Avoiding the dark energy coincidence problem with a cosmic vector
ABSTRACT We show that vector theories on cosmological scales are excellent candidates for dark energy. We consider two different examples, both are theories with no dimensional parameters nor potential terms, with natural initial conditions in the early universe and the same number of free parameters as LCDM. The first one exhibits scaling behaviour during radiation and a strong phantom phase today, ending in a "big-freeze" singularity. This model provides the best fit to date for the SNIa Gold dataset. The second theory we consider is standard electromagnetism. We show that a temporal electromagnetic field on cosmological scales generates an effective cosmological constant and that primordial electromagnetic quantum fluctuations produced during electroweak scale inflation could naturally explain, not only the presence of this field, but also the measured value of the dark energy density. The theory is compatible with all the local gravity tests, and is free from classical or quantum instabilities. Thus, not only the true nature of dark energy could be established without resorting to new physics, but also the value of the cosmological constant would find a natural explanation in the context of standard inflationary cosmology.
arXiv:0812.1970v1 [astro-ph] 10 Dec 2008
Avoiding the dark energy coincidence problem
with a cosmic vector
Jose Beltrán Jiménez and Antonio L. Maroto
Departamento de Física Teórica, Universidad Complutense de Madrid, 28040 Madrid, Spain
Abstract. We show that vector theories on cosmological scales are excellent candidates for dark
energy. We consider two different examples, both are theories with no dimensional parameters
nor potential terms, with natural initial conditions in the early universe and the same number of
free parameters as ΛCDM. The first one exhibits scaling behaviour during radiation and a strong
phantom phase today, ending in a "big-freeze" singularity. This model provides the best fit to date
for the SNIa Gold dataset. The second theory we consider is standard electromagnetism. We show
that a temporal electromagnetic field on cosmological scales generates an effective cosmological
constant and that primordial electromagnetic quantum fluctuations produced during electroweak
scale inflation could naturally explain, not only the presence of this field, but also the measured
value of the dark energy density. The theory is compatible with all the local gravity tests, and is
free from classical or quantum instabilities. Thus, not only the true nature of dark energy could be
established without resorting to new physics, but also the value of the cosmological constant would
find a natural explanation in the context of standard inflationary cosmology.
Keywords: Dark energy, cosmological vector fields
PACS: 95.36.+x, 98.80.-k, 98.80.Es
The fact that today matter and dark energy have comparable contributions to the energy
density, ρΛ∼ ρM∼ (2×10−3eV)4in natural units, poses one of the most important
problems for models of dark energy. Thus, if dark energy is a cosmological constant, its
energy density would remain constant throughout the history of the universe, whereas
those of the rest of components (matter and radiation) grow as we go back in time. Then
the question arises as to whether it is a coincidence (or not) that they have comparable
values today when they have differed by many orders of magnitude in the past. Notice
also that if Λ is a fundamental constant of nature, its scale (around 10−3eV) is more
than 30 orders of magnitude smaller than the natural scale of gravitation, G = M−2
with MP∼ 1019GeV. On the other hand, alternative models in which dark energy is a
dynamical component rather than a cosmological constant also require the introduction
of unnatural scales in their Lagrangians or initial conditions in order to account for the
present phase of accelerated expansion. Such models are usually based on new physics,
either in the form of new cosmological fields or modifications of Einstein’s gravity
[1, 2, 3, 4, 5] and they are generically plagued by additional problems such as classical
or quantum instabilities, or inconsistencies with local gravity constraints.
Therefore, we would like to find a description for dark energy without dimensional
scales (apart from Newton’s constant G), with the same number of free parameters as
ΛCDM, with natural initial conditions,with good fits to observations and no consistency
problems. In this work we will show that vector theories can do the job. With that
purpose, we present two examples of such theories which have been recently proposed
[6, 7] (for other vector models see references in those works).
SCALING VECTOR DARK ENERGY
The action in this case reads :
Notice that the theory contains no free parameters, the only dimensional scale being the
Newton’s constant. The numerical factor in front of the vector kinetic terms can be fixed
by the field normalization. Also notice that the "mass" term RµνAµAνcan be written as
a combination of derivative terms as ∇µAµ∇νAν−∇µAν∇νAµand therefore the theory
contains no potential terms. This action resembles that of Maxwell electromagnetism in
the Feynman gauge with a mass term.
The classical equations of motion derived from the action in (1) are the Einstein’s and
vector field equations:
where Tµνis the conserved energy-momentum tensor for matter and radiation and TA
is the energy-momentum tensor coming from the vector field. For the simplest isotropic
and homogeneous flat cosmologies, we assume that the spatial components of the vector
field vanish, so that Aµ= (A0(t),0,0,0) and that the space-time geometry will be given
For this metric (3) reads:
¨A0+3H˙A0−3?2H2+˙ H?A0= 0(5)
Assuming that the universe has gone through radiation and matter phases in which the
contribution from dark energy was negligible, we can easily solve this equation in those
periods. In that case, the above equation has a growing and a decaying solution:
A0(t) = A+
(−3±√33)/6 in the matter era. On the other hand, the (00) component of Einstein’s
0constants of integration and α±= −(1±1)/4 in the radiation era, and α±=
where the vector energy density is given by:
Figure 1: (Left) Evolution of energy densities for the best fit model. Dashed (red) for radiation, dotted
(green) for matter and solid (blue) for vector dark energy. We show also for comparison the cosmological
constant density in dashed-dotted line. (Right) Evolution of dark energy equation of state for the best fit
model. The lower panel shows the 1σ confidence interval.
Using the growing mode solution from (6), we obtain ρA= ρA0aκwith κ = −4 in
the radiation era and κ = (√33−9)/2 ≃ −1.63 in the matter era. Thus, the energy
density of the vector field starts scaling as radiation at early times, so that ρA/ρR=
const. However, when the universe enters its matter era, ρAstarts growing relative to
ρMeventually overcoming it at some point, in which the dark energy vector field would
becomethedominant component(see Fig. 1). Noticethat sinceA0is essentiallyconstant
during radiation era, solutions do not depend on the precise initial time at which we
specify it. Thus, once the present value of the Hubble parameter H0and the constant
A0during radiation (which fixes the total matter density ΩM) are specified, the model
is completely determined, i.e. this model contains the same number of parameters as
ΛCDM, which is the minimum number of parameters of a cosmological model with
dark energy. As seen from Fig.1 the evolution of the universe ends at a finite time tend
where a → aendwith aendfinite, A0(tend) = MP/(4√π), ρDE→ ∞ and pDE→ −∞. This
corresponds to a Type III (big-freeze) singularity according to the classification in .
We can also calculate the effective equation of state for dark energy as:
Again, using the approximate solutions in (6), we obtain: wDE=1/3 in the radiation era
and wDE≃ −0.457 in the matter era. As shown in Fig. 1, the equation of state can cross
the so called phantom divide, so that we can have wDE(z = 0) < −1.
In order to confront the predictions of the model with observations of high-redshift
supernovae type Ia, we have carried out a χ2statistical analysis for two supernovae
datasets, namely, the Gold set , containing 157 points with z < 1.7, and the more
recent SNLS data set , comprising 115 supernovae but with lower redshifts (z < 1).
Table 1: Best fit parameters with 1σ intervals for the vector model (VCDM) and the cosmological
constant model (ΛCDM) for the Gold (157 SNe) and SNLS (115 SNe) data sets. w0denotes the present
equation of state of dark energy. A0is the constant value of the vector field component during radiation.
zT is the deceleration-aceleration transition redshift. t0is the age of the universe in units of the present
Hubble time. tendis the duration of the universe in the same units.
In Table 1 we show the results for the best fit together with its corresponding 1σ
intervals for the two data sets. We also show for comparison the results for a standard
ΛCDM model. We see that the vector model (VCDM) fits the data considerably better
than ΛCDM (in more than 2σ) in the Gold set, whereas the situation is reversed in the
SNLS set. This is just a reflection of the well-known 2σ tension  between the two
data sets. Compared with ΛCDM, we see that VCDM favors a younger universe (in H−1
units) with larger matter density. In addition, the deceleration-acceleration transition
takes place at a lower redshift in the VCDM case. The present value of the equation of
state with w0= −3.53+0.46
Future surveys  are expected to be able to measure w0at the few percent level and
therefore could discriminate between the two models.
We have also compared with other parametrizations for the dark energy equation of
state . Since our one-parameter fit has a reduced chi-squared: χ2/d.o.f = 1.108,
−0.57which clearly excludes the cosmological constant value −1.
VCDM provides the best fit to date for the Gold data set.
We seethat unlikethecosmologicalconstantcase, throughoutradiation era ρDE/ρR∼
10−6inourcase. Moreoverthescaleofthevectorfield A0=3.71×10−4MPinthaterais
relativelyclose to the Planck scale and could arise naturally in the early universewithout
the need of introducing extremely small parameters (for instance in an inflationary
epoch), thus avoiding the coincidence problem.
In order to study the model stability we have considered the evolution of met-
ric and vector field perturbations. Thus, we obtain the dispersion relation and the
propagation speed of scalar, vector and tensor modes. For all of them we obtain
v = (1−16πGA2
does not exhibit exponential instabilities. As shown in , the fact that the propagation
speed is faster than c does not necessarily implies inconsistencies with causality. We
have also considered the evolution of scalar perturbations in the vector field generated
by scalar metric perturbations during matter and radiation eras, and, again, we do not
find exponentially growing modes.
If we are interested in extending the applicability range of the model down to so-
lar system scales then we should study the corresponding post-Newtonian parameters
(PPN). We can see that for the model in (10), the static PPN parameters agree with those
of General Relativity , i.e. γ = β = 1. For the parameters associated to preferred
frame effects we get: α1= 0 and α2= 8πA2
field at the solar system scale. Current limits α2<
fields during solar system formation) then impose a bound A2
der to determine whether such bounds conflict with the model predictions or not, we
should know the predicted value of the field at solar system scales, which in principle
does not need to agree with the cosmological value. Indeed, A2
the mechanism that generated this field in the early universe characterized by its pri-
mordial spectrum of perturbations, and the subsequent evolution in the formation of the
galaxy and solar system. Another potential difficulty arising generically in vector-tensor
modelsis the presence of negativeenergy modes for perturbations on sub-Hubblescales.
They are known to lead to instabilities at the quantum level, but not necessarily at the
classical level as we have shown previously. Work is in progress in order to determine
which are the necessary conditions to avoid the presence of such states in this model.
0)−1/2which is real throughout the universe evolution, since the value
0= (16πG)−1exactly corresponds to that at the final singularity. Therefore the model
⊙is the norm of the vector
∼10−7for static vector
P. In or-
⊙will be determined by
IS THE NATURE OF DARK ENERGY ELECTROMAGNETIC?
In the previous section, we have seen that a generic vector field whose action resembles
that of electromagnetism with a mass term could be a good candidate for dark energy,
but what about standard electromagnetism?. In a very recent work , it has been
shownusingthecovariant (Gupta-Bleuler)formalism that itis indeed possibleto explain
cosmic acceleration from the standard Maxwell’s theory.
We start by writing the standard electromagnetic action including a gauge-fixing term
in the presence of gravity:
The gauge-fixing term is required in order to define a consistent quantum theory for
the electromagnetic field , and we will see that it plays a fundamental role on
large scales. Still this action preserves a residual gauge symmetry Aµ→ Aµ+∂µφ with
2φ = 0. Electromagnetic equations derived from this action can be written as:
+ λ∇µ∇νAν= 0(11)
Notice that since we will be using the covariant Gupta-Bleuler formalism, we do not a
priori impose the Lorentz condition.
We shall first focus on the simplest case of a homogeneous electromagnetic field
Aµ= (A0(t),?A(t)) in a flat Robertson-Walker background. In this space-time, equations
¨A0+3H˙A0+3˙ HA0 = 0
¨?A+H˙?A = 0(12)
We can solve (12) during the radiation and matter dominated epochs when the Hubble
parameter is given by H = p/t with p = 1/2 for radiation and p = 2/3 for matter. In
such a case the solutions for (12) are:
A0(t) = A+
?A(t) = ?A+t1−p+?A−
component does not depend on the epoch being always proportional to the cosmic time
t, whereas the growing mode of the spatial component evolves as t1/2during radiation
and as t1/3during matter, i.e. at late times the temporal component will dominate over
the spatial ones.
The energy densities of the temporal and spatial components read:
0and?A±are constants of integration. Hence, the growing mode of the temporal
Notice that we need λ > 0 in order to have positive energy density for A0. In fact,
it is possible to show that imposing canonical normalization for the corresponding
creation and annihilation operators we get λ = 1/3 . Besides, when inserting the
growingmodes ofthefields intotheseexpressionsweobtain thatρA0=ρ0
and ∇µAµ= const. Thus, the field behaves as a cosmological constant throughout the
evolution of the universe since its temporal component gives rise to a constant energy
density whereas the energy density corresponding to?A always decays as radiation.
Moreover, this fact prevents the generation of a non-negligible anisotropy which could
spoil the highly isotropic CMB radiation. Finally, when the universe is dominated by
the electromagnetic field, both the Hubble parameter and A0become constant (one can
straightforwardly check that this is a solution of the complete system of equations) so
the energy density is also constant and the electromagnetic field behaves once again as
a cosmological constant leading therefore to a future de Sitter universe. As the observed
fraction of energy density associated to a cosmological constant today is ΩΛ≃ 0.7, we
obtain that the field value today must be A0(t0) ≃ 0.3MP.
The effects of the high electric conductivity σ can be introduced using the magneto-
hydrodynamical approximation and including the current term Ji= σ(∂0Ai−∂iA0) on
the r.h.s. of Maxwell’s equations. Notice that because of the universe electric neutrality,
conductivitydoes not affect the evolutionof A0(t).The infiniteconductivitylimitsimply
eliminates the growing mode of?A(t) in (14).
We still need to understand which are the appropriate initial conditions leading to the
present value of A0. In order to avoid the cosmic coincidence problem, such initial con-
ditions should have been set in a natural way in the early universe. In a very interesting
work , it was suggested that the present value of the dark energy density could be
related to physics at the electroweak scale since ρΛ∼ (M2
GeV. This relation offers a hint on the possible mechanism generating the initial am-
plitude of the electromagnetic fluctuations. Indeed, we see that if such amplitude is set
by the size of the Hubble horizon at the electroweak era, i.e. A0(tEW)2∼ H2
the correct scale for the dark energy density is obtained. Thus, using the Friedmann
equation, we find H2
A possible implementation of this mechanism can take place during inflation. Notice
that the typical scale of the dispersion of quantum field fluctuations on super-Hubble
scales generated in an inflationary period is precisely set by the almost constant Hubble
parameter during such period HI, i.e. ?A2
can then be naturally obtained if initial conditions for the electromagnetic fluctuations
are set during an inflationary epoch at the scale MI∼ MEW.
Despite the fact that the background evolution in the present case is the same as
in ΛCDM, the evolution of metric perturbations could be different, thus offering an
observational way of discriminating between the two models. In fact, the evolution of
the scalar perturbation Φkwith respect to the ΛCDM model gives rise to a possible
discriminating contribution to the late-time integrated Sachs-Wolfe effect . The
propagation speeds of scalar, vector and tensor perturbations are found to be real and
equal to the speed of light, so that the theory is classically stable. We have also checked
that the theory does not contain ghosts and it is therefore stable at the quantum level.
On the other hand, using the explicit expressions in  for the vector-tensor theory of
gravity corresponding to the action in (10), it is possible to see that all the parametrized
post-Newtonian (PPN) parameters agree with those of General Relativity, i.e. the theory
is compatible with all the local gravity constraints for any value of the homogeneous
background vector field .
EW/MP)4, where MEW∼ 103
EW/MP)4as commented before.
P, but according to (15), ρA0∼ H2A2
0∼ const., so that
0? ∼ H2
I. The correct dark energy density
The presence of large scale electric fields generated by inhomogeneitiesin theA0field
opens also the possibility for the generation of large scale currents which in turn could
light on the problem of explaining the origin of cosmological magnetic fields. Work is
in progress in this direction.
We have shown that vector theories offer a simple and accurate description of dark en-
ergy in which the coincidence problem could be easily avoided. In our first example, the
scaling behaviour during radiation and the natural initial conditions for the vector field
offer a neat way around the problem. Moreover, in our second example, the presence
of a cosmological electromagnetic field generated during inflation provides a natural
explanation for the cosmic acceleration. This result not only offers a solution to the
problem of establishing the true nature of dark energy, but also explains the value of
the cosmological constant without resorting to new physics. In this scenario the fact that
matter and dark energy densities coincide today is just a consequence of inflation taking
place at the electroweak scale. Present and forthcoming astrophysical and cosmologi-
cal observations will be able to discriminate these proposals from the standard ΛCDM
Acknowledgments: This work has been supported by DGICYT (Spain) project num-
bers FPA 2004-02602 and FPA 2005-02327, UCM-Santander PR34/07-15875 and by
CAM/UCM 910309. J.B. aknowledges support from MEC grant BES-2006-12059.
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