pynkowski.theory.spingaussian

  1"""This submodule contains the definition for spin field with isotropic Gaussian Q and U.
  2"""
  3import numpy as np
  4from .utils_th import get_μ, get_σ, lkc_ambient_dict, get_C2, flag, rho
  5from .gaussian import Gaussian
  6 
  7from .base_th import _prepare_lkc
  8
  9
 10    
 11def LKC_spin(j, us, mu, dim=3, lkc_ambient=lkc_ambient_dict["SO3"], Ks=[1., 0.31912, 0.7088, 1.]):
 12    """Compute the expected value of the Lipschitz–Killing Curvatures (LKC) of the excursion set for Gaussian Isotropic fields.
 13
 14    Parameters
 15    ----------
 16    j : int
 17        The index of the LKC.
 18        
 19    us : np.array
 20        The thresholds where the LKC is evaluated.
 21
 22    mu : float
 23        The value of the derivative of the covariance function at the origin for the U and Q fields.
 24        
 25    dim : int, optional
 26        The dimension of the ambient manifold, SO(3). Must be 3.
 27        
 28    lkc_ambient : np.array, optional
 29        An array of the Lipschitz–Killing Curvatures of the ambient manifold. Its lenght must be `dim+1` if `dim` is also given.
 30
 31    Ks : np.array, optional
 32        The normalisation factors for the Minkowski Functionals. 
 33        
 34    Returns
 35    ----------
 36    LKC : np.array
 37        The expected value of the Lipschitz–Killing Curvatures at the thresholds.
 38    """
 39    dim, lkc_ambient = _prepare_lkc(dim, lkc_ambient)
 40    assert dim==3, "The LKC for spin fields is only defined on SO(3), which has `dim=3`"
 41    result = np.zeros_like(us)
 42    KLs = Ks[::-1]              # To get the order as the LKC index, not the MF index
 43    KLs[-1] *= 2.**1.5 / 5.       # This is the determinant of the covariance matrix of the derivatives of the field
 44    KLs /= np.array([1., 1., mu**0.5, mu])
 45    for k in np.arange(0,dim-j+1):
 46        result += flag(k+j, k) * rho(k, us) * lkc_ambient[k+j] / KLs[k+j]
 47    return result * KLs[j] /lkc_ambient[-1]
 48
 49
 50
 51
 52class SpinGaussian(Gaussian):
 53    """Class for Spin Isotropic Gaussian fields in the SO(3) formalism.
 54
 55    Parameters
 56    ----------
 57    cls : np.array
 58        Angular power spectrum of the field (cls_E + cls_B).
 59    
 60    normalise : bool, optional
 61        If `True`, normalise the field to unit variance.
 62        Default : True
 63
 64    fsky : float, optional
 65        Fraction of the sky covered by the field, `0<fsky<=1`.
 66        Default : 1.
 67
 68    Ks : np.array, optional
 69        The normalisation constants for the MFs of the field: [K_0, K_1, K_2, K_3].
 70        Default : [1., 0.31912, 0.7088, 1.] as found in Carrón Duque et al (2023)
 71
 72    leading_order : bool, optional
 73        Whether to use only the leading order in μ for the computation of the MFs or the exact expression (with two terms).
 74        Default : False (exact expression)
 75    
 76    Attributes
 77    ----------
 78    cls : np.array
 79        Angular Power Spectrum of the field (cls_E + cls_B).
 80    
 81    fsky : float
 82        Fraction of the sky covered by the field.
 83
 84    dim : int
 85        Dimension of the space where the field is defined, in this case this is 3.
 86    
 87    name : str
 88        Name of the field, `"Spin Isotropic Gaussian"` by default.
 89    
 90    sigma : float
 91        The standard deviation of the field.
 92
 93    mu : float
 94        The derivative of the covariance function at the origin, times $-2$ (in the spatial coordinates θϕ). Equal to the variance of the first derivatives of the field.
 95
 96    nu : float
 97        The second derivative of the covariance function at the origin (in the spatial coordinates θϕ). 
 98        
 99    C2 : float
100        The second derivative of the angular covariance function at 1 (in the spatial coordinates θϕ). 
101        
102    lkc_ambient : np.array or None
103        The values for the Lipschitz–Killing Curvatures of the ambient space.
104
105    Ks : np.array
106        The normalisation constants for the MFs of the field: [K_0, K_1, K_2, K_3].
107        
108    leading_order : bool
109        Whether to use only the leading order in μ for the computation of the MFs or the exact expression (with two terms).
110
111    """   
112    def __init__(self, cls, normalise=True, fsky=1., Ks=[1., 0.31912, 0.7088, 1.], leading_order=True):
113        if normalise:
114            cls /= get_σ(cls)
115            self.sigma = 1.
116        else:
117            self.sigma = np.sqrt(get_σ(cls))
118        self.cls = cls
119        self.fsky = fsky
120        self.mu = get_μ(cls)
121        self.C2 = get_C2(cls)
122        self.nu = self.C2/4. - self.mu/24.
123        super().__init__(3, sigma=self.sigma, mu=self.mu, nu=self.nu, lkc_ambient=lkc_ambient_dict["SO3"]*self.fsky)
124        self.leading_order = leading_order
125        if leading_order:
126            self.lkc_ambient[1] = 0.
127        self.name = 'Spin Isotropic Gaussian'
128        self.Ks = Ks
129
130    def LKC(self, j, us):
131        """Compute the expected values of the Lipschitz–Killing Curvatures of the excursion sets at thresholds `us`, $\mathbb{L}_j(A_u(f))$.
132
133        Parameters
134        ----------
135        j : int
136            Index of the LKC to compute, `0 < j < dim`.
137            
138        us : np.array
139            The thresholds considered for the computation.
140
141        Returns
142        -------
143        lkc : np.array
144            Expected value of LKC evaluated at the thresholds.
145
146        """
147        us /= self.sigma
148        return LKC_spin(j, us, self.mu, dim=self.dim, lkc_ambient=self.lkc_ambient, Ks=self.Ks)
149
150
151__all__ = ["SpinGaussian"]