The Scretching–Planck Action and the Early Universe

Abstract

Research Description This research presents a critical mathematical review, correction, and extension of the twelve-page Part I investigation of early-universe cosmology using the Scretching–Planck (SP) formulation. The companion study revisits the mathematical foundations, physical interpretations, numerical estimates, and computational limitations of the original work, with particular emphasis on hydrogen recombination, cosmic microwave background (CMB) acoustic and damping scales, the 21-centimeter hydrogen transition, primordial nucleosynthesis, the proposed SP epoch, and ultralight dark-matter dynamics. A central objective is to distinguish exact mathematical reparameterizations of established physical laws from independently derived physical predictions. The SP formulation expresses familiar quantum and cosmological relationships through characteristic length, time, and energy variables. This investigation clarifies which results follow directly from algebraic transformations of conventional equations and which require additional physical assumptions, numerical modeling, or observational validation. The study independently recomputes fundamental constants, characteristic scales, and selected analytical formulas to assess their numerical consistency and dimensional correctness. Particular attention is given to correcting the sign of the Peebles recombination equation when expressed in redshift coordinates, recalibrating the comparison between the predicted CMB acoustic angular scale and Planck reference measurements, and ensuring that the baryonic quantum-pressure estimate consistently employs the physical wavenumber rather than the comoving wavenumber. These corrections address specific mathematical and numerical issues affecting the interpretation of the original calculations. The research also revisits the 24 solved problems presented in Part I, examining their assumptions, derivations, applicability, and limitations. The problems collectively address fundamental aspects of early-universe physics, including ionization equilibrium, photon scattering, cosmological expansion, acoustic propagation, diffusion damping, atomic transition scales, primordial light-element formation, and quantum-pressure effects in dark-matter models. By evaluating the calculations individually, the companion study provides a more transparent basis for distinguishing verified identities, controlled approximations, illustrative estimates, and results requiring additional numerical investigation. An important methodological contribution is the explicit separation of independently checked analytical calculations from unverified computational claims. Although selected formulas and constants are independently recalculated, the complete nonequilibrium recombination history, primordial nuclear reaction network, and coupled Einstein–Boltzmann perturbation hierarchy have not been independently executed or reproduced in this companion investigation. Accordingly, the study does not present these systems as independently verified numerical solutions or claim to reproduce the complete CMB power spectrum. The principal contribution is a more mathematically consistent, transparent, and reproducible presentation of the SP-variable approach to early-universe cosmology. By correcting identified errors, reassessing numerical comparisons, documenting the assumptions underlying 24 worked problems, and establishing clear boundaries between exact identities and physical modeling, the companion study strengthens the analytical reliability of the original framework. The investigation does not establish a modification of standard cosmology, introduce independently verified new cosmological dynamics, or demonstrate a complete first-principles numerical solution of the early universe. Instead, it provides a corrected and critically evaluated mathematical foundation for further research into SP characteristic scales, cosmic microwave background physics, primordial element formation, atomic transitions, and quantum effects in cosmological matter distributions. Keywords: Scretching–Planck formulation; early-universe cosmology; cosmic microwave background; hydrogen recombination; Peebles recombination equation; acoustic angular scale; Silk damping; 21-centimeter hydrogen line; primordial nucleosynthesis; SP epoch; ultralight dark matter; baryon quantum pressure; physical and comoving wavenumbers; Einstein–Boltzmann hierarchy; Planck cosmological parameters; numerical consistency; dimensional analysis; mathematical verification; cosmological reparameterization.