Context: Star formation efficiency (SFE) theories are currently based on statistical distributions of turbulent cloud structures and a simple model of star formation from cores. They remain poorly tested, especially at the highest densities. Aims: We
investigate the effects of gas density on the SFE through measurements of the core formation efficiency (CFE). With a total mass of $sim2times10^4$ M$_odot$, the W43-MM1 ridge is one of the most convincing candidate precursor of starburst clusters and thus one of the best place to investigate star formation. Methods: We used high-angular resolution maps obtained at 3 mm and 1 mm within W43-MM1 with the IRAM Plateau de Bure Interferometer to reveal a cluster of 11 massive dense cores (MDCs), and, one of the most massive protostellar cores known. An Herschel column density image provided the mass distribution of the cloud gas. We then measured the instantaneous CFE and estimated the SFE and the star formation rate (SFR) within subregions of the W43-MM1 ridge. Results: The high SFE found in the ridge ($sim$6% enclosed in $sim$8 pc$^3$) confirms its ability to form a starburst cluster. There is however a clear lack of dense cores in the northern part of the ridge, which may be currently assembling. The CFE and the SFE are observed to increase with volume gas density while the SFR steeply decreases with the virial parameter, $alpha_{vir}$. Statistical models of the SFR may well describe the outskirts of the W43-MM1 ridge but struggle to reproduce its inner part, which corresponds to measurements at low $alpha_{vir}$. It may be that ridges do not follow the log-normal density distribution, Larson relations, and stationary conditions forced in the statistical SFR models.
