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+ + + + +Journal of Open Source Software +JOSS + +2475-9066 + +Open Journals + + + +6888 +10.21105/joss.06888 + +GRBoondi: A code for evolving Generalized Proca theories +on arbitrary backgrounds + + + +https://orcid.org/0000-0002-8059-0359 + +Fell +Shaun David Brocus + + + + +https://orcid.org/0000-0002-5766-8242 + +Heisenberg +Lavinia + + + + + +Institute for Theoretical Physics, Universitaet Heidelberg, +Philosophenweg 12, 69120 Heidelberg, Germany + + + +9 +103 +6888 + +Authors of papers retain copyright and release the +work under a Creative Commons Attribution 4.0 International License (CC +BY 4.0) +2024 +The article authors + +Authors of papers retain copyright and release the work under +a Creative Commons Attribution 4.0 International License (CC BY +4.0) + + + +C++ +MPI +OpenMP +vector intrinsics +Generalized Proca +gravity +general relativity +numerical relativity +adaptive refinement + + + + + + Summary +

Proca theories are a simple massive extension of the typical + electromagnetic field. Such a simple modification of the underlying + theory can have tremendous phenomenological implications, particularly + in the case of early-universe high-energy physics, structure + formation, and various dark matter models. On the theoretical side, + they arise naturally in certain string theories and axionic + interactions. Phenomenologically, Proca fields can be produced in the + early universe from quantum fluctuations. This leads the Proca field + to be an excellent candidate for the dark matter particle. Naturally, + generalizations of the Proca field can lead to even richer + astrophysical implications, providing possible solutions to many + challenges across a wide range of astrophysical phenomena.

+ + Generalized Proca +

Generalized Proca theories + (Heisenberg, + 2014, + 2019) + provide a rich landscape to search for solutions to deep, + fundamental questions, such as the nature of dark energy and dark + matter. Moreover, strong gravity regimes, like those surrounding + dense, compact astrophysical entities, offer novel avenues to + explore fundamental fields. To effectively probe these domains, + precise models are imperative for sifting through vast quantities of + data. Crafting concrete models for the entirety of generalized Proca + theories represents a formidable endeavor, typically relying on + numerical methods. These numerical methods are usually written to be + case dependent, especially for a particular background. This makes + it difficult to generalize the computational code to account for + higher-order couplings or different types of backgrounds. The + difficulty is especially amplified when considering the full + landscape of generalized Proca theories and beyond + (Heisenberg + et al., 2016). Already at the level of the Lagrangian, the + equations are immense, + + ℒg.P.=−g(−14FμνFμν+∑n=26αnℒn), + where each sub-Lagrangian + + ℒn + contains terms of different orders in the derivatives of the Proca + field + (Heisenberg, + 2014, + 2019). + Even including solely + + ℒ2, + the equations of motion can be extremely cumbersome. Solving these + equations of motion analytically very quickly becomes + intractable.

+

GRBoondi provides a unified interface for computing the evolution + of any generalized Proca model on an arbitrary background. Given a + specific background with known expressions for the metric variables + and initial data for the Proca field, GRBoondi numerically computes + the time evolution of the Proca field, given user-specified + generalized Proca equations of motion. While the simultaneous + evolution of the metric and matter fields offers the most + comprehensive depiction of their temporal progression, there are + instances where the density of the Proca field is negligible in + comparison to the background curvature. In such cases, fixed + background evolution routines serve as an excellent approximation to + the complete mutual evolution. A core feature of GRBoondi is that it + simplifies much of the boilerplate code required to begin a + simulation from scratch. The only additions a user needs to input + are the initial conditions, the modifications to the equations of + motion (EOM), and the background functions. GRBoondi will + automatically compute various diagnostic quantities and plot files. + Moreover, GRBoondi is incredibly modular and modifications to any of + the in-built functions is effortless.

+ + + + Statement of need +

In practice, any numerical relativity (NR) software library can be + used to evolve generalized Proca theories. Prominent examples of NR + codes encompass the extensive Einstein Toolkit + (Brandt + et al., 2024) and its related Cactus framework + (Goodale + et al., 2003), Kranc + (Husa + et al., 2006), LEAN + (Sperhake, + 2007), and Canuda + (Witek + et al., 2023). Other libraries worth mentioning are the + non-public BAM + (Brügmann + et al., 2008), AMSS-NCKU + (Galaviz + et al., 2010), PAMR + (East + et al., 2012) and HAD + (Neilsen + et al., n.d.). The non-exhaustive list can be expanded with + SPeC + (Pfeiffer + et al., 2003), which is a pseudo-spectral code that uses + generalized harmonic coordinates. Similarly, SpECTRE + (Cao + et al., 2019; + Deppe + et al., 2021; + Kidder + et al., 2017) uses the Galerkin methods. NRPy + (Ruchlin + et al., 2018) is a Python library that aims for use on non-high + performance computing clusters. There are also cosmological simulation + codes like CosmoGRaPH + (Mertens + et al., 2016) and GRAMSES + (Barrera-Hinojosa + & Li, 2020). Simflowny + (Palenzuela + et al., 2018) is a magneto-hydrodynamic simulation software. + GR-Athena++ + (Daszuta + et al., 2021) uses the oct-tree AMR for maximum scaling. + ExaGRyPE is a NR program leveraging the ExaHyPRE PDE solver + (Zhang + et al., 2024). GRDzhadzha + (Aurrekoetxea + et al., 2024) is another fixed background code, based off the + GRChombo + (Andrade + et al., 2021) framework, which GRBoondi utilizes.

+

This extensive list underscores the abundance of NR libraries at + one’s disposal. However, none of them provide a tailored unified + interface for studying the vast landscape of gravity theories, like + generalized Proca. Using any of the numerous frameworks requires + significant work in order to evolve even a single generalized Proca + theory. GRBoondi tackles this problem by providing a collection of + specialized tools for computing the generalized Proca equations, + relying on existing tools which allows for rapid updating and + debugging. On top of this, GRBoondi offers catered plotting routines + for viewing data, leveraging the highly parallelizable VisIt + (Childs + et al., 2012) analysis tool. In addition, since the metric + variables and their derivatives are computed exactly at each grid + point, the adaptability of the AMR grid can be focused solely on the + matter variables.

+ + + Key features of GRBoondi + + +

Ease of use: A central pillar of GRBoondi is its + ease of use, relative to other NR software. Many of the basic + boilerplate code and complexities are kept within the source code, + allowing the user to have only a basic understanding of the code + in order to start researching their problem.

+ + +

Arbitrary choice of background spacetime: The main + parts of GRBoondi are extremely modular — users can swap in and + out backgrounds with ease. GRBoondi comes pre-equipped with four + background classes ready for use, along with testing suites to + verify convergence of each class. These include (starred + backgrounds are inherited from GRDzhadzha and are immediately + usable in GRBoondi):

+ + +

Minkowski space

+ + +

Kerr-de Sitter black hole

+ + +

*Boosted Schwarzschild black hole

+ + +

*Kerr black hole

+ + + + +

Arbitrary modifications to base Proca: GRBoondi + uses specific coding idioms that allow arbitrary modifications to + the hard-coded equations of motion. This allows for numerical + simulations of any generalized Proca theory. The base model is the + standard electromagnetic model, subject to the usual Proca + constraint. Any generalized Proca theory can be added on top of + this by simply adding the additional pieces from the generalized + Proca Lagrangian. Examples of simulating standard Proca and a + simple non-linear model + (Clough + et al., 2022; + Coates + & Ramazanoğlu, 2022; + Ünlütürk + et al., 2023) are included in the code repository.

+ + +

Diagnostics: GRBoondi comes equipped with a large + number of diagnostic quantities that can be calculated during the + course of a simulation and additional tools for users to calculate + their own quantities.

+ + +

Tailored post-processing tools: GRBoondi comes + equipped with custom-built post-processing scripts, leveraging + VisIt’s Python interface. Any of the diagnostic quantities + computed during the evolution can be plotted using these tools. It + also contains a simple integral plotter, for quickly visualizing + the various integrals computed during a simulation, leveraging + Python’s Matplotlib package + (Hunter, + 2007)

+ + +

Additional features of GRBoondi are elucidated on the + GitHub + Wiki.

+ +

Some examples of the usage of GRBoondi. The left image + is the energy density of a Proca cloud superradiantly excited around + a rapidly spinning black hole, superimposed with streamlines of the + spatial Proca vector field. The right image is a plot in the + xz-plane of the square of the Proca 4-vector in the same background. + The spin axis of the black hole is orientated along the z-axis. + Superimposed on both images is a slice of the computational grid, + clearly showing the refinement hierarchy.

+ + + + + Acknowledgements +

LH is supported by funding from the European Research Council (ERC) + under the European Unions Horizon 2020 research and innovation + programme grant agreement No 801781. LH further acknowledges support + from the Deutsche Forschungsgemeinschaft (DFG, German Research + Foundation) under Germany’s Excellence Strategy EXC 2181/1 - 390900948 + (the Heidelberg STRUCTURES Excellence Cluster).

+

SF is indebted to Katy Clough for her stimulating discussions and + extensive input to the code. Katy was an invaluable source of aid for + the nuances of NR simulations and some of the finer details of + GRChombo.

+

The simulations performed as part of this release were carried out + on the Baden-Württemberg high-performance computing cluster. The + authors acknowledge support by the state of Baden-Württemberg through + bwHPC.

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