamma}(a) = \Omega_m^{\gamma} (a)
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with :math: `\gamma` in LCDM, given approximately by:
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.. math::
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\gamma = 0.55
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see :cite:`2019:Euclid Preparation VII, eqn.32`
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"""
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if cosmo._flags["gamma_growth"]:
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return _growth_rate_gamma(cosmo, a)
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else:
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return _growth_rate_ODE(cosmo, a)
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def growth_rate_second(cosmo, a):
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"""Compute second order growth rate dD2/dlna at a given scale factor.
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Parameters
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----------
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cosmo: `Cosmology`
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Cosmology object
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a: array_like
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Scale factor
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Returns
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-------
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f2: ndarray, or float if input scalar
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Second order growth rate computed at requested scale factor
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Notes
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-----
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The growth computation will depend on the cosmology parametrization,
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as for the linear growth rate. Currently the second order growth
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rate is not implemented with $\gamma$ parameter.
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"""
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if cosmo._flags["gamma_growth"]:
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raise NotImplementedError(
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"Gamma growth factor is not implemented for second order growth!"
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)
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return None
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else:
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return _growth_rate_second_ODE(cosmo, a)
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def _growth_factor_ODE(cosmo, a, log10_amin=-3, steps=128, eps=1e-4):
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"""Compute linear growth factor D(a) at a given scale factor,
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normalised such that D(a=1) = 1.
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Parameters
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----------
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a: array_like
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Scale factor
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amin: float
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Mininum scale factor, default 1e-3
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Returns
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-------
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D: ndarray, or float if input scalar
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Growth factor computed at requested scale factor
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"""
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# Check if growth has already been computed
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if not "background.growth_factor" in cosmo._workspace.keys():
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# Compute tabulated array
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atab = np.logspace(log10_amin, 0.0, steps)
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def D_derivs(y, x):
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q = (
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2.0
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- 0.5
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* (
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Omega_m_a(cosmo, x)
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+ (1.0 + 3.0 * w(cosmo, x)) * Omega_de_a(cosmo, x)
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)
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) / x
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r = 1.5 * Omega_m_a(cosmo, x) / x / x
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g1, g2 = y[0]
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f1, f2 = y[1]
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dy1da = [f1, -q * f1 + r * g1]
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dy2da = [f2, -q * f2 + r * g2 - r * g1 ** 2]
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return np.array([[dy1da[0], dy2da[0]], [dy1da[1], dy2da[1]]])
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y0 = np.array([[atab[0], -3.0 / 7 * atab[0] ** 2], [1.0, -6.0 / 7 * atab[0]]])
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y = odeint(D_derivs, y0, atab)
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# compute second order derivatives growth
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dyda2 = D_derivs(np.transpose(y, (1, 2, 0)), atab)
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dyda2 = np.transpose(dyda2, (2, 0, 1))
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# Normalize results
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y1 = y[:, 0, 0]
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gtab = y1 / y1[-1]
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y2 = y[:, 0, 1]
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g2tab = y2 / y2[-1]
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# To transform from dD/da to dlnD/dlna: dlnD/dlna = a / D dD/da
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ftab = y[:, 1, 0] / y1[-1] * atab / gtab
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f2tab = y[:, 1, 1] / y2[-1] * atab / g2tab
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# Similarly for second order derivatives
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# Note: these factors are not accessible as parent functions yet
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# since it is unclear what to refer to them with.
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htab = dyda2[:, 1, 0] / y1[-1] * atab / gtab
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h2tab = dyda2[:, 1, 1] / y2[-1] * atab / g2tab
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cache = {
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"a": atab,
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"g": gtab,
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"f": ftab,
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"h": htab,
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"g2": g2tab,
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"f2": f2tab,
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"h2": h2tab,
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}
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cosmo._workspace["background.growth_factor"] = cache
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else:
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cache = cosmo._workspace["background.growth_factor"]
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return np.clip(interp(a, cache["a"], cache["g"]), 0.0, 1.0)
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def _growth_rate_ODE(cosmo, a):
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"""Compute growth rate dD/dlna at a given scale factor by solving the linear
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growth ODE.
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Parameters
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----------
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cosmo: `Cosmology`
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Cosmology object
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a: array_like
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Scale factor
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Returns
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-------
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f: ndarray, or float if input scalar
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Growth rate computed at requested scale factor
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"""
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# Check if growth has already been computed, if not, compute it
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if not "background.growth_factor" in cosmo._workspace.keys():
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_growth_factor_ODE(cosmo, np.atleast_1d(1.0))
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cache = cosmo._workspace["background.growth_factor"]
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return interp(a, cache["a"], cache["f"])
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def _growth_factor_second_ODE(cosmo, a):
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"""Compute second order growth factor D2(a) at a given scale factor,
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normalised such that D(a=1) = 1.
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Parameters
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----------
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a: array_like
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Scale factor
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amin: float
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Mininum scale factor, default 1e-3
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Returns
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-------
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D2: ndarray, or float if input scalar
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Second order growth factor computed at requested scale factor
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"""
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# Check if growth has already been computed, if not, compute it
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if not "background.growth_factor" in cosmo._workspace.keys():
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_growth_factor_ODE(cosmo, np.atleast_1d(1.0))
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cache = cosmo._workspace["background.growth_factor"]
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return interp(a, cache["a"], cache["g2"])
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def _growth_rate_ODE(cosmo, a):
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"""Compute growth rate dD/dlna at a given scale factor by solving the linear
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growth ODE.
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Parameters
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----------
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cosmo: `Cosmology`
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Cosmology object
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a: array_like
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Scale factor
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Returns
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-------
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f: ndarray, or float if input scalar
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Second order growth rate computed at requested scale factor
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"""
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# Check if growth has already been computed, if not, compute it
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if not "background.growth_factor" in cosmo._workspace.keys():
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_growth_factor_ODE(cosmo, np.atleast_1d(1.0))
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cache = cosmo._workspace["background.growth_factor"]
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return interp(a, cache["a"], cache["f"])
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def _growth_rate_second_ODE(cosmo, a):
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"""Compute second order growth rate dD2/dlna at a given scale factor by solving the linear
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growth ODE.
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Parameters
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----------
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cosmo: `Cosmology`
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Cosmology object
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a: array_like
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Scale factor
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Returns
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-------
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f2: ndarray, or float if input scalar
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Second order growth rate computed at requested scale factor
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"""
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# Check if growth has already been computed, if not, compute it
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if not "background.growth_factor" in cosmo._workspace.keys():
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_growth_factor_ODE(cosmo, np.atleast_1d(1.0))
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cache = cosmo._workspace["background.growth_factor"]
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return interp(a, cache["a"], cache["f2"])
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def _growth_factor_gamma(cosmo, a, log10_amin=-3, steps=128):
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r"""Computes growth factor by integrating the growth rate provided by the
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\gamma parametrization. Normalized such that D( a=1) =1
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Parameters
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----------
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a: array_like
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Scale factor
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amin: float
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Mininum scale factor, default 1e-3
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Returns
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-------
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D: ndarray, or float if input scalar
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Growth factor computed at requested scale factor
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"""
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# Check if growth has already been computed, if not, compute it
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if not "background.growth_factor" in cosmo._workspace.keys():
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# Compute tabulated array
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atab = np.logspace(log10_amin, 0.0, steps)
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def integrand(y, loga):
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xa = np.exp(loga)
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return _growth_rate_gamma(cosmo, xa)
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gtab = np.exp(odeint(integrand, np.log(atab[0]), np.log(atab)))
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gtab = gtab / gtab[-1] # Normalize to a=1.
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cache = {"a": atab, "g": gtab}
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cosmo._workspace["background.growth_factor"] = cache
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else:
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cache = cosmo._workspace["background.growth_factor"]
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return np.clip(interp(a, cache["a"], cache["g"]), 0.0, 1.0)
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def _growth_rate_gamma(cosmo, a):
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r"""Growth rate approximation at scale factor `a`.
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Parameters
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----------
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cosmo: `Cosmology`
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Cosmology object
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a : array_like
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Scale factor
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Returns
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-------
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f_gamma : ndarray, or float if input scalar
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Growth rate approximation at the requested scale factor
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Notes
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-----
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The LCDM approximation to the growth rate :math:`f_{\gamma}(a)` is given by:
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.. math::
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f_{\gamma}(a) = \Omega_m^{\gamma} (a)
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with :math: `\gamma` in LCDM, given approximately by:
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.. math::
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\gamma = 0.55
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see :cite:`2019:Euclid Preparation VII, eqn.32`
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"""
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return Omega_m_a(cosmo, a) ** cosmo.gamma
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def Gf(cosmo, a):
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r"""
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FastPM growth factor function
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Parameters
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----------
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cosmo: dict
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Cosmology dictionary.
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a : array_like
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Scale factor.
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Returns
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-------
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Scalar float Tensor : FastPM growth factor function.
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Notes
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-----
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The expression for :math:`Gf(a)` is:
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.. math::
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Gf(a)=D'_{1norm}*a**3*E(a)
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"""
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f1 = growth_rate(cosmo, a)
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g1 = growth_factor(cosmo, a)
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D1f = f1*g1/ a
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return D1f * np.power(a, 3) * np.power(Esqr(cosmo, a), 0.5)
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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 = E(cosmo, a)
|
||||
return (f2p * a**3 * E + D2f * a**3 * dEa(cosmo, a) +
|
||||
3 * a**2 * E * D2f)
|
||||
Loading…
Add table
Reference in a new issue