Schwarzschild-de Sitter black holes have two horizons that are at different temperatures for generic values of the black hole mass. Since the horizons are out of equilibrium the solutions do not admit a smooth Euclidean continuation and it is not immediately clear what role they play in the gravitational path integral. We show that Euclidean SdS is a genuine saddle point of a certain constrained path integral, providing a consistent Euclidean computation of the probability $\sim e^{-(S_{dS}-S_{SdS})}$ to find a black hole in the de Sitter bath.
We study the family of Einstein-Maxwell instantons associated with the Kerr-Newman metrics with a positive cosmological constant. This leads to a quantisation condition on the masses and charges of the resulting Euclidean solutions.
We discuss issues relating to the topology of Euclidean de Sitter space. We show that in (2+1) dimensions, the Euclidean continuation of the`causal diamond', i.e the region of spacetime accessible to a timelike observer, is a three-hemisphere. However, when de Sitter entropy is computed in a `stretched horizon' picture, then we argue that the correct Euclidean topology is a solid torus. The solid torus shrinks and degenerates into a three-hemisphere as one goes from the `stretched horizon' to the horizon, giving the Euclidean continuation of the causal diamond. We finally comment on the genera…
We compute the on-shell Euclidean action of Schwarzschild-de Sitter black holes, and take their contributions in the gravitational path integral into account using the formalism of constrained instantons. Although Euclidean de Sitter black hole geometries have conical singularities for generic masses, their on-shell action is finite and is shown to be independent of the Euclidean time periodicity and equal to minus the sum of the black hole and cosmological horizon entropy. We apply this result to compute the probability for a nonrotating, neutral arbitrary mass black hole to nucleate spontane…
By making use of the background field method, the one-loop quantization for Euclidean Einstein-Weyl quadratic gravity model on the de Sitter universe is investigated. Using generalized zeta function regularization, the on-shell and off-shell one-loop effective actions are explicitly obtained and one-loop renormalizability, as well as the corresponding one-loop renormalization group equations, are discussed. The so called critical gravity is also considered.
This paper is devoted to an investigation of Euclidean wormholes made by fuzzy instantons. We investigate the Euclidean path integral in anti-de Sitter space. In Einstein gravity, we introduce a scalar field with a potential. Because of the analyticity, there is a contribution of complex-valued instantons, so-called fuzzy instantons. If we have a massless scalar field, then we obtain Euclidean wormholes, where the probabilities become smaller and smaller as the size of the throat becomes larger and larger. If we introduce a non-trivial potential, then in order to obtain a non-zero tunneling ra…
We study the relation between two sets of correlators in interacting quantum field theory on de Sitter space. The first are correlators computed using in-in perturbation theory in the expanding cosmological patch of de Sitter space (also known as the conformal patch, or the Poincaré patch), and for which the free propagators are taken to be those of the free Euclidean vacuum. The second are correlators obtained by analytic continuation from Euclidean de Sitter; i.e., they are correlators in the fully interacting Hartle-Hawking state. We give an analytic argument that these correlators coincide…
We consider scalar field theory on the RP^3 de Sitter spacetime (RP3dS), which is locally isometric to de Sitter space (dS) but has spatial topology RP^3. We compare the Euclidean vacua on RP3dS and dS in terms of three quantities that are relevant for an inertial observer: (i) the stress-energy tensor; (ii) the response of an inertial monopole particle detector; (iii) the expansion of the Euclidean vacuum in terms of many-particle states associated with static coordinates centered at an inertial world line. In all these quantities, the differences between RP3dS and dS turn out to fall off exp…
We investigate Euclidean wormholes in Einstein gravity with a massless scalar field in de Sitter space. Euclidean wormholes are possible due to the analytic continuation of the time as well as complexification of fields, where we need to impose the classicality after the Wick-rotation to the Lorentzian signatures. For some parameters, wormholes are preferred than Hawking-Moss instantons, and hence wormholes can be more fundamental than Hawking-Moss type instantons. Euclidean wormholes can be interpreted in three ways: (1) classical big bounce, (2) either tunneling from a small to a large unive…
The holographic dual of a gravitational theory around the de Sitter background is argued to be a Euclidean conformal gravity theory in one fewer dimensions. The measure for the holographic theory naturally includes a sum over topologies as well as conformal structures.
We consider the perturbative treatment of the minimally coupled, massless, self-interacting scalar field in Euclidean de Sitter space. Generalizing work of Rajaraman, we obtain the dynamical mass m^2 \propto sqrt{lambda} H^2 of the scalar for non-vanishing Lagrangian masses and the first perturbative quantum correction in the massless case. We develop the rules of a systematic perturbative expansion, which treats the zero-mode non-perturbatively, and goes in powers of sqrt{lambda}. The infrared divergences are self-regulated by the zero-mode dynamics. Thus, in Euclidean de Sitter space the int…
We study the emergence of de Sitter space in Euclidean dynamical triangulations (EDT). Working within the semi-classical approximation, it is possible to relate the lattice parameters entering the simulations to the partition function of Euclidean quantum gravity. We verify that the EDT geometries behave semi-classically, and by making contact with the Hawking-Moss instanton solution for the Euclidean partition function, we show how to extract a value of the renormalized Newton's constant from the simulations. This value is consistent with that of our previous determination coming from the int…
Schwarzschild-de Sitter (SdS) black holes do not admit a completely smooth Euclidean continuation. We discuss some modifications of the gravitational path integral that give Euclidean SdS a semiclassical equilibrium interpretation. First we consider "gravity in a cavity," defining the canonical ensemble in a box that excises one horizon. However, this standard approach does not work for positive cosmological constant: the solution of lowest free energy has a negative heat capacity, which is inconsistent if it is to provide the leading semiclassical contribution to a canonical partition functio…
We consider an $O(N)$ scalar field model with quartic interaction in $d$-dimensional Euclidean de Sitter space. In order to avoid the problems of the standard perturbative calculations for light and massless fields, we generalize to the $O(N)$ theory a systematic method introduced previously for a single field, which treats the zero modes exactly and the nonzero modes perturbatively. We compute the two-point functions taking into account not only the leading infrared contribution, coming from the self-interaction of the zero modes, but also corrections due to the interaction of the ultraviolet…
We analyze infrared divergences arising in calculations involving light and massless fields in de Sitter space. We show that these arise from an incorrect treatment of the constant mode of the field, and show that a correct quantization leads to a well-defined and calculable perturbation expansion. We illustrate this by computing the first nontrivial loop correction in a theory of a massless scalar field with a quartic interaction.
We show that the Hartle-Hawking vacuum for theories of interacting massive scalars in de Sitter space is both perturbatively well-defined and stable in the IR. Correlation functions in this state may be computed on the Euclidean section and Wick-rotated to Lorentz-signature. The results are manifestly de Sitter-invariant and contain only the familiar UV singularities. More importantly, the connected parts of all Lorentz-signature correlators decay at large separations of their arguments. Our results apply to all cases in which the free Euclidean vacuum is well defined, including scalars with m…
We discuss the treatment of squeezed states as excitations in the Euclidean vacuum of de Sitter space. A comparison with the treatment of these states as candidate no-particle states, or alpha-vacua, shows important differences already in the free theory. At the interacting level alpha-vacua are inconsistent, but squeezed state excitations seem perfectly acceptable. Indeed, matrix elements can be renormalized in the excited states using precisely the standard local counterterms of the Euclidean vacuum. Implications for inflationary scenarios in cosmology are discussed.
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 black holes. We point out differences suggestive of non-standard thermal behaviour for the de Sitter horizon. We establish that geometries interpolating between an asymptotically AdS$_2 \times S^2$ space and a dS$_4$ interior are compatible with the null energy condition, albeit with a non-standard decreasing ra…
Pure de Sitter, anti de Sitter, and orthogonal gauge theories in four-dimensional Euclidean spacetime are studied. It is shown that, if the theory is asymptotically free and a dynamical mass is generated, then an effective geometry may be induced and a gravity theory emerges. The asymptotic freedom and the running of the mass might account for an Inönü-Wigner contraction which induces a breaking of the gauge group to the Lorentz group, while the mass itself is responsible for the coset sector of the gauge field to be identified with the effective vierbein. Furthermore, the resulting local isom…
For spaces which are not asymptotically anti-de Sitter where the asymptotic behavior is deformed by replacing the cosmological constant by a dilaton scalar potential, we show that it is possible to have well-defined boundary stress-energy tensors and finite Euclidean actions by adding appropriate surface counterterms. We illustrate the method by the examples of domain-wall black holes in gauged supergravities, three-dimensional dilaton black holes and topological dilaton black holes in four dimensions. We calculate the boundary stress-energy tensor and Euclidean action of these black configura…
The correlation functions of two or more Euclidean Wilson loops of various shapes in Euclidean anti-de Sitter space are computed by considering the minimal area surfaces connecting the loops. The surfaces are parametrized by Riemann theta functions associated with genus three hyperelliptic Riemann surfaces. In the case of two loops, the distance $L$ by which they are separated can be adjusted by continuously varying a specific branch point of the auxiliary Riemann surface. When $L$ is much larger than the characteristic size of the loops, then the loops are approximately regarded as local oper…
We study the thermodynamics of de Sitter black holes with a conformally coupled scalar field. The geometry is that of the ``lukewarm'' Reissner-Nordstrom-de Sitter black holes, with the event and cosmological horizons at the same temperature. This means that the region between the event and cosmological horizons can form a regular Euclidean instanton. The entropy is modified by the non-minimal coupling of the scalar field to the geometry, but can still be derived from the Euclidean action, provided suitable modifications are made to deal with the electrically charged case. We use the first law…
We compute the Euclidean action for constant curvature black holes (CCBHs), as an attempt to associate thermodynamic quantities to these solutions of Einstein anti-de Sitter (AdS) gravity. CCBHs are gravitational configurations obtained by identifications along isometries of a $D$-dimensional globally AdS space, such that the Riemann tensor remains constant. Here, these solutions are interpreted as extended objects, which contain a $(D-2)$-dimensional de-Sitter brane as a subspace. Nevertheless, the computation of the free energy for these solutions shows that they do not obey standard thermod…
Based on some recent work of the authors, we focus on the relationship between the Casimir energy of a Majorana spinor field for a Euclidean Einstein universe $S^4\times R$ and for a Euclidean de Sitter brane ($S^4$) embedded in AdS(5). This is for a conformally coupled massless field. Interestingly, the one brane effective potential is zero and the results are equivalent, as for the scalar case, when evaluated on the conformally related cylinder. However, using the actual metric this equivalence no longer holds because a non-trivial contribution from the path integral measure (known as the co…
We demonstrate that possession of a single negative mode is not a sufficient criterion for an instanton to mediate exponential decay. For example, de Sitter space is generically stable against decay via the Coleman-De Luccia instanton. This is due to the fact that the de Sitter Euclidean action is bounded below, allowing for an approximately de Sitter invariant false vacuum to be constructed.
We compute the spectrum of primordial gravitational wave perturbations in open de Sitter spacetime. The background spacetime is taken to be the continuation of an O(5) symmetric instanton saddle point of the Euclidean no boundary path integral. The two-point tensor fluctuations are computed directly from the Euclidean path integral. The Euclidean correlator is then analytically continued into the Lorentzian region where it describes the quantum mechanical vacuum fluctuations of the graviton field. Unlike the results of earlier work, the correlator is shown to be unique and well behaved in the …
We study causal diamonds in Minkowski, Schwarzschild, (anti) de Sitter, and Schwarzschild-de Sitter spacetimes using Euclidean methods. The null boundaries of causal diamonds are shown to map to isolated punctures in the Euclidean continuation of the parent manifold. Boundary terms around these punctures decrease the Euclidean action by $A_\diamond/4$, where $A_\diamond$ is the area of the holographic screen around the diamond. We identify these boundary contributions with the maximal entropy of gravitational degrees of freedom associated with the diamond.
We present an exact one-loop calculation of the tunneling process in Euclidean quantum gravity describing creation of black hole pairs in a de Sitter universe. Such processes are mediated by $S^2\times S^2$ gravitational instantons giving an imaginary contribution to the partition function. The required energy is provided by the expansion of the universe. We utilize the thermal properties of de Sitter space to describe the process as the decay of a metastable thermal state. Within the Euclidean path integral approach to gravity, we explicitly determine the spectra of the fluctuation operators,…
We investigate a lattice model for Euclidean quantum gravity based on discretization of the Palatini formulation of General Relativity. Using Monte Carlo simulation we show that while a naive approach fails to lead to a vacuum state consistent with the emergence of classical spacetime, this problem may be evaded if the lattice action is supplemented by an appropriate counter term. In this new model we find regions of the parameter space which admit a ground state which can be interpreted as (Euclidean) de Sitter space.
We introduce a simple coordinate system covering half of de Sitter space. The new coordinates have several attractive properties: the time direction is a Killing vector, the metric is smooth at the horizon, and constant-time slices are just flat Euclidean space. We demonstrate the usefulness of the coordinates by calculating the rate at which particles tunnel across the horizon. When self-gravitation is taken into account, the resulting tunneling rate is only approximately thermal. The effective temperature decreases through the emission of radiation.
We consider Einstein gravity with positive cosmological constant coupled with higher spin interactions and calculate Euclidean path integral perturbatively. We confine ourselves to the static patch of the 3 dimensional de Sitter space. This geometry, when Euclideanlized is equivalent to 3-sphere. However, infinite number of topological quotients of this space by discrete subgroups of the isometry group are valid Euclidean saddles as well. Pure Einstein gravity is known to diverge, when all saddles are included as contribution to the thermal partition functions (also interpreted as the Hartle H…
We consider the stochastic quantization method for scalar fields defined in a curved manifold. The two-point function associated to a massive self-interacting scalar field is evaluated, up to the first order level in the coupling constant $λ$, for the case of de Sitter Euclidean metric. Its value for the asymptotic limit of the Markov parameter $τ\to\infty$ is exhibited. We discuss in detail the covariant stochastic regularization to render the one-loop two-point function finite in the de Sitter Euclidean metric.
We study the holographic representation of the entanglement entropy, recently proposed by Ryu and Takayanagi, in a braneworld context. The holographic entanglement entropy of a de Sitter brane embedded in an anti-de Sitter (AdS) spacetime is evaluated using geometric quantities, and it is compared with two kinds of de Sitter entropy: a quarter of the area of the cosmological horizon on the brane and entropy calculated from the Euclidean path integral. We show that the three entropies coincide with each other in a certain limit. Remarkably, the entropy obtained from the Euclidean path integral …
We investigate the physical properties of the de Sitter spacetime and new type-de Sitter black holes in new massive gravity, a higher derivative gravity theory in three dimensions. We calculate thermodynamic quantities and check that the first law of thermodynamics is satisfied. In particular, we obtain the energies of the de Sitter spacetime and new type-de Sitter black holes from the renormalized Brown-York boundary stress tensor on the Euclidean surfaces at late temporal infinity. We also obtain the quasinormal modes and by using them we find that the entropy spectra are equally spaced via …
For a spherically symmetric vacuum model with a negative cosmological constant, a complex constrained instanton is considered as the seed for the quantum pair creation of Schwarzschild-anti-de Sitter black holes. The relative creation probability is found to be the exponential of the negative of the black hole entropy. The black hole entropy is known to be one quarter of the black hole horizon area. In the absence of a general no-boundary proposal for open creation, the constrained instanton approach is used in treating both the open and closed pair creations of black holes.
We analyze the renormalization properties of quantum field theories in de Sitter space and show that only two of the maximally invariant vacuum states of free fields lead to consistent perturbation expansions. One is the Euclidean vacuum, and the other can be viewed as an analytic continuation of Euclidean functional integrals on $RP^d$. The corresponding Lorentzian manifold is the future half of global de Sitter space with boundary conditions on fields at the origin of time. We argue that the perturbation series in this case has divergences at the origin, which render the future evolution of …
Field theory in de Sitter space admits a one-parameter family of vacua determined by a superselection parameter alpha. Of these vacua, the Euclidean vacuum uniquely extrapolates to the vacuum of flat Minkowski space. States which resemble the alpha-vacua can be constructed as excitations above the Euclidean vacuum. Such states have modes alpha(k) which decay faster that k^{(1-d)/2}. Fields in such states exhibit non-local correlations when examined from the perspective of fields in the Euclidean vacuum. The dynamics of such entangled states are fully consistent. If an alpha-state with properti…
We discuss the quantum creation scenario of a Kerr-de Sitter black hole in all dimensions. We show that its relative creation probability is the exponential to the entropy of the black hole, using a topological argument. The action of the regular Euclidean instanton can be calculated in the same way.
Quadratic Curvature Correction to the Euclidean Action of Rotating AdS Black Holes in General Dimensions
The International Joint Institute of Tianjin University, Fuzhou, Tianjin University（天津大学福州国际联合学院）；Center for Joint Quantum Studies, Department of Physics, School of Science, Tianjin University（天津大学理学院物理系联合量子研究中心）
We propose a Euclidean preparation of an asymptotically AdS$_2$ spacetime that contains an inflating dS$_2$ bubble. The setup can be embedded in a four dimensional theory with a Minkowski vacuum and a false vacuum. AdS$_2$ times 2-sphere approximate the near horizon geometry of a $4d$ near-extremal RN wormhole. Likewise, in the false vacuum the near-horizon geometry of a near-extremal black hole is approximately dS$_2$ times 2-sphere. We interpret the Euclidean solution as describing the decay of an excitation inside the wormhole to a false vacuum bubble. The result is an inflating region insi…
In the No-Boundary Universe a primordial black hole is created from a constrained gravitational instanton. The black hole created is immersed in the de Sitter background with a positive cosmological constant. The constrained instanton is characterized not only by the external parameters, the mass parameter, charge and angular momentum, but also by one more internal parameter, the identification period in the imaginary time coordinate. Although the period has no effect on the black hole background, its inverse is the temperature of the no-boundary state of the perturbation modes perceived by an…
We explore asymptotically AdS$_2$ solutions of a particular two-dimensional dilaton-gravity theory. In the deep interior, these solutions flow to the cosmological horizon of dS$_2$. We calculate various matter perturbations at the linearised and non-linear level. We consider both Euclidean and Lorentzian perturbations. The results can be used to characterise the features of a putative dual quantum mechanics. The chaotic nature of the de Sitter horizon is assessed through the soft mode action at the AdS$_2$ boundary, as well as the behaviour of shockwave type solutions.
We investigate the thermodynamic properties of 3+1 dimensional black holes in asymptotically de Sitter spacetimes, conformally coupled to a real scalar field. We use a Euclidean action approach, where boundary value data is specified at a finite radius `cavity' outside the black hole, working in the extended phase space where the cosmological constant is treated as a thermodynamic pressure. We examine the phase structure of these black holes through their free energy. For the MTZ subclass of solutions, we find Hawking-Page-like phase transitions from a black hole spacetime to thermal de Sitter…
Three exact solutions say $ϕ_0$ of massless scalar theories on Euclidean space, i.e. $D=6 ϕ^3$, $D=4 ϕ^4$ and $D=3 ϕ^6$ models are obtained which share similar properties. The information geometry of their moduli spaces coincide with the Euclidean ${AdS}_7$, ${AdS}_5$ and ${AdS}_4$ respectively on which $ϕ_0$ can be described as a stable tachyon. In D=4 we recognize that the SU(2) instanton density is proportional to $ϕ_0^4$. The original action $S[ϕ]$ written in terms of new scalars ${\tilde ϕ}=ϕ-ϕ_0$ is shown to be equivalent to an interacting scalar theory on $D$-dimensional de Sitter backg…
