  Context. The progenitors of short gamma-ray bursts (SGRBs) have not yet been
well identified. The most popular model is the merger of compact object
binaries (NS-NS/NS-BH). However, other progenitor models cannot be ruled out.
The delay-time distribution of SGRB progenitors, which is an important property
to constrain progenitor models, is still poorly understood. Aims. We aim to
better constrain the luminosity function of SGRBs and the delay-time
distribution of their progenitors with newly discovered SGRBs. Methods. We
present a low-contamination sample of 16 Swift SGRBs that is better defined by
a duration shorter than 0.8 s. By using this robust sample and by combining a
self-consistent star formation model with various models for the distribution
of time delays, the redshift distribution of SGRBs is calculated and then
compared to the observational data. Results. We find that the power-law delay
distribution model is disfavored and that only the lognormal delay distribution
model with the typical delay tau >= 3 Gyr is consistent with the data.
Comparing Swift SGRBs with T90 > 0.8 s to our robust sample (T90 < 0.8 s), we
find a significant difference in the time delays between these two samples.
Conclusions. Our results show that the progenitors of SGRBs are dominated by
relatively long-lived systems (tau >= 3 Gyr), which contrasts the results found
for Type Ia supernovae. We therefore conclude that primordial NS-NS systems are
not favored as the dominant SGRB progenitors. Alternatively, dynamically formed
NS-NS/BH and primordial NS-BH systems with average delays longer than 5 Gyr may
contribute a significant fraction to the overall SGRB progenitors.
