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.

Taking a comprehensive view, including a full range of boundary conditions, we reexamine QCD axion star solutions based on the relativistic Klein-Gordon equation (using the Ruffini-Bonazzola approach) and its non-relativistic limit, the Gross-Pitaevskii equation. A single free parameter, conveniently chosen as the central value of the wavefunction of the axion star, or alternatively the chemical potential with range −m < µ < 0 (where m is the axion mass), uniquely determines a spherically-symmetric ground state solution, the axion condensate. We clarify how the interplay of various terms of the Klein-Gordon equation determines the properties of solutions in three separate regions: the structurally stable (corresponding to a local energy minimum) dilute and dense regions, and the intermediate, structurally unstable transition region. From the Klein-Gordon equation, one can derive alternative equations of motion including the Gross-Pitaevskii and Sine-Gordon equations, which have been used previously to describe axion stars in the dense region. In this work, we clarify precisely how and why such methods break down as the binding energy increases, emphasizing the necessity of using the full relativistic Klein-Gordon approach. Finally, we point out that, even after including perturbative axion number violating corrections, solutions to the equations of motion, which assume approximate conservation of axion number, break down completely in the regime with strong binding energy, where the magnitude of the chemical potential approaches the axion mass.

Ultralight scalar dark matter has been proposed to constitute a component of dark matter, though the minimal scenarios have increasingly become constrained. In this work, we analyze scenarios where the dark matter consists of more than one ultralight boson, each with different masses. This potentially leads to formation of gravitationally-bound Bose-Einstein condensates with structures that are very different from condensates composed of a single scalar field. By generalizing from the well-understood single-flavor case, we explore a large range of input parameters, subject to stability criteria, and determine the allowed parameter space for two-flavor condensates as a function of particle physics parameters, paying particular attention to cases where such condensates could compose galactic cores. We also analyze single-flavor condensates subject to external gravity from massive inner bodies and find that such systems may mimic the size of galactic cores as well.

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Dark matter could potentially manifest itself in the form of asymmetric dark stars. In this paper we entertain the possibility of probing such asymmetric bosonic dark matter stars by the use of atomic clocks. If the dark sector connects to the standard model sector via a Higgs or photon portal, the interiors of boson stars that are in a Bose-Einstein condensate state can change the values of physical constants that control the timing of atomic clock devices. Dilute asymmetric dark matter boson stars passing through the Earth can induce frequency shifts that can be observed in separated Earth-based atomic clocks. This gives the opportunity to probe a class of dark matter candidates that for the moment cannot be detected with any different conventional method.