The research leading to these results has received funding from the ERC Starting Consolidator Grant (no. 307209). Some of the results in this paper have been derived using the HEALPix package. We acknowledge the use of the PSM package, developed by the Planck working group on component separation, for making the simulations used in this work. We acknowledge the use of the Ulysses cluster at SISSA. This research was partially supported by the RADIOFOREGROUNDS project, funded by the European Commission’s H2020 Research Infrastructures under the Grant Agreement 687312, and the INDARK INFN Initiative. GDZ acknowledges support by ASI/INAF agreement no. 2014-024-R.1. R.F.-C., E.M.-G., and P.V. acknowledge support from the Spanish MINECO project ESP2015-70646-C2-1-R (cofinanced with EU FEDER funds), Consolider-Ingenio 2010 project CSD2010-00064 and from the CSIC “Proyecto Intramural Especial” project 201550E091. JGN acknowledges financial support from the Spanish MINECO for a ‘Ramon y Cajal’ fellowship (RYC-2013-13256) and the I+D 2015 project AYA2015-65887-P (MINECO/FEDER). CJM is supported by an FCT Research Professorship, contract reference IF/00064/2012, funded by FCT/MCTES (Portugal) and POPH/FSE.