of the PEP catalog were extracted using the position of 24$\mu$m sources as a prior. This catalog is used only to select strong starbursts up to z$\sim$3. The 24$\mu$m prior should not induce any incompleteness of the strong-starburst sample, since their minimum expected 24$\mu$m flux is at least 2 times larger than the detection limit at this wavelength [^2].\

*Herschel*/SPIRE data
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We also used *Herschel* data at 250$\mu$m, 350$\mu$m, and 500$\mu$m taken by the spectral and photometric imaging receiver (SPIRE, @Griffin2010) as part of the *Herschel* multitiered extragalactic survey (HerMES, @Oliver2012). The FWHM of the PSF is 18.2“, 24.9”, and 36.3", the 1$\sigma$ instrumental noise is 1.6, 1.3, and 1.9mJy/beam, and the 1$\sigma$ confusion noise is 5.8, 6.3, and 6.8mJy/beam [@Nguyen2010] at 250$\mu$m, 350$\mu$m, and 500$\mu$m, respectively. In this paper, we used the sources catalog extracted using as a prior the positions, the fluxes, the redshifts, and mean colors measured by stacking of 24$\mu$m sources, as described in @Bethermin2012b.\

LABOCA data
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The COSMOS field was mapped at 870$\mu$m by the large APEX bolometer Camera (LABOCA) mounted on the Atacama Pathfinder Experiment (APEX) telescope[^3] (PI: Frank Bertoldi, Navarrete et al. in prep.). We retrieved the raw data from the 