Dynamical analysis approaches in spatially curved FRW spacetimes
PProceedings of
RAGtime 20–22, 15–19 Oct., 16–20 Sept., 19–23 Oct., 2018/2019/2020, Opava, Czech Republic Z. Stuchl´ık, G. T¨or¨ok and V. Karas, editors, Silesian University in Opava, 2020, pp. 1–11
Dynamical analysis approaches inspatially curved FRW spacetimes
Morteza Kerachian,
Giovanni Acquaviva and Georgios Lukes-Gerakopoulos Institute of Theoretical Physics, Faculty of Mathematics and Physics,Charles University, CZ-180 00 Prague, Czech Republic Astronomical Institute of the Academy of Sciences of the Czech Republic,Boˇcn´ı II 1401/1a, CZ-141 00 Prague, Czech Republic a [email protected] b [email protected] c [email protected] ABSTRACT
In this article, we summarize two agnostic approaches in the frameworkof spatially curved Friedmann-Robertson-Walker (FRW) cosmologies dis-cussed in detail in (Kerachian et al., 2020, 2019). The first case concernsthe dynamics of a fluid with an unspecified barotropic equation of state(EoS), for which the only assumption made is the non-negativity of thefluid’s energy density. The second case concerns the dynamics of a non-minimally coupled real scalar field with unspecified positive potential. Foreach of these models, we define a new set of dimensionless variables and anew evolution parameter. In the framework of these agnostic setups, we areable to identify several general features, like symmetries, invariant subsetsand critical points, and provide their cosmological interpretation.
Keywords:
Gravitation, Cosmology; Dynamical systems
The dynamical system analysis is a powerful tool that has broad applications indifferent fields of science. Dynamics itself was introduced by Newton through hislaws of motion and gravitation. These laws enabled Newton to tackle the two-body problem of the Earth’s motion around the Sun. Later on, when scientiststried to address the three-body problem of the Earth, the Moon and the Sun,they found it was too complicated to tackle it quantitatively. In the late 19thcentury, Henry Poincar´e suggested that celestial mechanics could be studied byconsidering qualitative features of a system rather than quantitative founding in thisway the branch of dynamical systems (Strogatz, 2018). In the context of cosmologydynamical systems analysis allows us to view the global evolution of a model, from © (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ? (cid:111) (cid:110) (cid:54) a r X i v : . [ g r- q c ] J a n M. Kerachian, G. Acquaviva and G. Lukes-Gerakopoulos its start near the initial singularity to its late-time evolution (Wainwright and Lim,2005).The observations indicate that the universe is homogeneous and isotropic (Aghanimet al., 2018), which makes the Friedmann-Robertson-Walker (FRW) spacetime therelevant metric to model its evolution. Even if the universe appears to be spatiallyflat, considering a non-zero spatial curvature is still observationally viable and mighthelp in alleviating some cosmological tensions (Ryan et al., 2019; Di Valentino et al.,2020). Therefore, in our work we used spatially curved FRW metrics.According to Planck Collaboration et al. (2020), the total energy density of theuniverse consist of ∼ .
5% dark energy, ∼ .
5% cold dark matter, and ∼ (cid:15) ≥ P of the fluid to attain negative values in order to beable to describe cosmological models with accelerated expansion. In these modelsthe speed of sound of the fluid is not necessarily less than the speed of light, whichimplies exotic EoS.The second type of models we analyse concerns a curved FRW geometry non-minimally coupled to a scalar field with generic positive potential (Kerachian et al.,2019). A similar analysis has been performed by Hrycyna and Szyd(cid:32)lowski (2010)in the presence of matter for flat FRW. Our formulation allows for several improve-ments in the aforementioned analysis by considering a generic spatially curved FRWmodel and a more general scalar field potential. The Friedmann and the Raychaudhuri equations for a FRW cosmology with onlyone fluid component are given by H + ka = (cid:15) , H + 3 H + ka = − P , (1)respectively and the continuity equation for the energy density reads˙ (cid:15) + 3 H ( P + (cid:15) ) = 0 . (2)In these equations, (cid:15) is the energy density, P is the pressure of the barotropic fluid, k is the spatial curvature, a is the scale factor, H = ˙ aa is the Hubble expansion rateand ˙ denotes derivative with respect to the coordinate time. (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ??
5% cold dark matter, and ∼ (cid:15) ≥ P of the fluid to attain negative values in order to beable to describe cosmological models with accelerated expansion. In these modelsthe speed of sound of the fluid is not necessarily less than the speed of light, whichimplies exotic EoS.The second type of models we analyse concerns a curved FRW geometry non-minimally coupled to a scalar field with generic positive potential (Kerachian et al.,2019). A similar analysis has been performed by Hrycyna and Szyd(cid:32)lowski (2010)in the presence of matter for flat FRW. Our formulation allows for several improve-ments in the aforementioned analysis by considering a generic spatially curved FRWmodel and a more general scalar field potential. The Friedmann and the Raychaudhuri equations for a FRW cosmology with onlyone fluid component are given by H + ka = (cid:15) , H + 3 H + ka = − P , (1)respectively and the continuity equation for the energy density reads˙ (cid:15) + 3 H ( P + (cid:15) ) = 0 . (2)In these equations, (cid:15) is the energy density, P is the pressure of the barotropic fluid, k is the spatial curvature, a is the scale factor, H = ˙ aa is the Hubble expansion rateand ˙ denotes derivative with respect to the coordinate time. (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ?? (cid:111) (cid:110) (cid:54) ynamical analysis approaches in spatially curved FRW spacetimes By introducing the normalization D = H + | k | /a , we are able to present well-defined dimensionless variables, i.e. the variables which are valid for k > k ≤
0. These new dimensionless variables areΩ (cid:15) = (cid:15) D , Ω H = HD , Ω P = PD , Ω ∂P = ∂P∂(cid:15) , Γ = ∂ P∂(cid:15) (cid:15). (3)In order to investigate the evolution of the dimensionless variables. we define anew evolution parameter τ as dτ = Ddt . This new evolution parameter is well-defined during the whole cosmic evolution. Taking the derivative of the dimension-less variables with respect to τ provides the autonomous systemΩ (cid:48) (cid:15) = − Ω H (cid:34) Ω p + Ω (cid:15) (cid:32) (cid:32) ˙ HD + Ω H − (cid:33)(cid:33)(cid:35) , (4)Ω (cid:48) H = (cid:0) − Ω H (cid:1) (cid:32) ˙ HD + Ω H (cid:33) , (5)Ω (cid:48) P = − Ω H (cid:34) ∂P (Ω P + 3Ω (cid:15) ) + 2Ω P (cid:32) ˙ HD + Ω H − (cid:33)(cid:35) , (6)Ω (cid:48) ∂P = − Ω H (cid:18) Ω P Ω (cid:15) + 3 (cid:19) Γ . (7) Positive curvature:
For positive curvature k >
0, in terms of the newvariables the Friedmann and Raychaudhuri equations (1) become respectivelyΩ (cid:15) = 1 , ˙ HD = −
12 (Ω P + 1) − Ω H . (8) Non-positive curvature:
For the non-positive spatial curvature k ≤ (cid:15) = 2 Ω H − , ˙ HD = −
12 (Ω P + 1) + (cid:0) − H (cid:1) . (9)From the definition of Ω H we have Ω H ≤ (cid:15) ≥
0, we getthat 0 ≤ Ω (cid:15) ≤ ≤ Ω H ≤ The next step is to investigate the critical points ( i.e. those points for which Ω (cid:48) = 0)of the autonomous system (4)- (7) and their stabilities. Once the critical points aredetermined, we can look for their cosmological interpretation. To do that a usefultool is the deceleration parameter q = − − ˙ HH = − − Ω − H ˙ HD , (10)in which we used the definition of Ω H . (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ??
0, we getthat 0 ≤ Ω (cid:15) ≤ ≤ Ω H ≤ The next step is to investigate the critical points ( i.e. those points for which Ω (cid:48) = 0)of the autonomous system (4)- (7) and their stabilities. Once the critical points aredetermined, we can look for their cosmological interpretation. To do that a usefultool is the deceleration parameter q = − − ˙ HH = − − Ω − H ˙ HD , (10)in which we used the definition of Ω H . (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ?? (cid:111) (cid:110) (cid:54) M. Kerachian, G. Acquaviva and G. Lukes-Gerakopoulos
Two de Sitter critical lines:
There are two critical lines with a deSitter behavior located at { Ω (cid:15) , Ω H , Ω P , Ω ∂P } = { , ± , − , ∀} . The critical linewith Ω H = 1 (called A + ) has the typical cosmological constant behaviour ( q = − { λ A + i } = {− , , − ∂P ) } , (11)while the critical line with Ω H = − A − ) describes an exponentially shrink-ing universe ( q = −
1) and its eigenvalues are { λ A − i } = { , , ∂P ) } . (12)Eq. (11) and Eq. (12) imply that for Ω ∂P < − A ± are saddle points. However, for Ω ∂P ≥ − A ± can not be determined even by the center manifold theorem. To discuss theirstability numerical examples for specific Γ have to be employed. Static universe critical line:
For positive spatial curvature, there isa critical line (called B ) located at { Ω (cid:15) , Ω H , Ω P , Ω ∂P } = { , , − , ∀} . This criticalline describes a static universe, i.e a = const. and its eigenvalues are { λ Bi } = { , − (cid:112) ∂P , (cid:112) ∂P } . (13)Eq. (13) implies that for 1 + 3Ω ∂P >
0, the critical points along the line B aresaddle; for 1 + 3Ω ∂P < ∂P = − / B ) corresponding toa static universe located at { Ω (cid:15) , Ω H , Ω P , Ω ∂P } = {− , , , ∀} , but as discussed inSec. 2.0.0.2, Ω (cid:15) < Γ : invariant subsets and critical points In this section let us assume that the function Γ has roots ˜Ω ∂P : this allows invariantsubsets lying on { Ω H , Ω P } planes. For each root of Γ, we get a pair of criticalpoints C ± located at { Ω H , Ω P } = {± , ∂P } . Note that, for any new invariantsubset { Ω H , Ω P } there might be an intersection with the critical lines A ± and B ;for simplicity we denote these resulting critical points with the same name as therespective critical lines.The scale factor for the critical point C + grows as a ∼ ( t − t )
23 (˜Ω ∂P +1) , whilefor the critical point C − it decreases as a ∼ ( t − t )
23 (˜Ω ∂P +1) . At these points thedeceleration parameter reduces to q = (3 ˜Ω ∂P + 1). C ± according to q representan accelerated universe when ˜Ω ∂P < − and a decelerated one when ˜Ω ∂P > − .The points C ± have eigenvalues { λ C ± i } = {± ∂P ) , ± (1 + 3 ˜Ω ∂P ) } . (14) (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ??
23 (˜Ω ∂P +1) . At these points thedeceleration parameter reduces to q = (3 ˜Ω ∂P + 1). C ± according to q representan accelerated universe when ˜Ω ∂P < − and a decelerated one when ˜Ω ∂P > − .The points C ± have eigenvalues { λ C ± i } = {± ∂P ) , ± (1 + 3 ˜Ω ∂P ) } . (14) (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ?? (cid:111) (cid:110) (cid:54) ynamical analysis approaches in spatially curved FRW spacetimes Based on these eigenvalues on the invariant subset { Ω H , Ω P } and one can see thatfor − < ˜Ω ∂P point C + ( C − ) is a source (sink). For the case − < ˜Ω ∂P < − instead C ± are saddle. Finally, for ˜Ω ∂P < − C + ( C − ) is a sink (source).These points can be seen in the examples shown in Figs. 1 and 2.Since the stability of the critical points ( A ± , B , and C ± ) of the system dependson the value of ˜Ω ∂P , we split our analysis into the following three ranges − < ˜Ω ∂P , − < ˜Ω ∂P < − , ˜Ω ∂P < − . (15)and we are going to depict the invariant subset { Ω H , Ω P } in these ranges. InFigs. 1, 2 we choose one representative value of ˜Ω ∂P for each range, since thetopology of the trajectories is independent of the specific value inside each range.For simplicity we assume that the function Γ has only one root.In order to be able to investigate the asymptotic behaviour of Ω P , i.e. Ω P = ±∞ ,in Figs. 1 and 2 we used the transformation X P = ζ Ω P (cid:112) ζ Ω P ∈ [ − , , (16)where ζ > X (cid:48) P = Ω H ζ (cid:113) − X P (cid:18) X P + 3 ζ (cid:113) − X P (cid:19) (cid:18) X P − ζ Ω ∂P (cid:113) − X P (cid:19) , (17)while for the non-positive curvature becomes X (cid:48) P = Ω H ζ (cid:113) − X P (9 ζ Ω ∂P (1 − H ) (1 − X P )+ ζ X P (cid:113) − X P (1 − ∂P + 2 Ω H ) + X P )) , (18)which along with the Eq. (5) define the compactified systems. Positive curvature:
Fig. 1 shows the invariant subsets { Ω H , X P } forthe positive curvature, on which two additional invariant subsets are located atΩ P = − P = 3 ˜Ω ∂P . Non-positive curvature
For the non-positive curvature there are addi-tional critical points once we consider the roots Γ( ˜Ω ∂P ) = 0. The locations of thesecritical points are { Ω H , Ω P } = {± √ , } and they represent a Milne universe, sincethe deceleration parameter q = 0 and the scale factor evolves as a = ± | k | ( t + c )for Ω H = ± √ .The critical point with Ω H = √ denoted as D + has eigenvalues { λ D + i } = {√ , − √ (cid:16) ∂P (cid:17) } , (19) (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ??
For the non-positive curvature there are addi-tional critical points once we consider the roots Γ( ˜Ω ∂P ) = 0. The locations of thesecritical points are { Ω H , Ω P } = {± √ , } and they represent a Milne universe, sincethe deceleration parameter q = 0 and the scale factor evolves as a = ± | k | ( t + c )for Ω H = ± √ .The critical point with Ω H = √ denoted as D + has eigenvalues { λ D + i } = {√ , − √ (cid:16) ∂P (cid:17) } , (19) (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ?? (cid:111) (cid:110) (cid:54) M. Kerachian, G. Acquaviva and G. Lukes-Gerakopoulos (a) ˜Ω ∂P = 0 . A - A + C - C + B - - - - Ω H X P (b) ˜Ω ∂P = − . A - A + C - C + B - - - - Ω H X P (c) ˜Ω ∂P = − . A - A + C - C + B - - - - Ω H X P Figure 1.
Invariant subsets for positive spatial curvature and ζ = 0 . ∂P in the ranges given in Sec. 2.2.0.1. The orange thick lines arethe separatrices of the system and the green shaded regions denote the part of the variablespace where the universe is accelerating. (a) ˜Ω ∂P = 0 . A - A + C - C + D + D - B - - - - Ω H X P (b) ˜Ω ∂P = − . A - A + C - C + D + D - B - - - - Ω H X P (c) ˜Ω ∂P = − . A - A + C - C + D + D - B - - - - Ω H X P Figure 2.
Invariant subsets for negative spatial curvature and ζ = 0 . ∂P in the ranges given in Sec. 2.2.0.2. The orange thick linesare the separatrices. The blue shaded areas are the regions excluded by our assumptionthat Ω (cid:15) >
0. The green shaded region are the part of the variable space where we haveaccelerating universe. (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ??
0. The green shaded region are the part of the variable space where we haveaccelerating universe. (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ?? (cid:111) (cid:110) (cid:54) ynamical analysis approaches in spatially curved FRW spacetimes in the invariant subset { Ω H , Ω P } , whiles the critical point denoted as D − has eigen-values { λ D − i } = {−√ , √ (cid:16) ∂P (cid:17) } . (20)Eqs. (19) and (20) show that for − < ˜Ω ∂P the critical points D ± are saddles, whilefor − > ˜Ω ∂P , D + is a source and D − is a sink. The action of a scalar field non-minimally coupled to gravity reads S = (cid:90) d x √− g (cid:18) R L ψ (cid:19) , (21)where L ψ is the Lagrangian for the scalar field ψ : L ψ = − (cid:0) g µν ∂ µ ψ ∂ ν ψ + ξRψ (cid:1) − V ( ψ ) , (22)and V ( ψ ) is a scalar field potential.By variation of the action (21) with respect to g µν , we arrive to the Einstein fieldequations R µν − R g µν = T ψµν . (23)where the stress-energy tensor T ψµν for the non-minimally coupled scalar field reads T ψµν = (1 − ξ ) ∇ µ ψ ∇ ν ψ + (cid:18) ξ − (cid:19) g µν ∇ α ψ ∇ α ψ − V ( ψ ) g µν + ξ (cid:18) R µν − g µν R (cid:19) ψ + 2 ξψ ( g µν ∇ α ∇ α − ∇ µ ∇ ν ) ψ. (24)By variation of the action with respect to the scalar field ψ we get the Klein-Gordonequation ∇ µ ∇ µ ψ − ξRψ − ∂V ( ψ ) ∂ψ = 0 . (25)The Friedmann and the Raychaudhuri equations for the non-minimally coupledscalar field in the FRW background read3 (cid:18) H + ka (cid:19) = (cid:15) ψ , (cid:18) H + 3 H + ka (cid:19) = − P ψ , (26) (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ??
0. The green shaded region are the part of the variable space where we haveaccelerating universe. (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ?? (cid:111) (cid:110) (cid:54) ynamical analysis approaches in spatially curved FRW spacetimes in the invariant subset { Ω H , Ω P } , whiles the critical point denoted as D − has eigen-values { λ D − i } = {−√ , √ (cid:16) ∂P (cid:17) } . (20)Eqs. (19) and (20) show that for − < ˜Ω ∂P the critical points D ± are saddles, whilefor − > ˜Ω ∂P , D + is a source and D − is a sink. The action of a scalar field non-minimally coupled to gravity reads S = (cid:90) d x √− g (cid:18) R L ψ (cid:19) , (21)where L ψ is the Lagrangian for the scalar field ψ : L ψ = − (cid:0) g µν ∂ µ ψ ∂ ν ψ + ξRψ (cid:1) − V ( ψ ) , (22)and V ( ψ ) is a scalar field potential.By variation of the action (21) with respect to g µν , we arrive to the Einstein fieldequations R µν − R g µν = T ψµν . (23)where the stress-energy tensor T ψµν for the non-minimally coupled scalar field reads T ψµν = (1 − ξ ) ∇ µ ψ ∇ ν ψ + (cid:18) ξ − (cid:19) g µν ∇ α ψ ∇ α ψ − V ( ψ ) g µν + ξ (cid:18) R µν − g µν R (cid:19) ψ + 2 ξψ ( g µν ∇ α ∇ α − ∇ µ ∇ ν ) ψ. (24)By variation of the action with respect to the scalar field ψ we get the Klein-Gordonequation ∇ µ ∇ µ ψ − ξRψ − ∂V ( ψ ) ∂ψ = 0 . (25)The Friedmann and the Raychaudhuri equations for the non-minimally coupledscalar field in the FRW background read3 (cid:18) H + ka (cid:19) = (cid:15) ψ , (cid:18) H + 3 H + ka (cid:19) = − P ψ , (26) (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ?? (cid:111) (cid:110) (cid:54) M. Kerachian, G. Acquaviva and G. Lukes-Gerakopoulos respectively, while the Klein-Gordon equation reads¨ ψ + 3 H ˙ ψ + ∂ ψ V + 6 ξ ψ (cid:18) ˙ H + 2 H + ka (cid:19) = 0 . (27)Here the (cid:15) ψ and P ψ are defined as (cid:15) ψ = 12 ˙ ψ + V ( ψ ) + 3 ξ ψ (cid:18) H ˙ ψ + ψ (cid:18) H + ka (cid:19)(cid:19) , (28) P ψ = (1 − ξ ) 12 ˙ ψ − V ( ψ ) − ξ (cid:18) H ψ ˙ ψ + 2 ψ ¨ ψ + ψ (cid:18) H + 3 H + ka (cid:19)(cid:19) . (29)We define a set of dimensionless variables which are well-defined for positive andnon-positive curvatures:Ω = ψ (cid:112) ξ ψ , Ω H = HD , Ω ψ = ˙ ψ √ D , (30)Ω V = √ V √ D , Ω ∂V = ∂ ψ VV ,
Γ = V · ∂ ψ V ( ∂ ψ V ) (31)where D = H + | k | a . Similarly as for the dynamical system in Sec. 2, for thesedimensionless variables the evolution parameter τ is defined as dτ = Ddt . By takingderivatives of the dimensionless variables with respect to the evolution parameterwe getΩ (cid:48) = √ ψ (cid:0) − ξ Ω (cid:1) / (32)Ω (cid:48) H = (cid:0) − Ω H (cid:1) (cid:32) ˙ HD + Ω H (cid:33) (33)Ω (cid:48) ψ = ¨ ψ √ D − Ω ψ Ω H (cid:32) ˙ HD + Ω H − (cid:33) (34)Ω (cid:48) V = Ω V (cid:34)(cid:114)
32 Ω ∂V Ω ψ − Ω H (cid:32) ˙ HD + Ω H − (cid:33)(cid:35) (35)Ω (cid:48) ∂V = √ ∂V Ω ψ (Γ − , (36)where Γ = V · ∂ ψ V / ( ∂ ψ V ) which is the so-called tracker parameter. This au-tonomous system of equations differs only in the ¨ ψ √ D and ˙ HD terms for k > k ≤
0. Namely for positive curvature we get from Klein-Gordon and Raychaudhuri (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ??
0. Namely for positive curvature we get from Klein-Gordon and Raychaudhuri (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ?? (cid:111) (cid:110) (cid:54) ynamical analysis approaches in spatially curved FRW spacetimes equations¨ ψ √ D = − H Ω ψ − (cid:114)
32 Ω ∂V Ω V − √ ξ Ω (cid:112) − ξ Ω (cid:32) ˙ HD + Ω H + 1 (cid:33) , ˙ HD + Ω H + 1 = − − ξ (1 − ξ ) Ω (cid:40) − (cid:0) − ξ Ω (cid:1) + ξ Ω (cid:112) − ξ Ω (cid:16) √ H Ω ψ + 3 Ω ∂V Ω V (cid:17) + 32 (cid:0) − ξ Ω (cid:1) (cid:104) (1 − ξ ) Ω ψ − Ω V (cid:105)(cid:41) , while for non-positive curvature these equations read¨ ψ √ D = − H Ω ψ − (cid:114)
32 Ω ∂V Ω V + √ ξ Ω (cid:112) − ξ Ω (cid:32) − ˙ HD − H (cid:33) , ˙ HD + Ω H = 12 − Ω H + 11 − ξ (1 − ξ ) Ω (cid:40) ξ Ω (cid:0) − H (cid:1) − ξ Ω (cid:112) − ξ Ω (cid:16) √ H Ω ψ + 3 Ω ∂V Ω V (cid:17) − (cid:0) − ξ Ω (cid:1) (cid:104) (1 − ξ ) Ω ψ − Ω V (cid:105)(cid:41) . The respective Friedmann equations differ as well, i.e. for k >
01 = 2 ξ Ω (cid:0) − Ω H (cid:1) + 3 ξ (cid:32)(cid:114)
23 Ω H Ω + Ω ψ (cid:112) − ξ Ω (cid:33) + (1 − ξ ) Ω ψ (cid:0) − ξ Ω (cid:1) + Ω V (cid:0) − ξ Ω (cid:1) , (37)while for k ≤
01 = 2 (cid:0) − ξ Ω (cid:1) (cid:0) − Ω H (cid:1) + 3 ξ (cid:32)(cid:114)
23 Ω H Ω + Ω ψ (cid:112) − ξ Ω (cid:33) + (1 − ξ ) Ω ψ (cid:0) − ξ Ω (cid:1) + Ω V (cid:0) − ξ Ω (cid:1) . (38) Symmetries.
The dynamical system (32)-(36) remains invariant underthe simultaneous transformation { Ω , Ω H , Ω ψ , Ω V , Ω ∂V } → {− Ω , Ω H , − Ω ψ , Ω V , − Ω ∂V } . (39)This symmetry, physically, is equivalent to the invariance under the transformation ψ → − ψ . Since Ω V is not affected by this transformation (39), then it must holdthat V ( ψ ) = V ( − ψ ) > (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ??
The dynamical system (32)-(36) remains invariant underthe simultaneous transformation { Ω , Ω H , Ω ψ , Ω V , Ω ∂V } → {− Ω , Ω H , − Ω ψ , Ω V , − Ω ∂V } . (39)This symmetry, physically, is equivalent to the invariance under the transformation ψ → − ψ . Since Ω V is not affected by this transformation (39), then it must holdthat V ( ψ ) = V ( − ψ ) > (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ?? (cid:111) (cid:110) (cid:54) M. Kerachian, G. Acquaviva and G. Lukes-Gerakopoulos
Table 1.
The critical elements of the system and their stability in the range 0 ≤ ξ ≤ / Ω ψ Ω H Ω Ω V Ω ∂V Curvature q w e stability A + A − − B + < Ω < ξ (cid:113) − ξ Ω − ξ Ω − ξ Ω √ − ξ Ω − ξ Ω flat -1 -1 sink B − − < Ω < ξ (cid:113) − ξ Ω − ξ Ω − ξ Ω √ − ξ Ω − ξ Ω flat -1 -1 source C ± ± ± √ ξ ∀ flat 1 saddle D ± ± √ ∀ ∀ negative 0 - saddle Singularities.
In this system there are singular points arising from thedecoupling of Raychaudhuri and Klein-Gordon equations, i.e. where the determi-nant of their Jacobian vanishes. These singular points, in terms of dimensionlessvariables, correspond to the vanishing ofΩ = ± (cid:112) ξ (1 − ξ ) . (40)By substituting the former relation into the Friedmann constraints and solving forΩ ψ one getsΩ ψ = √ ξ Ω H + (cid:112) (Ω H ∓ Ω V − ξ ± Ω V √ − ξ , (41)where the upper/lower sign corresponds to negative/positive curvature. In the range ξ ∈ (0 , / , Ω ψ ) of the singularity remainfinite . For ξ > /
6, Ω ψ is complex. In the case of a flat spacetime Ω H = ± S ± respectively. Invariant subsets.
For the dynamical system. (32)-(36), one can iden-tify some invariant subsets of the system. These invariant subsets are Ω H = ± V = 0 (free scalar field). Critical points.
Critical points and their physical interpretations of thissystem are summarized in the table 1.
This work introduces general frameworks to analyze dynamical systems of: • barotropic fluids with non-negative energy density and generic EoS, • non-minimally coupled real scalar fields with generic potential in the absence ofregular matter,both cases are treated in spatially curved FRW spacetimes without cosmologicalconstant. In both cases we have employed a general Γ function, which when specifiedreduces our general frameworks to specific models. We were able to identify criticalelements and basic features of the systems for unknown Γ functions. (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ??
This work introduces general frameworks to analyze dynamical systems of: • barotropic fluids with non-negative energy density and generic EoS, • non-minimally coupled real scalar fields with generic potential in the absence ofregular matter,both cases are treated in spatially curved FRW spacetimes without cosmologicalconstant. In both cases we have employed a general Γ function, which when specifiedreduces our general frameworks to specific models. We were able to identify criticalelements and basic features of the systems for unknown Γ functions. (cid:228)(cid:121) (cid:228) (cid:228) (cid:228)(cid:121) (cid:229) (cid:229) ?? (cid:111) (cid:110) (cid:54) ynamical analysis approaches in spatially curved FRW spacetimes REFERENCES
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