Constraints on the Majoron vev v₁ and mass m_J; see equation (18). The shaded region below the lines is excluded by cosmological observations because the Majoron would decay too fast. The thick (thin) line corresponds to the upper limit on the decay rate coming from Planck 2013, WMAP9 polarization, WiggleZ, and BOSS data (WMAP9 data) [61, 62]. We have assumed normal hierarchy and a vanishing mass for the lightest eigenstate. The dashed lines enclose the region where the observed dark matter density would be reproduced (within ﹩2\sigma ﹩ʼs) if the Majoron were produced thermally.