The Pipe Nebula is a massive, nearby dark molecular cloud with a low star-formation efficiency which makes it a good laboratory to study the very early stages of the star formation process. The Pipe Nebula is largely filamentary, and appears to be threaded by a uniform magnetic field at scales of few parsecs, perpendicular to its main axis. The field is only locally perturbed in a few regions, such as the only active cluster forming core B59. The aim of this study is to investigate primordial conditions in low-mass pre-stellar cores and how they relate to the local magnetic field in the cloud. We used the IRAM 30-m telescope to carry out a continuum and molecular survey at 3 and 1 mm of early- and late-time molecules toward four selected starless cores inside the Pipe Nebula. We found that the dust continuum emission maps trace better the densest regions than previous 2MASS extinction maps, while 2MASS extinction maps trace better the diffuse gas. The properties of the cores derived from dust emission show average radii of ~0.09 pc, densities of ~1.3x10^5 cm^-3, and core masses of ~2.5 M_sun. Our results confirm that the Pipe Nebula starless cores studied are in a very early evolutionary stage, and present a very young chemistry with different properties that allow us to propose an evolutionary sequence. All of the cores present early-time molecular emission, with CS detections toward all the sample. Two of them, Cores 40 and 109, present strong late-time molecular emission. There seems to be a correlation between the chemical evolutionary stage of the cores and the local magnetic properties that suggests that the evolution of the cores is ruled by a local competition between the magnetic energy and other mechanisms, such as turbulence.
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