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Notes on de Sitter space

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Beatrix Mühlmann

High Energy Physics - Theory · Submitted 16 Sep 2026 · 248 references

Pith's verdict: A thorough, well-structured lecture-note compilation on de Sitter space that delivers what it promises pedagogically, but suffers from a handful of corrupted equations and minor presentation defects that should be cleaned up before publication.

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arXiv ID2609.19454 (version 1)
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CitationBeatrix M\"uhlmann. "Notes on de Sitter space." arXiv:2609.19454 (2026). https://pith.science/paper/36FUPVRV
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The paper

The paper claims

A comprehensive review of de Sitter space that unifies geometry, quantum field theory, thermodynamics, and holography across two, three, and four dimensions.

The reading

This paper aims to present a self-contained, pedagogical overview of de Sitter space as the central arena for quantum gravity with a positive cosmological constant. It synthesizes classical geometry, quantum field theory on a fixed de Sitter background, the thermodynamics of cosmological horizons, and recent developments in low-dimensional quantum gravity. The author argues that understanding de Sitter space is essential for addressing major open problems such as the cosmological constant problem, the nature of vacuum states, and the microscopic origin of de Sitter entropy. By comparing the de Sitter horizon with black hole horizons and reviewing explicit solvable models in two, three, and four dimensions, the paper provides a unified framework for current research.

What carries the argument

The de Sitter isometry group SO(1,d+1) and its unitary irreducible representations (principal, complementary, and discrete series) which classify quantum fields and determine their Harish-Chandra characters. The Harish-Chandra character is used to compute quasinormal mode degeneracies and the one-loop sphere partition function, connecting group theory to thermodynamics.

Extended reading notes

Core claim

The paper systematically establishes that de Sitter space is the maximally symmetric solution of Einstein's equations with positive cosmological constant, and that its isometry group SO(1,d+1) governs the representation theory of quantum fields. It demonstrates that the Bunch-Davies/Hartle-Hawking state provides a natural vacuum, that the Gibbons-Hawking entropy arises from the Euclidean sphere partition function, and that low-dimensional models such as timelike Liouville theory, the complex Liouville string, and higher-spin de Sitter holography offer concrete realizations where conceptual puzzles - including the conformal mode problem, the algebraic definition of observables, and the counting of microstates - can be addressed exactly.

Load-bearing premise

The Bunch-Davies/Hartle-Hawking state is the correct initial state for quantum fields in de Sitter space, especially when discussing the wavefunction, entropy, and thermality.

Editorial extensions

If this is right

  • The explicit solution of the conformal mode problem in 2D timelike Liouville gravity and 4D higher-spin gravity provides a template for handling the phase of the Euclidean path integral in other contexts.
  • The algebraic description of the static patch in terms of type II1 von Neumann algebras offers a rigorous framework for de Sitter entropy and observer-dependent observables.
  • The dual matrix model for the complex Liouville string in 3D gives a microscopic count of states that matches the semiclassical de Sitter entropy, strengthening the case for a holographic description.
  • The gluing formula for the 4D higher-spin sphere partition function suggests a relation between the Euclidean path integral and the norm of the Hartle-Hawking wavefunction.
  • Low-dimensional models (2D and 3D) serve as tractable laboratories to test ideas about unitarity, Hilbert space structure, and interior observables that may extend to four dimensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper implicitly suggests that the absence of a conventional S-matrix in de Sitter is not a deficiency but a feature that can be accommodated by the in-in formalism and algebraic quantum field theory.
  • The comparison between black hole and de Sitter horizons, especially the Gao-Wald theorem and the anti-scrambling behaviour, highlights a potential tension: de Sitter causality and chaotic properties may differ fundamentally from those of asymptotically flat or AdS black holes.
  • If the Bunch-Davies state is not the unique de Sitter-invariant Hadamard state (as noted for the Schwinger model), then the thermal interpretation of the static patch might need revision, with implications for entropy counting.
Original abstract

We discuss classical and quantum features of spacetimes with a positive cosmological constant. After introducing basic aspects of de Sitter cosmology and the classical geometry of de Sitter space, we consider de Sitter space both as a rigid background, on which quantum fields propagate, and as a fluctuating spacetime in its own right. We review the wavefunction of an expanding universe, black holes in de Sitter space, and the de Sitter entropy, and compare the cosmological horizon with the black hole horizon. The final three sections are dedicated to concrete models of de Sitter space in two, three and four spacetime dimensions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,624 input tokens.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.22 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We discuss classical and quantum features of spacetimes with a positive cosmological constant. After introducing basic aspects of de Sitter cosmology and the classical geometry of de Sitter space, we consider de Sitter space both as a rigid background, on which quantum fields propagate, and as a fluctuating spacetime in its own right. We review the wavefunction of an expanding universe, black holes in de Sitter space, and the de Sitter entropy, and compare the cosmological horizon with the black hole horizon. The final three sections are dedicated to concrete models of de Sitter space in two, three and four spacetime dimensions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We discuss classical and quantum features of spacetimes with a positive cosmological constant. contribution 0.73 0.64 0.08 asserted 0.51
s1 After introducing basic aspects of de Sitter cosmology and the classical geometry of de Sitter space, we consider de Sitter space both as a rigid background, on which quantum fields propagate, and as a fluctuating spacetime in its own right. contribution 0.75 0.65 0.05 asserted 0.83
s2 We review the wavefunction of an expanding universe, black holes in de Sitter space, and the de Sitter entropy, and compare the cosmological horizon with the black hole horizon. contribution 0.93 0.41 0.08 asserted 0.80
s3 The final three sections are dedicated to concrete models of de Sitter space in two, three and four spacetime dimensions. contribution 0.75 0.58 0.04 asserted 0.96

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.69.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Summary of the equation of state parameter wi , dilution and current value for various components of our Universe. as the term −k/a 2 , not as matter. We use the definition Ωk ≡ − k (a0H0) 2 (2.17) 6
Figure 2. de Sitter space as the hyperboloid −(X0 ) 2 + ∑i (Xi ) 2 = ℓ 2 in R 1,d+1 . Slices of constant X0 are spheres S d , which shrink to the minimal sphere (red) at X0 = 0 and grow toward I ± . Coordinate Systems on de Sitter Space Global coordinates. A convenient parametrization of (3.2) is X 0 = ℓ sinh(τ ) , Xj = ℓ cosh(τ ) y j , yjy j = 1 , (3.13) so that the induced metric (3.3) becomes ds2 gl ℓ 2 = −dτ 2 + cosh2 τdΩ2 d , dΩ2 d = dθ 2 + sin2 θdΩ2 d−1 . (3.14) Here τ ∈ R and θ ∈ [0, π]. This is a closed Universe. Constant τ slices are compact d-spheres
Figure 3. The whole square is the Penrose diagram of global de Sitter. The overlap of past (blue) and future (orange) light rays is the de Sitter static patch. Each Cauchy slice of de Sitter, indicated by the red line is an S d . Each point on the red line is a S d−1 worth of points. Exceptions to this are only the north and south pole where sin θ = 0. These are single points. Moving along the red slice each S d−1 grows from a point at the SP to a maximal equator at θ = π/2 and shrinks back to a point at the NP. Each point on a horizontal line in the Penrose diagram represents an S d−1 , e
Figure 4. Penrose diagram of dS2 in conformal coordinates with its Killing vectors (3.17) (using σ = arctan(sinh τ )). Shaded regions are timelike (∥L∥ 2 < 0), white regions spacelike, dashed null (horizons) and dots = fixed points. 11
Figure 5. For dS2 it is often convenient to “open-up” the Penrose diagram. Instead of a square the Penrose diagram becomes a rectangle. Since S 0 is two points we can double the square, leading to the above rectangle where each point on the red S 1 is now indeed just a single point. Since the red line is a circle S 1 where we identify φ = 0 and φ = 2π we need to identify the two edges. Static patch. The exponential expansion in de Sitter restricts the region an observer can see or influence. An observer indeed has causal access only to a portion of global de Sitter space, known as the static…
Figure 6. Left: the static patch of the south-pole observer, foliated by constant-t (blue) and constant-r (red) slices; the diagonals are the past and future horizons. Right: the planar patch, foliated by constant-η slices; the red diagonal is the horizon η → −∞ of the flat slicing, and the slices approach I + as η → 0 − . The name static patch is chosen because the metrics such as (3.23) are time inde￾pendent, with Killing vector ∂t . Here t ∈ R and ρ ∈ [0, π/2) (for d ⩾ 2), while for d = 1, 1 In terms of embedding coordinates we obtain (3.23a) from X 0 = ℓ cos(ρ) sinh(t), X d+1 = ℓ cos
Figure 7. Both the static as well as the global patch of dSd+1 rotate in Euclidean signature to the (d + 1)-dimensional sphere S d+1 . The equator (red) is the S d at τE = 0, the throat of the hyperboloid in figure 2. 3.1 Asymptotically de Sitter Pure de Sitter is a highly symmetric instance of a vast solution space of Einstein’s equations. For convenience we restrict to d = 3. At late times the metric of a general asymptotically de Sitter spacetime can be organized in the Fefferman–Graham form as follows ds2 ℓ 2 = − dη 2 η 2 + 1 η 2 (g (0) ij + η 2 g (2) ij + η 3 g (3)
Figure 8. Left: ∆ in scalar UIRs. Right: ∆ in spinning UIRs. dS3 Representations. In three-dimensional de Sitter space, only the principal and com￾plementary series irreducible unitary representations occur: πν,s ∶ ∆ = 1 + iν , ν ∈ R+ , s ⩾ 0 , (4.25a) γ∆ ∶ ∆ ∈ (1, 2) , s = 0 . (4.25b) Spin s labels an SO(2) representation. The principal and complementary series irreducible representations are isomorphic to their shadow representations πν ≅ π−ν and γ∆ ≅ γ∆ , where ∆ ≡ 2 − ∆. Therefore we take ν ∈ R+ and ∆ ⩾ 1. Since 2 2 = 2 − 1 we infer from (4.23) that in dS3 there is no complementary series
Figure 9. Higuchi bound for dS4 along with some partially massless fields and their degeneracies. Harish-Chandra character is in fact a distribution, and its rigorous definition involves smearing over the group. Examples of Harish-Chandra characters are D ± ∆ in dS2: χ∆,0(t) = e −∆t (1 − e −t) , ∆ ∈ Z+ , (4.38a) F∆,0 in dSd+1: χ∆,0(t) = e −∆t + e −(d−∆)t (1 − e −t) d , ∆ = d 2 + iν , ν ∈ R or ∆ ∈ (0, d) , (4.38b) D ± 1,2 in dS4: χ3,2(t) = 5e−3t − 3e−4t (1 − e −t) 3 , (4.38c) D ± − 1 2 , 1 2 in dS4 ∶ χ− 1 2 , 1 2 (t) = e − 3t 2 (1 − e −t) 3 . (4.38d)
Figure 10. At I + operators are organized in terms of SO(1, d + 1) irreducible represen￾tations acting as the conformal group on the late time slice. The late time operators are generally non-commuting. where X(x) are the embedding coordinates in (3.7). Away from coincident points, the Wightman function satisfies the Klein–Gordon equation (−∇2 + m2 )GW (P) = 0 . (4.75) This leads to the differential equation (1 − P 2 )G ′′ W (P) − (d + 1)P G′ W (P) − m2 ℓ 2GW (P) = 0 , (4.76) The change of variables ξ = 1+P 2 maps (4.76) to a hypergeometric differential equation whose solutions are
Figure 11. The Hartle–Hawking construction. Left: the Euclidean vacuum ∣ΩE⟩ is prepared by gluing half of the Euclidean sphere (blue) onto Lorentzian de Sitter evolution (red). Right: the corresponding contour in the complex time plane, descending from τ = iπ/2 along the imaginary axis (Euclidean evolution, blue) and continuing along real time (Lorentzian evolution, red). gives K(x, x′ ;t) = ∑ E ψE(x)ψ ∗ E(x ′ ) e −iEt , (5.84) where ψE(x) = ⟨x∣E⟩. Taking t → ∞(1 − iϵ) suppresses the excited states and, up to normalization, projects onto the ground state, K(x, x′ ;t) ∼ ψE0 (x)ψ ∗ E
Figure 12. The Hartle–Hawking wavefunction fixes a single boundary (red); the manifold that is filled in can have arbitrary topology [71–74]. where SE is the Euclidean action and hij the induced metric on the boundary ∂M. The path integral is taken over manifolds that are regular in the interior and match hij at the boundary. In principle, this includes a sum over different topologies (see figure 12). Moreover the Euclidean gravitational action is Gaussian unsuppressed, this is the conformal mode problem, and we return to this in subsection 6.4. A simple example. As a simple example, we…
Figure 13. For conformal boundary conditions there is one real saddle, and the two caps are labelled by ±K. which in particular satisfies (see figure 13) Ψ[K,round S 3 ]Ψ[−K,round S 3 ] = e πℓ2 GN . (5.105) This is a saddle point result7 but it is indicative that the inner product should pair the wavefunctions with opposite K. We can interpret K as the York time [78] and perhaps we should see this as an indication that in the inner product we should only integrate over the conformal class of metric across the interface [79]. This might be a sharper version of the proposal in (5.72). 6 de
Figure 14. An observer at ρ = 0 is surrounded by an event horizon, created by the exponential accelerated expansion of spacetime. 6.1 Static Patch Density Matrix and Thermal Structure If one assumes that the Hilbert space factorizes as H = HL ⊗ HR , (6.6) one may define a reduced density matrix associated with an observer following a static patch worldline, ρˆL = trHR (∣0⟩⟨0∣) , (6.7) 44
Figure 15. Static patch decomposition of de Sitter space. An observer has access only to the left patch HL, while degrees of freedom beyond the cosmological horizon are traced over. To motivate a thermal structure of the partition function, it is useful to recall the Euclidean path integral representation of the thermal partition function in quantum me￾chanics. Wick rotating t → −iβ gives Z(β) = tr e−βH = ∫ PBC [Dx(τ )] e −SE[x(τ)] , (6.9) 45
Figure 16. Integration contours for one-loop sphere partition functions [67]. Examples We give two more examples 1. Gauge field in 2D 2. Graviton in 4D 1. Gauge field in 2D. We discuss a gauge field in dS2 with m2 ℓ 2 = −2. Examples of this are the fluctuations of the Weyl factor in Liouville theory, or the fluctuation of the Dilaton in JT gravity around an S 2 saddle. It is important that we view this particle not as a tachyonic scalar field but as a gauge field (it is part of the metric field) that transforms in the discrete series unitary irreducible representation of SO(1, 2). We conside
Figure 17. The stretched horizon. An observer at the pole ρ = 0 is surrounded by the cosmological horizon. Displacing the horizon slightly inward yields a timelike surface — the stretched horizon — on which one may imagine holographic qubit degrees of freedom residing. Thickened worldlines and timelike boundaries. Moreover we can imagine a family of timelike boundaries, that interpolate from a worldline to a thickened worldline to the stretched horizon [114–117]. These may serve as anchors for observables in the static patch. 7 de Sitter Black holes We will now turn to the study of black…
Figure 18. Penrose diagram of dS2. The Nariai limit of the Schwarzschild–de Sitter black hole is dS2 × S 2 . In the Penrose diagram we suppress the S 2 . The red lines are the cosmological (ρ = β) and black hole (ρ = 0) horizons. As discussed in section 3 the left and right edges are identified. These frequencies describe the characteristic damped oscillations of scalar perturbations in the near-horizon Nariai geometry and encode how perturbations decay between the black hole and cosmological horizons. Dimensional reduction of 4D Schwarzschild–de Sitter. Before moving on we re￾duce the four-d
Figure 19. The Sharkfin diagram of the Reissner–Nordstr¨om–de Sitter black hole. 7.3 Kerr–de Sitter black hole The Kerr–de Sitter black hole describes a rotating black hole in a spacetime with positive cosmological constant. Its line element is given by ds2 = − ∆r ρ 2 (dt − a Θ sin2 θ dϕ) 2 + ρ 2 ∆r dr 2 + ρ 2 ∆θ dθ 2 + ∆θ ρ 2 sin2 θ (a dt − r 2 + a 2 Θ dϕ) 2 , (7.33) where ∆r = (r 2 + a 2 )(1 − r 2 ℓ 2 ) − 2GNMr, Θ = 1 + a 2 ℓ 2 , (7.34a) ∆θ = 1 + a 2 ℓ 2 cos2 θ, ρ2 = r 2 + a 2 cos2 θ. (7.34b) The spacetime possesses the Killing vectors ∂
Figure 20. An observer at r = 0 is surrounded by an event horizon. Quasinormal modes are regular at r = 0 and purely outgoing at the future horizon. 8.2 Gao-Wald theorem The discussion so far neglects backreaction. The Gao–Wald theorem [129] states that a null geodesically complete, globally hyperbolic spacetime with compact Cauchy surface Σ, satisfying the Einstein equations with matter obeying the null energy condition and the null generic condition, cannot exhibit a particle horizon. In particular, at sufficiently late times the past light cone of a spacetime point in a slightly perturbed…
Figure 21. Left: Penrose diagram of the eternal black hole. Null signals (red) sent from the two exterior regions meet behind the horizon (red dot). Right: Including backreaction of an early perturbation, which sources a shockwave (double line), the Penrose diagram stretches horizontally and previously connected regions fall out of causal contact: the null signals (red, dashed) emitted by the two observers reach the singularity without ever meeting. The two-sided correlator between OL and OR — an out-of-time-order correlator (OTOC)— is computed in the geodesic approximation by the length of…
Figure 22. Penrose diagram of global de Sitter, with the static patch of the observer shaded in grey and the worldline in blue. A light ray emitted from the pole after I + cannot reach the antipodal worldline before I + . Right: The Gao–Wald effect for de Sitter space. Including backreaction, the Penrose diagram stretches vertically — it becomes taller than wide — and previously disconnected regions come into causal contact: a light ray (red) emitted after I − crosses the entire space and reaches the observer’s worldline before reaching I + . 8.3 State preparation In figures 23 and 24 we
Figure 23. From left to right: dS Penrose diagram. The static patch shaded in gray, the observer’s worldline in red. S d+1 viewed as the Euclidean continuation of the static patch. The worldline is mapped to a great circle on the sphere. The sphere is also the Euclidean continuation of global de Sitter. Cutting the Euclidean continued global de Sitter sphere – S d+1 – the boundary is S d , which corresponds to the blue line on the global Penrose diagram. This prepares the HH state. Analytically continuing to Lorentzian we obtain the Hartle-Hawking wavefunction ΨHH. Lorentzian continuation…
Figure 24. From left to right: Penrose diagram of global AdS2 with two Rindler wedges. Continuing the Rindler wedge to Euclidean signature we obtain the Poincare disk. Cutting the disk along the equator prepares the ∣TFD⟩. 8.4 Thermodynamics As already mentioned in section 6, for an AdS black hole the Euclidean path integral at inverse temperature β computes SBH − βE = log Z . (8.16) Euclidean de Sitter space is a sphere: both the spatial slices and the whole space, so there is no asymptotic region in which to anchor an ADM energy. Without boundaries there is no independently specified β…
Figure 25. Timelike (left) vs spacelike (right) regime of the momentum ˆp and p respec￾tively. 9.2 Timelike Liouville gravity Timelike Liouville theory provides a model of two-dimensional de Sitter quantum gravity with the following properties: 1. It exhibits a finite sphere partition function [94]. 2. It contains propagating degrees of freedom. 3. It incorporates the conformal mode problem [94, 95]. 4. It admits a semiclassical limit with small but nonvanishing quantum fluctuations [94, 140]. 5. It admits a “higher spin” extension [153]. 6. It allows for a supersymmetric extension [154]. 7
Figure 26. We show the main references for bosonic Liouville theory. We distinguish spacelike and timelike Liouville, as well as the CFT and the gravity aspect of the theory. From the CFT perspective we restrict to bulk correlators. c = 3 2 c = 27 2 c CFT [166–169] [170, 171] [154] [172, 173] Gravity N = 1 spacelike Liouville timelike Liouville
Figure 27. We show the main references for N = 1 Liouville theory. We distinguish spacelike and timelike Liouville, as well as the CFT and the gravity aspect of the theory. From the CFT perspective we restrict to bulk correlators. c = 3 c CFT [161–163] [174] [154] ? Gravity N = 2 spacelike Liouville timelike Liouville
Figure 28. We show the main references for N = 2 Liouville theory. We distinguish spacelike and timelike Liouville, as well as the CFT and the gravity aspect of the theory. From the CFT perspective we restrict to bulk correlators. 80
Figure 29. The wavefunction of 3D de Sitter quantum gravity is given by the Liouville correlator, with a Milne type singularity at t = 0. The surface at I + is a genus g Riemann surface with n punctures. In the example g = 2 and n = 3. The punctures extend to Wilson lines of non-dynamical particles in the past which fall into the singularity at t = 0. 10.3 Dual matrix integral The dual matrix model is a Hermitian two-matrix integral, ZMM = ∫ R2N2 dM1 dM2 exp [ − N tr(V1(M1) + V2(M2) − M1M2)] , (10.13) with M1,2 Hermitian N ×N matrices. The specific potentials V1,2 will not be needed: in the
Figure 30. Four point amplitude A (b) 0,4 (p1, p2, p3, p4) of the complex Liouville string. 11 de Sitter in 4D The ultimate target in the effort to understand our own Universe is quantum gravity in four-dimensional de Sitter space. Its importance notwithstanding, controlled avenues toward it are few. String constructions such as KKLT [231] are promising but to date do not admit a complete non-perturbative mathematical description. Inspired by holography in AdS, one may hope to gain control by introducing a worldtube — a timelike boundary at finite distance which is an actively researched topic
Figure 31. Norm of the Hartle-Hawking wavefunction versus the gluing formula (11.33). Acknowledgments It is a great pleasure to thank Dionysios Anninos, Tarek Anous, Andreas Blommaert, Jonah Kudler-Flam, Victor Ivo, David Kolchmeyer, Zimo Sun, as well as ChatGPT and Claude for useful discussions. I am particularly indebted to Dionysios Anninos for many illuminating discussions. I would also like to thank the students and organizers of the 31st 94

Figures reproduced from arXiv:2609.19454 by the authors.

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