In our previous work (Zhang and Shu in Eur Phys J. C 84(3):256, 2024), we have explored quantum gravity induced entanglement of masses (QGEM) in curved spacetime, observing entanglement formation between particles moving along geodesics in a Schwarzschild spacetime background. We find that long interaction time induces entanglement, even for particles with microscopic mass, addressing decoherence concerns. In this work, we build upon our previous work (Zhang and Shu 2024) by extending our investigation to a time-dependent spacetime. Specifically, we explore the entanglement induced by the mutual gravitation of massive particles in the Friedmann–Lemaître–Robertson–Walker (FLRW) universe. With the help of the phase shift and the QGEM spectrum, our proposed scheme offers a potential method for observing the formation of entanglement caused by the quantum gravity of massive particles as they propagate in the FLRW universe. Consequently, it provides insights into the field of entanglement in cosmology.

China

Chi Zhang

United States

Fu-Wen Shu

Gravity-induced entanglement between two massive microscopic particles in curved spacetime: II. Friedmann–Lemaître–Robertson–Walker universe

For a spatially Friedmann–Lemaître–Robertson–Walker cosmology, we propose a multi-scalar field gravitational model. Specifically, we consider a two-scalar field cosmological model in which the kinetic components of the scalar fields establish a two-dimensional sphere of Lorentzian signature. For our Chiral-Quintom model we choose a mixed potential term  $$V\left( \phi ,\psi \right) =V_{0}e^{\lambda \phi }+U_{0}e^{\kappa \phi }\psi ^{\frac{1}{\sigma } }$$  <math> <mrow> <mi>V</mi> <mfenced> <mi>ϕ</mi> <mo>,</mo> <mi>ψ</mi> </mfenced> <mo>=</mo> <msub> <mi>V</mi> <mn>0</mn> </msub> <msup> <mi>e</mi> <mrow> <mi>λ</mi> <mi>ϕ</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <msup> <mi>e</mi> <mrow> <mi>κ</mi> <mi>ϕ</mi> </mrow> </msup> <msup> <mi>ψ</mi> <mfrac> <mn>1</mn> <mi>σ</mi> </mfrac> </msup> </mrow> </math>   and we investigate the asymptotic limits of the cosmological parameters. This model for  $$U_{0}=0$$  <math> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math>  , provides a generalization of the hyperbolic inflation where the equation of the state parameter can cross the phantom divide line. When  $$U_{0}\ne 0$$  <math> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>≠</mo> <mn>0</mn> </mrow> </math>   we observe that this cosmological model exhibits asymptotic solutions that encompass accelerated universes, big rip singularities, and dust-like solutions. Hence, this multi-scalar field model it can be regarded as as a dark energy unify model which describes a variety of asymptotic cosmological scenarios.

Chile

South Africa

Andronikos Paliathanasis

Unified dark energy from Chiral-Quintom model with a mixed potential in Friedmann–Lemaître–Robertson–Walker cosmology

High-energy vector meson photoproduction is an important tool for studying the partonic structure of matter at low Bjorken <math><mi>x</mi></math>. In the Good-Walker (GW) paradigm, the cross section <math><mrow><mi>d</mi><mi>σ</mi><mo>/</mo><mi>d</mi><mi>t</mi></mrow></math> for coherent production of vector mesons or other final states depends the average transverse distribution of gluons, while the incoherent cross section depends on fluctuations in the nuclear structure, due to variations in nucleon positions, and/or gluonic hot spots. However, predictions of the the GW paradigm seemingly conflict with data from multiple experiments which observe coherent production of vector mesons accompanied by nuclear excitation, or in peripheral relativistic heavy-ion collisions. These data are consistent with a simpler, semiclassical approach. I will discuss this contradiction and explore how and why GW fails. I will also contrast the significant differences in incoherent photoproduction on <math><mmultiscripts><mi>Au</mi><mprescripts></mprescripts><none></none><mn>197</mn></mmultiscripts></math> and <math><mmultiscripts><mi>Pb</mi><mprescripts></mprescripts><none></none><mn>208</mn></mmultiscripts></math> targets in the GW approach with the much smaller expected differences in their low-<math><mi>x</mi></math> gluon content.

Spencer R. Klein

Challenges to the Good-Walker paradigm in coherent and incoherent photoproduction

Genly Leon

Sebastián Cuéllar

Esteban González

Samuel Lepe

Claudio Michea

Alfredo Millano

Erratum to: Averaging generalized scalar field cosmologies II: locally rotationally symmetric Bianchi I and flat Friedmann–Lemaître–Robertson–Walker models

Erratum to: Averaging generalized scalar field cosmologies I: locally rotationally symmetric Bianchi III and open Friedmann–Lemaître–Robertson–Walker models

Erratum to: Averaging generalized scalar-field cosmologies III: Kantowski–Sachs and closed Friedmann–Lemaître–Robertson–Walker models

Scalar-field cosmologies with a generalized harmonic potential and matter with energy density  $$\rho _m$$  <math> <msub> <mi>ρ</mi> <mi>m</mi> </msub> </math>  , pressure  $$p_m$$  <math> <msub> <mi>p</mi> <mi>m</mi> </msub> </math>  , and barotropic equation of state (EoS)  $$p_m=(\gamma -1)\rho _m, \; \gamma \in [0,2]$$  <math> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>γ</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>ρ</mi> <mi>m</mi> </msub> <mo>,</mo> <mspace width="0.277778em"></mspace> <mi>γ</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>]</mo> </mrow> </mrow> </math>   in Kantowski–Sachs (KS) and closed Friedmann–Lemaître–Robertson–Walker (FLRW) metrics are investigated. We use methods from non-linear dynamical systems theory and averaging theory considering a time-dependent perturbation function D. We define a regular dynamical system over a compact phase space, obtaining global results. That is, for KS metric the global late-time attractors of full and time-averaged systems are two anisotropic contracting solutions, which are non-flat locally rotationally symmetric (LRS) Kasner and Taub (flat LRS Kasner) for  $$0\le \gamma \le 2$$  <math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math>  , and flat FLRW matter-dominated universe if  $$0\le \gamma \le \frac{2}{3}$$  <math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>≤</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </math>  . For closed FLRW metric late-time attractors of full and averaged systems are a flat matter-dominated FLRW universe for  $$0\le \gamma \le \frac{2}{3}$$  <math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>≤</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </math>   as in KS and Einstein–de Sitter solution for  $$0\le \gamma &lt;1$$  <math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math>  . Therefore, a time-averaged system determines future asymptotics of the full system. Also, oscillations entering the system through Klein–Gordon (KG) equation can be controlled and smoothed out when D goes monotonically to zero, and incidentally for the whole D-range for KS and closed FLRW (if  $$0\le \gamma &lt; 1$$  <math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math>  ) too. However, for  $$\gamma \ge 1$$  <math> <mrow> <mi>γ</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math>   closed FLRW solutions of the full system depart from the solutions of the averaged system as D is large. Our results are supported by numerical simulations.

Averaging generalized scalar-field cosmologies III: Kantowski–Sachs and closed Friedmann–Lemaître–Robertson–Walker models

Scalar field cosmologies with a generalized harmonic potential and a matter fluid with a barotropic equation of state (EoS) with barotropic index  $$\gamma $$  <math> <mi>γ</mi> </math>   for the locally rotationally symmetric (LRS) Bianchi I and flat Friedmann–Lemaître–Robertson–Walker (FLRW) metrics are investigated. Methods from the theory of averaging of nonlinear dynamical systems are used to prove that time-dependent systems and their corresponding time-averaged versions have the same late-time dynamics. Therefore, the simplest time-averaged system determines the future asymptotic behavior. Depending on the values of  $$\gamma $$  <math> <mi>γ</mi> </math>  , the late-time attractors of physical interests are flat quintessence dominated FLRW universe and Einstein-de Sitter solution. With this approach, the oscillations entering the system through the Klein–Gordon (KG) equation can be controlled and smoothed out as the Hubble parameter H – acting as time-dependent perturbation parameter – tends monotonically to zero. Numerical simulations are presented as evidence of such behavior.

Averaging generalized scalar field cosmologies II: locally rotationally symmetric Bianchi I and flat Friedmann–Lemaître–Robertson–Walker models

Scalar field cosmologies with a generalized harmonic potential and a matter fluid with a barotropic Equation of State (EoS) with barotropic index  $$\gamma $$  <math> <mi>γ</mi> </math>   for locally rotationally symmetric (LRS) Bianchi III metric and open Friedmann–Lemaître–Robertson–Walker (FLRW) metric are investigated. Methods from the theory of averaging of nonlinear dynamical systems are used to prove that time-dependent systems and their corresponding time-averaged versions have the same late-time dynamics. Therefore, simple time-averaged systems determine the future asymptotic behavior. Depending on values of barotropic index  $$\gamma $$  <math> <mi>γ</mi> </math>   late-time attractors of physical interests for LRS Bianchi III metric are Bianchi III flat spacetime, matter dominated FLRW universe (mimicking de Sitter, quintessence or zero acceleration solutions) and matter-curvature scaling solution. For open FLRW metric late-time attractors are a matter dominated FLRW universe and Milne solution. With this approach, oscillations entering nonlinear system through Klein–Gordon (KG) equation can be controlled and smoothed out as the Hubble factor H – acting as a time-dependent perturbation parameter – tends monotonically to zero. Numerical simulations are presented as evidence of such behaviour.

Averaging generalized scalar field cosmologies I: locally rotationally symmetric Bianchi III and open Friedmann–Lemaître–Robertson–Walker models

This erratum concerns the corrections of Equation (41) and Equation (44), that should read: