A new Parametrization for Bulk Viscosity Cosmology as Extension of the ΛCDM Model Gabriel G omez1Guillermo Palma1Esteban Gonz alez2Angel Rinc on1and Norman Cruz1

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A new Parametrization for Bulk Viscosity Cosmology

as Extension of the ΛCDM Model

Gabriel G´omez,1, ∗Guillermo Palma,1, †Esteban Gonz´alez,2, ‡´

Angel Rinc´on,1, §and Norman Cruz1, ¶

1Departamento de F´ısica, Universidad de Santiago de Chile,

Avenida V´ıctor Jara 3493, Estaci´on Central, 9170124, Santiago, Chile

2Departamento de F´ısica, Universidad Cat´olica del Norte,

Avenida Angamos 0610, Casilla 1280, Antofagasta, Chile

(Dated: August 24, 2023)

Bulk viscosity in cold dark matter is an appealing feature that introduces distinctive phenomeno-

logical eﬀects in the cosmological setting as compared to the ΛCDM model. Under this view, we

propose a general parametrization of the bulk viscosity of the form ξ∼H1−2sρs

m, 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 quantita-

tively 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.

∗gabriel.gomez.d@usach.cl

†guillermo.palma@usach.cl

‡esteban.gonzalez@ucn.cl

§angel.rincon.r@usach.cl

¶norman.cruz@usach.cl

arXiv:2210.09429v2 [gr-qc] 23 Aug 2023

I. INTRODUCTION

The ΛCDM model describes a Universe with a dark energy (DE) component modeled by a positive cosmological

constant, which drives the recent accelerated expansion [1], and pressureless ﬂuid representing the up to date unknown

cold dark matter (DM) component, responsible for the structure formation in the Universe. This model has been

very successful to ﬁt very well the cosmological data [2–5]. Nevertheless, currently many tension are challenging the

physics behind this model, such as measurements of the Hubble parameter at the current time, H0, which exhibit a

discrepancy of 4.4σbetween the measurements obtained from Planck CMB and the locally measurements obtained

by A. G. Riess et al. [6]. Other tensions are the measurements of σ8−Ωm(where σ8is the r.m.s. ﬂuctuations of

perturbations at 8h−1Mpc scale) coming from large scale structure (LSS) observations and the extrapolated from

Planck CMB (dependent on the ΛCDM model) [7,8], and the results from the experiment EDGES to detect the

global absorption signal of 21 cm line during the dark ages, which reveal an excess of radiation in the reionization

epoch that is not predicted by the ΛCDM model, speciﬁcally at z≈17 [9].

One approach used to attempt to overcome some of the mentioned problems is the inclusion of viscosity in the

cosmological ﬂuids in order to have a more realistic description of their nature beyond the perfect ﬂuid idealization.

For example, in [10–12], the authors address the H0tension as an important guidance to construct new cosmological

models with viscous/inhomogeneous ﬂuids. Also, in [7] it is shown that the σ8−Ωmtension can be alleviated if one

assumes a small amount of viscosity in the DM component; even more, the excess of radiation observed by EDGES

experiment is explain in [13] by considering a viscous nature in DM. Nevertheless, due to the negative pressure

that characterizes dissipative process in cosmic ﬂuids, several authors have investigated the late time acceleration of

the Universe as a pure eﬀect of the bulk viscosity [14–25], as an alternative mechanism to the one provided by the

cosmological constant.

For a homogeneous and isotropic universe, the dissipative process can be characterized only by bulk viscosity, which

in the cosmic evolution has appealing eﬀects [11,26–28] and from the macroscopic point of view can be interpreted

as the existence of slow processes to restore the equilibrium state. Some authors have proposed that bulk viscosity

may be the result of non-conserving particle interactions [29] or it could be the result of diﬀerent cooling rates for

the components of the cosmic medium [30–32]. In addition, many observational properties of disk galaxies can be

described by a dissipative DM component [33,34]. At perturbative level, a viscous ﬂuid description is an accurate

approach for extending the description of cosmological perturbations into a non-linear regime [35]. In this same

direction several works have investigated the perturbative eﬀects of viscous DM models in the structure formation of

the Universe [7,36–39]. The inclusion of viscosity has been also investigated at early times aiming at describing the

primordial inﬂationary period [26,40] and, on the other hand, to evaluate the rate of cosmological entropy production

and its role in the survival of protogalaxies [41].

On the other hand, there are several microscopic models to explain how bulk viscosity could arise in cosmological

scenarios. Among them, we mention the inclusion of self-interacting scalar ﬁelds to describe dark energy (see [42]),

which gives rise to a contribution linearly proportional to the Hubble parameter to the ﬂuid pressure, as in Eckart’s

theory but in a more general setup within thermal ﬁeld theory, where the viscosity coeﬃcient becomes dynamical.

Still, an astringent and throughout analysis of the assumptions for the validity of the hydrodynamical description

used, including the eﬀects of cavitation, is still missing. Also from a microscopic point of view, the relation between

particle creation and bulk viscosity in the early universe is discussed in [43–46], which plays an important role in the

inﬂationary viscous model [47]. In addition to the discussion made by [42], a diﬀerent microscopic model that considers

a bulk viscosity induced by DM annihilation is discussed in [11,48], suited for the late time accelerated expansion.

The kinetic theory formalism has been also implemented to describe the viscous eﬀect within self-interacting DM

models [49,50]. In the context of neutralino CDM, an energy dissipation from the CDM ﬂuid to the radiation ﬂuid

is manifested in a collisional damping mechanism during the kinetic decoupling [51]. The examples mentioned above

highlight the importance of considering various dissipative processes and their potential eﬀects on the cosmic ﬂuid’s

dynamics. However, due to the lack of a general accepted model to include a microscopical motivated bulk viscosity,

we propose instead a general eﬀective parametrization that encompasses a wide class of possible models for the bulk

viscosity, which allows us to describe a richer cosmological dynamics beyond the standard ΛCDM model.

Previous considerations indicate that viscous eﬀect cannot be discarded at late times [52], where the unidentiﬁed

DM component is an essential protagonist, playing an important role from galactic dynamics to the formation of

large scale structures in the Universe. To describe viscous cosmological models it is needed a theory of relativistic

non-perfect ﬂuids out of equilibrium. Under this framework, Eckart was the ﬁrst to propose such a theory [53] with a

similar approach proposed by Landau and Lifshitz [28]. Nevertheless, it was shown later in [53,54] that the Eckart’s

theory is a non-causal theory. Subsequently and following the same spirit, Israel and Stewart (IS) in [55,56] introduced

the corresponding fully causal version which reduces to Eckart’s theory when the relaxation time for the bulk viscous

eﬀects are negligible [57].

Based on i) the richness of the physics behind the bulk viscosity, ii) the wide range of parameterizations proposed

for the bulk viscosity, and ﬁnally, iii) the observational implications in the cosmic evolution, we investigate in this

paper a more involved parameterization including simultaneously the eﬀects of both: ﬁrst, the Hubble parameter,

and second, the dark matter energy density. Our approach will be implemented within a viscous ΛCDM model.

Thus, the aim of this paper is to investigate the behavior of a concrete parameterization for the bulk viscosity of

the form

ξ∼H1−2sρs

m,(1)

by means of detailed dynamical system analysis. Albeit non-trivial, the goodness of this novel functional form is that

the bulk viscosity is consistently handled so that it naturally turns oﬀ when matter density vanishes.

Demanding solely complete cosmological dynamics, that is, from radiation era up to the present time, gives a

relevant information about what particular parameterization for the bulk viscosity is suitable and sets, therefore,

plausible dissipative cosmological models that must be subjected to the careful scrutiny requested for observational

constraints. We perform this analysis in the framework of Eckart’s theory as a ﬁrst approximation to the study of

relativistic non-perfect ﬂuids. Thus, the dynamical analysis will indicate, as a ﬁrst inquiry, what the values of the

model parameters are of the proposed parametrization that leads to a successful background cosmic evolution, and

this information is used a priori to constrain the available parameter space with the aid of Supernovae Ia (SNe Ia)

data and observational Hubble parameter data (OHD).

The present paper is organized as follows: After a concise introduction, we summarize a few essential features of

bulk viscosity and some explored parameterizations in dissipative cosmological models, discussing also the study of

their behavior in the framework of dynamical analysis, in section II. Then, in section III, we present the model to be

studied, as well as the concrete and new parameterization of the bulk viscosity that we will explore. Subsequently, in

section IV, we perform a dynamical system analysis to set the stability conditions of the ﬁxed points. We summarize

our main results in tables I and II for concrete values of the exponent s. A comprehensive analytical treatment is

provided separately in section V in order to explain fully the ﬁxed point structure found for arbitrary bulk viscosity

exponents. In addition, section VI displays numerical evolution of some bulk viscosity models along with estimation

of best-ﬁt values of their free parameters using the Markov Chain Monte Carlo method. Finally, main ﬁndings and a

general discussion of this work are presented in section VII.

II. THE BULK VISCOSITY AND ITS PARAMETRIZATIONS

Most of the approximations to describe a non-perfect ﬂuid, that can serve to account for the unknown DM compo-

nent, have implemented diﬀerent phenomenological parameterizations for the bulk viscosity. These parameterizations

have covered from the simplest scenario, i.e., ξ=ξ0, to more involved descriptions in terms of the Hubble rate and

its time derivatives [58]. Accordingly, most of the Ans¨atze used in the literature take into account any of these three

basic functional forms: i) a constant bulk viscosity coeﬃcient, ξ0, ii) a power of the (dark matter) energy density,

and iii) a lineal function of the Hubble parameter and/or its times derivatives. Nevertheless, other parameterizations

have considered even polynomial and hyperbolic functions of the redshift [59,60]. A ΛCDM model with this kind of

parameterization has been constrained in [61].

On the other hand, bulk viscosity is a concept closely connected with the equation of state (EoS) used to close the

cosmological system of equations. Thus, it is quite reasonable to expect that bulk viscosity can be written proportional

to (power of) the density of the full system or constituents of it. Even more, the kinetic inner state of a system also

contribute to the deﬁnition of viscosity. Such modiﬁcations are encoded into the temperature and then the viscosity

is, in general, written in terms of the density and/or the temperature [62]. Motivated by this physical argument, one

of the most common ways to parameterize the bulk viscosity is ξ=ξ0ρs, where ξ0>0 is a bulk viscous constant. This

particular type of parameterization has been widely investigated in [63–71]. Apart from these physical motivations,

the power-law form allows to obtain analytical cosmological solutions. In particular, for the dissipative ΛCDM model

with the speciﬁc election s= 1, an interesting exact solution was obtained which asymptotically tends to a de Sitter

expansion, in a certain region of the parameter space. Moreover it has not initial singularity, known as “soft-Big

Bang” [72,73].

Of course, in general, exact solutions can be obtained for some special values of sand the qualitatively study of

cosmological behavior, for arbitrary s, can be implemented using dynamical system analysis. For example, studies

of this type show that bulk-viscous inﬂation is possible in the truncated IS theory models [74], and in the full IS

theory a dynamical analysis was performed in [75]. More complex cosmological scenarios where Λ and Gwere taken

variables, in the causal framework, were analyzed in [76]. A cosmological model that considered a universe ﬁlled with

interacting DE and dissipative DM components, and radiation, was analyzed in the full IS formalism for the special

case s= 1/2 [77]. Other study indicates that a causal model of a universe ﬁlled with dissipative DM component shows

accelerated phase for the case s= 1/2, but the case with s < 1/2 and s > 1/2 are ruled out because they do not drive

accelerated expansions [78]. An interesting work analyzed a universe with viscous radiation and non-viscous dust in

the framework of the nonlinear IS, for arbitrary s[79].

The exploration in the context of the non-causal approach, has allowed to include more general expressions for the

bulk viscosity such a dependence with the Hubble parameter or combination including terms like ¨a/aH. However,

these two last cases have failed in displaying the conventional radiation dominated phase or a matter one [80,81].

A model with viscous DM, taking s= 1/2, and perfect ﬂuids for dark energy and radiation, which also included an

interaction term between DM and DE, was analyzed by performing stability analysis [82]. Taking advantage of some

of the aforementioned progresses in this viscous scenarios, we introduce a new and non-trivial parameterization of the

bulk viscosity able to produce exact analytic solutions. To be more precise, we introduce an Ansatz which combines

the dependence on both the dark matter density and the Hubble parameter simultaneously, i.e., ξ∼H1−2sρs

m. This

non-trivial form leads to a direct coupling between dark matter and all other components through the bulk viscosity. It

implies, by construction, that the bulk viscosity eﬀect becomes eﬀective only in cosmological stages when dark matter

is dominant and its energy density is proportional to the bulk viscosity coeﬃcient. The latter condition is largely

dictated by the dynamical behavior of the system. Another advantage of this functional form is the possibility of

studying, in a general way, the phenomenological implications for any s-exponent through dynamical system analysis

which is not possible for most of phenomenological parametrizations used in the literature.

III. THE MODEL

The main ingredients of the model are described by an eﬀective ﬂuid picture, containing radiation, bulk viscosity

dark matter with non-vanishing eﬀective pressure and the cosmological constant. According to this cosmological

setup, the Friedmann and the acceleration equations are respectively written as

3H2= 8πGN(ρr+ρm+ρΛ),(2)

3H2+ 2 ˙

H=−8πGNPr+Peﬀ

m+PΛ.(3)

Here the usual polytropic relations for radiation Pr=ρr/3 and for the cosmological constant, with energy density

ρΛ≡Λ

8πGN(where GNis the Newton’s constant), PΛ=−ρΛare set. Nevertheless, we assume for the dark matter

ﬂuid a bulk viscous pressure Π that gives place to a minimal extension of the ΛCDM model:

Peﬀ

m=Pm+ Π = −3Hξ, (4)

where ξis the usual bulk viscosity coeﬃcient that obeys the second law of thermodynamics provided that ξ > 0.

We want to realize a qualitative examination of the physical properties of general power-law bulk viscous models

demanding, besides, that the bulk viscosity eﬀect is consistently handled so that it naturally turns oﬀ when matter

density vanishes. For instance, viscous models with ξ∝Hsuﬀer from this problem on its own. To avoid this, we

propose the very convenient non-standard functional form to express the bulk viscosity coeﬃcient ξ

ξ≡ξ0

8πGN

H1−2sH2s

0ρm

ρm,0s

=ˆ

ξ0

8πGN

HΩs

m,(5)

which allows us to write, in turns, the system of equations Eqns. (3)-(5) in the form of autonomous system for any

value of the exponent swith the aid of the combination of the Hubble parameter and the matter energy density.

One may think a priori that the dependence of ξon the Hubble parameter leads in turn to an explicit dependence

on the other components. Nevertheless, the precise combination of Hand ρmmakes the bulk viscosity exist in an

eﬀective way only when matter energy density is dominant. it means that the bulk viscosity is eﬀectively turned oﬀ

when ρmvanishes. For instance, in the radiation domination era: ρm→0 which leads naturally to ξ→0. Hence,

the contribution of the other components to the bulk viscosity is unimportant at leading order in the cosmological

background evolution.

Notice that both ξ0and ˆ

ξ0=ξ0

(Ωm,0)s, with Ωm,0≡8πGNρm,0

3H2

, are dimensionless parameters within this setup,

describing the bulk viscosity eﬀect and it is the only new free parameter that accounts for the extension of the

ΛCDM model. This new parametrization provides the advantage of exploring unconventional values of sas negative

ones as shall be shown later. We remind that in the widely used parameterization ξ∝ρs, the most studied cases

s= 0 and s= 1/2 have been investigated in the framework of dynamical analysis, nevertheless for diﬀerent svalues

this analysis has not been carried out due to the diﬃculty in writing the resulting equations in the autonomous

form. It is interesting to see that well-known viscous models are enclosed within this new form and they are part

of one-parameter family of viscous cosmological solutions as the dynamical system analysis will reveal. For instance,

s= 0 leads evidently to the viscous model ξ∝H, while s= 1/2 leads to the particular case of the widely used

parametrization ξ∝ρ1/2

m, i.e. dependence on the energy density, ρm, only through a power law.

The conservation equations can be formulated in the simple form

˙ρr+ 4Hρr= 0,(6)

˙ρm+ 3H(ρm+ Π) = 0.(7)

In order to solve the system, smust be certainly speciﬁed, but, at which level of diﬃculty the system can be solved

for arbitrary large value of s? Are the corresponding ﬁxed points stable? If so, how diﬀerent are such solutions from

the ΛCDM model and from each other at the background level? These are ones of the features we want to investigate

in the present paper by studying the stability properties of the ﬁxed points by the standard linear stability theory in

the next section.

IV. DYNAMICAL SYSTEM ANALYSIS

We start by deﬁning the dimensionless variables that set the phase space of the system and allows us to rewrite it

in the form of an autonomous system. They are deﬁned so that they correspond to the energy density parameters

associated to each ﬂuid

Ωr≡8πGNρr

3H2; Ωm≡8πGNρm

3H2; ΩΛ≡8πGNρΛ

3H2.(8)

So, the Friedmann constraint takes the usual form

Ωr+ Ωm+ ΩΛ= 1.(9)

From the continuity equations Eqns. (6) and (7), the evolution equations for radiation and dark matter are, respec-

tively, derived with the help of the acceleration equation Eqn. (3) (or in its alternative form given by Eqn. (11) deﬁned

below) that introduces an explicit dependence on bulk viscosity. This also aﬀects the evolution of the energy density

parameter associated with the cosmological constant Eqn. (8) since it is normalized by the Hubble parameter1. After

some algebraic manipulations, the dynamical system is described as follows

Ω′

r= Ωr(−1−3ˆ

ξ0Ωs

m+ Ωr−3ΩΛ),

Ω′

m= 3ˆ

ξ0Ωs

m−3ˆ

ξ0Ω1+s

m+ Ωm(Ωr−3ΩΛ),(10)

Ω′

Λ= ΩΛ−3(−1+ΩΛ+ˆ

ξ0Ωs

m)+Ωr.

In this form, it is evidenced how the bulk viscosity may aﬀect non-trivially the dynamical behavior of all physical

quantities2(8). Here the prime denotes derivative with respect to N≡ln a. In the limit of ˆ

ξ0→0 the ΛCDM model

is recovered as can be plainly checked. We should mention that the autonomous system (10) clearly can be written

as part corresponding to the ΛCDM model, plus the bulk viscosity sector, which extends the standard cosmological

realization. From the functional form of the bulk viscosity, a non-linear “interaction”-like term emerge naturally. Less

evident is that the nonlinearity in the autonomous system aﬀects each equation diﬀerently as a consequence of our

Ansatz.

Notice that the evolution equation for radiation is an auxiliary equation that can be taken away from the system

by using the Friedmann constraint Eqn. (9). So the system is reduced to two dimensional phase space.

The eﬀective EoS parameter is deﬁned as

weﬀ =−2

H′

H−1,with H′

H=1

2(−3+3ˆ

ξ0Ωs

m−Ωr+ 3ΩΛ),(11)

where the new term related to the bulk viscosity appears here explicitly for a general exponent s. One expects,

from physical reasons, that ˆ

ξ0<1 (other than thermodynamics arguments ˆ

ξ0>0), as has been also conﬁrmed

by diﬀerent observational constraints. Nevertheless, one can adopt a less conservative position regarding the bulk

1It does not mean however that the cosmological constant evolves itself since the condition ˙ρΛ= 0 is preserved at any time.

2Notice however that all components are (minimally) coupled to gravity whereby the latter acts as a messenger between them. This is

an indirect way where the bulk viscosity eﬀects may be present in diﬀerent cosmological stages.

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AnewParametrizationforBulkViscosityCosmologyasExtensionoftheΛCDMModelGabrielG´omez,1,∗GuillermoPalma,1,†EstebanGonz´alez,2,‡´AngelRinc´on,1,§andNormanCruz1,¶1DepartamentodeF´ısica,UniversidaddeSantiagodeChile,AvenidaV´ıctorJara3493,Estaci´onCentral,9170124,Santiago,Chile2DepartamentodeF´ısica,Univer...

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A new Parametrization for Bulk Viscosity Cosmology as Extension of the ΛCDM Model Gabriel G omez1Guillermo Palma1Esteban Gonz alez2Angel Rinc on1and Norman Cruz1

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