By Antonio Alfonso-Faus
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Due to the flatness condition it must also be ημν = δαβ ∂y α ∂y β ∂ξμ ∂ξν (5) The y’s are Cartesian coordinates and the metric tensor ημν corresponds to an Euclidean geometry in N dimensions. For the natural manifold it will be ds 2 = gμν dx μ dx ν (6) Both line elements (5) and (6) can of course be expressed in the embedding space as ds 2 = δ ab dX a dX b (7) 32 Aspects of Today´s Cosmology Fig. 1. The embedding space with the two embedded manifolds. The figure represents a three-dimensional embedding of two bidimensional manifolds, but the scheme can be applied to any number of dimensions.
6. Acknowledgments Some of the perspectives presented here in this chapter incorporates the contributions I have developed with different colleagues. Particularly, I would like to thank Ramón Herrera and Diego Pavón, with whom I have been involved in the study of warm inﬂation. 710/2011. 7. ; Matarrese, S. , (2003). Second-order cosmological perturbations from inﬂation. Nucl. Phys. R. , (2011). PLANCK early results: The power spectrum Of cosmic infrared background anisotropies. 2028 [astro-ph] Albrecht, A.
The mode of the expansion affects the position of the acoustic peaks which depends on the expansion factor at the equality scale ae; in practice the position is influenced by the value of the boost factor for the radiation Xboost. The acoustic horizon formula will then be the same as for ΛCDM, but the equality scale factor is now boosted: ae = Xboostρr0/ρm0. As for the angular diameter distance, it depends on the total expansion history from the last scattering surface to present: DA ( zLS ) = zLS dz c 0 ( 1 + zLS ) H ( z ) (34) The final value for lA is not much sensitive to the choice of the cosmological model so we will make reference to the values obtained from WMAP-7 using ΛCDM (Komatsu, 2011).