When Fischler and Susskind proposed a holographic prescription based on the particle horizon, they found that spatially closed cosmological models do not verify it due to the apparently unavoidable recontraction of the particle horizon area. In this paper, after a short review of their original work, we expose graphically and analytically that spatially closed cosmological models can avoid this problem if they expand fast enough. It has also been shown that the holographic principle is saturated for a codimension one-brane dominated universe. The Fischler?Susskind prescription is used to obtain the maximum number of degrees of freedom per Planck volume at the Planck era compatible with the holographic principle.
Finally, two recent conjectures concerning holography in spatially closed universes deserve some comments. W. Zimdahl and D. Pavon @cite_17 claim that dynamics of the holographic dark energy in a spatially closed universe could solve the coincidence problem; however the cosmological scale necessary for the definition of the holographic dark energy seems to be incompatible with the particle horizon @cite_24 @cite_2 @cite_9 . In a more recent paper F. Simpson @cite_31 proposed an imaginative mechanism in which the non-monotonic evolution of the particle horizon over a spatially closed universe controls the equation of state of the dark energy. The abundant work in that line is still inconclusive but it seems to be a fairly promising line of work.
@cite_9 @cite_24 @cite_2 @cite_31 @cite_17
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A model for holographic dark energy is proposed, following the idea that the short distance cut-off is related to the infrared cut-off. We assume that the infrared cut-off relevant to the dark energy is the size of the event horizon. With the input Omega(Lambda) = 0.73, we predict the equation of state of the dark energy at the present time be characterized by w = -0.90. The cosmic coincidence problem can be resolved by inflation in our scenario, provided we assume the minimal number of e-foldings. (C) 2004 Elsevier B.V. All rights reserved. Here we consider a scenario in which dark energy is associated with the apparent area of a surface in the early universe. In order to resemble the cosmological constant at late times, this hypothetical reference scale should maintain an approximately constant physical size during an asymptotically de Sitter expansion. This is found to arise when the particle horizon—anticipated to be significantly greater than the Hubble length—is approaching the antipode of a closed universe. Depending on the constant of proportionality, either the ensuing inflationary period prevents the particle horizon from vanishing, or it may lead to a sequence of 'big rips'. We employ the holographic model of interacting dark energy to obtain the equation of state for the holographic energy density in non-flat (closed) universe enclosed by the event horizon measured from the sphere of horizon named L. The holographic bound states that the entropy in a region cannot exceed one quarter of the area (in Planck units) of the bounding surface. A version of the holographic principle that can be applied to cosmological spacetimes has recently been given by Fischler and Susskind. This version can be shown to fail in closed spacetimes and they concluded that the holographic bound may rule out such universes. In this paper I give a modified definition of the holographic bound that holds in a large class of closed universes. Fischler and Susskind also showed that the dominant energy condition follows from the holographic principle applied to cosmological spacetimes with @math . Here I show that the dominant energy condition can be violated by cosmologies satisfying the holographic principle with more general scale factors.