Emergent Universe, Phase Structure and de Sitter
Run the CDT path integral and the geometry organizes itself. In one specific phase of the lattice's phase diagram, what emerges is a smooth four-dimensional universe that expands the way our actual universe expands. The macroscale result that put CDT on the map.
Placeholder for a 3D visualisation of Causal Dynamical Triangulation. The interactive scene will land in Phase 3. Causal Dynamical Triangulation defines a quantum theory of gravity by summing over piecewise-linear Lorentzian geometries built from four-dimensional simplices, the higher-dimensional analogs of triangles, glued together along a strict causal ordering. The sum is the gravitational path integral, evaluated nonperturbatively by Monte Carlo simulation. Spacetime is not assumed; it emerges from the sum. After almost three decades of development the program has produced a clear catalog of results: a four-dimensional de Sitter-like universe self-organizes in the C phase of the lattice's phase diagram, the spectral dimension of the emergent spacetime runs from approximately 4 at large scales to approximately 2 at the Planck scale, the phase diagram has been mapped including a recently identified bifurcation phase, and modern extensions enforce causality locally and couple matter fields to the discrete geometry. The Ambjorn-Loll 1998 paper is the foundational reference, the 2012 Physics Reports review and the 2020 Classical and Quantum Gravity review are the canonical modern surveys, and a 2026 review preprint frames the program as a working lattice theory of quantum gravity. CDT is the most computationally productive of the three discrete-spacetime programs in this chapter, complementing Causal Set Theory's order-and-number kinematics and Loop Quantum Gravity's canonical and spin-foam quantization with a concrete path-integral construction that produces actual numbers.
In one sentence
Ambjorn, Jurkiewicz, and Loll showed in 2004 that the CDT path integral produces a phase, called the C phase, in which the sum over discrete geometries yields a smooth four-dimensional universe whose large-scale volume profile matches a de Sitter spacetime. The result was extended through 2008 with the explicit identification of the emergent geometry as Euclideanized de Sitter and through subsequent work with additional phases including the recently-identified bifurcation phase C_b.
The claim
The first thing a CDT simulation produces is a phase diagram. Depending on the values of the two free parameters in the action, the gravitational coupling and the cosmological constant, the sum over triangulated geometries lands in one of several qualitatively distinct phases, analogous to the way water can be ice, liquid, or vapor depending on temperature and pressure. CDT has at least four such phases. Phase A is highly crumpled, with effectively infinite Hausdorff dimension and no resemblance to a four-dimensional manifold. Phase B is a degenerate stalk-like structure in which spacetime collapses into thin polymer-like geometries. Phase C, the physically interesting phase, has extended four-dimensional geometry whose large-scale shape is smooth. A more recently identified phase, called the bifurcation phase or C_b, sits at a boundary of the C phase and is the focus of much of the program's modern phase-structure research because of its potential to host a second-order phase transition that would mark a true continuum limit.
In the C phase, the geometry is not put in by hand. The simulation starts from a fixed topology (typically a four-sphere or, in more recent work, a four-torus) and a fixed total number of simplices, and the path integral samples over how those simplices are glued. The output of the simulation, averaged over many configurations, is a measurement of the volume of the universe as a function of discrete time. The Ambjorn-Jurkiewicz-Loll 2004 Physical Review Letters showed that this volume profile, in the C phase, matches the volume profile of a four-dimensional de Sitter spacetime, the simplest expanding cosmological model. The match is not just qualitative; the long companion paper 'Reconstructing the universe' fits the simulated profile to a minisuperspace effective action and extracts a small number of effective cosmological parameters. The 2008 Physical Review D paper 'Nonperturbative Quantum de Sitter Universe' by Ambjorn, Gorlich, Jurkiewicz, and Loll, posted as arXiv:0807.4481, is the definitive extended treatment of the de Sitter identification.
More recent work has extended the phase-structure picture along two axes. First, the 2017 identification of the bifurcation phase C_b by Ambjorn, Gizbert-Studnicki, Gorlich, Jurkiewicz, and Klitgaard maps a new structural region of the phase diagram and locates candidate phase transitions in the C-to-C_b boundary that could mark a second-order critical point and a continuum limit. Second, the 2018 toroidal-topology paper by the same group (with Nemeth added) reruns the original 4D-emergence simulations on a four-torus rather than a four-sphere and confirms that the emergence is not a spherical-boundary artifact. The macroscale result is robust across topologies, which is one of the strongest internal-consistency arguments the program has produced.
The family stance
Spacetime is built from discrete geometric blocks glued along a strict causal ordering. Quantum gravity is defined nonperturbatively by summing over all such gluings via lattice Monte Carlo simulation, and the smooth four-dimensional spacetime of general relativity emerges from this sum in a specific phase of the lattice's phase diagram. The macroscale geometry that emerges in the C phase matches a de Sitter universe, and the microscale geometry shows a dimensional reduction from four dimensions at large scales to roughly two at the Planck scale. The same dimensional reduction appears independently in Asymptotic Safety and other quantum-gravity programs, which is taken in the literature as a possible clue that something real is happening to spacetime at the smallest scales. After three decades of development the program has produced concrete numerical results that no other quantum-gravity approach has produced at comparable detail, while remaining empirically unconfirmed and unresolved on its central technical question of whether a true continuum limit exists.
Predictions
- The CDT path integral has at least four phases in its parameter space, distinguished by qualitatively different geometric behavior. Three of these are accepted in the literature (A crumpled, B stalk-like, C extended four-dimensional); the bifurcation phase C_b is the focus of current research.
- In the C phase, the simulated spacetime has macroscale geometry matching a four-dimensional de Sitter universe, with a volume profile as a function of discrete time that fits the de Sitter form within numerical precision.
- The phase transition between the C phase and adjacent phases is a candidate location for a second-order critical point that would define a continuum limit. Current numerical work is mapping the critical behavior of the C-to-C_b boundary in this context.
Evidence
- The four-dimensional macroscale emergence is robust across multiple choices of topology (four-sphere and four-torus) and across truncations of the action, which suggests it is not an artifact of a particular boundary condition or parameter choice.
- The minisuperspace effective action extracted from the C-phase volume profile matches the expected form for a de Sitter universe to high precision, including the correct sign and order of magnitude of the effective cosmological constant.
- The phase diagram has been mapped in increasing detail over two decades, with the bifurcation phase C_b identified more recently as part of the same overall structure rather than as an artifact, supporting the view that CDT has a genuine phase structure rather than parameter-dependent finite-size effects.
Counterpoints
- The Euclidean signature of the de Sitter match. The simulated geometry is Euclideanized for computational tractability; the match to a Lorentzian de Sitter spacetime relies on an analytic continuation that, while standard, is not trivially justified in a nonperturbative context. Critics ask whether the Lorentzian and Euclidean results would agree in a strict sense.
- The macroscale emergence has been demonstrated for highly symmetric universes (de Sitter, with isotropic and homogeneous large-scale structure). Extending it to localized geometries (black holes, gravitational waves, inhomogeneous cosmological perturbations) is beyond current numerical resolution, which limits the program's contact with realistic gravitational phenomenology.
- The continuum limit remains unproven. The C-to-C_b transition is a candidate but not a settled result, and without a rigorous demonstration of a second-order critical point the program's macroscale results describe finite lattices rather than a continuum theory.
- The cosmological constant in the simulation is treated as a free parameter rather than predicted. The match to de Sitter geometry fixes its sign but the program does not, at present, derive its value or its small observed magnitude in our universe.
Variants in this family
▸Go deeperTechnical detail with proper terminology
The volume profile of the simulated C-phase universe is obtained by measuring the number of spatial three-simplices at each constant-time slice, averaged over many Monte Carlo configurations. The resulting profile as a function of discrete time fits a sin-like function of the form N3(t) ~ cos^3 of a rescaled time variable, which is the volume profile of a Euclideanized four-dimensional de Sitter spacetime. Quantum fluctuations around this average profile are also computable and can be matched to a minisuperspace effective action with one effective coupling and one effective cosmological constant.
The phase diagram of CDT is two-dimensional in the basic formulation, parameterized by the bare gravitational coupling and the bare cosmological coupling. Phase A (crumpled) sits at strong bare gravitational coupling, phase B (stalk-like) sits at weak bare gravitational coupling with small cosmological coupling, and phase C (extended four-dimensional) sits in an intermediate region with appropriately tuned cosmological coupling. The bifurcation phase C_b sits at a boundary of the C phase, in a region where the behavior of certain order parameters changes qualitatively, and is studied for its potential to host a continuum-limit phase transition.
The toroidal-topology study posted as arXiv:1802.10434 reruns the de Sitter emergence simulations with the topology of the lattice changed from the four-sphere to the four-torus. The result is that the four-dimensional emergence persists, with the macroscale geometry in the C phase still matching a de Sitter form modulo the topological difference. This is a nontrivial check because spherical and toroidal boundary conditions impose qualitatively different global constraints on the path integral, and a positive result on both topologies argues that the C-phase emergence is a property of the local dynamics rather than the global boundary.
Quantum fluctuations around the average de Sitter profile encode information about the effective action of the emergent theory. Fitting these fluctuations to a minisuperspace ansatz yields an effective cosmological action whose form is consistent with general relativity at leading order, with corrections suppressed at large scales. This is one of the closest contact points between CDT's nonperturbative output and the perturbative expectations of effective field theory; the agreement is not yet a derivation of general relativity from CDT but is treated in the literature as evidence that the C-phase semiclassical limit is consistent with continuum gravity.
References
- EstablishedAmbjorn, J., Jurkiewicz, J., Loll, R. (2004). 'Emergence of a 4D World from Causal Quantum Gravity.' Physical Review Letters 93, 131301
- EstablishedAmbjorn, J., Jurkiewicz, J., Loll, R. (2005). 'Reconstructing the universe.' Physical Review D 72, 064014
- EstablishedAmbjorn, J., Gorlich, A., Jurkiewicz, J., Loll, R. (2008). 'Planckian Birth of the Quantum de Sitter Universe.' Physical Review Letters 100, 091304
- EstablishedAmbjorn, J., Gorlich, A., Jurkiewicz, J., Loll, R. (2008). 'The Nonperturbative Quantum de Sitter Universe.' Physical Review D 78, 063544
- EstablishedAmbjorn, J., Gizbert-Studnicki, J., Gorlich, A., Jurkiewicz, J., Klitgaard, N. (2017). 'Characteristics of the new phase in CDT.' European Physical Journal C 77, 152
- EstablishedAmbjorn, J., Gizbert-Studnicki, J., Gorlich, A., Jurkiewicz, J., Nemeth, D. (2018). 'The phase structure of Causal Dynamical Triangulations with toroidal spatial topology.' JHEP 06, 111
Last reviewed May 19, 2026
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