General Relativity: An Introduction for Physicists · Problem 14.9

The Friedmann–Robertson–Walker geometry — Hobson, Efstathiou & Lasenby

Physics · flux-density-k-correction

Problem

The observed flux in the frequency range

[\nu_1, \nu_2]

received from some distant comoving object is given by

F_{\mathrm{obs}}(\nu_1, \nu_2) = \int_{\nu_1}^{\nu_2} f_{\mathrm{obs}}(\nu)\,d\nu,

where

f_{\mathrm{obs}}(v)

is the observed flux density (in

\mathrm{W\,m^{-2}\,Hz^{-1}}

) as a function of frequency. If

f_{\mathrm{em}}(\nu)

is the emitted (or intrinsic) flux density of the object, show that

f_{\mathrm{obs}}(\nu) = \frac{f_{\mathrm{em}}((1+z)\nu)}{1+z},

where

z

is the redshift of the object. If

f_{\mathrm{em}}(\nu) \propto \nu^{\alpha}

over a wide range of frequencies, show that

F_{\mathrm{obs}}(\nu_1, \nu_2) = K_z\, F_{\mathrm{em}}(\nu_1, \nu_2),

where the

K

-correction is given by

K_z = (1+z)^{\alpha-1}