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 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 ≤ S ≤ S max), the average energy < E > over one oscillation period is expressed as a function of S max 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 (S max = ∞) and never collapse to a point (S min = 0). S + describes one stable (S max is bounded) oscillating string and < E > is an increasing function of b for 0 ≤ b ≤ 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.