Alleviation of Cosmic Age Problem In Interacting Dark Energy Model

Abstract

We investigate the cosmic age problem associated with the old high-z quasar APM 08279 + 5255 and the oldest globular cluster M 107, both being difficult to accommodate in Λ\LambdaCDM model. By evaluating the age of the Universe in a model that has an extremely phantom like form of dark energy (DE), we show that simply introducing the dark energy alone does not remove the problem, and the interaction between dark matter (DM) and DE need to be taken into account. Next, as examples, we consider two interacting DE models. It is found that both these two interacting DE Models can predict a cosmic age much greater than that of Λ\LambdaCDM model at any redshift, and thus substantially alleviate the cosmic age problem. Therefore, the interaction between DM and DE is the crucial factor required to make the predicted cosmic ages consistent with observations.

Full text

arXiv:0809.3627v2 [astro-ph] 23 Sep 2008
Alleviation of Cosmic Age Problem In Interacting Dark Energy
Model
S. Wang∗and Y. Zhang
Astrophysics Center
University of Science and Technology of China
Hefei, Anhui, China
Abstract
We investigate the cosmic age problem associated with the old high-zquasar APM
08279 + 5255 and the oldest globular cluster M 107, both being difficult to accom-
modate in ΛCDM model. By evaluating the age of the Universe in a model that has
an extremely phantom like form of dark energy (DE), we show that simply introduc-
ing the dark energy alone does not remove the problem, and the interaction between
dark matter (DM) and DE need to be taken into account. Next, as examples, we
consider two interacting DE models. It is found that both these two interacting DE
Models can predict a cosmic age much greater than that of ΛCDM model at any
redshift, and thus substantially alleviate the cosmic age problem. Therefore, the in-
teraction between DM and DE is the crucial factor required to make the predicted
cosmic ages consistent with observations.
PACS numbers: 95.36.+x, 98.80.Es, 98.80.-k
∗Email: swang@mail.ustc.edu.cn
1

1. Introduction
The cosmic age problem [1] has been a longstanding issue. For instance, to accommodate with two old
galaxies 53W091 at z= 1.55 [2] and 53W069 at z= 1.43 [3], the cosmological constant Λwas employed
[4], which at the same time can be the candidate for the dark energy [5] driving the currently accelerating
expansion of Universe [6, 7, 8]. However, the recent discovery of an old high-zquasar APM 08279 +
5255 with an age (2.0∼3.0) Gyr at redshift z= 3.91 [9] declares the come-back of the cosmic age
problem. With Hubble parameter h= 0.72 and fractional matter density Ωm= 0.27, the ΛCDM model
would give an age t= 1.6Gyr at z= 3.91, much lower than the age lower limit 2.0Gyr. This so-called
“high-zcosmic age problem” [10] have attracted a lot of attention [10, 11, 12, 13, 14, 15, 16, 17, 18],
where various DE models are directly tested against this quasar. So far there is no DE model that can
pass this test, and most of these models perform even poorer than ΛCDM model in solving this issue
[10, 11]. Besides, the existence of the oldest globular cluster M 107 with an age 15 ±2Gyr [19] is also
a puzzle. Using ΛCDM model, the latest results of WMAP5 show that the present cosmic age should be
t0= 13.73 ±0.12 Gyr [20], which is lower than the central value of the formation time of M 107.
One can look at this issue from a different perspective. Instead of studying the age constraints on
the specific DE Model directly, one may firstly study in general what kind of DE models will be helpful
to alleviate cosmic age problem. In seeking DE models performing better than ΛCDM model in solving
high-zcosmic age problem, we find that simply introducing the dark energy alone does not remove the
problem, the scaling law of matter component need be modified, and the interaction between DM and
DE need to be taken into account. As examples, we consider the age problem in two interacting DE
models: our previously proposed coupled YMC DE model [21, 22] and a interacting scalar DE model. It
is found that both these two interacting DE Models can predict a cosmic age much greater than that of
ΛCDM model at any redshift, and thus substantially alleviate the cosmic age problem. So one may say
that the interaction between DM and DE is the crucial factor required to make the predicted cosmic ages
consistent with observations.
The organization of this paper is as follows. In section 2, we briefly introduce the cosmic age problem.
Since the high-zquasar APM 08279 + 5255 is more difficult to accommodate than the globular cluster M
107, the age constraints given by this quasar will be mainly discussed. In seeking DE models performing
better than ΛCDM, we find that simply introducing dark energy alone does not remove the problem, and
the interaction between DM and DE need to be taken into account. In section 3, we consider the age
problem in coupled YMC model. It is seen that the coupled YMC model predicts a cosmic age much
greater than that of ΛCDM model at redshift z= 3.91, and amply accommodates the high-zquasar APM
08279 + 5255. Besides, the age constraints given by the globular cluster M 107 is also discussed, and
it is found that coupled YMC model can also substantially accommodate this object. In Section 4, we
2

Table 1: The value T(3.91) depends on Ωmin ΛCDM model.
Ωm0.27 0.24 0.21 0.17
T(3.91) 0.118 0.125 0.133 0.148
consider the age problem in a simple interacting scalar DE model, and find that this interacting scalar
model can also substantially alleviate the cosmic age problem. In Section 5, we compare our results with
previous works, and discuss the reason why introducing interaction between DM and DE would be helpful
to resolve cosmic age problem. In this work, we assume today’s scale factor a0= 1, so the radshift z
satisfies z=a−1−1. Throughout the paper, the subscript “0” always indicates the present value of the
corresponding quantity, and the unit with c= ¯h= 1 is used.
2. The cosmic age in non-interaction DE models
The age t(z)of a flat Universe at redshift zis given by [11]
t(z) = Z∞
z
d˜z
(1 + ˜z)H(˜z),(1)
where H(z) is the Hubble parameter. It is convenient to introduce a dimensionless age parameter [11]
T(z)≡H0t(z) = Z∞
z
d˜z
(1 + ˜z)f(˜z),(2)
where
f(z)≡H(z)/H0=sΩm
ρm(z)
ρm0
+ (1 −Ωm)ρy(z)
ρy0
,(3)
H0= 100 hkm ·s−1·Mpc−1, and ρm(z),ρy(z)are energy density of matter and dark energy, ρm0,ρy0
are their present values, Ωmis the present fractional matter density, respectively. Here the radiation
component has been ignored since its contribution to T(z)at z < 5is only ∼0.1%. Once a specific DE
model is given, f(z)will be known, and so will be T(z). For ΛCDM model, one has ρm(z)
ρm0= (1 + z)3,
ρy(z)
ρy0= 1, and
T(z) = Z∞
z
d˜z
(1 + ˜z)pΩm(1 + ˜z)3+ (1 −Ωm),(4)
which gives a larger age for a smaller Ωm. WMAP3 gives Ωm= 0.268 ±0.018 [23], WMAP5 gives
Ωm= 0.258 ±0.030 (Mean) [24], and SDSS gives Ωm= 0.24 ±0.02 [25]. For ΛCDM model, we list the
values of T(z)at redshift z= 3.91 for various Ωmin Table 1.
At any redshift, the Universe cannot be younger than its constituents. Since the high-zquasar APM
08279 + 5255 is more difficult to accommodate than the globular cluster M 107, the age constraints given
3

Table 2: The value Tqua depends on hfor fixed tqua = 2.0 Gyr.
h0.72 0.64 0.58 0.57
Tqua 0.147 0.131 0.119 0.117
by this quasar will be mainly discussed. For this quasar, one must have
T(3.91) ≥Tqua ≡H0tqua,(5)
where tqua is the age of the quasar at z= 3.91, determined by the chemical evolution. The exact value of
tqua is not fully determined: by a high ratio Fe/O from the X-ray result Ref.[9] gives tqua = (2.0∼3.0)
Gyr at z= 3.91, while Ref.[10] gives tqua = 2.1Gyr at the same redshift by a detailed chemidynamical
modeling. The constraint of Eq.(5) sensitively depends on the value of the Hubble constant h. Using the
HST key project, Freedman, et. al. [26] give h= 0.72 ±0.08, and Sandage, et. al. [27] advocate a
lower h= 0.623 ±0.063. WMAP3 gives h= 0.732 ±0.031 [23], and WMAP5 gives h= 0.701 ±0.013
[24]. Adopting tqua = 2.0Gyr, the values of Tqua for different hare listed in Table 2. It is seen that if
h= 0.64,ΛCDM model can not accommodate this quasar unless Ωm≤0.21; if h= 0.72,ΛCDM model
can not accommodate this quasar unless Ωm≤0.17. Since WMAP3 [23], WMAP5 [24], and SDSS [25]
give Ωm= 0.24,0.25, and 0.24, respectively, ΛCDM has difficulty to accommodate the quasar even with
the age lower limit tqua = 2.0Gyr.
We seek possible DE models, which perform better than ΛCDM in solving the age problem. The key
factor is the function f(z)defined in Eq.(3), which should be smaller than that of ΛCDM. For the DE
models without interaction between DE and matter, the matter independently evolves as ρm(z)
ρm0= (1+z)3.
For the quintessence type DE models, ρy(z)is an increasing function of redshift z, then f(z)must be
greater than that of ΛCDM. Thus this class of models would not work. On the other hand, for the phantom
type DE models, ρy(z)is a decreasing function of redshift z, then a smaller f(z), a larger accelerating
expansion, and an older Universe could be achieved. However, since the value of ρy(z)cannot be negative,
the limiting case, as a toy model, is that ρy(z)decreases to zero very quickly with increasing z. Then
Eq.(2) would give a limiting age T(z) = 2
3(1 + z)−3/2Ω−1/2
mshown in Fig.1, where Ωm= 0.27 and
h= 0.64 are adopted. One can see that even this maximal possible age, which predicted by an extremely
phantom type DE model, is almost the same at z= 3.91 as that of ΛCDM. This is because the DE
becomes dominant at very late era, and thus is not important in calculating the age of early Universe.
Therefore, there is no non-interaction DE model that can perform better than ΛCDM. The main reason
is that the matter ρm(z)∝(1 + z)3growing too fast with redshift zin Eq.(3). To resolve the high-zage
problem, one need to modify the scaling law of matter component; and the only chance left is that some
4

DM-DE interaction might reduce ρm(z)and give a smaller f(z). The following is such two interacting
DE models that substantially alleviate the high-zcosmic age problem.
3. The cosmic age in Couple YMC Model
Based on the quantum effective Lagrangian of YM field [28], the energy density ρyand pressure pyof
YMC, up to the 3-loop, are given [22]
ρy=1
2bκ2eyh(y+ 1) + η(Y1+ 2Y2)−η2(Y3−2Y4)i,(6)
py=1
6bκ2eyh(y−3) + η(Y1−2Y2)−η2(Y3+ 2Y4)i,(7)
where y= ln |F /κ2|,F≡ −1
2Faµν Faµν =E2−B2plays the role of the order parameter of the YMC,
Y1= ln |y−1+δ|,Y2=1
y−1+δ,Y3=ln2|y−1+δ|−ln |y−1+δ|
y−1+δ,Y4=ln2|y−1+δ|−3 ln |y−1+δ|
(y−1+δ)2,δis a dimensionless
parameter representing higher order corrections, κis the renormalization scale with dimension of squared
mass, and for the gauge group SU(N)without fermions, b=11N
3(4π)2,η≃0.8. Setting the η2term to
zero gives the 2-loop model, and setting further the ηterm to zero gives the 1-loop model. The cosmic
expansion is determined by the Friedmann equations
(˙a
a)2=8πG
3(ρy+ρm),(8)
where the above dot denotes the derivative with respect to the cosmic time t. The dynamical evolutions
of DE and matter are given by
˙ρy+ 3 ˙a
a(ρy+py) = −Γρy,(9)
˙ρm+ 3 ˙a
aρm= Γρy,(10)
where Γ≥0is the decay rate of YMC into matter, a parameter of the model. Notice that the coupled
term Γρyreduces the increasing rate of ρm(z)with redshift z. To solve the set of equations (8) through
(10), it is convenience to replace the variables ρm,ρyand twith new variables x≡ρm/1
2bκ2,y, and
N≡ln a(t)(see Ref.[22] for details). The initial conditions are chosen at the redshift z= 1000. Actually,
the age of Universe in Eq.(2) is mainly contributed by the part of z≤100 of integration with an error
∼1%. We take the parameter δ= 4 and the initial yi= 14. Given the initial value of xi, the numerical
solution of Eqs.(8) through (10) follows, and so does T(z)in Eq.(2). To ensure the present fractional
densities Ωy= 0.73 and Ωm= 0.27, we take xi= 1.74×108for a decay rate Γ = 0.31H0,xi= 8.97×107
for Γ = 0.67H0, and xi= 5.28 ×107for Γ = 0.82H0.
We plot T(z)from YMC and from ΛCDM in Fig.2, where Ωm= 0.27 and tqua = 2.0Gyr are adopted.
The ΛCDM model has T(3.91) = 0.118 and can not accommodate the quasar for h≥0.58. YMC
5

with a large interaction Γwill yield a large T(z). For Γ = 0.31H0, YMC yields T(3.91) = 0.134 and
accommodates the quasar for h≤0.65. For a larger interaction Γ = 0.67H0, YMC yields T(3.91) = 0.172
and accommodates the quasar even for h≤0.83. We also check the 1-loop, and the 2-loop YMC models.
With Γ = 0.31H0and Ωm= 0.27 fixed, the 1-loop, and the 2-loop YMC gives T(3.91) = 0.128, and
0.133, respectively; both perform better than ΛCDM. Thus YMC with higher loops of quantum corrections
predicts an older Universe. In lack of explicit quantum corrections of 4-loops and higher, δhas been used
as a parameter to approximately represent that. A larger δwill yield a larger Ωmand a smaller T(3.91).
For instances, taking Γ = 0.31H0and δ= 5 yields Ωm≃0.31 and T(3.91) = 0.125.
In the flat Universe the age problem is mainly constrained by the observed data of (Ωm, h). Taking
a fixed Tqua and setting T(3.91) = Tqua, each specific model has its own critical curve h=h(Ωm). In
Fig.3, Adopting tqua = 2.0Gyr, we plot the critical curve h=h(Ωm)predicted by YMC and by ΛCDM
in the Ωmh2−hplane (Since the WMAP5 constraint on Ωmh2is much tighter than that on Ωm
alone, we choose Ωmh2as the variable of x-axis instead of Ωm), to directly confront with the current
observations. For each model, only the area below the curve h=h(Ωm)satisfies T(3.91) > Tqua. The
two rectangles denote the observational constraints: the first one filled with horizontal lines is given by
WMAP5 + Freedman [20, 26], and the second one filled with vertical lines is given by WMAP5 + Sandage
[20, 27]. In addition, the star symbol on the plot denotes the Max Like (ML) value of WMAP5 [20]. As is
seen, ΛCDM can only touch the bottom of WMAP5 + Sandage, but is below the ML value of WMAP5
and is out of the scope of WMAP5 + Freedman. So it is difficult for ΛCDM model to accommodate this
quasar. In contrast, YMC model with Γ = 0.67H0passes over all the rectangles, and amply satisfies the
observational constraints. Therefore, if the quasar has tqua = 2.0Gyr, the high-zage problem is amply
solved in YMC model, as shown in Fig.2 and Fig.3.
In our previous works [21, 22], it is found that a constant interaction Γcorresponds to a constant
EOS w0, and a larger Γyields a smaller w0. For instance, Γ = 0.31H0→w0=−1.05;Γ = 0.67H0→
w0=−1.15; and Γ = 0.82H0→w0=−1.21. In the recent works, Riess, et. al. advocate an observed
constraint w=−1.02±0.13
0.19 [29] that gives a lower limit of EOS w=−1.21. In this letter, we shall adopt
w≥ −1.21, and this constraint is equivalent to a range Γ = (0 ∼0.82)H0in coupled YMC model. Taking
the upper limit Γ = 0.82H0and adopting WMAP5 ML Ωm= 0.249 [24], we list the age of the Universe
t3.91 at z= 3.91 predicted by YMC for various observed hin Table 3. YMC accommodates the quasar
even with the age upper limit tqua = 3.0Gyr for h≤0.744; and even for the upper limit h= 0.80 given
by Freeman, YMC still accommodates the quasar if tqua ≤2.79 Gyr. Moreover, taking the WMAP3 result
(Ωm, h) = (0.238,0.734), YMC yields the age t3.91 = 3.16 Gyr at z= 3.91; taking the recent WMAP5
ML result (Ωm, h) = (0.249,0.724), YMC yields t3.91 = 3.08 Gyr at z= 3.91. For both sets of WMAP
data, YMC with maximal interaction can substantially resolves the high-zcosmic age problem.
Let us turn to the age constraints given by the globular cluster M 107 with an age 15 ±2Gyr [19],
6

Table 3: The age of the Universe t3.91 at z= 3.91 predicted by YMC with Γ = 0.82H0for various
observed hat fixed Ωm= 0.249.
Observations Sandage Freedman WMAP3 WMAP5
h0.623 ±0.063 0.72 ±0.08 0.732 ±0.031 0.701 ±0.013
t3.91 (Gyr) 3.25 ∼3.99 2.79 ∼3.49 2.93 ∼3.18 3.13 ∼3.24
Table 4: The age of the present Universe t0predicted by YMC with different interaction for the
observed data of WMAP3 and WMAP5.
Coupled YMC Γ = 0.31H0Γ = 0.67H0Γ = 0.82H0
WMAP3 14.6Gyr 16.8Gyr 18.5Gyr
WMAP5 14.6Gyr 16.7Gyr 18.4Gyr
which is also difficult to accommodate by ΛCDM model if the observed data of WMAP are adopted.
To resolve this puzzle, Sandage and collaborators claim that a lower Hubble constant h∼0.6should
be accepted [19, 27]. Although it can solve the problem of M 107, it still can not solve the high-z
cosmic age problem. Here we list the age of the present Universe t0predicted by YMC with various
interactions Γfor the observed data (Ωm, h)of WMAP3 and WMAP5 in Table 4. For the WMAP3 result
(Ωm, h) = (0.238,0.734), YMC with maximal interaction Γ = 0.82H0yields t0= 18.5Gyr; for the recent
WMAP5 ML result (Ωm, h) = (0.249,0.724), YMC with maximal interaction yields t0= 18.4Gyr. For
both sets of WMAP data, YMC with maximal interaction yields a present cosmic age that is greater than
the age upper limit of M 107, and thus amply accommodates this oldest globular cluster.
4. The cosmic age in a interacting scalar DE model
As seen above, when the interaction between DM and DE is included, YMC model can greatly alleviate
the cosmic age problem. One may ask how other interacting DE models will do. As a specific example,
in the following we shall consider a scalar DE model with the Lagrangian [30]
L=1
2∂µφ∂µφ−V(φ).(11)
We shall consider a simplest case here. Assuming the potential energy V(φ)is dominant, the energy
density and the pressure of DE are
ρφ=−pφ≃V. (12)
7

Table 5: The dimensionless age parameter T(3.91) at z= 3.91 predicted by coupled YMC and by
coupled ΛCDM model with different interaction.
Interacting Model Γ = 0.31H0Γ = 0.67H0Γ = 0.82H0
Coupled YMC 0.133 0.172 0.212
Coupled ΛCDM 0.136 0.185 0.246
The dynamical equations of DE and of matter are
˙ρφ=−Γρφ,˙ρm+ 3 ˙a
aρm= Γρφ(13)
where the coupled term Γρφis introduced. Since this model is quite similar to the ΛCDM model, we name
it as coupled ΛCDM model. It should be pointed that this scalar DE model can not resolve the coincidence
problem [31], which is naturally solved by the YMC model [21, 22]. Changing the time variable tto a new
variable N≡ln a(t)and making use of the Friedmann equations at z= 0, Eq.(13) becomes
dρφ
dN =−√ρφ0+ρm0
H0
Γρφ
√ρφ+ρm
,dρm
dN =√ρφ0+ρm0
H0
Γρφ
√ρφ+ρm−3ρm.(14)
As same as coupled YMC, the initial conditions are chosen at redshift z= 1000. To ensure the present
fractional densities Ωφ= 0.73 and Ωm= 0.27, we take (ρmi, ρφi) = (1.99 ×108,0.76) for Γ = 0.31H0,
(ρmi, ρφi ) = (1.32 ×108,1.52) for Γ = 0.67H0, and (ρmi, ρφi) = (7.12 ×107,2.04) for Γ = 0.82H0.
Here we list the dimensionless age parameter T(3.91) at z= 3.91 predicted by coupled YMC and by
coupled ΛCDM model with different interaction in Table 5, where Ωm= 0.27 are adopted. It is seen
that with the same interaction, coupled ΛCDM model performs a litter better than coupled YMC model,
and thus also substantially accommodates the old high-zquasar APM 08279 + 5255. Besides, we also
calculate the present cosmic age t0in coupled ΛCDM model. adopting Ωm= 0.27 and h= 0.72, for
Γ = 0.67H0and for Γ = 0.82H0, coupled ΛCDM yields t0= 16.3Gyr and t0= 18.0Gyr, respectively,
both accommodate the oldest globular cluster M 107. Therefore, one may say that the interaction between
DM and DE is the crucial factor to make the predicted cosmic ages consistent with observations.
5. Summary
We have investigated the cosmic age problem associated with the old high-zquasar APM 08279
+ 5255 and the oldest globular cluster M 107. To our present knowledge, there is no interacting DE
model performing on this issue had been studied before, and only non-interaction DE models had been
investigated. For instance, adopting the age lower limit of APM 08279 + 5255 tqua = 2.0Gyr and a lower
8

Hubble constant h= 0.64, Holographic DE model gives a constraint Ωm≤0.20 [17] and Agegraphic
DE model gives Ωm≤0.22 [18], both are not better than ΛCDM model. It means that in the frame
of previous works, even taking the age lower limit tqua = 2.0Gyr, the sets of WMAP data can not be
adopted any more. In this letter, it is shown that after introducing the interaction between DM and DE,
a much older Universe can be achieved, and the data of WMAP can be adopted again even for the age
upper limit tqua = 3.0Gyr. Therefore, our work greatly improve the previous obtained results.
The introduction of interaction between DM and DE alleviates cosmic age problem from the following
two aspects: firstly, it yields a EOS w < −1, which correspond to a larger accelerating expansion, and
thus give a larger cosmic age at present; secondly, it reduces the increasing rate of matter density ρm(z)
with redshift z, that would be very helpful to give a smaller f(z)defined in Eq.(3) and a larger cosmic
age at any redshift. As seen in Fig.1, although a larger accelerating expansion would be helpful to give a
larger cosmic age at z= 0 and solve the problem of M 107, it still can not solve the problem of high-z
quasar APM 08279 + 5255. This is because the DE becomes dominant at very late era, and thus is not
important in calculating the age of early Universe. Therefore, the most critical factor that alleviates cosmic
age problem is the second one, i.e. interaction between DM and DE can modify the scaling law of matter
component, and thus reduce the increasing rate of matter density with redshift z. It is well known that
interacting dark energy models can give an EOS crossing the phantom divide w=−1, as indicated by the
recent preliminary observations [29, 32]. In this work, we also demonstrate that the interaction between
DM and DE is the crucial factor required to make the predicted cosmic ages consistent with observations.
Is the DM-DE interaction a necessary factor in describing our real Universe? This issue still need to be
further investigated.
ACKNOWLEDGMENT: We are grateful the referee for helpful suggestions. Y.Zhang’s research work
is supported by the CNSF No.10773009, SRFDP, and CAS.
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Figure 1: Limiting age of the Universe in non-interaction dark energy models.
Figure 2: The predicted age T(z) by YMC and by ΛCDM. If tqua = 2.0 Gyr, the age problem is
amply solved by YMC, but not by ΛCDM.
11

Figure 3: The critical curve h=h(Ωm) predicted by YMC and by ΛCDM in the Ωmh2−hplane,
where tqua = 2.0 Gyr is adopted. YMC with Γ = 0.67H0amply satisfies the observational constraints.
12