Spherical couplings (couplings)

Compute spherical Fourier couplings coefficients for given survey spefications and cosmological models.

SphericalCoefficientWarning

Warning issued for poorly determined spherical coefficient.

Couplings(disc[, survey_specs, cosmo_specs, ...])

Angular, radial and RSD coupling coefficients computed for given survey and cosmological specifications.

Kernels

Coupling kernels are integrands without the coordinate Jacobian, which may include the following factors:

  • angular mask \(M(\hat{\mathbf{r}})\);

  • radial selection \(\phi(r)\);

  • radial weight \(w(r)\) and its derivative \(w'(r)\);

  • clustering evolution, \(D(z)\), which is the linear growth factor normalised to unity at the \(z = 0\) epoch;

  • linear bias + clustering evolution \(G(z, k) = b(z, k) D(z) / b(z_*, k) D(z_*)\) normalised to unity at a fiducial epoch \(z_*\), where \(b(z, k)\) is the scale-dependent linear bias;

  • linear growth rate + clustering evolution \(F(z) = f(z) D(z) / f(z_*) D(z_*)\) normalised to unity at a fiducial epoch \(z_*\), where \(f(z)\) is the linear growth rate;

  • differential Alcock–Paczynski (AP) distortion

    \[\gamma(z) = \frac{ \operatorname{d}\!\breve{\chi}(z) }{ \operatorname{d}\!\chi(z) } \,,\]

    where \(\breve{r} = \breve{\chi}(z)\) is the distance converted from redshift in a fiducial cosmology rather than from the true comoving distance–redshift correspondence \(z = \chi^{-1}(r)\).

Couplings

Coupling coefficients are computed by integrating the angular, radial and RSD coupling kernels

\[\begin{split}\begin{align*} M_{\mu\nu} &= \int \operatorname{d}^2\!\hat{\mathbf{r}} \, Y_{\ell_\mu m_\mu}^*(\hat{\mathbf{r}}) M(\hat{\mathbf{r}}) Y_{\ell_\nu m_\nu}(\hat{\mathbf{r}}) \,, \\ \Phi_{\mu\nu} &= \kappa_{\ell_\nu n_\nu} \int \operatorname{d}\!r \, r^2 w(\breve{r}) j_{\ell_\mu}(k_{\ell_\mu n_\mu} \breve{r}) j_{\ell_\nu}(k_{\ell_\nu n_\nu} r) G(z(r), k_{\ell_\nu n_\nu}) \phi(r) \,, \\ \Upsilon_{\mu\nu} &= \frac{\kappa_{\ell_\nu n_\nu}}{k_{\ell_\nu n_\nu}} \int \operatorname{d}\!r \, r^2 \frac{\operatorname{d}\!}{\operatorname{d}\!\breve{r}} \left[ w(\breve{r}) j_{\ell_\mu}(k_{\ell_\mu n_\mu} \breve{r}) \right] j'_{\ell_\nu}(k_{\ell_\nu n_\nu} r) \gamma(z(r)) F(z(r)) \phi(r) \,, \end{align*}\end{split}\]

over the spherical volume element, where \(k_{\ell n}\) is the discrete wavenumber.

When there is no angular masking (i.e. \(M(\hat{\mathbf{r}})\) is constant), the coupling coefficients reduce to \(M_{\mu\nu} = \delta_{\mu\nu}\); if in addition radial selection, weighting and evolutionary effects are all absent and there is no AP correction, then \(M_{\mu\nu} \Phi_{\mu\nu} = \delta_{\mu\nu}\).


exception harmonia.reader.couplings.SphericalCoefficientWarning[source]

Warning issued for poorly determined spherical coefficient.

class harmonia.reader.couplings.Couplings(disc, survey_specs=None, cosmo_specs=None, initialise=True, external_angular_couplings=None, pixelate=None, comm=None)[source]

Angular, radial and RSD coupling coefficients computed for given survey and cosmological specifications.

Notes

Survey specification functions must be given in spherical coordinates and may include the following:

  • ‘mask’ for angular mask;

  • ‘selection’, ‘weight’ and ‘weight_derivative’ for radial selection, weighting and weight derivative.

Cosmological specification functions must be given in redshift and/or the radial coordinate and may include the following:

  • ‘z_from_r’ and ‘chi_of_z’ for cosmological comoving distance-to-redshift conversion and fiducial redshift-to-comoving distance conversion;

  • ‘bias_evolution’, ‘growth_evolution’ for linear bias and linear growth rate, normalised to unity at a fiducial epoch;

  • ‘clustering_evolution’ for clustering evolution as a function of the redshift normalised to unity at the current epoch;

  • ‘differential_AP_distortion’ for differential AP distortion as a function of the redshift.

Parameters:

  • disc (DiscreteSpectrum) – Discrete spectrum associated with the couplings.

  • survey_specs (dict{str: callable or None} or None, optional) – Survey specification functions as detailed above.

  • cosmo_specs (dict{str: callable or None} or None, optional) – Cosmological specification functions as detailed above.

  • initialise (bool, optional) – If True, compile all coupling coefficients upon creation.

  • external_angular_couplings (dict{tuple(tuple, tuple): complex} or None, optional) – Pre-compute angular couplings (default is None).

  • pixelate (int or None, optional) – If not None (default), this sets the ‘NSIDE’ parameter of healpy pixelation for angular coupling integration.

  • comm (mpi4py.MPI.Comm or None, optional) – MPI communicator (default is None).

Variables:

  • disc (DiscreteSpectrum) – Discrete spectrum associated with the couplings.

  • couplings (dict{str: dict}) – Directory for all coupling coefficients of different coupling types.

__getitem__(key)[source]

Access coupling coefficient by key.

Notes

Only accessible if initialised with compile_couplings(). The access key is a tuple specifying the coupling type and the pair of triplet/double indices, e.g. ['angular', (0, 0), (3, -1, 1)] for \(M_{0,0,3,-1}\).

Parameters:

key (tuple) – Coefficient access key. Must contain elements of types specified above.

Returns:

Coupling coefficient.

Return type:

float or complex

Raises:

AttributeError – If the coupling coefficients have not been initialised.

load_angular_couplings(angular_couplings)[source]

Load pre-computed angular coupling coefficients which are independent of the cosmological model.

Parameters:

angular_couplings (dict{tuple(tuple, tuple): complex}) – Pre-compute angular couplings.

Raises:

ValueError – If the number of entries in angular_couplings do not match the class instance.

compile_couplings(pixelate=None)[source]

Compile all coupling coefficients.

Parameters:

pixelate (int or None, optional) – If not None (default), this sets the ‘NSIDE’ parameter of healpy pixelation for angular coupling integration.

compute_coefficient(mu, nu, coupling_type, pixelate=None)[source]

Compute coupling coefficients for given triplet indices.

Parameters:

  • mu, nu (tuple(int, int) or tuple(int, int, int)) – Coefficient triplet or reduced doublet index.

  • coupling_type ({‘angular’, ‘radial’, ‘RSD’}) – Coupling type.

  • pixelate (int or None, optional) – If not None (default), this sets the ‘NSIDE’ parameter of healpy pixelation for computing angular coupling coefficients.

Returns:

cofficient – Coupling coefficient of the specified type.

Return type:

float or complex numpy.ndarray

save(output_file)[source]

Save compiled couplings as a .npz file.

Parameters:

output_file (str or pathlib.Path) – Output file path.

classmethod load(input_file, comm=None)[source]

Load compiled couplings from a .npz file.

Parameters:

input_file (str or pathlib.Path) – Input file path.