Asymptotic Properties of a Supposedly Regular (Dirac-Born-Infeld) Modification of General Relativity

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by  Ricardo García-Salcedo

Asymptotic Properties of a Supposedly Regular (Dirac-Born-Infeld) Modification of
General Relativity.
Ricardo Garc´ıa-Salcedo,1,a Tame Gonzalez,2,b Claudia Moreno,3,c
Yunelsy Napoles,2,d Yoelsy Leyva,2,e and Israel Quiros4,f
1Centro de Investigacion en Ciencia Aplicada y Tecnologia Avanzada - Legaria del IPN, M´exico D.F., M´exico.
2Departamento de F´ısica, Universidad Central de Las Villas, 54830 Santa Clara, Cuba.
3Departamento de F´ısica y Matema´ticas, Centro Universitario de Ciencias Exa´ctas e Ingenier´ıas,
Av. Revoluci´on 1500 S.R., Universidad de Guadalajara, 44430 Guadalajara, Jalisco, M´exico.
4Divisio´n de Ciencias e Ingenier´ıa de la Universidad de Guanajuato, A.P. 150, 37150, Le´on, Guanajuato, M´exico.
(Dated: January 25, 2010)
Weapplythedynamicalsystemstoolstostudytheasymptoticpropertiesofacosmologicalmodel
based on a non-linear modification of General Relativity in which the standard Einstein-Hilbert
0 action is replaced by one of Dirac-Born-Infeld type. It is shown that thedynamics of this model is
1 extremely rich: there are found equilibrium points in the phase space that can be associated with
0 matter-dominated,matter-curvaturescaling,deSitter,andevenphantom-likesolutions. Depending
2
onthevalueoftheoverallparametersthedynamicsinphasespacecanshowmulti-attractorstructure

n into the future (multiple future attractors may co-exist). This is a consequence of bifurcations in
a control parameter space, showing strong dependence of the model’s dynamical properties on the
J freeparameters. Contrarytowhatisexpectedfromnon-linearmodificationsofgeneralrelativityof

5 this kind, removal of the initial spacetime singularity is not a generic feature of the corresponding
2 cosmological model. Instead, the starting point of the cosmic dynamics – the past attractor in the
phasespace –isastate ofinfinitelylarge valueoftheHubblerate squared,usuallyassociated with

] thebig bang singularity.
c
q
PACSnumbers: 04.20.-q,04.50.Kd,95.36.+x,98.80.-k,98.80.Bp,98.80.Cq,98.80.Jk
-
r
g
[ I. INTRODUCTION to be cured – is replaced by one of the DBI form:

4
v Attempts to modify the Einstein-Hilbert (EH) action 2L
= g µ 1+ 1 ,
8 of General Relativity (GR) L → LDBI | | s µ − !
4
p
50 SEH = 2κ12 d4x |g|(R−2Λ), where the scale µ sets the limit of energy density acces-
. Z sible to the theory.
2 p
where R is the Ricci curvature scalar, and Λ- the cos- A combination of the above possible modifications, i.
1 mological constant (κ2 = 8πm−2 = 8πG), have been e., a DBI-type action containing an EH term plus ad-
9 Pl
0 motivated by a number of reasons. In particular, renor- ditional higher curvature terms within the square root,
: malizationatone-loopdemandsthattheEinstein-Hilbert could supply an additional cosmological scenario where
v
action be supplemented by higher order curvature terms tolookforalternativeexplanationstoseveralphenomena
i
X [1].1 Besides,whenquantumcorrectionsorstringtheory such as inflation and the present speedup of the cosmic
r aretakenintoaccount,theeffectivelowenergyactionfor expansion.
a puregravityadmitshigherordercurvatureinvariants[3]. Several theories of gravity of this kind have been pro-
There are additional ways to modify the EH GR ac- posed since long ago in [7], and in more recent years,
tion. For instance, the one based on the Dirac-Born- for instance, in [8] (see also [9]). To be phenomenolog-
Infeld (DBI) procedure for smoothing out singularities ically viable, these modifications have to satisfy several
[4, 5].2 According to this procedure the original La- physically motivated requirements [8]:
grangian density = g L – whose singularities are
L | |
1. Reduction to EH action at small curvature,
p
2. Ghost freedom,
aElectronicaddress: [email protected]
3. Regularization of some singularities (as, for in-
bElectronicaddress: [email protected]
cElectronicaddress: [email protected] stance, the Coulomb-like Schwarzschild singular-
dElectronicaddress: [email protected] ity), and
eElectronicaddress: [email protected]
fElectronicaddress: iquiros@fisica.ugto.mx
1 Higher order actions are indeed renormalizable but not unitary
[2].
2 The proposal to remove initial as well as final singularities in that the addition of a R2 term to otherwise divergent modified
modifiedgravity has been given inRef. [6]. It was shownthere gravitymakesitregular.
2
4. Supersymmetrizability.
2
The later requirement is quite stringent and, for most f(R, )= 1 1 ακ2R+βκ4 . (3)
purposes, might be excluded. A DBI-type theory that G κ2 − − G
fulfills the above requirements can be based on the fol- (cid:16) p (cid:17)
Written in the form (2) the action (1) has the appear-
lowing action [9]:
anceofanf(R, )-theoryofthekindstudied,forinstance
G
in Ref.[13], so that we can safely apply known results of
1 these investigations.
S = d4x g 1 1 ακ2R+βκ4 , (1)
DBI κ4 | | − − G The equations of motion that can be derived from (2)
Z p (cid:16) p (cid:17) – plus the addition of a matter actionpiece – by varying
where R2 4R Rµν +R Rµνσυ is the Gauss- in respect to the metric g are the following [13]:
µν µνσυ µν
G ≡ −
Bonnet (GB) invariant. It has been demonstrated in
Ref.[9]that this actionhas the EHleadingterm atsmall
1
curvature, besides, for an appropriate region in the pa- fRRµν gµνf [ µ ν gµν(cid:3)]fR 4(cid:3)fG(Rµν
− 2 − ∇ ∇ − −
rameter space it shows indications for the cancellation
1
of the Coulomb-like Schwarzschild singularity, and it is gµνR)+4(Rσν σ µfG +Rσµ σ νfG
ghost-free. −2 ∇ ∇ ∇ ∇
Given that such a regular theory of gravity could gµνRστ σ τfG)+2[2Rµσντ σ τ
− ∇ ∇ ∇ ∇
have impact on the cosmology of our universe, in this −R∇µ∇ν]fG +2fG(RRµν −2RµσRσν
paper we aim at exploring the asymptotic properties +2RστR +R R στρ)=κ2T , (4)
of a Friedmann-Robertson-Walker (FRW) cosmological µστν µστρ ν µν
modelbasedonsuchanon-linearmodificationofGeneral where fR ∂Rf, fG ∂Gf, and (cid:3) gστ σ τ. Worth
Relativity.3 Our study will rely on the use of the stan- noticingth≡at,incaset≡hefunctionf w≡erea∇fun∇ctionofthe
dard tools of the dynamical systems (see, for instance, curvature scalar only, i. e., in case f = f(R) (formally
Ref. [11]). The present investigation represents a gener- this case corresponds to the limit 0), the above
alization of the study in Ref.[12] to include dependence field equations reduce to the equatiGons→of motion of an
notonlyontheRicciscalarbutalsoontheGauss-Bonnet f(R)-theory[14,15]. Forpurposeofcomparisonwiththe
term, whichis one of the few waysto overcomethe pres- equationsofEHGR,equations(4)canberecastintothe
ence of ghosts. Notwithstanding, we will not make a form that resembles Einstein’s equations [13]:
comparison of the results we obtain in this paper with
the results of [12], since the latter refer to a completely
1
differentnon-linearmodificationofgeneralrelativityhav- R g R=κ2 (T +Tcurv), (5)
ing nothing in common with the DBI modification stud- µν − 2 µν eff µν µν
ied here.
The paper has been organized as it follows. In the where κ2eff = κ2/(fR −4(cid:3)fG) is the effective gravita-
tional coupling, and we have introduced the following
next section the basic mathematics of the model (1) are
curvature(effective)stress-energytensoractingasanad-
exposed, including the derivation of the relevant FRW
ditional source of the field equations:
cosmological equations. The main section III is devoted
to the study of the asymptotic properties of the model.
AdetaileddiscussionoftheresultsisgiveninsectionIV, 1
κ2Tcurv =( g (cid:3))f + g (f Rf )
while brief conclusions are given in the last section V. µν ∇µ∇ν − µν R 2 µν − R
Here we use natural units (κ2 =8πG=~=c=1). 2(2Rµσντ σ τ R µ ν)fG 4(Rσν σ µfG
− ∇ ∇ − ∇ ∇ − ∇ ∇
+Rσµ σ νfG gµνRστ σ τfG) 2fG(RRµν
∇ ∇ − ∇ ∇ −
II. EQUATIONS OF MOTION −2RµσRσν +2RστRµστν +RµστρRνστρ). (6)
The trace of equation (4) leads to the following con-
We start by writing the action (1) in a more compact
straint:
form [9]:
S = 1 d4x g f(R, ), (2) 3(cid:3)fR+RfR−2f +2R(cid:3)fG
DBI 2κ2 | | G +2[ 2Rστ σ τ]fG =κ2T. (7)
Z G− ∇ ∇
p
where For a spatially flat FRW universe metric
ds2 = dt2+a(t)2(dr2+r2(dθ+sin2θdϕ2),
−
3 Theasymptoticpropertiesofabigclassoff(R,G)theories,with equations (4) can be written in the form of the following
theaimtostudyfuturesingularitieshasbeendiscussedin[10]. set of cosmologicalequations:
3
III. DYNAMICAL SYSTEMS STUDY
6H2f =Rf f 6Hf˙
R R− − R The dynamical systems tools offer a very useful ap-
24H3f˙G + fG +ρm, (8) proach to the study of the asymptotic properties of the
− G
4H˙fR =2Hf˙R+2f˙G(12H3 /3H) cosmological models [16]. In order to be able to apply
−G
−2f¨R−8H2f¨G −ρm, (9) tmheersaetteodoblseolonwe.hasto(unavoidably)followthestepsenu-
ρ˙ = 3Hρ , (10)
m m
−
First: to identify the phase space variables that
wherethe dotaccountsforderivativewithrespectto the •
allow writing the system of cosmologicalequations
cosmic time t, and for simplicity we are consideringdust
in the form of an autonomus system of ordinary
as background fluid (it could be, for instance, the dark
differential equations (ODE). There can be several
matter component).
different possible choices, however, not all of them
Starting with equation (3) it is straightforward to ob-
allow for the minimum possible dimensionality of
tainthe followingrelationshipbetweenthe derivativesof
the phase space.
the function f(R, ):
G
Next: withthehelpofthechosenphasespacevari-
•
ables, to build an autonomous system of ODE out
fG fGR A, (11) of the original system of cosmologicalequations.
f ≡ f ≡
R RR
Finally (some times a forgotten or under-
where the constant A κ2β/α = β/α (recall that •
appreciated step): to identify the phase space
we use the units system≡w−here κ2 = 1−). In the present
spanned by the chosen variables, that is relevant
paper,fordefiniteness,weshallchooseonlynon-negative
to the cosmologicalmodel under study.
α 0 andβ 0. Inconsequence,the constantA willbe
≥ ≥
a negative magnitude. After this one is ready to apply the standard tools of
After (11), and recalling that the(linear)dynamicalsystemsanalysis(see,forinstance
Ref. [17]).
=24H2(H˙ +H2), R=6(H˙ +2H2), We will split our study into two parts. First, in sub-
G
section A, we explore the asymptotic properties of the
we areableto rewrite equations(8,9) ina morecompact
simplest situation when in (1) we set = 0. Although
yet simpler form: G
in this case the corresponding DBI theory has a ghost,
spin-zerodegreeoffreedom,soitdoesnotadmitastable
6H2f (1+4AH2)=Rf (1+4AH2) de Sitter background [18], this case is easier to handle
R R
and could signal to relevant asymptotic behavior with
f 6Hf˙ (1+4AH2)+ρ , (12)
− − R m which to compare the results of the more general ghost-
4H˙f +16AHH˙f˙ =2Hf˙ (1+4AH2) free case ( =0), in the second part of our investigation
R R R
G 6
ρ 2f¨ (1+4AH2). (13) (subsection B).
m R
− −
Recall that, according to (5), the equation (12) can be
written in the form of a standard Friedmann equation: A. Einstein-Hilbert DBI Case ( =0)
G
1. Autonomous System of ODE
1
3H2 =κ2 (ρ +ρ ), κ2 = ,
eff m curv eff 2f (1+4AH2)
R
Our aim here is to write the following system of cos-
R f f˙
κ2 ρ = 3H R. (14) mological equations (basically equations (12), and (13)
eff curv 2 − 2fR(1+4AH2) − fR with A=0):
Notice that written in this latter form, due to the fact
that A < 0, the Friedmann equation blows up at H = 6H2f =Rf f 6Hf˙ +ρ ,
R R R m
1/(2√ A)= α/β/2(itisinfacttheeffectivegrav- − −
i±tational−couplin±g κ2 who does). However this is not 4H˙fR =2Hf˙R−2f¨R−ρm , (15)
peff
pathological since the above is just a convenient way to intheformofanautonomoussystemofordinarydifferen-
write the original equations. tialequation(ASODE).Tothatpurposeletus,following
Ourgoalwillbe to writethe abovecosmologicalequa- Ref.[19],tointroducethedimensionlessvariablesx,y,z:4
tions in the form of an autonomous system of ordinary
differential equations, so that we could apply the stan-
dard tools of the dynamical systems to find the equilib-
rium configurations that could be associated with rele- 4 Noticethatourchoiceofphasespacevariablesdiffersfromthat
vant cosmological solutions. ofRef. [19],inthedefinitionofoneofthevariables(y).
4
TABLE I:Properties of theequilibrium points of theautonomous system (17).
P x y z Existence Ω¯ Ω¯ w q
i m curv eff
P1 0 0 1 Always 0 1 1/3 1
P2 0 0 1 ” 2 1 1/3 1
− −
P3 1 0 3 ” 3 2 1 2
− − −
P4 2 4 1 ” 0 1 1 1
− − −
P5 2 4 3 ” 4 5 1 1
− − −
P6 2 9/2 0 ” 0 1 1 1
− −
TABLE II:Eigenvalues of thejacobian matrices corresponding to thecritical points in table I.
Pi λ1 λ2 λ3
P1 6 2 4
P2 2 2 4
−
P3 6 −23 +i√215 −23 −i√215
P4 2 4 4
− − −
P5 4 6 4
−
P6 −3 −32 + √221 −23 − √221
the curvature source in the right-hand-side of Einstein’s
equations.
R 1 f˙
x= , y = , z = R . (16) The first equation in (15) can be written in the form
6H2 3αH2 −HfR of the following (Friedmann) constraint:
Startingfromequations(12),(13),(16)and,afterabit ρ R f f˙
of algebra, it is uncomplicated to obtain the following 1= m + R ,
6H2f 6H2 − 6f H2 − Hf
autonomous system – the right-hand-side of the ODEs R R R
donotdependexplicitlyonthe time parameterτ –,that
or, after the above choice of phase space variables:
describes the cosmologicaldynamics of this model:
Ω¯ =1+x y z+ y2 2xy =1 Ω¯ , (18)
x′ = 2x(x 2) z(y 2x), m − − − − curv
− − − −
y′ = 2y(x 2), where we have convenienptly defined Ω¯ = ρ /6H2f ,
m m R
z′ =z−2 xz−+2(x 2)+ and Ω¯curv is the dimensionless curvature (effective or
− − “dark”) energy density.
+3(1+x y+ y2 2xy). (17)
Thenextstepistoidentifythephasespacerelevantfor
− −
Herethecommadenotesderivativeinprespecttothetime the cosmological model associated with the autonomous
system of ODE (17). Since we are considering positive
variable τ lna (basically the number of e-foldings of
inflation). ≡ α ≥ 0 ⇒ fR ≥ 0, then Ω¯m ≥ 0. Besides, since the
expressionunder the square root in the intermediate ex-
The following expressions will be useful:
pression in (18) must be positive, so that Ω¯ be real,
m
1 then the following conditions are relevant to define the
q =1 x, w = (2x 1),
− eff −3 − phase space:
where q = 1 H˙/H2 is the deceleration parameter,
− −
and w is the effective equation of state parameter for 1+x y z+ y2 2xy 0, y 2x, y 0, (19)
eff
− − − ≥ ≥ ≥
p
5
where the condition y 2x holds only for non-negative The past attractor in the phase space is the equilib-
≥
x 0. For x<0 the expressionunder the square root is rium point P = (0,0,1) in Tab. I. Although the corre-
1
≥
always positive. sponding cosmological phase is dominated by the curva-
Ourmostprecisedefinitionofthecorrespondingphase ture (Ω¯ = 1), the expansion is decelerated (q = 1).
curv
space is the following: The effective curvature energy density mimics radiation
(w = 1/3). This state is characterized by infinitely
eff
large values of the Hubble parameter squared H2 ,
→∞
Ψ = (x,y,z): y2 2xy y+z x 1, and f a(t) - the scale factor.
A R
{ − ≥ − − ∝
p y ≥0, y ≥2x}.(20) dleTehqeufiilxiberdiupmoinptoiPnt2. =It(i0s,0a,s−so1c)iaitnedTaalbs.o wIiitshadescaedl--
erated expansion (q = 1), but this time there exists a
scaling between the energy density of the background
fluid Ω¯ = 2 and that of the effective curvature com-
m
ponent Ω¯ = 1: Ω¯ /Ω¯ = 2. The fact that
0.8 curv − m curv −
the dimensionless energy density parameter for the dust
0.6
0.4 Ω¯ >1, should not bother us. Recall that we have cho-
m
0.2 sen a convenient definition Ω¯ = ρ /6H2f , so that,
z(t) 0 provided that 2f < Ω = ρm /3Hm2, we wiRll be faced
–0.2 R m m
–0.4 with Ω¯ >1. During this phase of the cosmic evolution
m
–0.6 the effective energy density of the universe also mimics
–0.8
–1 radiation.
TheequilibriumpointP =( 1,0, 3)isalsoasaddle
x(1t) 2 1 2y(t) 3 4 and is associated with a s3calin−g of b−oth energy density
components: Ω¯ /Ω¯ = 3/2. Inthiscasethedynam-
m curv
−
icsofthe expansionis“super”-decelerated(q =2),while
the effective equation of state parameter mimics that of
6 6
stiff fluid (w =1).
eff
4 4 ThefixedpointP =(2,4, 1)isthefuture(late-time)
4
t 2 t 2 attractor in ΨA. It is associa−ted with a curvature domi-
nated (Ω¯ = 1, Ω¯ = 0), inflationary (q = 1) solu-
0 0 curv m −
tion. The effective curvaturefluid mimics a cosmological
–2 –2
2 constant Λ (weff = 1). Another curvature-dominated,
x(1t) 2 1 2y(t) 3 4 x(1t) 0.80.60.40.20z–(0t).2 –0.6 –1 inflationary, with Λ −mimicry, twin-solution is associated
with the spiral saddle equilibrium point P = (2,9/2,0)
6
in Tab. I.
FIG. 1: Probe pathsin phase space originated from different The saddle fixed point P5 = (2,4,3) does not re-
setsofinitialconditionsforf(R)= 2 1 √1 ακ2R . The ally correspond to a meaningful cosmological scenario.
flux in time is shown in the lower pκa2n(cid:0)els.−The−trajecto(cid:1)ries in Actually, in this case Ω¯ = ρ /6H2f = 4, but
m m R
ΨA emerge from the past attractor (decelerated expansion since we are considering only non-negative α −0, then
phasecorrespondingtotheequilibriumpointP1 =(0,0,1)in f =α/√1 αR 0. Therefore, even if Ω¯ is≥an effec-
R m
Tab. I), and are attracted into the future by the curvature − ≥
tive(convenient)parametrization,itcannotbenegative.
dominated inflationary fixedpoint P4 =(2,4, 1).
−
B. Ghost-free Case ( =0)
G 6
2. Equilibrium points 1. Autonomous System of ODE
Thenextstepistofindtherootsofthesystemofequa- Now we turn to the more general case when in equa-
tions x′ = 0 , y′ = 0 , z′ = 0. Then one linearizes the tions (12), (13), the constant A = 0. In order to trans-
6
ASODE (17) – means that we expand (17) in the neigh- formtheseequationsintoanautonomoussystemofODE
borhood of the equilibrium point keeping only the linear we introduce the following phase space variables:
terms–andfindstheeigenvaluesofthelinearizationma-
trix.
R 1
The mostrelevantpropertiesofthe equilibriumpoints x= , y = ,
of the autonomous system of ODE (17) are summarized 6H2 3αH2
intableI. TableIIdisplaystheeigenvaluesoftherespec- f˙ 1
R
z = , v = . (21)
tive linearization or Jacobian matrices. −Hf −4AH2
R
6
Notice that this time the dimension of the phase space
has been increased from 3D to a 4-dimensional phase
Ψ = (x,y,z,v):y 0, v 0,
space spanned by the variables (x,y,z,v). However, as B { ≥ ≥
wewillshowquitesoon,wecanskiponeofthesevariables vy 2(v 1)x+2 . (24)
≥ − }
and, consequently, the phase space corresponding to the
Worth noticing that in the variables x,y,z,v, the par-
present case can be reduced also to a 3-dimensional one.
ticular case = 0 is recovered from the present case in
After the above choice of variables of the phase space, G
the formal limit v .
we can write the Friedmann constraint in terms of →∞
x,y,z,v:
2. Parametric ASODE
vy 2
Ω¯ =1+x z − +
m − − v 1 It is noticeable the similarity between 2nd and 4th
− equationsin (23). However,by lookingatthe definitions
v2y2 2v(v 1)xy 2vy
+ − − − =1 Ω¯ , (22) of the variables y and v one sees that this is not casual.
curv
v 1 −
p − Actually, from (21) one can see that
where now we have conveniently defined
4β
ρ v ρ y =µv , µ . (25)
Ω¯ = m = m . ≡ 3α2
m 6H2f (1+4AH2) v 1 6H2f
R (cid:18) − (cid:19) R Hence, the ASODE (23) is not really a 4-dimensional
Notice that, contrary to the former case, here the effec- system of equations but a 3-dimensional one. In fact
tivematterdensityparameterΩ¯ canbenegativewhen- it is a 3D, one-parameter ASODE. In the theory of the
m
ever v < 1. In this case constraints on the domain of dynamicalsystemstheparameterµ 4β/3α2iscalledas
≡
x,y,v arise from non-negativity of the expression under “control parameter”. It usually controls the occurrence
the square root: of bifurcations – a change of the value of the control
parameterproducesachangeinthetopologyofthephase
2(v 1)x+2 portrait.5 That such bifurcations arise in the present
y − .
≥ v study will be clear when we found that the asymptotic
structure of the DBI-EH model ( = 0 µ = 0) and
Other magnitudes of relevance for the analysis that G ⇒
that of the general, ghost-free case (µ = 0, µ > 0), are
can be straightforwardly put in terms of the above vari- 6
qualitatively different.
ablesaretheeffectiveequationofstate(EOS)parameter
In what follows we will skip the variable v, and will
ω = 1 2H˙/3H2, and the deceleration parameter
eff write the 4D autonomous system (23) as a 3D, one-
− −
q = (1+H˙/H2): parameter ASODE in the variables x,y,z:
−
1 2x
ωeff ≡ −3 , q =1−x, x′ = 2x(x 2)+ 2µ(x−2)2
− − y µ −
respectively. −
z[y2 2xy+2µ(x 1)]
Following the same procedure as before, after a bit of − − ,
− y µ
algebra,we canobtain the following autonomous system
−
of ordinarydifferential equations out of the cosmological y′ = 2y(x 2),
− −
equations (12), (13): 2y(x 2)
z′ =z2 xz+3z+ − +
− y µ
−
x′ =−2x(x−2)+ 2(vx−12)2 − +2µzy(x−µ2) +3Ω¯µm(x,y,z), (26)
− −
z[vy 2(v 1)x 2] where
− − − ,
− v 1
y′ = 2y(x 2), − y2 2µ
− − Ω¯µ(x,y,z)=1+x z − +
z′ =z2 xz+3z+ 2v(x−2) + m − − y−µ
− v 1
− y y2 2xy+2µ(x 1)
2z(x 2) + − − . (27)
+ − +3Ω¯ , y µ
v 1 m p −
−
v′ = 2v(x 2). (23)
− −
The 4D phase space relevant for the cosmological 5 Boundaries between regions of phase portrait with different
model of interest in this case: topologyarecalledas“bifurcationsets”.
7
TABLE III: Properties of the equilibrium points of the autonomous system (26). We have defined the parameter y =
3/2+k/2 (4µ 9)/2k, where k=(2µ2 18µ+27+2 µ4 2µ3)1/3. ∗
− − − p −
Equilibrium Point x y z Existence Ω¯ Ω¯ ω q
m curv eff
P1 1 0 0 µ>0 0 1 1/3 0
−
P2 1 0 1 ” 1 0 1/3 0
− −
P3 1/2 0 3 ” 5/2 3/2 0 1/2
− −
P4 3 0 2 ” 4 3 5/3 2
− − −
P5± 2 2±√4−2µ −1 0<µ≤2 0 1 −1 −1
P6± 2 2±√4−2µ 3 0<µ≤2 −4 5 −1 −1
P7 2 y 0 k>0(µ>2.25) 0 1 1 1
∗ − −
TABLEIV:Eigenvaluesofthelinearization maticescorrespondingtothecritical pointsintableIII. Fortheequilibriumpoint
P7 theanalyticexpressionsoftheeigenvaluesareextremelyhugeandcomplex,sothatwehavedecidedtoshownumericresults
for several values of thecontrol parameter µ instead.
Equilibrium Point µ λ1 λ2 λ3
P1 µ>0 1 2 4
P2 ” 1 2 2
−
P3 ” 3 3/4+i√71/4 3/4 i√71/4
− − −
P4 ” 2 4 5
− − −
P5± 0<µ≤2 −2 −4 −4
P6± 0<µ≤2 4 6 −4
P7 5 2.1 4.03+0.78i 4.03 0.78i
− − −
10 3.26 4.63+0.8i 4.63 0.8i
− − −
15 4.19 5.1+0.55i 5.1+0.55i
− −
Theone-parameter,3Dphasespacespannedbythevari- 3. Equilibrium points
ables x,y,z – relevant for the study of the asymptotic
properties of the cosmological model (12,13) – can be Therearefoundeightequilibriumpointscorresponding
finally defined in the following way: totheautonomoussystemofODE(26)inthephasespace
Ψµ. Theirmostrelevantpropertiesarelistedinthetable
B
III, while the eigenvalues of the linearization matrices
correspondingto these points – invaluable as they areto
Ψµ = (x,y,z):y 0, y2 2xy+2µ(x 1) 0 . (28)
B { ≥ − − ≥ } judge about their stability –, are displayed in table IV.
Notice that the point P in Tab. III is not really a
3
fixed point of (26), since the bound (29) is not satisfied.
We want to emphasize that the bound
Actually, for P =(1/2,0, 3);
3
−
(x,y,µ)= µ<0,
B −
(x,y,µ) y2 2xy+2µ(x 1) 0, (29)
B ≡ − − ≥ since we are considering positive µ-s.
Fixed points P , P , and P are associated with in-
1 2 4
has to be necessarily satisfied in order for the density finitely large values of the Hubble parameter H ,
→ ∞
parameter to be a real quantity. while points P -P correspond to de Sitter cosmological
5 7
8
be explained here as a curvature effect (no phan-
tom fields at all). Point P is a future attractor
4
in Ψµ (this time not a global one, see below). It
B
corresponds to a singular state (infinite curvature
3
R ) that could be identified with a catas-
→ ∞
trophic fate inherent in a large class of phantom
2
models.
z(t) 1 Curvature-dominated (Ω¯curv = 1, Ω¯m = 0) equi-
• librium points P± – corresponding to de Sitter
5
inflationary phases of the cosmic expansion since
0 q = 1 – are also future attractors in Ψµ. These
− B
points exist whenever µ 2.
≤
–1
4 3.5 3 x2(.t5) 2 1.5 2y(t) 0 • tPiooninatrsy)P6e±xp–anasssiooncia–taedreaslasdodwleitfihxedde pSoititnetrs(ininflthae-
phase space (28). These can be linked with scal-
ing solutions since Ω¯ /Ω¯ = 4/5. The fact
m curv
−
10 10 thattheeffectivedimensionlessmatterenergyden-
8 8 sity parameter Ω¯m is a negative parameter, is not
aproblemandiseasilyexplainedasduetoourdef-
6 6
t t inition:
4 4
2 2 v ρ y ρ
Ω¯ = m = m .
04 3.5 3 x2(.t5) 2 1.5 2y(t) 0 04 3.5 3 x2(.t5) 2 1.5 2 z(t) 0 m (cid:18)v−1(cid:19)6H2fR (cid:18)y−µ(cid:19)6H2fR
Actually, for y < µ (v < 1) Ω¯ < 0. Points P±
m 6
exist only if µ 2.
FIG. 2: Drawing of the phase space to show the “multi- ≤
attractor” structure (µ = 1). This future asymptotic struc-
It is found, additionally, another de Sitter sad-
tureisbetterseenintheτ-flowdrawingsinthelower panels. • dle equilibrium configuration (point P ) in Ψµ.
7 B
The fixed point P can be associated also with a
7
curvature-dominatedsolution(Ω¯ =1, Ω¯ =0)
curv m
ofthecosmologicalequations(12),(13). Thisequi-
phases with (different) finite Hubble rates.
libriumpointexistswheneverthefollowinginequal-
The main asymptotic properties of the model with
ity holds:
f(R) = 2(1 √1 αR+β ) can be summarized as it
− − G
follows:
(2µ2 18µ+27+2 µ4 2µ3)1/3 >0.
− −
• Tcuhrevateuqruei-ldibormiuinmatepdoi(nΩ¯t P1=is1,aΩs¯soci=ate0d), wnoitnh- Numerical investigatiopns show that this bound is
curv m equivalenttothefollowingrestrictiononthevalues
inflationary dynamics as long as q = 0. It is the
past attractor in Ψµ. the control parameter µ can take: µ>2.25.
B
We want to emphasize several interesting features of
Another non-inflationary phase can be associated
• the asymptotic structure of the model (12,13). Perhaps
with the fixed point P , corresponding to matter
2
domination (Ω¯ = 1, Ω¯ = 0). It is a saddle the most interesting fact is the “multi-attractor” struc-
m curv
equilibrium point in Ψµ. ture into the future of the τ-development of (26) for val-
B uesofthecontrolparameterµ 2. Inthiscasethereco-
• PT3he– siscaalsinsogcisaotleudtiwonithΩ¯dme/cΩe¯lceurravted=ex−p5a/n3sio–np(oqin=t eaxnidstPt5−hr)e.eOfuthtuerrweiastet,rfaocrtothrsis(einq≤tueirlivbarliuomf copnotinrotsl pPa4r,aPm5+e-,
1/2). The effective “curvature” fluid mimics dust ter space there is no a unique global late-time attractor.
(w =0). However,asalreadynoticedthisisnot Anyway, judging by the magnitudes of the eigenvalues,
eff
a fixed point of (26) since it does not satisfy the convergenceofphasespacetrajectoriestowardsthepoint
bound (29). P4 is stronger than towards points P5±.
Notoflessimportanceistheexistenceofasaddleequi-
Super-inflationarydynamicscanbeassociatedwith librium point correlated with matter dominance – point
• the equilibrium point P in Tab III (also a scaling P (Ω¯ = 1, Ω¯ = 0). The existence of this point is
4 2 m curv
solutionsinceΩ¯ /Ω¯ = 4/3). Actually,inthis crucial to explain the formation of structure. Existence
m curv
−
case w = 5/3< 1 (q = 2). Otherwise, the of an inflationary de Sitter saddle equilibrium point P
eff 7
− − −
effectivefluidmimicsphantombehavior,whichcan could be important to explain early time inflation.
9
4 4
2
3 2
2
1.5
1.5
z(t) 0.51 z(t) 0.51 2 0 0.2 0.4 0.6 0.8 m1u 1.2 1.4 1.6 1.8 2
0 0 0.1 1 –2
–0.5 3 –0.5 0 y(t)
0.1 0.05 y(0t) –0.05 –0.11 x2(t) 1 1.5 x2(t) 2.5 3 –0.1 0 0.2 0.4 0.6 0.8 m1u 1.2 1.4 1.6 1.8 2 –4
3
4 2 FIG. 4: Plots of the phase space coordinate y vs µ for the
23 t–011 eriqguhitlibprainueml iptoiisntsshoPw5±naandplPot6±o(fµth≤e 2e)ffe–ctlievfet pdaimneeln.siIonnltehses
t–011 –––423 hmaalft)t,eranddenPsi6±ty(pnaegraamtiveetehraΩ¯lfm). vAstµµf=or2ppooinintstsPP5±5+(apnodsitPiv5−e
–2 3 –5 0 0.5 z(t)1 1.5 3 2 (also P6+ and P6−) coincide, so that there are 7 equilibrium
–3 0.1 0.05 y(0t) –0.05 –0.11 x2(t) x(t) 1 patoianltls(ionnΨlyµB5inesqtueialidbroifu9m. Fpoorinµts>ar2ethfoeusendp)o.inTtshdisoinsoatcelxeiasrt
illustration of the bifurcation in thecontrol parameter.
FIG. 3: Probe paths in phase space originated
from different sets of initial conditions for f(R) =
µκ22=(cid:16)14−β/p3α12−hαaκs2Rbe+enβcGh(cid:17)o.sen toTbheeµ =con5trosol thpaatrapmoeintetsr tcrloascetrortopotihnets:rePa5d+er=),(2P,−3.4=1,−(21,)0.(6b,un1c)h(obfutnrcahjecttoortihees
5 −
P5± and P6± do not exist. The flux in time is shown in the center of the drawing), and P4 = (3,0,2) (set of trajec-
lower panels. The trajectories in ΨµB converge towards the tories at the end). This is more evident in the τ-flow
global (“super-inflationary”) future attractor – equilibrium diagramsinthe lowerpartofthefigure. Inthe leftpanel
point P4 = (3,0,2) in Tab. III. The past attractor is the the convergence towards three different points is appar-
curvature-dominated (non-inflationary) solution – point
ent. Inthe rightpanelonly twopointsseemto be points
P1 =(1,0,0) in Tab. III.
of convergence of the phase trajectories. Note, however,
that in this diagram only the coordinate axes x and z
are displayed. Hence, since the points P+ and P− differ
5 5
only in the coordinate y, in this drawing they appear as
Last but not least: the super-inflationary attractor
a single convergencepoint (the one closerto the reader).
point P (a global future attractor if µ > 2) represents
4
In figure 3 a different value of the control parameter
another example of how to mimic catastrophic phantom
µ=5has beenchosen. Inthis case,sinceµ>2,equilib-
behavior (w < 1) without any phantom fields. It is
eff − rium points P± (also P±) do not occur. Consequently
evidentthatinthemodelcrossingofthephantomdivide 5 6
thereisonlya(global)futureattractor: thepointP . As
is possible (it is in fact unavoidable for µ>2). 4
seenfromFig. 3(upperpart)probepathsinΨµ converge
B
into the past (decreasing τ) towards the past attractor
(equilibrium point P ), while these converge into the fu-
1
4. Bifurcations
ture towards the attractor point P . In the right panel
4
a view from a different angle is shown. This asymptotic
To get additional information on the asymptotic dy- structureisconfirmedbytheτ-flowdiagramsinthelower
namical properties of the model under study one can part of the figure.
rely on drawings of the phase space, where probe paths The above figures (Fig. 2 and Fig. 3) illustrate the
evolved from given (arbitrary) initial data probe the bifurcation in respect to the control parameter µ. De-
phase space and, eventually, uncover the nature of the pending on the magnitude of the parameter µ there are
relevant equilibrium points. Here, as in the former sub- 9 (µ < 2), 7 (µ = 2), or 5 (µ > 2) equilibrium points
section, we make a combined use of the standard tools in the phase space Ψµ. Therefore, µ = 2 is the critical
B
of the (linear) dynamical systems analysis and of phase value or the bifurcation point.
space drawings. This time the phase space drawingswill In the figure Fig. 4 we show bifurcation diagrams y
serve as a graphic illustration of what it has been said vs µ – left panel, and Ω¯ vs µ – right panel, at the
m
in the former subsection (what is shown in tables Tab. equilibrium points P± and P±. It is apparent that as
5 6
III and Tab. IV). They clearly illustrate, in particular, µ 2 y 2, points P+ and P− tend to be a single
→ ⇒ → 5 5
bifurcations in respect to the control parameter µ. equilibrium point (P =(2,2, 1)). The same is true for
5
In the upper part of figure 2 probe trajectories origi- the points P+ and P−: at the−critical value µ =2 these
6 6
nated from different initial data, with control parameter are a single fixed point P = (2,2,3). In this case the
6
µ = 1, converge (increasing τ) into three different at- four equilibrium points P± and P± reduce to just two
5 6
10
equilibrium points P and P . For values µ > 2 (not stability properties, co-exist together: i) the inflationary
5 6
drawn in the figure) even P and P are erased from the de Sitter phase linked with the saddle fixed point P ;
5 6 7
phase space.
2k
In the present model bifurcations in the control pa- H2 = ,
3α(k2+3k 4µ+9)
rameterspacedemonstrate the strongdependence ofthe
−
dynamic properties of the model on the values of the
wherek (2µ2 18µ+27+2 µ4 2µ3)1/3,andii)the
overallparameters α and β. ≡ − −
super-inflationary ghost-like phase (w < 1, q = 2)
eff
p − −
– point P – with infinite Hubble rate, is encouraging
4
as well since, one can try to accommodate both early
IV. DISCUSSION
and late time inflationary stages into a united picture,
whereinflationisdrivenbycurvatureeffects. Infact,the
Knowledgeoftheequilibriumpointsinthephasespace latter solution can be associated with late-time speedup
corresponding to a given cosmological model is a very of the cosmic expansion, while the former one might be
important information since, independent on the initial linked with early inflation as long as, due to its stabil-
conditions chosen, the orbits of the corresponding au- ity properties, it is a transient phase. By appropriately
tonomous system of ODE will always evolve for some choosing the value of the free parameter µ 3α2/4β,
time in the neighborhoodof these points. Besides, if the one can manage to produce the necessary am≡ount of in-
point were a stable attractor, independent on the initial flationatearlytimes,besides,sinceP isasaddlecritical
7
conditions, the orbits will always be attracted towards point, then, exit from this inflationary stage is natural.
it (either into the past or into the future). Going back Actually, fueled by the stability properties of the cor-
to the original cosmological model, the existence of the responding cosmological phase, the dynamics forces the
equilibriumpointscanbe correlatedwithgenericcosmo- model to leave the saddle critical point P to, eventu-
7
logicalsolutions thatmightreally decide the fate and/or ally, approach to the global late-time super-inflationary
the origin of the cosmic evolution. In a sense the knowl- attractor P (recall that we are considering µ > 2.25,
4
edgeoftheasymptoticpropertiesofagivencosmological so that the de Sitter attractors P± do not exist).6 For
5
modelismorerelevantthantheknowledgeofaparticular µ 2, the possibility to explain early and late time in-
analyticsolutionofthecorrespondingcosmologicalequa- fla≤tionina united scheme is basedonthe co-existenceof
tions. Whileinthelatercaseonemightevolvethemodel thesuper-inflationaryattractorP andoftheinflationary
4
fromgiveninitialdatagivingaconcretepicturethatcan (de Sitter) saddle critical points P±.7
6
be tested againstexisting observationaldata, the knowl- The existence of the non-inflationary, curvature dom-
edge of the asymptotic properties of the model gives the inated equilibrium point P = (1,0,0), being always the
1
powertorealizewhichwillbethegenericbehaviorofthe past attractor in the phase space, allows us to associate
modelwithoutsolvingthecosmologicalequations. Inthe thestartingpointofeveryprobepathinthe phasespace
dynamicalsystemslanguage,forinstance,agivenpartic- withastatecharacterizedbyinfinitelylargevaluesofthe
ular solution of the Einstein’s equations is just a single Hubble parameter, a state usually linked with the big
pointinthephasespace. Hence,phasespaceorbitsshow bangsingularity. Therefore,contrarytowhatoneshould
thewaythemodeldrivesthecosmologicalevolutionfrom expect, the initial singularity is a generic feature of the
one particular solution into another one. Equilibrium model sourced by the action (1).8 In case such a singu-
points in the phase space will correspond to solutions of larity were removed by the non-linear DBI dynamics, as
the cosmological (Einstein’s) equations that, in a sense, expected,thestartingpointofthecosmicdynamicswere
arepreferredbythemodel, i. e.,aregeneric. Thelackof to be linked with a regular solution resembling a finite
equilibrium points that could be correlated with a given big-bang event, which does not seem to be the case. It
analytic solution of the model, amounts to say that this has to be pointed out, however, that, even if the generic
solution is not generic, otherwise it can be attained un- behavior is associated with a big-bang singularity, there
der a verycarefullyarrangementofthe initial conditions
only.
The asymptotic structure of the model (12,13) (or
(8,9)) is extremely rich and complex. Contrary to what 6 Similar results about the possibility to have both inflationary
is believed (see, for instance, Ref. [19]), there are found quintessentialandsuper-inflationaryghost-likephasesinamodel
in this kind of f(R, )-theory, equilibrium points that havebeenobtainedformoreelaboratedmodels[20].
G 7 Notice that, even if there co-exist de Sitter attractors – points
can be associated with matter-dominated and matter-
curvature scaling solutions. This is a very nice feature P5±, and de Sitter solutions associated with saddle points P6±,
there is no room for a united description of early and late time
that makes the model attractive to do cosmology, since, inflationarystagesofthecosmicexpansioninthiscase,sincethe
by appropriately arranging the free parameters, there is Hubblerateisthesameforbothasymptoticstages.
room to accommodate the amount of structure obser- 8 This kind of statements has to be taken with caution since the
dynamical systems analysis uncovers the asymptotic properties
vations confirm it exists. The fact that, for µ > 2.25
ofthedynamics. Hence,forinstance,theasymptoticstatewhere
(see existence conditions in Tab. III), two inflationary H2blowsupmightbereachinaninfinitetime,inwhichcaseone
solutions with different Hubble rates, and with different wouldnotwanttosaythatthespace-timeisproperlysingular.