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

Abstract

We explore the impact of finite-temperature quantum gravity effects on cosmological parameters, particularly the effective vacuum-energy sector, by incorporating temperature-dependent quantum corrections into the Hubble parameter. To this end, we modify the Cosmic Linear Anisotropy Solving System and introduce new density parameters,

and

$$\Omega _{\Lambda _3}$$ Ω Λ 3

, arising from finite-temperature quantum gravity contributions. These parameters encode off-shell vacuum-bubble contributions from Standard Model fields—including heavy fields that cannot be treated as on-shell radiation—and may equivalently be viewed as quantum corrections to the effective equation of state of the early-Universe plasma. We analyze their influence on the cosmic microwave background power spectrum using machine learning techniques, including artificial neural networks and stochastic optimization. Our results reveal that

$$\Omega _{\Lambda _2}$$ Ω Λ 2

assumes a negative value, consistent with dimensional regularization in renormalization, and that the inclusion of

$$\Omega _{\Lambda _2}$$ Ω Λ 2

and

$$\Omega _{\Lambda _3}$$ Ω Λ 3

improves the fit to 2018 Planck data. The present approach is a phenomenological model-building exercise: a first-principles derivation that would rigorously separate on-shell kinetic-theory content from off-shell vacuum contributions has not yet been carried out, but the approach captures the leading effects. Although the Hubble tension persists, our findings highlight the potential of quantum gravitational corrections in refining cosmological models and motivate further investigation into higher-order thermal effects and polarization data constraints.