Table Of ContentQuintessence arising from exponential potentials
T. Barreiro,E. J. Copeland and N. J. Nunes
Centre for Theoretical Physics, University of Sussex, Falmer, Brighton BN1 9QJ, U. K.
(February 1, 2008)
Wedemonstratehowtheproperties oftheattractorsolutions ofexponentialpotentialscan lead to
models of quintessence with the currently observationally favored equation of state. Moreover, we
showthatthesepropertiesholdforawiderangeofinitialconditionsandfornaturalvaluesofmodel
parameters.
PACS numbers: 98.80.Cq SUSX-TH-99-016 astro-ph/9910214
1
I. INTRODUCTION fluid. However,the latter regime does not provide a fea-
0
0 sible scenario either, as there is a tight constrainton the
2 Measurements of the redshift-luminosity distance re- allowed magnitude of ΩQ at nucleosynthesis [17,18]. It
n lation using high redshift type Ia supernovae combined turnsoutthatitmustsatisfyΩQ(1MeV)<0.13. Onthe
a with cosmic microwave background (CMB) and galaxy otherhand,wemustallowtimeforformationofstructure
J clusters data appear to suggest that the present Uni- before the Universe starts accelerating. For this scenario
6 verseisflatandundergoingaperiodofΛdriveninflation, tobepossiblewewouldhavetofinetunetheinitialvalue
1 withtheenergydensitysplitintotwomaincontributions, of ρQ, but this is precisely the kind of thing we want to
2 fiΩnmdaitntegr h≈as1n/a3tuarnadllyΩΛled≈th2e/o3ris[1ts–3t]o. pSruocphosae estxaprltalninag- avoAidn.umber of authors have proposed potentials which
v
tions for such a phenomenon. One such possibility that will lead to Λ dominance today. The initial sugges-
4
1 has attracted a great deal of attention is the suggestion tion was an inverse power law potential (“trackertype”)
2 thataminimallycoupledhomogeneousscalarfieldQ(the V Q−α [5,12,19], which can be found in models of
∝
0 “quintessence” field), slowly rolling down its potential, supersymmetric QCD [20,21]. Here the ratio of energy
1 could provide the dominant contribution to the energy densities is no longer a constant but ρQ scales slower
9 density today thanks to the specialformof the potential than ρB (the background energy density) and will even-
9 [4,5]. Non-minimally coupled models have also been in- tually dominate. This epoch can be set conveniently
/
h vestigated [6–11]. The advantage of considering a more to be today by tuning the value of only one parame-
p generalcomponentthatevolvesintimesoastodominate ter in the potential. However, although appealing, these
o- the energy density today, as opposed to simply insert- models suffer in that their predicted equation of state
r ing the familiar cosmological constant is that the latter wQ = pQ/ρQ is marginally compatible with the favored
st would require a term ρΛ 10−47 GeV4 to be present at values emerging from observationsusing SNIa and CMB
a allepochs,arathersmall≈valuewhencomparedtotypical measurements, considering a flat universe [22–24]. For
v: particle physics scales. On the other hand, quintessence example,atthe2σconfidencelevelintheΩM wQplane,
−
i modelspossessattractorsolutionswhichallowforawide the data prefer wQ < 0.6 with a favored cosmological
X −
range of initial conditions, all of which can correspond constantwQ = 1(seee.g.[24]),whereasthevaluesper-
−
r to the same energy density today simply by tuning one mitted by these tracker potentials (without fine-tuning)
a
overallmultiplicative parameter in the potential. have wQ > 0.7 [25]. For an interpretation of the data
−
There is a long history to the study of scalarfield cos- which allows for wQ < 1 see Ref. [26].
−
mology especially related to time varying cosmological Since this initial proposal, a number of authors have
constants. Some of the most influential early work is made suggestions as to the form the quintessence poten-
to be found in Refs. [12–14]. One particular case which tial could take [27–33]. In particular, Brax and Mar-
at first sight appears promising is the one involving ex- tin [28] constructed a simple positive scalar potential
ponential potentials of the form V exp(λκQ), where motivated from supergravity models, V exp(Q2)/Qα,
κ2 = 8πG [12–19]. These have two∝possible late-time and showed that even with the conditio∝n α 11, the
≥
attractors in the presence of a barotropic fluid: a scal- equation of state could be pushed to wQ 0.82, for
≈ −
ing regime where the scalar field mimics the dynamics ΩQ = 0.7. A different approach was followed by the au-
of the background fluid present, with a constant ratio thors of [30,33]. They investigated a class of scalar field
between both energy densities, or a solution dominated potentials where the quintessence field scalesthroughan
by the scalar field. The former regime cannot explain exponential regime until it gets trapped in a minimum
the observed values for the cosmologicalparameters dis- withanon-zerovacuumenergy,leadingtoaperiodofde
cussed above; basically it does not allow for an acceler- Sitter inflation with wQ 1.
→−
ating expansion in the presence of a matter background In this Brief Report we investigate a simple class of
potentials which lead to striking results. Despite previ-
1
ous claims, exponential potentials by themselves are a
promising fundamentaltool to build quintessence poten-
tials. Inparticular,weshowthatpotentialsconsistingof
sums of exponential terms can easily deliver acceptable 1.2 Observational constraints −0.80
models of quintessence in close agreement with observa- 1 −0.85
tions for natural values of parameters.
−0.90
0.8
β nd
II. MODEL 0.6 ou −0.95
b
s
si
0.4 he
We first recall some of the results presented in nt
y
[14,17,18]. Consider the dynamics of a scalar field Q, 0.2 eos wQ= −0.99
withanexponentialpotentialV exp(λκQ). Thefieldis ucl
∝ N
evolving in a spatially flat Friedmann-Robertson-Walker 0
0 5 10 15 20 25 30
(FRW) universe with a background fluid which has an α
equation of state p = w ρ . There exists just two FIG.1.
B B B
possible late time attractorsolutionswith quite different Contour plot of wQ(today) as a function of (α,β), with
properties, depending on the values of λ and wB: the constraint ΩQ(today) ≈ 0.7. The region α <
(1) λ2 > 3(w + 1). The late time attractor is 5.5 is excluded because of the nucleosynthesis bound,
B
one where the scalar field mimics the evolution of the ΩQ(1MeV) < 0.13, and the upper region due to 1σ ob-
barotropic fluid with w = w , and the relation Ω = servational constraints.
Q B Q
3(w +1)/λ2 holds.
B
(2)λ2 <3(w +1). Thelatetimeattractoristhescalar
B thenalltheparametersbecomeoftheorderofthePlanck
fielddominatedsolution(Ω =1)withw = 1+λ2/3.
Q Q − scale. Since the scaling regime of exponential potentials
Given that single exponentialterms canleadto one of
does not depend upon its mass scale [i.e. M in Eq. (1)],
the above scaling solutions, then it should follow that a
Aisactuallyafreeparameterthatcan,forsimplicity,be
combinationof the above regimes should allow for a sce-
set to M or even to zero. On the other hand, just like
Pl
nariowhere the universe canevolvethrougha radiation-
before for M, B needs to be such that today we obtain
matter regime (attractor 1) and at some recent epoch
the right value of ρ . In other words, we require M4
Q
evolve into the scalar field dominated regime (attractor M4e−βB ρ . This turns out to be B = (100)M ∼,
2). We will show that this does in fact occur for a wide Pl ∼ Q O Pl
depending on the precise values of α, β and A.
rangeofinitialconditions. Toprovideaconcreteexample
There is another important advantage to the poten-
consider the following potential for a scalar field Q:
tials of the form in Eq.(1) or Eq.(2); namely, we obtain
acceptable solutions for a wider range of initial energy
V(Q)=M4(eακQ+eβκQ), (1)
densitiesofthequintessencefieldthanwewouldwithsay
the inverse power law potentials. For example, in Fig. 2
where we assume α to be positive (the case α < 0 can
we show that it is perfectly acceptable to start with the
always be obtained taking Q Q). We also require
→ − energy density of the quintessence field abovethat of ra-
α > 5.5, a constraint coming from the nucleosynthesis
diation,andstillenterintoasubdominantscalingregime
bounds on Ω mentioned earlier [17,18].
Q
at later times; however, this is an impossible feature in
First, we assume that β is also positive. In order to
the context of inverse power law type potentials [25].
have an idea of what the value of β should be, note that
Another manifestation of this wider class of solutions
if today we were in the regime dominated by the scalar
can be seen by considering the case where the field evo-
field (i.e. attractor 2), then in order to satisfy observa-
lution began at the end of an initial period of inflation.
tional constraints for the quintessence equation of state
In that case, as discussed in Ref. [25], we could expect
(i.e. w < 0.8), we must have β < 0.8. We are not
Q
− that the energy density of the system is equally divided
obviously in the dominant regime today but in the tran-
amongallthethousandsofdegreesoffreedominthecos-
sition between the two regimes so this is just a central
mologicalfluid. Thisequipartitionofenergywouldimply
value to be considered. In Fig. 1 we show that accept-
that just after inflation Ω 10−3. If this were the case,
able solutions to Einstein’s equations in the presence of i
≈
for inverse power law potentials, the power could not be
radiation, matter and the quintessence field can be ac-
smaller than 5 if the field was to reach the attractor by
commodated for a large range of parameters (α, β).
matter domination. Otherwise, Q would freeze at some
ThevalueofM inEq.(1)ischosensothattodayρ
Q
ρ 10−47GeV4. This then implies M 10−31M ≈ valueandsimplyactasacosmologicalconstantuntilthe
10c−≈3eV. However,notethatifwegeneraliz≈ethepotenPtlia≈l present (a perfectly acceptable scenario of course, but
not as interesting). Such a bound on the power implies
in Eq. (1) to
w > 0.44 for Ω = 0.7. With an exponential term,
Q Q
−
V(Q)=M4(eακ(Q−A)+eβκ(Q−B)), (2) this constraint is considerably weakened. Using the fact
Pl
2
that the field is frozen at a value Q Q √6Ω /κ,
f i i
≈ −
where Q is the initial value of the field [25], we can see
i
that the equivalent problem only arises when
0.2
ρ
α 6Ωi 2lnα>ln Qi , (3) 0
p − ∼ (cid:18)2ρeq(cid:19)
−0.2
where ρQi is the initial energy density of the scalar field wQ
and ρ is the background energy density at radiation-
eq −0.4
matter equality. For instance, for our plots with a =
i
10−14, a =10−4, this results in a bound α<103. −0.6
eq
∼
−0.8
10
−1
−6 −5 −4 −3 −2 −1 0
log a
0
FIG.3.
The late time evolution of the equation of state for pa-
−10 rameters(α,β):dashedline(20,0.5);solidline(20, 20)
4V) and Ω 0.7. (a =1 today). −
e Q 0
G ≈
ρ (−20
g
o
l−30 In [25], a quantity Γ V′′V/(V′)2 is proposed as an
≡
indicatorofhowwellagivenmodelconvergestoatracker
−40 solution. Ifitremainsnearlyconstant,thenthesolutions
can converge to a trackersolution. It is easy to see from
Eq. (1) that apart from the transient regime where the
−50
−14 −12 −10 −8 −6 −4 −2 0
log a solutionevolvesfromattractor1 to attractor2, Γ=1 to
FIG.2. a high degree of accuracy.
Plot of the energy density, ρ , for α = 20, β = 0.5 Itisimportanttonotethatforthismechanismtowork,
Q
and severalinitial conditions admitting an Ω =0.7 flat we are not limited to potentials containing only two ex-
Q
universe today. The solid line represents the evolution ponential terms and one field. Indeed, all we require of
which emerges from equipartition at the end of inflation the dynamics is to enter one period like regime 1, which
and the dotted line represents ρ +ρ . caneitherbefollowedbyoneregimelike2,orbythefield
matter radiation
settling in a minimum with a non-zero vacuum energy.
A new feature arises when we consider potentials of We can consider as an example the case of a potential
theformgiveninEq.(1)withthenucleosynthesisbound depending on two fields of the form
α > 5.5 but taking this time β < 0. In this case the
potential has a minimum at κQmin =ln( β/α)/(α β) V(Q1,Q2)=M4(eα1κQ1+α2κQ2 +eβ1κQ1+β2κQ2), (4)
with a corresponding value V =M4β−−α( β)α/(α−−β).
min β −α whereallthecoeficientsarepositive. Thisleadstosimilar
Far from the minimum, the scalar field scales as de-
results to Eq.(1) for a single field Q, with effective early
scribed above (attractor 1). However, when the field
andlateslopesgivenbyα2 =α2+α2andβ2 =β2+β2,
reachestheminimum,theeffectivecosmologicalconstant eff 1 2 eff 1 2
respectively. Sucharesultisnotsurprisingandiscaused
V will quickly take over the evolution as the oscilla-
min bytheassistedbehaviorthatcanoccurformultiplefields
tions are damped, driving the equation of state towards
[35]. Note that for this type of multiple field examples
w = 1. ThisscenarioisillustratedinFig.3,wherethe
Q − theeffectiveslopesinthe resultingeffectivepotentialare
evolutionoftheequationofstateisshownandcompared
larger than the individual slopes, a useful feature since
tothepreviouscasewithβ >0. Inmanywaysthisisthe
we require α to be large.
key result of the paper, as in this figure it is clearly seen eff
that the field scales the radiation (w = 1/3) and mat-
ter (w = 0) evolutions before settling in an accelerating
III. DISCUSSION
(w < 0) expansion. Once again, as a result of the scal-
ing behavior of attractor 1, it is clear that there exists
a wide range of initial conditions that provide realistic So far, we have presented a series of potentials that
results. The feature resembles the recent suggestions of can lead to the type of quintessence behavior capable
Albrecht and Skordis [30]. The same mechanism can be of explaining the current data arising from high redshift
used to stabilize the dilaton in string theories where the typeIasupernovas,CMBandclustermeasurements. The
minimum of the potential is fine-tuned to be zero rather beautiful property of exponential potentials is that they
than the non-zero value it has in these models [34].
3
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