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Emergent Universe, Phase Structure and de Sitter vs Foundational CDT Program

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Causal Dynamical Triangulation· within family

	Emergent Universe, Phase Structure and de Sitter Frontier	Foundational CDT Program Frontier
Proposed	2004	1998
Key figures	Jan Ambjorn, Jerzy Jurkiewicz, Renate Loll, Andrzej Gorlich	Jan Ambjorn, Renate Loll
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.	Ambjorn and Loll proposed in 1998 that the gravitational path integral can be defined nonperturbatively by summing over discrete Lorentzian geometries built from simplices, with one structural rule: every building block must agree on which direction is the past and which is the future. The causal-foliation constraint is what rescued lattice quantum gravity from the pathological geometries that defeated earlier Euclidean approaches.
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.	The gravitational path integral can be defined nonperturbatively on a lattice without picking a background metric, by summing over all causal triangulations weighted by the Regge action. Imposing a global causal ordering on the simplicial geometries suppresses the crumpled and branched-polymer geometries that dominate the Euclidean Dynamical Triangulation path integral, allowing a smooth four-dimensional phase to exist. The continuum limit of the lattice theory, if it exists, should approach a true renormalized quantum field theory of gravity, possibly connected to the ultraviolet fixed point of the Asymptotic Safety program.
Where it breaks	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.	The causal-foliation construction picks a global time direction at the microscopic level. Critics argue this breaks full background independence and is at odds with the diffeomorphism invariance that any quantum theory of gravity should preserve. The Locally Causal Dynamical Triangulation extension is the program's response, but most published LCDT work is in lower dimensions because the local-causality construction is computationally much more expensive than the global one. The continuum limit of the lattice theory has not been rigorously established. Without it, the program's results are statements about finite-size simulations rather than about a true continuum quantum theory of gravity. Recent work on phase transitions in the C-to-bifurcation region is encouraging but does not yet close the argument. The bare action contains only the Einstein-Hilbert term and a [[cosmological constant]]. Whether the results survive the inclusion of higher-derivative operators or matter sectors is the subject of the program's ongoing extensions.
Key unresolved problem	The wrong-clock problem: the matching emerging universe is computed in a math-friendly version of time where time behaves like space, and whether the result carries over to real flowing time, the Lorentzian signature, is unproven.	The zoom-out problem: the results come from a finite grid of building blocks, and no one has proven that shrinking the blocks to nothing, the continuum limit, yields a genuine quantum theory of gravity rather than a grid artifact.
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Emergent Universe, Phase Structure and de Sitter

2004 · Frontier

Foundational CDT Program

1998 · Frontier

Proposed

2004

1998

Key figures

Jan Ambjorn, Jerzy Jurkiewicz, Renate Loll, Andrzej Gorlich

Jan Ambjorn, Renate Loll

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.

Ambjorn and Loll proposed in 1998 that the gravitational path integral can be defined nonperturbatively by summing over discrete Lorentzian geometries built from simplices, with one structural rule: every building block must agree on which direction is the past and which is the future. The causal-foliation constraint is what rescued lattice quantum gravity from the pathological geometries that defeated earlier Euclidean approaches.

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.

The gravitational path integral can be defined nonperturbatively on a lattice without picking a background metric, by summing over all causal triangulations weighted by the Regge action.
Imposing a global causal ordering on the simplicial geometries suppresses the crumpled and branched-polymer geometries that dominate the Euclidean Dynamical Triangulation path integral, allowing a smooth four-dimensional phase to exist.
The continuum limit of the lattice theory, if it exists, should approach a true renormalized quantum field theory of gravity, possibly connected to the ultraviolet fixed point of the Asymptotic Safety program.

Where it breaks

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.

The causal-foliation construction picks a global time direction at the microscopic level. Critics argue this breaks full background independence and is at odds with the diffeomorphism invariance that any quantum theory of gravity should preserve. The Locally Causal Dynamical Triangulation extension is the program's response, but most published LCDT work is in lower dimensions because the local-causality construction is computationally much more expensive than the global one.
The continuum limit of the lattice theory has not been rigorously established. Without it, the program's results are statements about finite-size simulations rather than about a true continuum quantum theory of gravity. Recent work on phase transitions in the C-to-bifurcation region is encouraging but does not yet close the argument.
The bare action contains only the Einstein-Hilbert term and a [[cosmological constant]]. Whether the results survive the inclusion of higher-derivative operators or matter sectors is the subject of the program's ongoing extensions.

Key unresolved problem

The wrong-clock problem: the matching emerging universe is computed in a math-friendly version of time where time behaves like space, and whether the result carries over to real flowing time, the Lorentzian signature, is unproven.

The zoom-out problem: the results come from a finite grid of building blocks, and no one has proven that shrinking the blocks to nothing, the continuum limit, yields a genuine quantum theory of gravity rather than a grid artifact.

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