Volume 260, number 1,2 PHYSICS LETTERS B 9 May 1991 Clustering of cosmological defects at the time of formation Robert Leese and Tomislav Prokopec Department of Physics, Brown University, Providence, RI 02912, USA Received 11 December 1990 A simple model for the formation of global monopoles is considered. It is shown that they naturally form in clusters, with monopoles adjacent to antimonopoles, and vice-versa. The strong attraction between pole and antipole causes the clusters to collapse very rapidly, leading to the annihilation of most (62% in our model) of the original defects within a time z, where z is of the order of the correlation length. 1. Introduction Various topological defects may form at the phase transitions that accompany spontaneous symmetry breaking in the early universe [ 1 ]. Recently, in- creased attention has been given to the defects aris- ing from the breaking of global, rather than local, symmetries. Both global monopoles [ 2 ] and textures [3] have been proposed as possible seeds for large- scale structure. This paper is concerned with the for- mation of global monopoles, and in particular with the way they naturally form in clusters. The cluster- ing of monopoles formed at the time of equal matter and radiation gives distinct features to the large-scale structures that they seed. In general, when a symmetry group G is sponta- neously broken to a smaller group H, defects can form if the resulting vacuum manifold J//, equal to G/H, has suitable nontrivial homotopy. Monopoles arise when 7t2(J¢'):#0. The simplest such model has G=O(3) and H=O(2), i.e. J//=S 2. It is given by the lagrangian S= ½0~0" 0~0- ~;~(0"0- C) 2, ( 1 ) where ~i is a triplet of real scalar fields. According to the Kibble mechanism [ 1 ], the phase transition gives rise to domains of size the order of the correlation length ~~ (22r/) -1. In each domain, 0 takes some value in ~', and is roughly constant. However, the values of ¢ in different domains are uncorrelated. Near the boundaries of several domains, 0 may be topologically forced away from Jg, in order to ensure continuity. It is in such regions that monopoles can form. Although at a monopole ¢¢~g, there is a closed surface 5 ~, enclosing the pole, over which ¢ does lie in ~g. Moreover, as ~ is traced out, ¢ winds around ~g exactly once. For an antipole, ¢ also winds around ~g, but with the opposite orientation. The winding number is interpreted as the magnetic charge. Bearing in mind that four domains generally meet in a point, a natural way to model this process is to take cj to be the surface of a tetrahedron, with each vertex corresponding to a different domain. A ran- dom value of¢ is assigned to each vertex. Then each face of the tetrahedron is naturally mapped to a spherical triangle Y-- on ~g = S 2. The area of W relative to the total area of J/defines a fractional winding number. The overall winding (i.e. the magnetic charge) is found by summing the contributions from all four faces, taking proper account of orientation. The result is always either zero, + 1 (corresponding to a pole) or - 1 (corresponding to an antipole). The mechanism for clustering is now easily under- stood. A positive winding for some face becomes a negative winding when viewed from an adjacent tet- rahedron. Since a cell containing a pole has higher than average positive winding, the adjacent cells have higher than average negative winding and are there- fore more likely to contain antipoles. Clustering pre- sumably occurs for both local and global defects, but we concentrate here on the latter, since the strong at- tractive forces between global defects lead to rapid 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland ) 27