Cosmological models of the Universe without a Beginning and without a singularity

A new type of cosmological models for the Universe that has no Beginning and evolves from the infinitely distant past is considered.
These models are alternative to the cosmological models based on the Big Bang theory according to which the Universe has a finite age and was formed from an initial singularity.
In our opinion, there are certain problems in the Big Bang theory that our cosmological models do not have.
In our cosmological models, the Universe evolves by compression from the infinitely distant past tending a finite minimum of distances between objects of the order of the Compton wavelength $\lambda_C$ of hadrons and the maximum density of matter corresponding to the hadron era of the Universe. Then it expands progressing through all the stages of evolution established by astronomical observations up to the era of inflation.
The material basis that sets the fundamental nature of the evolution of the Universe in the our cosmological models is a nonlinear Dirac spinor field $\Psi(x^k)$ with nonlinearity in the Lagrangian of the field of type $\beta(\bar{\Psi}\Psi)^n$ ($\beta=const$, $n$ is a rational number), where $\Psi(x^k)$ is the 4-component Dirac spinor, and $\bar{\Psi}$ is the conjugate spinor.
In addition to the spinor field $\Psi$ in cosmological models, we have other components of matter in the form of an ideal liquid with the equation of state $p=w\epsilon$ ($w=const$) at different values of the coefficient $w$ ($-1<w<1$). Additional components affect the evolution of the Universe and all stages of evolution occur in accordance with established observation data. Here $p$ is the pressure, $\epsilon=\rho c^2$ is the energy density, $\rho$ is the mass density, and $c$ is the speed of light in a vacuum.
We have shown that cosmological models with a nonlinear spinor field with a nonlinearity coefficient $n=2$ are the closest to reality.
In this case, the nonlinear spinor field is described by the Dirac equation with cubic nonlinearity.
But this is the Ivanenko–Heisenberg nonlinear spinor equation which W. Heisenberg used to construct a unified spinor theory of matter.
It is an amazing coincidence that the same nonlinear spinor equation can be the basis for constructing a theory of two different fundamental objects of nature — the evolving Universe and physical matter.
The developments of the cosmological models are supplemented by their computer researches the results of which are presented graphically in the work.