The specific intensity
*I*^{crb}_{}(_{R})
of the CRB at the observed radio frequency
_{R} is given by the
integral over sources (e.g.
Peebles 1993):

(4) |

where _{}(, *z*) is the specific radio
luminosity per comoving volume element at redshift *z*,
=
_{R}(1 + *z*) is
the frequency in the rest frame of the luminous objects and
|*dt* / *dz*| is given by
(Hogg 1999)

(5) |

where *H*_{0} is the Hubble constant, *h* its value in
units of 100 km s^{-1} Mpc^{-1},
_{m}
_{m} /
_{c} is
the present mass density of the universe normalized to the critical density
_{c} =
1.88 × 10^{-29} *h*^{2} g cm^{-3}, and
_{}
/
3*H*^{2}_{0} is the dimensionless cosmological
constant.

We will assume that the radio-IR correlation, expressed at a given rest
frame frequency _{0},
holds for all redshifts and that the radio spectrum of the individual
galaxies follows a ~ ^{-} power law with
spectral index
, so that
*P*_{}(,
*L*_{IR}, *z*) =
( /
_{0})^{-}
*P*_{}(_{0},
*L*_{IR}, *z*). The spectral luminosity density
_{} can then be expressed as:

(6) |

where
(*L*_{IR},
*z*)
[*d*(*L*_{IR}, *z*) /
*dL*_{IR}] is the differential IR luminosity function,
d(*L*_{IR}, *z*) is the comoving number
density of galaxies with luminosities between L_{IR} and
L_{IR} + dL_{IR} in the {*L*_{min},
*L*_{max}} luminosity interval, and
_{0},
_{0}
are the parameters of the radio-IR correlation at frequency
_{0}.

The specific intensity of the CRB can now be written as:

(7) |

where _{IR}(z) is the
comoving IR luminosity density at redshift *z*:

(8) |

and *f* (*z*) is defined as:

(9) |

The same star forming galaxies give rise to the CIB with an intensity
*I*_{CIB} that is given by:

(10) |

For a linear correlation,
_{0}
= 1, and *f* (*z*) = _{0}(Hz^{-1}). The spectral intensity of the
CRB then becomes:

(11) |

In this case the relation between
*I*^{crb}_{R} and *I*_{CIB} is considerably
simplified and given by:

(12) |

where

(13) |

and *T*_{crb}(_{R}) is the CRB brightness temperature at frequency
_{R}.