# Baryogenesis: A Symmetry Breaking in the Primordial Universe Revisited

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## Abstract

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## 1. Introduction

## 2. Standard Cosmology

## 3. Thermodynamics in the Primordial Universe

#### 3.1. Thermal History of the Universe

- Quantum Gravity? ($T>{T}_{Pl}\sim {10}^{19}$ GeV)—GR is rendered invalid at this scale so that quantum corrections are necessary. In inflationary scenarios, this is usually called pre-inflationary cosmology.
- Inflation—This is an epoch of the accelerated expansion of the Universe, probably exponential. In such a case, the Universe can be described by the de Sitter solution, which is characterized by Equation (16). Adiabaticity fails, and equilibrium thermodynamics is not valid in such a period of the history of the Universe.
- End of inflation and particle production—In this period, dark expanding “emptiness” filled by a scalar (or some other) field, possible inflaton, creates radiation and other elementary particles. This is the period of reheating.
- Start of the radiation-dominated Universe—The Universe becomes well described by equilibrium thermodynamics, being adiabatically cooled down. If current ideas are correct, during this epoch the Universe underwent several phase transitions. During the stages at which phase transitions occur, adiabaticity could be broken.
- Grand unification phase transition ($T\sim {10}^{15}$ GeV–${10}^{17}$ GeV)—The main goal of the Grand Unification Theories (GUTs) is to attempt to unify the electromagnetic, electroweak and strong interaction into a simple gauge group that should be a valid symmetry at the highest energies. As energy is lowered, the theory undergoes a hierarchy of spontaneous symmetry breakings (SSBs) into successive subgroups. From the cosmological point of view, these SSBs should have been reflected into phase transitions occurring during the evolution of the Universe, with the possible formation of topological defects.
- EW phase transition ($T\sim 100$ GeV)—Some of the gauge bosons and other particles acquire mass via the Higgs mechanism.
- QCD phase transition ($T\sim 150$ MeV)—Quarks lose their so-called asymptotic freedom, which they have at high energies, and there are no more free quarks and gluons, rather the quark–gluon plasma becomes a hadron gas. Combinations of three quarks (baryons) and quark–antiquark pairs (mesons) are formed.
- Neutrino decoupling ($T\sim 1$ MeV)—Previous to this point in the history of the Universe, neutrinos were kept in thermal equilibrium via the weak interactions of the type $\overline{\nu}\nu \leftrightarrow {e}^{+}{e}^{-}$, where $\nu $ and $\overline{\nu}$ represent, respectively, generic neutrinos and antineutrinos, and ${e}^{+}$ and ${e}^{-}$ represent positrons and electrons, respectively.
- Electron–positron annihilation ($T\sim 0.5$ MeV)—Shortly after the neutrino decoupling, the temperature drops below the mass of the electron/positron, which thereby become non-relativistic.
- Big Bang Nucleosynthesis ($T\simeq 10$–0.1 MeV)—At about 1 MeV, the ratio between the neutron number and the proton number (usually termed neutron-to-proton) ratio freezes out. Shortly thereafter ($T\sim 0.1$ MeV), the synthesis of light elements begins.
- Matter–radiation equality—After nucleosynthesis, the Universe reaches a point after which matter begins to dominate over the radiation, relativistic particles that, within FLRW models, have an energy density proportional to ${a}^{-4}$ or to ${T}^{4}$ since $T\propto {a}^{-1}$ (as the scale factor grows, the temperature decreases). According to the stages of the Universe previously defined, such a point marks the entrance of the Universe in its adolescence, and it is usually called the matter–radiation equality.
- Photon decoupling and recombination ($T\sim 0.2$ MeV–$0.3$ MeV)—The interactions between photons and electrons are rapid relative to the expansion rate of the Universe, so radiation (photons) and matter (electrons, protons and nuclei) are kept in good thermal contact, thus remaining in equilibrium. However, eventually, the Universe reaches a point were the thermal contact between these two components is no longer maintained and radiation decouples from matter.
- The Dark Age—Here, stars are not yet formed, and with the expansion of redshift cosmic microwave background photons towards the infrared, they become invisible heat radiation.
- Formation of the first stars and reionization—In the dark, the seeds of structure formation were already settled, and as masses were gathering together, one by one the first stars lit up.
- Present time ${t}_{0}$.

#### 3.2. Kinetic Equilibrium: Distribution Functions for Bosons and for Fermions

#### 3.3. Chemical Equilibrium

#### 3.4. Interactions in an Expanding Universe

#### 3.5. Entropy in the Expanding Universe and Baryon Number

#### 3.6. Equilibrium in an Expanding Universe

## 4. Sakharov Criteria

- Baryon number violation;
- Violation of C (charge conjugation symmetry) and CP (the composition of parity and C);
- Departure from the equilibrium.

#### 4.1. Baryon Number Violation

#### 4.2. Violation of C and CP

#### 4.3. Departure from Equilibrium

## 5. Eletroweak Baryogenesis

#### 5.1. The EWB Mechanism

- The thin wall regime or nonadiabatic regime: ${L}_{\omega}/l\le 1$;
- The thick wall regime or adiabatic regime: ${L}_{\omega}/l\ge 1$;

#### 5.2. Necessary Conditions for a Successful EWB: The Sphaleron Bound

#### 5.3. Departure from Equilibrium

## 6. GUT Baryogenesis

#### 6.1. Departure from Equilibrium

#### 6.2. Non-Inflationary GUT-Baryogenesis Mechanism

## 7. Scalar–Tensor Theories

#### 7.1. Action and Field Equations

#### 7.2. Scalar–Tensor Cosmology: 3-Epoch Model

## 8. Scalar–Tensor Baryogenesis

#### 8.1. Scalar–Tensor EWB

#### 8.2. Scalar–Tensor GUT Baryogenesis

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EWB | Electroweak Baryogenesis |

GUT | Grand Unification Theories |

STT | Scalar–Tensor Theories |

FLRW | Friedmann–Lemaıtre–Robertson–Walker |

FD | Fermi–Dirac |

BE | Bose–Einstein |

JBD | Jordan–Brans–Dicke |

BAU | Baryon asymmetry of the Universe |

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Pereira, D.S.; Ferraz, J.; Lobo, F.S.N.; Mimoso, J.P.
Baryogenesis: A Symmetry Breaking in the Primordial Universe Revisited. *Symmetry* **2024**, *16*, 13.
https://doi.org/10.3390/sym16010013

**AMA Style**

Pereira DS, Ferraz J, Lobo FSN, Mimoso JP.
Baryogenesis: A Symmetry Breaking in the Primordial Universe Revisited. *Symmetry*. 2024; 16(1):13.
https://doi.org/10.3390/sym16010013

**Chicago/Turabian Style**

Pereira, David S., João Ferraz, Francisco S. N. Lobo, and José P. Mimoso.
2024. "Baryogenesis: A Symmetry Breaking in the Primordial Universe Revisited" *Symmetry* 16, no. 1: 13.
https://doi.org/10.3390/sym16010013