Asymptotically de Sitter Spacetimes

Asymptotically de Sitter geometry describes how a spacetime with positive cosmological constant can approach a regular spacelike conformal boundary. The subject is global: the same local de Sitter metric admits a compact global presentation, an observer’s static patch, and an expanding planar patch with different causal reach and boundary topology. This chapter builds the definitions, horizon geometry, boundary data, conditional charges, and stability interfaces needed to keep those descriptions distinct.

Required background. The Einstein Equation supplies the $\Lambda\gt 0$ field equation and its constraint content.

Helpful background. Pseudo-Riemannian Metrics and Signature fixes causal character; Affine and Variational Geodesics supplies completeness; Causality and Global Structure supplies conformal boundaries and domains of dependence; Exact Solutions of Einstein’s Equations supplies comparison solutions; and Formulations of General Relativity supplies constrained initial and boundary data.

Enter this chapter

The basic sign is decisive. In four spacetime dimensions, write the physical metric as $\widehat g_{ab}=\Omega^{-2}g_{ab}$ near $\Omega=0$, assume $d\Omega\ne0$ there, and impose the Einstein equation with $\Lambda\gt 0$. Under the regular matter falloff used in this chapter, the conformal field equations imply at the boundary

$$g^{ab}\nabla_a\Omega\nabla_b\Omega\mathrel{=}-\frac{\Lambda}{3}.$$

With signature $(-,+,+,+)$, the conformal normal is timelike and the induced metric $q_{ab}$ on $\mathscr I$ is Riemannian. Thus de Sitter infinity is spacelike, unlike asymptotically flat null infinity and asymptotically anti-de Sitter timelike infinity. This single distinction changes what boundary data mean, which asymptotic symmetries are natural, and why a global particle S-matrix is not canonically available. The derivation and its precise regularity assumptions appear in Ashtekar, Bonga, and Kesavan 2015, §II.B, pp. 7–9.

Our baseline is a smooth four-dimensional conformal completion. We require $\Omega^{-1}\widehat T_{ab}$ to extend smoothly when matter is present, unless a page names a different model. We do not assume that $q_{ab}$ is conformally flat. Boundary topologies such as $S^3$, $\mathbb R^3$, and $S^2\times\mathbb R$ are kept explicit because completeness, symmetry groups, and the relation to a physical patch depend on them. Results about other dimensions, finite differentiability, polyhomogeneous expansions, or slower matter decay require separate hypotheses.

From the global cylinder to observer patches

Begin with Spacelike Conformal Infinity in de Sitter Geometry. Global de Sitter space is conformal to a finite slab of the Einstein cylinder, with round $S^3$ components $\mathscr I^-$ and $\mathscr I^+$. The page derives the compactification rather than inferring the boundary from a coordinate limit. It also shows why a null geodesic can reach the conformal boundary at finite unphysical parameter while having infinite physical affine length.

The next step, Definitions of Asymptotically de Sitter Spacetimes, separates several logically different properties. A spacetime can be future asymptotically de Sitter without having a regular past boundary; it can possess a smooth conformal boundary without being asymptotically simple; and completeness of the induced boundary metric is an additional global condition. The definition records dimension, conformal regularity, topology, matter falloff, and whether the completion is future, past, or two-sided. These data are not decorative qualifications: changing any one can change the theorem being stated.

Then Observer Patches and Cosmological Horizons compares global, static, and planar descriptions. A static observer occupies a causal diamond bounded by cosmological Killing horizons. The surface $r=\ell$, where $\ell^2=3/\Lambda$, is a null horizon inside the conformal completion, not the spacelike conformal boundary. The planar expanding chart covers only part of global de Sitter space and reaches a noncompact $\mathbb R^3$ component of future infinity. Local isometry therefore does not identify global domains, boundary topologies, or observer access.

This distinction is especially important when the word “horizon” is used. A cosmological event horizon is defined relative to an observer’s future endpoint or worldline and concerns which events can signal that observer. Conformal infinity is an added boundary of the entire chosen completion. One may intersect or delimit the other in a conformal diagram, but they have different causal character and answer different questions.

Boundary data are not final-state observables

Asymptotic Data at de Sitter Infinity identifies the leading geometric information on a smooth spacelike boundary. In four-dimensional vacuum gravity it may be represented, after conformal gauge choices, by a Riemannian conformal metric $[q_{ab}]$ and a symmetric transverse trace-free tensor related to the electric part $\mathcal E_{ab}$ of the leading rescaled Weyl curvature. These are constrained conformal data. On a spacelike boundary they play a Cauchy-like role for conformal evolution; they are not automatically particle states or measurements available to one timelike observer. Friedrich’s construction of asymptotically simple solutions from such free data is given in Friedrich 1986, pp. 101–117.

Matter modifies the divergence constraint and may alter the permitted expansion. The regularity condition $\Omega^{-1}\widehat T_{ab}$ smooth at $\mathscr I$ is an assumption on decay, not a consequence of $\Lambda\gt 0$. Likewise, specifying data on $\mathscr I^+$ gives a future conformal problem; it does not by itself prove that the backward development has a smooth $\mathscr I^-$ or avoids a singularity. A two-boundary map exists only on a class of developments regular in both directions.

The magnetic part $\mathcal B_{ab}$ of the leading rescaled Weyl tensor measures the Cotton curvature of $q_{ab}$. In three boundary dimensions,

$$q_{ab}\text{ conformally flat} \quad\Longleftrightarrow\quad \mathcal B_{ab}=0.$$

This equivalence is useful, but imposing it restricts the data. It removes half of the radiative degrees of freedom even though the electric transverse trace-free tensor need not vanish. Consequently, conformal flatness must always be labeled as an additional boundary condition, never treated as part of the definition of asymptotically de Sitter spacetime.

Symmetries and charges depend on restrictions

The minimal boundary conditions permit a general Riemannian metric on $\mathscr I$. de Sitter Asymptotic Symmetries and Charges explains that the universal group is then essentially $\operatorname{Diff}(\mathscr I)$, with no canonically preferred translation subgroup. Exact de Sitter’s ten Killing fields do not extend to a canonical ten-dimensional symmetry group for every nearby radiative solution.

A finite de Sitter-like group emerges after one imposes conformal flatness and boundary completeness. Topology then matters: $S^3$ admits the ten-dimensional de Sitter group, while complete $\mathbb R^3$ and $S^2\times\mathbb R$ boundaries retain smaller subgroups. The topology-sensitive classification is developed in Ashtekar, Bonga, and Kesavan 2015, §IV.C, pp. 20–25.

There is an unavoidable price. In the ABK framework the condition $\mathcal B_{ab}=0$ makes the associated gravitational de Sitter-momentum flux through $\mathscr I$ vanish. This is not evidence that generic gravitational waves carry no asymptotic information; it is a consequence of restricting the phase space so that half of the freely specifiable radiative data are absent. The electric part can remain nonzero, so “no flux” is also not equivalent to “the Weyl curvature vanishes.” The exact restriction and flux statement are derived in Ashtekar, Bonga, and Kesavan 2015, §V.C.4, pp. 33–36.

Charges are therefore conditional constructions. Before naming an energy or momentum, one must state the boundary topology, falloff, conformal frame or equivalence class, allowed symmetry field, matter contribution, and cuts on which the charge is evaluated. In a generic phase space there is no universally preferred positive global energy. Observer-patch energies based on a static Killing field solve a different, quasi-local problem and need not define a charge on all of global $\mathscr I$.

Stability and scattering boundaries

Finally, Stability and Scattering Interfaces states the nonlinear result without enlarging its scope. For smooth vacuum constraint data on compact $S^3$, sufficiently close in the required Sobolev norms to global de Sitter data, the maximal development is null and timelike geodesically complete in both directions and possesses smooth spacelike $\mathscr I^-$ and $\mathscr I^+$. This is a two-sided small-data theorem; it does not cover arbitrary topology, matter, large data, or black-hole perturbations. A modern statement with the hypotheses and causal conclusion appears in Friedrich 2024, §2.

Three questions must not be conflated. Decay of expansion-normalized quantities along cosmological observers is one form of no-hair behavior. Smooth future conformal completion is a stronger geometric statement but can coexist with a singular past. Two-sided nonlinear stability gives regular evolution between two conformal boundaries for a neighborhood of specified initial data. Wald’s classic result concerns expanding spatially homogeneous Bianchi models under energy assumptions, not generic inhomogeneous cosmologies Wald 1983, pp. 2118–2120.

Classical conformal evolution may define a map from admissible geometric data at $\mathscr I^-$ to data at $\mathscr I^+$ within a two-sided regular class. That is not the ordinary asymptotically flat S-matrix. Spacelike infinity, the absence of a preferred global timelike translation, and observer horizons obstruct a canonical global positive-frequency particle interpretation. Linear fields, quantum states, static-patch response, horizon scattering, and black-hole exterior scattering each require model-specific equations and boundary conditions.

For proofs of nonlinear decay, black-hole stability, or detailed hyperbolic estimates, continue to Mathematical Relativity. For quantum states and observables, continue to Quantum Field Theory in Curved Spacetime. This chapter supplies the causal and conformal framework those analyses must respect; it does not import their conclusions.

Review the chapter

Diagnose each statement below.

1. At a smooth four-dimensional conformal boundary with $\Lambda\gt 0$ and the stated matter falloff, $n^an_a=-\Lambda/3$. What is the causal character of $\mathscr I$?
2. In static de Sitter coordinates the lapse vanishes at $r=\ell$. Is this surface $\mathscr I^+$?
3. A completion allows arbitrary smooth $q_{ab}$ on $S^3$. Does it possess a canonical ten-dimensional de Sitter asymptotic symmetry group?
4. One imposes $\mathcal B_{ab}=0$ to recover that group. What physical restriction has been introduced?
5. Small smooth vacuum data on $S^3$ satisfy the hypotheses of the de Sitter stability theorem. Does this establish an ordinary global particle S-matrix?
6. An expanding planar cosmology has smooth $\mathscr I^+\cong\mathbb R^3$ but a big-bang singularity. Is it two-sided asymptotically simple?

Solution

1. The normal is timelike in signature $(-,+,+,+)$, so $\mathscr I$ is spacelike. 2. No. The surface $r=\ell$ is the observer’s null cosmological horizon; $\mathscr I^+$ is the spacelike conformal boundary of the global completion. 3. No. With arbitrary $q_{ab}$ the universal asymptotic group is $\operatorname{Diff}(S^3)$; the finite de Sitter group requires extra structure. 4. The condition is conformal flatness of $q_{ab}$ and sets the leading magnetic Weyl data to zero. It removes half of the radiative data and forces the corresponding ABK gravitational flux to vanish, although electric data may remain. 5. No. The theorem establishes two-sided smooth completeness for this small-data development. It does not supply a preferred global time translation, positive-frequency splitting, or observer-independent particle states. 6. No. It is future asymptotically de Sitter only; past asymptotic simplicity fails at the singularity.

References

• Abhay Ashtekar, Béatrice Bonga, and Aruna Kesavan, “Asymptotics with a Positive Cosmological Constant: I. Basic Framework,” Classical and Quantum Gravity 32 (2015), 025004. doi:10.1088/0264-9381/32/2/025004. Open PDF
• Helmut Friedrich, “Cosmologies with Positive $\lambda$: Hierarchies of Future Behaviour,” Philosophical Transactions of the Royal Society A 382 (2024), 20230044. doi:10.1098/rsta.2023.0044
• Helmut Friedrich, “Existence and Structure of Past Asymptotically Simple Solutions of Einstein’s Field Equations with Positive Cosmological Constant,” Journal of Geometry and Physics 3 (1986), 101–117. doi:10.1016/0393-0440(86)90004-5
• Robert M. Wald, “Asymptotic Behavior of Homogeneous Cosmological Models in the Presence of a Positive Cosmological Constant,” Physical Review D 28 (1983), 2118–2120. doi:10.1103/PhysRevD.28.2118