Model of an Oscillating Cosmos which Rejuvenates during Contraction

For an oscillating cosmos we assume time‐symmetric initial conditions, invariant under t→ —t and t→ T + t, where t = 0, ±T, … are the times of maximal cosmos contraction. In particular we specify the same value U₀ for the cosmos' ``internal energy'' U at all these times. The expectation value of an Heisenberg operator becomes now $$\mathrm{〈Q(t)〉\hspace{0.17em}=\hspace{0.17em}}\mathrm{Tr}\mathrm{\hspace{0.17em}\{Q(t)\hspace{0.17em}D\},\hspace{0.17em}}\mathrm{with}{\text{\hspace{0.17em}D=[A(T),A(0)]}}_{+}\text{/2\hspace{0.17em}}\mathrm{Tr}\text{\hspace{0.17em}{A(T)A(0)}}$$ . Here A(0) is the projector into the space of the eigenvectors of U with eigenvalue U₀, and A(t) has the time dependence of an Heisenberg operator. The time symmetry implies that the cosmos' oscillations are periodic and that expansions and contractions occur (except for local statistical fluctuations) time‐symmetrically to each other: If the entropy increases during expansion then it must decrease during contraction. The general theory is applied to a highly simplified cosmos model in which no star condensation occurs. Assuming here that the initial state contains equal numbers of particles and antiparticles, the theory predicts a slightly inhomogeneous distribution of these particles such that during the cosmos' expansion not all particles annihilate, but a realistic density of particles and antiparticles (in different space regions) survives the expansion.