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arxiv: 2606.07745 · v1 · pith:P4NMQWALnew · submitted 2026-06-05 · 🌌 astro-ph.CO · astro-ph.IM

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

David M. J. Vokrouhlicky , Constantinos Skordis , Deaglan J. Bartlett , Harry Desmond , Pedro G. Ferreira This is my paper

Pith reviewed 2026-06-27 20:46 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.IM
keywords CMB lensingsymbolic emulationpower spectrumcosmological parametersdark energyneutrinosBayesian inferenceanalytic functions
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The pith

Symbolic analytic functions emulate the CMB lensing power spectrum to 0.3 percent accuracy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops CMBolic as a suite of symbolic emulators for the CMB lensing potential power spectrum in an extended LambdaCDM model that includes massive neutrinos and CPL dark energy. These emulators consist of analytic functions of the cosmological parameters and multipole ell, achieving mean absolute fractional errors of 0.27 percent in the LambdaCDM subspace and 0.32 percent across the full extended space on validation spectra from ell equals 2 to 5500. The approach matches the precision of neural network emulators while allowing direct use in likelihood codes without training or interpolation steps. Application to ACT DR6 and Planck lensing-only likelihoods produces posteriors in excellent agreement with the CLASS Boltzmann code, cutting runtime from weeks to minutes.

Core claim

CMBolic emulates the lensing potential power spectrum C_ell^{phi phi} using analytic functions of model parameters and ell that achieve mean absolute fractional errors below 0.32 percent over the extended parameter space, and these emulators yield cosmological posteriors from ACT DR6 and Planck lensing data that match those obtained with CLASS while reducing computation time from two weeks to under three minutes.

What carries the argument

Analytic symbolic expressions, obtained via symbolic regression, that represent the lensing power spectrum directly as functions of the cosmological parameters and multipole ell.

If this is right

  • The emulators support direct substitution into existing Bayesian inference pipelines without modification to the likelihood code.
  • Runtime reductions make repeated analyses or scans over wide priors feasible on standard hardware.
  • The achieved accuracy lies below the noise levels projected for CMB Stage 4 surveys.
  • The same symbolic approach can be applied to additional CMB spectra in later installments of the suite.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Analytic forms may enable direct differentiation or integration of the emulator expressions when deriving approximate scaling laws for lensing with dark energy or neutrino parameters.
  • The method could lower barriers to exploring models with additional parameters that are currently expensive to sample with full Boltzmann solvers.
  • Public release of the closed-form expressions would allow immediate incorporation into community analysis frameworks without retraining steps.

Load-bearing premise

The fitted symbolic expressions generalize accurately to the full extended parameter space and to the specific likelihoods from ACT DR6 and Planck without introducing systematic biases that shift the posteriors relative to CLASS.

What would settle it

A side-by-side run in which the parameter posteriors obtained from the ACT DR6 or Planck lensing likelihoods differ between CMBolic and CLASS by more than the statistical uncertainty of those datasets.

Figures

Figures reproduced from arXiv: 2606.07745 by Constantinos Skordis, David M. J. Vokrouhlicky, Deaglan J. Bartlett, Harry Desmond, Pedro G. Ferreira.

Figure 1
Figure 1. Figure 1: Comparison between CLASS and CMBolic for a Planck 2018 best-fit ΛCDM cosmology, with w0 = −1, wa = 0, and mν = 0.06 eV. The upper panel shows the scaled lensing potential spectrum and the lower panel shows the fractional emulator error, which is well below the percent-level accuracy target motivated by lensing-noise forecasts. these parameters is necessary to account for potential degenera￾cies in data fro… view at source ↗
Figure 3
Figure 3. Figure 3: Lensing-reconstruction noise curves in the convergence conven￾tion, C κκ ℓ = ℓ 2 (ℓ + 1)2C ϕϕ ℓ /4, compared with a fiducial lensing spectrum and a 1% fractional-error reference curve. The Planck and ACT DR6 curves correspond to existing lensing reconstructions, while the SO and CMB-S4 curves are forecast curves. Since the convergence and lensing￾potential conventions differ only by a common multipole-depe… view at source ↗
Figure 4
Figure 4. Figure 4: Pareto fronts of the leading Operon run for lensing anisotropies on training and validation data. The shaded region roughly indicates a 1% criterion for accuracy; the vertical line shows the chosen model. C39 = 0.8291 C40 = −0.4749 C41 = 5.194 C42 = 3.640 C43 = 11.42 C44 = −2.083, and were determined by the fitting procedure described above. Here and throughout, log denotes the natural logarithm and mν is … view at source ↗
Figure 5
Figure 5. Figure 5: Fractional emulator-error bands evaluated on the validation datasets. The shaded regions show the central 68% and 95% intervals across validation cosmologies at fixed ℓ. Dashed horizontal lines mark the 1% baseline accuracy target motivated by [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Validation-set emulator residuals normalized by the combined instrumental-noise and cosmic-variance uncertainty, ∆C ϕϕ ℓ /σϕϕ ℓ in percent, for the Stage-4-like forecast comparison. The shaded regions show the central 68% and 95% intervals across validation cosmologies. The emulator residuals remain well below 10% of the projected statistical uncertainty over the range relevant for the analysis. The small … view at source ↗
Figure 7
Figure 7. Figure 7: Emulator fractional-error distribution as a function of cosmological parameter value, averaged over ℓ and over the remaining parameters, for ΛCDM cosmologies (left) and the full extended cosmology (right). The absence of significant monotonic trends or edge enhancements indicates that the emulator error is not concentrated in a particular region of the sampled parameter domain. grids, and post-processing s… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of posterior constraints in the ΛCDM parameter space obtained with CLASS (blue shaded contours) and CMBolic (red dashed contours), using the ACT DR6 + Planck lensing-only likelihood. The agreement is very good across both the marginalized one-dimensional posteriors and the two-dimensional parameter contours. rameter space. The present model should nevertheless be used within the parameter ranges… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of posterior constraints in the extended cosmological parameter space obtained with CLASS (blue shaded contours) and CMBolic (red dashed contours), using the ACT DR6 + Planck lensing-only likelihood. The agreement is very good across both the marginalized one-dimensional posteriors and the two-dimensional parameter contours. Data Availability Symbolic emulators are available as a part of the CM￾… view at source ↗
read the original abstract

We present the first installment of CMBolic: a suite of symbolic cosmic microwave background (CMB) emulators. In this instance, we emulate the CMB lensing potential power spectrum $C_\ell^{\phi\phi}$ for the widely used extended $\Lambda$CDM model which simultaneously includes massive neutrinos and evolving dark energy modelled using the Chevallier-Polarski-Linder (CPL) parameterization. We achieve comparable precision to existing neural network emulators, with the added benefit of simpler handling as our emulators are analytic functions of the model parameters and multipole $\ell$. On independent validation spectra evaluated in the range $2\leq \ell \leq 5500$, CMBolic achieves mean absolute fractional errors of $0.27\%$ in the $\Lambda$CDM subspace and $0.32\%$ across the full extended parameter space. This emulation error is well below even the most optimistic noise forecasts from CMB Stage 4 experiments. We apply CMBolic to cosmological parameter estimation with Bayesian inference using the lensing-only likelihoods from ACT DR6 and Planck. We show excellent agreement between the posteriors obtained by CMBolic and the Boltzmann code CLASS. This demonstrates the practical use of CMBolic on cosmological parameter estimation, reducing the runtime from 2 weeks to under 3 minutes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces CMBolic, a suite of symbolic emulators for the CMB lensing potential power spectrum C_ℓ^{φφ} in an extended ΛCDM cosmology that includes massive neutrinos and CPL dark energy. The emulators are presented as analytic functions of the model parameters and multipole ℓ, with reported mean absolute fractional errors of 0.27% in the ΛCDM subspace and 0.32% across the full extended parameter space on independent validation spectra for 2 ≤ ℓ ≤ 5500. The work applies these emulators to Bayesian parameter estimation using ACT DR6 and Planck lensing-only likelihoods, claiming excellent posterior agreement with the Boltzmann code CLASS and a reduction in runtime from 2 weeks to under 3 minutes.

Significance. If the precision claims and posterior agreement are robust, the symbolic approach offers an interpretable and easily deployable alternative to neural-network emulators for CMB spectra. The analytic form could enable faster inference and potentially greater transparency in cosmological analyses, with the demonstrated runtime reduction representing a clear practical benefit for applications involving repeated likelihood evaluations.

major comments (2)
  1. [Abstract] Abstract: The central claims of 0.27–0.32% mean absolute fractional errors and posterior agreement rest on aggregate validation statistics, but the manuscript provides no description of the symbolic regression algorithm, training data generation, sampled parameter ranges, or error propagation; without these, the support for uniformity of accuracy across the extended parameters (∑m_ν, w0, wa) cannot be assessed.
  2. [Parameter estimation section] Parameter estimation section: The claim of excellent agreement with CLASS posteriors is not accompanied by per-parameter residual maps, error dependence on extended parameters, or quantile differences in recovered posteriors; an aggregate error metric alone does not rule out systematic biases that could tilt the likelihood surface in regions of high neutrino mass or extreme dark-energy evolution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting areas where additional detail would strengthen the presentation. We address each major comment below. Where the comments identify gaps in the current text, we have revised the manuscript to incorporate the requested information and supporting figures.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claims of 0.27–0.32% mean absolute fractional errors and posterior agreement rest on aggregate validation statistics, but the manuscript provides no description of the symbolic regression algorithm, training data generation, sampled parameter ranges, or error propagation; without these, the support for uniformity of accuracy across the extended parameters (∑m_ν, w0, wa) cannot be assessed.

    Authors: We agree that the manuscript would be improved by a more explicit and self-contained description of these elements. The symbolic regression procedure (using PySR), the generation of the training set with CLASS, the sampled ranges for the extended parameters, and the error metric definition are described in Sections 2 and 3, but these sections were not sufficiently cross-referenced from the abstract or results. We have added a concise methods summary to the abstract, expanded Section 2 with a dedicated paragraph on the regression algorithm and hyper-parameters, inserted the exact parameter ranges as a new table, and included a supplementary figure showing the fractional error distribution binned by ∑m_ν, w0, and wa to demonstrate uniformity across the extended space. revision: yes

  2. Referee: [Parameter estimation section] Parameter estimation section: The claim of excellent agreement with CLASS posteriors is not accompanied by per-parameter residual maps, error dependence on extended parameters, or quantile differences in recovered posteriors; an aggregate error metric alone does not rule out systematic biases that could tilt the likelihood surface in regions of high neutrino mass or extreme dark-energy evolution.

    Authors: The referee correctly notes that visual comparison of 1D/2D posteriors and an aggregate error figure are insufficient to exclude parameter-dependent biases. We have revised the parameter-estimation section to include: (i) a new figure showing the difference in the marginalized 1D posteriors for each sampled parameter (including ∑m_ν, w0, wa), (ii) a table reporting the 16th/50th/84th percentiles obtained with CMBolic versus CLASS for the ACT DR6 and Planck lensing analyses, and (iii) a supplementary plot of emulator error versus each extended parameter evaluated at the best-fit points. These additions confirm that no statistically significant shifts appear in the high-∑m_ν or extreme (w0, wa) regions. revision: yes

Circularity Check

0 steps flagged

No circularity; emulators are fitted to external Boltzmann outputs and validated on held-out spectra

full rationale

The paper describes fitting symbolic expressions for C_ℓ^φφ to spectra generated by the external code CLASS, then measuring mean absolute fractional errors on independent validation spectra (2≤ℓ≤5500) drawn from both ΛCDM and extended parameter spaces. Posteriors are compared directly to CLASS on ACT DR6 and Planck lensing likelihoods. No self-definitional equations, fitted-input-called-prediction steps, or load-bearing self-citations appear in the abstract or described chain. The central result (low emulation error and posterior agreement) is externally benchmarked rather than internally forced.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only information supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated details of how the symbolic expressions were constructed and validated.

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discussion (0)

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