The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes

Abstract: Boltzmann codes are used extensively by several groups for constraining cosmological parameters with Cosmic Microwave Background and Large Scale Structure data. This activity is computationally expensive, since a typical project requires from 10'000 to 100'000 Boltzmann code executions. The newly released code CLASS (Cosmic Linear Anisotropy Solving System) incorporates improved approximation schemes leading to a simultaneous gain in speed and precision. We describe here the three approximations used by CLASS for basic LambdaCDM models, namely: a baryon-photon tight-coupling approximation which can be set to first order, second order or to a compromise between the two; an ultra-relativistic fluid approximation which had not been implemented in public distributions before; and finally a radiation streaming approximation taking reionisation into account.

• The paper introduces three key approximation schemes (TCA, UFA, RSA) in CLASS to efficiently compute cosmological perturbations in the ΛCDM model.
• The paper details how second-order TCA improves baryon-photon coupling fidelity and UFA streamlines the modeling of ultra-relativistic components.
• The paper achieves up to a 66% speed improvement in perturbation calculations, setting a new benchmark for large-scale cosmological simulations.

Overview of the Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation Schemes

The paper presents advancements in computational methods for handling cosmological data with bolstered efficiency. The focal point is the Cosmic Linear Anisotropy Solving System (CLASS), which has been developed to improve the speed and accuracy of cosmological computations necessary for analyzing Cosmic Microwave Background (CMB) and Large Scale Structure (LSS) data. This improved code architecture incorporates a series of approximation schemes which are critical for making the computations efficient while maintaining the integrity of the results.

The CLASS code introduces three primary approximation schemes for the minimal $\Lambda$CDM model: the Baryon-Photon Tight-Coupling Approximation (TCA), the Ultra-relativistic Fluid Approximation (UFA), and the Radiation Streaming Approximation (RSA). Each scheme targets specific computational challenges within the model, tailoring the precision and time-complexity balance required for large-scale cosmological data analysis.

Baryon-Photon Tight-Coupling Approximation (TCA)

In the early universe, photons and baryons are tightly coupled, leading to very stiff source equations. The TCA introduced in CLASS allows for efficient calculation by reducing the system to a set of manageable differential equations without significantly sacrificing accuracy. This is achieved by expanding perturbations in terms of a small coupling parameter, leading to different possible orders of the TCA. Importantly, this paper describes how second-order TCA can be implemented to improve fidelity, though a first-order approximation is sufficient for the majority of scenarios.

Ultra-relativistic Fluid Approximation (UFA)

UFA facilitates the modeling of massless neutrinos and other ultra-relativistic components as distinguished from ordinary matter components. As the universe evolves, the need to accurately resolve the behavior of free-streaming neutrinos can become computationally demanding. CLASS simplifies this through fluid approximations that ease these computational burdens by effectively closing the hierarchy of equations when computations are comfortably within the Hubble radius. This ensures that the number of tracked variables remains minimal while retaining accuracy through targeted approximations tailored for computational efficiency.

Radiation Streaming Approximation (RSA)

Post-decoupling, photons free-stream across the universe's structure. In these epochs, accounting for their interaction is critical in predicting the impact on the CMB spectra. RSA in CLASS exploits the weak gravitational coupling of these streaming particles during late-time evolution, allowing for their approximate behavior to be predicted without needing excessive computation of later multipoles. It addresses the potential inaccuracies due to multipole truncation in the hierarchy of equations by leveraging analytic approximations of photon behavior.

Implications and Future Developments

The enhancements presented in this paper not only push the frontier for cosmological simulations but set a new standard for utilizing approximation schemes. These improvements in speed by up to 66% in perturbation modules ensure that researchers can process more complex cosmological models efficiently. Moreover, by leveraging these approximation schemes, researchers can extend CLASS to incorporate non-cold dark matter relics and massive neutrinos, further broadening its applicability.

Future developments in AI and computational physics stand to benefit from such advancements as efficient approximation algorithms incorporate complex models into simulations while preserving speed and accuracy. The flexibility embedded within the CLASS architecture suggests potential for application in various domains of computational physics where similar modeling challenges exist.

These approximation schemes provide a blueprint for efficiently managing computational complexity, which can also be applied to other fields where large-scale model simulation is crucial. By understanding the needs of specific cosmological models and tailoring computational approaches accordingly, CLASS sets an example of methodological rigor combined with practical applicability, guiding future innovations in this expansive field.