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5.2 Brane-world instanton

The creation of an inflating brane-world can be modelled as a de Sitter instanton in a way that closely follows the 4D instanton, as shown in [105Jump To The Next Citation Point]. The instanton consists of two identical patches of AdS5 joined together along a de Sitter brane (dS4) with compact spatial sections. The instanton describes the “birth” of both the inflating brane and the bulk spacetime, which are together “created from nothing”, i.e., the point at the south pole of the de Sitter 4-sphere. The Euclidean AdS5 metric is
[ ] (5)ds2 = dr2 + ℓ2 sinh2(r∕ ℓ) dχ2 + sin2 χ dΩ2 , (221 ) euclid (3)
where dΩ2(3) is a 3-sphere, and r ≤ r0. The Euclidean instanton interpolates between r = 0 (“nothing”) and r = r0 (the created universe), which is a spherical brane of radius
−1 H0 ≡ ℓsinh(r0∕ℓ). (222 )
After creation, the brane-world evolves according to the Lorentzian continuation, χ → iH0t + π ∕2,
(5) 2 2 2 2 [ 2 −2 2 2 ] ds = dr + (ℓH0 ) sinh (r∕ℓ) − dt + H 0 cosh (H0t )dΩ (3) (223 )
(see Figure 7View Image).
View Image

Figure 7: Brane-world instanton. (Figure taken from [105Jump To The Next Citation Point].)

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