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(Abridged) In this paper, we present analyses of images taken with theHerschel ESA satellite from 70mu to 500mu. We first constructed column densityand dust temperature maps. Next, we identified compact cores in the maps, andcharacterize the cores using modified blackbody fits to their SEDs: weidentified 684 starless cores, of which 199 are bound and potential prestellarcores, and 132 protostars. We also matched the Herschel-identified young starswith GAIA sources to model distance variations across the Perseus cloud. Wemeasure a linear gradient function with right ascension and declination for theentire cloud. From the SED fits, mass and temperature of cores were derived.The core mass function can be modelled with a log-normal distribution thatpeaks at 0.82~$M_\sun$ suggesting a star formation efficiency of 0.30. Thehigh-mass tail can be modelled with a power law of slope $\sim-2.32$, close tothe Salpeter's value. We also identify the filamentary structure of Perseus,confirming that stars form preferentially in filaments. We find that themajority of filaments where star formation is ongoing are transcritical againsttheir own internal gravity because their linear masses are below the criticallimit of 16~$M_\sun$pc$^{-1}$ above which we expect filaments to collapse. Wefind a possible explanation for this result, showing that a filament with alinear mass as low as 8~$M_\sun$pc$^{-1}$ can be already unstable. We confirm alinear relation between star formation efficiency and slope of dust probabilitydensity function and a similar relation is also seen with the core formationefficiency. We derive a lifetime for the prestellar core phase of$1.69\pm0.52$~Myr for Perseus but different regions have a wide range inprestellar core fractions, hint that star-formation has started only recentlyin some clumps. We also derive a free-fall time for prestellar cores of0.16~Myr.

Physical properties of the ambient medium and of dense cores in the Perseus star-forming region derived from Herschel Gould Belt Survey observations