Stochastic analysis of finite-temperature effects on cosmological parameters by artificial neural networks

A phenomenological study combining a modified cosmological Boltzmann code with artificial neural networks found that adding two finite-temperature quantum-gravity parameters produced a closer numerical match between a theoretical cosmic microwave background spectrum and the 2018 Planck reference curve. The parameters, denoted ΩΛ2 and ΩΛ3, represent temperature-dependent vacuum contributions in the model’s expansion history. Across the reported searches, the preferred ΩΛ2 values were small and negative, while ΩΛ3 remained positive. The eight-parameter model reached a lower distance from the Planck spectrum than a seven-parameter comparison that omitted ΩΛ2. The study did not resolve the Hubble tension, did not provide the posterior uncertainties associated with a full Bayesian cosmological analysis, and did not establish that the new parameters had been directly detected.

The theoretical framework begins with the one-particle irreducible effective action for the Standard Model coupled to gravity. Vacuum-to-vacuum diagrams in this framework generate temperature-dependent corrections to the cosmological constant. At leading order, the corrections scale with the fourth and second powers of temperature, so the effective cosmological constant can be written schematically as Λ = Λ1 + Λ2a⁻⁴ + Λ3a⁻² + …, where a is the cosmic scale factor. The associated density parameters ΩΛ2 and ΩΛ3 resemble radiation and curvature in their scale-factor dependence, but the study assigns them a different physical origin. Standard radiation is the energy density of physical particles such as photons and relativistic neutrinos, whereas the new parameters summarize off-shell vacuum-bubble contributions from Standard Model fields and, in the complete framework, gravitons.

Heavy fields are central to that distinction. The Higgs boson, top quark, and electroweak gauge bosons have masses far above the recombination-era temperature and do not behave as ordinary relativistic radiation at those temperatures. Nevertheless, virtual vacuum bubbles can contribute to the finite-temperature effective action. The paper describes two equivalent ways of organizing these terms: as time-dependent shifts in the effective cosmological-constant sector, or as quantum corrections to the effective equation of state of the early-Universe plasma. The numerical analysis is independent of which of these interpretations is preferred, because the same T⁴ and T² terms enter the background expansion rate. The new components are treated as spatially homogeneous, background-only effects in the approximation studied; no separate perturbation equations are added for them, unlike for standard radiation, baryons, cold dark matter, or curvature.

To implement the proposal, the authors modified the Cosmic Linear Anisotropy Solving System, or CLASS, which evolves a specified cosmological model and calculates observables including cosmic microwave background power spectra. The eight parameters explored were ΩΛ2, ΩΛ3, the dimensionless Hubble parameter h, physical baryon density ωb, physical cold-dark-matter density ωcdm, scalar amplitude As, spectral index ns, and optical depth τreio. The modified Hubble parameter included the ordinary matter contribution together with terms proportional to ΩΛ2 and ΩΛ3. At the level used in the study, ΩΛ2 and ΩΛ3 alter the homogeneous expansion history but do not source perturbations. The authors also report numerical checks indicating that ΩΛ3 is not redundant with curvature density, despite their similar appearance in the Hubble expression, because curvature enters the action and perturbation equations as a geometric quantity whereas ΩΛ3 originates in a quantum correction to the cosmological-constant sector.

The first stage of the numerical search consisted of broad scans over the eight-dimensional parameter space, producing tens of millions of points, followed by increasingly focused searches in regions that yielded small Euclidean distances to a theoretical Planck 2018 CMB spectrum. The reference file was an unbinned, noise-free theoretical spectrum sampled at integer multipoles and did not include a covariance matrix. The authors therefore used a deterministic comparison rather than a conventional Bayesian likelihood, describing the procedure as an iterative form of trial and error rather than a substitute for formal statistical inference. They reported that adding tools such as Cobaya or MontePython increased the runtime by a factor of six to eight, while their external script could rank spectra almost immediately once CLASS had calculated them. The data collection alone took more than three months, making computational efficiency central to the search strategy.

Artificial neural networks were then used to approximate the otherwise unavailable function connecting cosmological parameters to distance from the Planck curve. Separate networks were trained for the seven-parameter and eight-parameter populations. Each had four hidden layers of 16 neurons, with rectified linear activation functions, and received standardized inputs so that parameters with very different numerical scales could be processed more effectively. The networks were trained using the Adam optimizer with a learning rate of 0.001 and mean squared error loss. The main datasets contained 459,912 points for the seven-parameter model and 335,985 for the eight-parameter model. Despite having fewer samples and a coarser grid, the eight-parameter population had a smaller average distance to the 2018 Planck result, and its test loss converged more closely to its training loss than the seven-parameter model’s.

Repeated training provided a measure of variability in the neural-network estimator. Across 500 independent replications of the eight-parameter fitting procedure, with each model trained for 1,000 epochs, the test mean squared error had a median of 0.55 and an empirical 95% confidence interval from 0.31 to 1.14. A larger run produced a global average test mean squared error of 4.95 for a combined dataset of 795,897 points. The distance function ranged from 0 to 1,491, and the source cautions that not every digit in the theoretical Planck file carries statistical significance. These statistics characterize how accurately the neural networks reproduced the calculated distance surface. They are not probability distributions for the cosmological parameters and do not quantify uncertainty in the underlying finite-temperature theory.

Two optimization schemes searched the learned distance functions. Broyden–Fletcher–Goldfarb–Shanno optimization, using numerical gradients derived from the neural-network surrogate, found a minimum distance of 29.24 in the eight-dimensional model, compared with 32.09 in the seven-dimensional model. The corresponding eight-parameter values were ΩΛ2 = −2.069 × 10⁻⁸, ΩΛ3 = 0.00430, h = 0.6752, ωb = 0.02240, ωcdm = 0.1196, As = 2.1007 × 10⁻⁹, ns = 0.9660, and τreio = 0.05424. An artificial-network-guided stochastic simulated-annealing run, using random-walk Metropolis–Hastings transitions, converged to a slightly lower distance of 28.8111. Its values were ΩΛ2 = −2.061 × 10⁻⁸, ΩΛ3 = 0.00425, h = 0.6762, ωb = 0.0223, ωcdm = 0.1196, As = 2.1 × 10⁻⁹, ns = 0.9664, and τreio = 0.056. These are best-fit coordinates under the study’s distance metric, not posterior means or confidence-limited parameter estimates.

Feature-ablation experiments indicated that ΩΛ2 helped the neural networks reproduce the fine structure of the Planck spectrum. In an analysis restricted to the 5,000 points with the shortest distances, removing ΩΛ2 increased the average test mean squared error by approximately 25%, with 50-fold cross-validation used to construct confidence intervals for predictive performance. A global leave-one-out analysis also found that removing ΩΛ2 caused a larger error increase than removing some established cosmological parameters, although the large-scale shape of the curve remained primarily controlled by the conventional parameters. The investigators interpreted the negative ΩΛ2 estimates as consistent with dimensional regularization in their renormalization procedure. The sign is therefore a feature of the model, parameterization, and renormalization choice rather than an observationally established property of quantum gravity.

Several limitations restrict how the result can be interpreted. The authors explicitly characterize the work as a phenomenological model-building exercise, not a first-principles derivation of finite-temperature hydrodynamics from the Standard Model coupled to gravity. Their present approach couples the one-particle irreducible effective action to a separately specified hydrodynamic sector, leaving the full separation between on-shell kinetic content and off-shell vacuum contributions as an open theoretical task. The adopted T⁴ and T² expansion is not uniformly appropriate for every combination of field mass and cosmic temperature. For a heavy field, a low-temperature treatment may be required, and its leading contribution can scale as T⁶/m², implying an additional term such as ΩΛ4a⁻⁶. The framework also omits more complete in-medium effects, including Debye screening and collective excitations. Follow-up work would need a larger parameter set, formal Bayesian inference, thermal expansions that change appropriately across cosmic history, and tests against CMB polarization data. The defensible conclusion is thus a computationally promising improvement in fitting within a specified phenomenological model, not a detection of finite-temperature quantum corrections or a solution to the Hubble tension.

Subject of Research: Phenomenological analysis of finite-temperature quantum-gravity corrections to cosmological parameters and the Planck CMB spectrum

Article Title: Stochastic analysis of finite-temperature effects on cosmological parameters by artificial neural networks

Article References: Hatefi, A., Hatefi, E., & Park, I. Y. (2026). Stochastic analysis of finite-temperature effects on cosmological parameters by artificial neural networks. The European Physical Journal C, 86(10), Article 1203. https://doi.org/10.1140/epjc/s10052-026-16214-w

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16214-w

Keywords: finite-temperature cosmology, quantum gravity, cosmological constant, Planck CMB spectrum, artificial neural networks, CLASS Boltzmann code, vacuum energy, Hubble tension, stochastic optimization, Stochastic, analysis, finite-temperature