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CAMB v2: cosmological power spectra for high-precision surveys

This paper introduces CAMB v2, a substantially updated cosmological code featuring a new hyperspherical Bessel function treatment and improved integrators to deliver fast, high-precision power spectra for future surveys, with the notable distinction that its core algorithms were developed using LLMs and AI agents under human supervision.

Original authors: Antony Lewis

Published 2026-07-17

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CC BY 4.0

Original authors: Antony Lewis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, expanding balloon covered in a faint, ancient glow. This glow, known as the Cosmic Microwave Background (CMB), is the leftover heat from the Big Bang, a baby picture of the cosmos taken when the universe was just 380,000 years old. Astronomers treat this glow like a cosmic treasure map; by measuring the tiny ripples and patterns in its temperature and polarization, they can deduce the universe's shape, its age, and what it's made of. But to read this map correctly, scientists need a perfect translator. They use complex computer programs to simulate how light and matter should behave according to the laws of physics, creating a "theoretical map" to compare against the real one. If the computer's translation is even slightly off, it could look like the universe is made of different ingredients than it actually is. As our telescopes get sharper and more powerful, the margin for error shrinks to almost nothing, demanding a translator that is faster, smarter, and more precise than ever before.

This paper introduces a major upgrade to one of the most important translators in the field, a software tool called CAMB (Code for Anisotropies in the Microwave Background). Think of CAMB as the engine room of modern cosmology, crunching numbers to predict what the universe's power spectra (the statistical fingerprints of its structure) should look like. The authors, led by Antony Lewis, have completely overhauled the way CAMB handles the tricky math of curved space, updated how it simulates the birth of atoms, and even figured out how to use Artificial Intelligence to help write and test the code itself. The result is a version of the software that is not only significantly faster but also accurate enough to handle the demands of the next generation of giant telescopes, which are about to start scanning the sky with unprecedented detail.

The core of this upgrade tackles a specific mathematical headache: how to calculate the behavior of light traveling through a universe that might not be perfectly flat. In a flat universe, the math is like drawing straight lines on a piece of paper. But if the universe is curved (like the surface of a sphere or a saddle), the math requires "hyperspherical Bessel functions," which are notoriously difficult to compute quickly and accurately. The authors developed a new method that acts like a clever shortcut. Instead of trying to solve the complex curved equation from scratch every time, they map it onto a simpler, flat version of the problem. They use a technique called an "Olver construction," which is essentially a way of stretching and squishing the flat solution so it fits the curved universe perfectly, especially around the "turning point" where the behavior of the light changes. This new method is exact when the universe is flat, smooths out the rough edges where the math usually breaks, and allows the computer to use a fast, efficient stepping method (called Numerov integration) to move between points without losing accuracy.

Beyond the math of curved space, the paper details several other critical improvements. The team updated the model for "recombination," the moment when the universe cooled enough for atoms to form and light to travel freely. They recalibrated this process using more complete physical models to ensure the predictions are accurate to within one part in a thousand. They also refined how the software handles massive neutrinos (ghostly particles that barely interact with anything) and stabilized the equations for "dark energy," the mysterious force driving the universe's acceleration. A particularly unique aspect of this work is the development process itself: almost all of the new algorithms and code were written and tested with the help of AI agents under human supervision. The authors used AI to suggest mathematical approximations, write custom-precision code routines, and run thousands of tests to find and fix bugs. This "human-in-the-loop" approach allowed them to create highly specialized tools that are tuned exactly to the accuracy needed for the physics problem, rather than relying on generic, slower libraries.

The authors rigorously tested their new software against a grid of over one hundred different cosmological scenarios, including universes with different amounts of curvature and massive neutrinos. They found that the new default settings meet a strict accuracy target of 0.1% (or $10^{-3}$ ) across the most important ranges for future surveys. In fact, the errors in typical runs are even smaller than this target. When compared to the previous version of the software and a competing tool called CLASS, the new CAMB is shown to be vastly more accurate at high resolutions while running at comparable or even faster speeds, especially for curved universe models where the old software was painfully slow. The paper concludes that this updated tool is ready to power the analysis of data from upcoming experiments like the Simons Observatory, ensuring that as we look deeper into the cosmos, our theoretical maps are precise enough to reveal the true secrets of the universe.

Technical Summary: CAMB 2: Cosmological Power Spectra for High-Precision Surveys

Problem Statement
Upcoming cosmic microwave background (CMB) and large-scale-structure surveys, such as the Simons Observatory and its Advanced SO extension, demand theoretical power spectra with numerical errors significantly below observational uncertainties. Previous versions of CAMB (v1.x) and CLASS, while sufficient for Planck-era analyses (requiring $\sim 0.1\%$ accuracy at $\ell \lesssim 2000$ ), face challenges in meeting the stricter per-mille ( $10^{-3}$ ) accuracy requirements across a wider range of scales and in non-flat cosmologies. Specifically, the evaluation of hyperspherical Bessel functions in curved spacetime, the treatment of recombination history, massive neutrino sampling, and dark-energy perturbations required substantial updates to ensure robustness and speed without sacrificing precision.

Methodology and Key Developments
The paper describes a comprehensive update to CAMB (v2.0.0), focusing on the lensed CMB and matter power spectra. The development philosophy shifts from relying on general-purpose special-function libraries with fixed accuracy/speed trade-offs toward custom-precision numerical methods. These methods are explicitly calibrated to the accuracy required by the specific physical problem and the grid being sampled. A significant portion of this development was conducted using Large Language Models (LLMs) and AI agents under human supervision to suggest algorithms, debug, and validate numerical changes.

The core technical contributions include:

Hyperspherical Bessel Functions (Line-of-Sight Integration):
- The Innovation: A new treatment of hyperspherical Bessel functions ( $\phi^\nu_l$ ) used for line-of-sight integration in non-flat cosmologies. Instead of mapping the curved radial equation to an Airy function (as in previous WKB treatments), the authors map it onto the flat spherical Bessel equation.
- The Mechanism: An Olver action map matches the action integral of the curved equation to that of the flat equation through the turning point. This defines a coordinate transformation $z(\chi)$ and an amplitude factor $A(\chi)$ .
- Properties: The approximation is exact in the flat limit, remains smooth through the turning point, and reduces to a simple rescaling of the flat Bessel argument and amplitude in near-flat regimes.
- Implementation: The map provides accurate starting values for efficient Numerov propagation between source points, avoiding expensive pointwise evaluations at every step. Fallback mechanisms (exact recurrence, Airy approximations) are used for low multipoles or invalid corners.
Recombination and Background Evolution:
- Recombination Model: The default RECFAST effective few-level approximation is recalibrated against the high-accuracy CosmoRec reference. A new phenomenological correction factor is added to the Helium I rate equation to remove coherent residuals in the ionization history, particularly in the Helium recombination window ( $1500 \lesssim z \lesssim 3000$ ).
- BBN Consistency: The code now interpolates primordial helium abundance ( $Y_p$ ) from PRIMAT (2024) based on $\Omega_b h^2$ and $\Delta N_{eff}$ , replacing older PArthENoPE-based tables.
- Integration: The stiff early phase of recombination is handled by an implicit Rosenbrock method (ROS2) before handing off to the adaptive Dormand–Prince integrator.
Massive Neutrinos:
- The background density and pressure are calculated using direct low-order fits rather than precomputed tables.
- Momentum Sampling: The code automatically adjusts the number of sampled momentum points (from 3 to 4 or 5) based on the neutrino mass to ensure accurate integration of Fermi–Dirac kernels.
- Hierarchy Switching: The code employs perturbative relativistic expansions for highly relativistic species and non-relativistic velocity-integrated hierarchies for late-time evolution, switching between them based on mass and redshift.
Dark Energy (PPF):
- A stabilized Parameterized Post-Friedmann (PPF) evolution is implemented for dark energy fluids crossing the phantom divide ( $w = -1$ ).
- Regularization: To avoid numerical stiffness at high $k/H$ (where the relaxation rate grows as $k^2$ ), the relaxation rate is capped at a threshold $\beta_\star = 30$ . This removes stiffness and integration tolerance sensitivity while maintaining continuity and avoiding the artifacts of hard cutoffs.
Lensing and Accuracy:
- The lensing calculation uses a curved-sky correlation-function method with a new $C^2$ -continuous (quintic) taper to suppress ringing from truncation.
- The code automatically determines the maximum wavenumber ( $k_{max}$ ) required for the lensing potential to ensure convergence up to the requested $\ell_{max}$ .

Results and Validation
The authors validate the new defaults against a "boosted" reference calculation (with accuracy parameters doubled) across a grid of over 100 cosmological configurations, including variations in curvature, neutrino mass, dark energy, and recombination models.

Convergence: The unboosted default settings meet conservative $10^{-3}$ pointwise convergence targets over the main lensed-CMB range ( $600 \le \ell \le 3500$ ) and the quasi-linear matter power spectrum.
Comparison with CLASS: When run with matched physical assumptions and high-precision settings, CAMB v2 and CLASS v3.3.4 agree at the $\sim 10^{-4}$ level for flat $\Lambda$ CDM.
Performance:
- Flat Models: The new defaults are slightly faster or comparable to the previous version when matched for high- $k$ support, while providing significantly higher accuracy (reducing errors from $\sim 1\%$ to $< 0.1\%$ at high $\ell$ ).
- Non-Flat Models: The new hyperspherical Bessel scheme reduces the computational cost for non-flat models by a factor of 17 compared to CAMB v1.6.6 at matched accuracy levels. Near-flat models ( $|\Omega_K| \le 10^{-3}$ ) incur only a modest overhead over the flat case.
Likelihood Impact: The grid-level agreement results in a median $\Delta\chi^2 \approx 0.4$ relative to the boosted reference, well below the acceptance envelope, indicating that numerical errors are negligible for parameter inference in next-generation surveys.

Significance and Claims
The paper claims that CAMB v2.0.0 provides the necessary speed and numerical precision for upcoming high-precision CMB and large-scale-structure surveys without requiring users to manually tune accuracy parameters. The central contribution is the flat-Bessel comparison method for hyperspherical functions, which resolves the technical bottleneck of curved-space integration.

Furthermore, the paper highlights a methodological shift: the development of custom-precision numerical routines calibrated specifically to the physical problem's needs, rather than relying on general-purpose libraries. The authors present this work as a proof-of-concept for using AI agents under human supervision to develop, test, and validate complex numerical algorithms, demonstrating that this approach can make the laborious task of fitting and validating numerous small numerical changes practical. The paper does not claim to solve physical mysteries but rather to provide a robust, high-fidelity computational tool for the next generation of cosmological data analysis.

Technical Summary: CAMB 2: Cosmological Power Spectra for High-Precision Surveys

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