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A formulation of cosmology driven by fermions $\psi $ is studied. Assumption
that the expectation value of the fermion bilinear is non-zero simplifies the
homogeneous solution of the Dirac equations and connects the spinor field with
the scale parameter of the universe. With coupling between the Einstein term
and spinor field $1 - \frac{\xi }{6} (\bar{\psi }\psi )^{-l}$, the possibility
for a late time interaction emerges. In that way, the early universe agrees
with $\Lambda $CDM model, but for the late universe the new integrating term
dominates.
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