Compare · The Nature of Space & Time

Emergent Universe, Phase Structure and de Sitter vs Locally Causal CDT and Matter Coupling

← Back to Emergent Universe, Phase Structure and de Sitter

Causal Dynamical Triangulation· within family

	Emergent Universe, Phase Structure and de Sitter Frontier	Locally Causal CDT and Matter Coupling Frontier
Proposed	2004	2013
Key figures	Jan Ambjorn, Jerzy Jurkiewicz, Renate Loll, Andrzej Gorlich	Steven Jordan, Renate Loll, Ben Ruijl
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.	Jordan and Loll demonstrated in 2013 that the causality constraint that makes CDT work can be enforced locally rather than globally, removing the preferred-foliation construction that is the family's central foundational concern. The Locally Causal Dynamical Triangulation extension is the active 2020-2026 research frontier of the program, alongside matter-coupled CDT and the continuum-limit search.
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 causality constraint that makes the CDT path integral well-defined can be enforced locally rather than globally, recovering the original construction as a special case while permitting a wider class of triangulated geometries. The local construction is computationally much more expensive than the global one; most LCDT results to date are in two dimensions, with three-dimensional investigations active and four-dimensional results still out of reach at the precision needed for full phase-structure mapping. Matter-coupled CDT modifies the phase diagram in computable ways, with the modification dependent on the matter content (scalar, gauge, fermion) and the matter-matter and matter-gravity couplings. The pattern of modification is a direct probe of the gravitational degrees of freedom in the presence of matter and is the most direct CDT-side contact with Asymptotic Safety's matter-coupled fixed-point calculations.
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.	Most of the post-2013 LCDT literature lives in two dimensions because the local-causality construction is computationally much more expensive than the global one. Four-dimensional LCDT, where the program's most important questions live, remains computationally limited. Critics argue that the LCDT response to the preferred-foliation concern is real in principle but not yet realized in the dimension that matters. Matter coupling at present requires simplified matter sectors (single scalar fields, abelian gauge fields, fermions in restricted regimes) because of the same computational cost. Realistic Standard Model matter content coupled to CDT is beyond current resources and may require substantial algorithmic innovation rather than just more compute time. The continuum-limit search has identified candidate second-order phase transitions but has not closed the rigorous argument that the CDT lattice theory defines a continuum quantum field theory of gravity. Without this the LCDT and matter-coupling extensions remain extensions of a lattice construction whose long-distance status is not yet settled. The cross-program contact with Asymptotic Safety is qualitative at present rather than quantitative. CDT and Asymptotic Safety see similar dimensional reduction at short scales and similar features of matter-coupling, but a direct numerical match between the discrete and continuum approaches is still work in progress.
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 too-slow-to-compute problem: the upgraded version that drops any built-in time direction, locally causal CDT, is so demanding to simulate that no one can yet run it in the full four dimensions where the theory must finally be tested.
Reader vote	No votes yet	No votes yet

Emergent Universe, Phase Structure and de Sitter

2004 · Frontier

Locally Causal CDT and Matter Coupling

2013 · Frontier

Proposed

2004

2013

Key figures

Jan Ambjorn, Jerzy Jurkiewicz, Renate Loll, Andrzej Gorlich

Steven Jordan, Renate Loll, Ben Ruijl

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.

Jordan and Loll demonstrated in 2013 that the causality constraint that makes CDT work can be enforced locally rather than globally, removing the preferred-foliation construction that is the family's central foundational concern. The Locally Causal Dynamical Triangulation extension is the active 2020-2026 research frontier of the program, alongside matter-coupled CDT and the continuum-limit search.

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 causality constraint that makes the CDT path integral well-defined can be enforced locally rather than globally, recovering the original construction as a special case while permitting a wider class of triangulated geometries.
The local construction is computationally much more expensive than the global one; most LCDT results to date are in two dimensions, with three-dimensional investigations active and four-dimensional results still out of reach at the precision needed for full phase-structure mapping.
Matter-coupled CDT modifies the phase diagram in computable ways, with the modification dependent on the matter content (scalar, gauge, fermion) and the matter-matter and matter-gravity couplings. The pattern of modification is a direct probe of the gravitational degrees of freedom in the presence of matter and is the most direct CDT-side contact with Asymptotic Safety's matter-coupled fixed-point calculations.

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.

Most of the post-2013 LCDT literature lives in two dimensions because the local-causality construction is computationally much more expensive than the global one. Four-dimensional LCDT, where the program's most important questions live, remains computationally limited. Critics argue that the LCDT response to the preferred-foliation concern is real in principle but not yet realized in the dimension that matters.
Matter coupling at present requires simplified matter sectors (single scalar fields, abelian gauge fields, fermions in restricted regimes) because of the same computational cost. Realistic Standard Model matter content coupled to CDT is beyond current resources and may require substantial algorithmic innovation rather than just more compute time.
The continuum-limit search has identified candidate second-order phase transitions but has not closed the rigorous argument that the CDT lattice theory defines a continuum quantum field theory of gravity. Without this the LCDT and matter-coupling extensions remain extensions of a lattice construction whose long-distance status is not yet settled.
The cross-program contact with Asymptotic Safety is qualitative at present rather than quantitative. CDT and Asymptotic Safety see similar dimensional reduction at short scales and similar features of matter-coupling, but a direct numerical match between the discrete and continuum approaches is still work in progress.

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 too-slow-to-compute problem: the upgraded version that drops any built-in time direction, locally causal CDT, is so demanding to simulate that no one can yet run it in the full four dimensions where the theory must finally be tested.

Reader vote

No votes yet