Velocity estimators

The velocities estimators from the Hubble diagram residulas \(\Delta\mu\) are implemeted in the redshift_dependence_velocity(). This function return the coefficient \(J(z)\) such that \(\hat{v} = J(z)\Delta\mu\).

Watkins estimator

Watkins estimator from Watkins and Feldman 2015 is such that

\[J(z) = \frac{c\ln10}{5} \frac{z}{1+z}\]

Low z estimator

The low-z estimator is such as:

\[J(z) = \frac{c\ln10}{5} z\]

Hubble highorder estimator

The Hubble highorder estimator use an order 3 expansion with respect to \(z\) of the Hubble law:

\[J(z) = \frac{\ln10}{5}\frac{z}{1 + z}\left[ 1 + \frac{1}{2} (1 - q_0)z - \frac{1}{6}(1 - q_0 - 3 q_0^2 + j_0) z^2\right]\]

When using this estimator you need to pass the deceleration \(q_0\) and jerk \(j_0\) parameters.

Example with the VelFromHDres class:

from flip import data_vector

DataVel = data_vector.VelFromHDres(data, velocity_estimator="hubble highorder", q0=-0.55,j0=-1)

Full estimator

The Full estimator need to assume a cosmology it is such as:

\[J(z) = \frac{c\ln10}{5}\left(c\frac{1 + z}{r(z)H(z)} -1\right)^{-1}\]

where \(r(z)\) is the comoving distance and \(H(z)\) the hubble function.

When using this estimator your data need to contain the hubble_norm and rcom_zobs fields such that hubble_norm is \(h(z) = H(z) / 100\) and rcom_zobs is the comoving distance in Mpc \(h^{-1}\).