Virial Condition Research Articles

The scaling relations of galaxies back in time: The road toward virialization

Context. The structural scaling relations (SSRs) of galaxies, that is, the observed correlations between effective radius, effective surface intensity, and velocity dispersion, are important tools for understanding how evolution proceeds. Aims. In this paper, we aim to demonstrate that the evolution of the SSRs back in time is governed by a combination of virial theorem (VT) and the relation L = L0′(t)σβ(t), where the parameters β and L0′ vary with time and from galaxy to galaxy. Methods. Using the WINGS database for the galaxies at redshift z = 0 and the Illustris-1 and IllustrisTNG databases of artificial galaxies, for the galaxies up to redshift z = 4, we analyse the SSRs back in time and, by means of simple algebraic expressions for L0′ and β (functions of time and other physical quantities), we derive the expected paths followed by galaxies in the various SSRs toward the distributions observed at z = 0. Results. The distribution of galaxies in the SSRs is ultimately related to the evolution in luminosity and velocity dispersion, which are empirically mirrored by the L = L0′(t)σβ(t) law. Furthermore, the β parameter works as a thermometer of the virialization of a galaxy. This parameter can assume either positive or negative values, and its absolute value attains high values when the galaxy is close to the virial condition, while it tends to zero when the galaxy is far from this condition. Conclusions. As the SSRs change with time, the method proposed in this paper allows us to decipher the temporal evolution of galaxies.

On particle scattering in Gross-Pitaevskii theory and implications for dark matter halos

Bose-Einstein-condensed dark matter (BEC-DM), also called scalar field dark matter (SFDM), has become a popular alternative to the standard, collisionless cold dark matter (CDM) model, due to its long-held potential to resolve the small-scale crisis of CDM. Halos made of BEC-DM have been modelled using the Gross-Pitaevskii (GP) equation coupled to the Poisson equation; the so-called GPP equations of motion. These equations are based on fundamental microphysical conditions that need to be fulfilled in order for the equations to be valid in the first place, related to the diluteness of the DM gas and the nature of the particle scattering model. We use these conditions in order to derive the implications for the BEC-DM parameters, the 2-particle self-interaction coupling strength g and the particle mass m. We compare the derived bounds with the constraint that results from the assumption of virial equilibrium of the central cores of halos, deriving a relationship that connects g and m. We find that the GPP conditions are greatly fulfilled, for BEC-DM particle masses of interest, if such models also obey the virial condition that turns out to be the strongest constraint. We also derive the implications for the elastic scattering cross section (per particle mass) in BEC-DM halos, based on the scattering model of GPP, and find a huge range of possible values, depending on the self-interaction regime. We put our results into context to recent literature which predicts sub-kpc core size in BEC-DM halos.

Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups. II

In this paper, we improve and generalize the results of [1] on large-time expansion of the heat and Schrödinger semigroups with sub-exponential time-decay estimates on the remainder for a class of non-selfadjoint Schrödinger operators. For the Schrödinger semigroup, the large-time expansion is established for distortion analytic potentials satisfying a damped virial condition outside some compact and the contributions from threshold eigenvalue and outgoing positive resonances are explicitly calculated.

Effect of interacting dark energy on mass-temperature relation in galaxy clusters

There are a variety of cosmological models for dark matter and dark energy in which a possible interaction is considered between these two significant components of the universe. We focus on five suggested models of interacting dark matter and dark energy and derive the modified virial theorem for them by developing a previous approach. It provides an opportunity to study the evolution of this modified virial theorem with time and interacting constants for different interacting models. Then we use this obtained virial condition to investigate the modified mass-temperature relation in galaxy clusters via three various methods. It reveals that the effect of interaction between dark matter and dark energy merely appears in the normalization factor of $M\propto T^{\frac{3}{2}}$. This relation also leads to a new constraint on the constants of interacting models, which only depends on the concentration parameter and density profile of the cluster. Then we use five observational data sets to check some proposed figures for the constants of interaction which have been resulted from other observational constraints. Finally, by fitting the observational results to the modified mass-temperature relation, we obtain values for interacting constants of three models and four specific cases of the two remained models. In agreement with many other observational outcomes, we find that according to observational data for masses and temperatures of the galaxy clusters, energy transfer occurs from dark matter to dark energy in the seven investigated models.

Dynamics to the universal structure of one-dimensional self-gravitating systems in the quasi-equilibrium state

We investigate the quasi-equilibrium state of one-dimensional self-gravitating systems. If the null virial condition is satisfied at initial time, it is found that the number density around the center of the system at the quasi-equilibrium state has the universality similar to two- and three-dimensional self-gravitating systems reported in Refs. [1,2]. The reason why the null virial condition is sufficient for the universality is unveiled by the envelope equation. We present a phenomenological model to describe the universal structure by using a special Langevin equation with a distinctive random noise to self-gravitating systems. Additionally, we unveil a mechanism which decides the radius of the system.

Coupled DM Heating in SCDEW Cosmologies

Strongly-Coupled Dark Energy plus Warm dark matter (SCDEW) cosmologies admit the stationary presence of ∼1% of coupled-DM and DE, since inflationary reheating. Coupled-DM fluctuations therefore grow up to non-linearity even in the early radiative expansion. Such early non-linear stages are modelized here through the evolution of a top-hat density enhancement, reaching an early virial balance when the coupled-DM density contrast is just 25–26, and the DM density enhancement is ∼10 % of the total density. During the time needed to settle in virial equilibrium, the virial balance conditions, however, continue to modify, so that “virialized” lumps undergo a complete evaporation. Here, we outline that DM particles processed by overdensities preserve a fraction of their virial momentum. Although fully non-relativistic, the resulting velocities (moderately) affect the fluctuation dynamics over greater scales, entering the horizon later on.

Deuterium fractionation and H2D+ evolution in turbulent and magnetized cloud cores

High-mass stars are expected to form from dense prestellar cores. Their precise formation conditions are widely discussed, including their virial condition, which results in slow collapse for super-virial cores with strong support by turbulence or magnetic fields, or fast collapse for sub-virial sources. To disentangle their formation processes, measurements of the deuterium fractions are frequently employed to approximately estimate the ages of these cores and to obtain constraints on their dynamical evolution. We here present 3D magneto-hydrodynamical simulations including for the first time an accurate non-equilibrium chemical network with 21 gas-phase species plus dust grains and 213 reactions. With this network we model the deuteration process in fully depleted prestellar cores in great detail and determine its response to variations in the initial conditions. We explore the dependence on the initial gas column density, the turbulent Mach number, the mass-to-magnetic flux ratio and the distribution of the magnetic field, as well as the initial ortho-to-para ratio of H2. We find excellent agreement with recent observations of deuterium fractions in quiescent sources. Our results show that deuteration is rather efficient, even when assuming a conservative ortho-to-para ratio of 3 and highly sub-virial initial conditions, leading to large deuterium fractions already within roughly a free-fall time. We discuss the implications of our results and give an outlook to relevant future investigations.

Universal structure of two- and three-dimensional self-gravitating systems in the quasiequilibrium state.

We study a universal structure of two- and three-dimensional self-gravitating systems in the quasiequilibrium state. It is shown numerically that the two-dimensional self-gravitating system in the quasiequilibrium state has the same kind of density profile as the three-dimensional one, especially when null virial conditions are fulfilled. It is unveiled why the conditions are necessary for the universal structure by the envelope equation. We develop a phenomenological model to describe this universal structure by using a special Langevin equation with a distinctive random noise to self-gravitating systems. We find that the density profile derived theoretically is very consistent with results of observations and simulations.

Trends of stellar entropy along stellar evolution

This paper is devoted to discussing the difference in the thermodynamic entropy budget per baryon in each type of stellar object found in the Universe. We track and discuss the actual decrease of the stored baryonic thermodynamic entropy from the most primitive molecular cloud up to the final fate of matter in black holes, passing through evolved states of matter as found in white dwarfs and neutron stars. We then discuss the case of actual stars with different masses throughout their evolution, clarifying the role of the virial equilibrium condition for the decrease in entropy and related issues. Finally, we discuss the role of gravity in driving the composition and the structural changes of stars with different Main Sequence masses during their evolution up to the final product. Particularly, we discuss the entropy of a black hole in this context arguing that the dramatic increase in its entropy, differently from the other cases, is due to the gravitational field itself.

Ergodicity breaking and quasistationary states in systems with long-range interactions.

In the thermodynamic limit, systems with long-range interactions do not relax to equilibrium, but become trapped in quasistationary states (qSS), the life time of which diverges with the number of particles. In this paper we will explore the relaxation of the Hamiltonian Mean-Field model to qSS for a class of initial conditions of the multilevel water-bag form. We will show that if the initial distribution satisfies the virial condition, thereby reducing mean field changes, the final distribution in the qSS can be predicted very accurately using a reduced exactly integrable model. The calculated distribution functions obtained using this approach are found to be more accurate than the ones predicted by the Lynden-Bell theory.

Sommerfeld Radiation Condition at Threshold

We prove Besov space bounds of the resolvent at low energies in any dimension for a class of potentials that are negative and obey a virial condition with these conditions imposed at infinity only. We do not require spherical symmetry. The class of potentials includes in dimension ≥3 the attractive Coulomb potential. There are two boundary values of the resolvent at zero energy which we characterize by radiation conditions. These radiation conditions are zero energy versions of the well-known Sommerfeld radiation condition.

Topology of collisionless relaxation.

Using extensive molecular dynamics simulations we explore the fine-grained phase space structure of systems with long-range interactions. We find that if the initial phase space particle distribution has no holes, the final stationary distribution will also contain a compact simply connected region. The microscopic holes created by the filamentation of the initial distribution function are always restricted to the outer regions of the phase space. In general, for complex multilevel distributions it is very difficult to a priori predict the final stationary state without solving the full dynamical evolution. However, we show that, for multilevel initial distributions satisfying a generalized virial condition, it is possible to predict the particle distribution in the final stationary state using Casimir invariants of the Vlasov dynamics.

Ergodicity Breaking and Parametric Resonances in Systems with Long-Range Interactions

We explore the mechanism responsible for the ergodicity breaking in systems with long-range forces. In thermodynamic limit such systems do not evolve to the Boltzmann-Gibbs equilibrium, but become trapped in an out-of-equilibrium quasi-stationary-state. Nevertheless, we show that if the initial distribution satisfies a specific constraint-a generalized virial condition-the quasistationary state is very close to ergodic and can be described by Lynden-Bell statistics. On the other hand, if the generalized virial condition is violated, parametric resonances are excited, leading to chaos and ergodicity breaking.

Optimized scattering functions and amplitudes for electron impact ionization and the post–prior symmetry

The electron impact ionization of hydrogen, in the all-s-wave approximation, is treated by the improved distorted wave theory. The leading corrections to the distorted wave Born amplitude are calculated in the POST form, using simple variational trial functions. The localized virial conditions are imposed for the determination of nonlinear parameters, thus optimizing the scattering function of the initial state. It is shown that the singly differential cross section can be adequately described by including up to three linear parameters. The calculated cross sections agree with that of the PRIOR form, and also compare well with the existing theoretical data. Furthermore, the procedure for accuracy estimates based on the post–prior comparison is critically re-examined, showing that the DWBA comparison can lead to erroneous conclusion. The fully optimized, distorted wavefunctions are extremely simple, and should be suitable for various applications.

Theoretical measures for optimization of collision amplitudes

A set of new theoretical measures for the optimal determination of collision amplitudes is formulated in two parts: (i) the localized virial conditions, which optimize the scattering solutions, and (ii) the temperate function criteria, which identify and discard spurious solutions. Thus, for the first time the quality of approximations introduced on the scattering function can be systematically improved, and the collision amplitudes are then evaluated with the improved wave functions. The measures proposed are not an extre{mu}m principle, so that the improvement of the amplitude is not monotonic. But, in its various forms, the theory should be applicable generally to most collision systems, at all scattering energies and even above the breakup threshold.

Improved distorted wave theory with the localized virial conditions

The distorted wave theory is operationally improved to treat the full collision amplitude, such that the corrections to the distorted wave Born amplitude can be systematically calculated. The localized virial conditions provide the tools necessary to test the quality of successive approximations at each stage and to optimize the solution. The details of the theoretical procedure are explained in concrete terms using a collisional ionization model and variational trial functions. For the first time, adjustable parameters associated with an approximate scattering solution can be fully determined by the theory. A small number of linear parameters are introduced to examine the convergence property and the effectiveness of the new approach.

Collisionless relaxation in gravitational systems: From violent relaxation to gravothermal collapse

Theory and simulations are used to study collisionless relaxation of a gravitational N -body system. It is shown that when the initial one-particle distribution function satisfies the virial condition--potential energy is minus twice the kinetic energy--the system quickly relaxes to a metastable state described quantitatively by the Lynden-Bell distribution with a cutoff. If the initial distribution function does not meet the virial requirement, the system undergoes violent oscillations, resulting in a partial evaporation of mass. The leftover particles phase-separate into a core-halo structure. The theory presented allows us to quantitatively predict the amount and the distribution of mass left in the central core, without any adjustable parameters. On a longer time scale tauG-N , collisionless relaxation leads to a gravothermal collapse.

A Priori Estimation of the Resolvent on Approximation of Born-Oppenheimer

In this study, we estimate the resolvent of the two bodies Shrodinger operator perturbed by a potential of Coulombian type on Hilbert space when h tends to zero. Using the Feschbach method, we first distorted it and then reduced it to a diagonal matrix. We considered a case where two energy levels cross in the classical forbidden region. Under the assumption that the second energy level admits a non degenerate point well and virial conditions on the others levels, a good estimate of the resolvent were observed.

Besov estimates in the high-frequency Helmholtz equation, for a non-trapping and [formula omitted] potential

We study the high-frequency Helmholtz equation, for a potential having C 2 smoothness, and satisfying the non-trapping condition. We prove optimal Morrey–Campanato estimates that are both homogeneous in space and uniform in the frequency parameter. The homogeneity of the obtained bounds, together with the weak assumptions we require on the potential, constitute the main new result in the present text. Our result extends previous bounds obtained by Perthame and Vega, in that we do not assume the potential satisfies the virial condition, a strong form of non-trapping.

Counting of galaxy clusters with the S-Z effect of triaxial ellipsoidal model

In the near future a large number of galaxy clusters will be observed in surveys based on the Sunyaev-Zel'dovich effect (SZE) . Both the spherical and ellipsoidal models of the dark matter of galaxy clusters are adopted to make theoretical predictions of the number of observable S-Z galaxy clusters, and to asses the effect of the cluster shape on the calculation. By means of the virial equilibrium condition the mass lower limit of the observable galaxy clusters is determined from given survey flux lower limit and the prediction of the number count of galaxy clusters is carried out by the mass function. The calculated results show that if the cosmological parameters are to be constrained by the S-Z surveys, then the uncertainty in the shape of the clusters should be taken into account.