Based on CHANDRA and ASCA observations of nearby starburst galaxies and RXTE/ASM , ASCA , and MIR-KVANT/TTM studies of high mass X-ray binary ( HMXB ) populations in the Milky Way and Magellanic Clouds , we propose that the number and/or the collective X-ray luminosity of HMXBs can be used to measure the star formation rate ( SFR ) of a galaxy .
We show that , within the accuracy of the presently available data , a linear relation between HMXB number and star formation rate exists .
The relation between SFR and collective luminosity of HMXBs is non-linear in the low SFR regime , L _ { X } \propto \text { SFR } ^ { \sim 1.7 } , and becomes linear only for sufficiently high star formation rate , SFR \ga 4.5 M _ { \odot } yr ^ { -1 } ( for M _ { * } > 8 M _ { \odot } ) .
The non-linear L _ { X } - SFR dependence in the low SFR limit is not related to non-linear SFR-dependent effects in the population of HMXB sources .
It is rather caused by the fact , that we measure the collective luminosity of a population of discrete sources , which might be dominated by the few brightest sources .
Although more subtle SFR-dependent effects are likely to exist , in the entire range of SFRs the data are broadly consistent with the existence of a universal luminosity function of HMXBs which can be roughly described as a power law with a differential slope of \sim 1.6 , a cutoff at L _ { X } \sim { few } \times 10 ^ { 40 } erg s ^ { -1 } and a normalisation proportional to the star formation rate .

We apply our results to ( spatially unresolved ) starburst galaxies observed by CHANDRA at redshifts up to z \sim 1.2 in the Hubble Deep Field North and show that the calibration of the collective luminosity of HMXBs as a SFR indicator based on the local sample agrees well with the SFR estimates obtained for these distant galaxies with conventional methods .