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CMBolic: Symbolic emulators for the Cosmic Microwave Background. I. Lensing

The paper introduces CMBolic, a suite of symbolic emulators that accurately and efficiently predicts the CMB lensing potential power spectrum for extended $\Lambda$ CDM models, achieving precision comparable to neural networks while reducing cosmological parameter estimation runtime from weeks to minutes.

Original authors: David M. J. Vokrouhlicky, Constantinos Skordis, Deaglan J. Bartlett, Harry Desmond, Pedro G. Ferreira

Published 2026-06-09

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CC BY 4.0

Original authors: David M. J. Vokrouhlicky, Constantinos Skordis, Deaglan J. Bartlett, Harry Desmond, Pedro G. Ferreira

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 Cosmic Microwave Background (CMB) as the "baby photo" of the universe, a faint glow left over from when the universe was just a baby. But this photo isn't perfectly clear; it's been distorted by gravity, much like looking at a landscape through a wavy glass window. This distortion is called gravitational lensing. To understand the universe's secrets (like how much dark energy exists or how heavy neutrinos are), scientists need to calculate exactly how this "glass" bends the light.

Traditionally, calculating this bending is like trying to solve a massive, complex math puzzle every single time you want to check a new theory. It takes powerful computers weeks to run these calculations, which makes testing new ideas incredibly slow.

Enter "CMBolic": The Shortcut

The authors of this paper have created a new tool called CMBolic. Think of it as a magic calculator or a GPS shortcut for these cosmic calculations. Instead of solving the complex puzzle from scratch every time, CMBolic uses a clever trick called Symbolic Regression.

Here is how they did it, using simple analogies:

The Problem: Imagine you have a library of 150 different "universe recipes" (each with slightly different amounts of dark energy, neutrino mass, etc.). For each recipe, you need to know exactly how the CMB light bends. Doing the math for all of them takes forever.
The Solution: The authors fed these 150 recipes into a computer program (called Operon) that acts like a creative detective. Instead of using a "black box" (like a neural network, which is great at guessing but hard to understand), this detective searches for a single, neat mathematical formula that fits all the data.
The Result: They found a formula that acts as a perfect stand-in for the complex physics. It's like finding a simple rule of thumb that predicts the weather just as accurately as a supercomputer model, but you can write it down on a napkin.

Why is this special?

It's a "Real" Formula: Unlike many modern AI tools that are opaque "black boxes" full of millions of hidden numbers, CMBolic is a closed-form equation. It's a clean, readable mathematical expression with only 45 numbers in it. You can plug it into any computer language (C++, Python, Fortran) without needing heavy software libraries.
It's Incredibly Fast: The paper claims that while the old method (using a program called CLASS) took two weeks to run a specific analysis, CMBolic did the exact same job in under 3 minutes (specifically 164 seconds). That's a speed-up of over 7,000 times!
It's Accurate: The authors tested this shortcut against the "gold standard" calculations. They found that the shortcut is wrong by less than 0.32% on average. To put that in perspective, the "noise" in future telescopes (like the upcoming CMB-S4 experiment) will be much larger than this tiny error. So, the shortcut is precise enough for the most advanced experiments coming up.

What does it actually do?

This specific paper focuses on lensing (the bending of light). The tool can handle a "universe recipe" that includes:

Dark Energy: The mysterious force pushing the universe apart (modeled with two variables, $w_0$ and $w_a$ ).
Neutrinos: Tiny, ghostly particles with mass (modeled by their total mass, $\Sigma m_\nu$ ).

The authors tested CMBolic using real data from the ACT and Planck telescopes. They found that the results (the "posters" of what the universe parameters likely are) were virtually identical to the results from the slow, traditional method.

The Bottom Line

CMBolic is a new, ultra-fast, and highly accurate "symbolic emulator" for the cosmic microwave background. It replaces weeks of heavy computing with a simple mathematical formula that anyone can use. It allows scientists to test complex theories about the universe's composition in minutes rather than weeks, making the process of discovering the secrets of the cosmos much more efficient.

Note: This paper is the first in a series. The authors plan to use this same "shortcut" approach to model other parts of the cosmic background (like temperature and polarization) in future work, but this specific paper only covers the lensing aspect.

Technical Summary: CMBolic: Symbolic Emulators for the Cosmic Microwave Background (I. Lensing)

Problem Statement
Cosmological parameter estimation using Cosmic Microwave Background (CMB) data, particularly from next-generation Stage 4 experiments, requires the evaluation of theoretical power spectra millions of times within Markov Chain Monte Carlo (MCMC) frameworks. Standard Einstein-Boltzmann solvers like CLASS or CAMB, while precise, are computationally expensive, often requiring days or weeks to converge on extended cosmological models. While neural network (NN) emulators have been developed to accelerate this process, they often function as "black boxes," require specific software environments (e.g., TensorFlow, PyTorch), and frequently lack a unified framework for joint extended parameter spaces (specifically combining dynamical dark energy and massive neutrinos). Furthermore, existing NN approaches may not offer the same level of portability or analytical differentiability as closed-form mathematical expressions.

Methodology
The authors present CMBolic, a suite of symbolic emulators for CMB observables. This first installment focuses on the CMB lensing potential power spectrum, $C^{\phi\phi}_\ell$ , within an extended $\Lambda$ CDM model that includes the Chevallier-Polarski-Linder (CPL) parameterization for dynamical dark energy ( $w_0, w_a$ ) and the sum of neutrino masses ( $\Sigma m_\nu$ ).

Symbolic Regression (SR): Instead of using NNs, the authors employ SR to discover explicit analytical functions that fit the data. They utilize Operon, a genetic programming (GP) code based on linear tree representation, which allows for efficient evolution of mathematical expressions.
Dataset Generation: Training data was generated using CLASS v3.3.0 with ACT DR6-inspired accuracy settings and the HMcode 2020 nonlinear matter power spectrum prescription. A Latin Hypercube (LH) sampling was performed over a pragmatic parameter space covering current constraints and future priors (Table 1). The dataset included 150 cosmologies, with spectra evaluated at 500 logarithmically spaced multipoles ( $2 \le \ell \le 5500$ ) per cosmology, resulting in 75,000 training points.
Target and Optimization: The emulator targets the quantity $\log[10 \ell(\ell+1)C^{\phi\phi}_\ell / (2\pi A_s)]$ . The optimization was a multi-objective problem balancing complexity (tree size) and Root Mean Square Error (RMSE). The authors factored out the primordial amplitude $A_s$ explicitly, treating the remaining dependence as a function of other parameters.
Post-Processing: To ensure physical plausibility and reduce complexity, the authors manually simplified the best-fit expressions by removing negligible terms, rounding coefficients, and verifying that trigonometric factors acted as smooth curvature corrections rather than rapid oscillations.

Key Contributions

Unified Extended Emulator: CMBolic is the first single symbolic emulator covering the joint parameter space of $\{w_0, w_a, \Sigma m_\nu\}$ for CMB lensing. Previous emulators often treated these extensions separately or relied on modular NN families.
Analytic Form: The resulting emulator is a closed-form symbolic expression containing only 45 fitted scalar coefficients. It relies on standard mathematical operations (log, cos, square roots, analytic quotients) and does not require external machine learning libraries.
Portability and Differentiability: The analytic nature of the emulator allows for seamless integration into any programming language (C++, Fortran, Python) and enables the use of gradient-based inference methods (e.g., Hamiltonian Monte Carlo) without relying on finite-difference approximations.
Validation: The emulator was rigorously tested against independent validation sets and compared against the Boltzmann code CLASS in a full Bayesian inference pipeline using real ACT DR6 + Planck lensing likelihoods.

Results

Accuracy: On independent validation spectra ( $2 \le \ell \le 5500$ ), CMBolic achieves mean absolute fractional errors of 0.27% in the standard $\Lambda$ CDM subspace and 0.32% across the full extended parameter space. These errors are well below the 1% target motivated by Stage 4 noise forecasts and are significantly smaller than discrepancies arising from different nonlinear matter power spectrum prescriptions (e.g., HMcode 2020 vs. Halofit).
Inference Agreement: When used in Cobaya for MCMC analysis with ACT DR6 + Planck lensing-only data, the posterior distributions obtained with CMBolic show excellent agreement with those from CLASS.
- The one-dimensional posterior overlap is $>0.98$ for all parameters.
- The maximum shift in posterior means is only $0.038\sigma_{CLASS}$ .
- Constraints on neutrino mass ( $\Sigma m_\nu < 0.286$ eV) are identical between the two methods at the relevant precision.
Computational Speed: Replacing CLASS with CMBolic in a full extended inference run reduced the wall-clock time from over 2 weeks to 164 seconds (a speed-up factor of $>7.4 \times 10^3$ ) on the same hardware. A single-spectrum evaluation speed-up of $2.1 \times 10^5$ (without fast-math optimizations) was also demonstrated.

Significance and Claims
The paper claims that CMBolic provides a practical, high-precision alternative to both slow Boltzmann codes and complex neural network emulators. Its primary significance lies in the combination of Stage-4-relevant precision, portability (no dependency on ML frameworks), and differentiability. The authors emphasize that while raw speed is a benefit, the ability to perform fast, portable inference over a joint extended cosmological parameter space with a single, compact symbolic expression is the distinct advantage.

The authors remain modest regarding the scope, noting that the emulator is valid strictly within the trained parameter ranges and multipole interval ( $2 \le \ell \le 5500$ ) and relies on the specific HMcode 2020 prescription used for training. They state that changes to nonlinear prescriptions, baryonic feedback models, or significantly broader priors would require retraining or dedicated correction factors. This work serves as the first installment of the CMBolic series, with temperature and polarization spectra reserved for future publications.

Technical Summary: CMBolic: Symbolic Emulators for the Cosmic Microwave Background (I. Lensing)

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