Symbolic Emulators for Cosmology: Accelerating Cosmological Analyses Without Sacrificing Precision

1. Problem Statement

Cosmological parameter inference requires evaluating complex physical models (e.g., matter power spectra, growth factors) thousands of times to explore high-dimensional parameter spaces.

• The Bottleneck: Direct numerical simulations (N-body) are computationally prohibitive for likelihood-based inference.
• Current Solutions: Numerical machine learning emulators (e.g., AEMULUS, BACCO, COSMOPOWER) are faster but suffer from interpretability issues, dependency on specific software packages, and often lack portability.
• Limitations of Existing Symbolic Methods: Previous symbolic regression approaches (e.g., SYREN) offered interpretable, fast equations but were restricted to narrow prior ranges, making them unsuitable for modern, broad-scope cosmological surveys (Stage-III and IV).
• Goal: Develop symbolic emulators that cover the broad parameter ranges required for current analyses (like the Dark Energy Survey) while maintaining sub-percent accuracy and offering significant gains in speed and memory efficiency.

2. Methodology

The authors utilized Symbolic Regression via the genetic programming code OPERON to discover analytic approximations for key cosmological quantities.

A. Training Data & Parameter Space

• Model: $\Lambda$CDM cosmology.
• Parameters: $\Omega_m, \Omega_b, h, n_s, \sigma_8$ (and $A_s$).
• Priors: Expanded significantly compared to previous works to match current survey needs (e.g., $\Omega_m \in [0.1, 0.5]$, $z \in [0, 3]$).
• Generation: Training and validation datasets were generated using Latin Hypercube sampling and evaluated using standard codes like CAMB and HALOFIT.

B. Key Symbolic Approximations Developed

The authors derived new analytic expressions for four critical components:

1. Radial Comoving Distance ($\chi$):

   • Challenge: Requires evaluating the Gaussian hypergeometric function $_2F_1$, which is computationally expensive and lacks native support in differentiable frameworks like JAX.
   • Solution: An approximation for $_2F_1$ accurate to $<0.001\%$ for all redshifts and $\Omega_m \in [0.1, 0.5]$.
   • Feature: The error decreases when extrapolating to higher redshifts (beyond the training range).

2. Linear Growth Factor ($D(z)$):

   • Challenge: Also relies on a hypergeometric function.
   • Solution: An analytic approximation accurate to $<0.05\%$.
   • Feature: Similar to the distance emulator, it exhibits excellent extrapolation behavior at high redshifts.

3. Conversion between $A_s$ and $\sigma_8$:

   • Solution: A symbolic expression for $\log(\sqrt{10^9 A_s}/\sigma_8)$ as a function of $\Omega_m, \Omega_b, h, n_s$.
   • Accuracy: RMSE of $\sim 1.3 \times 10^{-3}$ on training data.

4. Linear Matter Power Spectrum ($P_{lin}$):

   • Strategy: Instead of fitting $P_{lin}$ directly, the authors modeled the ratio $F(k, \theta)$ between the true CAMB spectrum and the Eisenstein & Hu (E&H) "no-wiggle" approximation: $P_{lin} = D^2 P_{no-wiggle} F$.
   • Solution: A compact symbolic form for $\log F$ involving oscillatory terms (cosine) to capture Baryon Acoustic Oscillations (BAOs).
   • Accuracy: Achieves a 0.6% RMSE across the parameter space, significantly outperforming the standard E&H formula (3% RMSE).

5. HALOFIT Variables ($k_\sigma, n_{eff}, C$):

   • Challenge: Calculating these derived variables requires numerical integration, root-finding, and spline fitting, which is slow.
   • Solution: Direct symbolic approximations for the nonlinear scale ($k_\sigma$), effective slope ($n_{eff}$), and curvature ($C$).
   • Accuracy: All variables achieved sub-percent accuracy (e.g., $n_{eff}$ error $<0.1\%$).

3. Validation & Results

A. Mock Cosmological Analysis (DES-Y1 3$\times$2pt)

The authors embedded their symbolic emulators into a mock Dark Energy Survey Year 1 (DES-Y1) analysis combining cosmic shear, galaxy-galaxy lensing, and galaxy clustering.

• Comparison: They compared the symbolic pipeline against:
  1. COSMOSIS: Using CAMB + HALOFIT (the "exact" benchmark).
  2. JAX-COSMO: Using E&H + HALOFIT.
  3. COSMOPOWER: A numerical neural network emulator.
• Inference Accuracy:
  • The Symbolic Emulator results were statistically indistinguishable from the COSMOSIS/CAMB benchmark for well-constrained parameters ($\Omega_m, n_s, A_s, S_8$).
  • In contrast, the standard E&H + HALOFIT combination introduced significant biases (shifts in posterior peaks for $\Omega_m$ and $n_s$).
  • Nuisance parameters (bias, shear calibration, photo-z shifts) also showed no bias when using the symbolic approach.

B. Performance Metrics

• Speed:
  • The symbolic pipeline was ~60 times faster than the COSMOSIS/CAMB inference (1.2 hours on a single GPU vs. 72 hours on 128 CPU cores).
  • It was ~3 times faster than the COSMOPOWER numerical emulator (1.2h vs 3.5h).
  • The speedup was driven primarily by the analytic approximations for the comoving distance and growth factor, which eliminated the need for numerical integration/differential equation solving.
• Memory Efficiency:
  • Evaluating gradients (essential for efficient sampling like NUTS) required 50% less memory with the symbolic emulators compared to the JAX-COSMO pipeline. This allows for more parallel MCMC chains per GPU.

4. Key Contributions

1. Extended Parameter Validity: Successfully expanded symbolic emulators from narrow research ranges to the broad priors required for Stage-III cosmological surveys.
2. High-Precision Hypergeometric Approximations: Introduced new analytic forms for $_2F_1$ used in distance and growth calculations, achieving accuracy better than $10^{-5}$ and $5 \times 10^{-4}$ respectively, with superior extrapolation properties.
3. End-to-End Validation: Demonstrated for the first time that a fully symbolic pipeline can replace numerical methods in a realistic 3$\times$2pt analysis without introducing cosmological biases.
4. Portability and Interpretability: Provided equations that are language-agnostic, easily differentiable (compatible with JAX/PyTorch), and do not depend on heavy external libraries.

5. Significance

This work establishes symbolic regression as a viable, and often superior, alternative to numerical machine learning emulators for cosmology.

• Scalability: The massive reduction in compute time and memory usage makes these emulators ideal for the upcoming flood of data from Stage-IV surveys (e.g., Euclid, Rubin, DESI).
• Robustness: The ability to analytically differentiate these expressions improves the efficiency of high-dimensional sampling algorithms (like Hamiltonian Monte Carlo).
• Future Outlook: While currently validated for Stage-III, the authors note that the framework can be extended to Stage-IV requirements by training on more accurate power spectrum models (e.g., SYREN-NEW) with wider priors.

In conclusion, the paper proves that one can accelerate cosmological inference by orders of magnitude without sacrificing precision, provided that the underlying physical models are approximated via robust symbolic regression rather than black-box numerical surrogates.