Malcolm Longair has described how the Cavendish Laboratory spent the 1960s practicing human sacrifice in order to determine the extragalactic radio-source background, with the following approximate result:
to within an uncertainty of about 20% in amplitude and 0.1 in spectral index. This background dominates over the CMB for 1 m, and is consistent with the integrated contribution of discrete sources.
On the other hand, it is also not ruled out that a genuine continuum background might exist at up to 10% or so of the above level. What would this mean if it was really so? The hope would be to learn something about diffuse intergalactic gas, and there are two standard emission mechanisms to which we might appeal: synchrotron radiation and bremsstrahlung. The parameters available are the density of the emitting plasma, parameterized by its contribution to (in the case of synchrotron radiation, the electrons would have an assumed power-law energy distribution), plus the local value of either the magnetic field, B or temperature T - both of which should scale as (1 + z)2. The resultant background can then be worked out in the standard way (see Longair 1978). For synchrotron radiation, we get
What is a plausible value for the intergalactic magnetic field? It is worth recalling that magnetic fields are very much a skeleton in the closet of cosmology, since we cannot easily rule out rather large values - which would significantly change our ideas about structure formation, for example. A nice review of the issue is given by Coles (1992); he argues that B could be as large as 10-4 nT. This would allow observed magnetic fields in astrophysical sources to be made via compression, rather than dynamo effects, and would greatly alter the progress of galaxy clustering. For such a field, the observed background would be produced with h ~ 10-3. This is an implausibly high density for a plasma with fully relativistic electrons, but it is perhaps surprising that the effect is this close to being interesting.
Turning to bremsstrahlung, one can simply try scaling old solutions for the X-ray background in which a `low'-energy flux of around 10-3 Jy sr-1 is produced by models with T 108 K and h2 0.1. Since bremsstrahlung emissivity scales as T-1/2, this implies
If we ignore the difficulty in keeping plasma at such temperatures ionized, this seems the closest that the radio background is likely to get to setting constraints on Cold Dark Matter...