  We address the optimal estimation of quantum parameters, in the framework of
local quantum estimation theory, for a massive scalar quantum field in the
expanding Robertson-Walker universe exhibiting Lorentz invariance violation
(LIV). The information about the history of the expanding spacetime in the
presence of LIV can be extracted by taking measurements on the entangled state
of particle modes. We find that, in the estimation of cosmological parameters,
the ultimate bounds to the precision of the Lorentz-invariant massive scalar
field can be improved due to the effects of LIV under some appropriate
conditions. We also show that, in the Lorentz-invariant massive scalar field
and massless scalar field due to LIV backgrounds, the optimal precision can be
achieved by choosing the particles with some suitable LIV, cosmological and
field parameters. Moreover, in the estimation of LIV parameter during the
spacetime expansion, we prove that the appropriate momentum mode of field
particles and larger cosmological parameters can provide us a better precision.
Particularly, the optimal precision of the parameters estimation can be
obtained by performing projective measurements implemented by the projectors
onto the eigenvectors of specific probe states.
