Tawfik Abdel Nasser, Greiner Carsten
Egyptian Center for Theoretical Physics, Juhayna Square off 26th-July-Corridor, Giza 12588, Egypt.
Institute for Theoretical Physics (ITP), Goethe University Frankfurt, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany.
Entropy (Basel). 2021 Feb 28;23(3):295. doi: 10.3390/e23030295.
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, H(a(t)), where H(t) is the Hubble parameter and a(t) is the scale factor, but none of them so far could to be directly expressed in terms of either proper or cosmic time . For Eckart-type viscous background, especially at finite cosmological constant, non-singular H(t) and a(t) are obtained, where H(t) diverges for QCD/EW and asymptotic EoS. For non-viscous background, the dependence of H(a(t)) is monotonic. The same conclusion can be drawn for an ideal EoS. We also conclude that the rate of decreasing H(a(t)) with increasing a(t) 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.
基于近期采用近夸克味的微扰和非微扰晶格计算以及光子、中微子、轻子、电弱粒子和标量希格斯玻色子的热贡献，在净重子密度为零的情况下，各种热力学量，如压力、能量密度、体黏滞系数、弛豫时间和温度，已计算至 TeV 尺度，即涵盖了早期宇宙中的强子、夸克胶子等离子体（QGP）和电弱（EW）相。这一显著进展促使本研究确定体黏滞系数在早期宇宙中可能产生的影响，并了解其如何随时间变化。我们考虑了相对论性宇宙流体的一阶（埃卡特）和二阶（以色列 - 斯图尔特）理论，并将黏性状态方程纳入弗里德曼方程。得到了非线性非齐次微分方程的解析解。对于以色列 - 斯图尔特理论，微分方程非常复杂难以求解。在此将其作为未来研究的路线图进行概述。对于埃卡特理论，唯一可能的解是函数 H(a(t))，其中 H(t) 是哈勃参数，a(t) 是标度因子，但到目前为止它们都无法直接用固有时间或宇宙时间表示。对于埃卡特型黏性背景，特别是在有限宇宙学常数的情况下，得到了非奇异的 H(t) 和 a(t)，其中 H(t) 在 QCD/EW 和渐近状态方程时发散。对于非黏性背景，H(a(t)) 的依赖性是单调的。对于理想状态方程也可得出相同结论。我们还得出结论，在宇宙学常数为零和有限的情况下，H(a(t)) 随 a(t) 增加而减小的速率随时间变化。这些结果显然有助于增进我们对核合成和宇宙学大尺度结构的理解。
