Analytical Approximations for Families of Numerical Boson Star Solutions
Core Concepts
This work proposes a simple strategy to build analytical proxies for families of numerical solutions to the Einsteinmatter field equations, using spherical miniboson stars as a case study. The approach involves a double expansion of the unknown functions in an appropriately chosen basis, which can facilitate studying properties across the parameter space, data compression, and wider usage of such solutions.
Abstract
The authors tackle the need for analytical proxies for whole families of numerical solutions, rather than just individual solutions within a family. They use spherical, fundamental miniboson stars as an exploratory case study to illustrate the feasibility of their approach.
The key highlights and insights are:

The authors present two different methods for building analytical approximations  the "Standard" and "AccelerationInformed" methods. The "AccelerationInformed" method, which optimizes the approximation of the second derivative of the functions, outperforms the "Standard" method in terms of accurately approximating the spacetime metric functions.

For a specific boson star solution with maximum mass, the "AccelerationInformed" approximation can satisfy the field equations up to an error of ~10^5 for the KleinGordon equation, and ~10^6 for the Einstein field equations.

The authors then extend the analytical approximation to cover a family of boson star solutions, focusing on two specific regions near the maximum mass configuration. By using a double expansion approach, they are able to represent the entire family with a relatively compact set of coefficients, offering significant data compression compared to storing the numerical solutions directly.

While the methodology is successful in approximating boson star solutions in the targeted regions, the authors note challenges in expanding the domain to cover the full parameter space, especially towards the Newtonian limit where the solutions become more diluted.
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Analytical proxy to families of numerical solutions: the case study of spherical miniboson stars
Stats
"The nontrivial field equations, when written in terms of the metric functions Fk(x), are obtained by equating the terms {Ett, Err, Eθθ, EKG} to zero."
"For the BS solution with maximal mass, the "AccelerationInformed" approximation can satisfy the field equations up to an error of ~10^5 for the KleinGordon equation, and ~10^6 for the Einstein field equations."
Quotes
"Whereas analytic approximations to each individual solution within a numerical family have been proposed, proxies for whole families are missing, which can facilitate studying properties across the parameter space, data compression and a wider usage of such solutions."
"To the best of our knowledge, however, these have focused on individual solutions within a family, typically spanned by (say) m nontrivial parameters."
"We show how to perform a double expansion of the unknown functions on a given chosen basis, to obtain a proxy for the solutions along the 1D family of solutions, or a subset thereof."
Deeper Inquiries
How could the methodology be extended to handle boson star solutions in the Newtonian limit more effectively?
To extend the methodology for approximating boson star solutions in the Newtonian limit, several strategies could be employed. First, a new definition of the radial coordinate could be introduced, which would better accommodate the characteristics of solutions with larger frequencies (i.e., where ω/µ approaches 1). This could involve a different compactification technique that allows for a more gradual transition from the compact to the dilute regime, thereby capturing the behavior of the scalar field over greater radial distances.
Additionally, the use of higherorder polynomial expansions in the Chebyshev basis could enhance the accuracy of the approximations in this limit. By increasing the number of basis functions, the method could better capture the steep gradients and variations in the metric functions that are typical in the Newtonian limit. Furthermore, implementing advanced numerical techniques, such as adaptive mesh refinement or Gaussian quadrature, could improve the precision of the numerical integration involved in the approximation process. These enhancements would collectively allow for a more robust analytical framework capable of addressing the complexities associated with boson stars in the Newtonian limit.
What other types of numerical solutions, beyond boson stars, could benefit from this analytical approximation approach?
The analytical approximation approach developed for boson stars could be beneficial for a variety of other numerical solutions in the realm of general relativity and beyond. For instance, neutron stars, which are also compact objects governed by similar field equations, could leverage this methodology to provide analytical proxies for their equilibrium configurations. Additionally, black hole solutions, particularly those that are not easily expressible in closed form, could benefit from this approach, allowing for a continuous study of their properties across different parameters.
Moreover, the methodology could be applied to other exotic compact objects, such as gravastars or dark energy stars, which are characterized by complex field dynamics. The ability to create analytical proxies for these solutions would facilitate a deeper understanding of their stability, dynamics, and interactions with surrounding matter. Furthermore, the approach could extend to cosmological models that involve scalar fields, such as those found in inflationary scenarios, where analytical approximations could aid in exploring the parameter space of various models.
Could the analytical proxies developed here be integrated into a broader framework for studying the properties and dynamics of compact objects described by numerical solutions?
Yes, the analytical proxies developed for boson stars can indeed be integrated into a broader framework for studying the properties and dynamics of compact objects. By establishing a systematic approach to create analytical approximations for various families of solutions, researchers could develop a comprehensive database of compact object metrics that are easily accessible and computationally efficient.
This framework could include tools for visualizing the properties of these objects, such as their mass, stability, and response to perturbations, across different parameter spaces. Additionally, the integration of these proxies into numerical relativity simulations could enhance the understanding of dynamical processes, such as gravitational wave emissions from binary systems involving compact objects.
Furthermore, the analytical proxies could serve as initial data for numerical simulations, providing a more accurate starting point that could lead to improved convergence and stability in numerical methods. By combining analytical and numerical techniques, researchers could gain insights into the complex interactions and evolution of compact objects, ultimately contributing to a more unified understanding of their role in the universe.