Evolution of the early universe in Einstein-Cartan theory

Einstein--Cartan theory is a generalization of general relativity that introduces spacetime torsion. In this paper, we perform phase space analysis to investigate the evolution of the early universe in Einstein--Cartan theory. By studying the stability of critical points in the dynamical system, we find that there exist two stable critical points which represent an Einstein static solution and an expanding solution, respectively. After analyzing the phase diagram of the dynamical system, we find that the early universe may exhibit an Einstein static state, an oscillating state, or a bouncing state. By assuming the equation of state $\omega$ can decrease over time $t$, the universe can depart from the initial Einstein static state, oscillating state, or bouncing state and then evolve into an inflationary phase. Then, we analyze four different inflationary evolution cases in Einstein--Cartan theory and find that a time-variable equation of state $\omega$ cannot yield values of $n_{s}$ and $r$ consistent with observations, while a time-invariant equation of state $\omega$ is supported by the Planck 2018 results. Thus, in Einstein--Cartan theory, the universe likely originates from a bouncing state rather than an Einstein static state or an oscillating state.