Hubble radius

R. M. Kiehn, paper presented at Gravitation and Cosmology From the Hubble Radius to the Planck Scale, Vigier 2000 Symp. Univ. of California, Berkeley, USA (Aug. 21-25, 2000) (to be published in Proceedings edited by R. L. Amoroso, G. Hunter, M. Kafatos, and J.-P. Vigier). 

In other word, as any physical length, the curvature radius grows as a, that is more slowly than the Hubble radius. In order for the curvature radius to be larger than the Hubble radius today it has to be immensely larger than the Hubble radius at earlier times. Again, this is unexpected. One would for example prefer that at Planck epoch, both the curvature radius and the Hubble radius were of same order of magnitude. 

As we have seen above, the horizon and flatness problems can be solved if we suppose that at some epoch, the quantity has become negative. This is in particular the case if — 1 scale factor, but this time, they grow faster than the Hubble radius. 

What is the necessary number of e-folds to solve the horizon and flatness problems For the horizon problem, we want that the observable universe today was inside the Hubble radius at the beginning of inflation. Let us assume for definiteness than the Universe was matter dominated since zeq 104 till now then radiation dominated before. The relation between the comoving horizon today rjo and the comoving horizon rjf at some early epoch deep in the radiation era is given by Eq. (7.24) ... 

Let us first consider the asymptotic solution for this equation. During an accelerated phase of expansion, any physical length exits the Hubble radius at some stage during inflation the ratio k/H decreases with time. Therefore at late times, the evolution equation of the scalar perturbations becomes... 

Although we are in the quasi de Sitter case, all quantities such that H, V slowly vary with time. Therefore, for a given mode, we evaluate them at the epoch when the mode under consideration exits the Hubble radius. This is what the subscript k means in the above equation. Therefore, we obtain the power spectrum of the Bardeen potential in the matter era... 

The study of the inflationary perturbations that we have presented here is of course far from complete. We did not derive the calculation of the spectral indices as a function of the slow-roll parameters, nor of the running of the spectral indices. However we hope to have made it clear that this can been done analytically within the slow-roll approximation (as well as numerically, of course). Let us emphasize that many alternative models such as the ekpy-rotic universe or the pre Big-Bang scenario can also be studied within this framework, as the core ingredient (quantum fluctuations which are expelled from the Hubble radius) are present. What changes is the matter content of the Universe during this phase as well as the dynamics of the scale factor. 

Earlier speculations about the effect of the curvature of space on elemental synthesis and the stability of nuclides (2.4.1) are consistent with the interface model. The absolute curvature of the closed double cover of projective space, and the Hubble radius of the universe, together define the golden mean as a universal shape factor [233], characteristic of intergalactic space. This factor regulates the proton neutron ratio of stable nuclides and the detail of elemental periodicity. The self-similarity between material structures at different levels of size, such as elementary particles, atomic nuclei, chemical... 

Global universe out to the Hubble radius and beyond Type la supemovae, cosmic shear, largest structures, fluctuations in 3K background (35, 36, 37, 38) 270 0.27... 

The significance of this number is its close correspondence with another large number, obtained as the ratio of the Hubble radius c/H) of the universe and the radius that encloses the rest mass of an electron, e /meC, ... 

The term k in this metric is a constant that determines the spacial curvature of the cosmology. For k = 1 the cosmology is a closed spherical universe, for k = 0 the cosmology is flat, and for k - — 1 the cosmology is open. The Einstein field equations give a constraint equation and a dynamical equation for the rate the radius changes with time. If we define a velocity as v = (R/R)H(t)r, where H (t) is the Hubble parameter, a constant locally, the constraint equations is... 

An equivalent linear relationship Av = Hx may be assumed in curved space. In the limit Av oc, H —> l/to as before, but now l/H = Tq/c is interpreted to define an upper bound to some radius of the curved universe, rather than its age. Unlike the Hubble age, the predicted radius is not in conflict with any known observation. 

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