  The ARCADE 2 balloon bolometer along with a number of other instruments have
detected what appears to be a radio synchrotron background at frequencies below
about 3 GHz. Neither extragalactic radio sources nor diffuse Galactic emission
can currently account for this finding. We use the locally measured Cosmic ray
electron population, demodulated for effects of the Solar wind, and other
observational constraints combined with a turbulent magnetic field model to
predict the radio synchrotron emission for the Local Bubble. We find that the
spectral index of the modelled radio emission is roughly consistent with the
radio background. Our model can approximately reproduce the observed antenna
temperatures for a mean magnetic field strength B between 3-5 nT. We argue that
this would not violate observational constraints from pulsar measurements.
However, the curvature in the predicted spectrum would mean that other, so far
unknown sources would have to contribute below 100 MHz. Also, the magnetic
energy density would then dominate over thermal and cosmic ray electron energy
density, likely causing an inverse magnetic cascade with large variations of
the radio emission in different sky directions as well as high polarisation. We
argue that this disagrees with several observations and thus that the magnetic
field is probably much lower, quite possibly limited by equipartition with the
energy density in relativistic or thermal particles (B = 0.2-0.6 nT). In the
latter case, we predict a contribution of the Local Bubble to the unexplained
radio background at most at the per cent level.
