We argue that the notion of entanglement in de Sitter space arises naturally from the non-trivial Lorentzian geometry of the spacetime manifold, which consists of two disconnected boundaries and a causally disconnected interior. In four bulk dimensions, we propose an holographic description of an inertial observer in terms of a thermofield double state in the tensor product of the two boundaries Hilbert spaces, whereby the Gibbons-Hawking formula arises as the holographic entanglement entropy between the past and future conformal infinities. When considering the bulk entanglement between the two causally disconnected Rindler wedges, we show that the corresponding entanglement entropy is given by one quarter of the area of the pair of codimension two minimal surfaces that define the set of fixed points of the dS 4 /Z q orbifold.1 10]. However, several proposals have been made. These essentially follow two related approaches, both relying on asymptotic symmetries arguments and their possible central extensions, but differing on the spacetime region where these symmetries are centrally enhanced.Near-horizon symmetries. One of these approaches has been motivated by Carlip's derivation [11] of the Bekenstein-Hawking formula for the BTZ black hole [12,13], and it is based on the algebraic nature and affine (typically Virasoro) extensions of the near-horizon symmetries.Following this approach and making use of the Chern-Simons formulation of three-dimensional gravity, Maldacena and Strominger [14] showed that the underlying symmetries at the dS horizon corresponds to an SL(2, C) current algebra at the boundary of a spatial disk, where the Chern-Simon theory reduces to an SL(2, C) Wess-Zumino-Witten model, and where the dS entropy arises as the entropy of a highly excited thermal state, at an energy level roughly given by the (imaginary part of the complex) Chern-Simons level.Later on, the analysis of the near-horizon symmetries was extended to any dimension and to arbitrary type of Killing horizon, including the dS horizon. By using covariant phase space methods [15,16] and under a suitable set of boundary conditions, Carlip showed [17] that modelling the boundary of a manifold locally as a Killing horizon, the constraint algebra of general relativity acquires a non-trivial central extension given in terms of the gravity coupling constants, which determines the density of states at the boundary and gives rise, via Cardy's formula [18,19], to the quarter of the area formula.Asymptotic symmetries at spacelike infinity. A second route to the dS entropy problem, originally proposed by Strominger in the context of the microstates counting of the threedimnsional black hole [20] and inspired by the Brown-Henneaux [21] construction of aymptotic symmetries of three-dimensional anti de Sitter (AdS) space, treats as the symmetry enhancement region the spacelike infinity of dS space. According to the dS/CFT correspondence [8,22], its precursors [23-30] and refinements [31][32][33][34][35][36][37][38], the microscopic degrees of fre...

We model the back-reaction of a massive static observer in four-dimensional de Sitter spacetime by means of a singular Z q quotient. The set of fixed points of the Z q action consists of a pair of codimension two minimal surfaces given by 2-spheres in the Euclidean geometry. The introduction of an orbifold parameter q > 1 permits the construction of an effective action for the bulk gravity theory with support on each of these minimal surfaces. The effective action corresponds to that of Liouville field theory on a 2-sphere with a finite vacuum expectation value of the Liouville field.The intrinsic Liouville theory description yields a thermal Cardy entropy that we reintrepret as a modular free energy at temperature T = q −1 , whereupon the Gibbons-Hawking entropy arises as the corresponding modular entropy. We further observe that in the limit q → ∞ the fourdimensional geometry reduces to that of global dS 3 spacetime, where the two original minimal surfaces can be mapped to the future and past infinities of dS 3 by means of a double Wick rotation. In this limit, the Liouville theories on the two conformal boundaries become free bosons with background charge whose total central charge equals that computed using the dS 3 /CFT 2 correspondence. 1

It has been argued that the entropy of de Sitter space corresponds to the entanglement between disconnected regions computable by switching on a replica parameter q modeled by the quotient dS/ℤq. Within this framework, we show that the centrally-extended asymptotic symmetry algebra near the cosmic horizon is a single copy of the Virasoro algebra. The resulting density of states matches the semi-classical result of Gibbons and Hawking up to an undetermined constant that is chosen to reproduce the entanglement entropy previously found in the literature. It follows that the logarithmic quantum corrections to the Cardy entropy reproduces the known one-loop result computed in the bulk in the presence of a cutoff. The resulting entanglement entropy follows the divergent area law, where the UV cutoff is now a function of the replica parameter. Thus, as the near-horizon CFT fixes the cutoff in units of the Planck scale, the model can be viewed as a probe into whether the defect Hilbert space has a finite dimension; indeed, the limit q → 0, reproduces Banks’ formula. We also study the quantum corrections of the effective description of the horizon entropy by means of Liouville field theory, where the large q limit corresponds to a realization of dS3/CFT2 correspondence matching the logarithmic corrections to three-dimensional de Sitter space obtained by computing the one-loop contribution to the quantum gravity partition function in the round three-sphere.

No abstract

No abstract