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A commentary on cosmological entropy bounds. (English) Zbl 1097.83029

The holographic principle states the maximum number of degrees of freedom in a volume should be proportional to the surface area. The translation of the concept of holography into a cosmological setting has been an ongoing concern of the theoretical community [L. Susskind, J. Math. Phys. 36, 6377–6396 (1995; Zbl 0850.00013); R. Easther, D. Lowe, Phys. Rev. Lett. 82, 4967–4970 (1999; Zbl 0951.83061); R. Brustein, G. Veneziano, Phys. Rev. Lett. 84, 5695–5698 (2000)]. For the most part, the challenge of the holographic paradigm has been to find an innovative way of placing a meaningful and well–defined upper bound on the entropy in various cosmological situations [R. Bousso, Light sheets and Bekenstein’s entropy bound. Phys. Rev. Lett. 90, 121392–1 to 121302–4 (2003)]. Progress in the direction of establishing and justfying entropy bounds via the holographic principle has been impeded by the ambiguous nature of both entropy and the notion of holography in cosmology. An especially pertinent question is what the concept of holography can contribute to the knowledge about inflationary fluctuations. This could be of interest at early times in the era when inflation can be modeled as a quantum field theoretic scenario within the horizon as well as at later times after many of the fluctuations have been classically frozen at the super-horizon scales of inflationary modes. For instance, it has been asked if holographic bounds can allow to place an upper limit on the ultraviolet-cutoff scale of the effective field theory [A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Rev. Lett. 82, 4971–4974 (1999; Zbl 0949.83045)]. One might also inquire as to the viability of holographically implicating the dark energy source that mimics a cosmological constant and is responsible for the current period of acceleration [S. Thomas, Holography stabilizes the vacuum energy. Phys. Rev. Lett. 89, 081301–1 to 081301–4 (2002); N. Straumann, On the cosmological constant problems and the astronomical evidence for a homogeneous energy density with negative pressure Poincaré Seminar 2002, Vacuum Energy, Renormalization, Birkhäuser, 7–51 (2003; Zbl 1054.83044)]. Therefore it has become a tendency, as of late, to call upon holographic arguments for the purpose of rationalizing the inexplicably small magnitude of the dark energy. Because the early–universe inflation also represents a cosmic period of acceleration, there appears to be some sort of relationship between the inflationary potential and the dark energy. It is therefore quite natural to ask if this notion of connection could be strengthened by appealing to the holographic paradigm in order to encompass both of these cosmic events occurring in a flat universe.
The purpose of the paper under review is to discuss the concept of a holographic energy content which represents a non-conventional matter source for which the energetics can be described in terms of an ultraviolet- and infrared-cutoff so that the holographic picture boils down to just a single cutoff parameter \(\Lambda\). It is then the holographically induced dynamics of \(\Lambda\) that enables the connection between the two temporally distant eras. Another consequence is that the super–horizon inflationary modes which can be regarded as classically frozen also know about the holographic cutoffs. This takes place in spite of the fact that the cutoff parameters are intrinsic to a quantum field theory which is confined to the horizon interior. Such a reasoning has the esoteric appeal of a truly holographic universe with one of the space-time dimensions, the cosmic time, being an emergent holographically induced construct. This outcome is consistent with the notion that the fundamental quantum field theory of gravity should ultimately have a background–independent meaning.

MSC:

83F05 Relativistic cosmology
85A40 Astrophysical cosmology
94A17 Measures of information, entropy
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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