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# Behavior of γ ef f given by Eq. (41) as a function of T for the solution with m = 0 and ∆0 = 0, for the particular value of γ = 1 and for ΩΛ given by Eq. (35), for different values of Ω ξ . Tc and Ts are given by (38) and Eq. (39), respectively. We also plotted the γ ef f for the ΛCDM model.

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In this paper we explore the different types of singularities that arise in the ΛCDM model when dissipative processes are considered, in the framework of the Eckart’s theory. In particular, we study the late-time behavior of ΛCDM model with viscous cold dark matter (CDM) and an early-time viscous radiation domination era with cosmological constant...

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## Citations

... Motivated by this physical argument, one of the most common ways to parameterize the bulk viscosity is ξ = ξ 0 ρ s , where ξ 0 > 0 is a bulk viscous constant. This particular type of parameterization has been widely investigated in [49][50][51][52][53][54][55][56][57]. Apart from these physical motivations, the power-law form allows to obtain analytical cosmological solutions. ...

Bulk viscosity in cold dark matter is an appealing feature that introduces distinctive phenomenological effects in the cosmological setting as compared to the $\Lambda$CDM model. Under this view, we propose a general parametrization of the bulk viscosity of the form $\xi\sim H^{1-2s} \rho_{m}^{s}$, that covers intriguingly some well-known cases in the Eckart's theory. Some advantages of this novel parametrization are: first, it allows to write the resulting equations of cosmological evolution in the form of an autonomous system for any value of $s$, so a general treatment of the fixed points and stability can be done, and second, the bulk viscosity effect is consistently handled so that it naturally turns off when matter density vanishes. As a main result we find, based on detailed dynamical system analysis, one-parameter family of de-Sitter-like asymptotic solutions with non-vanishing bulk viscosity coefficient during different cosmological periods. Numerical computations are performed jointly along with analytical phase space analysis in order to assess more quantitatively the bulk viscosity effect on the cosmological background evolution. Finally, as a first contact with observation we derive constraints on the free parameters of some bulk viscosity models with specific $s$-exponents from Supernovae Ia and observations of the Hubble parameter, by performing a Bayesian statistical analysis thought the Markov Chain Monte Carlo method.

... The aim of this paper is to explore the thermodynamic consistency in the description of a dissipative WDM component, as there is not enough investigated up to date, and if the constraints from the present cosmological data on the model that we propose are compatible with the consistency criteria found. Our model is described by an analytical solution obtained in [90] for a flat Friedman-Lemaître-Robertson-Walker (FLRW) maximally symmetric universe, dominated by a dissipative DM modeled by the barotropic EoS p = (γ − 1)ρ, where ρ is the energy density of the dissipative DM and γ is known as a barotropic index, and DE is modeled by the CC, in the framework of the Eckart's theory. This solution was obtained using the expression ξ = ξ 0 ρ m for the bulk viscosity with the particular choice of m = 1, and it was studied in the context of the late and early-times singularities. ...

... The outline of this paper is as follows: In Section 2, we summarize a solution that was found in [90], which represents the model of our study. In Section 3, we present general results about the near-equilibrium condition and the entropy production of the viscous fluid. ...

... In this section, we briefly resume a de Sitter-like solution and an analytical solution found in [90] for a flat FLRW universe composed of a dissipative DM ruled by the barotropic EoS p = (γ − 1)ρ, with a bulk viscosity of the form ξ = ξ 0 ρ m , and a DE given by the CC; the field equations, in the framework of Eckart's theory, are given by [66,90] ...

Extensions to a ΛDM model have been explored in order to face current tensions that occur within its framework, which encompasses broadening the nature of the dark matter (DM) component to include warmness and a non-perfect fluid description. In this paper, we investigated the late-time cosmological evolution of an exact solution recently found in the literature, which describes a viscous warm ΛDM model (ΛWDM) with a DM component that obeys a polytropic equation of state (EoS), which experiences dissipative effects with a bulk viscosity proportional to its energy density, with proportionality constant ξ0. This solution has the particularity of having a very similar behavior to the ΛCDM model for small values of ξ0, evolving also to a de Sitter type expansion in the very far future. We explore firstly the thermodynamic consistences of this solution in the framework of Eckart’s theory of non-perfect fluids, focusing on the fulfillment of the two following conditions: (i) the near-equilibrium condition and (ii) the positiveness of the entropy production. We explore the range of parameters of the model that allow to fulfill these two conditions at the same time, finding that a viscous WDM component is compatible with both ones, being in this sense, a viable model from the thermodynamic point of view. Furthermore, we constrained the free parameters of the model with the observational data coming from supernovae Ia (SNe Ia) and the observational Hubble parameter data (OHD), using these thermodynamics analyses to define the best priors for the cosmological parameters related to the warmness and the dissipation of the DM, showing that this viscous ΛWDM model can describe the combined SNe Ia+OHD data in the same way as the ΛCDM model. The cosmological constraint at 3σ CL gives us an upper limit on the bulk viscous constant of order ξ0∼106 Pa·s, which is in agreement with some previous investigations. Our results support that the inclusion of a dissipative WDM, as an extension of the standard cosmological model, leads to a both thermodynamically consistent and properly fitted cosmological evolution.

... ‡ jose.jovel@usach.cl discussed in [10][11][12][13][14], describing universes that start in an eternal physical past time that comes from a static universe. Other models correspond to the so-called emergent and bouncing universes [15][16][17][18][19][20][21], with the particularity of avoiding also future singularities like the "Big-Rip" singularities. ...

... To avoid this non-physical issue, a causal theory was proposed by Israel and Stewart (IS) in [50,51], which is capable to reduce to Eckart's theory when the relaxation time for the bulk viscous effects is negligible [52]. Accordingly, Eckart's theory is widely considered in the literature as a first approximation to describe viscous cosmology since the IS theory presents a much greater mathematical difficulty [10,[53][54][55][56][57][58][59]. It is important to mention that, both theories are constructed assuming the so-called near equilibrium condition, which was widely discussed by Maartens in the context of dissipative inflation [52]. ...

... As a matter of fact, this viscous pressure Π is proportional to the bulk viscosity coefficient ξ, which depends, particularly, on the temperature and pressure of the viscous fluid [7]. Therefore, a natural election for the bulk viscous coefficient for the dissipative fluid is a dependency proportional to the power of their energy density, an election that has also been explored in the literature in the context of singularities [10,20,22,57,[62][63][64]. For a universe dominated only by perfect fluids, there is no entropy production since the thermodynamics of these fluids are reversible. ...

In this paper, we study the thermodynamical and mathematical consistencies for a non-singular early-time viscous cosmological model known as soft-Big Bang, which was previously found in [N. Cruz, E. Gonz\'alez, and J. Jovel, Phys. Rev. D \textbf{105}, 024047 (2022)]. This model represents a flat homogeneous and isotropic universe filled with a dissipative radiation fluid and a cosmological constant $\Lambda$, which is small but not negligible, in the framework of Eckart's theory. In particular, we discuss the capability of the solution in the fulfillment of the three following conditions: (i) the near equilibrium condition, which is assumed in Eckart's theory of non-perfect fluids, (ii) the mathematical stability of the solution under small perturbations, and (iii) the positiveness of the entropy production. We have found that this viscous model can describe the radiation domination era of the $\Lambda$CDM model and, at the same time, fulfill the three conditions mentioned by the fulfillment of a single constraint on the bulk viscous coefficient $\xi_{0}$, finding also that this non-singular model has a positive energy density in the infinity past witch is infinity hotter with a constant entropy.

... The aim of this paper is to explore the thermodynamic consistency in the description of a dissipative WDM component, not enough investigated up to date, and if the constraints from the present cosmological data on the model that we propose are compatible with the consistency criteria found. Our model is described by an analytical solution obtained in [90], for a flat Friedman-Lemaître-Robertson-Walker (FLRW) universe, dominated by a dissipative DM modeled by the barotropic EoS p = (γ − 1)ρ, where ρ is the energy density of the dissipative DM and γ is known as barotropic index, and a DE modeled by the CC, in the framework of the Eckart's theory. This solution was obtained using the expression ξ = ξ 0 ρ m for the bulk viscosity with the particular choice of m = 1, and was studied in the context of the late and early-times singularities. ...

... The outline of this paper is as follows: In Sec. II we summarize a solution that was found in [90] which represent the model of our study. In Sec. ...

... In this section, we briefly resume a de Sitter-like solution and an analytical solution found in [90]. For a flat FLRW universe composed with a dissipative DM ruled by the barotropic EoS p = (γ − 1)ρ, with a bulk viscosity of the form ξ = ξ 0 ρ m , and DE given by the CC, the field equations, in the framework of Eckart's theory, are given by [64,90] ...

Extensions to a ΛDM model have been explored to face current tensions that occur within its framework, which encompasses broadening the nature of the dark matter (DM) component to include warmness and a non-perfect fluid description. In this paper, we investigated the late-time
cosmological evolution of an exact solution recently found in [N. Cruz, E. Gonz´alez, and J. Jovel,
Phys. Rev. D 105, 024047 (2022)], which describe a viscous warm ΛDM model (ΛWDM) with a
DM component that obeys a polytropic equation of state (EoS), which experience dissipative effects
with a bulk viscosity proportional to its energy density, with proportionality constant ξ0. This solution has the particularity of having a very similar behavior to the ΛCDM model for small values of
ξ0, evolving also to a de Sitter type expansion in the very far future. We explore firstly the thermodynamic consistences of this solution in the framework of the Eckart’s theory of non-perfect fluids,
focusing on the fulfillment of the two following conditions: (i) the near equilibrium condition and (ii)
the positiveness of the entropy production. We explore the range of parameters of the model that
allow to fulfilling these two conditions at the same time, finding that a viscous WDM component
is compatible with both ones, being in this sense, a viable model from the thermodynamic point
of view. Besides, we constraint the free parameters of the model with the observational data coming form supernovae Ia (SNe Ia) and the observational Hubble parameter data (OHD), using these
thermodynamics analysis to define the best priors for the cosmological parameters related to the
warmness and the dissipation of the DM, showing that this viscous ΛWDM model can describe the
combined SNe Ia + OHD data in the same way as the ΛCDM model. The cosmological constraint
at 3σ CL give us an upper limit on the bulk viscous constant of the order ξ0 ∼ 106P a×s, which is in
agreement with some previous investigations. Our results support that the inclusion of a dissipative
WDM, as an extension of the standard cosmological model, leads to a both thermodynamically
consistent and properly fitted cosmological evolution.

We present a further observational analysis of the $\Lambda_{\rm s}$CDM model proposed in Akarsu \textit{et al.} [\href{https://doi.org/10.1103/PhysRevD.104.123512}{Phys. Rev. D 104, 123512 (2021)}]. This model is based on the recent conjecture suggesting the universe has transitioned from anti-de Sitter vacua to de Sitter vacua (viz., the cosmological constant switches sign from negative to positive), at redshift ${z_\dagger\sim2}$, inspired by the graduated dark energy (gDE) model proposed in Akarsu \textit{et al.} [\href{https://doi.org/10.1103/PhysRevD.101.063528}{Phys. Rev. D 101, 063528 (2020)}]. $\Lambda_{\rm s}$CDM was previously claimed to simultaneously relax five cosmological discrepancies, namely, the $H_0$, $S_8$, and $M_B$ tensions along with the Ly-$\alpha$ and $\omega_{\rm b}$ anomalies, which prevail within the standard $\Lambda$CDM model as well as its canonical/simple extensions. In the present work, we extend the previous analysis by constraining the model using the Pantheon data (with and without the SH0ES $M_B$ prior) and/or the \textit{completed} BAO data along with the full \textit{Planck} CMB data. We find that $\Lambda_{\rm s}$CDM exhibits a better fit to the data compared to $\Lambda$CDM, and relaxes the six discrepancies of $\Lambda$CDM, namely, the $H_0$, $M_B$, $S_8$, Ly-$\alpha$, $t_0$, and $\omega_{\rm b}$ discrepancies, each one discussed in detail. When the $M_B$ prior is included in the analyses, $\Lambda_{\rm s}$CDM performs significantly better in relaxing the $H_0$, $M_B$, and $S_8$ tensions with the constraint ${z_\dagger\sim1.8}$ even when the Ly-$\alpha$ data (which imposed the $z_\dagger\sim2$ constraint in the previous studies) are excluded. In contrast, the presence of the $M_B$ prior causes only negligible improvements for $\Lambda$CDM. Thus, the $\Lambda_{\rm s}$CDM model provides remedy to various cosmological tensions simultaneously, only that the galaxy BAO data hinder its success to some extent.