Clustering modification (clustering_modification)

Compute modifications to the Newtonian power spectrum in the distant-observer and plane-parallel limits.

Standard Kaiser RSD model

Redshift-space distortions induce anisotropy in the power spectrum multipoles

\[\begin{split}\begin{align*} P_0(k, z) &= \left[ b_1(z)^2 + \frac{2}{3} b_1(z) f(z) + \frac{1}{5} f(z)^2 \right] P_\mathrm{m}(k, z) \,, \\ P_2(k, z) &= \left[ \frac{4}{3} b_1(z) f(z) + \frac{4}{7} f(z)^2 \right] P_\mathrm{m}(k, z) \,, \\ P_4(k, z) &= \frac{8}{35} f(z)^2 P_\mathrm{m}(k, z) \, \end{align*}\end{split}\]

where \(P_\mathrm{m}\) is the matter power spectrum, \(b_1(z)\) is the linear bias and \(f(z)\) is the linear growth rate.

standard_kaiser_factor(order, bias, redshift)

Compute the standard Kaiser power spectrum multipoles as multiples of the matter power spectrum, i.e.

Non-Gaussianity scale-dependent modifications

Local primordial non-Gaussianty \(f_\mathrm{NL}\) induces scale dependence in the linear bias,

\[b_1(z) \mapsto b_1(z) + \Delta b_k(z) \,, \quad \Delta b(k, z) = f_\mathrm{NL} [b_1(z) - p] \frac{A(k, z)}{k^2} \,,\]

where the scale-dependence kernel is

\[A(k, z) = 3 \left( \frac{H_0}{\mathrm{c}} \right)^2 \frac{1.27 \varOmega_\mathrm{m,0} \delta_\mathrm{c}}{D(z)T(k)} \,.\]

Here \(H_0\) is the Hubble parameter at the current epoch (in km/s/Mpc), \(\mathrm{c}\) the speed of light, \(\varOmega_\mathrm{m,0}\) the matter density parameter at the current epoch, and \(\delta_\mathrm{c}\) the critical over-density in spherical gravitational collapse. The growth factor \(D(z)\) is normalised to unity at the current epoch (thus the numerical factor 1.27), the transfer function \(T(k)\) is normalised to unity as \(k \to 0\), and \(p\) is a tracer-dependent parameter.

Modifications to power spectrum multipoles as a result of local primordial non-Gaussianty are

\[\begin{split}\begin{align*} \Delta P_0(k, z) &= \left[ \left( 2 b_1 + \frac{2}{3} f \right) \Delta b(k, z) + \Delta b(k, z)^2 \right] P_\mathrm{m}(k, z) \,, \\ \Delta P_2(k, z) &= \frac{4}{3} f \Delta b(k, z) P_\mathrm{m}(k, z) \,. \end{align*}\end{split}\]

scale_dependence_kernel(redshift[, cosmo])

Return the scale-dependence kernel \(A(k, z)\) in the presence of local primordial non-Gaussianity.

non_gaussianity_correction_factor(…[, …])

Compute modifications to the power spectrum multipoles by local primordial non-Gaussianity as multiples of the matter power spectrum, i.e.

Relativistic corrections

Relativistic corrections to the Newtonian clustering mode

\[\delta(\mathbf{k}, z) = b_1(z) \delta_\mathrm{m}(\mathbf{k}, z) + g(z) v_{\parallel}(\mathbf{k}, z) = b_1(z) \delta_\mathrm{m}(\mathbf{k}, z) + \mathrm{i} \frac{\mathcal{H}}{k} g(z) f(z) \mu \delta_\mathrm{m}(\mathbf{k}, z)\]

are parametrised by the redshift-dependent, dimensionless quantity

\[g(z) = \frac{\mathcal{H}'}{\mathcal{H}^2} + 5s + \frac{2 - 5s}{\mathcal{H} \chi} - b_\mathrm{e}\]

with evolution bias \(b_\mathrm{e}(z)\) and magnification bias \(s(z)\), where \(v_{\parallel}\) is the line-of-sight peculiar velocity, \(\chi(z)\) is the comoving distance and \('\) denotes derivatives with respect to the conformal time. This can be written as the sum of three contributions

\[g(z) = \underbrace{\left[ 1 - \frac{3}{2} \frac{H_0^2}{H(z)^2} \varOmega_\mathrm{m,0} (1 + z)^3 + \frac{2}{\mathcal{H}\chi} \right]}_{\text{background expansion}} \underbrace{- b_\mathrm{e}(z)}_{\text{evolution}} + \underbrace{ 5s(z) \left( 1 - \frac{1}{\mathcal{H}\chi} \right) }_{\text{magnification}} \,.\]

Modifications to power spectrum multipoles from the relativistic corrections are

\[\begin{split}\begin{align*} \Delta P_0(k, z) &= \frac{1}{3} \frac{\mathcal{H}^2}{k^2} g(z)^2 f(z)^2 P_\mathrm{m}(k,z) \,,\\ \Delta P_2(k, z) &= \frac{2}{3} \frac{\mathcal{H}^2}{k^2} g(z)^2 f(z)^2 P_\mathrm{m}(k,z) \,, \end{align*}\end{split}\]

relativistic_correction_func([cosmo, …])

Return the relativistic correction function as \(\mathcal{H} g(z)\).

relativistic_correction_value(redshift[, …])

Evaluate the relativistic correction function \(\mathcal{H} g(z)\) at a given redshift.

relativistic_correction_factor(wavenumber, …)

Compute modifications to the power spectrum multipoles by relativistic corrections as multiples of the matter power spectrum, i.e.

horizonground.clustering_modification.FIDUCIAL_COSMOLOGY = <nbodykit.cosmology.cosmology.Cosmology object>

Default Planck15 cosmology.

Type

nbodykit.cosmology.Cosmology

horizonground.clustering_modification.standard_kaiser_factor(order, bias, redshift, cosmo=<nbodykit.cosmology.cosmology.Cosmology object>)[source]

Compute the standard Kaiser power spectrum multipoles as multiples of the matter power spectrum, i.e. \(P_\ell/P_\mathrm{m}\).

Parameters

  • order (int) – Order of the multipole, order >= 0.

  • bias (float) – Scale-independent linear bias at redshift.

  • redshift (float) – Redshift.

  • cosmo (nbodykit.cosmology.Cosmology, optional) – Cosmological model (default is FIDUCIAL_COSMOLOGY).

Returns

factor – Power spectrum multipoles as multiples of the matter power spectrum.

Return type

float

horizonground.clustering_modification.scale_dependence_kernel(redshift, cosmo=<nbodykit.cosmology.cosmology.Cosmology object>)[source]

Return the scale-dependence kernel \(A(k, z)\) in the presence of local primordial non-Gaussianity.

Parameters

  • redshift (float) – Redshift.

  • cosmo (nbodykit.cosmology.Cosmology, optional) – Cosmological model (default is FIDUCIAL_COSMOLOGY).

Returns

Scale-dependence kernel as a function of wavenumber (in \(h/\mathrm{Mpc}\)).

Return type

callable

horizonground.clustering_modification.non_gaussianity_correction_factor(wavenumber, order, local_png, bias, redshift, cosmo=<nbodykit.cosmology.cosmology.Cosmology object>, tracer_p=1.0)[source]

Compute modifications to the power spectrum multipoles by local primordial non-Gaussianity as multiples of the matter power spectrum, i.e. \(\Delta P_\ell/P_\mathrm{m}\).

Parameters

  • wavenumber (float, array_like) – Wavenumber (in \(h/\mathrm{Mpc}\)).

  • order (int) – Order of the multipole, order >= 0.

  • local_png (float) – Local primordial non-Gaussianity.

  • bias (float) – Scale-independent linear bias at redshift.

  • redshift (float) – Redshift.

  • cosmo (nbodykit.cosmology.Cosmology, optional) – Base cosmological model (default is FIDUCIAL_COSMOLOGY).

  • tracer_p (float, optional) – Tracer parameter (default is 1.).

Returns

factor – Power spectrum multipole modifications as multiples of the matter power spectrum.

Return type

float numpy.ndarray

horizonground.clustering_modification.relativistic_correction_func(cosmo=<nbodykit.cosmology.cosmology.Cosmology object>, geometric=True, evolution_bias=None, magnification_bias=None)[source]

Return the relativistic correction function as \(\mathcal{H} g(z)\).

Parameters

  • cosmo (nbodykit.cosmology.Cosmology, optional) – Cosmological model (default is FIDUCIAL_COSMOLOGY).

  • geometric (bool, optional) – If True (default), include geometric contributions from the background expansion.

  • evolution_bias, magnification_bias (callable or None, optional) – Evolution bias or magnification bias as a function of redshift (default is None).

Returns

Relativistic correction function as a function of redshift (in \(h\)/Mpc).

Return type

callable

horizonground.clustering_modification.relativistic_correction_value(redshift, cosmo=<nbodykit.cosmology.cosmology.Cosmology object>, geometric=True, evolution_bias=None, magnification_bias=None)[source]

Evaluate the relativistic correction function \(\mathcal{H} g(z)\) at a given redshift.

Parameters

  • redshift (float) – Redshift.

  • cosmo (nbodykit.cosmology.Cosmology, optional) – Base cosmological model (default is FIDUCIAL_COSMOLOGY).

  • geometric (bool, optional) – If True (default), include geometric contributions from the background expansion.

  • evolution_bias, magnification_bias (float or None, optional) – Evolution bias or magnification bias evaluated at input redshift (default is None).

Returns

correction_value – Relativistic correction function value (in \(h\)/Mpc) at redshift.

Return type

float

horizonground.clustering_modification.relativistic_correction_factor(wavenumber, order, redshift, correction_value=None, cosmo=<nbodykit.cosmology.cosmology.Cosmology object>, geometric=True, evolution_bias=None, magnification_bias=None)[source]

Compute modifications to the power spectrum multipoles by relativistic corrections as multiples of the matter power spectrum, i.e. \(\Delta P_\ell/P_\mathrm{m}\).

Parameters

  • wavenumber (float, array_like) – Wavenumber (in \(h/\mathrm{Mpc}\)).

  • order (int) – Order of the multipole, order >= 0.

  • redshift (float) – Redshift.

  • correction_value (float or None, optional) – If not None (default), this is directly used as \(\mathcal{H} g(z)\) at redshift in calculations, and cosmo, geometric, evolution_bias and magnification_bias are ignored.

  • cosmo (nbodykit.cosmology.Cosmology, optional) – Cosmological model (default is FIDUCIAL_COSMOLOGY).

  • geometric (bool, optional) – If True (default), include geometric contributions from the background expansion.

  • evolution_bias (float or callable or None, optional) – Evolution bias as a function of redshift, or evaluated at input redshift (default is None). If callable/float, magnification_bias must also be callable/float.

  • magnification_bias (float or callable or None, optional) – Magnification bias as a function of redshift, or evaluated at input redshift (default is None). If callable/float, evolution_bias must also be callable/float.

Returns

factor – Power spectrum multipoles as multiples of the matter power spectrum.

Return type

float numpy.ndarray