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arXiv:astro-ph/9910214v2 16 Jan 2001
Quintessence 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)
We demonstrate how the properties of the attractor solutions of exponential potentials can lead to
models of quintessence with the currently observationally favored equation of state. Moreover, we
show that these properties hold for a wide range of initial conditions and for natural values of model
parameters.
PACS numbers: 98.80.Cq SUSX-TH-99-016 astro-ph/9910214
I. INTRODUCTION
Measurements of the redshift-luminosity distance re-
lation using high redshift type Ia supernovae combined
with cosmic microwave background (CMB) and galaxy
clusters data appear to suggest that the present Uni-
verse is flat and undergoing a period of Λ driven inflation,
with the energy density split into two main contributions,
Ωmatter ≈1/3 and ΩΛ≈2/3 [1–3]. Such a startling
finding has naturally led theorists to propose explana-
tions for such a phenomenon. One such possibility that
has attracted a great deal of attention is the suggestion
that a minimally coupled homogeneous scalar field Q(the
“quintessence” field), slowly rolling down its potential,
could provide the dominant contribution to the energy
density today thanks to the special form of the potential
[4,5]. Non-minimally coupled models have also been in-
vestigated [6–11]. The advantage of considering a more
general component that evolves in time so as to dominate
the energy density today, as opposed to simply insert-
ing the familiar cosmological constant is that the latter
would require a term ρΛ≈10−47 GeV4to be present at
all epochs, a rather small value when compared to typical
particle physics scales. On the other hand, quintessence
models possess attractor solutions which allow for a wide
range of initial conditions, all of which can correspond
to the same energy density today simply by tuning one
overall multiplicative parameter in the potential.
There is a long history to the study of scalar field cos-
mology especially related to time varying cosmological
constants. Some of the most influential early work is
to be found in Refs. [12–14]. One particular case which
at first sight appears promising is the one involving ex-
ponential potentials of the form V∝exp(λκQ), where
κ2= 8πG [12–19]. These have two possible late-time
attractors in the presence of a barotropic fluid: a scal-
ing regime where the scalar field mimics the dynamics
of the background fluid present, with a constant ratio
between both energy densities, or a solution dominated
by the scalar field. The former regime cannot explain
the observed values for the cosmological parameters dis-
cussed above; basically it does not allow for an acceler-
ating expansion in the presence of a matter background
fluid. However, the latter regime does not provide a fea-
sible scenario either, as there is a tight constraint on the
allowed magnitude of ΩQat nucleosynthesis [17,18]. It
turns out that it must satisfy ΩQ(1MeV) <0.13. On the
other hand, we must allow time for formation of structure
before the Universe starts accelerating. For this scenario
to be possible we would have to fine tune the initial value
of ρQ, but this is precisely the kind of thing we want to
avoid.
A number of authors have proposed potentials which
will lead to Λ dominance today. The initial sugges-
tion was an inverse power law potential (“tracker type”)
V∝Q−α[5,12,19], which can be found in models of
supersymmetric QCD [20,21]. Here the ratio of energy
densities is no longer a constant but ρQscales slower
than ρB(the background energy density) and will even-
tually dominate. This epoch can be set conveniently
to be today by tuning the value of only one parame-
ter in the potential. However, although appealing, these
models suffer in that their predicted equation of state
wQ=pQ/ρQis marginally compatible with the favored
values emerging from observations using SNIa and CMB
measurements, considering a flat universe [22–24]. For
example, at the 2σconfidence level in the ΩM−wQplane,
the data prefer wQ<−0.6 with a favored cosmological
constant wQ=−1 (see e.g. [24]), whereas the values per-
mitted by these tracker potentials (without fine-tuning)
have wQ>−0.7 [25]. For an interpretation of the data
which allows for wQ<−1 see Ref. [26].
Since this initial proposal, a number of authors have
made suggestions as to the form the quintessence poten-
tial could take [27–33]. In particular, Brax and Mar-
tin [28] constructed a simple positive scalar potential
motivated from supergravity models, V∝exp(Q2)/Qα,
and showed that even with the condition α≥11, the
equation of state could be pushed to wQ≈ −0.82, for
ΩQ= 0.7. A different approach was followed by the au-
thors of [30,33]. They investigated a class of scalar field
potentials where the quintessence field scales through an
exponential regime until it gets trapped in a minimum
with a non-zero vacuum energy, leading to a period of de
Sitter inflation with wQ→ −1.
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 fundamental tool to build quintessence poten-
tials. In particular, we show that potentials consisting of
sums of exponential terms can easily deliver acceptable
models of quintessence in close agreement with observa-
tions for natural values of parameters.
II. MODEL
We first recall some of the results presented in
[14,17,18]. Consider the dynamics of a scalar field Q,
with an exponential potential V∝exp(λκQ). The field is
evolving in a spatially flat Friedmann-Robertson-Walker
(FRW) universe with a background fluid which has an
equation of state pB=wBρB. There exists just two
possible late time attractor solutions with quite different
properties, depending on the values of λand wB:
(1) λ2>3(wB+ 1). The late time attractor is
one where the scalar field mimics the evolution of the
barotropic fluid with wQ=wB, and the relation ΩQ=
3(wB+ 1)/λ2holds.
(2) λ2<3(wB+1). The late time attractor is the scalar
field dominated solution (ΩQ= 1) with wQ=−1 + λ2/3.
Given that single exponential terms can lead to one of
the above scaling solutions, then it should follow that a
combination of the above regimes should allow for a sce-
nario where the universe can evolve through a radiation-
matter regime (attractor 1) and at some recent epoch
evolve into the scalar field dominated regime (attractor
2). We will show that this does in fact occur for a wide
range of initial conditions. To provide a concrete example
consider the following potential for a scalar field Q:
V(Q) = M4(eακQ +eβκQ ),(1)
where we assume αto be positive (the case α < 0 can
always be obtained taking Q→ −Q). We also require
α > 5.5, a constraint coming from the nucleosynthesis
bounds on ΩQmentioned earlier [17,18].
First, we assume that βis also positive. In order to
have an idea of what the value of βshould be, note that
if today we were in the regime dominated by the scalar
field (i.e. attractor 2), then in order to satisfy observa-
tional constraints for the quintessence equation of state
(i.e. wQ<−0.8), we must have β < 0.8. We are not
obviously in the dominant regime today but in the tran-
sition between the two regimes so this is just a central
value to be considered. In Fig. 1 we show that accept-
able solutions to Einstein’s equations in the presence of
radiation, matter and the quintessence field can be ac-
commodated for a large range of parameters (α, β).
The value of Min Eq. (1) is chosen so that today ρQ≈
ρc≈10−47GeV4. This then implies M≈10−31MPl ≈
10−3eV. However, note that if we generalize the potential
in Eq. (1) to
V(Q) = M4
Pl(eακ(Q−A)+eβκ(Q−B)),(2)
0 5 10 15 20 25 30
0
0.2
0.4
0.6
0.8
1
1.2
α
β
−0.99
−0.95
−0.90
−0.85
−0.80
wQ=
Observational constraints
Nucleosynthesis bound
FIG. 1.
Contour plot of wQ(today) as a function of (α, β), with
the constraint ΩQ(today) ≈0.7. The region α <
5.5 is excluded because of the nucleosynthesis bound,
ΩQ(1MeV) <0.13, and the upper region due to 1σob-
servational constraints.
then all the parameters become of the order of the Planck
scale. Since the scaling regime of exponential potentials
does not depend upon its mass scale [i.e. Min Eq. (1)],
Ais actually a free parameter that can, for simplicity, be
set to MPl or even to zero. On the other hand, just like
before for M,Bneeds to be such that today we obtain
the right value of ρQ. In other words, we require M4∼
M4
Ple−βB ∼ρQ. This turns out to be B=O(100)MPl ,
depending on the precise values of α,βand A.
There is another important advantage to the poten-
tials of the form in Eq. (1) or Eq. (2); namely, we obtain
acceptable solutions for a wider range of initial energy
densities of the quintessence field than we would with say
the inverse power law potentials. For example, in Fig. 2
we show that it is perfectly acceptable to start with the
energy density of the quintessence field above that of ra-
diation, and still enter into a subdominant scaling regime
at later times; however, this is an impossible feature in
the context of inverse power law type potentials [25].
Another manifestation of this wider class of solutions
can be seen by considering the case where the field evo-
lution began at the end of an initial period of inflation.
In that case, as discussed in Ref. [25], we could expect
that the energy density of the system is equally divided
among all the thousands of degrees of freedom in the cos-
mological fluid. This equipartition of energy would imply
that just after inflation Ωi≈10−3. If this were the case,
for inverse power law potentials, the power could not be
smaller than 5 if the field was to reach the attractor by
matter domination. Otherwise, Qwould freeze at some
value and simply act as a cosmological constant until the
present (a perfectly acceptable scenario of course, but
not as interesting). Such a bound on the power implies
wQ>−0.44 for ΩQ= 0.7. With an exponential term,
this constraint is considerably weakened. Using the fact
2
that the field is frozen at a value Qf≈Qi−√6 Ωi/κ,
where Qiis the initial value of the field [25], we can see
that the equivalent problem only arises when
αp6Ωi−2 ln α>
∼ln ρQi
2ρeq ,(3)
where ρQiis the initial energy density of the scalar field
and ρeq is the background energy density at radiation-
matter equality. For instance, for our plots with ai=
10−14,aeq = 10−4, this results in a bound α<
∼103.
−14 −12 −10 −8 −6 −4 −2 0
−50
−40
−30
−20
−10
0
10
log a
log ρ (GeV4)
FIG. 2.
Plot of the energy density, ρQ, for α= 20, β= 0.5
and several initial conditions admitting an ΩQ= 0.7 flat
universe today. The solid line represents the evolution
which emerges from equipartition at the end of inflation
and the dotted line represents ρmatter +ρradiation.
A new feature arises when we consider potentials of
the form given in Eq. (1) with the nucleosynthesis bound
α > 5.5 but taking this time β < 0. In this case the
potential has a minimum at κ Qmin = ln(−β/α)/(α−β)
with a corresponding value Vmin =M4β−α
β(−β
α)α/(α−β).
Far from the minimum, the scalar field scales as de-
scribed above (attractor 1). However, when the field
reaches the minimum, the effective cosmological constant
Vmin will quickly take over the evolution as the oscilla-
tions are damped, driving the equation of state towards
wQ=−1. This scenario is illustrated in Fig. 3, where the
evolution of the equation of state is shown and compared
to the previous case with β > 0. In many ways this is the
key result of the paper, as in this figure it is clearly seen
that the field scales the radiation (w= 1/3) and mat-
ter (w= 0) evolutions before settling in an accelerating
(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
results. The feature resembles the recent suggestions of
Albrecht and Skordis [30]. The same mechanism can be
used to stabilize the dilaton in string theories where the
minimum of the potential is fine-tuned to be zero rather
than the non-zero value it has in these models [34].
−6 −5 −4 −3 −2 −1 0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
log a
wQ
FIG. 3.
The late time evolution of the equation of state for pa-
rameters (α, β) : dashed line (20,0.5); solid line (20, −20)
and ΩQ≈0.7. (a0= 1 today).
In [25], a quantity Γ ≡V′′V /(V′)2is proposed as an
indicator of how well a given model converges to a tracker
solution. If it remains nearly constant, then the solutions
can converge to a tracker solution. It is easy to see from
Eq. (1) that apart from the transient regime where the
solution evolves from attractor 1 to attractor 2, Γ = 1 to
a high degree of accuracy.
It is important to note that for this mechanism to work,
we are not limited to potentials containing only two ex-
ponential terms and one field. Indeed, all we require of
the dynamics is to enter one period like regime 1, which
can either be followed by one regime like 2, or by the field
settling in a minimum with a non-zero vacuum energy.
We can consider as an example the case of a potential
depending on two fields of the form
V(Q1, Q2) = M4(eα1κQ1+α2κQ2+eβ1κQ1+β2κQ2),(4)
where all the coeficients are positive. This leads to similar
results to Eq. (1) for a single field Q, with effective early
and late slopes given by α2
eff =α2
1+α2
2and β2
eff =β2
1+β2
2,
respectively. Such a result is not surprising and is caused
by the assisted behavior that can occur for multiple fields
[35]. Note that for this type of multiple field examples
the effective slopes in the resulting effective potential are
larger than the individual slopes, a useful feature since
we require αeff to be large.
III. DISCUSSION
So far, we have presented a series of potentials that
can lead to the type of quintessence behavior capable
of explaining the current data arising from high redshift
type Ia supernovas, CMB and cluster measurements. The
beautiful property of exponential potentials is that they
3
lead to scaling solutions which can either mimic the back-
ground fluid or dominate the background dynamics de-
pending on the slope of the potential. We have used
this to develop a picture where by simply combining po-
tentials of different slopes, it is easy to obtain solutions
which first enter a period of scaling through the radiation
and matter domination eras and then smoothly evolve to
dominate the energy density today. We have been able
to demonstrate that the quintessence behavior occurred
for a wide range of initial conditions of the field, whether
ρQbe initially higher or lower than ρmatter +ρradiation.
We have also shown that the favored observational val-
ues for the equation of state wQ(today) <−0.8 can
be easily reached for natural values of the parameters
in the potential. This is a big improvement in respect
to most quintessence models as they usually give either
wQ>
∼−0.8 or wQ=−1.
We have to ask, how sensible are such potentials? Can
they be found in nature and, if so, can we make use of
them here? The answer to the first question seems to
be, yes they do arise in realistic particle physics models
[36–41], but the current models do not have the correct
slopes. Unfortunately, the tight constraint emerging from
nucleosynthesis, namely α > 5.5, is difficult to satisfy in
the models considered to date which generally have α≤
1. It remains a challenge to see if such potentials with
the required slopes can arise out of particle physics. One
possibility is that the desirable slopes will be obtained
from the assisted behavior when several fields are present
as mentioned above.
It is encouraging that the quintessence behavior re-
quired to match current observations occurs for such sim-
ple potentials.
ACKNOWLEDGMENTS
We would like to thank Orfeu Bertolami, Robert Cald-
well, Thomas Dent, Jackie Grant, Andrew Liddle, Jim
Lidsey and David Wands for useful discussions. E.J.C.
and T.B. are supported by PPARC. N.J.N. is sup-
ported by FCT (Portugal) under contract PRAXIS XXI
BD/15736/98. E.J.C is grateful to the staff of the Isaac
Newton Institute for their kind hospitality during the pe-
riod when part of this work was being completed.
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