  Polycyclic Aromatic Hydrocarbons (PAHs) are widely accepted as the carriers
of the Aromatic Infrared Bands (AIBs), but an unambiguous identification of any
specific interstellar PAH is still missing. For polar PAHs, pure rotational
transitions can be used as fingerprints for identification. Combining dedicated
experiments, detailed simulations and observations, we explored the mm domain
to search for specific rotational transitions of corannulene (C20H10). We
performed high-resolution spectroscopic measurements and a simulation of the
emission spectrum of UV-excited C20H10 in the environment of the Red Rectangle,
calculating its synthetic rotational spectrum. Based on these results, we
conducted a first observational campaign at the IRAM 30m telescope towards this
source to search for several high-J rotational transitions of (C20H10). The
laboratory detection of the J = 112 <- 111 transition of corannulene showed
that no centrifugal splitting is present up to this line. Observations with the
IRAM 30m telescope towards the Red Rectangle do not show any corannulene
emission at any of the observed frequencies, down to a rms noise level of Tmb =
8 mK for the J =135 -> 134 transition at 137.615 GHz. Comparing the noise level
with the synthetic spectrum, we are able to estimate an upper limit to the
fraction of carbon locked in corannulene of about 1.0x10(-5) relative to the
total abundance of carbon in PAHs. The sensitivity achieved shows that radio
spectroscopy can be a powerful tool to search for polar PAHs. We compare this
upper limit with models for the PAH size distribution, emphasising that small
PAHs are much less abundant than predicted. We show that this cannot be
explained by destruction but is more likely related to the chemistry of their
formation in the environment of the Red Rectangle.
