Cosmological Distances

Author

NumCosmo developers

Cosmological Distances

This page describes the distance and time-related quantities implemented by NcDistance. The object computes the dimensionless versions of these distances; the dimensionful results are recovered by multiplying by the appropriate Hubble radius, as described below.

Hubble Radius

The Hubble radius (or scale) is the inverse of the Hubble function \(H(z)\), \[ \RH = \frac{c}{H(z)}, \qquad \RH_0 = \frac{c}{H_0}, \] where \(c\) is the speed of light, \(z\) is the redshift, and \(H_0 \equiv H(0)\) is the Hubble parameter. The comoving Hubble radius is \[ \RHc(z) = \frac{c}{aH(z)} = \frac{c(1+z)}{a_0H(z)}, \qquad \RHc_0 = \frac{c}{a_0H_0}, \] where the \({}_0\) subscript denotes evaluation at the present time, and the redshift \(z\) is defined by \[ 1 + z = \frac{a_0}{a}. \]

Comoving Distance

The comoving distance \(d_c\) is \[ d_c(z) = \RHc_0\int_0^z \frac{\mathrm{d}z^\prime}{E(z^\prime)}, \] where \(E(z)\) is the normalized Hubble function, \[ E(z) \equiv \frac{H(z)}{H_0}. \]

The object works with the dimensionless comoving distance, \[ D_c(z) \equiv \frac{d_c(z)}{\RHc_0}. \] Note that \(D_c(z)\) also coincides with the proper distance today, \(r(z) \equiv a_0 d_c(z)\), in units of the Hubble radius: \(D_c(z) = r(z) / \RH_0\). The comoving distance and the proper distance today are therefore recovered by multiplying \(D_c(z)\) by \(\RHc_0\) and \(\RH_0\), respectively.

Transverse, Luminosity, and Angular Diameter Distances

The transverse comoving distance \(D_t\) and its derivative with respect to \(z\) are \[ D_t(z) = \frac{\sinh\left[\sqrt{\Omega_{k0}}\,D_c(z)\right]}{\sqrt{\Omega_{k0}}}, \qquad \frac{\mathrm{d}D_t}{\mathrm{d}z}(z) = \frac{\cosh\left[\sqrt{\Omega_{k0}}\,D_c(z)\right]}{E(z)}, \] where \(\Omega_{k0}\) is the curvature density parameter today. The luminosity and angular diameter distances follow as \[ D_l = (1 + z)\,D_t(z), \qquad D_A = D_t(z) / (1 + z). \]

Distance Modulus

The distance modulus is \[ \delta\mu(z) = 5\log_{10}(D_l(z)) + 25. \] The conventional distance modulus is \[ \mu(z) = 5\log_{10}[\RH_0 D_l(z)/(1\,\mathrm{Mpc})] + 25, \] which differs from the definition above by a constant factor \(5\log_{10}[\RH_0/(1\,\mathrm{Mpc})]\): \[ \mu(z) = \delta\mu(z) + 5\log_{10}[\RH_0/(1\,\mathrm{Mpc})], \] where \(\mathrm{Mpc}\) is the megaparsec.

Distances Between Two Redshifts

For a pair \(z_1 < z_2\), the transverse comoving and angular diameter distances are \[ \begin{aligned} D_t(z_1, z_2) &= \frac{\sinh\left\{\sqrt{\Omega_{k0}}\left[D_c(z_2)-D_c(z_1)\right]\right\}}{\sqrt{\Omega_{k0}}}, \\ D_A(z_1, z_2) &= D_t(z_1, z_2) / (1 + z_2). \end{aligned} \]

API Reference

See NcDistance for the full class reference. The underlying cosmology enters through the Hubble function and density parameters of the NcHICosmo model, notably nc_hicosmo_H, nc_hicosmo_H0, nc_hicosmo_E, and nc_hicosmo_Omega_k0.

The most relevant methods are: