Abstract
By analyzing the energy-momentum tensor and equations of state of ideal gas, scalar, spinor and vector potential in detail, we find that the total mass density of all matter is always positive, and the initial total pressure is negative. Under these conditions, by qualitatively analyzing the global behavior of the dynamical equation of cosmological model, we get the following results: (i) K=1, namely, the global spatial structure of the universe should be a three-dimensional sphere S3; (ii) 0≤Λ<10−24ly−2, the cosmological constant should be zero or an infinitesimal; (iii) a(t)>0, the initial singularity of the universe is unreachable, and the evolution of the universe should be cyclic in time. Since the matter components considered are quite complete and the proof is very elementary and strict, these conclusions are quite reliable in logic and compatible with all observational data. Obviously, these conclusions will be very helpful to correct some popular misconceptions and bring great convenience to further research other problems in cosmology such as the properties of dark matter and dark energy. In addition, the macroscopic Lagrangian of fluid model is derived.
Highlights
In cosmology, we have two important constants to be determined
In [23,24], the authors use (ΩK, a) phase plane to discuss the dynamical behavior of the universe, and conclude that a cyclical universe is reasonable from a dynamical systems perspective, and requires in addition to standard cosmological assumptions, only two conditions: (i) the spatial sections must have positive spatial curvature K = +1, and (ii) the late time effective cosmological “constant” must decay fast enough as a function of the scale factor
In [27], we showed that the global behavior of Friedmann equation is mainly determined by two hypotheses of positive mass density and initial negative pressure
Summary
We have two important constants to be determined. They are cosmic curvature K and cosmological constant Λ. In [23,24], the authors use (ΩK , a) phase plane to discuss the dynamical behavior of the universe, and conclude that a cyclical universe is reasonable from a dynamical systems perspective, and requires in addition to standard cosmological assumptions, only two conditions: (i) the spatial sections must have positive spatial curvature K = +1, and (ii) the late time effective cosmological “constant” must decay fast enough as a function of the scale factor Both of these conditions are consistent with all current observations to date. Under two ordinary hypotheses for functions of state, namely the total mass density of all matter is always positive, and the initial total pressure is negative, by qualitatively analyzing the global behavior of the dynamical equation of cosmological model, we get the following results: the global spatial structure of the universe should be a closed 3-d sphere. These conclusions will be very helpful to correct some popular misconceptions and bring great convenience in further researching other problems in cosmology such as the properties of dark matter and dark energy
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