Einstein-Klein-Gordon Research Articles

Gravitating Scalarons with Inverted Higgs Potential

Previously, a class of regular and asymptotically flat gravitating scalar solitons (scalarons) has been constructed in the Einstein–Klein–Gordon (EKG) theory by adopting a phantom field with Higgs-like potential where the kinetic term has the wrong sign and the scalaron possesses the negative Arnowitt–Deser–Misner (ADM) mass as a consequence. In this paper, we demonstrate that the use of the phantom field can be avoided by inverting the Higgs-like potential in the EKG system when the kinetic term has a proper sign, such that the corresponding gravitating scalaron can possess the positive ADM mass. We systematically study the basic properties of the gravitating scalaron, such as the ADM mass, the energy conditions, the geodesics of test particles, etc. Moreover, we find that it can be smoothly connected to the counterpart hairy black hole solutions from our recent work in the small horizon limit.

Oscillating soliton stars with network of domain walls

In this work, we study oscillating soliton stars (oscillatons) with network of domain walls. We consider a Lagrangian with three scalar fields coupled among themselves by a specific potential. We choose an appropriate potential to admit the formation of network of domain walls on the oscillatons. With small perturbations applied to this potential, we then compute the Einstein–Klein–Gordon (EKG) equations numerically and analyze the mass profile of this new object. From the results we discuss how the stability of the oscillatons is affected by the network. At some conditions, the network provides a ‘bouncing stability’ to the oscillatons.

On the constant roll complex scalar field inflationary models

In this paper we wish to point out the possibility of using a complex scalar field in a constant roll inflationary model, as needed for observational viability. We extend the idea of real field inflaton with constant rate of roll to a complex field, showing the feasibility of solving Einstein Klein-Gordon equations constrained by an appropriate form of constant roll definition. As compared to the well known (two-parametric class of) real field models, there is one more degree of flexibility in constant roll inflationary solutions which is represented by an arbitrary function of time, γ(t). We work with an arbitrary but constant function γ (where γ = 0 refers to the corresponding real field model) and find new inflationary class of potentials. In this class of models, the behavior of real and complex field models are similar in some aspects, for example the solutions with large constant roll parameter are not stable and should be considered as early time transients. These field solutions relax at late time on a dual attractor trajectory. However, complex fields phase space trajectories reach this stable regime after real fields. We performed the stability analysis on γ function space solutions and found that dynamically stable trajectories in phase space are stable under γ variations. We extended this study by considering multifield models of constant roll inflation with non-canonical kinetic terms. By enlarging the size of field space, we showed that a multifield constant roll model is dynamically a single field effective theory. If field space is parametrized by N non-canonical fields, there will be N free parameters in the potential that can be attributed to the interaction between the fields.

Spinning gauged boson and Dirac stars: A comparative study

Scalar boson stars and Dirac stars are solitonic solutions of the Einstein–Klein-Gordon and Einstein-Dirac classical equations, respectively. Despite the different bosonic vs. fermionic nature of the matter field, these solutions to the classical field equations have been shown to have qualitatively similar features [1]. In particular, for spinning stars the most fundamental configurations can be in both cases toroidal, unlike spinning Proca stars that are spheroidal [2]. In this paper we gauge the scalar and Dirac fields, by minimally coupling them to standard electromagnetism. We explore the impact of the gauge coupling on the resulting solutions. One of the most relevant difference concerns the gyromagnetic ratio, which for the scalar stars takes values around 1, whereas for Dirac stars takes values around 2.

Two-scalar Bose-Einstein condensates: from stars to galaxies

We study the properties of Bose-Einstein Condensate (BEC) systems consisting of two scalars, focusing on both the case where the BEC is stellar scale as well as the case when it is galactic scale. After studying the stability of such systems and making contact with existing single scalar limits, we undertake a numerical study of the two interacting scalars using Einstein-Klein-Gordon (EKG) equations, including both non-gravitational self-interactions and interactions between the species. We show that the presence of extra scalars and possible interactions between them can leave unique imprints on the BEC system mass profile, especially when the system transitions from being dominated by one scalar to being dominated by the other. At stellar scales (nonlinear regime), we observe that a repulsive interaction between the two scalars of the type +ϕ1 2 ϕ2 2 can stabilize the BEC system and support it up to high compactness, a phenomenon only known to exist in the +ϕ4 system. We provide simple analytic understanding of this behavior and point out that it can lead to interesting gravitational wave signals at LIGO-Virgo. At galactic scales, on the other hand, we show that two-scalar BECs can address the scaling problem that arises when one uses ultralight dark matter mass profiles to fit observed galactic core mass profiles. In the end, we construct a particle model of two ultralight scalars with the repulsive ϕ1 2 ϕ2 2 interaction using collective symmetry breaking. We develop a fast numerical code that utilizes the relaxation method to solve the EKG system, which can be easily generalized to multiple scalars.

A stability analysis of the static Einstein‐Klein‐Gordon Boson Stars

AbstractThe stability of the recently proposed static solutions for boson stars is analyzed. These solutions of Einstein‐Klein‐Gordon (EKG) equations are obtained by considering the interaction of a real scalar field with matter. We assume that the inclusion of the scalar field in addition to matter allows one to justify that stability implies that the total mass of the solution should grow when the initial condition for the density of matter at the origin is also increased. Employing numerical values for the static boson star based on a linear relation between the source and the energy density and between this and the pressure, we found the relation that linked the scalar field at the origin with the matter energy density at the same point. We also determine the behavior of the total mass with the matter energy density in the origin by obtaining through this and the weak energy condition two possible ranges for stable solutions of static boson stars.

Real scalar field solutions of the EKG equations including matter

Static (not stationary) solutions of the Einstein–Klein–Gordon (EKG) equations including matter are obtained for real scalar fields. The scalar field interaction with matter is considered. The introduced coupling allows the existence of static solutions in contraposition with the case of the simpler EKG equations for real scalar fields and gravity. Surprisingly, when the considered matter is a photon-like gas, it turns out that the gravitational field intensity at large radial distances becomes nearly a constant, exerting an approximately fixed force to small bodies at any distance. The effect is clearly related with the massless character of the photon-like field. It is also argued that the gravitational field can generate a bounding attraction, that could avoid the unlimited increase in mass with the radius of the obtained here solution. This phenomenon, if verified, may furnish a possible mechanism for explaining how the increasing gravitational potential associated to dark matter, finally decays at large distances from the galaxies. A method for evaluating these photon bounding effects is just formulated in order to be further investigated.

Multipolar boson stars: Macroscopic Bose-Einstein condensates akin to hydrogen orbitals

Boson stars are often described as macroscopic Bose-Einstein condensates. By accommodating large numbers of bosons in the same quantum state, they materialize macroscopically the intangible probability density cloud of a single particle in the quantum world. We take this interpretation of boson stars one step further. We show, by explicitly constructing the fully non-linear solutions, that static (in terms of their spacetime metric, gμν) boson stars, composed of a single complex scalar field, Φ, can have a non-trivial multipolar structure, yielding the same morphologies for their energy density as those that elementary hydrogen atomic orbitals have for their probability density. This provides a close analogy between the elementary solutions of the non-linear Einstein–Klein-Gordon theory, denoted Φ(N,ℓ,m), which could be realized in the macrocosmos, and those of the linear Schrödinger equation in a Coulomb potential, denoted Ψ(N,ℓ,m), that describe the microcosmos. In both cases, the solutions are classified by a triplet of quantum numbers (N,ℓ,m). In the gravitational theory, multipolar boson stars can be interpreted as individual bosonic lumps in equilibrium; remarkably, the (generic) solutions with m≠0 describe gravitating solitons [gμν,Φ(N,ℓ,m)] without any continuous symmetries. Multipolar boson stars analogue to hybrid orbitals is also constructed.

Classical and quantum exact solutions for the anisotropic Bianchi type I in multi-scalar field cosmology with an exponential potential driven inflation

The anisotropic Bianchi type I in multi-scalar field cosmology is studied with a particular potential of the form V=V0e−[λ1ϕ1+⋯+λnϕn], which emerges as a condition between the time derivatives of their corresponding momenta. Using the Hamiltonian formalism for the inflation epoch with a quintessence framework we find the exact solutions for the Einstein-Klein-Gordon (EKG) system with different scenarios specified by the parameter λ2=∑i=1nλi2. For the quantum scheme of this model, the corresponding Wheeler-DeWitt (WDW) equation is solved by applying an appropriate change of variables and suitable ansatz.

Future stability of the 1 + 3 Milne model for the Einstein–Klein–Gordon system

We study the small perturbations of the 1 + 3-dimensional Milne model for the Einstein–Klein–Gordon (EKG) system. We prove the nonlinear future stability, and show that the perturbed spacetimes are future causally geodesically complete. For the proof, we work within the constant mean curvature (CMC) gauge and focus on the 1 + 3 splitting of the Bianchi–Klein–Gordon equations. Moreover, we treat the Bianchi–Klein–Gordon equations as evolution equations and establish the energy scheme in the sense that we only commute the Bianchi–Klein–Gordon equations with spatially covariant derivatives while normal derivative is not allowed. We propose some refined estimates for lapse and the hierarchies of energy estimates to close the energy argument.

Balancing a static black ring with a phantom scalar field

All known five dimensional, asymptotically flat, static black rings possess conical singularities. However, there is no fundamental obstruction forbidding the existence of balanced configurations, and we show that the Einstein–Klein-Gordon equations admit (numerical) solutions describing static asymptotically flat black rings, which are regular on and outside the event horizon. The scalar field is ‘phantom’, which creates the self-repulsion necessary to balance the black rings. Similar solutions are likely to exist in other spacetime dimensions, the basic properties of a line element describing a four dimensional, asymptotically flat black ring geometry being discussed.

Hamilton’s approach in cosmological inflation with an exponential potential and its observational constraints

The Friedmann-Robertson-Walker (FRW) cosmology is analyzed with a general potential $\rm V(\phi)$ in the scalar field inflation scenario. The Bohmian approach (a WKB-like formalism) was employed in order to constraint a generic form of potential to the most suited to drive inflation, from here a family of potentials emerges; in particular we select an exponential potential as the first non trivial case and remains the object of interest of this work. The solution to the Wheeler-DeWitt (WDW) equation is also obtained for the selected potential in this scheme. Using Hamilton's approach and equations of motion for a scalar field $\rm \phi$ with standard kinetic energy, we find the exact solutions to the complete set of Einstein-Klein-Gordon (EKG) equations without the need of the slow-roll approximation (SR). In order to contrast this model with observational data (Planck 2018 results), the inflationary observables: the tensor-to-scalar ratio and the scalar spectral index are derived in our proper time, and then evaluated under the proper condition such as the number of e-folding corresponds exactly at 50-60 before inflation ends. The employed method exhibits a remarkable simplicity with rather interesting applications in the near future.

Classical and Quantum Exact Solutions for a FRW Multiscalar Field Cosmology with an Exponential Potential Driven Inflation

A flat Friedmann-Robertson-Walker (FRW) multiscalar field cosmology is studied with a particular potential of the form V(ϕ,σ)=V0e-λ1ϕ-λ2σ, which emerges as a relation between the time derivatives of the scalars field momenta. Classically, by employing the Hamiltonian formalism of two scalar fields (ϕ,σ) with standard kinetic energy, exact solutions are found for the Einstein-Klein-Gordon (EKG) system for different scenarios specified by the parameter λ2=λ12+λ22, as well as the e-folding function Ne which is also computed. For the quantum scheme of this model, the corresponding Wheeler-DeWitt (WDW) equation is solved by applying an appropriate change of variables.

Geodesics around oscillatons made of exponential scalar field potential

Oscillatons are spherically symmetric solutions to the Einstein Klein Gordon (EKG) equations for soliton stars made of real time dependent scalar fields. These equations are non singular and satisfy flatness conditions asymptotically with periodic time dependency. In this paper, we investigate the geodesic motion of particles moving around an oscillaton related to a time dependent scalar field. Bound orbital is found for these particles under the condition of particular values of angular momentum L and initial radial position. We discuss this topic for an exponential scalar field potential which could be of the exponential form with a scalar field and investigate whether the radial coordinates of such particles oscillate in time or not and thereby we could predict the corresponding oscillating period as well as amplitude. It is necessary to recall, in general relativity, a geodesic generalizes the notion of a straight line to curved space time. Importantly, the world line of a particle free from all external, non gravitational forces, is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic. In general relativity, gravity can be regarded as not a force but a consequence of a curved space time geometry where the source of curvature is the stress energy tensor (representing matter, for instance). Thus, for example, the path of a planet orbiting around a star is the projection of a geodesic of the curved 4D space time geometry around the star onto 3D space.

Boson star solutions of the Einstein–Klein–Gordon equations in the sense of Colombeau–Egorov's theory of generalized functions

We present static, spherically symmetric solutions of the equations of motion of a scalar field interacting with gravity through the Einstein–Klein–Gordon (EKG) equations in the Colombeau–Egorov sense. The scalar fields are confined within the interior region of the solution, and the exterior fields are purely gravitational and coinciding with the Schwarzschild ones. The solution resembles the so‐called gravastars discussed in the literature. The presented solutions of the EKG equations open the possibility for the existence of static boson stars. The argument is based on defining an infinitely differentiable one‐parameter (ϵ‐dependent) family of radial dependencies of the metric and the scalar field. It is shown that in the limit ϵ → 0, the EKG equations are satisfied in the sense of the generalized functions. The solutions exhibit properties that qualitatively support their physical meaning. For example, close to the boundary at the interior, the scalar field energy density piles up toward the limit surface. On the other hand, also close to the separation surface but on the outside, the known “non‐hair” theorem clearly indicates that any scalar field perturbation also tends to be attracted to the boundary. The work also suggests the possibility for obtaining a regular gravastar, after using the singular configuration found as a first step in an iterative solution of the quantum EKG equations.

Oscillatons described by self-interacting quartic scalar fields

Abstract In this paper, we will study some properties of oscillatons, spherically symmetric object made of a real time-dependent scalar field, using a self-interaction quartic scalar field potential as a class of what is called Self-Interacting Dark Matter (SIDM) models. Since the oscillatons can be regarded as models for astrophysical objects which play the role of dark matter, therefore investigation of their properties could have an important place in present time of cosmology researches. We investigate the properties of these objects by Numerical Solving the system of differential equations obtained from the Einstein–Klein–Gordon (EKG) equations to show their importance as new candidates for dark matter in the galactic scales.

Stability of Schwarzschild-AdS for the Spherically Symmetric Einstein-Klein-Gordon System

In this paper, we study the global behavior of solutions to the spherically symmetric coupled Einstein-Klein-Gordon (EKG) system in the presence of a negative cosmological constant. We prove that the Schwarzschild-AdS spacetimes (the trivial black hole solutions of the EKG system for which $\phi=0$ identically) are asymptotically stable: Small perturbations of Schwarzschild-AdS initial data again lead to regular black holes, with the metric on the black hole exterior approaching a Schwarzschild-AdS spacetime. The main difficulties in the proof arise from the lack of monotonicity for the Hawking mass and the asymptotically AdS boundary conditions, which render even (part of) the orbital stability intricate. These issues are resolved in a bootstrap argument on the black hole exterior, with the redshift effect and weighted Hardy inequalities playing the fundamental role in the analysis. Both integrated decay and pointwise decay estimates are obtained.

Evolution and stability Φ4 oscillatons

We solve numerically the Einstein–Klein–Gordon (EKG) system, assuming spherical symmetry, for a real scalar field endowed with a quartic self-interaction potential, and obtain the so-called oscillatons: oscillating soliton stars. We obtain the equilibrium configurations for Φ4oscillatons. Also, we analyze numerically the evolution of the EKG equations to study the stability of such oscillatons. We present the influence of the quartic potential on the behavior of both the stable (S-oscillatons) and unstable (U-oscillatons) branches, under the influence of small and large radial perturbations.

Evolution of the Schrödinger-Newton system for a self-gravitating scalar field

Using numerical techniques, we study the collapse of a scalar field configuration in the Newtonian limit of the spherically symmetric Einstein--Klein--Gordon (EKG) system, which results in the so called Schr\"odinger--Newton (SN) set of equations. We present the numerical code developed to evolve the SN system and topics related, like equilibrium configurations and boundary conditions. Also, we analyze the evolution of different initial configurations and the physical quantities associated to them. In particular, we readdress the issue of the gravitational cooling mechanism for Newtonian systems and find that all systems settle down onto a 0--node equilibrium configuration.

Inside oscillatons

Non-singular self-gravitating objects can be found by solving the coupled Einstein–Klein–Gordon (EKG) equations for a real scalar field. Such objects are generically known as oscillatons, in which the scalar field and the metric are fully time-dependent. In this paper, we describe a numerical procedure to minimize the nonlinearities present in the EKG equations, in the case of spherical symmetry, which permits us to find accurate numerical solutions. In order to gain physical insight of relativistic oscillatons, we study oscillatons in flat space, in the weak field limit, the so-called Newtonian oscillatons, using a fixed Schwarzschild background. This last case may be related to the ejected scalar field during a gravitational collapse of scalar field configurations.