Introduction to the application of the dynamical systems theory in the study of the dynamics of cosmological models of dark energy

0.67 MB·English

by  Ricardo García-Salcedo

Introduction to the application of the dynamical systems theory in the study of the
dynamics of cosmological models of dark energy
Ricardo Garc´ıa-Salcedo,1,a Tame Gonzalez,2,b Francisco A.
Horta-Rangel,2,c Israel Quiros,2,d and Daniel Sanchez-Guzm´an1,e
1CICATA - Legaria del Instituto Polit´ecnico Nacional, 11500, M´exico, D.F., M´exico.
2Departamento de Ingenier´ıa Civil, Divisio´n de Ingenier´ıa, Universidad de Guanajuato, Gto., M´exico.
(Dated: January 21, 2015)
Thetheoryofthedynamicalsystemsisaverycomplexsubjectwhichhasbroughtseveralsurprises
in the recent past in connection with the theory of chaos and fractals. The application of the
tools of the dynamical systems in cosmological settings is less known in spite of the amount of
published scientific papers on this subject. In this paper a – mostly pedagogical – introduction
5 to the application in cosmology of the basic tools of the dynamical systems theory is presented.
1 It is shown that, in spite of their amazing simplicity, these allow to extract essential information
0 on the asymptotic dynamics of a wide variety of cosmological models. The power of these tools
2 is illustrated within the context of the so called ΛCDM and scalar field models of dark energy.
This paper is suitable for teachers, undergraduate and postgraduate students from physics and
n
mathematics disciplines.
a
J
PACSnumbers: 01.40.E,01.40.Ha,04.20.-q,05.45.-a,98.80.-k
0
2

I. INTRODUCTION Inordertodeterminethescalefactoritisnecessaryto
]
c solve the cosmological Einstein’s field equations, which
q relate the spacetime geometry with the distribution of
Cosmologyisaboutthestudyofthestructureandevo-
- matter in the Universe. The maximal symmetry implied
r lution of the universe as a whole. Einstein’s general rel-
g by the cosmological principle allows for great simplifica-
ativity (GR) stands as its mathematical basis. In gen-
[ tionoftheEinstein’sequations. Inthispaper,forfurther
eral, Einstein’s GR equations are a very complex sys-
simplification, we shall consider FRW spacetimes with
1 tem of coupled nonlinear differential equations,but it is
v simplified by the underlying spacetime symmetries. The flat spatial sections (k = 0). The resulting cosmological
1 equations are written as it follows:3
Friedmann-Robertson-Walker(FRW)spacetimemodelis
5
depicted by the line element1
8
4
0 ds2 = dt2+a(t)2 dr2 +r2(dθ2+sin2θdφ2) , 3H2 = ρx+ρX,
1. − 1 kr2 Xx
(cid:20) − (cid:21)
0 1
H˙ = γ ρ +γ ρ ,
5 where t is the cosmic time, r, θ and φ, are co-moving x x X X
−2 !
1 spherical coordinates, a(t) is the cosmological scale fac- Xx
v: tor,andtheconstantk = 1,0,+1parametrizesthecur- ρ˙x+3γxHρx =0, ρ˙X +3γXHρX =0, (1)
−
i vature of the spatial sections, is based on what is known
X
asthecosmologicalprinciple: inthelargescalestheUni- where H a˙/a is the Hubble parameter, ρx and ρX are
≡
r verse is homogeneous and isotropic. Homogeneity and the energy densities of the gravitating matter and of the
a
isotropy imply that there is no preferred place and no X-fluid (the dark energy in the present paper) respec-
preferred direction in the universe.2 The observations tively, and the sum is over all of the gravitating mat-
confirmthevalidityofthecosmologicalprincipleatlarge ter species living in the FRW spacetime (dark matter,
scales( 100Mpc 1026 cm). Thedynamicsoftheuni- baryons, radiation, etc.). The last two equations above
∼ ≈
verseisfullygivenbytheexplicitformofthescalefactor aretheconservationequationsforthegravitatingmatter
which depends on the symmetry properties and on the degreesoffreedomandforthe X-fluidrespectively.4 Be-
matter content of the Universe. sides, it is adopted that the following equations of state
are obeyed:
p =(γ 1)ρ , p =(γ 1)ρ ,
x x x X X X
aElectronicaddress: [e-mail] − −
bElectronicaddress: [e-mail]
cElectronicaddress: [e-mail]
dElectronicaddress: [e-mail]
eElectronicaddress: [e-mail] 3 Foradidacticderivationofthecosmologicalequationsnotbased
1 Forapedagogicalexpositionofthegeometricalandphysicalsig- ongeneralrelativitysee[1–3].
nificanceoftheFRWmetricsee[1]. 4 Itshouldbepointedoutthatoneequationin(1),saythesecond
2 Hereweadopttheunitswhere8πG=c=1. (Raychaudhuri) equation, isredundant.
2
wherep isthe barotropicpressureofthe x-matterfluid, the pace of the cosmological expansion. In particular,
x
p istheparametricpressureofX-component,5 whileγ ifwechoosetheX-componentintheformofvacuumen-
X x
and γ are their barotropic parameters.6 ergy (γ ,ρ ) (γ ,ρ ), usually identified with the
X vac vac X X
→
According to our current understanding of the expan- energy density of the cosmological constant (see section
sion history, the Universe was born out of a set of initial III), since p = ρ , then γ = 0 < 2/3. Due to
vac vac vac
−
conditionsknownas“big-bang.” Furtherevolutionledto the knownfact thatthe vacuum energyis notdiluted by
a stage of primeval inflation driven by a scalar field [4]. the cosmic expansion,8 eventually this component of the
The inflation era was followed by a stage dominated by cosmic budget will dominate the cosmologicaldynamics,
the radiation,followed,inturn,byanintermediatestage leading to late-time acceleration of the expansion
dominatedbynon-relativisticcolddarkmatter(CDM).7
During this stage of the cosmic expansion most of the a¨ 1
ρ >0.
structureweobservewasformed. Theendofthematter- a ≈ 3 vac
domination era can be traced back to a recent moment
of the cosmic expansion where a very peculiar form of A related (dimensionless) parameter which measures
matter which does not interact with baryons, radiation whether the expansion is accelerated or decelerated, is
or any other form of “visible” matter – known as “dark the so called deceleration parameter which is defined as
energy” (DE) – started dominating [5]. This alien form
of matter antigravitatesand causes the presentUniverse
toexpandatanacceleratingpaceinsteadofdecelerating, H˙ aa¨
q 1 = . (3)
asitwouldforamatter–eitherCDMorbaryons–orra- ≡− − H2 −a˙2
diation dominated Universe. Actually, if one substitutes
The pace of the expansion accelerates if q < 0. On the
a¨ contrary, positive q >0 means that the expansion is de-
=H˙ +H2,
a celerating. Thisparameterwillbeusefulinthediscussion
in the next sections.
back into the second equation in (1), and taking into
Inspiteoftheapparentsimplicityoftheequations(1),
accountthe Friedmann equation – first equationin (1) –
aswithanysystemofnon-linearsecond-orderdifferential
then, one is left with the following equation:
equations, it is a very difficult (and perhaps unsuccess-
ful) task to find exact solutions. Even when an analytic
solutioncanbefounditwillnotbeuniquebutjustonein
a¨ 1 2 1 2
= γ ρ γ ρ . (2) alargesetofthem[6]. Thisisinadditiontothequestion
x x X X
a −2 − 3 − 2 − 3
x (cid:18) (cid:19) (cid:18) (cid:19) aboutstabilityofgivensolutions. Inthiscasethedynam-
X
icalsystems tools come to our rescue. These verysimple
It is known that for standard (gravitating) forms of tools give us the possibility to correlate such important
matter the barotropic pressure is a non-negative quan- conceptslikepastandfutureattractors(alsosaddleequi-
tity: px = (γx 1)ρx 0 γx 1. In particular, libriumpoints)inthephasespace,withgenericsolutions
− ≥ ⇒ ≥
for CDM and baryons the (constant) barotropic index of the set of equations (1) without the need to analyti-
γm = 1, while for radiation γr = 4/3. Hence, since cally solve them.
obviously γ > 2/3 – if forget for a while about the
x In correspondencewith the above mentioned stages of
second term in the RHS of (2) – it is seen that a cos-
the cosmic expansion, one expects that the state space
mic background composed of standard matter will ex-
(also, phase space) of any feasible cosmological model
pand at a decelerated pace (a¨ < 0). In contrast, for
should be characterized by – at least one of – the fol-
unconventional matter with γ < 2/3 (the parametric
X lowing equilibrium configurations in the space of cos-
pressure p is obviously negative), the second term in
X mological states: (i) a scalar field dominated (inflation-
the RHS of Eq. (2) contributes towards accelerating
ary) past attractor, (ii) saddle critical points associated
withradiation-domination,radiation-matterscaling,and
matter-domination, and either (iii) a matter/DE scaling
or (iv) a de Sitter9 future attractors. Any heteroclinic
5 By parametric pressure we understand that it might not corre-
spond to standard barotropic pressure with the usual thermo-
dynamic properties but that it is rather a conveniently defined
parameter which obeys similar equations than its matter coun-
terpart. 8 Werecallthat,incontrastwiththevacuumenergydensitywhich
6 In several places in this paper we use the so called equation of doesnotdilutewiththeexpansionρvac=Λ=const.,theenergy
state(EOS)parameterωX =pX/ρX =γX−1,interchangeably densityofCDMandbaryonsdiluteslike∝a−3,whilethedensity
withthebarotropicparameterγX. ofradiationdecays veryquickly∝a−4.
7 Dark matter is a conventional form of matter in the sense that 9 A de Sitter cosmological phase is a stage of the cosmological
it gravitates as any other known sort of standard matter such expansionwhosedynamicsisdescribedbythelineelementds2=
as radiation, baryons, etc. However, unlike the latter forms of −dt2 +eH0tdx2, where H0 is the Hubble constant. Since the
matter,CDMdoesnotinteractwithradiation. Thisiswhyitis deceleration parameter q ≡ −1−H˙/H2 = −1 is a negative
calledas“darkmatter.” quantity, thedeSitterexpansionisinflationary.
3
orbit joining these critical points then representsa feasi- to understand the material in this section. The exposi-
ble image of the featured cosmic evolution in the phase tion in the appendix is enough to understand how the
space.10 computations in sections III and IV are done. We rec-
Whilethetheoryofthedynamicalsystemshasbrought ommend those readers which are not familiar with the
several surprises in the recent past in connection with dynamical systems theory to start with this section.
the theory ofchaos[7,8] andfractals[9], the application Our goal is to keep the discussion as general as pos-
of the tools of the dynamical systems in cosmological sible while presenting the exposition in as much as pos-
settingsislessknowninspiteoftheamountofpublished sible pedagogical way. The paper contains many foot-
scientific papers on this subject (just to cite few of them notes with comments and definitions which complement
see [2, 5, 10–19] and the related references therein). the main text. Since the subject is exposed with some
This paper aims at an introduction to the application degree of technical details and a basic knowledge of cos-
of the simplest tools of the theory of the dynamical sys- mology is assumed, this paper is suitable for teachers,
tems – those which can be explained with only previous undergraduate and postgraduate students from physics
knowledge of the fundamentals of linear algebra and of and mathematics disciplines.
ordinary differential equations – within the context of
the so called “concordance” or ΛCDM (section III) and
scalar field models (section IV) of dark energy [5, 11– II. ASYMPTOTIC COSMOLOGICAL
16, 18–22]. The dynamical systems tools play a central DYNAMICS: A MODEL-INDEPENDENT
role in the understanding of the asymptotic structure of ANALYSIS
these models. I. e., these tools are helpful in looking
for an answer to questions like: where does our Universe In this section the asymptotic structure in the phase
come from? or, what would be its fate? An specific ex- space of a general DE model will be discussed in detail.
ample: the cosh potential, is explored in the section V, For simplicity and compactness of the exposition here
in order to show the power of these tools in the search weshallassumeonlyatwo-componentcosmologicalfluid
for generic dynamical behavior. composedofcolddarkmatter–labeledherebytheindex
We want to stress that there are very good introduc- “m,”i. e.,(ρ ,p ) (ρ ,p )–andofdarkenergy(the
m m x x
toryandreviewpapersontheapplicationofthedynami- X-component). →
calsystemsincosmology–see,forinstance,[5,10–12,16] In order to derive an autonomous system of ordinary
–so,whatisthepointofwritinganotherintroductorypa- differential equations (ASODE) out of (1) it is useful to
perontheissue? Inthisregardwewanttopointoutthat, introduce the so called dimensionless energy density pa-
in the present paper, we aim not only at providing just rameters of matter and of the DE
another introductory exposition of the essentials of the
subjectbut,also,atfillingseveralgapsregardingspecific
ρ ρ
topics usually not covered in similar publications exist- m X
Ω = , Ω = , (4)
inginthebibliography. Herewepayspecialattention,in m 3H2 X 3H2
particular,tothenecessary(yetusuallyforgotten)defini-
respectively, which are always non-negativequantities.11
tion of the physically meaningful phase space, i. e., that
In terms of these parameters the Friedmann equation –
regionofthephasespacewhichcontainsphysicallymean-
first equation in (1) – can be written as the following
ingfulcosmologicalsolutions. Wealsoincludethederiva-
constraint:
tionofrelevantformulasinordertofacilitatethewayfor
beginners to concrete computations. We shall explain,
in particular, a method developed in [23] to consider ar-
Ω +Ω =1, (5)
bitrary potentials within the dynamical systems study m X
of scalar field cosmological models. This subject is not
whichentailsthatnoneofthenon-negativedimensionless
usuallycoveredintheexistingintroductorybibliography.
energy density components alone may exceed unity: 0
At the end of the paper, for completeness, we include an ≤
Ω 1, 0 Ω 1.
appendix section VII, where an elementary introduction m ≤ ≤ X ≤
Given the above constraints, if one thinks of the di-
to the theory of the dynamical systems is given and use-
mensionlessdensityparametersasvariablesofsomestate
ful comments on the interplay between the cosmological
space, only one of them is linearly independent, say Ω .
field equations and the equivalent phase space are also X
Let us write a dynamical equation for Ω , for which
provided. Only knowledge of the fundamentals of linear X
purpose we rewrite the Raychaudhuri equation – second
algebra and of ordinary differential equations is required
equation in (1) – and the conservation equation for the
X-component, in terms of Ω :
X
10 For a very concise introductory knowledge of the theory of the
dynamical systems and the related concepts like critical points
(attractors and saddle points), heteroclinic orbits, etc, see the 11 Inagreementwithconventionalnon-negativityofenergywecon-
appendixVII. sidernon-negative energydensitiesexclusively.
4
Besides, if γ > γ , the matter-dominated solution
m X
2 H˙ = 3γ Ω 3γ Ω Ωsinmce=th1e(iΩniXtia=lly0)s:maal(lt)pe∝rtut2r/b3aγmtio,nisδt0h(τe)paset3a(γtmtr−aγcXto)τr
H2 − m m− X X increases exponentially with τ, thus takin∝g the system
=−3γm+3(γm−γX)ΩX, awayfromtheconditionΩm =1. Otherwise,ifγm <γX,
ρ˙ the X-dominatedsolutionis the past attractorwhile the
X
= 3γ HΩ . (6)
3H2 − X X matter-dominated solution is the future attractor. How-
ever, this last situation is not consistent with the known
Then, if we substitute (6) into cosmic history of our Universe.
The interesting thing about the above result is that
Ω˙ = ρ˙X 2H˙ Ω , it is model-independent, i. e., no matter which model
X 3H2 − H X to adopt for the DE (X-component), given that γ is a
X
constant, the above mentioned critical points – together
and consider the constraint (5), we obtain
with their stability properties – are obtained. Besides,
since a non-constant γ obeys an autonomous ODE of
X
Ω′ =3(γ γ )Ω (1 Ω ), (7) the general form γX′ = f(γX,ΩX), at any equilibrium
X m− X X − X point where, necessarily
wherethetildedenotesderivativewithrespecttothenew
′ ′
variable τ lna (Ω˙ =HΩ′, etc.). ΩX =0, γX =0 ⇒ γX =γ¯X =const.
≡
Ingeneral,forvaryingγ ,anotherordinarydifferential
X provided that γ¯ = γ , either Ω = 1, or Ω = 0
equation(ODE)forγX isneededinordertohaveaclosed (Ω = 1). HencXe, i6ndepmendent of thXe model adopXted to
system of differential equations (the barotropic index of m
accountfortheX-component,matter-dominatedandX-
matter γ is usually set to a constant). The problem
m fluid dominated solutions are always equilibrium points
is that, unlike the ODE (7) which is model-independent,
in the phase space of the two-fluids cosmological model
theODEforγ =(ρ +p )/ρ requiresofcertainspec-
X X X X depicted by equations (1). This is a generic result.
ificationswhichdependonthechosenmodel(seebelow).
Anyway, certain important results may be extracted
from (7) under certain assumptions. For instance, if we
III. ΛCDM MODEL
assume a constant γ , the mentioned ODE is enough
X
to uncover the asymptotic structure of the model (1) in
the phase space. In this case the phase space is the 1- Let us now to show with the help of a specific cos-
dimensional segment Ω [0,1]. The critical points are mological model of dark energy, how it is possible that
X
∈ a 2-dimensional system of ODE may come out of the
Ω (1 Ω )=0 Ω =1, Ω =0, 3-dimensional system of second-order field equations (1)
X X X X
− ⇒
where, we recall, one of these equations is redundant.
where we are assuming that γX = γm. The first fixed For this purpose we shall focus in the so called Λ-cold
6
pointΩX =1iscorrelatedwithdarkenergydomination, dark matter (ΛCDM) model [24] whose phase space is a
whilethesecondoneΩX =0(Ωm =1)isassociatedwith subspace of the phase plane.12 Here – only for the ex-
CDM dominated expansion. tent of this section – in addition to the CDM and to the
IflinearizeEq. (7)aroundthecriticalpoints: 1 δ1 DE (the cosmologicalconstant) we shall consider a radi-
− →
1, and 0+δ0 → 0 (δ1 and δ0 are small perturbations), ationfluidwithenergydensityρr andpressurepr =ρr/3
one obtains (γ = 4/3). The conservation equation for radiation
r
reads
δ1′ =−3(γm−γX)δ1 ⇒ δ1(τ)=δ¯1e−3(γm−γX)τ, ρ˙r+4Hρr =0 ⇒ ρr ∝a−4.
δ′ =3(γ γ )δ δ (τ)=δ¯ e3(γm−γX)τ. (8)
0 m− X 0 ⇒ 0 0 Meanwhile, for the cosmological constant term – as for
As seen the DE-dominated solution: any vacuum fluid – we have that pΛ = ρΛ (γΛ = 0),
−
which means that ρ˙ = 0, so that this term does not
Λ
Ω =1 3H2 =ρ a−3γX a(t) t2/3γX, evolve during the course of the cosmic expansion.
X X
⇒ ∝ ⇒ ∝
Inwhatfollows,sinceweadoptedtheunitssystemwith
is a stable (isolated) equilibrium point whenever γm > 8πG = c = 1, we write ρΛ = Λ. Besides, since we deal
γX since, in this case, the perturbation herewithCDM,whichismodeledbyapressurelessdust,
δ (τ) e−3(γm−γX)τ,
1
∝
decreasesexponentially. Hence, giventhatγ >γ , the
m X 12 For adynamical systemsstudy ofthe FRWcosmological model
darkenergydominatedsolutionΩ =1,isthefutureat-
X with spatial curvature k 6= 0 and with a cosmological constant
tractorsolutionwhichdescribes the fate ofthe Universe. see[2].
5
we set γ =1. After these assumptions, and identifying where we have to take in mind the Raychaudhuri equa-
m
ρ ρ =Λinequations(1),theresultingcosmological tion and the conservation equation for radiation in (9)
X Λ
≡
equations are written in terms of the variables of the phase space
H2
1 4 H˙ = (3+x 3y),
3H2 =ρ +ρ +Λ, H˙ = ρ +ρ , − 2 −
r m r m
−2 3
(cid:18) (cid:19)
andthenwehavetogotoderivativeswithrespecttothe
ρ˙ +3Hρ =0, ρ˙ +4Hρ =0. (9)
m m r r variable τ =lna(t) (ξ′ H−1ξ˙). We obtain:
→
Let us at this point to make a step aside to explain
how it is that the cosmological constant can explain the
′ ′
present speed-up of the cosmic expansion. If we con- x = x(1 x+3y), y =(3+x 3y)y. (13)
− − −
veniently combine the first two equations above, taking
into account that straightforward integration of the last It is remarkable the simplicity of the system of two
equations in (9) yield ρ = 6C /a3 and ρ = 3C /a4 ordinary differential equations (13) as compared with
m m r r
respectively (C and C are constants), we get: the system of three second-order cosmological equations
m r
(9). The critical points of (13) in Ψ are easily found
Λ
a¨ =H˙ +H2 = Cr Cm + Λ. by solving the following system of algebraic equations:
a −a4 − a3 3 x(1 x+3y)=0, (3+x 3y)y =0.
− −
We see that acceleration of the expansion a¨ > 0 may In the present case, in terms of the variables x, y, the
occur in this model thanks to the third term above deceleration parameter q = 1 H˙/H2 (equation (3))
− −
(the DE, i. e., the cosmological constant). Besides, can be written as:
the accelerated pace of the expansion is a recent phe-
q =(1+x 3y)/2,
nomenon: during the course of the cosmic evolution the −
initially dominating radiation component dilutes very
which means that those fixed points which fall in that
quickly a−4 until the matter component starts dom-
∝ region of the phase space laying above the line y =
inating. As the cosmic expansion further proceeds the
1/3+ x/3, correspond to cosmological solutions where
matter component also dilutes a−3 until very recently
∝ the expansion is accelerating.
when the cosmological constant started to dominate to
yield to positive a¨>0.
Following the same procedure as in the former section
A. Critical points
we write the Friedmann constraint
1. Radiation-dominated critical point
Ω +Ω +Ω =1, (10)
r m Λ
The critical point P : (1,0), Ω = 1 3H2 = ρ
where Ωr = ρr/3H2 and ΩΛ = Λ/3H2. In what follows a(t) √t, correspronds to therradiat⇒ion-dominatedr
we choose the following variables of the phase space: ⇒ ∝
cosmic phase. This solution depicts decelerated expan-
sionsince q =1. The linearizationmatrixfor the system
x Ω , y Ω , (11) of ODE (13) – for details see the appendix – is
r Λ
≡ ≡
where,forsakeofsimplicityofwriting,weadoptxandy 1+2x 3y 3x
inplaceofthedimensionlessenergydensitiesofradiation J = − y − 3+−x 6y .
Ω andofthecosmologicalconstantΩ respectively. But (cid:18) − (cid:19)
r Λ
we warn the reader that the same symbols x and y will The roots of the algebraic equation
be used in the next sections to mean different variables
of the phase space. 1 λ 3
We have that Ω = 1 x y, and, since 0 x 1, det|J(Pr)−λU|=det −0 4− λ =0,
m − − ≤ ≤ (cid:18) − (cid:19)
0 y 1,and0 Ω 1,thephysicallyrelevantphase
m
≤ ≤ ≤ ≤ arethe eigenvaluesλ =1andλ =4. Since both eigen-
spaceisdefinedasthefollowing2Dtriangularregion(see 1 2
values are positive reals, then P is a source point (past
the top panel of FIG. 1): r
attractor). This means that in the ΛCDM model de-
scribed by the cosmological equations (9) the radiation-
ΨΛ = (x,y): x+y 1, 0 x 1, 0 y 1 . (12) dominated solution a(t) √t is a privileged solution.
{ ≤ ≤ ≤ ≤ ≤ } ∝
Fromthe inspectionofthephaseportraitof(13)–top
As before, in order to derive the autonomous ODE-s,
panel of FIG. 1 – it is apparent that any viable pattern
we first take the derivative of the variables x and y with
of cosmological evolution should start in a state where
respect to the cosmic time t:
the mattercontentofthe Universeis dominatedbyradi-
ρ˙ ρ H˙ Λ H˙ ation. Ofcoursethisisadrawbackofourclassicalmodel
r r
x˙ = 2 , y˙ = 2 , whichcannotexplainthe veryearlystagesofthecosmic
3H2 − 3H2H − 3H2H
6
conveniently chosen initial conditions the given orbits in
Ψ approachcloseenoughtoP . Sincethisisaunstable
Λ m
(metastable) critical point, it can be associated with a
transientstageofthecosmicexpansiononly. Thisisgood
sinceastagedominatedbythedarkmattercanbeonlya
transientstate, lasting for enoughtime as to account for
the observedamount ofcosmic structure. A drawbackis
that we need to fine tune the initial conditions in order
for feasible orbits in the phase space to get close enough
to P .
m
3. de Sitter phase
The critical point P : (0,1), Ω = 1 3H2 = Λ
dS Λ
⇒
a(t) e√Λ/3t, corresponds to a stage of inflationary
⇒ ∝
(q = 1)deSitterexpansion. Theeigenvaluesofthema-
−
trix J(P ) are λ = 3 and λ = 4. This means that
dS 1 2
− −
P isafutureattractor(seetheclassificationofisolated
dS
equilibriumpointsintheappendixVII).Thisentailsthat
independent onthe initial conditions chosenΩ0, Ω0, the
r Λ
orbits in Ψ are always attracted towards the de Sit-
Λ
ter state which explains the actual accelerated pace of
thecosmicexpansioninperfectfitwiththeobservational
data. Thisiswhy,inspiteoftheseriousdrawbackincon-
nection with the cosmologicalconstant problem [25, 26],
the very simple ΛCDM model represents such a success-
FIG. 1: Phase portrait of the ASODE (13) corresponding to ful descriptionofthe presentcosmologicalparadigmand
the ΛCDM model (top panel). The orbits of (13) emerge it is, therefore, coined the “concordance model”. Any
from thepastattractorPr :(1,0)–theradiation domination other DE model, regardless of its nature, ought to be
state – and end up in the future attractor P : (0,1) (de
dS compared to the ΛCDM predictions as a first viability
Sitter phase). For a non-empty set of initial conditions the
test.
corresponding orbits approach close enough to the matter-
Additional refinement of the above model is achieved
dominated saddle point at the origin P :(0,0). In the bot-
m
if we add the energy density of baryons.
tom panel the phase portrait of the ASODE (31) – showing
twoorbitsgeneratedbytwodifferentsetsofinitialconditions
– is shown. Here the free parameter µ has been arbitrarily
set µ=10. IV. SCALAR FIELD MODELS OF DARK
ENERGY: THE DYNAMICAL SYSTEMS
PERSPECTIVE
evolution where quantum effects of gravity play a role.
The cosmological constant problem can be split into
This includes the early inflation. In a model that would
two questions [25]: (i) why the vacuum energy ρ =Λ
vac
account for such early period of the cosmic dynamics,
is not very much larger? – old cosmological constant
inflation should be the past attractor in the equivalent
problem – and (ii) why it is of the same order of magni-
phase space.
tudeasthepresentmassdensityoftheuniverse? whichis
acknowledgedasthenewcosmologicalconstantproblem.
In orderto avoidthe oldcosmologicalconstantproblem,
2. Matter-dominated critical point
whichis exclusiveofthe ΛCDM model, scalarfieldmod-
els of dark energy are invoked.13 In this last case an ef-
The critical point Pm : (0,0), Ωm = 1 3H2 = ρm fective“dynamical”cosmologicalconstantisdescribedby
⇒
a(t) t2/3, corresponds to the matter-dominated the scalarfield’s X self-interactionpotentialV(X). This
⇒ ∝
solution, which is associated with decelerated expansion may be arranged in such a way that at early times the
(q =1/2). Followingthe sameprocedureabove– seethe vacuum energy ρ = V(X ) (X˙ = 0) is large enough
vac,0 0
appendix VII – onefinds the followingeigenvaluesofthe
linearization matrix J(P ) evaluated at the hyperbolic
m
equilibrium point P : λ = 1, λ = 3. Hence this is
m 1 2
−
a saddle critical point. As seen from the inspection of 13 The new cosmological problem is inherent also inseveral scalar
the phase portrait in the top panel of FIG. 1, only for fieldmodelsofdarkenergy.
7
as to produce the desired amount of inflation, while at One of these constraints is given by (18):
late times ρ = V(X ) is of the same order of the
vac,f f
CDM energy density ρm. 0≤Ωm ≤1 ⇒ 0≤x2+y2 ≤1.
In this section, we shall explore the asymptotic struc-
Besides,since0 Ω 1,then x 1. Additionallywe
ture of general scalar field models of DE [5, 11–16, 18– X
≤ ≤ | |≤
shallbeinterestedincosmicexpansionexclusivelyH 0,
22, 27, 28]. These represent a viable alternative to the
≥
sothaty 0. Hence,forinstance,forpotentialsofoneof
ΛCDM model explored above. Among them “exponen-
the follow≥ing kinds (see below): V = V , V = V e±µX,
tial quintessence” V(X) = Mexp( µX) [11, 29] is one 0 0
− the physical phase plane is defined as the upper semi-
of the most popular scalar field models of dark energy.
disk,
The specification of a scalar field model for the DE
means that the energy density and the parametric pres-
sure in the cosmologicalfield equations (1) are given by
Ψ := (x,y):0 x2+y2 1, x 1, y 0 . (19)
X
{ ≤ ≤ | |≤ ≥ }
However,asweshallsee,ingeneral–arbitrarypotentials
ρ =X˙2/2+V(X), p =X˙2/2 V(X), (14)
X X − – the phase space is of dimension higher than 2 [20].
respectively. In these equations V = V(X) is the self-
interacting potential of the scalar field X. Besides, for
B. Autonomous system of ODE-s
the EOS parameter ω =γ 1, one has
X X
−
In order to derive the autonomous ordinary equations
p X˙2 2V forthephasespacevariablesx,yoneproceedsinasimilar
X
ωX ≡ ρX = X˙2−+2V . (15) fashionthaninsectionII.Firstwe writte the Raychaud-
huri equation in (1)
Aftertheabovechoice,theconservationequation(1)can
be written in the form of the following Klein-Gordon
1
equation: H˙ = (γmρm+ρX +pX)
−2 ⇒
H˙ X˙2
2 =3γ Ω + ,
X¨ +3HX˙ = V . (16) − H2 m m H2
,X
−
in terms of the new variables:
It happens that deriving an autonomous ODE for the
variableω –seesectionII –canbeaverydifficulttask.
X
Henceitcouldbebettertochooseadifferentsetofphase H˙
space variables which do the job. Here, instead of the −2H2 =3γm 1−x2−y2 +6x2, (20)
phase space variables Ω and ω we shall choose the
X X (cid:0) (cid:1)
following variables [11]: where we have taken into account the Friedmann con-
straint (18). Then, given the definition (17), let us find
X˙ √V
x , y , (17) X¨ X˙ H˙
≡ √6H ≡ √3H x˙ = , (21)
√6H − √6H H
so that the dimensionless energy density of the scalar
field X is given by Ω = x2 +y2, and the Friedmann or if one substitutes (16) and (20) back into (21), then
X
constraint (5) can be written as
x′ = 3x(1 x2)+
− −
Ωm =1−x2−y2. (18) 3γm x 1 x2 y2 3V,X y2, (22)
2 − − − 2 V
r
(cid:0) (cid:1)
A. Determination of the physical phase space wherewe replacedderivativeswithrespectto the cosmic
time t by derivatives with respect to the variable τ
≡
lna(t). Applying the sameprocedurewiththe variabley
The first step towardsa complete study of the asymp-
one obtains
totic structure of a given cosmological model is the rig-
orousdeterminationofthe phasespacewhereto lookfor
1V X˙ H˙
the relevant equilibrium points. In the present model y′ =y ,X ,
there are several constraints on the physical parameters 2 V H − H2!
which help us to define the physically meaningful phase
space. wherewehavetakenintoaccountthatV˙ =V X˙,hence
,X
8
Correspondence f(z) V(X)
⇔
Function f(z) z Self-interaction potential V(X) Reference
f(z)=0 µ V =V0e±µX [11]
f(z)= αβ+(α+β)z z2 α+βe±(β−α)X V =V eαX +eβX [21]
− − 1+e(β−α)X 0(cid:0) (cid:1)
f(z)=(p2µ2 z2)/p pµcotanh(µX) V =V sinhp(µX) [31]
0
−
f(z)=(p2µ2 z2)/p pµtanh(µX) V =V coshp(µX) [31]
0
−
f(z)=(p2µ2 z2)/2p pµ sinh(µX) V =V [cosh(µX) 1]p [32]
− cosh(µX) 1 0 −
−
TABLE I:The functions f(z) in Eq. (27) for different self-interaction potentials.
y′ =y 3γm 1 x2 y2 +3x2 + 3V,X xy. (23) f(z)≡z2[Γ−1], Γ≡VV,XX/V,2X, (26)
2 − − 2 V
(cid:20) (cid:21) r
(cid:0) (cid:1) and the main assumptionhas been that the aboveΓ is a
Unless V = V V /V = 0, or V = V e±µX function of the variable z: Γ=Γ(z). Notice that, since
0 ,X 0
⇒ ⇒
V /V = µ = const., the ASODE (22), (23) is not a
,X
closed syst±em of equations since one equation involving VV,XX V,XX/V 1 V,XX
Γ = = ,
the derivative of V,X/V with respect to τ is lacking. ≡ V,2X (V,X/V)2 z2 V
One example where the ASODE (22), (23) is indeed
a closed system of ODE-s is given by the so called “ex- the left-hand equation in (26) can be rewritten also as
ponentialquintessence.” This casehas been investigated
in all detail in [11] so that we submit the reader to that
V
referencetolookforaveryinterestingexampleoftheap- f(z)= ,XX z2. (27)
plicationofthedynamicalsystemstoolsincosmology. As V −
amatteroffact,itresultsaveryusefulexerciseforthose
In the event that Γ can not be explicitly written as a
who want to learn how to apply the dynamical systems function of z, then, an additional ODE: Γ′ =..., is to be
tools in cosmologicalsettings to reproduce the results of
considered. However, this case is by far more complex
the dynamical systems study in the seminal work [11].
and does not frequently arise.
In order to be able to consider self-interaction poten-
Before going further we want to point out that, as a
tialsbeyondtheconstantandtheexponentialpotentials,
matter of fact, the function Γ and the variable z were
in addition to the variables x, y, one needs to adopt a
first identified in [30]. Besides, the dynamical system
new variable [14, 23, 30]
(22), (23), (25) was explored in [14] several years before
it was studied in the reference [23]. However, it was in
[23] where the possibility that for several specific self-
z V /V, (24)
≡ ,X interaction potentials Γ can be explicitly written as a
function of z, was explored for the first time. In the ref-
sothatz =0correspondstotheconstantpotential,while
z = µ is for the exponential potential.14 By taking the erence [14], although the correct dynamical system was
± identified,theauthorswerenotinterestedinspecificself-
derivative of the new variable z with respect to τ one
interaction potentials. The cost of the achieved general-
obtains
ity of the analysis was that the authors had to rely on
the obscure notion of “instantaneous critical points”.
z′ =√6xf(z), (25)
where we have defined (as before take into account that V. AN EXAMPLE: THE COSH-LIKE
V˙ =V X˙ and V˙ =V X˙, etc.) POTENTIAL
,X ,X ,XX
Inordertoshowwithaconcreteexamplehowthefunc-
tionf(z)canbe obtainedforanspecific potential, letus
14 For non-exponential potentials, besides the increase in the di- choose the cosh potential [31, 32]
mensionalityofthephasespace,thereisanotherproblem. Typ-
icallythephasespacebecomesunbounded,sothattwodifferent
sets of variables are required to cover the entire phase space
V =V cosh(µX). (28)
[20]. In the present paper, however, the chosen example of the 0
cosh-likepotential does not lead to unbounded phase space. In
This potential – as well as a small variation of it in the
consequence a single set of phase space variables is enough to
describethewholeasymptotic(andintermediate)dynamics. lastrowofTAB. I – has avery interestingbehaviornear
9
oftheminimumatX =0. Actually,intheneighborhood
of the minimum, the potential (28) approaches to
3 3
x′ = x 1 x2+y2 y2z,
V(X) V + m2X2, m2 =V µ2. −2 − −r2
≈ 0 2 0 y′ = 3y 1(cid:0)+x2 y2 +(cid:1) 3xyz,
2 − 2
r
The quadratic term is responsible for oscillations of the
z′ =√6x(cid:0)(µ2 z2). (cid:1) (31)
scalar field around the minimum which play the role of −
cold dark matter, hence, at late times when the system
Since z = µtanh(µX) µ z µ, the phase space
is going to stabilize in the minimum of V(X) the cosh ⇒ − ≤ ≤
where to look for critical points of (31) is the bounded
potential makes the scalar field model of DE to be in-
semi-cylinder (see Eq. (19)):
distinguishablefromthe standardΛCDMmodel(seethe
properties of the equivalent model of Ref. [15]).
Taking derivatives of (28) with respect to the scalar Ψ := (x,y,z):0 x2+y2 1,
cosh
{ ≤ ≤
field X we have that V = µV sinh(µX), and V =
,X 0 ,XX x 1, y 0, z <µ . (32)
µ2V. Hence, | |≤ ≥ | | }
Nine critical points P : (x ,y ,z ) of the autonomous
i i i i
system of ODE (31) and one critical manifold are found
z =V,X/V =µtanh(µX), f(z)=µ2 z2. (29) in Ψcosh. These together with their main properties are
−
shown in TAB. II. For sake of conciseness, here we shall
Notice that the z-s which solve f(z) = 0, i. e., z = concentrate only in a few of them which have some par-
µ, which correspond to the exponential potentials, are ticular interest.
±critical points of (25). In TAB. I the functions f(z) for The de Sitter critical point PdS : (0,1,0), for which
several well-known potentials are displayed. ΩX =1(Ωm =0),isassociatedwithacceleratingexpan-
sion since the deceleration parameter
Let us to collect all of the already found autonomous
ODE-s which correspond to a scalar field model of DE
with arbitrary self-interaction potential: H˙ 1 3
q = 1 = + x2 y2 = 1, (33)
− − H2 2 2 − −
(cid:0) (cid:1)
x′ = 3x(1 x2)+ is a negative quantity. Since z = 0 V = V0, and
− − since x=0 X˙ =0, and ⇒
3γm x 1 x2 y2 3y2z, ⇒
y′ =y 23γm(cid:0)1− x2− y2(cid:1)−+r3x22 + 3xyz, y =1 ⇒ H =rV30 ⇒a(t)∝eHt,
2 − − 2
(cid:20) (cid:21) r this point correspondsto a de Sitter (inflationary)phase
z′ =√6xf(z(cid:0)), (cid:1) (30) ofthe cosmic expansion. We underline the fact that to a
givenpointinthephasespace–inthiscaseP Ψ –
dS cosh
∈
where it is evident that the equations for x and y are itcorrespondsaspecificcosmologicaldynamics(deSitter
independent of the self-interaction potential, and that expansion in the present case).
details of the given model are encoded in the function NextwehavethescalarfieldX-dominatedequilibrium
f(z)– Eq. (27)– whichdepends onthe concreteformof points
the potential (see TAB. I).
µ µ2 µ2
±
P : , 1 , µ , Ω =1, q = 1+ .
A. Critical points X ±√6 r − 6 ∓ ! X − 2
The first thing one has to care about is the existence of
Ifweapplythesimplesttoolsofthedynamicalsystems the critical point. By existence we mean that the given
theory to the the above example with the cosh poten- point actually belongs in the phase space (Ψ in the
cosh
tialthe correspondingasymptoticdynamics inthe phase present case). Hence, since x 1 µ2 6. This
spaceisrevealed. Giventhat,attheminimumofthecosh is precisely the condition for|ex|is≤tence⇒of the p≤oints P±:
X
potential,themodelbehavesasΛCDM,hereweshallas- µ2 6. This warrants, besides, that y = 1 µ2/6 be
sumethatthematterfluidiscomposedmainlyofbaryons ≤ −
a real number. If one substitutes (17) back into (15) on
so that it behaves like pressureless dust (γ =1). p
m gets
The following closed system of autonomous ODE-s is
obtained(it is just Eq. (30) with the appropriatesubsti- x2 y2
ω = − ,
tutions): X x2+y2
10
Critical Points x y z Existence Ω Ω λ λ λ
X m 1 2 3
0 0 z Always 0 1 3/2 3/2 0
Z −
PdS 0 1 0 ” 1 0 −32 + 32q1− 4µ32 −32 − 23q1− 4µ32 −3
PK± ±1 0 µ ” 1 0 3±q32µ ∓2√6µ 3
P¯K± ±1 0 −µ ” 1 0 3∓q32µ ±2√6µ 3
PX±/m ±q32µ1 q32µ1 ∓µ µ2 ≥ 32 µ32 1− µ32 −34 + 34qµ242 −7 −43 − 43qµ242 −7 6
PX± ±√µ6 q1− µ62 ∓µ µ2 ≤6 1 0 −3+µ2 −3+ µ22 2µ2
TABLE II: Critical points of theautonomous system of ODE (31) and their properties.
sothat,forthepresentcasetheEOSparameteroftheX- one needs to linearize the system of ODE – in this case
fieldω =µ2/3 1 γ =µ2/3. Thenthe continuity (31) – which means, in the end, that we have to find the
X X
− ⇒ ±
equation (1) can be readily integrated at P to obtain: eigenvaluesofthelinearizationorJacobianmatrixJ (see
X
ρX = Ma−3γX = Ma−µ2, where M is an integration Eq. (35) of the appendix)
constant. After this the Friedmann constraint
∂x′/∂x ∂x′/∂y ∂x′/∂z
Ω =1 3H2 =ρ a˙ = M a−µ2, J =∂y′/∂x ∂y′/∂y ∂y′/∂z.
X X
⇒ ⇒ a r 3 ∂z′/∂x ∂z′/∂y ∂z′/∂z
 
can be integrated also to obtain the following cosmolog-  
ical dynamics a(t) t2/µ2, which is to be associated Next J is to be evaluated ateachnon-hyperbolic critical
∝ ± point, for instance, J(P ), J(P ), etc. The eigenvalues
with the critical points P . We want to point out that dS X
whenever 2 < µ2 6 (qX> 0) the present scalar field of the resulting numerical matrices is what we use to
≤ judge about the stability of the given points.
dominated solution represents decelerated expansion.
Inthecaseofthematter-dominatedcriticalmanifold15 Take, for instance, the de Sitter point PdS : (0,1,0).
In this case one has to find the roots of the following
= (0,0,z):z R , Ω =1, q =1/2, algebraic equation for the unknown λ:
m
M { ∈ }
instead of an isolated equilibrium point this is actually 3 λ 0 3/2µ
− −
a 1-dimensionalmanifold which is extended along the z- det 0 3 λ 0 =0.
direction. Thismeansinturnthatthematter-dominated  − − p 
√6µ 0 λ
solution: Ω =1 − − 
m
⇒  
3H2 =ρ =Ma−3 a(t) t2/3, The resulting eigenvalues are
m
⇒ ∝
may coexist with the scalar field domination no matter 3 3 4
λ = 1 µ2, λ = 3.
1,2 3
whether V cosh(µX) intermediate X-s, V = V −2 ± 2 − 3 −
0 r
X = 0, or∝V e±µX → very large X-s (X →).
This entails tha∝t it can b→e found an heterocli|nic| o→rbi∞t in Forµ2 3/4the criticalpointis anisolatedfocus, while
thephaseportrait–anorbitconnectingtwoormoredif- for µ2 >≤ 3/4, since the eigenvalues λ1 and λ2 are com-
ferentequilibriumpoints–joiningthematter-dominated plex numbers with negative real parts, the point PdS is
equilibrium state with a scalar field domination critical a stable spiral. Hence the de Sitter solution is always a
point, be it originatedeither by a cosmologicalconstant, future attractor (see the bottom panel of FIG. 1). This
or by exponential or cosh-like potentials. is consistent with the fact that the cosh potential is a
minimum at X =0 V =V .
0
⇒ ±
The eigenvalues of the linearization matrix J(P ) are
X
B. Stability
λ = 3+µ2/2, λ = 3+µ2, λ =µ2.
1 2 3
− −
Inordertojudgeaboutthestabilityofgivenhyperbolic
equilibrium points (as explained in the appendix VII) Hence, since as requiredby the existence µ2 <6, PX± are
alwayssaddlecriticalpoints(atleastoneoftheeigenval-
ues is of a different sign).
If one computes the eigenvalues of J( ) one finds
M
15 Here by critical manifold we understand a curve in the phase λ1,2 = 3/2, λ3 =0. The vanishing eigenvalue is due to
±
spaceallofwhosepointsarecriticalpoints. the fact that the 1-dimensional manifold extends along