Cosmic Holography

A version of holographic principle for the cosmology is proposed, which dictates that the particle entropy within the cosmological apparent horizon should not exceed the gravitational entropy associated with the apparent horizon. It is shown that, in the Friedmann-Robertson-Walker (FRW) cosmology, the open Universe as well as a restricted class of flat cases are compatible with the principle, whereas closed Universe is not. It is also found that inflationary universe after the big-bang is incompatible with the cosmic holography. 1 Work supported in part by BK-21 Initiative Program and KRF International Collaboration Grant 1998010-192. The holographic principle in quantum gravity is first suggested by ’t Hooft[1] and, later, extended to string theory by Susskind[2]. The most radical part of the principle is that the degrees of freedom of a spatial region reside not in the bulk but in the boundary. Further, the number of boundary degrees of freedom per Planck area should not exceed unity. Recently, the holographic principle is applied to the standard cosmological context by Fischler and Susskind[3]. This Fischler-Susskind version of cosmological holographic principle demands that the particle entrophy contained in a volume of particle horizon should not exceed the area of the horizon in Planck units. The string cosmology is also tested by the Fischler-Susskind holographic principle[4]. In both cases, the matter contents as well as the spacetime geometry of the universe are restricted by the holographic principle alone and the results appear to be consistent with the recent measurement of the redshift-to-distance relation and the theory of the large scale structure formation[5]. In applying the holography in the cosmological context, several outstanding questions still remain unanswered. One of them is concerning a natural choice of the holographic boundary. Fischler and Susskind have chosen it to be the particle horizon, but it is not clear if the choice is consistent with other physical principles. In this note, we will propose a simple choice of the boundary surface based on the concept of cosmological apparent horizon that is a boundary hypersurface of an anti-trapped region and has a topology of S. It turns out that there is natural gravitational entropy associated with the apparent horizon and the associated holographic principle demands that the particle entrophy inside the apparent horizon should not exceed the apparent-horizon gravitational entropy. Moreover, the holography based on the apparent horizon obeys the first law of thermodynamics, in sharp contrast to that based on the particle horizon. We shall apply the proposed principle to the FRW cosmology and show that, in both the standard cosmology and the string cosmology, the open universe as well as restricted class of flat universe are compatible, while the closed universe is not. We shall further show that the inflationary senario for the standard cosmology is not compatible with the cosmic holography. Cosmological Apparent Horizon and Gravitational Entropy: We shall consider the spatially homogeneous and isotropic universe described by the FRW metric, ds = −dt + a(t) dr 2 1− kr2 + a (t)rdΩd−1 , (1) where k = 0,−1,+1 correspond to a flat, open or closed Universe respectively. Using the spherical symmetry, the metric can be rewritten as ds = habdx dx + r̃(x)dΩd−1 , (2) where x = t, x = r and the two metric hab = diag [ −1, a2 1−kr2 ] is introduced. A dynamical apparent horizon is defined by h∂ar̃∂br̃ = 0, which implies that the vector ∇r̃ is null (or 1 degenerate) on the apparent horizon surface. The explicit evaluation of the condition reads