  We consider theoretically ${}^{13}$C-hyperfine interaction induced dephasing
in carbon nanotubes double quantum dots with curvature induced spin-orbit
coupling. For two electrons initially occupying a single dot, we calculate the
average return probability after separation into the two dots, which have
random nuclear-spin configurations. We focus on the long time saturation value
of the return probability, $P_\infty$. Because of the valley degree of freedom,
the analysis is more complex than in, for example, GaAs quantum dots, which
have two distinct $P_\infty$ values depending on the magnetic field. Here the
prepared state and the measured state is non-unique because two electrons in
the same dot are allowed in six different states. Moreover, for one electron in
each dot sixteen states exist and therefore are available for being mixed by
the hyperfine field. The return probability experiment is found to be strongly
dependent on the prepared state, on the external magnetic field---both Zeeman
and orbital effects - and on the spin-orbit splitting. The lowest saturation
value, being $P_\infty$=1/3, occurs at zero magnetic field for nanotubes with
spin-orbit coupling and the initial state being the groundstate, this situation
is equivalent to double dots without the valley degree of freedom. In total, we
report nine dynamically different situations that give $P_\infty$=1/3, 3/8,
2/5, 1/2 and for valley anti-symmetric prepared states in an axial magnetic
field, $P_\infty$=1. When the groundstate is prepared the ratio between the
spin-orbit splitting and the Zeeman energy due to a perpendicular magnetic
field can tune the effective hyperfine field continuously from being three
dimensional to two dimensional giving saturation values from $P_\infty$=1/3 to
3/8.
