Abstract
We consider a cosmological model starting from (1) the(1+3+6)-dimensional space-times consisting of the outer space (the 3-dimensional expanding section) and the inner space (the 6-dimensional section) and reaching (2) the Friedmann model after the decoupling between the outer space and the inner space, and derive fluctuations of the background radiation appearing in the above 10-dimensional space-times. For this purpose we first derive the fluid-dynamical perturbations in the above 10-dimensional space-times, corresponding to two kinds of curvature perturbations (in the scalar mode) in the non-viscous case, and next study the quantum fluctuations in the scalar and tensor modes, appearing at the stage when the perturbations are within the horizon of the inflating outer space. Lastly we derive the wave-number dependence of fluctuations (the power spectrum) in the two modes, which formed at the above decoupling epoch and are observed in the Friedmann stage. It is found that it can be consistent with the observed spectra of the cosmic microwave background radiation.
Highlights
In order to derive the observed fluctuations of cosmic microwave background radiation, we study the cosmological evolution of the (1 + 3 + 6)-dimensional space-times, in which it is assumed that our universe was born as an isotropic and homogeneous 10-dimensional spacetimes and evolved to the state consisting of the 3-dimensional inflating outer space and the 6-dimensional collapsing inner space
Quantum fluctuations are created at the epoch of r(τ )H/k < 1 in the outer space of the 10-dimensional space-time, and the fluctuations are observed as the fluctuations of the background radiation at the epochs of a(tf )H/k < 1 at the Friedmann stage
The gauge-invariant quantities Φh and Φ(Ar) in the outer space correspond to the gaugeinvariant perturbations defined by Bardeen[7] in the (1 + 3)-dimensional usual universes, and ΦH and Φ(AR) in the inner space are similar to the above quantities
Summary
In order to derive the observed fluctuations of cosmic microwave background radiation, we study the cosmological evolution of the (1 + 3 + 6)-dimensional space-times, in which it is assumed that our universe was born as an isotropic and homogeneous 10-dimensional spacetimes and evolved to the state consisting of the 3-dimensional inflating outer space and the 6-dimensional collapsing inner space. Our 4-dimensional Friedmann universe appeared after decoupling of the outer space from the inner space. It is expected that the inner space is so homogeneous and quietly evolves without violent phenomena This is because in both the inner and outer spaces the perturbations are assumed to be caused by quantum fluctuations before the decoupling, grow gravitationally, and remain very small and at the linear stage, before the decoupling. At the decoupling epoch tdec and (tf )dec, the entropy is assumed to be conserved in the 10-dimensional space-time and the Friedmann model. Quantum fluctuations are created at the epoch of r(τ )H/k < 1 in the outer space of the 10-dimensional space-time, and the fluctuations are observed as the fluctuations of the background radiation at the epochs of a(tf )H/k < 1 at the Friedmann stage
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