Limits on the star formation rates of z > 2 damped Ly alpha systems from Halpha spectroscopy

We present the results of a long-slit K-band spectroscopic search for H alp ha emission from eight damped Ly alpha absorbers (DLAs) at z > 2 with the g oal of measuring the star formation rates in these systems. For each system we searched for compact sources of H alpha emission within a solid angle 1 1 X 2.5 arcsec(2) (44 x 10 h(-2) kpc(2), for q(0) = 0.5, H-0 = 100 hkm s(-1 ) Mpc(-1)). No H alpha emission was detected above 3 sigma limits in the ra nge (6.5-16) X 10(-20) W m(-2), equivalent to star formation rates of 5.6-1 8 h(-2) M. yr(-1), for a standard initial mass function, assuming the lines are spectrally unresolved (<650 km s(-1) FWHM). We compare these results a gainst the predictions of the models of Pei & Fall of the global history of star formation, under two different simplifying hypotheses: (i) the space density of DLAs at z = 2.3 is equal to the space density of spiral galaxies today (implying DLA discs were larger in the past, the 'large-disc' hypoth esis); (ii) the sizes of DLAs at z = 2.3 were the same as the gas sizes of spiral galaxies today (implying DLA discs were more common in the past, the 'hierarchical' hypothesis). Compared with the previous most sensitive spec troscopic search, our sample is twice as large, our limits are a factor gre ater than two deeper, and the solid angle surveyed is over three times as g reat. Despite this, our results are not in conflict with either the large-d isc hypothesis, because of the limited solid angle covered by the slit, or the hierarchical hypothesis, because of the limited sensitivity.