Inverse Oscillations: Symbolic Regression Discovers a New Rhythm of the Early Universe
The search for oscillatory features in the primordial power spectrum, pointing to new physics in the early Universe, is typically conducted using fixed templates. In this study, a template-free search was performed for the first time using symbolic regression. Analysis of Planck data and the combination Planck+ACT+SPT-3G independently singled out an inverse k-oscillation: cos(B/k) with B≃4 Mpc⁻¹ as the most simple interpretable structure. Comparison with linear and logarithmic oscillating models showed that the inverse cosine better describes the observations, demonstrating a mild preference for a nonzero amplitude. The results demonstrate the effectiveness of symbolic regression as an interpretable, model-independent tool for cosmological discoveries.
Context
Primordial oscillations are key to the physics right after the Big Bang. Georges Lemaître's theory of the expanding universe was confirmed by spectroscopy of the cosmic microwave background, predicted by Ralph Alpher. Deviations from a simple power-law spectrum could indicate exotic processes inaccessible to particle accelerators.
Methods
Instead of templates, the authors used symbolic regression—the PySR algorithm, which searches through analytic expressions, minimizing the discrepancy with cosmic microwave background data. Spectra were computed with the CLASS code at fixed cosmological parameters. Two datasets were used: Planck alone and the combination Planck+ACT+SPT-3G.
Results
Both datasets independently selected an oscillation of the form cos(B/k) with B≈4 Mpc⁻¹. For Planck, the amplitude was −0.03, and for SPA, −0.02. MCMC comparison showed that the inverse template gives a chi-squared improvement of ~12.6 versus ~7.7 for linear and ~5.8 for logarithmic. The amplitude of the inverse oscillation has weak nonzero significance: A = 0.0163+0.0099−0.0071, while the others are consistent with zero. The primordial origin of the signal is supported by polarization data on scales 800≲ℓ≲2000.
Implications
The inverse oscillation may point to a period of super-fast expansion with violation of the null energy condition (w < −1), proposed in some models. This links observational cosmology to quantum gravity and string theory, opening a window to the Planck era.
Future development
Future experiments like Simons Observatory and CMB-S4, with improved resolution on small scales, will confirm or refute the inverse oscillation. Moreover, the spectroscopic approach with symbolic regression will be increasingly applied to search for non-template signals.
Impact
The results touch upon early universe physics, modified gravity, and machine learning methods in astrophysics.
Next steps
Detailed theoretical elaboration of models that generate inverse oscillations is planned, along with their verification on new data.
Key open problems
The work aligns with the search for inflationary physics beyond the Standard Model, linking the problem of dark energy's nature and the initial singularity of the Big Bang.
🎯 Symbolic regression allowed the algorithm to 'invent' the function cos(B/k) without clues—much like it previously rediscovered Kepler's laws from data.
🎬 Oscillations on cosmological scales echo the imagery of a pulsating universe in the works of Ursula Le Guin.
Key numbers
- Frequency B (inverse oscillation): ≈ 4.1 Mpc⁻¹
- Amplitude A (Planck): ≈ 0.03
- Amplitude A (SPA): ≈ 0.02
- Δχ² improvement (inverse vs linear): ~12.7
- Δχ² improvement (inverse vs logarithmic): ~5.8