arXiv subjects

Steffen Gielen

Publications and source records attributed to Steffen Gielen.

At least 19 recordsLinked to original sources

arXiv2026

Inflation in unimodular loop quantum cosmology

We study inflation in the setting of unimodular loop quantum cosmology, where time evolution is defined in unimodular time rather than with respect to a free, massless scalar field as is standard in loop quantum cosmology. The unimodular setting leads to a natural Schrödinger time evolution in a time coordinate with clear geometric meaning, defined independently of any particular matter content; an inflaton can be included but is not needed as a clock. We review the unimodular version of loop quantum cosmology and comment on possible connections to full (unimodular) loop quantum gravity. Then, focusing on semiclassical effective equations, we derive analytical solutions in simple cases such as a constant potential, emphasising the use of a unimodular time coordinate. We also discuss numerical solutions for phenomenologically interesting cases such as a quadratic potential and Starobinsky inflation, comparing different possible choices of initial conditions. In particular, we show that choosing an $α$-attractor potential allows for models of a bounce either dominated by kinetic or potential energy, which are compatible with observations while potentially including observable imprints of the quantum-gravity regime.

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arXiv2026

Boundary conditions and Hilbert spaces in no-roll quantum cosmology

We construct Hilbert spaces for the minisuperspace quantum cosmology of a closed Universe in the limit of extreme slow-roll inflation, in which the scalar field is approximated as constant. In this setting, the potential energy in the scalar field is an integration constant depending on initial conditions, equivalent to the cosmological constant as it appears in unimodular gravity. If one fixes the value of this integration constant, the Wheeler-DeWitt equation admits two independent solutions, and a natural inner product picks out one of them (essentially Vilenkin's tunnelling wavefunction) with positive norm. The physical Hilbert space is then one-dimensional, in agreement with some recent discussions of closed universes in quantum gravity. However, if the potential energy is left arbitrary, the theory allows for an infinite-dimensional Hilbert space corresponding to energy eigenstates of an effective Hamiltonian. Requiring that this Hamiltonian be represented as a self-adjoint operator leads to a one-parameter family of boundary conditions at the singularity, generalising the DeWitt criterion of a vanishing wavefunction. The boundary condition always leads to a mixture of Hartle-Hawking (no-boundary) and tunnelling wavefunctions, but a particular choice "almost" singles out the Hartle-Hawking wavefunction, with exponentially suppressed corrections.

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arXiv2026

Cyclic Kruskal Universe: a quantum-corrected Schwarzschild black hole in unitary unimodular gravity

We analyse the physical properties of an analytical, nonsingular quantum-corrected black hole solution recently derived in a minisuperspace model for unimodular gravity under the assumption of unitarity in unimodular time. We show that the metric corrections compared to the classical Schwarzschild solutions only depend on a single new parameter, corresponding to a minimal radius where a black hole-white hole transition occurs. While these corrections substantially alter the structure of the spacetime near this minimal radius, they fall off rapidly towards infinity, and we show in various examples how physical properties of the exterior spacetime are very close to those of the Schwarzschild solution. We derive the maximal analytic extension of the initial solution, which corresponds to an infinite sequence of Kruskal spacetimes connected via black-to-white hole transitions, and compare with some other proposals for non-singular black hole metrics. The metric violates the achronal averaged null energy condition, which indicates that we are capturing physics beyond the semiclassical approximation. Finally, we include some thoughts on how to go beyond the simple eternal black hole-white hole model presented here.

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arXiv2025

Bouncing Bianchi Models with Deformed Commutation Relations

We study the anisotropic Bianchi I and Bianchi II models in vacuum in the framework of deformed commutation relations (DCRs). Working in a parametrisation of the spatial metric by a volume and two anisotropy variables, we propose modified Poisson brackets that for the volume alone reproduce the bounce dynamics of effective loop quantum cosmology (LQC), with additional modifications for anisotropy degrees of freedom. We derive effective Friedmann equations and observe cosmological bounces both in Bianchi I and Bianchi II. For Bianchi II, we find that the cosmological bounce now interacts with the usual reflection seen in the Kasner indices in various interesting ways, in close similarity again with what had been seen in LQC. This suggests that the DCR framework could model more general quantum-gravity inspired bounce scenarios in a relatively straightforward way.

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arXiv2025

Cosmological scalar perturbations for a metric reconstructed from group field theory

While homogeneous cosmologies have long been studied in the group field theory (GFT) approach to quantum gravity, including a quantum description of cosmological perturbations is highly non-trivial. Here we apply a recent proposal for reconstructing an effective spacetime metric in GFT to the case of a metric with small inhomogeneities over a homogeneous background. We detail the procedure and give general expressions for cosmological scalar perturbations defined in terms of the GFT energy-momentum tensor. These include all the scalar components of standard perturbation theory and hence can be used to define gauge-invariant quantities. This is a major advantage of the effective metric approach compared to previous GFT studies limited to volume perturbations. We compute these perturbations explicitly for a particular Fock coherent state. While it was previously shown that such a state can be interpreted as an approximately flat homogeneous cosmology at late times, here we find that, in a very simple example, inhomogeneities do not follow the dynamics of general relativity in the semiclassical regime. More specifically, restricting ourselves to a specific coherent state in a simple (free) GFT, we study two types of perturbative GFT modes, squeezed and oscillating modes. For squeezed modes we find perturbation equations with Euclidean signature and a late-time limit that differs from general relativistic perturbation equations. Oscillating modes satisfy different dynamical equations that also differ from those of general relativity, but show a Lorentzian signature. Considering that our results were obtained within a number of simplifying assumptions [...], we discuss how going beyond these assumptions could lead to a more desirable phenomenology. Overall, our analysis should be understood as a first step in understanding cosmological perturbations within the effective GFT metric.

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arXiv2025

Quantum Schwarzschild-(A)dS Black Holes: Unitarity and Singularity Resolution

We consider the canonical quantisation of spherically symmetric spacetimes within unimodular gravity, leaving sign choices in the metric general enough to include both the interior and exterior Schwarzschild-(Anti-)de Sitter spacetime. In unimodular gravity the cosmological constant appears as an integration constant analogous to a total energy, and the quantum Wheeler-DeWitt equation takes the form of a Schrödinger equation in unimodular time. We discuss self-adjoint extensions of the Schrödinger-like Hamiltonian arising from the requirement of unitarity in unimodular time, and identify a physically motivated one-parameter family of extensions. For semiclassical states we are able to derive analytical expressions for expectation values of the metric, representing a quantum-corrected, nonsingular extension of the classical Schwarzschild-(A)dS geometry which describes a quantum transition between asymptotic black hole and white hole states. The sign of the self-adjoint extension parameter corresponds to the allowed sign of the black hole/white hole mass, and so it can be chosen to ensure that this mass is always positive. We also discuss tunnelling states which allow for a change in the sign of the mass, but which are not semiclassical in high-curvature regions. Our mechanism for singularity resolution and the explicit form of the quantum-corrected metric can be compared to other proposals for black holes in quantum gravity, and in the asymptotically AdS case can be contrasted with holographic arguments.

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arXiv2025

Hilbert space formalisms for group field theory

Group field theory is a background-independent approach to quantum gravity whose starting point is the definition of a quantum field theory on an auxiliary group manifold (not interpreted as spacetime, but rather as the finite-dimensional configuration space of a single "atom" of geometry). Group field theory models can be seen as an extension of matrix and tensor models by additional data, and are traditionally defined through a functional integral whose perturbative expansion generates a sum over discrete geometries. More recently, some efforts have been directed towards formulations of group field theory based on a Hilbert space and operators, in particular in applications to cosmology. This is an attempt to review some of these formulations and their main ideas, to disentangle these constructions as much as possible from applications and phenomenology, and to put them into a wider context of quantum gravity research.

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arXiv2025

Low-curvature quantum corrections from unitary evolution of de Sitter space

We study the quantum dynamics of de Sitter space formulated as a minisuperspace model with flat spatial hypersurfaces in unimodular gravity, both in the Wheeler-DeWitt approach and in loop quantum cosmology (LQC). Time evolution is defined naturally in unimodular time, which appears as conjugate to the cosmological (integration) constant. We show that requiring unitary time evolution "resolves" the de Sitter horizon where the flat slicing breaks down and leads to strong quantum effects there, even though locally nothing special happens at this surface. For a cosmological constant that is far below the Planck scale, loop quantum gravity corrections do not alter the main results in any substantial way. This model illustrates the fundamental clash between general covariance and unitarity in quantum gravity.

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arXiv2025

Canonical analysis of unimodular Plebański gravity

We present the canonical analysis of different versions of unimodular gravity defined in the Plebański formalism, based on a (generally complex) SO(3) spin connection and set of (self-dual) two-forms. As in the metric formulation of unimodular gravity, one can study either a theory with fixed volume form or work in a parametrised formalism in which the cosmological constant becomes a dynamical field, constrained to be constant by the field equations. In the first case, the Hamiltonian density contains a part which is not constrained to vanish, but rather constrained to be constant, again as in the metric formulation. We also discuss reality conditions and challenges in extracting Lorentzian solutions.

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arXiv2025

Black hole singularity resolution in unimodular gravity from unitarity

We study the quantum dynamics of an interior planar AdS (anti-de Sitter) black hole, requiring unitarity in the natural time coordinate conjugate to the cosmological "constant of motion" appearing in unimodular gravity. Both the classical singularity and the horizon are replaced by a non-singular highly quantum region; semiclassical notions of spacetime evolution are only valid in an intermediate region. For the singularity, our results should be applicable to general black holes: unitarity in unimodular time always implies singularity resolution.

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arXiv2025

Relational dynamics and Page-Wootters formalism in group field theory

Group field theory posits that spacetime is emergent and is hence defined without any background notion of space or time; dynamical questions are formulated in relational terms, in particular using (scalar) matter degrees of freedom as time. Unlike in canonical quantisation of gravitational systems, there is no obvious notion of coordinate transformations or constraints, and established quantisation methods cannot be directly applied. As a result, different canonical formalisms for group field theory have been discussed in the literature. We address these issues using a parametrised version of group field theory, in which all (geometry and matter) degrees of freedom evolve in a fiducial parameter. There is a constraint associated to the freedom of reparametrisation and the Dirac quantisation programme can be implemented. Using the "trinity of relational dynamics", we show that the resulting "clock-neutral" theory is entirely equivalent to a deparametrised canonical group field theory, interpreted in terms of the Page-Wootters formalism. Our results not only show that the deparametrised quantisation is fully covariant and can be seen as encoding the dynamics of joint quantum matter and geometry degrees of freedom, they also appear to be the first application of the Page-Wootters formalism directly to non-perturbative quantum gravity. We show extensions to a setting in which many independent gauge symmetries are introduced, which connects to the "multi-fingered time" idea in quantum gravity and provides a somewhat novel extension of the Page-Wootters formalism.

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arXiv2024

Reconstructing the metric in group field theory

We study a group field theory (GFT) for quantum gravity coupled to four massless scalar fields, using these matter fields to define a (relational) coordinate system. We exploit symmetries of the GFT action, in particular under shifts in the values of the scalar fields, to derive a set of classically conserved currents, and show that the same conservation laws hold exactly at the quantum level regardless of the choice of state. We propose a natural interpretation of the conserved currents which implies that the matter fields always satisfy the Klein--Gordon equation in GFT. We then observe that in our matter reference frame, the same conserved currents can be used to extract all components of an effective GFT spacetime metric. Finally, we apply this construction to the simple example of a spatially flat homogeneous and isotropic universe, where we derive an effective Friedmann equation directly from this metric. The Friedmann equation displays a bounce and a late-time limit equivalent to general relativity with a single scalar field. Our proposal goes substantially beyond the GFT literature in which only specific geometric quantities such as the total volume or volume perturbations could be defined, opening up the possibility to study more general geometries as emerging from GFT.

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arXiv2024

Quantum Cosmology

Invited contribution to the Encyclopedia of Mathematical Physics (2nd edition), providing an overview over some main ideas and results in quantum cosmology. Key points: Canonical quantisation of homogeneous, isotropic cosmology; discussion of ambiguities in this quantisation; Construction of explicit solutions, attempts at physical interpretation; conceptual issues (time evolution, unitarity, probability interpretation); Introduction of scalar fields or perfect fluids; Discussion of whether classical singularities are resolved; Addition of inhomogeneities, possible predictions for primordial cosmology; Connections to string theory and loop quantum gravity.

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arXiv2024

Generalised Gaussian states in group field theory and $\mathfrak{su(1,1)}$ quantum cosmology

We define generalised Gaussian states for quantum cosmological models based on the $\mathfrak{su(1,1)}$ algebra, with particular emphasis on its realisation in group field theory for a single field mode, and study their semiclassical properties. These states are generalisations of coherent, squeezed and thermal states considered previously. As two possible characterisations of semiclassicality, we contrast the requirement of small relative fluctuations in volume and energy with the saturation of the Robertson--Schrödinger uncertainty principle. We find that for the most general class of states the appearance of small relative fluctuations, which we take as the main criterion relevant for the emergence of cosmology, is mostly determined by the amount of displacement used to define the state. We also observe that defining such generalised Gaussian states is less straightforward in the algebraic approach to canonical quantisation of group field theory, and discuss special cases.

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arXiv2024

Unimodular Plebański Gravity

We present new action principles for unimodular gravity, defined in the chiral Plebański formulation based on (complex) two-forms and a complex ${\rm SO}(3)$ connection. In these theories, just as in their analogues in the metric formulation, the cosmological constant does not take a prescribed value but is an integration constant whose value can differ between different (classical) solutions. We discuss some subtleties when identifying Lorentzian solutions in the generally complex theory, and show how these theories can be reduced to a ``pure connection'' form similar to Krasnov's pure connection formalism for general relativity.

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arXiv2023

Overall signature of the metric and the cosmological constant

We consider a little known aspect of signature change, where the overall sign of the metric is allowed to change, with physical implications. We show how, in different formulations of general relativity, this type of classical signature change across boundaries with a degenerate metric can be made consistent with a change in sign (and value) of the cosmological constant $Λ$. In particular, the separate "mostly plus" and "mostly minus" signature sectors of Lorentzian gravity are most naturally associated with different signs of $Λ$. We show how this general phenomenon allows for classical solutions where the open dS patch can arise from a portion of AdS space time. These can be interpreted as classical "imaginary space" extensions of the usual Lorentzian theory, with $a^2<0$.

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arXiv2023

Quantum cosmology of pure connection general relativity

We study homogeneous cosmological models in formulations of general relativity with cosmological constant based on a (complexified) connection rather than a spacetime metric, in particular in a first order theory obtained by integrating out the self-dual two-forms in the chiral Plebanski formulation. Classical dynamics for the Bianchi IX model are studied in the Lagrangian and Hamiltonian formalism, where we emphasise the reality conditions needed to obtain real Lorentzian solutions. The solutions to these reality conditions fall into different branches, which in turn lead to different real Hamiltonian theories, only one of which is the usual Lorentzian Bianchi IX model. We also show the simpler case of the flat Bianchi I model, for which both the reality conditions and dynamical equations simplify considerably. We discuss the relation of a real Euclidean version of the same theory to this complex theory. Finally, we study the quantum theory of homogeneous and isotropic models, for which the pure connection action for general relativity reduces to a pure boundary term and the path integral is evaluated immediately, reproducing known results in quantum cosmology. An intriguing aspect of these theories is that the signature of the effective spacetime metric, and hence the interpretation of the cosmological constant, are intrinsically ambiguous.

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arXiv2023

Towards anisotropic cosmology in group field theory

In cosmological group field theory (GFT) models for quantum gravity coupled to a massless scalar field the total volume, seen as a function of the scalar field, follows the classical Friedmann dynamics of a flat Friedmann-Lemaître-Robertson-Walker (FLRW) Universe at low energies while resolving the Big Bang singularity at high energies. An open question is how to generalise these results to other homogeneous cosmologies. Here we take the first steps towards studying anisotropic Bianchi models in GFT, based on the introduction of a new anisotropy observable analogous to the $β$ variables in Misner's parametrisation. In a classical Bianchi I spacetime, $β$ behaves as a massless scalar field and can be used as a (gravitational) relational clock. We construct a GFT model for which in an expanding Universe $β$ initially behaves like its classical analogue before "decaying" showing a previously studied isotropisation. We support numerical results in GFT by analytical approximations in a toy model. One possible outcome of our work is a definition of relational dynamics in GFT that does not require matter.

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