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

Abstract

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 dot >0): E ≠ 0 even at S = 0, E decreases for small S and increases α 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 string of the S − solution are unstable ( S max = ∞) and never collapse to a point ( S min ≠ 0).v S + describes one stable ( S max is bounded) oscillating string and 〈 E〉 is an increasing function of b for 0⩽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.