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## 1. Inflation in non-SUSY GUTs 

The original idea of inflation was built on SU (5) GUT, and the motivation was to dilute the unwanted relics, i.e. GUT monopoles, besides predicting the universe as large and as homogeneous on the largest scales as possible. Guth [1] first suggested (see also [494, 681]) that the GUT phase transition is first order, driven by the potential of the GUT Higgs field Φ in the adjoint representation 24 . Once the finite temperature effects are taken into account, [494] 4 2 2 

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where φ 2 ≡ TrΦ 2 and α$_{i}$ and γ$_{i}$ are constants of the theory. Assuming an initial temperature much larger than µ and the GUT scale, the cooling of the universe induces the GUT phase transition SU (5) → G$_{S}$$_{M}$ , which could be used to generate a false vacuum inflation if the universe was trapped in a local minimum of the potential, typically at φ = 0. The false vacuum leads to the so called “old inflation”, which is terminated by the formation of bubbles of true vacuum, as the phase transition takes place. This idea was abandoned because it was plagued with many problems. In particular, inside a bubble of new vacuum, the energy density of the false vacuum is transfered into kinetic energy, inducing the bubble expansion and the collisions due to bubble walls. This leads to a highly inhomogeneous and anisotropic universe in strong contradiction with the observations. 

In Refs. [3, 5], the “new inflation” scenario was proposed to avoid such problems and also implemented within SU (5), as it was suggested that the field φ trapped in the false vacuum φ = 0 slowly rolls down its potential described by the Coleman-Weinberg potential [360] at finite high temperature T ≫ M$_{X}$ , [494] 

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where the second expression describes the effective inflationary potential close to the origin when the temperature has dropped to T ∼ H . In this expression, V (0) = 9 M 4 $_{X}$/ 32 π 2 and λ ≃ 25 g $^{4}$/ 32 π $^{2}$(ln H/φ$_{0}$ − 1 / 4). This model evaded the old inflation problem but predicted the wrong amplitude of anisotropies δρ/ρ ∼ $_{110}$√$_{λ}$ ≫ 10 − $^{4}$. 

An improved version of this idea followed, the Shafi-Vilenkin model and the Pi model [682–685], based on an SU (5) theory containing an additional SU (5) singlet χ , which 

