  We investigate spin-incoherent Luttinger liquid of a one-dimensional spin-1
Bose gas in a harmonic trap. In this regime highly degenerate spin
configurations emerge since the spin exchange energy is much less than the
thermal energy of the system, while the temperature is low enough that the
lowest energetic orbitals are occupied. As an example we numerically study the
momentum distribution of a one-dimensional spin-1 Bose gas in Tonks- Girardeau
gas limit and in the sector of zero magnetization.We find that the momentum
distributions broaden as the number of atoms increase due to the averaging of
spin function overlaps. Large momentum ($p$) asymptotic is analytically
derived, showing the universal $1/p^4$ dependence. We demonstrate that the
spin-incoherent Luttinger liquid has a momentum distribution also distinct from
spinless bosons at finite temperature.
