Early Universe Thermodynamics and Evolution in Nonviscous and Viscous Strong and Electroweak Epochs: Possible Analytical Solutions

• Main Authors: Abdel Nasser Tawfik, Carsten Greiner
• Format: Article
• Language: English
• Published: MDPI AG 2021-02-01
• Series: Entropy
• Subjects: viscous cosmology; particle-theory and field-theory models of the early universe; mathematical and relativistic aspects of cosmology; thermodynamic functions and equations of state
• Online Access: https://www.mdpi.com/1099-4300/23/3/295

Based on recent perturbative and non-perturbative lattice calculations with almost quark flavors and the thermal contributions from photons, neutrinos, leptons, electroweak particles, and scalar Higgs bosons, various thermodynamic quantities, at vanishing net-baryon densities, such as pressure, energy density, bulk viscosity, relaxation time, and temperature have been calculated up to the TeV-scale, i.e., covering hadron, QGP, and electroweak (EW) phases in the early Universe. This remarkable progress motivated the present study to determine the possible influence of the bulk viscosity in the early Universe and to understand how this would vary from epoch to epoch. We have taken into consideration first- (Eckart) and second-order (Israel–Stewart) theories for the relativistic cosmic fluid and integrated viscous equations of state in Friedmann equations. Nonlinear nonhomogeneous differential equations are obtained as analytical solutions. For Israel–Stewart, the differential equations are very sophisticated to be solved. They are outlined here as road-maps for future studies. For Eckart theory, the only possible solution is the functionality, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>a</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the Hubble parameter and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the scale factor, but none of them so far could to be directly expressed in terms of either proper or cosmic time <i>t</i>. For Eckart-type viscous background, especially at finite cosmological constant, non-singular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> are obtained, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> diverges for QCD/EW and asymptotic EoS. For non-viscous background, the dependence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>a</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> is monotonic. The same conclusion can be drawn for an ideal EoS. We also conclude that the rate of decreasing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>a</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> with increasing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> varies from epoch to epoch, at vanishing and finite cosmological constant. These results obviously help in improving our understanding of the nucleosynthesis and the cosmological large-scale structure.