  A plausible formation scenario for the Galactic globular clusters 47 Tuc and
Omega Cen is that they are tidally stripped remnants of dwarf galaxies, in
which case they are likely to have retained a fraction of their dark matter
cores. In this study, we have used the ultra-wide band receiver on the Parkes
telescope (Murriyang) to place upper limits on the annihilation rate of exotic
Light Dark Matter particles (X) via the XX/$e^+e^-$ channel using measurements
of the recombination rate of positronium (Ps). This is an extension of a
technique previously used to search for Ps in the Galactic Centre. However, by
stacking of spectral data at multiple line frequencies, we have been able to
improve sensitivity. Our measurements have resulted in 3-sigma flux density
(recombination rate) upper limits of 1.7 mJy (1.4x10$^{43}$ s$^{-1}$) and 0.8
mJy (1.1x10$^{43}$ s$^{-1}$) for 47 Tuc and Omega Cen, respectively. Within the
Parkes beam at the cluster distances, which varies from 10 to 23 pc depending
on the frequency of the recombination line, we calculate upper limits to the
dark matter mass and rms dark matter density of <1.2-1.3x10$^5$ fn$^{-0.5}$
(mX/MeV c$^{-2}$) solar masses and <48-54 fn$^{-0.5}$ (mX/MeV c$^{-2}$) solar
masses pc$^{-3}$ for the clusters, where fn=Rn/Rp is the ratio of Ps
recombination transitions to annihilations, estimated to be 0.01. The radio
limits for Omega Cen suggest that, for a fiducial dark/luminous mass ratio of
0.05, any contribution from Light Dark Matter is small unless the cross section
<7.9x10$^{-28}$ (mX/MeV c$^{-2}$)$^2$ cm$^3$ s$^{-1}$. Owing to the compactness
and proximity of the clusters, archival 511-keV measurements suggest even
tighter limits than permitted by CMB anisotropies, <8.6x10$^{-31}$ (mX/MeV
c$^{-2}$)$^2$ cm$^3$ s$^{-1}$. Our recombination rate limits substantially
improve on previous radio limits for the Milky Way.
