  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 $t$. 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.
