cosmicpy.cosmology – Cosmology module¶

class cosmicpy.cosmology.cosmology(**kwargs)[source]¶

Bases: object

Stores all cosmological parameters and performs the computation of derived quantities (distances, matter power spectra, etc...)

Parameters:

h (float) – Reduced hubble constant (def: 0.7) Omega_b (float) – Baryonic matter density (def: 0.045) Omega_m (float) – Matter density (def: 0.25) Omega_de (float) – Dark energy density (def: 0.75) wa (w0,) – Dark energy equation of state parameters (def: w0 = -0.95, wa = 0.0) n (float) – Scalar spectral index (def: 1.0) tau (float) – Reionization optical depth (def: 0.09) sigma8 (float) – Fluctuation amplitude at 8 Mpc/h (def: 0.8)

References

[EfstathiouBond99]

(1, 2) G. Efstathiou and J. R. Bond. Cosmic confusion: degeneracies among cosmological parameters derived from measurements of microwave background anisotropies. MNRAS, 304:75–97, March 1999. arXiv:arXiv:astro-ph/9807103, doi:10.1046/j.1365-8711.1999.02274.x.

[EisensteinHu98]

(1, 2, 3) D. J. Eisenstein and W. Hu. Baryonic Features in the Matter Transfer Function. ApJ, 496:605, March 1998. arXiv:arXiv:astro-ph/9709112, doi:10.1086/305424.

[Linder03]

E. V. Linder. Exploring the Expansion History of the Universe. Physical Review Letters, 90(9):091301, March 2003. arXiv:arXiv:astro-ph/0208512, doi:10.1103/PhysRevLett.90.091301.

[Percival05]

(1, 2, 3, 4) W. J. Percival. Cosmological structure formation in a homogeneous dark energy background. AA, 443:819–830, December 2005. arXiv:arXiv:astro-ph/0508156, doi:10.1051/0004-6361:20053637.

[SmithPeacockJenkins+03]

R. E. Smith, J. A. Peacock, A. Jenkins, S. D. M. White, C. S. Frenk, F. R. Pearce, P. A. Thomas, G. Efstathiou, and H. M. P. Couchman. Stable clustering, the halo model and non-linear cosmological power spectra. MNRAS, 341:1311–1332, June 2003. arXiv:astro-ph/0207664, doi:10.1046/j.1365-8711.2003.06503.x.

Esqr(a)[source]¶

Square of the scale factor dependent factor E(a) in the Hubble parameter.

Parameters:	a (array_like) – Scale factor
Returns:	E^2 – Square of the scaling of the Hubble constant as a function of scale factor
Return type:	ndarray, or float if input scalar

Notes

The Hubble parameter at scale factor a is given by \(H^2(a) = E^2(a) H_o^2\) where \(E^2\) is obtained through Friedman’s Equation (see [Percival05]) :

\[E^2(a) = \Omega_m a^{-3} + \Omega_k a^{-2} + \Omega_{de} a^{f(a)}\]

where \(f(a)\) is the Dark Energy evolution parameter computed by f_de().

G(a)[source]¶

Compute Growth factor at a given scale factor, normalised such that G(a=1) = 1.

Parameters:	a (array_like) – Scale factor
Returns:	G – Growth factor computed at requested scale factor
Return type:	ndarray, or float if input scalar

H(a)[source]¶

Hubble parameter [km/s/(Mpc/h)] at scale factor a

Parameters:	a (array_like) – Scale factor
Returns:	H – Hubble parameter at the requested scale factor.
Return type:	ndarray, or float if input scalar

Omega_b[source]¶

baryon density

Omega_de_a(a)[source]¶

Dark Energy density at scale factor a.

Parameters:	a (array_like) – Scale factor
Returns:	Omega_de – Dark Energy density at the requested scale factor
Return type:	ndarray, or float if input scalar

Notes

The evolution of Dark Energy density \(\Omega_{de}(a)\) is given by:

\[\Omega_{de}(a) = \frac{\Omega_{de} a^{f(a)}}{E^2(a)}\]

where \(f(a)\) is the Dark Energy evolution parameter computed by f_de() (see [Percival05] Eq. (6)).

Omega_m_a(a)[source]¶

Matter density at scale factor a.

Parameters:	a (array_like) – Scale factor
Returns:	Omega_m – Non-relativistic matter density at the requested scale factor
Return type:	ndarray, or float if input scalar

Notes

The evolution of matter density \(\Omega_m(a)\) is given by:

\[\Omega_m(a) = \frac{\Omega_m a^{-3}}{E^2(a)}\]

see [Percival05] Eq. (6)

T(k, type='eisenhu_osc')[source]¶

Computes the matter transfer function.

Parameters:	k (array_like) – Wave number in h Mpc^{-1} type (str, optional) – Type of transfer function. Either ‘eisenhu’ or ‘eisenhu_osc’ (def: ‘eisenhu_osc’)
Returns:	T – Value of the transfer function at the requested wave number
Return type:	array_like

Notes

The Eisenstein & Hu transfer functions are computed using the fitting formulae of [EisensteinHu98]

a2chi(a)[source]¶

Radial comoving distance in [Mpc/h] for a given scale factor.

Parameters:	a (array_like) – Scale factor
Returns:	chi – Radial comoving distance corresponding to the specified scale factor.
Return type:	ndarray, or float if input scalar

Notes

The radial comoving distance is computed by performing the following integration:

\[\chi(a) = R_H \int_a^1 \frac{da^\prime}{{a^\prime}^2 E(a^\prime)}\]

chi2a(chi)[source]¶

Scale factor for the radial comoving distance specified in [Mpc/h].

Parameters:	chi (array_like) – Radial comoving distance in [Mpc/h]
Returns:	a – Scale factor corresponding to the specified radial comoving distance.
Return type:	ndarray, or float if input scalar

d_A(a)[source]¶

Angular diameter distance in [Mpc/h] for a given scale factor.

Parameters:	a (array_like) – Scale factor
Returns:	d_A
Return type:	ndarray, or float if input scalar

Notes

Angular diameter distance is expressed in terms of the transverse comoving distance as:

\[d_A(a) = a f_k(a)\]

dchioverda(a)[source]¶

Derivative of the radial comoving distance with respect to the scale factor.

Parameters:	a (array_like) – Scale factor
Returns:	dchi/da – Derivative of the radial comoving distance with respect to the scale factor at the specified scale factor.
Return type:	ndarray, or float if input scalar

Notes

The expression for \(\frac{d \chi}{da}\) is:

\[\frac{d \chi}{da}(a) = \frac{R_H}{a^2 E(a)}\]

dzoverda(a)[source]¶

Derivative of the redshift with respect to the scale factor.

Parameters:	a (array_like) – Scale factor
Returns:	dz/da – Derivative of the redshift with respect to the scale factor at the specified scale factor.
Return type:	ndarray, or float if input scalar

Notes

The expression for \(\frac{d z}{da}\) is:

\[\frac{d z}{da}(a) = \frac{1}{a^2}\]

f_de(a)[source]¶

Evolution parameter for the Dark Energy density.

Parameters:	a (array_like) – Scale factor
Returns:	f – The evolution parameter of the Dark Energy density as a function of scale factor
Return type:	ndarray, or float if input scalar

Notes

For a given parametrisation of the Dark Energy equation of state, the scaling of the Dark Energy density with time can be written as:

\[\rho_{de}(a) \propto a^{f(a)}\]

(see [Percival05]) where \(f(a)\) is computed as \(f(a) = \frac{-3}{\ln(a)} \int_0^{\ln(a)} [1 + w(a^\prime)] d \ln(a^\prime)\). In the case of Linder’s parametrisation for the dark energy in Eq. (?) \(f(a)\) becomes:

\[f(a) = -3(1 + w_0) + 3 w \left[ \frac{a - 1}{ \ln(a) } - 1 \right]\]

f_k(a)[source]¶

Transverse comoving distance in [Mpc/h] for a given scale factor.

Parameters:	a (array_like) – Scale factor
Returns:	f_k – Transverse comoving distance corresponding to the specified scale factor.
Return type:	ndarray, or float if input scalar

Notes

The transverse comoving distance depends on the curvature of the universe and is related to the radial comoving distance through:

\[\begin{split}f_k(a) = \left\lbrace \begin{matrix} R_H \frac{1}{\sqrt{\Omega_k}}\sinh(\sqrt{|\Omega_k|}\chi(a)R_H)& \mbox{for }\Omega_k > 0 \\ \chi(a)& \mbox{for } \Omega_k = 0 \\ R_H \frac{1}{\sqrt{\Omega_k}} \sin(\sqrt{|\Omega_k|}\chi(a)R_H)& \mbox{for } \Omega_k < 0 \end{matrix} \right.\end{split}\]

g(a, a_s)[source]¶

Lensing efficiency kernel computed a distance chi for sources placed at distance chi_s

pk(k, a=1.0, nl_type='smith2003', **kwargs)[source]¶

Computes the full non linear matter power spectrum.

Parameters:	k (array_like) – Wave number in h Mpc^{-1} a (array_like, optional) – Scale factor (def: 1.0) nl_type (str, optional) – Type of non linear corrections. Only ‘smith2003’ is implemented type (str, optional) – Type of transfer function. Either ‘eisenhu’ or ‘eisenhu_osc’ (def: ‘eisenhu_osc’)
Returns:	pk – Non linear matter power spectrum at the specified scale and scale factor.
Return type:	array_like

Notes

The non linear corrections are implemented following [2003:smith]

pk_lin(k, a=1.0, **kwargs)[source]¶

Computes the linear matter power spectrum.

Parameters:	k (array_like) – Wave number in h Mpc^{-1} a (array_like, optional) – Scale factor (def: 1.0) type (str, optional) – Type of transfer function. Either ‘eisenhu’ or ‘eisenhu_osc’ (def: ‘eisenhu_osc’)
Returns:	pk – Linear matter power spectrum at the specified scale and scale factor.
Return type:	array_like

pk_prim(k)[source]¶

Primordial power spectrum Pk = k^n

pl(l, a, **kwargs)[source]¶

Computes the non linear matter power spectrum at a given angular scale using the Limber approximation

pl_lin(l, a, **kwargs)[source]¶

Computes the linear matter power spectrum at the specified scale and scale factor

sh_d[source]¶

Sound horizon at drag epoch in Mpc/h

Computed from Equation (6) in [EisensteinHu98] :

\[r_s(z_d) = \frac{2}{3 k_{eq}} \sqrt{ \frac{6}{R_{eq}} } \ln \frac{ \sqrt{1 + R_d} + \sqrt{R_d + R_{eq}}}{1 + \sqrt{R_{eq}}}\]

where \(R_d\) and \(R_{eq}\) are respectively the ratio of baryon to photon momentum density at drag epoch and equality epoch (see Equation (5) in [EisensteinHu98]) and \(k_{eq}\) is the scale of the scale of the particle horizon at equality epoch.

sh_r[source]¶

Sound horizon at recombination in Mpc/h

Computed from Equation (19) in [EfstathiouBond99] :

\[r_s(z_r) = \frac{4000 \sqrt{a_{equ}}}{\sqrt{\omega_b (1 + \eta_\nu)}} \ln \frac{ \sqrt{1 + R_r} + \sqrt{R_r + R_{eq}}}{1 + \sqrt{R_{eq}}}\]

where \(R_r\) and \(R_{eq}\) are respectively the ratio of baryon to photon momentum density at recombination epoch and equality epoch (see Equation (18) in [EfstathiouBond99]) and \(\eta_{\nu}\) denotes the relative densities of massless neutrinos and photons.

sigmasqr(R, **kwargs)[source]¶

Computes the energy of the fluctuations within a sphere of R h^{-1} Mpc

\[\sigma^2(R)= \frac{1}{2 \pi^2} \int_0^\infty \frac{dk}{k} k^3 P(k,z) W^2(kR)\]

where

\[W(kR) = \frac{3j_1(kR)}{kR}\]

update(**kwargs)[source]¶

Updates the current cosmology based on the parameters specified in input.

Parameters:	h (float) – Reduced hubble constant Omega_b (float) – Baryonic matter density Omega_m (float) – Matter density Omega_de (float) – Dark energy density wa (w0,) – Dark energy equation of state parameters n (float) – Scalar spectral index tau (float) – Reionization optical depth sigma8 (float) – Fluctuation amplitude at 8 Mpc/h

w(a)[source]¶

Dark Energy equation of state parameter using the Linder parametrisation.

Parameters:	a (array_like) – Scale factor
Returns:	w – The Dark Energy equation of state parameter at the specified scale factor
Return type:	ndarray, or float if input scalar

Notes

The Linder parametrization [Linder03] for the Dark Energy equation of state \(p = w \rho\) is given by:

\[w(a) = w_0 + w (1 -a)\]