Cosmology

These helpers encode the fixed background cosmology used by the semi-analytic model and its derived distance and volume factors.

EE

EE(z)

Compute the dimensionless Hubble parameter \(E(z)\).

The cosmology used by fastropop is a fixed flat \(\Lambda\)CDM background with \(\Omega_{\rm m} = 0.3\), \(\Omega_\Lambda = 0.7\), \(\Omega_k = 0\), and \(h = 0.7\). The corresponding expansion function is

\[ E(z) = \sqrt{\Omega_{\rm m}(1+z)^3 + \Omega_k (1+z)^2 + \Omega_\Lambda}. \]

Parameters:

• z (float or Array) –

  Redshift.

Returns:

• float or Array –

  Dimensionless Hubble parameter \(E(z)\).

dtodz

dtodz(z)

Compute \(dt_r/dz\), the derivative of cosmic time with redshift.

This helper is used throughout the semi-analytic model when converting merger-rate densities written in redshift into time-based quantities:

\[ \frac{dt_r}{dz} = \frac{1}{H_0 (1+z) E(z)}. \]

Parameters:

• z (float or Array) –

  Redshift.

Returns:

• float or Array –

  Derivative \(dt_r/dz\) in seconds.

Dc_interp

Dc_interp(z)

Interpolate the comoving distance on a precomputed redshift grid.

Parameters:

• z (float or Array) –

  Redshift.

Returns:

• float or Array –

  Line-of-sight comoving distance in metres.

dVcdz

dVcdz(z)

Compute the comoving volume element \(dV_c/dz\).

The quantity returned is

\[ \frac{dV_c}{dz} = \frac{4 \pi c}{H_0} \frac{D_c(z)^2}{E(z)}, \]

with \(D_c(z)\) the comoving distance.

Parameters:

• z (float or Array) –

  Redshift.

Returns:

• float or Array –

  Comoving volume element in SI units.

DL

DL(z)

Compute the luminosity distance.

Parameters:

• z (float or Array) –

  Redshift.

Returns:

• float or Array –

  Luminosity distance \(D_L(z) = (1+z) D_c(z)\) in metres.