What Is the Gravitational Field of a Mass in a Spatially Nonlocal Quantum Superposition?

The study of the gravitational field produced by a spatially nonlocal, superposed quantum state of a massive particle is an interesting and active area of research. One outstanding issue is whether the gravitational field behaves like the classical superposition of the gravitational field of two particles separated by a spatial distance with half the mass at each position. Alternatively, does the gravitational field behave as a quantum superposition with a far more interesting and subtle behavior than a simple classical superposition? Quantum field theory is ideally suited to probe exactly this kind of question. We study the scattering of a massless scalar on a spatially nonlocal quantum superposition of a massive particle. We compute the differential scattering cross section corresponding to one-graviton exchange. We find that the scattering cross section disagrees with the Newton-Schrödinger picture of potential scattering from two localized sources with half the mass at each source. This suggests that experimental observation of gravitational scattering could inform the viability of the semiclassical treatment of the gravitational field, as in the Newton-Schrödinger description, vs the fully quantum mechanical treatment adopted here. We comment on the experimental feasibility of observing such effects in systems with many particles such as Bose-Einstein condensates.

Schrödinger-Poisson systems under gradient fields

A singularity-free generalisation of Newtonian gravity can be constructed (Lazar in Phys Rev D 102:096002, 2020) within the framework of gradient field theory. This procedure offers a straightforward regularisation of Newtonian gravity and remains equally well applicable to other fields, such as electromagnetic fields. Here, with the aim of finding potentially measurable effects of gradient fields on the dispersion properties of various media, we present a quantum kinetic treatment of matter under gradient fields. The method is based on the application of the Wigner–Moyal procedure to the modified Schrödinger–Poisson equation emerging in the framework of gradient field theory. This allows us to treat, on equal footing, three different scenarios, namely self-gravitating systems, plasmas, and cold atoms in magneto-optical traps. We address the signature of gradient fields in the elementary excitations of these media. In particular, we estimate this effect to be accessible in state-of-the-art plasma-based experiments. We discuss in detail the classical kinetic and hydrodynamic limits of our approach and obtain a class of generalised Lane–Emden equations, in the context of gradient field theory, which remain valid in the three scenarios discussed here.

Deep Zoom-In Simulation of a Fuzzy Dark Matter Galactic Halo

Fuzzy dark matter (FDM) made of ultralight bosonic particles is a viable alternative to cold dark matter with clearly distinguishable small-scale features in collapsed structures. On large scales, it behaves gravitationally like cold dark matter deviating only by a cutoff in the initial power spectrum and can be studied using N-body methods. In contrast, wave interference effects near the de Broglie scale result in new phenomena unique to FDM. Interfering modes in filaments and halos yield a stochastically oscillating granular structure which condenses into solitonic cores during halo formation. Investigating these highly nonlinear wave phenomena requires the spatially resolved numerical integration of the Schrödinger equation. In previous papers we introduced a hybrid zoom-in scheme that combines N-body methods to model the large-scale gravitational potential around and the mass accretion onto pre-selected halos with simulations of the Schrödinger-Poisson equation to capture wave-like effects inside these halos. In this work, we present a new, substantially improved reconstruction method for the wave function inside of previously collapsed structures. We demonstrate its capabilities with a deep zoom-in simulation of a well-studied sub-L_{*}-sized galactic halo from cosmological initial conditions. With a particle mass of m=2.5×10^{-22}  eV and halo mass M_{vir}=1.7×10^{11}  M_{⊙} in a (60  h^{-1} comoving Mpc)^{3} cosmological box, it reaches an effective resolution of 20 comoving pc. This pushes the values of m and M accessible to simulations significantly closer to those relevant for studying galaxy evolution in the allowed range of FDM masses.

Self-gravitating clusters of Bose-Einstein gas with planar, cylindrical, or spherical symmetry: Gaseous density profiles and onset of condensation

We calculate density profiles for self-gravitating clusters of an ideal Bose-Einstein gas with nonrelativistic energy-momentum relation and macroscopic mass at thermal equilibrium. Our study includes clusters with planar symmetry in dimensions D=1,2,3, clusters with cylindrical symmetry in D=2,3, and clusters with spherical symmetry in D=3. Wall confinement is imposed where needed to prevent escape. The length scale and energy scale in use for the gaseous phase render density profiles for gaseous macrostates independent of total mass. Density profiles for mixed-phase macrostates have a condensed core surrounded by a gaseous halo. The spatial extension of the core is negligibly small on the length scale tailored for the halo. The mechanical stability conditions as evident in caloric curves permit multiple macrostates to coexist. Their status regarding thermal equilibrium is examined by a comparison of free energies. The onset of condensation takes place at a nonzero temperature in all cases. The critical singularities and the nature of the phase transition vary with the symmetry of the cluster and the dimensionality of the space.

Incoherent localized structures and hidden coherent solitons from the gravitational instability of the Schrödinger-Poisson equation

The long-term behavior of a modulationally unstable conservative nonintegrable system is known to be characterized by the soliton turbulence self-organization process. We consider this problem in the presence of a long-range interaction in the framework of the Schrödinger-Poisson (or Newton-Schrödinger) equation accounting for the gravitational interaction. By increasing the amount of nonlinearity, the system self-organizes into a large-scale incoherent localized structure that contains "hidden" coherent soliton states: The solitons can hardly be identified in the usual spatial or spectral domains, but their existence can be unveiled in the phase-space representation (spectrogram). We develop a theoretical approach that provides the coupled description of the coherent soliton component [governed by the Schrödinger-Poisson equation (SPE)] and of the incoherent structure [governed by a wave turbulence Vlasov-Poisson equation (WT-VPE)]. We demonstrate theoretically and numerically that the incoherent structure introduces an effective trapping potential that stabilizes the hidden coherent soliton and we show that the incoherent structure belongs to a family of stationary solutions of the WT-VPE. The analysis reveals that the incoherent structure evolves in the strongly nonlinear regime and that it is characterized by a compactly supported spectral shape. By relating the analytical properties of the hidden soliton to those of the stationary incoherent structure, we clarify the quantum-to-classical (i.e., SPE-to-VPE) correspondence in the limit ℏ/m→0: The hidden soliton appears as the latest residual quantum correction preceding the classical limit described by the VPE. This study is of potential interest for self-gravitating Boson models of fuzzy dark matter. Although we focus our paper on the Schrödinger-Poisson equation, we show that the regime of hidden solitons stabilized by an incoherent structure is general for long-range wave systems featured by an algebraic decay of the interacting potential. This work should stimulate nonlinear optics experiments in highly nonlocal nonlinear (thermal) media that mimic the long-range nature of gravitational interactions.

Coherent Soliton States Hidden in Phase Space and Stabilized by Gravitational Incoherent Structures

We consider the problem of the formation of soliton states from a modulationally unstable initial condition in the framework of the Schrödinger-Poisson (or Newton-Schrödinger) equation accounting for gravitational interactions. We unveil a previously unrecognized regime: By increasing the nonlinearity, the system self-organizes into an incoherent localized structure that contains "hidden" coherent soliton states. The solitons are hidden in the sense that they are fully immersed in random wave fluctuations: The radius of the soliton is much larger than the correlation radius of the incoherent fluctuations, while its peak amplitude is of the same order of such fluctuations. Accordingly, the solitons can hardly be identified in the usual spatial or spectral domains, while their existence is clearly unveiled in the phase-space representation. Our multiscale theory based on coupled coherent-incoherent wave turbulence formalisms reveals that the hidden solitons are stabilized and trapped by the incoherent localized structure. Furthermore, hidden binary soliton systems are identified numerically and described theoretically. The regime of hidden solitons is of potential interest for self-gravitating Boson models of "fuzzy" dark matter. It also sheds new light on the quantum-to-classical correspondence with gravitational interactions. The hidden solitons can be observed in nonlocal nonlinear optics experiments through the measurement of the spatial spectrogram.

Fingerprints of nonequilibrium stationary distributions in dispersion relations

Distributions different from those predicted by equilibrium statistical mechanics are commonplace in a number of physical situations, such as plasmas and self-gravitating systems. The best strategy for probing these distributions and unavailing their origins consists in combining theoretical knowledge with experiments, involving both direct and indirect measurements, as those associated with dispersion relations. This paper addresses, in a quite general context, the signature of nonequilibrium distributions in dispersion relations. We consider the very general scenario of distributions corresponding to a superposition of equilibrium distributions, that are well-suited for systems exhibiting only local equilibrium, and discuss the general context of systems obeying the combination of the Schrödinger and Poisson equations, while allowing the Planck's constant to smoothly go to zero, yielding the classical kinetic regime. Examples of media where this approach is applicable are plasmas, gravitational systems, and optical molasses. We analyse in more depth the case of classical dispersion relations for a pair plasma. We also discuss a possible experimental setup, based on spectroscopic methods, to directly observe these classes of distributions.

Jeans Instability of Dissipative Self-Gravitating Bose–Einstein Condensates with Repulsive or Attractive Self-Interaction: Application to Dark Matter

We study the Jeans instability of an infinite homogeneous dissipative self-gravitating Bose–Einstein condensate described by generalized Gross–Pitaevskii–Poisson equations [Chavanis, P.H. Eur. Phys. J. Plus2017, 132, 248]. This problem has applications in relation to the formation of dark matter halos in cosmology. We consider the case of a static and an expanding universe. We take into account an arbitrary form of repulsive or attractive self-interaction between the bosons (an attractive self-interaction being particularly relevant for the axion). We consider both gravitational and hydrodynamical (tachyonic) instabilities and determine the maximum growth rate of the instability and the corresponding wave number. We study how they depend on the scattering length of the bosons (or more generally on the squared speed of sound) and on the friction coefficient. Previously obtained results (notably in the dissipationless case) are recovered in particular limits of our study.

A Quantum Model for the Dynamics of Cold Dark Matter

A model for cold dark matter is given by the solution of a coupled Schrödinger–Poisson equation system. We present a numerical scheme for integrating these equations, discussing the problems arising from their nonlinear and nonlocal character. After introducing and testing our numerical approach, we illustrate key features of the system by numerical examples in 1 + 1 dimensions. In particular, we study the properties of asymptotic states to which the numerical solutions converge for artificial initial conditions.

First Star-Forming Structures in Fuzzy Cosmic Filaments

In hierarchical models of structure formation, the first galaxies form in low-mass dark matter potential wells, probing the behavior of dark matter on kiloparsec scales. Even though these objects are below the detection threshold of current telescopes, future missions will open an observational window into this emergent world. In this Letter, we investigate how the first galaxies are assembled in a "fuzzy" dark matter (FDM) cosmology where dark matter is an ultralight ∼10^{-22}  eV boson and the primordial stars are expected to form along dense dark matter filaments. Using a first-of-its-kind cosmological hydrodynamical simulation, we explore the interplay between baryonic physics and unique wavelike features inherent to FDM. In our simulation, the dark matter filaments show coherent interference patterns on the boson de Broglie scale and develop cylindrical solitonlike cores, which are unstable under gravity and collapse into kiloparsec-scale spherical solitons. Features of the dark matter distribution are largely unaffected by the baryonic feedback. On the contrary, the distributions of gas and stars, which do form along the entire filament, exhibit central cores imprinted by dark matter-a smoking gun signature of FDM.