Stability Analysis of Bulk Viscous Cosmology

In this paper, we study phase space analysis of FRW universe model by taking a power-law model for bulk viscosity coefficient. An autonomous system of equations is developed by deÞning normalized dimensionless variables. We Þnd corresponding critical points for di↵erent values of the parameters to investigate stability of the system. It is found that the presence of power-law model of bulk viscosity appears as an e↵ective ingredient to enhance the stability of the respective universe model.


Introduction
Many astronomical observations (type Ia supernova and cosmic microwave background radiation) have suggested that our universe is undergoing an accelerated expansion [1].An unusual antigravitational force, known as dark energy (DE), is considered to be responsible for the current cosmic expansion.There are various alternative candidates for DE to study its mysterious characteristics among which the cosmological constant (⇤) is the simplest one.It has also been suggested that bulk viscous ßuids play an important role for accelerated cosmic expansion [2].Bulk viscosity is one of the favorable dissipative contributions in homogeneous universe scenarios.
A phase space analysis is useful to study di↵erent patterns of evolution by transforming the system of equations to an autonomous one.Guo et al. [3] studied phase space analysis for FRW universe model Þlled with barotropic ßuid as well as phantom scalar Þeld and found that phantom dominated solution is a stable late-time attractor.Yang and Gao [4] explored this analysis for k-essence cosmology and obtained that stability of the critical points play a substantial role for the cosmic evolution.Recently, we have discussed the stability of accelerated expansion via phase space analysis for both isotropic [5] as well as anisotropic [6] universe models.
This work is devoted to study stability of FRW universe model by taking bulk viscous ßuid through phase space analysis.The plan of the paper is as follows.In Sect.2, we provide some basic equations and a power-law of bulk viscosity parameter.We study phase space analysis for both matter as well as radiation dominated universe model.We conclude our results in the last section.

General Equations and Phase Space Analysis
We consider homogeneous and isotropic universe model deÞned by the line element where a(t) is the scale factor.The matter distribution for the cosmic ßuid is given by where ⇢ and p correspond to the energy density and total pressure, respectively.The e↵ective bulk viscous pressure is deÞned by the Eckart formalism as [7] where ⌘ represents bulk viscosity coefficient.The bulk viscosity gives dissipative e↵ects to the cosmic ßuid.The Friedmann equations describing the evolution of the respective universe model are where H = ú a a is the Hubble parameter and dot means derivative with respect to time.The conservation of energy-momentum tensor yields ú We consider a power-law model for bulk viscosity [8] where ⇠ and δ are free parameters.

Bulk Viscous Matter Dominated Universe
Her we study stability of FRW model through phase space analysis dominated by matter (neglecting radiation).We deÞne the dimensionless bulk viscous parameters as which takes the form where H 0 deÞnes the current value of Hubble parameter.The constraint and Raychaudhuri equations turn out to be where ⇢ m corresponds to the matter density.The conservation equation for bulk viscous dust becomes The deceleration and EoS parameter are deÞned by To Þnd analytical solution of the evolution equations, we deÞne normalized dimensionless variables which can reduce this system to an autonomous one.The system of Eqs.( 11) and ( 12) in terms of normalized variables become We deÞne a new variable ⌧ for time such that dt d⌧ = 1 H 2 .The corresponding derivative will be represented by prime.In order to determine the critical points (x c , z c ), we need to solve the respective dynamical system by imposing the condition x 0 = z 0 = 0. We evaluate two critical points for the autonomous Eqs.( 17) and (18).
For P 1 = (1, 1), both the eigenvalues are positive showing unstable past attractor.Here z = 1 implies either H 0 = 0 or H ! 1.Since H 0 cannot be zero therefore it shows the initial singular cosmic epoch characterized by H.We plot the dynamical behavior of critical points corresponding to di↵erent values of ⇠ and δ as shown in Þgure 1.In these numerical plots, the white region shows decelerated expansion with q > 0 while the green region corresponds to q < 0 showing accelerated expansion of the universe.We observe that both eigenvalues are positive indicating the point P 1 as an unstable past attractor.This point lies in the physical phase space outside the green region showing decelerated cosmic expansion for all choices of parameters.The point P 2 = (1, 0) corresponds to a saddle point since the eigenvalues have opposite sign.For

◆
, the corresponding eigenvalues are negative showing a stable future attractor which lies in green region showing an expanding universe model dominated by viscous matter.

Bulk Viscous Matter and Radiation
We discuss the phase space structure of the model by adding radiation as a new cosmic constituent to study whether the bulk viscous model indicates a radiation dominated epoch.The constraint and Raychaudhuri equations take the form The conservation equation for radiation case can be written as We generalize the density parameter for radiation by r = ⇢ r 3H 2 such that Eq.(20) becomes The deceleration and EoS parameters are computed as We deÞne dimensionless variables as through which, the dynamical system of equations for phase space coordinates become We evaluate three critical points by setting x 0 = y 0 = z 0 = 0.For P 1 = (x c , y c , z c ) = (0, 1, 1), the corresponding eigenvalues show an unstable past attractor.This point lies in decelerated expanding phase.The point P 2 = (u, 0, 1) corresponds to a saddle node showing decelerated cosmic expansion.

!
, the corresponding eigenvalues are negative indicating stable late attractor in deceleration phase for di↵erent choices of parameters.The summary of the corresponding results is given in table 1.

Summary
We have studied phase space analysis for homogeneous and isotropic universe model by taking a mixture of viscous dust and radiating ßuid.We have considered a power-law model for the bulk viscous coefficient.Firstly, we have discussed stability of the universe dominated by bulk viscous matter through their eigenvalues corresponding to di↵erent values of ⇠ and δ (Þgure 1).We have found an unstable initial matter dominated state undergoing decelerated expansion for P 1 with all choices of parameters.The point P 2 is a saddle point in matter dominated universe.The point P 3 corresponds to a stable matter dominated universe undergoing accelerated expansion.It is observed that the green region (accelerated expansion) tends to increase by increasing ⇠ while this region gets smaller by increasing δ.It is worth mentioning here that stable solutions (undergoing accelerated expansion) exist in the presence of power-law model of bulk viscosity.Secondly, we have studied stability of the universe model with bulk viscous radiation and matter to check whether it also indicates a prior radiation dominance.It is found that the critical points P 1 and P 2 are saddle/unstable showing initially the matter dominated phase (without acceleration).The point P 3 is a stable attractor showing decelerated cosmic expansion.We conclude that no stable point (undergoing accelerated expansion) exists with bulk viscous radiation and matter.

Figure 1 .
Figure 1.Plots for the phase plane evolution of FRW universe model with di↵erent values of ⇠ and δ.

Table 1 .
Stability Analysis for Matter and Radiation Case