G. Symmetry breaking and topological defects 1. Formation of cosmic defects during or after inflation in 4 D The formation of topological defects during symmetry breaking has been found by Kib- ble [449], and has rapidly gained popularity in the 80’s and the 90’s as it was realized that the cosmic strings - the line-like topological defects - formed at the GUT scale could generate temperature anisotropies at the level of 10 − 5 as observed by the COBE. They form during symmetry breaking, if generated via the Higgs mechanism, then their nature depend on the actual symmetry breaking and the fields involved. As presented earlier in this chapter, these symmetry breaking are assumed to have occurred in the SM of particle physics as well as many of its extensions: MSSM, (SUSY) GUTs, etc. Even string theories often give rise to a 4 D effective theories that possesses symmetries that are larger than those of the SM, requir- ing then some symmetry breaking [40, 41]. In this section we will give a rapid presentation on some general properties of topological defects useful for inflation model builders, and we refer the reader to the more specialized literature for more details [449–456], for a review see [457]. As we will describe in the chapter IV, many semi-realistic models of inflation within regular 4 D field theory consider the interaction between the inflaton φ and a second scalar field ψ that acquires during or at the end of inflation a non-vanishing VEV. As a consequence, depending on the precise model and inflation dynamics, it is frequent to break symmetries during or at the end of inflation, if the Higgs-type field ψ is charged under some symmetries. The hybrid inflation models is the most common class of such models (see Sec. IV). Let us for a moment forget about the phase of inflation and illustrate the formation of topological defects. If ψ is non-trivially charged under some gauge symmetry G , a non- vanishing VEV of ψ will realize the symmetry breaking G → H . The manifold of all the vacua accessible to ψ is given by the quotient group M = G/H . For example, for the simplest abelian Higgs model, the symmetry breaking is U (1) → I and the manifold of vacua is M = U (1), corresponding to the circle of constant radius in the complex plane | φ | = constant. What govern the formation and the type of topological defects are the topological properties of M [449, 452]. The homotopy groups π$_{n}$ of order n are the most 71