By David J. Eicher
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Additional resources for 50 Greatest Mysteries of the Universe (Astronomy Magazine Collectors Edittion)
That it is very flat during inflation, and becomes more and more steep later on, in order to exit the phase of inflationary dynamics. This scenario is realized in the “slow-roll” approximation, which we will discuss in Sect. 4. 33) in terms of the scalar field quantities. 8) 1 ϕ ∂σ ϕ ∂ σ ϕ + V (ϕ) gμν . 52) Tμν 2 For a homogeneous scalar field ϕ(t), we obtain by means of the FRW decomposition ϕ = Tμν 1 2 ϕ˙ + V (ϕ) δμ0 δν0 + 2 1 2 ϕ˙ − V (ϕ) 2 δμi δνj qi j . e. e ϕ equations a˙ a ωϕ = ϕ 1 2 ˙ 2ϕ 1 2 ˙ 2ϕ + V (ϕ) − V (ϕ) .
12 Slow-roll potential with small perturbations of the inflaton field. The accelerated and delayed decay of ϕ due to small quantum corrections translates perturbations of the inflaton field δϕ into density perturbations δρ The fluctuations δϕ(x, t) are space-dependent and should be approximately random. Therefore, it is assumed that they obey (nearly) Gaussian statistics. This in turn means that the fluctuations are at each point independent objects and thus each Fourier mode δϕk (t) represents an independent random variable.
10 Different slices of the de Sitter hyperboloid embedded in M(5) . The red lines correspond to hypersurfaces of constant time t = (d − 2)(d − 1) = 3 H 2. 63) The slices shown in Fig. 61) γμν := ∂XI ∂X J ηI J , ∂x μ ∂x ν I, J, . . = 0, . . , 4. 64) This does not mean that de Sitter space is different for different coordinate choices X I , it just means that de Sitter space is covered by different sets of coordinates x μ . Not all choices for X I ∈ M(5) lead to a set of coordinates x μ ∈ d S(4) that covers the whole de Sitter space.