  We present limits on the molecular gas content of Lyman Break Galaxies (LBGs)
at z~5 from observations targetting redshifted CO(1-0) and CO(2-1) line
emission. We observed a single field containing eight
spectroscopically-confirmed z~5 LBGs, seven of which are contained within a
narrow (z=4.95\pm 0.08) redshift range and the eighth is at z=5.2. No source
was individually detected. Assuming the CO to H2 conversion factor for vigorous
starbursts, we place upper limits on the molecular gas content of individual
z~5 LBGs of M(H2)<~10^10 Msun. From a stacking analysis combining all of the
non-detections, the typical z~5 LBG has an H2 mass limit comparable to their
stellar mass, < 3.1 x 10^9 Msun. This limit implies that, given the star
formation rates of these systems (measured from their UV emission), star
formation could be sustained for at most ~100Myr, similar to the typical ages
of their stellar populations. The lack of a substantially larger reservoir of
cold gas argues against the LBGs being UV luminous super starbursts embedded in
much larger UV-dark systems and as a result increases the likelihood that at
least those LBGs with multiple components are starbursts triggered by mergers.
The sources responsible for reionization are expected to be starbursts similar
to these systems, but with lower luminosities, masses and consequently with
star formation timescales far shorter than the recombination timescale. If so,
the ionized bubbles expected in the IGM during the reionization era will
infrequently have UV-luminous sources at their centres.
