07outlook.tex

\section{Summary and Discussion}\label{ part6}
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%\subsection{Discussion}
We have presented a compendium of theoretical models addressing the particle and plasma content of the primordial Universe. The Universe at a temperature above 10\keV\ is dominated by `visible' matter, and dependence on unknown dark matter and dark energy is minimal. However any underlying dark component will later surface, thus the understanding of this primordial epoch also as a source of darkness (including neutrinos background) in the present day Universe is among our objectives.

Select introductory material addressing kinetic theory, statistical physics, and general relativity has been presented. Kinetic and plasma theory are described in greater detail. Einstein's gravity theory, found in many other sources, is limited to the minimum required in the study of the primordial Universe within the confines of the FLRW cosmology model. 

This work connects several of our prior and ongoing studies of cosmic particle plasma in the primordial Universe. The three primary eras, radiation, matter, and dark energy dominance, can be recognized in terms of the acceleration parameter $q$. We introduce this tool in the cosmology primer \rsec{sec:flrw}, connecting these distinct epochs smoothly in \rsec{sec:dynamic}. Detailed results concerning time and temperature relation allowing for the reheating of the Universe were shown. Entropy transfer (reheating) inflates the Universe's expansion whenever the ambient temperature is too low to support the massive particle abundance.

In detailed studies, we explored particle abundances and plasma properties, which improved our comprehensive understanding of the Universe and its evolution. {\color{black} Many of the methods we presented and used in the study of the early Universe were developed for relativistic heavy-ion collision applications. However, the hot Universe environment differs from relativistic heavy-ion laboratory experiments, and this is best appreciated considering the evolution of particle inventory: In the evolving Universe, we allow for a full adjustment of particle yields to the ambient temperature of the dynamically expanding Universe, and we follow these yields as a function of temperature with chemical potential(s) constrained by the Universe baryon asymmetry. Each particle, irrespective of its interaction, has individual decoupling temperature, with unstable particles usually rapidly disappearing after decoupling, while stable decoupled particles free-streaming.} 

{\color{black}In contrast to this, in the laboratory heavy-ion experiments: a) Only strongly interacting degrees of freedom are typically observed; b) Particle yields follow from a single freeze-out near to QGP-hadron gas phase cross-over; c) Detailed particle yields allow us to measure the dynamically created chemical potentials; d) An analysis of laboratory experimental data creates a snap-shot image of the one high $T$ freeze-out instant. Clearly, a direct, detailed comparison between the early Universe particle inventory and laboratory experiments cannot be attempted. Similar remarks apply to all other observables: we learn in laboratory experiments the required methodology which we use in the study of the dynamic primordial Universe.}

{\color{black} This conceptual connection between heavy ion experiments and the early Universe has profound importance beyond the study of particle inventory. Known probes of the primordial Universe, such as the cosmic microwave background and primordial light nuclei abundances, are indirectly driven by macroscopic EM fields and transport properties in primordial plasma, including the QGP, which underpin response to primordial magnetic fields and eventual early structure formation. In~\rsec{chap:QCD}, we presented the electromagnetic properties of QGP in heavy ion collisions. This insight leads us to derive an analytic formula that predicts the freeze-out magnetic field that governs the micro-bang in the laboratory, potentially enabling experimental determination of the QGP electromagnetic conductivity, as was recently proposed~\cite{STAR:2023jdd}, which determines the  understanding of the primordial QGP.}

One important aspect of the laboratory study of the hot primordial Universe is the experimental access in ultra relativistic heavy-ion collision experiments to the process of melting of matter into constituent quarks at high enough temperature. The idea that one could recreate this Big-Bang condition in laboratory was indeed the beginning of the modern interest in better understanding the structure of the primordial Universe. 

We recalled the 50 years of effort that began with the recognition of novel structures in the primordial Universe beyond the Hagedorn temperature and the exploration of this high-temperature deconfined quark-gluon phase. Moreover, the study of the phase transformation between confined hadrons and deconfined quark-gluon plasma in the laboratory facilitates the understanding of the primordial Universe dating to the earliest instants after its birth, about 20-30\,$\mu$s after the Big-Bang. The question of how we can recognize the quark-gluon plasma observed in the laboratory to be different from the hadron Universe content was mentioned.

Many interesting phenomena in the primordial Universe depend on nonequilibrium conditions, and this topic is at the core of our theoretical interest. Nuance differences between kinetic and chemical equilibrium, dynamic but stationary detailed balance, and non-stationary phenomena recur as topics of interest in our discussion. For bottom quarks in \rsec{Bottom} we recognize in detail the deviations from thermal equilibrium, particle freeze-out, and detailed balance away from the thermal equilibrium condition and isolate the non-stationary components. These nonequilibrium concepts developed for more esoteric purposes are pivotal, in our opinion, in recognizing any remnant observable of the primordial Universe. 

The experimental study in the laboratory of the dynamic micro-bang stimulates the development of detailed models of the strongly interacting hadron era of the Universe. We use some of the tools created for laboratory experiment interpretation to study properties of hadronic matter in the Universe and strangeness flavor freeze-out in particular in \rsec{Strangeness}. 

These kinetic and dynamic insights drive our interest, leading beyond our interest in strangeness and bottom quarks to all heavy PP-SM particles. We question the potential that primordial QGP era harbors opportunity for baryogenesis, we look both for the bottom quarks and the Higgs particle induced reactions, \rsec{HiggsQGP}. This work will continue.

The different epochs in the Universe evolution are often considered as being distinctly separate. However, we have shown that this is not always the case. We note the `squeeze' of neutrino decoupling between: The electron-positron annihilation reheating of photons at the low temperature edge at about $T=1\MeV$; and heavy lepton (muon) disappearance on the high-$T$ edge at about $T=4.5\MeV$, \rsec{Electron}. 

This fine-tuning into a narrow available domain prompted our investigation of neutrino decoupling as a function of the magnitude of the governing natural constants, \rsec{ch:param:studies}. This characterization of neutrino freeze-out constrains the time variation of natural constants. We present in \rapp{ch:boltz:orthopoly} a novel computationally efficient moving-frame numerical method we developed to obtain the required results.

Our in-depth study of the neutrino background shows future potential to reconcile observational tensions that arise between the reported present day speed of Universe expansion $H_0$ (Hubble parameter in present epoch) and extrapolations from the recombination epoch. One can question how $H_0$ could depend on a better understanding of the dynamics of the free-streaming quantum neutrinos\index{neutrino!massive free-streaming quantum liquid} across mass thresholds. We recently laid a relevant theoretical foundation, allowing us to develop further this very intricate topic~\cite{Birrell:2024bdb}.

In~\rsec{sec:BoltzmannEinstein}, we provided background on the Boltzmann-Einstein equation, including proofs of the conservation laws and the Boltzmann's H-theorem for interactions between any number of particles; this is of interest as the evolution of the Universe often requires detailed balance involving more than two particle scattering. To our knowledge, proof for general numbers $m$, $n$ with $m\to n$-particle interactions is not available in other references on the subject.

Following on the neutrino decoupling we encounter in the temporal evolution of the Universe another example of two era overlap, this time potentially much more consequential: The era of electron-positron pair plasma annihilation begins immediate after neutrino decoupling and yet the primordial nucleosynthesis at a temperature that is 15 times lower proceeds amidst a dense $e^+e^-$-pair plasma background, which fades out well after BBN ends. 

This effect is clearly visible but maybe is not fully appreciated when inspecting in~\rf{fig:energy:frac}: We see that the line for the $e^+e^-$-component is a ``small'' $e^+e^-$-energy fraction during the marked BBN epoch. It seems that the $e^+e^-$-pair plasma is in process of disappearance and does not matter. This is, however, a wrong first impression: The $e^+e^-$-energy fraction is starting with a giant $10^9$ pair ratio over nucleon dust. Dropping by three orders of magnitude there remains a huge $e^+e^-$-pair abundance left with millions of pairs per each nucleon at the onset of the BBN era.

We studied the ratio of $e^+e^-$-pair abundance to baryon number in detail in \rf{fig:densityratio} (see also \rf{BBN:Electron} right ordinate): As a curious tidbit let us note that as long as there are more than a few thousand $e^+e^-$-pairs per nucleon the antimatter content in the primordial Universe is practically symmetric with the matter content in any applicable measure. The nuclear dust is not tilting the balance as matter are electrons and antimatter are positrons. Thus it is not entirely correct to consider the disappearance of of antibaryons, see \rf{Baryon:fig}, at $T\simeq 38.2\MeV$, as the end of antimatter epoch. It is instead correct to view the temperature\index{antimatter!disappearance} $T=30\keV$ as the onset of the antimatter disappearance which completes at $T=20.3\keV$, as is seen in~\rf{fig:densityratio}.

Investigation of the dense charged particle plasma background during BBN constitutes a major part of this work. In~\rsec{part4} we develop a covariant kinetic plasma theory to analyze the influence of $e^+e^-$-pair plasma polarization. We solve the dynamic phase space equations using linear response method considering both spatial and temporal dispersion. We are focusing our attention on the understanding how the covariant polarization tensor, which includes collisional damping, shapes the self-consistent electromagnetic fields within the medium. This approach allows us to elucidate the intricate dynamics introducing QED damping effects that characterize the behavior of the $e^+e^-$-pair plasma.

We explore the damped-dynamic screening effects between reacting nucleons and light elements in $e^+e^-$-pair plasma during the Big-Bang Nucleosynthesis (BBN). Our results indicate that the in plasma screening can modify inter nuclear potentials and thus also nuclear fusion reaction rates in an important manner. However, the effect during the accepted BBN temperature range is found to remain a minor correction to the usually used effective screening enhancement. Despite the significant perturbatively evaluated damping, and high temperatures characteristic of BBN, the enhancement in nuclear reaction rates remains relatively small, around $10^{-5}$, yet it provides a valuable refinement to our understanding of the primordial Universe's conditions. We also show a very significant impact of non-perturbative self-consistent evaluation of damping in \rsec{section:electron}. We have not yet had an opportunity to explore how the non-perturbative damping impacts BBN epoch fusion rates. 

The long lasting (in relative terms) antimatter $e^+e^-$-pair plasma offers an opportunity to consider a novel mechanism of magneto-genesis in primordial Universe: Extrapolating the intergalactic fields observed in the current era back in time to the $e^+e^-$-pair plasma era, magnetic field strengths are encountered which approach the strength of the surface magnetar fields~\rsec{sec:theory}.

This has prompted our interest to study the primordial $e^+e^-$-pair plasma as the source of Universe magnetization. We studied the temperature range of $2000\keV$ to $20\keV$ where all of space was filled with a hot dense electron-positron plasma (up to 450 million pairs per baryon) still present in primordial Universe within the first few minutes after the Big-Bang. We note that our chosen period also includes the BBN era.

We found that subject to a primordial magnetic field, the primordial Universe electron-positron plasma has a significant paramagnetic response, see~\rf{fig:magnet} due to magnetic moment polarization. We considered the interplay of charge chemical potential, baryon asymmetry, anomalous magnetic moment, and magnetic dipole polarization on the nearly homogeneous medium. We presented a simple model of self-magnetization of the primordial electron-positron plasma which indicates that only a small polarization asymmetry is required to generate significant magnetic flux when the primordial Universe was very hot and dense.

Our novel approach to high temperature magnetization, see Chapter~\ref{sec:mag:universe} shows that the $e^{+}e^{-}$-plasma paramagnetic response (see \req{g2magplus} and \req{g2magminus}) is dominated by the varying abundance of electron-positron pairs, decreasing with decreasing $T$ for $T\!<\!m_{e}c^2$. This is unlike conventional laboratory cases where the magnetic properties emerge with the number of magnetic particles being constant.  As the number of pairs depletes while the primordial Universe cools the electron-positron spin magnetization clearly cannot be maintained. However, once created magnetic fields want to persist. How the transit from Gilbertian to Amperian magnetism proceeds will be topic of future investigation: This presents an opportunity for understanding formation of space-time persistent  induced currents helping to facilitate magnetic and potentially matter inhomogeneity in the primordial Universe. 
 
Outside of the scope of our report we can also check for era overlaps at temperature below 10\keV: Inspecting \rf{fig:energy:frac} one can wonder about the coincidental multiple crossing of different visible energy components in the Universe seen near to $T=0.25\meV$. This means at condition of recombination there is an unexpected component coincidence. This special situation depends directly on the interpretation of our current era in terms of specific matter and darkness components. The analysis of cosmic background microwave (CBM) data which underpins this, is not retold here. However, the present day conditions propagate on to the primordial times in the particles and plasma Universe and provide for the era overlaps we reported in regard of earlier eras.

Skeptics could interpret the appearance of several such coincidences as indicative of a situation akin to pre-Copernican epicycles. Are we seeing odd `orbits' because we do not use the `solar' centered model? We note that current standard model of cosmology is being challenged by Fulvio Melia~\cite{Melia:2022itm} ``One cannot avoid the conclusion that the standard model needs a complete overhaul in order to survive.'' or by the same author~\cite{Melia:2024rzy} ``\ldots the timeline in $\Lambda$CDM is overly compressed at $z\ge 6$, while strongly supporting the expansion history in the early Universe predicted by\ldots" the Melia model of cosmology. 

This well could be the case. However, we believe that in order to argue for or against different models of primordial cosmology we need first to establish the Universe particles and plasma model properties very well as we presented in coherent fashion for the first time in the wide $130\GeV\le T\le 10\keV$ range. Without this any declarations about the cosmological context of particles and plasma Universe based on a few atomic, molecular, stellar phenomena observed at in comparison tiniest imaginable redshift $z=6\simeq7$ are not compelling. Similarly we view with some hesitance the many hypothesis about the properties of the Universe prior to the formation of the PP-SM particles with properties we have explored in laboratory.

The search to understand the grand properties of the primordial Universe without understanding is particle and plasma content has a much longer historical backdrop, which we noted and which had to evolve: Before about year 1971 there was no inkling about the particle physics standard model; we were attempting to understand the primordial Universe based on a thermal hadron model. Hagedorn's bootstrap approach~\cite{Rafelski:2016hnq} was particularly welcome as the exponential mass spectrum of hadronic resonances generated divergent energy density for point-sized hadrons. This well known result allowed the hypothesis that there is a maximum (Hagedorn) temperature in the Universe. 

This argument had excellent and convincing footing and yet it was not lasting: We needed to accommodate the energy content we observe in the infinite Universe. A divergence of energy at a singular starting point converts to a divergence, inflation in space size. However, as soon as experiments in laboratory clarified our understanding of fundamental particle physics, this narrative collapsed within weeks as one of us (JR) saw in late 70s at CERN, working with Hagedorn in his office long hours developing non-divergent models of hadrons. 

The outcome of more than 50 years of ensuing effort is seen in these pages, and yet with certainty this is just a tip of an iceberg. We presented here the primordial Universe within the realm of the known laws of physics. As the reader will note while  turning pages, there are many `loose' ends: we show and tell clearly about any and all we recognize. We cannot tell as yet what happened `before' our PP-SM begins at $T\simeq 130\GeV$. Many further key dynamic details characterizing evolution before recombination at $T=0.25\eV$ need to be resolved. The particles and plasma Universe based on PP-SM spans a 12 orders of magnitude temperature window $ 130\GeV > T > 0.25\eV $. And, there is the need to understand the ensuing atomic and molecular Universe which presents another challenge we did not mention. We believe that there is a lot more work to do, which will be much helped by gaining better insights into the riddles of the present day Universe dynamics.


\section*{Declarations}
\textbf{Author Contributions} All authors participated in every stage of the development of this work.\\
\textbf{Data Availability} No datasets were generated or analyzed during the current study.\\
\textbf{Competing Interests} The authors declare no competing interests.