adds growth functions from Chirag

jaxpm/growth.py

@@ -0,0 +1,608 @@
+import jax.numpy as np
+
+from jax_cosmo.scipy.interpolate import interp
+from jax_cosmo.scipy.ode import odeint
+from jax_cosmo.background import *
+
+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 = E(cosmo, a)
+    return (f2p * a**3 * E + D2f * a**3 * dEa(cosmo, a) +
+            3 * a**2 * E * D2f)