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என்றா கூபிடுகிறீர்க்ச்ள்.
___________________________________________________________________________________________
https://youtu.be/lOpaux5WD54
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EPJ manuscript No.
(will be inserted by the editor)
Rip/singularity free cosmology models with bulk viscosity
Xin-he Meng1,2 a and Zhi-yuan Ma1 b
1 Department of physics, Nankai University, Tianjin 300071, China
2 Kavli Institute of Theoretical Physics China, CAS, Beijing 100190, China
Received: date / Revised version: date
Abstract. In this paper we present two concrete models of non-perfect fluid with bulk viscosity to interpret
the observed cosmic accelerating expansion phenomena, avoiding the introduction of exotic dark energy.
The first model we inspect has a viscosity of the form ζ = ζ0 + (ζ1 − ζ2q)H by taking into account of the
decelerating parameter q, and the other model is of the form ζ = ζ0 + ζ1H + ζ2H
2
. We give out the exact
solutions of such models and further constrain them with the latest Union2 data as well as the currently
observed Hubble-parameter dataset (OHD), then we discuss the fate of universe evolution in these models,
which confronts neither future singularity nor little/pseudo rip. From the resulting curves by best fittings
we find a much more flexible evolution processing due to the presence of viscosity while being consistent
with the observational data in the region of data fitting. With the bulk viscosity considered, a more
realistic universe scenario is characterized comparable with the ΛCDM model but without introducing the
mysterious dark energy.
1 Introduction
The independent discovery respectively in 1998 and 1999
indicates that the current universe is in accelerating expansion [1]. To accommodate this surprisingly exotic phenomenon, dark energy, varieties of models are proposed.
The basic idea of dark energy comes up in the context of
supposing the general theory of relativity works precisely
well in cosmological scale, a perfect fluid with effectively
large enough negative pressure is required to speed the
universe expansion up. According to the Wilkinson Microwave Anisotropy Probe (WMAP) 7-year dataset analysis [2] it makes up about 72.8% of the universe’s contents. In the past decade, many attempts are made to
understand the accelerating mechanism of dark energy,
the most mysterious component of the universe hitherto
as envisioned. The ΛCDM is such a model and receives
great attention because it is well consistent with the observational data although suffering from some serious fundamental physics problems [3]. Another popular theory of
dark energy is referred as quintessence [4]. However, an
alternatively instructive idea is that the general theory of
relativity may fail in large scale, therefore modification of
gravity theory should be introduced to drive an accelerating phase of universe expansion. The f(R) gravity, for
example, which generalizes the Einstein-Hilbert action by
invoking an arbitrary function of Ricci scalar, is of this
class [5]. More details and references are available in the
recent reviews [6].
a
e-mail: xhm@nankai.edu.cn
b
e-mail: mazhiyuan@mail.nankai.edu.cn
In the context of perfect fluid, models with different
equation of state are extensively studied (see Ref. [7]),
such as Chaplygin gas, generalized Chaplygin gas, inhomogeneous equation of state, and barotropic fluid dark
energy, etc. However, the perfect fluid assumption is assertive since it suggests no dissipation, which actually exists widely and intuitively plays a critical role in the universe evolution, especially in the early hot stages. To be
more realistic, models of imperfect fluid are invoked by
introducing viscosity into the investigation. The simplest
theory of the kind is constant bulk viscosity, regarded as
equivalence to constant dark energy with cold dark matter model. A widely investigated case is that bulk viscosity
with the form as a linear function of the Hubble parameter, which is proved to be well consistent with the observed
late-time acceleration and can recover both the matter and
dark energy dominant eras. Another case is that bulk viscosity behaves as a parameterized power-law form with
respect to the matter density, which can be shown to be
similar to the Chaplygin gas model [8]. Furthermore, viscosity term contains red-shift and higher derivatives of
scale factor is also considered as a more general theory [9].
On the other hand, as subsequence of viscosity and dissipation, turbulence is considered as a cosmic component in
Ref. [10]. In this paper we focus on the bulk viscosity and
propose two such new models, which can be dealt with
analytically and performed well when comparing with observational data.
It is well known that significant number of dark energy models suffer from the finite-time singularity problem. The classification of the (four) finite-time future singularities has been proposed by S. Nojiri et al. (2005) [11],
2 Xin-he Meng, Zhi-yuan Ma: Rip/singularity free cosmology models with bulk viscosity
while the big rip (Type I) and sudden rip (Type II) are
discussed previously in Ref. [12] and [14], respectively. Recently, a novel concept has been proposed in Ref. [15], the
so-called “Little Rip”, in which for some kinds of models
the non-viscous dark energy density increases with equation of state parameter ω < −1, but that ω → −1 asymptotically is required to avoid the future singularity, thereby
leading to a scenario that bound structure is disassembled
(see Ref. [16] for more concrete models of this class). Likewise, “Pseudo Rip”, which is introduced as another case
of the rips, occurs when an upper bound Finert where F
refers to the inertial force is confronted during the increasing of dark energy density with scale factor. Several models have been studied as this type [17]. Further more, models reconstructed to realize or avoid such rip/singularity
in scheme of modified gravity are also studied recently in
Ref. [18]. The presence of rips causes destruction of cosmos
structure, therefore it is an important aspect for a cosmological viable model building that a natural scenario can
be yielded to cure such problem [13]. In the scheme of
viscous cosmology, singularity as a vital feature attracts
great attention (see most of the models in Ref. [7][8]).
Recently, the rips confronted in viscosity models are also
studied in Ref. [19]. Generally, it implies that the presence
of particular viscosity could mitigate the singularness of
cosmos quantities in some extends.
In this letter, we propose and investigate two different
viscosity models, both of which are constructed under the
assumption that our universe is bound more tightly therefore the bulk viscosity varies with respect to not only the
conventional momentum term H, but also the accelerating
status (H˙
, model I) and total energy (H2
, model II). Arrangement for this article is as follows. In the next section,
we derive the explicit solution of the models, respectively,
and calculate analytically the corresponding cosmic quantities. In Section 3, we briefly review the statistics analysis
employed for constraining model parameters by SNe Type
Ia date and OHD dataset. In Section 4 figures and tables
are listed as our results, in which the comparison with
the ΛCDM model is also performed. In Section 5, as our
conclusion, we discuss the result of best-fitting and the
fate of universe evolution accordingly by means of the singularity and rip analysis, as well as the advantages and
disadvantages of such viscosity models. For the benefit to
the related astrophysics and cosmology community we list
the latest 18 OHD in the appendix.
2 Cosmology Models of Imperfect Fluid
In the scheme of imperfect fluid, the energy stress tensor
reads,
Tµν = ρUµUν + (p−θζ)hµν −2ησµν +QµUν +QνUµ, (1)
where ρ is the mass density, p the isotropic pressure, U
µ =
(1, 0, 0, 0) the four-velocity of the cosmic fluid in comoving
coordinates, hµν = gµν + UµUν the projection tensor, θ ≡
θ
µ
µ = h
α
µh
β
µU(α;β) = U
µ
;µ
the expansion scalar, σµν = θµν −
1
3
hµνθ the shear tensor, ζ the bulk viscosity, η the shear
viscosity, and Qµ = −κhµν(T,ν + T UνUµ;ν ) the heat flux
density four-vector with κ the thermal conductivity.
In the case of thermal equilibrium, Qµ = 0. Moreover,the term of shear viscosity vanished when a completely isotropic unverse is assumed. In the isotopic and
homogeneous Friedmann-Robertson-Walker (FRW) metric,
ds2 = −dt2 + a
2
(t)
dr2
1 − kr2
+ r
2
dΩ2
, (2)
where k = −1, 0, 1 being the curvature parameter, Eq. (1)
can be rewritten as
Tµν = ρUµUν + (p − θζ)hµν . (3)
Here θ = 3H where H =
a˙
a
is the Hubble parameter and
the dot denotes differential with respect to cosmic time t.
Therefore the corresponding Friedmann equation, considering the case of flat space-time, is of the form
H2 =
8πG
3
ρ, (4)
H˙ + H2 =
4πG
3
(ρ + 3˜p), (5)
as well as the equation of energy conservation for a complete dynamics system,
ρ˙ + (ρ + ˜p) θ = 0, (6)
where ˜p = p−θζ is the effective pressure. If we assume the
matter presented is cold thereby pressureless, ˜p will have
a simple form of −θζ. So far we can principally solve the
matter density ρ with respect to cosmic time t when the
explicit form of ζ is specified.
Combining equations (4), (6) and the current value of
Hubble parameter H0, we obtain the following dimensionless equation,
h˙
H0
+
3
2
h
2 = ξh, (7)
in which h ≡
H
H0
and ξ ≡
2
3
H0ζ
8πG are the dimensionless
hubble parameter and viscosity, respectively. Using the
relation
d
dt
=
a˙
a
d
d ln a
(8)
and let ′ denote the differential respect to conformal time
ln a, we obtain a dimensionless equation,
h
′ +
3
2
h = ξ. (9)
2.1 Model I: Viscosity with Decelerating Parameter q
In this subsection we consider the model by taking into account of the decelerating parameter q. The bulk viscosity
reads,
ζ = ζ0 + (ζ1 − ζ2q)H, (10)
Xin-he Meng, Zhi-yuan Ma: Rip/singularity free cosmology models with bulk viscosity 3
in which q ≡ − a¨
aH2 is referred as the decelerating parameter, and ζ1,ζ2,ζ3 are constant, respectively. By introducing decelerating parameter q into the viscosity, the model
now can take account of the role played by the transition
of cosmic evolution phases.
By transformation ζ0 =
12πG
H0
ξ0, ζ1 =
12πG
H2
0
(ξ1 − ξ2)
and ζ2 = 12πG
H2
0
ξ2, we obtain the dimensionless form of
viscosity,
ξ = ξ0 + ξ1h + ξ2h
′
, (11)
where the prime denotes d
d ln a
. Considering the condition
h(a0) = 1, Eq. (9) can be solved analytically when ξ1 6= 3
2
and ξ2 6= 1 as
h(a) = 2ξ0
3 − 2ξ1
+
1 −
2ξ0
3 − 2ξ1
(a/a0)
−
3−2ξ1
2−2ξ2 . (12)
After integration we obtain the corresponding scale factor,
a(t) = a0
3 − 2ξ1
2ξ0
e
ξ0
1−ξ2
H0(t−t0) −
3 − 2ξ1
2ξ0
+ 12−2ξ2
3−2ξ1
.
(13)
Hence the Hubble parameter with respect to t reads
H (t) = H0
2ξ0
3 − 2ξ1
e
ξ0
1−ξ2
H0(t−t0)
e
ξ0
1−ξ2
H0(t−t0) −
3−2ξ0−2ξ1
3−2ξ1
, (14)
and the decelerating parameter q is
q = −1 +
3 − 2ξ0 − 2ξ1
2 − 2ξ2
e
−
ξ0
1−ξ2 H0(t−t0)
. (15)
Substitute Eq. (14), (15) to Eq. (10) we obtain the evolution of bulk-viscosity,
ζ(t) = 12πG
H0
3ξ0
3 − 2ξ1
+
ξ0(2ξ1 − 3ξ2)(3 − 2ξ0 − 2ξ1)
(3 − 2ξ1)
2(1 − ξ2)
×
1
e
ξ0
1−ξ2
H0(t−t0) −
3−2ξ0−2ξ1
3−2ξ1
. (16)
2.2 Model II: Viscosity with H2
In this subsection we consider the viscosity model with
Hubble parameter up to quadratic order,
ζ = ζ0 + ζ1H + ζ2H2
. (17)
The corresponding dimensionless viscosity is given as
ξ = ξ0 + ξ1h + ξ2h
2
, (18)
We should note that it is another case of stronger coupling
between matter and dark components since the term H2 ∼
h
2
in proportion to velocity square (kinetic energy) takes
into account of the energy transference. We will discuss
the detailed feature of this model in Section 4 and 5
The solution of this model depends on the sign of ∆ ≡
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