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The specific intensity Icrbnu(nuR) of the CRB at the observed radio frequency nuR is given by the integral over sources (e.g. Peebles 1993):

Equation 4 (4)

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

Equation 5 (5)

where H0 is the Hubble constant, h its value in units of 100 km s-1 Mpc-1, Omegam ident rhom / rhoc is the present mass density of the universe normalized to the critical density rhoc = 1.88 × 10-29 h2 g cm-3, and OmegaLambda ident Lambda / 3H20 is the dimensionless cosmological constant.

We will assume that the radio-IR correlation, expressed at a given rest frame frequency nu0, holds for all redshifts and that the radio spectrum of the individual galaxies follows a ~ nu-alpha power law with spectral index alpha, so that Pnu(nu, LIR, z) = (nu / nu0)-alpha Pnu(nu0, LIR, z). The spectral luminosity density Enu can then be expressed as:

Equation 6 (6)

where Phi(LIR, z) ident [dphi(LIR, z) / dLIR] is the differential IR luminosity function, dphi(LIR, z) is the comoving number density of galaxies with luminosities between LIR and LIR + dLIR in the {Lmin, Lmax} luminosity interval, and kappa0, beta0 are the parameters of the radio-IR correlation at frequency nu0.

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

Equation 7 (7)

where LIR(z) is the comoving IR luminosity density at redshift z:

Equation 8 (8)

and f (z) is defined as:

Equation 9 (9)

The same star forming galaxies give rise to the CIB with an intensity ICIB that is given by:

Equation 10 (10)

For a linear correlation, beta0 = 1, and f (z) = kappa0(Hz-1). The spectral intensity of the CRB then becomes:

Equation 11 (11)

In this case the relation between IcrbnuR and ICIB is considerably simplified and given by:

Equation 12 (12)


Equation 13 (13)

and Tcrb(nuR) is the CRB brightness temperature at frequency nuR.

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