Abstract: The vacuum expectation value v(s) of a Higgs triplet field Delta carrying two units of lepton number L induces neutrino masses alpha v(s). The neutral component of Delta gives rise to two Higgs particles, a pseudoscalar A and a scalar S. The most general renormalizable Higgs potential V for Delta and the Standard-Model Higgs doublet Phi does not permit the possibility that the mass of either A or S is small, of order v(s), while the other mass is heavy enough to forbid the decay Z -> AS to comply with LEP 1 data. We present a model with additional dimension-6 terms in V, in which this feature is absent and either A or S can be chosen light. Subsequently we propose the model as a remedy to cosmological anomalies, namely the tension between observed and predicted tensor-to-scalar mode ratios in the cosmic microwave background and the different values of the Hubble constant measured at different cosmological scales. Furthermore, if Delta dominantly couples to the third-generation doublet L-tau = (v(tau), tau), the deficit of v(tau) events at IceCube can be explained. The singly and doubly charged triplet Higgs bosons are lighter than 280 GeV and 400 GeV, respectively, and could be found at the LHC.