Interacting Agegraphic Dark Energy

0.19 MB·

by  Hao Wei

arXiv:0707.4052
Interacting Agegraphic Dark Energy
Hao Wei∗
Department of Physics, Beijing Institute of Technology, Beijing 100081, China
Department of Physics and Tsinghua Center for Astrophysics,
Tsinghua University, Beijing 100084, China
Rong-Gen Cai†
Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China
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0 ABSTRACT
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A new dark energy model, named “agegraphic dark energy”, has been proposed recently, based
n
on the so-called K´arolyh´azy uncertainty relation, which arises from quantum mechanics together
a
J with general relativity. In this note, we extend the original agegraphic dark energy model by
including the interaction between agegraphic dark energy and pressureless (dark) matter. In the
0
interacting agegraphic dark energy model, there are many interesting features different from the
1
original agegraphic dark energy model and holographic dark energy model. The similarity and

difference between agegraphic dark energy and holographic dark energy are also discussed.
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t PACSnumbers: 95.36.+x,98.80.Qc,98.80.-k
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∗emailaddress:[email protected]
†emailaddress:[email protected]
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I. INTRODUCTION
Dark energy [1] has been one of the most active fields in modern cosmology since the discovery of
accelerated expansion of our universe [2, 3, 4, 5, 6, 7, 8, 9]. The simplest candidate of dark energy is a
tinypositivecosmologicalconstant. However,asiswellknown,itisplaguedbytheso-called“cosmological
constant problem” and “coincidence problem” [1]. The cosmological constant problem is essentially a
problem in quantum gravity, since the cosmological constant is commonly considered as the vacuum
expectation value of some quantum fields. Before a completely successful quantum theory of gravity is
available, it is more realistic to combine quantum mechanics with general relativity directly.
Following the line of quantum fluctuations of spacetime, in Refs. [11, 12, 17], by using the so-called
K´arolyh´azy relation δt=λt2p/3t1/3 [10] and the well-known time-energy uncertainty relation Eδt3 t−1,
∼
it was argued that the energy density of metric fluctuations of Minkowski spacetime is given by
ρ Eδt3 1 m2p. (1)
q ∼ δt3 ∼ t2t2 ∼ t2
p
We use the units h¯ = c = k = 1 throughout, whereas l = t = 1/m with l , t and m being
B p p p p p p
the reduced Planck length, time and mass, respectively. It is worth noting that in fact the K´arolyh´azy
relation and the corresponding energy density (1) have been independently rediscovered later for many
times in the literature (see e.g. [30, 31, 32]).
In [17], one of us (R.G.C.) proposeda new dark energy model basedon the energy density (1). As the
most natural choice, the time scale t in Eq. (1) is chosen to be the age of our universe
a da
T = , (2)
0 Ha
Z
whereaisthescalefactorofouruniverse;H a˙/aistheHubbleparameter;adotdenotesthederivative
≡
with respect to cosmic time. Therefore, we call it “agegraphic” dark energy. The energy density of
agegraphic dark energy is given by [17]
3n2m2
p
ρ = , (3)
q T2
2
where the numerical factor 3n is introduced to parameterize some uncertainties, such as the species of
quantum fields in the universe, the effect of curved spacetime (since the energy density is derived for
Minkowski spacetime), the non-saturation of the quantum fluctuations, and so on. Obviously, since the
presentage of the universeT0 H0−1 (the subscript “0”indicates the presentvalue ofthe corresponding
∼
quantity; we set a0 = 1), the present energy density of the agegraphic dark energy explicitly meets the
observed value, provided that the numerical factor n is of order unity. In addition, by choosing the age
of the universe rather than the future event horizon as the length measure, the drawback concerning
causality in the holographic dark energy model does not exist in the agegraphic dark energy model [17].
Considering a flat Friedmann-Robertson-Walker (FRW) universe with agegraphic dark energy and
pressureless matter, the corresponding Friedmann equation reads
1
2
H = (ρ +ρ ). (4)
3m2 m q
p
It is convenient to introduce the fractional energy densities Ω ρ /(3m2H2) for i = m and q. From
i ≡ i p
Eq. (3), it is easy to find that
n2
Ω = , (5)
q H2T2
whereas Ω = 1 Ω from Eq. (4). By using Eqs. (3), (5) and the energy conservation equation
m q
−
ρ˙ +3Hρ =0, we obtain the equation of motion for Ω [17],
m m q
2
Ω′ =Ω (1 Ω ) 3 Ω , (6)
q q − q − n q
(cid:18) (cid:19)
p
3
where a prime denotes the derivative with respect to the e-folding time N lna. From the energy
≡
conservation equation ρ˙ +3H(ρ + p ) = 0, as well as Eqs. (3) and (5), it is easy to find that the
q q q
equation-of-state parameter (EoS) of the agegraphic dark energy, w p /ρ , is given by [17]
q q q
≡
2
w = 1+ Ω . (7)
q q
− 3n
p
Obviously, the EoS of the agegraphic dark energy is always larger than 1, and cannot cross the so-
−
called phantom divide w = 1. The total EoS w p /ρ = Ω w . To accelerate the expansion
de tot tot tot q q
of our universe, w < 1/3 i−s necessary. Thus, n > 2≡Ω3/2(3Ω 1)−1 follows. It is easy to see that if
tot q q
Ω > 1/3 (nb. Ω 0.7−today), the minimum of 2Ω3/2(3Ω 1)−−1 is 1 at Ω = 1. Therefore, n > 1 is
q q q q q
≃ −
necessary to drive the (present) accelerated expansion of our universe.
Inaddition,itisofinteresttocompareEqs.(6)and(7)withtheonesoftheholographicdarkenergy[13,
15, 18]. Obviously, they are fairly similar. Of course, there are some differences. Except for the slight
differencesofthenumericalconstant,themostimportantdifferenceisthesignbeforetheterm√Ω (the
de
subscriptde=qandΛfortheagegraphicdarkenergyandholographicdarkenergy,respectively). Infact,
this sign is opposite in these two models. We will see that this difference brings about some interesting
features to the agegraphic dark energy.
In fact, soon after the appearance of [17], it is found that the agegraphic dark energy model cannot
have a matter-dominated phase if n > 1 and if there is no interaction between the dark components in
the universe [33]. In [33, 34], a so-called “new agegraphic dark energy” model was proposed to remove
the inconsistency by replacing the time scale T in Eq. (3) with the conformal time η. Of course, there
existotherwaysoutofthedifficultyintheoriginalagegraphicdarkenergymodel(seee.g. Sec.2of[33]).
Therefore, it is still worthwhile to study the original version of the agegraphic dark energy model. In
this note, we will see that the interaction between the original agegraphic dark energy and pressureless
(dark)mattercansignificantlychangethecosmologicalevolution. Thus,theinconsistencyintheoriginal
version without interaction can also be removed in the interacting agegraphic dark energy model.
II. INTERACTING AGEGRAPHIC DARK ENERGY
Inthisnote,weextendtheoriginalagegraphicdarkenergymodelbyincludingtheinteractionbetween
agegraphic dark energy and pressureless (dark) matter. Given the unknown nature of both dark energy
and dark matter, it seems very peculiar that these two major components in the universe are entirely
independent [27, 28]. In fact, the models with interaction between dark energy and dark matter have
been studied extensively in the literature. For a complete list of references concerning the interacting
dark energy models, we refer to e.g. [19, 20, 29] and references therein.
We assume that the agegraphic dark energy and pressureless (dark) matter exchange energy through
an interaction term Q, namely
ρ˙ +3H(ρ +p )= Q, (8)
q q q
−
ρ˙ +3Hρ =Q, (9)
m m
which preserves the total energy conservation equation ρ˙ + 3H(ρ +p ) = 0. In this work, we
tot tot tot
consider the three most familiar forms of interaction [19, 20, 27, 28, 29], namely
Q=3αHρ , 3βHρ , 3γHρ , (10)
q m tot
where α, β and γ are dimensionless constants. In fact, the interaction forms in Eq. (10) are given by
hand. Although agegraphic dark energy is the quantum fluctuation of spacetime, it might decay into
matter, similar to the Λ(t)CDM model in which the vacuum fluctuations can decay into matter. This
effect could be described by the interaction term Q phenomenologically. From Eq. (5), we get
H˙ 2
Ω′ =Ω 2 Ω . (11)
q q − H2 − n q!
p
4
Differentiating Eq. (4) and using Eqs. (9), (3) and (5), it is easy to find that
H˙ 3 Ω3/2 Q
q
= (1 Ω )+ . (12)
− H2 2 − q n − 6m2H3
p
Therefore, we obtain the equation of motion for Ω ,
q
2 Q
Ω′ =Ω (1 Ω ) 3 Ω , (13)
q q − q − n q − 3m2H3
(cid:20) (cid:18) (cid:19) p (cid:21)
p
where
3αΩ for Q=3αHρ
Q q q
= 3β(1 Ω ) for Q=3βHρ . (14)
3m2H3  − q m
p 3γ for Q=3γHρ
 tot
If Q=0, Eq. (13) reduces to Eq. (6). From Eqs. (8), (3) and (5), we get the EoS of the agegraphicdark
energy, namely
2 Q
w = 1+ Ω , (15)
q q
− 3n − 3Hρ
q
p
where
α for Q=3αHρ
3HQρq =βγΩ(cid:0)Ω−q−q11−1(cid:1) ffoorr QQ==33βγHHρρtmqot . (16)
Again, if Q=0, Eq. (15) reduces to Eq. (7). Using Eq. (12), the deceleration parameter is given by
a¨a H˙ 1 3 Ω3/2 Q
q
q = 1 = Ω + , (17)
≡−a˙2 − − H2 2 − 2 q n − 6m2H3
p
wherethelasttermcanbefoundfromEq.(14). ThetotalEoSw p /ρ =Ω w ,wherew isgiven
tot tot tot q q q
≡
in Eq. (15). On the other hand, from the Friedmann equation and the Raychaudhuri equation, we have
w = 1 2 H˙ = 1/3+2q/3. As mentioned above, in the case of Q = 0 (i.e. without interaction),
tot − − 3H2 −
n > 1 is necessary to drive the (present) accelerated expansion of our universe. In the case of Q = 0,
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the situation is changed. For example, if Q = 3αHρ , to drive the (present) accelerated expansion of
q
our universe, we should have w =Ω w < 1/3, which means that n>2Ω3/2[3(1+α)Ω 1]−1. It is
tot q q q q
easy to see that the minimum of the right ha−nd side of this inequality is (1+α)−3/2 at Ω =−(1+α)−1,
q
if Ω > [3(1+α)]−1 (nb. Ω 0.7 today). For α >0, this minimum (1+α)−3/2 is smaller than 1. For
q q
≃
instance, if the present Ωq0 =0.7, to drive the accelerated expansion of our universe today, n>0.89414
is enough for α=0.1. In other words, n can be smaller than 1 to drive the accelerated expansion of our
universe in the case of Q=0. We will see this point explicitly in the following (e.g. Figs. 4 and 5).
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To get illustrations for the behaviors of Ω , w , q and w , we show some numerical plots by using
q q tot
Eqs.(13)—(17)andw =Ω w . However,tobebrief,wedonotpresentplotsforallformsofinteraction
tot q q
Q. Inthefollowing,wemainlyfocusonthecaseofQ=3αHρ asanexample. Notethatinthenumerical
q
integration of Eq. (13) we use the initial condition Ωq0 =0.7 for demonstration.
InFig.1,weshowtheevolutionofΩ fordifferentmodelparametersnandαinthecaseofQ=3αHρ .
q q
It is easy to see that for the fixed α which describes the interaction between the agegraphic dark energy
andthe pressureless(dark)matter, the agegraphicdarkenergystartsto be effectiveearlierandΩ tends
q
to a lower value at the late time when n is smaller. On the other hand, for fixed n, the agegraphic
dark energy starts to be effective earlier and Ω tends to a lower value at the late time when α is larger.
q
Interestinglyenough,these behaviorsareexactlyoppositetothe onesoftheinteractingholographicdark
energy model [21]. As mentioned above, this is mainly due to the opposite sign before √Ω in the
de
equation of motion for Ω (the subscript de=q and Λ for the agegraphic dark energy and holographic
de
dark energy respectively).
5
0.9 0.9
0.8 Α=0.1 0.8 n=1.2
0.7 0.7
0.6 0.6
Wq Wq
0.5 n=0.9 0.5 Α=0.0
n=1.2 Α=0.1
0.4 n=2.0 0.4 Α=0.2
0.3 0.3
0.5 0.75 1 1.25 1.5 1.75 2 0.5 0.75 1 1.25 1.5 1.75 2
a a
FIG. 1: Evolution of Ωq for various model parameters n and α in thecase of Q=3αHρq.
-0.4 n=0.9 -0.4 Α=0.0
-0.6 n=1.2 -0.6 Α=0.1
Α=0.2
wq n=2.0 wq-0.8
-0.8
-1
-1 n=1.2
Α=0.1
-1.2
0.5 1 1.5 2 0.5 1 1.5 2
a a
FIG. 2: Evolution of wq for various model parameters n and α in thecase of Q=3αHρq.
InFig.2,weshowtheevolutionofw fordifferentmodelparametersnandαinthecaseofQ=3αHρ .
q q
Obviously, the EoS of the agegraphic dark energy w can cross the phantom divide w = 1. In the
q de
−
case of Q = 0 (i.e. without interaction), as mentioned above, w is always larger than 1 and cannot
q
−
cross the phantom divide. By the help of interaction Q=0 between the agegraphicdark energy and the
6
pressureless (dark) matter, the situation is changed. From Eq. (15) with the first line of Eq. (16), it is
easytounderstandthatw convergestothevalue 1 αattheearlytimeinthecaseofQ=3αHρ . Of
q q
− −
course,the mostinteresting observationfromFig.2 is that w crossesthe phantomdivide fromw < 1
q q
−
to w > 1. This makes it distinguishable from many other dark energy models whose w can cross
q de
−
the phantom divide. It can be categorized into the so-called Quintom B type model, in the terminology
of[22,23]. Tomakethispointmorerobust,wealsoplottheevolutionofw forvariousmodelparameters
q
n and γ in the case of Q = 3γHρ . The results are presented in Fig. 3. Clearly, the observation that
tot
w crosses the phantom divide from below to above still holds. By the way, the w tends to at
q q
−∞
early time for γ =0; this is due to the last term in Eq. (15) with the last line of Eq. (16) in the case of
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Q=3γHρ . Itisanticipatedthatthe behaviorofw inthecaseofQ=3βHρ issimilartothe caseof
tot q m
Q=3γHρ ,sincethe lasttermsintheversionsofEq.(15)forthesetwocasesaresimilar[cf. Eq.(16)].
tot
It is worth noting that these results are for the cases of positive α, β and γ. In the cases of negative α,
β and γ, from Eq. (15) with Eq.(16), one cansee that w is alwayslargerthan 1 and cannot cross the
q
−
phantom divide. Obviously, the cases of positive α, β and γ are more interesting since the w can cross
q
the phantom divide from w < 1 to w > 1.
q q
− −
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-0.4 n=0.9 -0.4 Γ=0.0
-0.6 n=1.2 -0.6 Γ=0.1
-0.8 n=2.0 -0.8 Γ=0.15
-1 -1
wq wq
-1.2 -1.2
-1.4 -1.4
-1.6 Γ=0.1 -1.6 n=1.2
-1.8 -1.8
0.5 1 1.5 2 0.5 1 1.5 2
a a
FIG. 3: Evolution of wq for various model parameters n and γ in thecase of Q=3γHρtot.
One of the benefits of w > 1 at late time in the interacting agegraphic dark energy model is that
q
−
the universe can avoid the big rip singularity [24, 25]. Of course, the direct condition for the avoidance
of big rip should be w > 1 instead. Since w = Ω w and 0 Ω 1, the condition w > 1 is
tot tot q q q tot
− ≤ ≤ −
automatically satisfiedwhen w > 1. This can be seenclearly from the plot of the evolutionof w for
q tot
−
various model parameters n and α in the case of Q=3αHρ for example, which is shown in Fig. 4.
q
The other thing one can see from Fig. 4 is that w > 1/3 at the early time and w < 1/3 at
tot tot
− −
the late time. This implies thatthe universeundergoesdeceleratedexpansionatthe earlytime andlater
starts accelerated expansion. To see this point clearly, we show the deceleration parameter q in Fig. 5.
Obviously, q crosses the boundary q = 0 from q > 0 to q < 0. Some remarks on Figs. 4 and 5 are in
order. First,the similaritybetweenthesetwofiguresisduetotherelationw = 1/3+2q/3mentioned
tot
−
above. Second,attheearlytimew andq convergeto0and1/2,respectively;thisisbecauseΩ canbe
tot q
neglectedatthe earlytime, whereasthe universe is dominatedby the pressureless(dark)matter. This is
in the case of Q=3αHρ . In the cases of Q=3βHρ and Q=3γHρ , however, from Eqs. (17), (15)
q m tot
andw =Ω w , onecanseethatatthe earlytime w andq convergetoother constantsratherthan0
tot q q tot
and 1/2. For instance, in the case of Q=3βHρ , at the early time w β and q 1/2 3β/2. In
m tot
→− → −
the case of Q=3γHρ , at early time w γ and q 1/2 3γ/2. Third, for fixed n, the universe
tot tot
→− → −
starts accelerated expansion earlier when α is larger (see the right panels of Figs. 4 and 5). Fourth, the
universewillundergoacceleratedexpansionatthelatetime foreverandcannotcomebacktodecelerated
expansion,asshowninFigs.4and5. Afterall,inthecaseofQ=3αHρ ,wenoticethatforα=0.1,the
q
universe can undergo acceleratedexpansion for n=0.95<1. One can see this point from Figs. 4 and 5.
As mentioned above, this is impossible in the case of Q = 0 (i.e. without interaction). The interaction
Q=0 changes the situation.
6
Before the end of this work, we would like to mention another interesting feature of the interacting
agegraphic dark energy model. The equation of motion for Ω , Eq. (13), can be viewed as an one-
q
dimensional dynamical system [26]. The critical points of this autonomous equation are determined by
Ω′ =0. They are Ω =0 and the solutions of the equation
q q
2 Q
(1 Ω ) 3 Ω = , (18)
− q − n q 3m2H3
(cid:18) (cid:19) p
p
wheretherighthandsideisgiveninEq.(14). InthecaseofQ=0(i.e.withoutinteraction),thephysical
solution of Eq. (18) is only Ω = 1, whereas the other solution Ω = 3n/2 > 1 is unphysical because
q q
n > 1 is required by the accelerated expansion of our universe, as mentioned above. Thus, there is no
p
scalingsolutioninthecasewithoutinteraction. Again,thissituationischangedinthecasesofQ=0. For
instance,inthecaseofQ=3βHρ ,thecriticalpointsareΩ =0,1and9n2(1 β)2/4. Noteth6atinthe
m q
−
case of Q=0, n>1 is not necessary to drive the accelerated expansion of our universe. We can choose
appropriat6e model parameters n and β to ensure 0<Ω =9n2(1 β)2/4=const.<1. Thus, there is a
q
−
7
0.1
0
0
Α=0.1 n=1.2
-0.1
-0.1
-0.2 -0.2
wtot-0.3 n=0.95 wtot-0.3
-0.4
n=1.2 Α=0.0
-0.4
-0.5
Α=0.1
-0.6 n=2.0 -0.5 Α=0.2
0.5 1 1.5 2 0.5 1 1.5 2
a a
FIG. 4: Evolution of wtot for various model parameters n and α in thecase of Q=3αHρq.
0.4 Α=0.1 0.4 n=1.2
0.2
0.2
q 0 n=0.95 q
n=1.2 0
-0.2 Α=0.0
Α=0.1
-0.4 n=2.0 -0.2
Α=0.2
0.5 1 1.5 2 0.5 1 1.5 2
a a
FIG.5: EvolutionofthedecelerationparameterqforvariousmodelparametersnandαinthecaseofQ=3αHρq.
scalingsolution. Inthe casesofQ=3αHρ andQ=3γHρ , the situationis similar. The solutionsare
q tot
fairly complicated and we do not present them here, since Eq. (18) is a cubic equation of Ω in these
q
twocases. Infact, the flattails ofsomecurvesinFigs.1—5 perhapshint the scalingsolutionsinthe late
p
time. ThescalingsolutionsinthecasesofQ=0canhelptoalleviatethecoincidenceproblem. Asiswell
6
known, for a dynamical system [26], the universe will enter the attractors in the late time, regardless of
the initial conditions. If the attractors are scaling solutions, both Ω and Ω = 1 Ω are fixed values
q m q
−
overthere. Ifnandα,β,γ areoforderunity,itisnotsurprisingthatΩ andΩ =1 Ω arecomparable
q m q
−
at the late time. In fact, this is just the essential point of the literature (see e.g. [19, 20, 27, 28, 29]) to
alleviate (rather than solve) the coincidence problem using the method of dynamical system [26].
III. CONCLUDING REMARKS
Insummary,wehaveextendedthe agegraphicdarkenergymodelby includingthe interactionbetween
the agegraphic dark energy and the pressureless (dark) matter. The original agegraphic dark energy
model was proposed in [17] based on the K´arolyh´azy uncertainty relation, which arises from quantum
mechanics together with general relativity. In the interacting agegraphic dark energy model, there are
many interesting features differentfrom the originalagegraphicdarkenergy modeland holographicdark
8
energy model. In the cases with interaction Q = 0, the parameter n > 1 is no longer necessary to drive
6
the accelerated expansion of our universe; the EoS of agegraphic dark energy can cross the phantom
divide, whereas the big rip can be avoided; the universe undergoes decelerated expansion at early time
and then starts accelerated expansion later; there are scaling solutions which can help to alleviate the
coincidence problem. In particular, the difficulty in the original version of the agegraphic dark energy
model [17] can be avoided here, thanks to the interaction between the dark components.
Itisofinteresttodiscussthesimilarityanddifferencebetweenagegraphicdarkenergyandholographic
dark energy. It is shown that the agegraphic dark energy naturally obeys the holographic black hole
entropybound[12,17],justlikeholographicdarkenergy. Bychoosingtheageoftheuniverseratherthan
the future event horizon as the length measure, the drawback concerning causality in the holographic
dark energy model does not exist in the agegraphic dark energy model [17]. It is worth noting that
the agegraphic energy density Eq. (1) is similar to the one of holographic dark energy [13, 14, 15, 16],
i.e., ρΛ ∼ lp−2l−2. The similarity between ρq and ρΛ might reveal some universal features of quantum
gravity, although they arise in different ways. In addition, the sign before the term √Ω (the subscript
de
de = q and Λ for agegraphic dark energy and holographic dark energy, respectively) is opposite in
the equation of motion for Ω , the EoS of the agegraphic dark energy w , the total EoS w and the
q q tot
deceleration parameter q. This difference brings about some interesting features to the agegraphic dark
energydifferentfromtheonesofholographicdarkenergy. Insomesense,therelationbetweenagegraphic
dark energy and holographic dark energy is similar to the one between phantom and quintessence.
Finally, some remarks are in order. First, we admit that a sufficiently strong interaction might be
required to relax the condition n > 1 for an accelerated expansion and to allow that w crosses the
q
phantomdivide. However,astronginteractionmightencounterfairlytightconstraintsfromlocalgravity
tests. Second, after the appearance of our relevant works on the (new) agegraphic dark energy, it was
found that the original agegraphic dark energy model proposed in [17] is difficult to reconcile with the
big bang nucleosynthesis (BBN) constraint [35]. On the other hand, as shown in [36], the situation is
better in the new agegraphic dark energy model [33, 34]. In addition, the (new) agegraphic dark energy
modelfacestheproblemofinstabilities[37],while theholographicdarkenergymodelalsofacesthesame
problem [38]. Third, the quintessence reconstructions of the (new) agegraphic dark energy have been
studied in [39]. The statefinder diagnostic and w w′ analysis for the agegraphic dark energy models
−
wereperformedin[40]. Inaddition,the (new)agegraphicdarkenergywasextendedwiththe generalized
uncertainty principle in [41]. Furthermore,it was arguedthat the holographicdarkenergy models might
share the same origin with the (new) agegraphic dark energy models [42]. So, we consider the (new)
agegraphic dark energy model to deserve further investigation in future work.
ACKNOWLEDGMENTS
We are grateful to Prof. Miao Li, Prof. Shuang Nan Zhang and Prof. Yi Ling for helpful discussions.
WealsothankMinziFeng,aswellasHuiLi,YiZhang,XingWu,XinZhang,JingfeiZhang,Jian-PinWu,
Jian Wang and Bin Hu, for kind help and discussions. This workwas supported in part by a grant from
China Postdoctoral Science Foundation, a grant from Chinese Academy of Sciences (No. KJCX3-SYW-
N2), and by NSFC under grants No. 10325525,No. 10525060and No. 90403029.
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