We present a catalogue of 147 serendipitous X-ray sources selected to have
hard spectra (alpha <0.5) from a survey of 188 ROSAT fields. Such sources m
ust be the dominant contributors to the X-ray background at faint fluxes. W
e have used Monte Carlo simulations to verify that our technique is very ef
ficient at selecting hard sources: the survey has greater than or equal to
10 times as much effective area for hard sources as it has for soft sources
above a 0.5-2 keV flux level of 10(-14) erg cm(-2) s(-1). The distribution
of best-fitting spectral slopes of the hard sources suggests that a typica
l ROSAT hard source in our survey has a spectral slope alpha similar to0. T
he hard sources have a steep number flux relation (dN/dS proportional toS(-
gamma) with a best-fitting value of gamma =2.72 +/-0.12) and make up about
15 per cent of all 0.5-2 keV sources with S > 10(-14) erg cm(-2) s(-1). If
their N(S) continues to fainter fluxes, the hard sources will comprise simi
lar to 40 per cent of sources with 5x10(-15)<S < 10(-14) erg cm(-2) s(-1).
The population of hard sources can therefore account for the harder average
spectra of ROSAT sources with S < 10(-14) erg cm(-2) s(-1). They probably
make a strong contribution to the X-ray background at faint fluxes and coul
d be the solution to the X-ray background spectral paradox.