De Sitter Horizon Research Articles

Remarks on 2D quantum cosmology

We consider two-dimensional quantum gravity endowed with a positive cosmological constant and coupled to a conformal field theory of large and positive central charge. We study cosmological properties at the classical and quantum level. We provide a complete ADM analysis of the classical phase space, revealing a family of either bouncing or big bang/crunch type cosmologies. At the quantum level, we solve the Wheeler-DeWitt equation exactly. In the semiclassical limit, we link the Wheeler-DeWitt state space to the classical phase space. Wavefunctionals of the Hartle-Hawking and Vilenkin type are identified, and we uncover a quantum version of the bouncing spacetime. We retrieve the Hartle-Hawking wavefunction from the disk path integral of timelike Liouville theory. To do so, we must select a particular contour in the space of complexified fields. The quantum information content of the big bang cosmology is discussed, and contrasted with the de Sitter horizon entropy as computed by a gravitational path integral over the two-sphere.

Cosmological Mass of the Photon Related to Stueckelberg and Higgs Mechanisms

I consider the electro-weak (EW) masses and interactions generated by photons using vacuum expectation values of Stueckelberg and Higgs fields. I provide a prescription to relate their parametric values to a cosmological range derived from the fundamental Heisenberg uncertainty principle and the Einstein–de Sitter cosmological constant and horizon. This yields qualitative connections between microscopic ranges acquired by W± or Z0 gauge Bosons and the cosmological scale and minimal mass acquired by g-photons. I apply this procedure to an established Stueckelberg–Higgs mechanism, while I consider a similar procedure for a pair of Higgs fields that may spontaneously break all U(1) × SU(2) gauge invariances. My estimates of photon masses and their additional parity-breaking interactions with leptons and neutrinos may be detectable in suitable accelerator experiments. Their effects may also be observable astronomically through massive g-photon condensates that may contribute to dark matter and dark energy.

Entropy-Area Law and Temperature of de Sitter Horizons from Modular Theory

Abstract We derive an entropy-area law for the future horizon of an observer in diamonds inside a static patch of de Sitter space-time, taking into account the back reaction of quantum matter fields. We prove the positivity and convexity of the relative entropy for coherent states using Tomita–Takesaki modular theory, from which the quantum null energy condition for diamonds follows. Furthermore, we show that the generalized entropy conjecture holds. Finally, we reveal that the local temperature that is measured by an observer at rest exhibits subleading quantum corrections with respect to the well known cosmological horizon temperature H/(2π).

DS2 supergravity

We construct two-dimensional supergravity theories endowed with a positive cosmological constant, that admit de Sitter vacua. We consider the cases of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 as well as N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetry, and couple the supergravity to a superconformal field theory with the same amount of supersymmetry. Upon fixing a supersymmetric extension of the Weyl gauge, the theories are captured, at the quantum level, by supersymmetric extensions of timelike Liouville theory with N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 and N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetry respectively. The theories exhibit good ultraviolet properties and are amenable to a variety of techniques such as systematic loop expansions and, in the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 case, supersymmetric localization. Our constructions offer a novel path toward a precise treatment of the Euclidean gravitational path integral for de Sitter, and in turn, the Gibbons-Hawking entropy of the de Sitter horizon. We argue that the supersymmetric localization method applied to the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theory must receive contributions from boundary terms in configuration space. We also discuss how these theories overcome several obstructions that appear upon combining de Sitter space with supersymmetry.

The minus sign in the first law of de Sitter horizons

Due to a well-known, but curious, minus sign in the Gibbons-Hawking first law for the static patch of de Sitter space, the entropy of the cosmological horizon is reduced by the addition of Killing energy. This minus sign raises the puzzling question how the thermodynamics of the static patch should be understood. We argue the confusion arises because of a mistaken interpretation of the matter Killing energy as the total internal energy, and resolve the puzzle by introducing a system boundary at which a proper thermodynamic ensemble can be specified. When this boundary shrinks to zero size the total internal energy of the ensemble (the Brown-York energy) vanishes, as does its variation. Part of this vanishing variation is thermalized, captured by the horizon entropy variation, and part is the matter contribution, which may or may not be thermalized. If the matter is in global equilibrium at the de Sitter temperature, the first law becomes the statement that the generalized entropy is stationary.

Finite features of quantum de Sitter space

We consider degrees of freedom for a quantum de Sitter spacetime. The problem is studied from both a Lorentzian and a Euclidean perspective. From a Lorentzian perspective, we compute dynamical properties of the static patch de Sitter horizon. These are compared to dynamical features of a black hole. We point out differences suggestive of non-standard thermal behaviour for the de Sitter horizon. We establish that geometries interpolating between an asymptotically AdS space and a dS4 interior are compatible with the null energy condition, albeit with a non-standard decreasing radial size of S 2. The putative holographic dual of an asymptotic AdS2 spacetime is comprised of a finite number of underlying degrees of freedom. From a Euclidean perspective we consider the gravitational path integral for fields over compact manifolds. In two-dimensions, we review Polchinski’s BRST localisation of Liouville theory and propose a supersymmetric extension of timelike Liouville theory which exhibits supersymmetric localisation. We speculate that localisation of the Euclidean gravitational path integral is a reflection of a finite number of degrees of freedom in a quantum de Sitter Universe.

Stationary black holes and stars in the Brans-Dicke theory with Λ>0 revisited

It was shown a few years back that for a stationary regular black hole or star solution in the Brans-Dicke theory with a positive cosmological constant $\mathrm{\ensuremath{\Lambda}}$, endowed with a de Sitter or cosmological event horizon in the asymptotic region, not only there exists no nontrivial field configurations, but also the inverse Brans-Dicke parameter ${\ensuremath{\omega}}^{\ensuremath{-}1}$ must be vanishing. This essentially reduces the theory to Einstein's general relativity. The assumption of the existence of the cosmological horizon was crucial for this proof. However, since the Brans-Dicke field $\ensuremath{\phi}$, couples directly to the $\mathrm{\ensuremath{\Lambda}}$-term in the energy-momentum tensor as well as $\mathrm{\ensuremath{\Lambda}}$ acts as a source in $\ensuremath{\phi}$'s equation of motion, it seems reasonable to ask: can $\ensuremath{\phi}$ become strong instead and screen the effect of $\mathrm{\ensuremath{\Lambda}}$, at very large scales, so that the asymptotic de Sitter structure is replaced by some alternative, yet still acceptable boundary condition? In this work we analytically argue that no such alternative exists, as long as the spacetime is assumed to be free of any naked curvature singularity. We further support this result by providing explicit numerical computations. Thus we conclude that in the presence of a positive $\mathrm{\ensuremath{\Lambda}}$, irrespective of whether the asymptotic de Sitter boundary condition is imposed or not, a regular stationary black hole or even a star solution in the Brans-Dicke theory always necessitates ${\ensuremath{\omega}}^{\ensuremath{-}1}=0$, and thereby reducing the theory to general relativity. The qualitative differences of this result with that of the standard no hair theorems are also pointed out.

Cosmological constant problem on the horizon

We revisit the quantum cosmological constant problem and highlight the important roles played by the de Sitter (dS) horizon of zero-point energy. We argue that fields which are light enough to have a dS horizon of zero-point energy comparable to the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) Hubble radius are the main contributors to dark energy. On the other hand, the zero-point energy of heavy fields develop nonlinearities on sub-Hubble scales and cannot contribute to dark energy. We speculate that our proposal may provide a resolution for both the old and new cosmological constant problems by noting that there exists a field, the (lightest) neutrino, which happens to have a mass comparable to the present background photon temperature. The proposal predicts multiple transient periods of dark energy in the early and late expansion history of the Universe, yielding a higher value of the current Hubble expansion rate which can resolve the ${H}_{0}$ tension problem.

Cosmological mass of the photon and dark energy as its Bose–Einstein condensate in de Sitter space

I develop a physical picture of dark energy (DE) based on fundamental principles and constants of quantum mechanics (QM) and general relativity (GR) theories. It derives from a conjecture of nonzero masses for nearly standard-model photons or gluons, based on QM localization at a cosmological scale. Dark energy is associated with de Sitter space and that has a fundamentally invariant event horizon, which provides the basis for my DE model. I conceive of DE as a Bose–Einstein condensate (BEC) of cosmologically massive photons, and I estimate fundamentally the binding energy per particle originating from an effectively attractive QM potential in that BEC. Since massive photons may stand at rest in a de Sitter universe with flat spatial geometry, I solve the time-independent Schrödinger equation for a nonrelativistic attractive spherical-well potential self-confining at the de Sitter horizon. The minimal critical potential depth that binds a particle state at the top of that well, combined with the prototypical condition of dark energy-pressure relation in the standard flat Lambda -CDM model, provides an estimate of the photon mass, m_g. That is supported by an independent calculation of the vacuum energy of the BEC in a de Sitter static metric with coordinate-time slicing. I also consider classical gravitational collapse for a uniform dark energy density, approaching Schwarzschild condition. These QM and GR estimates provide compatible accounts of dark energy condensation, bridging a chasm between nuclear and cosmological scales. I then investigate statistical properties of equilibrium between the g-BEC phase and the ordinary ‘vapor’ phase of m_g photons. Resulting corrections to the Planck spectrum of the CMB are too small to be detectable, at least currently. Most notably, I consider a system of cosmological units, or ‘g-units,’ that complements the fundamental system of Planck units in various ways. The geometric mean of Planck and g-mass turns out to be remarkably close to current estimates of neutrino masses, suggesting that even masses of the lightest known fermions may be deeply related to both GR and QM fundamental constants Lambda , G, c and h.

Spacetime entanglement entropy: covariance and discreteness

We review some recent results on Sorkin’s spacetime formulation of the entanglement entropy (SSEE) for a free quantum scalar field both in the continuum and in manifold-like causal sets. The SSEE for a causal diamond in a 2d cylinder spacetime has been shown to have a Calabrese–Cardy form, while for de Sitter and Schwarzschild de Sitter horizons in dimensions $$d>2$$ , it matches the mode-wise von-Neumann entropy. In these continuum examples the SSEE is regulated by imposing a UV cut-off. Manifold-like causal sets come with a natural covariant spacetime cut-off and thus provide an arena to study regulated QFT. However, the SSEE for different manifold-like causal sets in $$d=2$$ and $$d=4$$ has been shown to exhibit a volume rather than an area law. The area law is recovered only when an additional UV cut-off is implemented in the scaling regime of the spectrum which mimics the continuum behaviour. We discuss the implications of these results and suggest that a volume-law may be a manifestation of the fundamental non-locality of causal sets and a sign of new UV physics.

Thermodynamic ensembles with cosmological horizons

The entropy of a de Sitter horizon was derived long ago by Gibbons and Hawking via a gravitational partition function. Since there is no boundary at which to define the temperature or energy of the ensemble, the statistical foundation of their approach has remained obscure. To place the statistical ensemble on a firm footing we introduce an artificial “York boundary”, with either canonical or microcanonical boundary conditions, as has been done previously for black hole ensembles. The partition function and the density of states are expressed as integrals over paths in the constrained, spherically reduced phase space of pure 3+1 dimensional gravity with a positive cosmological constant. Issues related to the domain and contour of integration are analyzed, and the adopted choices for those are justified as far as possible. The canonical ensemble includes a patch of spacetime without horizon, as well as configurations containing a black hole or a cosmological horizon. We study thermodynamic phases and (in)stability, and discuss an evolving reservoir model that can stabilize the cosmological horizon in the canonical ensemble. Finally, we explain how the Gibbons-Hawking partition function on the 4-sphere can be derived as a limit of well-defined thermodynamic ensembles and, from this viewpoint, why it computes the dimension of the Hilbert space of states within a cosmological horizon.

Expanding edges of quantum Hall systems in a cosmology language: Hawking radiation from de Sitter horizon in edge modes

Expanding edge experiments are promising to open new physics windows of quantum Hall systems. In a static edge, the edge excitation, which is described by free fields decoupled with the bulk dynamics, is gapless, and the dynamics preserve conformal symmetry. When the edge expands, such properties need not be preserved. We formulate a quantum field theory in 1+1 dimensional curved spacetimes to analyze the edge dynamics. We propose methods to address the following questions using edge waveforms from the expanding region: Does the conformal symmetry survive? Is the nonlinear interaction of the edge excitations induced by edge expansion? Do the edge excitations interact with the bulk excitations? We additionally show that the expanding edges can be regarded as expanding universe simulators of two-dimensional dilaton-gravity models, including the Jackiw-Teitelboim gravity model. As an application, we point out that our theoretical setup might simulate emission of analog Hawking radiation with the Gibbons-Hawking temperature from the future de Sitter horizon formed in the expanding edge region.

Möbius mirrors

An accelerating boundary (mirror) acts as a horizon and black hole analog, radiating energy with some particle spectrum. We demonstrate that a Möbius transformation on the null coordinate advanced time mirror trajectory uniquely keeps invariant not only the energy flux but the particle spectrum. We clarify how the geometric entanglement entropy is also invariant. The transform allows generation of families of dynamically distinct trajectories, including -symmetric ones, mapping from the eternally thermal mirror to the de Sitter horizon, and different boundary motions corresponding to Kerr or Schwarzschild black holes.

Holographic complexity and de Sitter space

We compute the length of spacelike geodesics anchored at opposite sides of certain double-sided flow geometries in two dimensions. These geometries are asymptotically anti-de Sitter but they admit either a de Sitter or a black hole event horizon in the interior. While in the geometries with black hole horizons, the geodesic length always exhibit linear growth at late times, in the flow geometries with de Sitter horizons, geodesics with finite length only exist for short times of the order of the inverse temperature and they do not exhibit linear growth. We comment on the implications of these results towards understanding the holographic proposal for quantum complexity and the holographic nature of the de Sitter horizon.

Spacetime entanglement entropy of de Sitter and black hole horizons

We calculate Sorkin’s manifestly covariant entanglement entropy for a massive and massless minimally coupled free Gaussian scalar field for the de Sitter horizon and Schwarzschild de Sitter horizons, respectively, in d > 2. In de Sitter spacetime we restrict the Bunch–Davies vacuum in the conformal patch to the static patch to obtain a mixed state. The finiteness of the spatial norm in the static patch implies that is well defined for each mode. We find that for this mixed state is independent of the effective mass of the scalar field, and matches that of Higuchi and Yamamoto (2018 Phys. Rev. D 98 065014), where, a spatial density matrix was used to calculate the horizon entanglement entropy. Using a cut-off in the angular modes we show that , where A c is the area of the de Sitter cosmological horizon. Our analysis can be carried over to the black hole and cosmological horizon in Schwarzschild de Sitter spacetime, which also has finite spatial norm in the static regions. Although the explicit form of the modes is not known in this case, we use the boundary conditions of Qiu and Traschen (2020 Class. Quantum Grav. 37 135012) for a massless minimally coupled scalar field, to find the mode-wise , where b, c denote the black hole and de Sitter cosmological horizons, respectively. As in the de Sitter calculation we see that after taking a cut-off in the angular modes.

Three-dimensional de Sitter horizon thermodynamics

We explore thermodynamic contributions to the three-dimensional de Sitter horizon originating from metric and Chern-Simons gauge field fluctuations. In Euclidean signature these are computed by the partition function of gravity coupled to matter semi-classically expanded about the round three-sphere saddle. We investigate a corresponding Lorentzian picture — drawing inspiration from the topological entanglement entropy literature — in the form of an edge-mode theory residing at the de Sitter horizon. We extend the discussion to three-dimensional gravity with positive cosmological constant, viewed (semi-classically) as a complexified Chern-Simons theory. The putative gravitational edge-mode theory is a complexified version of the chiral Wess-Zumino-Witten model associated to the edge-modes of ordinary Chern-Simons theory. We introduce and solve a family of complexified Abelian Chern-Simons theories as a way to elucidate some of the more salient features of the gravitational edge-mode theories. We comment on the relation to the AdS4/CFT3 correspondence.

A group theoretical approach to quantum gravity in (A)dS

This thesis is devoted to developing a group-theoretical approach towards quantum gravity in (Anti)-de Sitter spacetime. We start with a comprehensive review of the representation theory of de Sitter (dS) isometry group, focusing on the construction of unitary irreducible representations and the computation of characters. The three chapters that follow present the results of novel research conducted as a graduate student. Chapter 4 is based on [1]. We provide a general algebraic construction of higher spin quasinormal modes of de Sitter horizon and identify the boundary operator insertions that source the quasinormal modes from a local QFT point of view. Quasinormal modes of a single higher spin field in dSD furnish two nonunitary lowest-weight representations of the dS isometry group SO(1,D). We also show that quasinormal mode spectrums of higher spin fields are precisely encoded in the Harish-Chandra characters of the corresponding SO(1,D) unitary irreducible representations. Chapter 5 is based on work with D. Anninos, F. Denef and A. Law [2]. With potential application to constraining UV-complete microscopic models of de Sitter quantum gravity, we compute de Sitter entropy as the logarithm of the sphere path integral, for any possible low energy effective field theory containing a massless graviton, in arbitrary dimensions. The path integral is performed exactly at the one-loop level. The one-loop correction to the dS entropy is found to take a universal “bulk−edge” form, with the bulk part being an integral transformation of a Harish-Chandra character encoding quasinormal modes spectrum in a static patch of dS and the edge part being the same integral transformation of an edge character encoding degrees of freedom frozen on the dS horizon. In 3D de Sitter spacetime, the one-loop exact entropy is promoted to an all-loop exact result for truncated higher spin gravity, the latter admitting an SL(n,C) Chern-Simons formulation with n being the spin cut-off. Chapter 6 is based on [3]. Inspired by [2], we revisit the one-loop partition function of any higher spin field in (d + 1)-dimensional Anti-de Sitter spacetime and show that it can be universally expressed as an integral transform of an SO(2, d) bulk character and an SO(2, d − 2) edge character. We apply this character integral formula to various higherspin Vasiliev gravities and find miraculous (almost) cancellations between bulk and edge characters, leading to striking agreement with the predictions of higher spin holography. We also comment on the relation between our character integral formula and Rindler-AdS [4] thermal partition functions.

Heat shield for de Sitter space

We investigate a quantum vacuum state for $1 + 1$ dimensional de Sitter space, corresponding to a region within a perfectly reflecting symmetric box of fixed physical size. We find that a particle detector inside the box registers zero response, implying that the walls of the box screen out the thermal effects of the de Sitter horizon. The box thus creates a `quiescent oasis' with a temperature below the de Sitter horizon temperature, an unexpected feature that opens the way to an analysis of heat and entropy exchange between the box and the horizon in the context of the generalized second law of thermodynamics. We also calculate the stress-energy-momentum tensor of the region within the box, showing that the total energy in the box is less than the same volume of de Sitter space in the presence of thermal fluctuations.

Black holes in ultraviolet-complete Hořava gravity

Horava gravity is a proposal for completing general relativity in the ultraviolet by interactions that violate Lorentz invariance at very high energies. We focus on (2+1)-dimensional projectable Horava gravity, a theory which is renormalizable and perturbatively ultraviolet-complete, enjoying an asymptotically free ultraviolet fixed point. Adding a small cosmological constant to regulate the long distance behavior of the metric, we search for all circularly symmetric stationary vacuum solutions with vanishing angular momentum and approaching the de Sitter metric with a possible angle deficit at infinity. We find a two-parameter family of such geometries. Apart from the cosmological de Sitter horizon, these solutions generally contain another Killing horizon and should therefore be interpreted as black holes from the viewpoint of the low-energy theory. Contrary to naive expectations, their central singularity is not resolved by the higher derivative terms present in the action. It is unknown at present if these solutions form as a result of gravitational collapse. The only solution regular everywhere is just the de Sitter metric devoid of any black hole horizon.

The radial Teukolsky equation for Kerr–Newman–de Sitter geometry: revisited

We use Heun-type solutions given in Suzuki et al. (Prog Theor Phys 100:491, 1998) for the radial Teukolsky equation written in the background metric of the Kerr–Newman–de Sitter geometry to calculate reflection coefficient for waves coming from the de Sitter horizon and reflected at the outer horizon of the black hole.