Approximation methods in the study of boson stars

Abstract

We analyze the accuracy of the variational method in computing physical quantities relevant for gravitationally bound Bose-Einstein condensates. Using a variety of variational ansätze found in existing literature, we determine physical quantities and compare them to exact numerical solutions. We conclude that a "linear+exponential" wavefunction proportional to (1 + ξ) exp(−ξ) (where ξ is a dimensionless radial variable) is the best fit for attractive self-interactions along the stable branch of solutions, while for small particle number N it is also the best fit for repulsive self-interactions. For attractive self-interactions along the unstable branch, a single exponential is the best fit for small N , while a sech wavefunction fits better for large N . The Gaussian wavefunction ansatz, which is used often in the literature, is exceedingly poor across most of the parameter space, with the exception of repulsive interactions for large N . We investigate a "double exponential" ansatz with a free constant parameter, which is computationally efficient and can be optimized to fit the exact solutions in different limits. We show that the double exponential can be tuned to fit the sech ansatz, which is computationally slow. We also show how to generalize the addition of free parameters in order to create more computationally efficient ansätze using the double exponential. Determining the best ansatz, according to several comparison parameters, will be important for analytic descriptions of dynamical systems. Finally, we examine the underlying relativistic theory, and critically analyze the Thomas-Fermi approximation often used in the literature.

Global view of QCD axion stars

“…The effective coupling of the field Y (x) to gravity is given by κ ≡ 8πδ/∆ 2 . These equations are exactly equivalent [29] to the nonrelativistic Gross-Pitaevskii-Poisson (GPP) equations, which we will discuss in Section III B 1, and the solutions have been discussed many times in the literature [16,24,25,30,31]. For completeness, we reproduce the well-known massradius relation for axion stars in the dilute region in the top panel of Figure 1 for different decay constants, f .…”

“…(III.8) is equivalent to the standard GP equation used to analyze axion stars [21,30]. This is made transparent by identifying, as in [29], the relationship between Z(y) and the standard Schrödinger wavefunction ψ:…”

“…A recent surge in studies of boson stars [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] stems, in part, from the renewed interest in determining whether dark matter (DM) could consist of condensates of ax-ions or other axion like particles. A particularly wellmotivated scalar DM candidate is the QCD axion, parametrized by a decay constant f = 6 × 10 11 GeV and particle mass m = 10 −5 eV; 2 as a result, bound states of QCD axions (which we will call QCD axion stars) have received special attention.…”

“…For QCD axions, the EKG formalism was applied with appropriate scaling relations to determine the maximum mass, M max = 10M P f /m ≈ 10 19 kg [16]. In this dilute branch of solutions the radius scales inversely as the mass [16,17,[29][30][31]. 3 The BB solutions for QCD axion stars had masses much lower than the maximum.…”

A conjecture on the minimal size of bound states

“…4 For a review on boson stars (updated in 2017) see [29]. A selection of recent work can be found in [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] the heaviest elementary state has mass m , no bound state with radius below the scale 1/m set by the Compton wavelength exists. Thus, gravity may be dropped, the connection to the standard notion of the Swampland and the scalar WGC becomes remote, but we may have an interesting statement about quantum field theories.…”