Infinitely many strings in de Sitter spacetime: Expanding and oscillating elliptic function solutions

The exact general evolution of circular strings in (2+1)-dimensional de Sitter spacetime is described closely and completely in terms of elliptic functions. The evolution depends on a constant parameter b, related to the string energy, and falls into three classes depending on whether b<[1]/[4] (oscillatory motion), b=[1]/[4] (degenerated, hyperbolic motion) or b>[1]/[4] (unbounded motion). The novel feature here is that one single world-sheet generically describes infinitely many (different and independent) strings. The world-sheet time τ is an infinite-valued function of the string physical time; each branch yields a different string. This phenomenon has no analogue in flat spacetime. We compute the string energy E as a function of the string proper size S, and analyze it for the expanding and oscillating strings. For expanding strings (S>0): E≠0 even at S=0, E decreases for small S and increases ∝S for large S. For an oscillating string (0≤qslantS≤qslantSmax), the average energy left angle E right angle over one oscillation period is expressed as a function of Smax as a complete elliptic integral of the third kind. For each b, the two independent solutions S+ and S- are analyzed. For b<[1]/[4], all the strings of the S- solution are unstable (Smax=∞) and never collapse to a point (Smin≠0). S+ describes one stable (Smax is bounded) oscillating string and left angle E right angle is an increasing function of b for 0≤qslantb≤qslant[1]/[4]. For b>[1]/[4], all strings (for both S+ and S-) are unstable and have a collapse during their evolution. For b=[1]/[4], S- describes two strings (one stable and one unstable for large de Sitter radius), while S+ describes one stable non-oscillating string. ((orig.))