We compute the critical density of collapse for spherically symmetric overdensities in a class of f(R) modified gravity models. For the first time we evolve the Einstein, scalar field and non-linear fluid equations, making the minimal simplifying assumptions that the metric potentials and scalar field remain quasi-static throughout the collapse. Initially evolving a top hat profile, we find that the density threshold for collapse depends significantly on the initial conditions imposed, specifically the choice of size and shape. By imposing `natural' initial conditions, we obtain a fitting function for the spherical collapse delta_c as a function of collapse redshift, mass of the overdensity and f_{R0}, the background scalar field value at z=0. By extending delta_c into drifting and diffusing barrier within the context of excursion set theory, we obtain a realistic mass function that might be used to confront this class of scalar-tensor models with observations of dark matter halos. The proposed analytic formula for the halo mass function was tested against Monte Carlo random walks for a wide class of moving barriers and can therefore be applied to other modified gravity theories.
In the excursion set approach to structure formation initially spherical regions of the linear density field collapse to form haloes of mass $M$ at redshift $z_{\rm id}$ if their linearly extrapolated density contrast, averaged on that scale, exceeds some critical threshold, $δ_{\rm c}(z_{\rm id})$. The value of $δ_{\rm c}(z_{\rm id})$ is often calculated from the spherical or ellipsoidal collapse model, which provide well-defined predictions given auxiliary properties of the tidal field at a given location. We use two cosmological simulations of structure growth in a $Λ$ cold dark matter scen…
Cold dark matter halos form within a smoothly distributed background of relic neutrinos -- at least some of which are massive and non-relativistic at late times. We calculate the accumulation of massive neutrinos around spherically collapsing cold dark matter halos in a cosmological background. We identify the physical extent of the "neutrino halo" in the spherical collapse model, which is large in comparison with the virial radius of the dark matter, and conditions under which neutrinos reaching the cold dark matter halo will remain bound to the halo at late times. We calculate the total neut…
We use two cosmological simulations of structure formation in the LambdaCDM scenario to study the evolutionary histories of dark-matter haloes and to characterize the Lagrangian regions from which they form. We focus on haloes identified at redshift z_id=0 and show that the classic ellipsoidal collapse model systematically overestimates their collapse times. If one imposes that halo collapse takes place at z_id, this model requires starting from a significantly lower linear density contrast than what is measured in the simulations at the locations of halo formation. We attempt to explain this …
We explore the real solutions to the Spherical Collapse Model in a non-flat Universe with a Cosmological Constant, and observe a possible situation for a fake or Ghost Collapse, in which an expanding overdense spherical region, turns around and begins to collapse, turns around again after a finite time and starts expanding. To make such a situation of spurious collapse feasible, we make a linear redshift dependent correction to the standard Dark Energy density term which is originally in the form of a cosmological constant. There is good reason to believe in such a correction based on recent r…
We present and test a new halo finder based on the spherical overdensity (SO) method. This new adaptive spherical overdensity halo finder (ASOHF) is able to identify dark matter haloes and their substructures (subhaloes) down to the scales allowed by the analysed simulations. The code has been especially designed for the adaptive mesh refinement cosmological codes, although it can be used as a stand-alone halo finder for N-body codes. It has been optimised for the purpose of building the merger tree of the haloes. In order to verify the viability of this new tool, we have developed a set of be…
The nonlinear collapse for viscous generalized Chaplygin gas Model (VGCG) was analyzed in the framework of spherical top-hat collapse. As the VGCG and baryons are essential to form the large scale structure, we focused on their nonlinear collapse in this paper. We discussed the influence of model parameters $α$ and $ζ_{0}$ on the spherical collapse by varying their values and compare with $ΛCDM$. The results show that, for the VGCG model, smaller $ζ_{0}$ and larger $α$ make the structure formation earlier and faster, and the collapse curves of VGCG model is almost distinguished with the $ΛCDM$…
We study the evolution of inhomogeneous spherical perturbations in the universe in a way that generalizes the spherical top hat collapse in a straightforward manner. For that purpose we derive a dynamical equation for the evolution of the density contrast in the context of a Lemaitre-Tolman-Bondi metric and construct solutions with and without a cosmological constant for the evolution of a spherical perturbation with a given initial radial profile.
In this paper, we test the spherical collapse of a viscous unified dark fluid (VUDF) which has constant adiabatic sound speed and show the nonlinear collapse for VUDF, baryons, and dark matter which are important to form the large scale structure of our Universe. By varying the values of model parameters $α$ and $ζ_{0}$, we discuss their effects on the nonlinear collapse of the VUDF model, and compare its result to $ΛCDM$ model. The analyzed results show that, within the spherical top-hat collapse framework, larger values of $α$ and smaller values of $ζ_{0}$ make the structure formation earlie…
The spherical collapse model is often used to follow the evolution of overdensities into the nonlinear regime. We describe the correct approach to be used in coupled dark energy cosmologies, where a fifth force, different from gravity and mediated by the dark energy scalar field, influences the collapse. We reformulate the spherical collapse description by deriving it directly from the set of nonlinear hydrodynamical Navier Stokes equations. By comparing with the corresponding relativistic equations, we show how the fifth force should be taken into account within the spherical collapse picture…
We study spherical collapse in the Quartic and Quintic Covariant Galileon gravity models within the framework of the excursion set formalism. We derive the nonlinear spherically symmetric equations in the quasi-static and weak-field limits, focusing on model parameters that fit current CMB, SNIa and BAO data. We demonstrate that the equations of the Quintic model do not admit physical solutions of the fifth force in high density regions, which prevents the study of structure formation in this model. For the Quartic model, we show that the effective gravitational strength deviates from unity at…
We investigate the collapse and internal structure of dark matter halos. We consider halo formation from initially scale-free perturbations, for which gravitational collapse is self-similar. Fillmore and Goldreich (1984) and Bertschinger (1985) solved the one dimensional (i.e. spherically symmetric) case. We generalize their results by formulating the three dimensional self-similar equations. We solve the equations numerically and analyze the similarity solutions in detail, focusing on the internal density profiles of the collapsed halos. By decomposing the total density into subprofiles of pa…
In this paper, we test the spherical collapse of a unified dark fluid (UDF) which has constant adiabatic sound speed. By choosing the different values of model parameters $B_s$ and $α$, we show the nonlinear collapse for UDF and baryons which are considered for their formation of the large scale structure of our Universe. The analyzed results show that larger values of $α$ and $B_s$ make the structure formation faster and earlier.
We reconsider the ellipsoidal-collapse model and extend it in two ways: We modify the treatment of the external gravitational shear field, introducing a hybrid model in between linear and non-linear evolution, and we introduce a virialisation criterion derived from the tensor virial theorem to replace the ad-hoc criterion employed so far. We compute the collapse parameters delta_c and Delta_v and find that they increase with ellipticity e and decrease with prolaticity p. We marginalise them over the appropriate distribution of e and p and show the marginalised results as functions of halo mass…
The influence of the shear stress and angular momentum on the nonlinear spherical collapse model is discussed in the framework of the Einstein-de Sitter (EdS) and $Λ$CDM models. By assuming that the vacuum component is not clustering within the homogeneous nonspherical overdensities, we show how the local rotation and shear affects the linear density threshold for collapse of the non-relativistic component ($δ_\mathrm{c}$) and its virial overdensity ($Δ_\mathrm{V}$). It is also found that the net effect of shear and rotation in galactic scale is responsible for higher values of the linear over…
We describe an algorithm for identifying ellipsoidal haloes in numerical simulations, and quantify how the resulting estimates of halo mass and shape differ with respect to spherical halo finders. Haloes become more prolate when fit with ellipsoids, the difference being most pronounced for the more aspherical objects. Although the ellipsoidal mass is systematically larger, this is less than 10% for most of the haloes. However, even this small difference in mass corresponds to a significant difference in shape. We quantify these effects also on the initial mass and deformation tensors, on which…
We study, for the first time, how shear and angular momentum modify typical parameters of the spherical collapse model, in dark energy dominated universes. In particular, we study the linear density threshold for collapse $δ_\mathrm{c}$ and the virial overdensity $Δ_\mathrm{V}$, for several dark-energy models and its influence on the cumulative mass function. The equations of the spherical collapse are those obtained in Pace et al. (2010), who used the fully nonlinear differential equation for the evolution of the density contrast derived from Newtonian hydrodynamics, and assumed that dark ene…
We present a detailed study of the collapse of a spherical perturbation in DGP braneworld gravity for the purpose of modeling simulation results for the halo mass function, bias and matter power spectrum. The presence of evolving modifications to the gravitational force in form of the scalar brane-bending mode lead to qualitative differences to the collapse in ordinary gravity. In particular, differences in the energetics of the collapse necessitate a new, generalized method for defining the virial radius which does not rely on strict energy conservation. These differences and techniques apply…
The abundance of massive dark matter halos hosting galaxy clusters provides an important test of the masses of relic neutrino species. The dominant effect of neutrino mass is to lower the typical amplitude of density perturbations that eventually form halos, but for neutrino masses $> 0.4eV$ the threshold for halo formation can be changed significantly as well. We study the spherical collapse model for halo formation in cosmologies with neutrino masses in the range $m_{νi} =0.05eV$- $1eV$ and find that halo formation is differently sensitive to $Ω_ν$ and $m_ν$. That is, different neutrino hier…
Traditionally the spherical collapse of objects is studied with respect to a uniform background density, yielding the critical over-density $δ_\mathrm{c}$ as key ingredient to the mass function of virialized objects. Here we investigate the shear and rotation acting on a peak in a Gaussian random field. By assuming that collapsing objects mainly form at those peaks, we use this shear and rotation as external effects changing the dynamics of the spherical collapse, which is described by the Raychaudhuri equation. We therefore assume that the shear and rotation have no additional dynamics on top…
We calculate the non-linear virialization density, $Δ_c$, of halos under spherical collapse from peaks with an arbitrary initial and final density profile. This is in contrast to the standard calculation of $Δ_c$ which assumes top-hat profiles. Given our formalism, the non-linear halo density can be calculated once the shape of the initial peak's density profile and the shape of the virialized halo's profile are provided. We solve for $Δ_c$ for halos in an Einstein de-Sitter and $Λ$CDM universe. As examples, we consider power-law initial profiles as well as spherically averaged peak profiles c…
Characterizing the level of primordial non-Gaussianity (PNG) in the initial conditions for structure formation is one of the most promising ways to test inflation and differentiate among different scenarios. The scale-dependent imprint of PNG on the large-scale clustering of galaxies and quasars has already been used to place significant constraints on the level of PNG in our observed Universe. Such measurements depend upon an accurate and robust theory of how PNG affects the bias of galactic halos relative to the underlying matter density field. We improve upon previous work by employing a mo…
We derive approximated, yet very accurate analytical expressions for the abundance and clustering properties of dark matter halos in the excursion set peak framework; the latter relies on the standard excursion set approach, but also includes the effects of a realistic filtering of the density field, a mass-dependent threshold for collapse, and the prescription from peak theory that halos tend to form around density maxima. We find that our approximations work excellently for diverse power spectra, collapse thresholds and density filters. Moreover, when adopting a cold dark matter power spectr…
We report the findings of new exact analytical solutions to the cosmological fluid equations, namely for the case where the initial conditions are perturbatively close to a spherical top-hat profile. To do so we enable a fluid description in a Lagrangian-coordinates approach, and prove the convergence of the Taylor-series representation of the Lagrangian displacement field until the time of collapse ("shell-crossing"). This allows the determination of the time for quasi-spherical collapse, which is shown to happen generically earlier than in the spherical case. For pedagogical reasons, calcula…
Here we present a simple, parameter-free, non-perturbative algorithm that gives low-redshift cosmological particle realizations accurate to few-Megaparsec scales, called MUSCLE (MUltiscale Spherical ColLapse Evolution). It has virtually the same cost as producing N-body-simulation initial conditions, since it works with the 'stretch' parameter ψ, the Lagrangian divergence of the displacement field. It promises to be useful in quickly producing mock catalogs, and to simplify computationally intensive reconstructions of galaxy surveys. MUSCLE applies a spherical-collapse prescription on multiple…
The smallest dark matter haloes are the first objects to form in the hierarchical structure formation of cold dark matter (CDM) cosmology and are expected to be the densest and most fundamental building blocks of CDM structures in our universe. Nevertheless, the physical characteristics of these haloes have stayed illusive, as they remain well beyond the current resolution of N-body simulations (at redshift zero). However, they dominate the predictions (and uncertainty) in expected dark matter annihilation signal, amongst other astrophysical observables. Using the conservation of total energy …
Comparing the excursion set and CUSP formalisms for the derivation of the halo mass function, we investigate the role of the mass definition in the properties of the multiplicity function of cold dark matter (CDM) haloes. We show that the density profile for haloes formed from triaxial peaks that undergo ellipsoidal collapse and virialisation is such that the ratio between the mean inner density and the outer local density is essentially independent of mass. This causes that, for suited values of the spherical overdensity $Δ$ and the linking length $b$, SO and FoF masses are essentially equiva…
We have studied the merger rate of primordial black holes (PBHs) in the ellipsoidal-collapse model of halo to explain the dark matter abundance by the PBH merger estimated from the gravitational waves detections via the Advanced LIGO (aLIGO) detectors. We have indicated that the PBH merger rate within each halo for the ellipsoidal models is more significant than for the spherical models. We have specified that the PBH merger rate per unit time and per unit volume for the ellipsoidal-collapse halo models is about one order of magnitude higher than the corresponding spherical models. Moreover, w…
We study the gravitational clustering of spherically symmetric overdensities and the statistics of the resulting dark matter halos in the "symmetron model", in which a new long range force is mediated by a $Z_2$ symmetric scalar field. Depending on the initial radius of the overdensity, we identify two distinct regimes: for small initial radii the symmetron mediated force affects the spherical collapse at all redshifts; for initial radii larger than some critical size this force vanishes before collapse because of the symmetron screening mechanism. In both cases overdensities collapse earlier …
Testing the isothermal Jeans model for self-interacting dark matter halos in the collapse phase
School of Physics and Astronomy, Beijing Normal University（北京师范大学物理与天文学院）；National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）；School of Astronomy and Space Science, University of Chinese Academy of Sciences（中国科学院大学天文与空间科学学院）；Donostia International Physics Center (DIPC)（圣塞巴斯蒂安国际物理中心）；Department of Astronomy, Peking University（北京大学天文学系）；Kavli Institute for Astronomy and Astrophysics, Peking University（北京大学科维理天文与天体物理研究所）；Department of Physics and Astronomy, University of California, Riverside（加州大学河滨分校物理与天文学系）
We use 1-dimensional numerical simulations to study spherical collapse in the f(R) gravity models. We include the nonlinear coupling of the gravitational potential to the scalar field in the theory and use a relaxation scheme to follow the collapse. We find an unusual enhancement in density near the virial radius which may provide observable tests of gravity. We also use the estimated collapse time to calculate the critical overdensity $δ_c$ used in calculating the mass function and bias of halos. We find that analytical approximations previously used in the literature do not capture the compl…
Gravitational collapse of a massive primordial gas cloud is thought to be a promising path for the formation of supermassive blackholes in the early universe. We study conditions for the so-called direct collapse (DC) blackhole formation in a fully cosmological context. We combine a semianalytic model of early galaxy formation with halo merger trees constructed from dark matter $N$-body simulations. We locate a total of 68 possible DC sites in a volume of $20\;h^{-1}\;\mathrm{Mpc}$ on a side. We then perform hydrodynamics simulations for 42 selected halos to study in detail the evolution of th…
Primordial black holes (PBHs), as a potential macroscopic candidate for dark matter, can encounter other compact objects in dark matter halos because of their random distribution. Besides, the detection of gravitational waves (GWs) related to the stellar-mass black hole-neutron star (BH-NS) mergers raises the possibility that the BHs involved in such events may have a primordial origin. In this work, we calculate the merger rate of PBH-NS binaries within the framework of ellipsoidal-collapse dark matter halo models and compare it with the corresponding results derived from spherical-collapse d…
The mass distribution of dark matter halos is a sensitive probe of primordial non-Gaussianity (NG). We derive an analytical formula of the halo mass function by perturbatively computing excursion set path-integrals for a non-Gaussian density field with non-vanishing skewness, fnl. We assume a stochastic barrier model which captures the main features of the ellipsoidal collapse of halos. Contrary to previous results based on extensions of the Press-Schechter formalism to NG initial conditions, we find that the non-spherical collapse of halos directly alter the signature of primordial NG. This p…
We give a set of exact nonlinear closed--form solutions for the non-spherical collapse of pressure-less matter in Newtonian gravity, and indicate their possible cosmological applications. Keywords: Newtonian gravitation: free collapse, large-scale cosmic structures
Role of prompt cusps in driving the core collapse of SIDM halos
Massachusetts Institute of Technology（麻省理工学院）；University of Chicago（芝加哥大学）；Carnegie Observatories（卡内基观测站）；Center for Astrophysics, Harvard & Smithsonian（哈佛-史密森天体物理中心）；Rice University（莱斯大学）
In this article we compare the halo mass function predicted by the excursion set theory with a drifting diffusive barrier against the results of N-body simulations for several cosmological models. This includes the standard LCDM case for a large range of halo masses, models with different types of primordial non-Gaussianity, and the Ratra-Peebles quintessence model of Dark Energy. We show that in all those cosmological scenarios, the abundance of dark matter halos can be described by a drifting diffusive barrier, where the two parameters describing the barrier have physical content. In the cas…
We discuss an idealized model of halo formation, in which a collapsing halo node is tetrahedral, with a filament extruding from each of its four faces, and with a wall connecting each pair of filaments. In the model, filaments generally spin when they form, and the halo spins if and only if there is some rotation in filaments. This is the simplest-possible fully three-dimensional halo collapse in the 'origami approximation,' in which voids are irrotational, and the dark-matter sheet out of which dark-matter structures form is allowed to fold in position-velocity phase space, but not stretch (i…
A sample of roughly 1,800 halo subdwarf stars with radial velocities and proper motions is assembled, using the repeated multi-band Sloan Digital Sky Survey photometric measurements in Stripe 82. Our sample of halo subdwarfs is extracted via a reduced proper motion diagram and distances are obtained using photometric parallaxes, thus giving full phase space information. The tilt of the velocity ellipsoid with respect to the spherical polar coordinate system is computed and found to be consistent with zero for two of the three tilt angles, and very small for the third. We prove that if the inne…
We study the spherical collapse in the Parametrized Post-Friedmannian (PPF) scheme. We use a general form of the PPF parameter related to the Poisson equation and found the equations to solve that includes a non-trivial fifth force coming from the convolution of the modified gravity term in the k-space. In order to compute a concrete model, we use the parametrization proposed by Bertschinger and Zukin. The equations of the spherical collapse are solved assuming a Gaussian density profile and we show there is no shell crossing before reaching the turn around point. We show that the fifth force …
We construct axisymmetric mass models for dwarf spheroidal (dSph) galaxies in the Milky Way to obtain plausible limits on the non-spherical structure of their dark halos. This is motivated by the fact that the observed luminous parts of the dSphs are actually non-spherical and Cold Dark Matter (CDM) models predict non-spherical virialized dark halos. Our models consider velocity anisotropy of stars $\bar{v^2_R} / \bar{v^2_ϕ}$, which can vary with the adopted cylindrical coordinates under the assumption $\bar{v^2_z}=\bar{v^2_R}$ for simplicity, and also include an inclination of the system as a…
We use a semianalytic approach that is calibrated to N-body simulations to study the evolution of self-interacting dark matter cores in galaxies. We demarcate the regime where the temporal evolution of the core density follows a well-defined track set by the initial halo parameters and the cross section. Along this track, the central density reaches a minimum value set by the initial halo density. Further evolution leads to an outward heat transfer, inducing gravothermal core collapse such that the core shrinks as its density increases. We show that the time scale for the core collapse is high…
We study the gravitational collapse of an overdensity of nonrelativistic matter under the action of gravity and a chameleon scalar field. We show that the spherical collapse model is modified by the presence of a chameleon field. In particular, we find that even though the chameleon effects can be potentially large at small scales, for a large enough initial size of the inhomogeneity the collapsing region possesses a thin shell that shields the modification of gravity induced by the chameleon field, recovering the standard gravity results. We analyse the behaviour of a collapsing shell in a co…
Under the commonly used spherical collapse model, we study how dark energy affects the growth of large scale structures of the Universe in the context of agegraphic dark energy models. The dynamics of the spherical collapse of dark matter halos in nonlinear regimes is determined by the properties of the dark energy model. We show that the main parameters of the spherical collapse model are directly affected by the evolution of dark energy in the agegraphic dark energy models. We compute the spherical collapse quantities for different values of agegraphic model parameter $α$ in two different sc…
This paper presents a kinetic theory model of gravitational collapse due to a small perturbation. Solving the relevant equations yields a pattern of entropy destruction in a spherical core around the perturbation, and entropy creation in a surrounding halo. This indicates collisional "de-relaxation" in the core, and collisional relaxation in the halo. Core-halo patterns are ubiquitous in the astrophysics of gravitational collapse, and are found here without any of the prior assumptions of such a pattern usually made in analytical models. Motivated by this analysis, the paper outlines a possibl…
High-redshift quasar observations imply that supermassive black holes (SMBHs) larger than $\sim 10^9 ~ M_\odot$ formed before $z=6$. That such large SMBHs formed so early in the Universe remains an open theoretical problem. One possibility is that gas in atomic cooling halos exposed to strong Lyman-Werner (LW) radiation forms $10^4-10^6 ~ M_\odot$ supermassive stars which quickly collapse into black holes. We propose a scenario for direct collapse black hole (DCBH) formation based on synchronized pairs of pristine atomic cooling halos. We consider halos at very small separation with one halo b…
Accurately simulating core-collapse self-interacting dark matter halos
Ludwig-Maximilians-Universität München（慕尼黑大学）；Excellence Cluster ORIGINS（ORIGINS卓越集群）；University of California, Riverside（加州大学河滨分校）；Max-Planck-Institut für Astrophysik（马克斯·普朗克天体物理学研究所）
Considering perturbation growth in spherical Top-Hat model of structure formation in a generalized Chaplygin gas dominated universe, we want to study this scenario with a modified Chaplygin gas model. The evolution of background and collapsed region parameters are found for different cases. The stability of the model and the collapse time rate are considered in different cases. The turn-around redshifts are also computed; the results are in relatively good agreement with current observational data.
We study the spherical collapse of non-top-hat matter fluctuations in the presence of dark energy with arbitrary sound speed. The model is described by a system of partial differential equations solved using a pseudo-spectral method with collocation points. This method can reproduce the known analytical solutions in the linear regime with an accuracy better than $10^{-6}\%$ and better than $10^{-2}\%$ for the virialization threshold given by the usual spherical collapse model. We show the impact of nonlinear dark energy fluctuations on matter profiles, matter peculiar velocity and gravitationa…
We use the Excursion Set formalism to compute the properties of the halo mass distribution for a stochastic barrier model which encapsulates the main features of the ellipsoidal collapse of dark matter halos. Non-markovian corrections due to the sharp filtering of the linear density field in real space are computed with the path-integral technique introduced by Maggiore & Riotto (2010). Here, we provide a detailed derivation of the results presented in Corasaniti & Achitouv (2011) and extend the mass function analysis to higher redshift. We also derive an analytical expression for the linear h…
We comment on the general solution of the scalar field dark matter provided in the paper "Remarks on the spherical scalar field halo in galaxies" by Kamal K. Nandi, Ildar Valitov and Nail G. Migranov. The authors made a mistake in the general form of the tangential pressure profile p_t(r), which deviates from the correct profile, especially when r is small. Although this mistake does not alter significantly the value of w(r) when the integration constant D is small, we found that it does result in an overestimate of w(r) when D is large.
We study gravitational collapse of a spherically symmetric inhomogeneous dust cloud in the Lovelock theory without cosmological constant. We show that the final fate of gravitational collapse in this theory depends on the spacetime dimensions. In odd dimensions the naked singularities formed are found to be massive. In the even dimensions,on the other hand, the naked singularities are found to be massless. We also show that the curvature strength of naked singularity is independent of the spacetime dimensions in odd dimensions. However, it depends on the spacetime dimensions in even dimension.
Mergers Matter: Gravothermal Collapse in Dwarf Halos with Self-Interacting Dark Matter
University of California, Irvine（加州大学尔湾分校）；DARK, Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔研究所DARK）；Massachusetts Institute of Technology（麻省理工学院）；University of Washington（华盛顿大学）；Flatiron Institute（平顿研究所）；Princeton University（普林斯顿大学）；Northeastern University（东北大学）
We study the mechanisms of the gravitational collapse of the Bose-Einstein condensate dark matter halos, described by the zero temperature time-dependent nonlinear Schrödinger equation (the Gross-Pitaevskii equation), with repulsive inter-particle interactions. By using a variational approach, and by choosing an appropriate trial wave function, we reformulate the Gross-Pitaevskii equation with spherical symmetry as Newton's equation of motion for a particle in an effective potential, which is determined by the zero point kinetic energy, the gravitational energy, and the particles interaction e…
