Classical transitions with the topological number changing in the early
  Universe
  We consider classical dynamics of two real scalar fields within a model with
the potential having a saddle point. The solitons of such model are field
configurations that have the form of closed loops in the field space. We study
the formation and evolution of these solitons, in particular, the conditions at
which they could be formed even when the model potential has only one minimum.
These non-trivial field configurations represent domain walls in the
three-dimensional physical space. The set of these configurations can be split
into disjoint equivalence classes. We provide a simple expression for the
winding number of an arbitrary closed loop in the field space and discuss the
transitions that change the winding number. We also show that non-trivial field
configurations could be responsible for the energy density excess that could
evade the CMB constraints but could be important at scales which are
responsible for the formation of galaxies and the massive primordial black
holes.
