of the C$\nu$B, which can leave an imprint on the observed Cosmic Microwave Background (CMB). Strong constraints on $\nu$SI have been set in Refs. [@Cyr-Racine:2013jua; @Archidiacono:2013dua]. In the mediator mass range we focus on, the constraint was derived assuming a heavy mediator, and is $G \lesssim 100 \;{\rm GeV}^{-2}$, which is much more stringent than the SN 1987A limit. In Ref. [@Cyr-Racine:2013jua; @Archidiacono:2013dua], the C$\nu$B is constrained to be free-streaming untill redshift $\sim 2\times10^{5}$, where $\sqrt{s}\sim 10^{-4}\,{\rm MeV}$. Therefore, the domain of applicability of the CMB limit would end at a vertical boundary (not shown) at $M\sim10^{-4}\,{\rm MeV}$.

Limits on $\nu$SI can also be set by observations of laboratory processes. Even if the neutrinos are not detected directly, their presence can be inferred by precise measurements of other particles. For example, in the presence of $\nu$SI, a mediator can be produced by bremsstrahlung from an external neutrino [@Barger:1981vd; @Lessa:2007up; @Laha:2013xua]; a massive mediator will then decay into other particles [@Bilenky:1992xn; @Bilenky:1994ma; @Bilenky:1999dn]. In Majoron-type models, the best laboratory constraints come from meson and lepton decays, e.g., Refs. [@Barger:1981vd; @Lessa:2007up], but they depend on the particular flavor coupling, $g_{\alpha\beta}$, where $\alpha,\beta = e,\mu,\tau$, and are valid only up to the mass 