Testing $Λ$CDM with ANN-Reconstructed Expansion History from Cosmic Chronometers

Abstract: In modern cosmology, the rapid growth of high-precision observational data, along with significant theoretical advances, has intensified the challenge of identifying a robust, model-independent framework to probe the expansion history of the Universe. In this work, we propose a novel artificial neural network (ANN)-based framework for the non-parametric reconstruction of the late-time cosmic expansion. The framework is trained and validated through a three-stage screening pipeline prior to its application to real observational data. As a demonstration of its effectiveness, we reconstruct the Hubble parameter $H(z)$ using the latest cosmic chronometer measurements. Our results show that the reconstructed expansion history aligns with the predictions of the $Λ$CDM model within observational uncertainties, thereby supporting the robustness and reliability of the proposed approach.

• The paper introduces a three-stage ANN pipeline for nonparametric reconstruction of the Hubble parameter using Cosmic Chronometer data, reducing model bias.
• It employs mock data-driven architecture selection and ensemble methods to quantify uncertainty, showing consistency with ΛCDM predictions across various H₀ priors.
• The approach decouples network design from real data, ensuring transparency and reproducibility, and setting a foundation for future multi-probe cosmological analyses.

Testing $\Lambda$CDM with ANN-Reconstructed Expansion History from Cosmic Chronometers

Introduction

This work presents a nonparametric cosmological reconstruction framework based on artificial neural networks (ANNs) for recovering the late-time expansion history of the universe using Cosmic Chronometer (CC) measurements. The authors address the critical limitations of parametric cosmological modeling and propose a staged, reproducible process to infer the Hubble parameter $H(z)$ from direct differential-age data. The study’s essential contribution is the explicit and methodical separation between model design (activation/loss/architecture selection) and application to observational data, mitigating a posteriori selection bias and enhancing methodological transparency.

Theoretical Context and Methodological Advances

In the standard cosmological model, i.e., spatially flat $\Lambda$CDM, the redshift-dependent Hubble parameter is defined by the Friedmann equations incorporating contributions from matter, radiation, and dark energy. Numerous model-independent approaches for $H(z)$ reconstruction have been established (e.g., Gaussian processes, genetic algorithms, neural regressors), but a significant challenge has persisted: robust and auditable architecture selection and uncertainty quantification in ANN-based reconstructions.

The key methodological innovation here is a three-stage pipeline:

• Stage 1: Mock Data Generation – Synthetic CC-like samples sampled from a fiducial $\Lambda$CDM background with realistic redshift and error distributions are used for unbiased design selection.
• Stage 2: Model Selection – Activation functions (ELU, $\tanh$, SiLU, TanhExp) and network architectures (width, depth) are systematically screened over repeated mock realizations and random seeds, with the performance quantified by scores incorporating truth-recovery and smoothness metrics.
• Stage 3: Application to Real Data – The selected architecture (fixing all hyperparameters a priori) is trained on actual CC compilations, with three distinct external $H_0$ priors (Planck 2018, TRGB, SH0ES R21) to assess the effect of low-redshift anchoring.

This pipeline’s salient property is that it fully decouples the architecture and training decision process from the observed data, making the ANN a genuine prediction tool rather than a curve-fitting mechanism subject to over-tuning.

Data, Architecture, and Training Protocols

CC data directly measure $H(z)$ via the differential-age of passively evolving galaxies and cover $0.07 \leq z \leq 1.97$. The mock pipeline faithfully matches the observed redshift distribution and error profile, creating high-fidelity testbeds for architecture benchmarking.

A simple feed-forward ANN with a single hidden layer of 128 SiLU-activated units, determined via mock-based architecture selection, constitutes the main reconstructive engine. The training employs an L1 weighted objective with aleatoric uncertainty prediction (‘sigma head’), Adam optimization, and ensemble methods for epistemic uncertainty quantification (100 independently initialized members per $H_0$ prior). No validation set or a posteriori hyperparameter drift is permitted, aligning with the demand for auditable pipeline integrity.

Main Results and Numerical Benchmarks

Stage 2: Design Search

The performance ranking over five mock realizations and 100 seeds each reveals the SiLU activation to yield the optimal average selection score ($H(z)$0), with the best architectures being shallow (one hidden layer, 128 units). The findings underscore that moderate network complexity suffices for current CC data and that model selection cannot be trivially abstracted from either the loss or activation function.

Stage 3: CC Reconstruction and Comparison with $H(z)$1CDM

Applying the fixed architecture to the observed CC sample, the recovered $H(z)$2 history aligns with the $H(z)$3CDM prediction at the $H(z)$4 level for all three $H(z)$5 priors. Notably, the ensemble-mean $H(z)$6 values reconstructed at $H(z)$7 accurately reflect their imposed priors:

• Planck 2018: $H(z)$8
• TRGB: $H(z)$9
• SH0ES R21: $\Lambda$0 (all units km s$\Lambda$1 Mpc$\Lambda$2)

The corresponding inferred matter densities for flat-$\Lambda$3CDM fits also track the $\Lambda$4 prior: $\Lambda$5 transitions from 0.33 (Planck) through 0.30 (TRGB) down to 0.25 (SH0ES), quantitatively consistent with the expectation from the Friedmann equation.

The RMSE on the empirical data lies in the interval $\Lambda$6–$\Lambda$7 km s$\Lambda$8 Mpc$\Lambda$9, and the 68% predictive coverage is slightly conservative (0.909 across priors). The reconstructed $H(z)$0 statistic remains flat and matches the $H(z)$1 inferred by parametric fits for each prior, reinforcing consistency with the flat-$H(z)$2CDM framework.

No statistically significant deviation from $H(z)$3CDM is detected by this CC-only, prior-anchored ANN, given current uncertainties and sample sizes.

Implications, Limitations, and Future Work

The formal structure of this pipeline introduces reproducibility and mitigation of implicit look-elsewhere effects previously pervasive in nonparametric cosmological ANN literature. The explicit prior anchoring at $H(z)$4 clarifies that current CC data cannot independently arbitrate among the $H(z)$5 tension scenarios, instead propagating the chosen low-$H(z)$6 anchor through the reconstructed expansion history.

Limitations include the restriction to CC-only and single-anchor analyses—multi-probe (e.g., SNe Ia, BAO), covariance-aware, and simulation-based extensions are a logical next step. Additionally, while the ANN ensemble framework robustly represents aleatoric uncertainty, hierarchical Bayesian architectures capable of more comprehensively propagating both statistical and systematic uncertainty would further augment the interpretive power of such reconstructions.

Extending this staged design to (i) joint CC+SNe+BAO, (ii) covariance-structured losses, and (iii) null test diagnostics or direct reconstructions of derived quantities such as $H(z)$7 or $H(z)$8 will be essential for thorough $H(z)$9CDM stress-testing in forthcoming high-precision cosmological data regimes.

Conclusion

This work demonstrates that carefully benchmarked, mock-validated, and ensemble-averaged ANN architectures can deliver nonparametric, data-driven, and reproducible reconstructions of the Hubble parameter’s redshift evolution directly from CC measurements. With appropriate external prior selection, all results are fully consistent with the predictions of spatially flat $\Lambda$0CDM. The technical blueprint laid out here constitutes a robust foundation for systematic, multi-probe, and covariance-sensitive AI accelerants to cosmological model selection, with significant relevance for future high-precision expansion history inference.