aa r X i v : . [ g r- q c ] M a r Fixed points and FLRW cosmologies:Flat case
Adel Awad Centre for Theoretical Physics, Zewail City of Science and Technology,Sheikh Zayed, 12588, Giza, Egypt.Department of Physics, Faculty of Science, Ain Shams University, Cairo, 11566, EGYPT
Abstract
We use phase space method to study possible consequences of fixed points in flat FLRWmodels. One of these consequences is that a fluid with a finite sound speed, or a differentiablepressure, reaches a fixed point in an infinite time and has no finite-time singularities of types I,II and III described in hep-th/0501025 . It is impossible for such a fluid to cross the phantomdivide in a finite time. We show that a divergent dp/dH , or a speed of sound is necessary butnot sufficient condition for phantom crossing. We use pressure properties, such as asymptoticbehavior and fixed points, to qualitatively describe the entire behavior of a solution in flatFLRW models. We discuss FLRW models with bulk viscosity η ∼ ρ r , in particular, solutionsfor r = 1 and r = 1 / [email protected] Introduction
Various cosmological observations [1, 3, 4, 5] have provided us with a strong evidence foraccelerating expansion of the universe. Although, the component that causes this accelerationis not known yet, the best model that fits dark energy and other cosmological data is the ΛCDMmodel, which is a Friedmann-Lematre-Robertson-Walker (FLRW) universe with a cosmologicalconstant. Unfortunately, this model does not provide a physical picture of dark energy. Inorder to describe dark energy in the realm of general relativity, we need to consider some exoticfluid with an unusual equation of state that has a negative pressure and violates the strong-energy condition (see for example [9, 10]). The existence of this exotic fluid not only opensthe door for reexamining the constituents of our universe but also evades some nonsingularitytheorems through relaxing the strong-energy condition. This revives interest in nonsingularcosmologies, especially those describing the universe in early and late times. Having a moregeneral class of equations of state, that violates the strong-energy condition, creates a widerclass of singularities beyond that of Big Bang and Big Crunch [6, 7, 8]. In FLRW models thesesingularities are classified as follows [25]: • Type I (”Big Rip”): t → t s , a → ∞ , ρ → ∞ , and | P | → ∞• Type II (”Sudden”): t → t s , a → a s , ρ → ρ s , and | P | → ∞• Type III : t → t s , a → a s , ρ → ∞ , and | P | → ∞• Type IV : t → t s , a → a s , ρ →
0, and | P | →
0, but higher derivative of H diverges.It is of interest to build cosmological models, which are free from the above singularities.In the last decade, there have been several proposals to describe dark energy/matter or darkenergy alone as a single barotropic fluid in FLRW models with various equations of state (EoS)such as; Chaplygin gas, Van der Waal, linear, and quadratic EoS[13, 14, 15, 16, 17, 18, 19, 20,21, 22, 23, 24]. Notice that, most of these models have fixed points, or de Sitter solutions.In this article we use a phase space method to study general solutions of single fluid FLRWmodels with fixed points and a pressure p ( H ), where H is the hubble parameter. We discusspossible consequences of having these fixed points. Some of these consequences are; (i) if wedescribe our universe as a single component fluid and model the late time acceleration by afuture fixed point, then the resulting cosmology does not have future-time singularities of typesI, II and III described in [25], (ii) cosmologies with a future and a past fixed points are freeof types I, II and III singularities, (iii) one can use a simple argument to show the phantomdivide [27, 28], or in a single fluid FLRW models it is impossible for a physical solution to crossthe phantom divide line in a finite time, and (iv) in these models, the only way to get bouncesolutions is to have a nonvanishing pressure as ρ →
0. The phase space method can be used toconstruct nonsingular late time model, in particular, unified dark fluid (UDF) and dark energymodels. We use this qualitative analysis to describe the entire behavior of a FLRW cosmology1ith bulk viscousity η ∼ ρ r , presenting two exact solutions, with r = 1 and r = 1 / In this section, we use a phase space method in flat FLRW models with fixed points to arguethat a fluid with a continuous and differentiable pressure always reaches a fixed point in aninfinite time and has no finite-time singularities of types I, II and III described in [25].Let us start with a FLRW universe, with an equation of state p = p ( H ) , (1)where the pressure p ( H ) is a continuous function of the hubble parameter H . Using the unitconvention, 8 πG = c = 1, Einstein field equations lead to Friedmann equation and Raychaudhriequation H = ρ − ka , (2)˙ H = − H −
16 ( ρ + 3 p )) H = ˙ a/a. (3)For energy-momentum conservation, we obtain˙ ρ = − H ( p + ρ ) (4)which is not independent of Eqn. (3). In this work we are interested in the flat FLRW case,the spatially curved case will be discussed elsewhere [30]. For a flat FLRW universe, Eqn.(3)becomes ˙ H = −
12 ( p ( H ) + ρ ) = f ( H ) , (5)which can be expressed in terms of a dimensionless hubble parameter and time h = H/H ∗ and τ = c H ∗ t , as follows dhdτ = F ( h ) . (6)Where H ∗ is a parameter that depends on the equation of state parameters and c is somenumber. A solution, h ( τ ) of Eqn. (6) is subject to an initial condition h (0) = h .Motivated by the fact that many of the proposed models for dark energy and unified darkmatter/energy models do have fixed points, we assume the existence of fixed points for Eqn.(6),which are the zeros of the function F ( h ) = − / ρ + p ). Let us call the zeros of F ( h ), h , h , ... ,where h < h < h .. . 2ixed points are classified according to their stability as follows; stable, unstable, or half-stable, depending on the sign of their tangents as shown in Figure (1) . Unstable fixed pointsare represented by arrows emanating out of them and stable points are represented by arrowspointing toward them as shown in Figure (1). Half-stable points are stable from one side andunstable from the other side or vice versa. unstable Fixed Points half-stable stable F(h) h Figure 1:
Types of fixed points
The arrows determine how the solution develops with time. Notice that, a fixed pointsatisfies equation (6), i.e., h ( τ ) = h , therefore, it is a solution. This constant solution isnothing but a de Sitter space. If the system started from an initial value, h = h , i.e., at afixed point, then it will remain at this point forever. But when the value of h is close to thatof h , the solution h ( τ ) develops towards h , if h is a stable fixed point, or away from it, if thepoint is unstable. This technique has been used in literature to study particular equations ofstate and their solutions (e.g., [18]), but here we try to keep our discussion general assuming apressure p ( H ).Using this tool, one can determine how a solution behaves upon knowing the nature of thesefixed points. For example, if we start with a value for h between the stable and the unstablefixed points in Figure (1), then the solution will develop to the left, i.e., will take values withsmaller h till it reaches the stable point. Therefore, the solution in this case interpolates betweentwo de Sitter spaces. If h takes values larger than the unstable fixed point in Figure(1), thenthe solution develops to the right and might reach a singularity if the time to reach h → ∞ isfinite.Fixed points and the asymptotic behavior of F ( h ) enable us to predict the behavior of thesystem without knowing the form of the solution. But, in order to have a reliable qualitative Here I am following the terminology of Strogatz [26]. h → ∞ . Using the argument below we will see that the continuityand differentiability of the pressure, p ( h ) determine if the time to reach a fixed point is finiteor infinite. Also, in the coming section we use the asymptotic behavior of F ( h ) to determinethe time taken by a solution to reach a point where F ( h ) is diverging. We are going to usepressure properties to describe the behavior of a solution qualitatively without the need forexact or approximate solutions.Here we argue that for a flat FLRW fluid with a pressure p ( h ), which satisfies;i) F ( h ) is continoues and differentiable, andii) F ( h ) has a future fixed point h , i.e., h < h if F ( h ) >
0, (or h > h if F ( h ) < h ( τ ) which is defined for times τ >
0, and has no future finite-time singularities of types I, II and III.Notice that, the first assumption is needed to ensure the existence of a unique local solutionaround some initial value h by the existence-uniqueness theorem. In addition, it leaves thesolution free from type-II singularities, which might contradict casuality since the sound speed dp/dρ = c s will diverge as well. The second assumption ensures the extendibility of this localsolution to all values τ > h (0) = h ,then one obtains G ( h ) = Z hh dyF ( y ) = τ (7)Since F ( h ) = 0, its either F ( h ) > F ( h ) <
0, let us choose F ( h ) >
0, and h < h from ii). As a result G ( h ) is a monotone near h . For any solution φ ( τ ), we have G ( φ ( τ )) = τ. (8)Since G ( h ) is a monotone near h , the above relation can be inverted G ( τ ) − = φ ( τ ) , (9)where G − is the inverse map of G . This is a local solution by construction (unique since F ( h )is differentiable by the existence-uniqueness theorem) that can be extended by looking for amaximal interval in which G ( h ) is a monotonic function. Since F ( h ) > h < h < h , then G ( h ) is a monotone in this interval. This imply that themaximum interval in which we can extend the solution is [ h , h ]. The solution can be definedfor all values τ > τ + = Z h h dhF ( h ) = ∞ . (10)4ow, let us show that τ + = ∞ , when i) and ii) are given. Since F ( h ) is differentiable, the slopeof the tangent at any point is finite. Let us choose a number M < F ′ ( h ) , ∀ h ∈ [ h , h ]. Theexistence of M enables us to define a linear function Y(h) such that Y ( h ) ≥ F ( h ) , (11)where Y ( h ) = M ( h − h ). This leads to Z h h dhF ( h ) ≥ Z h h dhF ′ ( h min ) ( h − h ) = ∞ , (12)therefore, τ = Z h h dhF ( h ) = ∞ . (13)We have shown that for the first-order system of Eqn.(6) there exists a unique solution, h ( τ )defined for times τ > h . It is clear that the solution is bounded, i.e., h ( τ ) ∈ [ h , h ] for times τ >
0. Therefore, the density, ρ ( τ ) is bounded for times τ >
0, as aresult, there is no future singularities of type I and III. The pressure p = − F ( h ) − h H ∗ is bounded too in this interval, since F ( h ) and h are bounded , which means, no future singu-larities of type II in these spacetimes. As a result this class of FLRW solutions are free fromfuture finite-time singularities of type I, II and III .The above argument can be generalized by relaxing the differentiability condition, in thiscase we have one of the following;a) F ′ ( h ) has a jump discontinuity: F ( h ) is not continuous, but its left and right derivatives are finite for h ∈ [ h , h ]. In this case, one can still construct the linear function Y ( h ), whichcan be used to show that the time to reach h is always infinite and the solution has no futurefinite-time singularities of types, I, II, and III.b) F ( h ) has infinite discontinuity: F ( h ) is not continuous and at least the left or right derivativeof F ( h ) is infinite for some h ∈ [ h , h ]. This case is not physical, since the divergence of F ′ ( h )leads to a divergent speed of sound, dp/dρ = c s , which has to be less than unity for the modelto be causal. We are going to discuss this issue in more details in the coming section whenwe discuss phantom crossing, but through out this paper we are going to assume no infinitediscontinuities in F ′ ( h ). Since any continuous function on a closed interval is bounded Left derivative is defined as d − F ( a ) dh = lim h → a − F ( h ) − F ( a ) h − a and the right derivative is defined in a similar way. Consequences of Fixed Points
Here we discuss consequences of fixed points and how can we use this dynamical method todescribe the entire behavior of a single fluid in flat FLRW without knowing the form of thesolution.
Consequences of using phase space method to study fixed points can be listed as follows; i) If the late-time behavior of our universe is described by a single fluid, as in unified darkfluid models, and the late-time acceleration is developing towards a de Sitter universe, thenthere is no future-time singularity of types I, II and III in this solution. Since the pressure p ( h ), is a differentiable function of h , or at most has finite discontinuities, it takes the universeinfinite time to reach the de Sitter space. ii) If we combine a future fixed point with a past fixed point, by applying the argument insection 2 we get a nonsingular solution, which is free from types I, II and III singularities. Thetime taken by the solution to reach h or to come from h starting from an initial value h isinfinite. In addition, the hubble parameter h ( τ ) interpolates between h and h in a monotonicmanner. iii) As one might notice, the solutions we have so far, still, admit a weaker type of singularity,namely, type-IV in the classification given in [25]. But even these weaker singularities can beavoided by requiring p ( h ) to be a smooth function, i.e., a C ∞ function. One can show that asfollows; The n-time derivative of h ( τ ) can be written as; h ( n ) = F ( h ) ddh ! n − F ( h ) . (14)which means, unless F ( h ) or one of its derivatives (up to the ( n − h ( n ) is always bounded in [ h , h ]. This leads to the conclusion: In a flat FLRW universe, if a) p ( h ) is a smooth function, i.e., arbitrarily differentiable, and b) F ( h ) has a future and a pastfixed point, then the spacetime is free from singularities of types, I, II, III and IV [25] when h ∈ [ h , h ]. iv) A single fluid in flat FLRW with a pressure p ( h ) admits bouncing solutions only if thereare no fixed points between h = 0 and h = h , i.e., F (0) = 0. These solutions either have abounce or turnaround at h = 0 depending on sign of F (0). At h = 0, ¨ a/a = F (0), therefore, if F (0) > F (0) < v) Another consequence of this phase space method is a simple and transparent way to show theno-go theorem of phantom divide [27, 28] (see also, [39, 29]), which can be stated as follows; In6 single fluid FLRW cosmology it is impossible for a causal solution to go from a region where ω ( h ) < − ω ( h ) > −
1. In other words, a solution in one of the mentionedregions has no access to the other region. Let us explain this in more details using the analysiswe have in section 2. First, let us assume that the pressure p ( h ) is differentiable. In this case, ifa solution approaches a fixed point, where w ( h ) = −
1, starting from a region where w ( h ) < − p ( h ) is notdifferentiable, then p ′ ( h ) has either a finite or an infinite discontinuity. If the discontinuity isfinite, the time to reach the crossing point or the fixed point is infinite, as we showed in theprevious section, therefore, the crossing will not occur. If p ′ ( h ) has an infinite discontinuity,a solution will reach w ( h ) = − F ( h ):i) lim h → h F ( h ) = 0,ii) lim h → h dF ( h ) /dh = ±∞ andiii) τ = R h h /F ( h ) < ∞ .A class of functions which satisfy the above conditions is F ( h ) = F ( h − h ) s , where 0 < s < F >
0. A solution is given by the following expression; h ( τ ) = h − h ( h − h ) − s − F (1 − s ) τ i − s , τ ≤ τ ∗ = h τ > τ ∗ (15)where, h (0) = h and τ ∗ = ( h − h ) − s /F (1 − s ). It is clear from Eqn. (15) that the solutionstays at the fixed point for τ ≥ τ ∗ . Also, notice that h is a stable fixed point. One can seethat by assuming a small perturbation away from h , i.e., h ( τ ) = h + δ ( τ ), it leads to δ ( τ ) = − h ( h − h ) − s − F (1 − s ) τ i − s . (16)As a result, if we extend the definition of F ( h ) for h > h , e.g., F ( h ) = − F ( h − h ) s as shownin Figure (2)-a), we will not have a phantom crossing. But if we define F ( h ) = ± F ( h − h ) s ,i.e., a double valued function as in Figure (2)-b), the solution can cross the phantom dividein a finite time. Therefore, in addition to its infinite discontinuity F ′ ( h ), need to be a doublevalued function to have a phantom crossing solution. Our conclusion is that a general causalsolution of any flat FLRW model, with a continuous pressure p ( H ), can not cross the phantomdivide line in a finite time. It is clear that infinite discontinuity of F ( h ) is necessary but notsufficient for a phantom crossing. As we mentioned earlier, to have a complete qualitative description of a general solution inflat FLRW cosmology we need to know the fixed points as well as the asymptotic behavior of7
Figure 2: a) F ( h ) = (1 − h ) / , for h ≤
1, and F ( h ) = − ( h − / , for h > b) F ( h ) = ± (1 − h ) / , for h ≤ F ( h ). The later property enables us to determine the time to reach a point where F ( h ) → ±∞ ,starting from some initial value h (0) = h . Let us first show the relation between the asymptoticbehaviors of F ( h ) and finite-time singularities. Asymptotic Behavior of F ( h ) and Singularities: The dimensionless Hubble parameter h ,in a flat FLRW cosmology is controlled by a one-dimensional phase space evolution function F ( h ). In this dynamical system a solution develops towards either a fixed point, where F ( h ) →
0, or a point where F ( h ) → ±∞ . A solution approaching a point where F ( h ) → ±∞ does notnecessarily mean that the it has a finite-time singularity. It is crucial to know how fast F ( h )reaches infinity. This enables us to determine if the singularity is reached in a finite time ornot [32]. Considering the integral τ = Z hh dh ′ F ( h ′ ) , (17)it is easy to see that if lim h →±∞ F ( h ) ∼ h s , where 0 ≤ s ≤
1, the integral diverges. As a result,the solution takes an infinite time to reach the singular point, therefore, it has no finite-timesingularities. One can observe that if F ( h ) grows as a linear function or slower , as h → ±∞ ,the solution will have no finite-time singularities. One can show this rigourously following thesame argument in section 2. An example of a singular asymptotic behavior is a quadraticfunction F ∼ h , which leads to a Big Bang singularity.It is intriguing to notice that there is a class of F ( h ) that grows faster than a linear function,but still, does not have finite-time singularities. An example of this is F ( h ) ∼ h ln h , or h ln h ln ln h, and so on [31]. These functions grow faster than a linear function, but leadto time τ ( h ), that depends on h logarithmically, therefore, leads to nonsingular solutions. Ingeneral, if F ( h ) can be expressed as F ( h ) = g/g ′ , where g ( h ) is any function such that g ( h ) → ∞ This is a known fact in the literature see for example [32] h → ±∞ , then τ ( h ) will diverge logarithmically as h → ±∞ , which leads to a nonsingularsolution. To conclude, except for F ( h ) with a special form (as in the above mentioned cases),the solution reaches a point where F ( h ) → ±∞ in a finite time if F ( h ) grows faster than alinear function. Qualitative Description:
In this section we take the pressure p ( h ) to be a continuous functionof h , or at most has finite discontinuities. Therefore, we are not going to allow any infinitediscontinuities for p ′ ( h ), since this leads to a divergent speed of sound that violates causality.One can qualitatively describe a solution as follows;a) A solution starts from an initial value h , then, develops towards either a future fixed pointor a point where F ( h ) → ±∞ .b) If it develops towards a fixed point, then, the time taken by a solution to reach this point isalways infinite according to the argument in section 2 and the solution has no future finite-timesingularities.c) If it develops towards a point where F ( h ) → ±∞ , and the asymptotic behavior is linear orslower, then, the time to reach this point is infinite and the solution has no future finite-timesingularities.d) If it develops towards a point where F ( h ) → ±∞ , and the asymptotic behavior is growingfaster than a linear function but F ( h ) has the asymptotic form F ( h ) ∼ g/g ′ , where g ( h ) is anyfunction such that g ( h ) → ∞ as h → ±∞ , then, the time to reach this point is infinite and thesolution has no future finite-time singularities.e) If it develops towards a point where F ( h ) → ±∞ , and the asymptotic behavior is growingfaster than a linear function and F ( h ) has the asymptotic form F ( h ) ∼ g/g ′ , but g ( h ) does notgrow as g ( h ) → ∞ when h → ±∞ , then, the time to reach this point is finite and the solutionis singular.f) In all the above cases the solution is a monotonic function of time as it develops from h towards either a fixed point or a point where F ( h ) → ±∞ .The same analysis can be followed backward in time but with past fixed points and pointswhere F ( h ) → ±∞ . The solution has no finite time singularities if it has no future and no pasttime singularities. In this section we use the qualitative method developed in the previous sections to describe thegeneral behavior of viscous fluids in FLRW models and compare it with exact solutions. Thelast exact solution with r = 1 / .1 Examples: Nonsingular viscous fluids Consider the following equation of state (EoS) p ( H ) = ( γ − ρ − η ( ρ ) H, (18)where η ( ρ ) = η ρ r . The above EoS can describe a fluid with bulk viscosity η ( ρ ) (see e.g., [33]), apolytropic fluid [37], or a fluid with adiabatic particle production (see e.g., [36]). Although thesedifferent interpretations of the above EoS produce the same dynamics, their thermodynamicscould be different. Several solutions for the above EoS with different values of r , including thecases discussed here, are known in the literature, see for example [35]. Here we are going todiscuss two viscous solutions, the one with r = 1 which is well known in the literature [34] andanother with r = 1 /
4, which is less known and express them in terms of Lambert-W function.This clearly shows how the density and the scale factor behave as functions of time.Now using the above pressure in Eqn. 3 we obtain˙ H = −
12 ( ρ + p ) = −
32 ( γ H − r η H r +1 ) (19)Taking h = H/H ∗ and τ = c H ∗ t , where H ∗ = ( γ r η ) r − and c = 3 γ/ dhdτ = − h (1 − h r − ) , c a dadτ = h, (20)Notice that we have two fixed points, h , = 0 ,
1, for this equation. The nature of these fixedpoints depends on the value of r . Solutions with r > / r < /
2. In the first case F ( h ) is negative during its interpolation between thetwo fixed points, while the reverse is true for the second case. It is interesting to notice thatthe first case describes a nonsingular universe which has an EoS parameter w ( ρ ) ≥ − F ( h ) = − / ρ + p ) while the second case describes a nonsingular universe dominated by aphantom component with EoS parameter w ( ρ ) ≤ −
1. It is interesting to notice that fluids with r < / h > p = ( γ − ρ . One can see that by writing Eqn.(4) as a dρ ( a ) da = − ρ ( a ) + p ( a )) = − γ ρ (1 − η ′ ρ r − / ) , (21)where η ′ = √ η solving the above equation for a general viscous fluid with r − / − s < ρ ( a ) = ( γ/η ′ ) − /s h C a − γs i /s (22)10 r > 1/2 r < 1/2F(h) 1h Figure 3:
Fluids with r > / r < / If ρ > ( η ′ /γ ) /s = ρ ∗ , the integration constant C is positive, sice C = a γs [( ρ /ρ ∗ ) s − , (23)then for small a , i.e., ( a /a ) γs [( ρ /ρ ∗ ) s − >>
1, we get ρ ∼ a − γ (24)which describe a fluid with an EoS p = ( γ − ρ , and for large a we get ρ ∼ ρ ∗ (25)which describes an empty space with a cosmological constant ρ ∗ . Another interesting featureof these models is that they show that a normal matter, i.e., γ >
0, with bulk viscosity, behavesas a phantom matter. Furthermore, in the r < / p = γ ρ (1 − ( H ∗ /H ) s ) − ρ, (26)which leads to an effective EoS parameter w eff = γ (1 − ( H ∗ /H ) s ) − , (27)since γ >
0, and H ∗ /H >
1, we always have w eff ≤ −
1, which breaks all energy conditions.Since H ( t ), in this case, interpolates between two fixed points ( H ∗ and 0) and the pressure p ( H ) is continuous and differentiable, then according to the argument in section 2, the solutionis nonsingular and takes an infinite time to reach a fixed point. We are going to see an explicitexample of this behavior for the r = 1 / luids with r=1 : This case was first discussed in [34] as nonsingular solution for a viscousfluid cosmology. In literature this solution usually expressed in terms of time as a function ofthe scale factor or the density. Here we express the energy density and the scale factor asfunctions of time in terms of Lambert W-function. First, let us analyze this case qualitativelytaking subsection 3.2 in consideration. The asymptotic behavior has the form F ( h ) ∼ h whichleads to singular solutions unless there are fixed points. F ( h ) has two fixed points h = 0 and h = 1, as shown in Figure (4). The first is half-stable point and the second is an unstablepoint. These points divide possible solutions into three types; i) a solution where h ∈ ( −∞ , h < h , ii) a solution where h ∈ [0 , h > h > h , and iii) a solution where h ∈ [1 , ∞ ),if h < h . Notice that, in case b), since p ( h ) is differentiable then by the argument in section2, it takes the solution an infinite time to reach h starting from some initial value, h , where h < h < h . The same is true if we calculate the time taken by the solution to go from h to h . Therefore, for h < h < h the solution is nonsingular and interpolates smoothlybetween h , and h . If h < h or h > h the solution has a finite time singularity sincethe asymptotic behavior of F ( h ) ∼ h is growing faster than a linear behavior. For h > h the solution describes a universe that starts from a de Sitter space and endes with a Big Ripsingularity. The above equations can be solved exactly in terms of Lambert W-function, which unstable half-stable Fixed Points, r=1F(h) h Figure 4:
Fixed points for r = 1 case is the solution of the equation W e W = x . The hubble parameter and the scale factor in termsof the time ”t” are given by H ( t ) = H ∗ [ W ( c e (3 γ/ H ∗ t ) + 1] − ,a ( t ) = c [ W ( c e (3 γ/ H ∗ t )] / γ (28)Having a ( t ) = a , and ρ ( t ) = 3 H at t = t , we get ρ ( t ) = 3 H ∗ [ W ( β e (3 γ/ H ∗ ( t − t ) ) + 1] − , (29)12 ( t ) = a " W ( β e (3 γ/ H ∗ ( t − t ) ) W ( β ) (2 / γ ) (30)where β is given by β = (cid:18) H ∗ H − (cid:19) e ( H ∗ H − (31)It is crucial at this point to know the sign of β since it controls the behavior of the W-function.The h = H /H ∗ < β , which leads to a smoothbehavior in all times for the energy density. For the scale factor, the moments at which iteither diverges or goes to zero are when t = + ∞ and t = −∞ respectively. Therefore, we haveno finite-time singularities in this case. In early times, this model describes an empty universewith a cosmological constant Λ ∼ H ∗ , which evolves to a universe with an EoS p = ( γ − ρ in late times. The behavior of H ( t ) and a ( t ) as a function of time t , is shown in Figure(5),which is clearly monotonic. If the initial value h = H /H ∗ > β < Hubble Parameter00.20.40.60.81H(t)–30 –20 –10 10 20 30 40t
Scale Factor
Figure 5:
Hubble parameter H ( t ), in units of H ∗ , and scale factor a ( t ), in units of a , versus time t in unitsof 1 /H ∗ , r = 1 and γ = 1. have a singularity of type-I, or a Big Rip singularity. In this case the effective EoS parameter w eff = p/ρ < −
1, therefore the fluid is phantom. Notice that, if h < h , the r < / /H ∗ = λ <<
1, or the cosmological constant of the early times is larger than that of latetimes, then the fixed points become h ∼ √ λ, h ∼ − λ, (32)This model describes a universe that interpolates between two de Sitter spaces one with largecosmological constant in early times and another with small cosmological constant in late timeswhich can model the inflation and late time acceleration periods.13 luid with r=1/4: Here we discuss the solution for the r = 1 / r < / h > h . If h < h , the solution describesa nonsingular phantom matter with w eff ≤ −
1. To analyze this case qualitatively, let us startwith the asymptotic behavior of F ( h ) which has the form F ( h ) ∼ h . It clearly leads to singularsolutions unless we have fixed points. F ( h ) has two fixed points h = 0 and h = 1, as shownin Figure (8). The first is an unstable point and the second is a stable point. These pointsdivide possible solutions into two types; i) a solution where h ∈ [0 , h > h > h , andii) a solution where h ∈ [1 , ∞ ), if h < h . Notice that, in the first case, i), since p ( h ) isdifferentiable then by the argument in section 2, it takes the solution an infinite time to reach h starting from some initial value, h , where h < h < h . The same is true if we calculatethe time to go from h to h , therefore, the solution is nonsingular and interpolates smoothlybetween h , and h . The second case ii) is singular, since the asymptotic behavior F ( h ) ∼ h is growing faster than a linear behavior. In fact, this solution describes a universe that startsfrom a finite-time singularity in the past (Big Bang type) and evolves to a de Sitter space afteran infinite time. stable unstable Fixed Points, r=1/4F(h) h Figure 6:
Fixed points for r = 1 / Eqn.(19) for r = 1 / H ( t ) = H ∗ [ W ( c e ( − γ/ H ∗ t ) + 1] − ,a ( t ) = c " W ( c e ( − γ/ H ∗ t ) + 1 W ( c e ( − γ/ H ∗ t ) / γ . (33)Using initial conditions, the integration constants are c = e γ H ∗ t β ′ , c = a " W ( β ′ ) + 1 W ( β ′ ) γ , (34)14here β ′ = (cid:18)q H ∗ /H − (cid:19) e (cid:16) √ H ∗ /H − (cid:17) . (35)Notice that β ′ is positive for h < h >
1. For initial value h < H ( t ) and the scale factor a ( t ), as functions of time ”t”, are shownin Figure(7). As we mentioned in the beginning of the section, this class of solutions shows how Hubble Parameter00.20.40.60.81H(t)–30 –20 –10 10 20 30 40t
Scale Factor
Figure 7:
Hubble parameter H ( t ), in units of H ∗ , and scale factor a ( t ), in units of a , versus time t in unitsof 1 /H ∗ , r = 1 / γ = 1. a viscous fluid with a usual EoS (i.e., γ >
0) behaves as phantom component. Similar to the r = 1, one can consider adding a small cosmological constant. A small cosmological constantΛ /H ∗ = λ << h ∼ λ / , h ∼ − λ. (36)The solution with initial value h < h < h described a universe filled with a phantom matterinterpolating between a de Sitter space with a small cosmological constant at early times andanother with a large cosmological constant in late times. Here we list all possible future scenarios of the universe as a single fluid in FLRW cosmologywithout assuming any fixed points but imposing the causality and stability constraints. It isknown that, if p > − / ρ , destiny of the universe is tied to geometry (see for example [38]) andthe value of k is important to predict the fate of the universe. On the other hand, if p < − / ρ destiny is not tied to geometry but controlled by the behavior of the energy density ρ . This canbe shown if we consider the known mechanical model for a , by rewriting Friendmann equationas ˙ a = 13 a ρ ( a ) − k = − V eff ( a ) . (37)15aking the EoS p = w ρ leads to ρ ( a ) = C a − w ) ⇒ V eff ( a ) = − C/ a − (1+3 w ) + k (38)For large a , if w > − / k controls the existence of vanishing velocities, or turning points,but for w > − /
3, the potential gets a small contribution from k compared to that comingfrom a ρ . This breaks the connection between the geometry and destiny. In fact, this simpleargument is also suggesting that our universe will keep on expanding because of the dominationof the dark energy component. We will see next that this is not generally the case.Here we are going to use the above constraints to list possible scenarios for the future of theuniverse. Let us model our universe using a general single barotropic fluid, which is a reasonableassumption since dark energy is dominating in late times.˙ H = 1 / H + p ( H )) . (39)First, it is easy to show that we are in a region in the phase space (i.e., ˙ H − H space ) where˙ H < . It is known that the deceleration parameter has changed sign from positive to negativeas the universe evolved from a matter dominating era to a dark energy dominating era. To seehow this crossing happened consider first the zero acceleration curve which is given by¨ aa = ˙ H + H = 0 ⇒ ˙ H = − H , (40)then, as an example of dark energy, consider an accelerating universe with a cosmologicalconstant and matter ˙ H = F ( H ) = − / H − ρ Λ ) . (41)By plotting both functions − H and − / H − ρ Λ ) in Figure(8) one can see how this crossinghappened, i.e., going from ¨ a < a >
0. The point at which the acceleration vanishes can beused as a reference point to draw the constraints on possible evolutions of the universe. Usingthe causality and stability constraints, dp/dρ ≤ dp/dρ ≥
0, we get 0 ≥ dp ( H ) /dH ≤ H which in turn leads to − H ≥ dF ( H ) /dH ≤ − H . Integrating this inequality leads to − / H + C ≥ F ( H ) ≤ − H + C , where the integration constants C and C are fixed bythe initial values of H and ˙ H at the reference point. Notice that, C and C have to be positivenumbers, otherwise crossing the zero-acceleration curve will not occur. The last inequalityconstrains the future behavior of F ( H ) to lie between these two parabolas, as a result, F ( H )must meet the H -axes in a future time and ends as a de Sitter universe after infinite time.In addition, this evolution starting from the point of zero-acceleration till the end of time isnonsingular since it ends with de Sitter universes. One can directly calculate ˙ H using the best measured values for w D , Ω m and Ω D . –3/2 H -H 22 + C1 + C22 –3 H–1–0.500.51F(H)–0.2 0.2 0.4 0.6 0.8H –1–0.500.51F(H)–0.2 0.2 0.4 0.6 0.8H Figure 8:
The − H curve is the zero-acceleration curve. The black region between the two curves, − / H and − / H satisfies the two constraints on barotropic fluids and any curve in this region must end with afixed point on positive H -axes As one might notice, although, the causality constraint is essential for any physical model.It is not clear if we should insist on having the stability constraint, since we do not know thephysics of the dark energy component. Now, if we relax the stability constraint, dp/dρ ≥ H -axes. Thelast possibility is interesting since according to the discussion in subsection 3.1 it describes aturnaround behavior, therefore, the universe in a future finite time reaches a maximum size,then, recollapses, since the hubble parameter H changes sign. In this work, we used a phase space method to study possible consequences of having fixedpoints in a single fluid flat FLRW models. Some of these are; (i) if we describe our universe asa single component fluid with a future fixed point, then the resulting cosmology does not havefuture-time singularities of types I, II and III in [25], (ii) cosmologies with a future and a pastfixed points are free of of types I, II and III singularities, (iii) one can use a simple argumentto show the phantom divide [27, 28], or in a single fluid FLRW models it is impossible for aphysical solution to cross the phantom divide line in a finite time, and (iv) in these models, theonly way to get bounce solutions is to have a nonvanishing pressure as ρ →
0. This method canbe used to construct nonsingular late-time models, in particular, unified dark fluid and darkenergy models. We use this method to qualitatively describe any flat FLRW model with fixedpoints. We discussed FLRW cosmology with bulk viscosity η ∼ ρ r , and presented two exactsolutions with r = 1 and r = 1 /
4, which are expressed in terms of Lambert-W function. Thelast solution describes either a nonsingular phantom dark energy or a unified dark fluid model.17he phantom solution is interesting since it shows how a viscous normal fluid behaves verysimilar to a phantom matter without Big Rip singularities. In addition, it interpolates betweentwo de Sitter spaces with small and large cosmological constants. Possible future scenarios ofour universe include; a de Sitter space, an empty universe with vanishing cosmological constant,or a turn a round solution that reaches a maximum size, then collapses.
Acknowledgement
I would like to thank P. Argyres, S. Das, A. Shapere, E. Lashin and A. El-Zant for severaldiscussions and comments.
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