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arXiv:1006.0368v2 [astro-ph.CO] 6 Sep 2010
Dark energy from primordial inflationary quantum fluctuations
Christophe Ringeval∗
Institute of Mathematics and Physics, Centre for Cosmology,
Particle Physics and Phenomenology, Louvain University,
2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium
Teruaki Suyama†
Institute of Mathematics and Physics, Centre for Cosmology,
Particle Physics and Phenomenology, Louvain University,
2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium and
Research Center for the Early Universe, Graduate School of Science,
The University of Tokyo, Tokyo 113-0033, Japan
Tomo Takahashi‡
Department of Physics, Saga University, Saga 840-8502, Japan
Masahide Yamaguchi§
Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
Shuichiro Yokoyama¶
Department of Physics and Astrophysics, Nagoya University, Aichi 464-8602, Japan
(Dated: September 7, 2010)
We show that current cosmic acceleration can be explained by an almost massless scalar field
experiencing quantum fluctuations during primordial inflation. Provided its mass does not exceed
the Hubble parameter today, this field has been frozen during the cosmological ages to start dom-
inating the universe only recently. By using supernovae data, completed with baryonic acoustic
oscillations from galaxy surveys and cosmic microwave background anisotropies, we infer the energy
scale of primordial inflation to be around a few TeV, which implies a negligible tensor-to-scalar ratio
of the primordial fluctuations. Moreover, our model suggests that inflation lasted for an extremely
long period. Dark energy could therefore be a natural consequence of cosmic inflation close to the
electroweak energy scale.
PACS numbers: 98.80.Cq
INTRODUCTION
In the past decade, various cosmological observations
have accumulated evidence that the universe is currently
undergoing accelerated expansion [1–5]. Although cosmic
acceleration can be triggered by a non-vanishing cosmo-
logical constant, one still has to explain why it is so small
and why it has started to dominate the energy content
of the universe only recently [6]. These questions have
motivated the exploration of alternative explanations all
referred to as dark energy. Among them, the quintessence
models consider a scalar field rolling down a potential in
a way similar to the mechanism at work during primor-
dial inflation [7–9]. A quintessence field yields a time-
dependent equation of state w(t) = P/ρ > −1, P and
ρ being its pressure and energy density. A cosmologi-
cal constant being equivalent to w = −1, quintessence
models generically predict a different expansion history
of the universe, and this can be tested by observations.
According to Ref. [10], the quintessence models can be
divided into “freezing” and “thawing” types. The for-
mer yields a decreasing w(t) which approaches −1 at low
redshifts. This type of model has been intensively used
in the literature to address the coincidence problem. In-
deed, assuming the potential to be of the Ratra–Peebles
kind, the quintessence field tracks a cosmological attrac-
tor which erases memory of the initial conditions [11].
Once the field energy scale is adjusted to the current dark
energy value, its recent domination is automatic.
the other hand, the thawing type gives w ≃ −1 at high
redshifts but can evolve and start deviating from this
value at low redshifts, exactly as during the inflationary
graceful exit. These models have been less explored than
On
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the freezing type due to their dependence on the initial
field values. Initial conditions are not washed out by an
attractor mechanism and therefore constitute additional
and unwanted model parameters. In this letter, we show
that inflationary cosmology solves this problem by giv-
ing natural initial values for the quintessence field which
can explain the current acceleration. They are set by the
quantum generation of field fluctuations during inflation.
Assuming the accelerated expansion today to be sourced
by the quintessence field, we use the supernovae data to
infer the energy scale of inflation which ends up being
at a few TeV, i.e. close to the electroweak symmetry
breaking energy scale. Such a scenario implies a negligi-
ble tensor-to-scalar ratio of the primordial cosmological
perturbations and a reheating temperature also around a
few TeV. Combined with cosmic microwave background
(CMB) bounds, the class of allowed inflationary mod-
els is thus severely constrained [12]. Moreover, the total
number of e-folds of inflation has to be extremely large,
as it could be in a self-reproducing inflationary scenario.
Linking primordial inflation and dark energy has been
originally discussed in the context of anthropic selection
effects [13–16]. In these approaches, inflationary quan-
tum fluctuations are used to randomize the possible dark
energy density values while anthropic selection effects
favour the ones compatible with our own existence. In-
flationary stochastic effects in freezing-type quintessence
models have been explored in Ref. [17] to determine how
inflation influences the likely initial conditions of the
field. As shown in Ref. [18], they indeed have a tendency
to push the freezing type quintessence field away from the
region where the tracker behavior can efficiently wash out
the initial conditions. As a result, freezing quintessence
on the tracker today suggests that inflation lasted a low
number of e-folds.
In our model, the initial value of the thawing
quintessence field is directly determined by the en-
ergy scale of inflation and keeps its initial value until
low redshifts. Therefore today’s energy density of the
quintessence field is almost equal to its initial energy
density, and hence is directly related to the inflationary
energy scale. Checking the consistency of the model pro-
vides us a way to probe primordial inflation from dark
energy observations. This is our main point and an es-
sential difference from the freezing type quintessence sce-
nario.
INITIAL FIELD VALUES FROM INFLATION
Let us consider a (real) free massive scalar field ϕ of
mass m such that m is less than the present Hubble scale
H0.Clearly, since the inflationary Hubble parameter
Hinf ≫ H0, we are in the presence of an almost mass-
less scalar field which will acquire quantum fluctuations
during inflation. By decomposing the scalar field as a ho-
mogeneous mode plus linear perturbations (at the onset
of inflation),
ϕ(x,t) = φ(t) + δφ(x,t),(1)
and assuming an almost constant value for Hinf, its
Fourier modes after Hubble exit read [19–23]
|δφk|2≃H2
inf
2k3
?
k
aHinf
?2m2/(3H2
inf)
.(2)
The comoving wavevector is k and a stands for the scale
factor. If inflation starts at a = as, after N = ln(a/as) e-
folds, super-Hubble fluctuations induce a real space field
variance given by
?δφ2?≃
?aHinf
asHinf
3H4
8π2m2
d3k
(2π)3|δφk|2
=
inf
?
1 − exp
?
−2m2
3H2
inf
N
??
→
3H4
8π2m2,
inf
(3)
where the last limit is reached if inflation lasts long
enough for the exponential term to cancel.
sult can be reproduced using the stochastic inflation
formalism [24–27]. The homogeneous mode evolves as
φ = φsexp?−Nm2/(3H2
suppressed compared to the field fluctuations. The same
holds for the mean squared field derivative (in e-folds)
which is suppressed by a factor m4/H4
Eq. (3). For m ≪ Hinf the required number of e-folds
could actually be extremely large. Here, one should no-
tice that the long-wave fluctuation δφ is almost homo-
geneous and determines the typical value of the classical
field ϕ. Notice that backreaction effects induced by the
field fluctuations over the expansion rate can produce
O(1) variations in the total number of e-folds.
ever, one can show that they induce a correction factor
to Eq. (3) given by 1+O?H2
completely negligible as soon as Hinf ≪ MP. Since af-
ter inflation H ≫ m, Hubble damping prevents the field
from rolling down the potential until the time at which
m ≃ H ? H0. Before going into a detailed comparison
with observations, let us derive some order of magnitude
results. The present energy density of this quintessence
field is roughly
This re-
inf)?
and becomes completely
infcompared to
How-
inf/M2
P
?which ends up being
V (φ) ≃1
2m2?δφ2? ≃3H4
inf
16π2,(4)
and does not depend on m. In order to explain dark
energy, one needs
Hinf≃ (ΩΛ)1/4?4πH0MP,(5)
where MPstands for the reduced Planck mass and ΩΛis
the current density parameter associated with a cosmo-
logical constant. By using fiducial values for H0and ΩΛ,
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one gets Hinf ≃ 6 × 10−3eV [28]. The energy scale of
inflation is thus1
Einf≡ ρ1/4
inf=?3M2
PH2
inf
?1/4≃ 5TeV. (6)
As a result, in the context of our scenario, we rephrase the
question on the smallness of the dark energy density into
the relatively small energy scale (TeV) of inflation, which
may be more tractable to address and surprisingly close
to the electroweak energy scale. Notice that although the
mass m does not appear in Eq. (4) and (6), it still deter-
mines when the quintessence field starts rolling down the
potential and how the equation of state of dark energy
will deviate from w = −1. For TeV scale inflation, our
model predicts a negligible tensor-to-scalar ratio.
ing the above estimates, with m ? H0, the number of
e-folds required to reach the Bunch–Davies fluctuations
is N ≃ H2
self-reproducing inflationary model [29, 30]. In the fol-
lowing, we assume that N > H2
a large N, the homogeneous mode of the scalar field is
also suppressed significantly so that the typical value of
the classical field ϕ is completely determined by the long-
wave fluctuation δφ.
Us-
inf/m2? 1060, which may therefore suggest a
inf/m2> 1060. For such
OBSERVATIONAL CONSTRAINTS
In order to constrain our model from current observa-
tions, we have numerically solved the Einstein and Klein–
Gordon equations
H2=
1
3M2
P
(ρr+ ρm+ ρφ),
¨φ + 3H˙φ + m2φ = 0,
(7)
for various values of the parameters (m,φini,H0). The
value φinidenotes one realisation of the quintessence field
fluctuations in our Hubble patch and ρrand ρmare the
energy density associated with radiation and matter, re-
spectively. The integration is started at high redshift,
when the dark energy is sub-dominant, to the time at
which H = H0. We have fixed the current radiation
energy density to Ωrh2= 4.15 × 10−5and dropped the
decaying mode for the field. By solving the evolution
equations, we obtain H as a function of the redshift z
which can be used to compare the model with obser-
vational data. We have used type Ia supernovae (SN)
redshift-distance modulus relations given in Ref. [3] com-
plemented with the redshift-distance relations at z = 0.2
1However, we should keep in mind that this constraint is based
on the assumption that the cosmological constant is adjusted to
zero by some unknown mechanism.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5
φini/MP
6 7 8 9 10
m/(100km/Mpc/sec)
FIG. 1: One and two-sigma contours of the two-dimensional
marginalised probability distribution in the plane (φini,m)
from SN+BAO+CMB data.
and z = 0.35 coming from the baryon acoustic oscilla-
tions (BAO) data [31]. Finally, we have used the angu-
lar scale and height of the first peak of the CMB power
spectrum given in Ref. [28]. Our likelihood has therefore
been defined as the product of those three, namely by
summing the chi-squared of SN, BAO and CMB data.
In order to test how the model can fit cosmic acceler-
ation, let us first consider a Jeffreys’ prior for the initial
field values φini ≥ MP. Here we are not yet assuming
that the initial conditions are set during inflation, but
simply requiring super-Planckian field values (necessary
for triggering acceleration). Together with flat priors on
m ≤ 100km/s/Mpc and H0 around the current mea-
sured value, we have explored the three-dimensional pa-
rameter space with gridding methods. In Fig. 1, we have
represented the one and two-sigma contours of the two-
dimensional marginalised probability distribution (over
H0) in the plane (φini,m). By marginalising over φini, we
find the mass of the quintessence field to be constrained
by m < 75km/s/Mpc (at 95% of confidence), which is
consistent with our earlier estimation that it cannot ex-
ceed H0 too much. We also find, as expected, that the
data favour arbitrarily high super-Planckian initial field
values. In order to check our results, we have also derived
the field energy density parameter Ωφand its equation of
state wφ, both evaluated at the present time. The con-
tours plotted in Fig. 1 end up being centered around the
cosmological constant case Ωφ= 0.73, wφ= −1 and are
compatible with the results of Ref. [3].
We are now in a position to infer the energy scale of
inflation when the initial field values are set by infla-
tionary quantum fluctuations. Compared to the above
data analysis, our mechanism gives a prior probability
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4
6
8
10
12
14
16
18
20
0.1 0.2 0.3 0.4
m /(100km/Mpc/sec)
0.5 0.6 0.7 0.8 0.9 1
Einf/TeV
2 4 6 8 10
Einf/ TeV
12 14 16 18 20
posterior
FIG. 2: One and two-sigma contours in the plane (m,Einf)
from SN+BAO+CMB data.
marginalised posterior for Einf.
find 3.8TeV < Einf < 12.1TeV.
The lower panel shows the
At 95% of confidence, we
distribution of φini which is Gaussian2with a standard
deviation given by Eq. (3)
σ(Hinf,m) =
?3
8
H2
πm.
inf
(8)
Denoting by D our data sets, and I the prior space, using
Bayes’ theorem and marginalising over φiniand H0yields
P(Hinf,m|D,I) ∝ P(Hinf,m|I)
?
2σ2
??
dH0
dφini
√2π σ
× exp−φ2
ini
?
P(H0|I)L(D|m,φini,H0,I),
(9)
where L is the overall SN+BAO+CMB likelihood and
P(Hinf,m|I) the prior probability distribution on Hinf
2Notice that embedding our mechanism into an explicit eternal
inflationary model may change this prior, either due to volume
effects or by a choice of peculiar probability measure. The scale
factor cut-off measure might however preserve such a prior [15,
32–34].
and m. The scale of inflation being a priori unknown, we
have chosen a flat prior on the logarithm of Hinf while
the other parameters assume the same prior as before.
In Fig. 2, we have represented the one and two-sigma
confidence intervals of this two-dimensional posterior, up
to the change of variable Hinf → Einf. The lower panel
is the fully marginalised probability distribution for Einf
and the energy scale of inflation verifies
3.8TeV < Einf< 12.1TeV, (10)
at 95% of confidence. Though this constraint depends on
the priors on Hinf and m, the dependence is small. As
discussed below, our model works for a more general po-
tential, which can slightly change the constraint. Thus,
it is safe to say that our model suggests the energy scale
of inflation Einfto be around a few TeV for a wide class
of thawing quintessential models.
DISCUSSION
Although we have focused on a free scalar field so far,
the mechanism discussed here could also be applied to
any potential having an absolute minimum by using the
inflationary stochastic formalism [26]. However, the prior
probability distribution for the field φini will no longer
be Gaussian and this could therefore shift the favoured
values of Einf. Since our scenario needs an extremely
large e-folding number N ≃ H2
ning of Hinf along the inflaton potential must be ex-
tremely small. Thus, as a candidate of such a TeV scale
inflation, new inflation might be preferred, which can in-
deed realize a self-reproducing era. We also would like
to mention the case that the total number of e-folds is
much smaller than H2
Hinf/MP∝
bound Einf> TeV and would have to explain why the ho-
mogeneous value φs≃ 0 after inflation. Notice that CMB
anisotropies imposing Hinf/MP< 10−5, we get an abso-
lute lower limit for the total number of e-folds N > 109,
under which the scale of inflation would actually be too
high. Finally, it is worth stressing that we are in pres-
ence of a transient dark energy model. By rolling down
its potential, the thawing quintessence field will acquire
kinetic energy and the current cosmic acceleration will
come to an end.
We are grateful to Takeshi Chiba, Andr´ e F¨ uzfa,
KiyotomoIchiki, Masahiro
dama, J´ erˆ ome Martin, Shinji Mukohyama and Jun’ichi
Yokoyama for enlightening discussions. This work is par-
tially supported by the Belgian Federal Office for Scien-
tific, Technical, and Cultural Affairs through the Inter-
University Attraction Pole Grant No. P6/11 (C. R. and
T. S.); by the Grant-in-Aid for Scientific research from
the Ministry of Education, Science, Sports, and Culture,
inf/m2? 1060, the run-
inf/m2, in which case Eq. (3) yields
?ΩΛ/N. In this limit, we have only a lower
Kawasaki,HideoKo-
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Japan, No. 19740145(T. T.) and 21740187(M. Y.). T. S.
is supported by the JSPS. The work of S. Y. is supported
in part by Grant-in-Aid for Scientific Research on Prior-
ity Areas No. 467 “Probing the Dark Energy through
an Extremely Wide and Deep Survey with Subaru Tele-
scope”. He also acknowledges support from the Grant-
in-Aid for the Global COE Program “Quest for Funda-
mental Principles in the Universe: from Particles to the
Solar System and the Cosmos” from MEXT, Japan.
∗Electronic address: christophe.ringeval@uclouvain.be
†Electronic address: teruaki.suyama@uclouvain.be
‡Electronic address: tomot@cc.saga-u.ac.jp
§Electronic address: gucci@phys.titech.ac.jp
¶Electronic address: shu@a.phys.nagoya-u.ac.jp
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