Impact of a light stabilized radion in supernovae cooling

In the Randall-Sundrum model where the Standard Model fields are confined to the TeV brane located at the orbifold point $\theta = \pi$ and the gravity peaks at the Planck brane located at $\theta = 0$, the stabilized modulus (radion) field is required to stabilize the size of the fifth spatial dimension. It can be produced copiously inside the supernova core due to nucleon-nucleon bremstrahlung, electron-positron and plasmon-plasmon annihilations, which then subsequently decays to neutrino-antineutrino pair and take away the energy released in SN1987A explosion. Assuming that the supernovae cooling rate $\dot{\varepsilon} \le 7.288\times 10^{-27} \rm{GeV}$, we find the lower bound on the radion vev $\vphi \sim 9.0$ TeV, 2.2 TeV and 0.9 TeV corresponding to the radion mass $m_\phi = 5$ GeV, 20 GeV and 50 GeV, respectively.