We propose a revised formulation of General Relativity for cosmological settings, in which the Einstein constant varies with the energy density of the Universe. We demonstrate that this modification has only phenomenological impact of providing an effective dark energy density expression. Assuming a state close to vacuum, here defined by the vanishing product of the Einstein coupling constant and the Universe’s energy density, we perform a Taylor expansion of the theory and hence extend it to the whole domain. In this framework, the (renormalized) vacuum energy problem is studied, and an additional constant pressure term, which induces a Chaplygin-like contribution to the dark energy sector, arises in the late-time dynamics. The correction to the late-time Hubble parameter is investigated by comparing theoretical predictions with the late Universe observational data. Our findings indicate that the current value of the stated vacuum energy is consistent with zero within 1 $$\sigma $$ . Implications of the modified $$\Lambda $$ CDM model with respect to the Hubble tension are also discussed.

Italy

Giovanni Montani

Giulia Maniccia

Elisa Fazzari

Alessandro Melchiorri

Running Einstein constant and a possible vacuum state of the universe

In this work, we show that it is possible to define a classical system associated with a Generalized Uncertainty Principle (GUP) theory via the implementation of a consistent symplectic structure. This provides a solid framework for the classical Hamiltonian formulation of such theories and the study of the dynamics of physical systems in the corresponding deformed phase space.By further characterizing the functions that govern non-commutativity in the configuration space using the algebra of angular momentum, we determine a general form for the rotation generator in these theories and crucially, we show that, under these conditions, unlike what has been previously found in the literature at the quantum level, this requirement does not lead to the superselection of GUP models at the classical level.Finally, we postulate that a properly defined GUP theory can be correctly interpreted classically if and only if the corresponding quantum commutators satisfy the Jacobi identities, identifying those quantization prescriptions for which this holds true.

Matteo Bruno

Sebastiano Segreto

Generalized uncertainty principle theories and their classical interpretation

Gabriele Barca

Erratum: Non-singular gravitational collapse through modified Heisenberg algebra

We investigate the emergence of a quantum Big-Bounce in the context of an isotropic Universe, filled by a self-interacting scalar field, which plays the role of a physical clock. The bouncing cosmology is the result of a scattering process, driven by the scalar field potential, which presence breaks down the frequency separation of the Wheeler-DeWitt equation, treated in strict analogy to a relativistic quantum system. Differently from previous analyses, we consider a really perturbative self-interaction potential, affecting the dynamics in a finite range of the time labeled by the scalar clock (and in particular we remove the divergent character previously allowed). The main result of the present analysis is that, when the Relativistic Quantum Mechanics formalism is properly implemented in the Mini-superspace analogy, the probability amplitude for the bounce is, both in the standard and polymerized case, characterized by a maximum in correspondence of the quasi-classical condition of a Universe minimum volume.

Simone Lo Franco

Quantum Big-Bounce as a phenomenology of RQM in the Mini-superspace

We analyse a n-dimensional Generalized Uncertainty Principle (GUP) quantization framework, characterized by a non-commutative nature of the configurational variables. First, we identify a set of states which are maximally localized only along a single direction, at the expense of being less localized in all the other ones. Subsequently, in order to recover information about localization on the whole configuration space, we use the only state of the theory which exhibits maximal localization simultaneously in every direction to construct a satisfactory quasi-position representation, by virtue of a suitable translational operator. The resultant quantum framework is then applied to model the dynamics of the Bianchi I cosmology. The corresponding Wheeler–DeWitt equation is reduced to Schrödinger dynamics for the two anisotropy degrees of freedom, using a WKB representation for the volume-like variable of the Universe, in accordance with the Vilenkin scenario. The main result of our cosmological implementation of the constructed quantum theory demonstrates how the dynamics of a wave packet peaked at some point in the configuration space represented in the quasi-position variables favours as the most probable configuration exactly the initial one for a relatively long time, if compared with the ordinary quantum theory. This preference arises from the different dynamical behavior exhibited by wave packets in the two quantum theories.

n-Dimensional non-commutative GUP quantization and application to the Bianchi I model

We analyze inhomogeneous cosmological models in the local Universe, described by the Lemaître–Tolman–Bondi (LTB) metric and developed using linear perturbation theory on a homogeneous and isotropic Universe background. Focusing on the different evolution of spherical symmetric inhomogeneities, we compare the  $$\Lambda $$  <math> <mi>Λ</mi> </math>  LTB model, in which the cosmological constant  $$\Lambda $$  <math> <mi>Λ</mi> </math>   is included in the LTB formalism, with inhomogeneous cosmological models based on  $$f\left( R\right) $$  <math> <mrow> <mi>f</mi> <mfenced> <mi>R</mi> </mfenced> </mrow> </math>   modified gravity theories viewed in the Jordan frame. We solve the system of field equations for both inhomogeneous cosmological models adopting the method of separation of variables: we integrate analytically the radial profiles of local perturbations, while their time evolution requires a numerical approach. The main result of the analysis concerns the different radial profiles of local inhomogeneities due to the presence of a non-minimally coupled scalar field in the Jordan frame of  $$f\left( R\right) $$  <math> <mrow> <mi>f</mi> <mfenced> <mi>R</mi> </mfenced> </mrow> </math>   gravity. While radial perturbations follow a power-law in the  $$\Lambda $$  <math> <mi>Λ</mi> </math>  LTB model, Yukawa-like contributions appear in the  $$f\left( R\right) $$  <math> <mrow> <mi>f</mi> <mfenced> <mi>R</mi> </mfenced> </mrow> </math>   theory. Interestingly, this latter peculiar behavior of radial profile is not affected by the choice of the  $$f\left( R\right) $$  <math> <mrow> <mi>f</mi> <mfenced> <mi>R</mi> </mfenced> </mrow> </math>   functional form. The numerical solution of time-dependent perturbations exhibits a non-diverging profile. This work suggests that investigations about local inhomogeneities in the late Universe may allow us to discriminate if the present cosmic acceleration is caused by a cosmological constant term or a modified gravity effect.

Portugal

Tiziano Schiavone

Signature of f(R) gravity via Lemaître–Tolman–Bondi inhomogeneous perturbations

We study the effects of cut-off physics, in the form of a modified algebra inspired by Polymer Quantum Mechanics and by the Generalized Uncertainty Principle representation, on the collapse of a spherical dust cloud. We analyze both the Newtonian formulation, originally developed by Hunter, and the general relativistic formulation, that is the Oppenheimer–Snyder model; in both frameworks we find that the collapse is stabilized to an asymptotically static state above the horizon, and the singularity is removed. In the Newtonian case, by requiring the Newtonian approximation to be valid, we find lower bounds of the order of unity (in Planck units) for the deformation parameter of the modified algebra. We then study the behaviour of small perturbations on the non-singular collapsing backgrounds, and find that for certain range of the parameters (the polytropic index for the Newtonian case and the sound velocity in the relativistic setting) the collapse is stable to perturbations of all scales, and the non-singular super-Schwarzschild configurations have physical meaning.

Non-singular gravitational collapse through modified Heisenberg algebra

We investigate the quantum dynamics of the quasi-isotropic inflationary solution. This is achieved by deriving the Lagrangian and Hamiltonian for both the FLRW background and the inhomogeneous correction, via an expansion of the Einstein–Hilbert action up to second order in the perturbation amplitudes. Then we implement a semiclassical WKB scenario for which the inhomogeneous component of the Universe is treated as a “small” quantum subsystem, evolving on the classical isotropic background. Starting from the Wheeler–DeWitt equation, we recover a Schrödinger dynamics for the perturbations, in which the time dependence of the wave function emerges thanks to the classicality of the background, and we solve it for an inflationary phase. The main result of this paper is to show that, while the scalar component of the power spectrum has the standard scale invariant profile, the tensor one results to be not constrained by the inflationary expansion, apart from an overall normalization factor which guarantees a small tensor-to-scalar ratio. This means that the spatial distribution of the quasi-isotropic correction to the metric remains preserved, with the consequence that some information about the pre-inflationary Universe survives to the de Sitter expansion.

Nicola Bortolotti

Inflationary quantum spectrum of the quasi-isotropic Universe

We study the implementation of Polymer Quantum Mechanics (PQM) to a system decomposed into a quasi-classical background and a small quantum subsystem, according to the original Vilenkin proposal. We develop the whole formalism in the momentum representation that is the only viable in the continuum limit of the polymer paradigm and we generalize the fundamental equations of the original Vilenkin analysis in the considered context. Then, we provide a Minisuperspace application of the theory, first considering a Bianchi I cosmology and then extending the analysis to a Bianchi IX model in the limit of small anisotropies. In both these cases, the quasi-classical background is identified with an isotropic bouncing Universe whereas the small quantum subsystem contains the anisotropic degrees of freedom. When the Big Bounce scenario is considered, we obtain that in the Bianchi I model the anisotropies standard deviation is regular at  $$t=0$$  <math> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </math>   but still increases indefinitely, whereas in the presence of the harmonic Bianchi IX potential such same quantity is bounded and oscillate around a constant value. As a consequence, we demonstrate that the picture of a semiclassical isotropic Bounce can be extended to more general cosmological settings if the spatial curvature becomes relevant when the anisotropic degrees of freedom are still small quantum variables.

France

Eleonora Giovannetti

The role of spatial curvature in constraining the Universe anisotropies across a Big Bounce

We investigate the well-known phenomenon of the beam–plasma instability in the gravitational sector when a fast population of particles interacts with the massive scalar mode of a Horndeski theory of gravity, resulting in linear growth of the latter amplitude. Following the approach used in the standard electromagnetic case, we start from the dielectric representation of the gravitational plasma, as introduced in a previous analysis of the Landau damping for the scalar Horndeski mode. We then set up the modified Vlasov–Einstein equation, using a Dirac delta function to describe the fast beam distribution. We thus provide an analytical expression for the dispersion relation, and we demonstrate the existence of a nonzero growth rate for the linear evolution of the Horndeski scalar mode. A numerical investigation is then performed with a trapezoidal beam distribution function, which confirms the analytical results and allows us to demonstrate how the growth rate decreases as the beam spread increases.