producing again no topologically stable strings, but only embedded (or semi-local) strings. Note that their stability in this case depends on various parameters; the ratio of the scalar mass over the U (1)$_{ξ}$ gauge field mass, the existence of zero modes and the details of the symmetry breaking [655]. As a conclusion, it is not excluded that these objects have an interesting impact on cosmology. Both of these ingredients can be found in string theory [462, 655]. Other mechanisms have been proposed to eliminate the impact of cosmic strings by breaking the U (1) symmetry before inflation. For example, authors in [659] modified the D -term superpotential with the introduction of a pair of vector-like fields [under U (1)$_{ξ}$] Ψ , ¯ Ψ and two singlets X, σ [659] Inflation then takes place with 〈 X 〉 = 〈 σ 〉 = 0, 〈 Ψ 〉 = 〈 ¯ Ψ 〉 = M , therefore breaking U (1)$_{ξ}$ before or during inflation, and diluting the strings. The Refs. [656, 660], explored various ays to reduce the impact of cosmic strings on the CMB, proposing the “sneutrino modified D -term inflation”. The superpotential is modified by introducing a coupling of the inflaton superfield S to the right-handed sneutrino N$_{R}$ field [656, 660] where the second and third terms are responsible for generating neutrino masses via see- saw mechanism. The model also requires a coupling through a non-renormalizable K¨ ahler potential: which affects the main conclusions of the model concerning the impact of cosmic strings via the predictions for the spectral index. The potential in Eq. (251) is modified by an additional term, − κs $^{2}$/ 2, where κ = ( c − 1) M 2 R | N$_{R}$ | $^{2}$/ Λ $^{2}$, if the coupling is assumed to be λ > ∼ 0 . 1. For such large couplings, inflation is realized with z ≫ 1 and the effective potential reduces to V ≃ V$_{0}$ + g $^{4}$ξ 16 π 2 ln( s $^{2}$/ Λ $^{2}$) − κs $^{2}$/ 2. For c > 1, this reduces the spectral index and allows for a better fit to the observations. In addition, the normalization of the power spectrum of primordial fluctuations is enhanced, leading to a lowering the energy scale of inflation, therefore a reducing the energy per unit length for cosmic strings. W = λ Φ$_{+}$$^{Φ}$− + X (Ψ ¯ Ψ − M ) + σ Φ$_{+}$Ψ (263) W = λS Φ$_{+}$$^{Φ}$− + λ$_{ν}$ Φ H$_{u}$L + M$_{R}$ 2 N 2 R , (264) K$_{min}$ → K$_{min}$ + c S † $_{SN}$† $_{R}$N$_{R}$ Λ 2 , (265) 122