Modave lectures on de Sitter space & holography

Abstract

These lecture notes provide an overview of different aspects of de Sitter space and their plausible holographic interpretations. We start with a general description of the classical spacetime. We note the existence of a cosmological horizon and its associated thermodynamic quantities, such as the Gibbons-Hawking entropy. We discuss geodesics and shockwave solutions, that might play a role in a holographic description of de Sitter. Finally, we discuss different approaches to quantum theories of de Sitter space, with an emphasis on recent developments in static patch holography.

Static sphere observers and geodesics in Schwarzschild-de Sitter spacetime

“…This modified structure plays an important role when considering protocols for information recovery and exchange in de Sitter space times [30,31]. In particular, the outward bending of de Sitter future infinity implies that signals crossing the de Sitter exterior have more time to be intersect a (positive energy) shockwave emitted [10,47] from the conjugate static sphere observer, and be scattered into the causal region of that same conjugate static sphere observer. Black holes play a natural role of providing an energy reservoir for the positive energy shockwaves in these scenarios and it would be interesting to study the details of those information exchange protocols [26,27], especially regarding the question of how much information can be transmitted this way [31].…”

“…Recent studies of quantum gravity in de Sitter [1][2][3][4][5][6][7][8][9][10][11][12] suggest that, inspired by recent progress in our understanding of quantum gravity in Anti-de Sitter space [13][14][15][16], non-perturbative contributions in the (Euclidean) path integral could play an important role [7,[17][18][19][20]. The simplest, most elementary, solutions of relevance in this context are black holes in de Sitter space, whose contribution to the Euclidean path integral can be (partially) understood using the formalism of constrained instantons [9,21].…”

de Sitter space, extremal surfaces, and time entanglement

“…This is not a spacelike brane here localized on a spacelike RT/HRT surface but a timeevolving brane localized on the part Euclidean, part Lorentzian no-boundary dS extremal surface. The area is complex (with ( 9), (10), for dS 4 and dS 3 respectively). This is a replica formulation of the entropy of the Wavefunction (restricted to this boundary Euclidean slice): this is not hermitian, unlike an ordinary density matrix.…”

“…Since Ψ dS = Z CF T via dS/CF T this amounts to a replica formulation on Z CF T which is an analytic continuation of Z CF T = Z AdS bulk . The analytic continuations ( 9), (10), of the extremal surface areas in dS 4 , dS 3 , respectively can be seen to be the analytic continuations of the entanglement entropy for the relevant maximal subregions in AdS (with sphere boundary). This is broadly consistent with the early discussions on holographic entanglement in dS/CF T by analytic continuations [16,17], [18], and with appropriate analytic continuations of entanglement entropy in 2-dim CFT [78,79,80].…”

Gravitational observatories

“…Relatedly, in the stretched horizon picture one imagines a timelike surface that approaches the horizon of a black hole [14][15][16][17][18][19][20] or a cosmological horizon [21][22][23][24][25][26][27][28]. Notably, a worldtube perspective appears as an important ingredient in recent literature incorporating methods of AdS/CFT to analyse the de Sitter static patch in two [29][30][31][32] and three [9,33] dimensional models (see [34] for a recent review). Moreover, the presence of a worldline plays an important role in the definition of a sensible von Neumann entropy for quantum fields weakly coupled to gravity in the static patch [35].…”