APPLICABILITY OF BULK VISCOUS MODELS
Aroon Beesham
Department of Mathematical Sciences, University of Zululand, South Africa
November 29, 2016
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2
3
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Figure 3: Unizulu.
Figure 4: Hall.
PLAN OF TALK
1. Introduction
2. Theories of Viscosity
3. Field Equations
4. Solutions
5. Conclusion
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6. References
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1
INTRODUCTION
• Explain photon to baryon ratio
• Generate inflation without additional scalar field
• Models without initial singularity
• Description of particle creation in strong gravitational
field
• Decoupling of photons during radiation era
• Can mimic variable cosmological parameter - cosmological constant problem
• Equations equivalent to those of Chaplygin gas
• Candidate for dark energy
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2
THEORIES OF VISCOSITY
In this section, we give a brief review of some theories of
bulk viscosity.
2.1
ECKART THEORY
The total pressure of the fluid is given by
p̄ = p + Π
(1)
where p is the equilibrium pressure (usually given by a barotropic
linear equation of state p = ωρ, where ω is constant, and ρ
is the energy density) and Π is the viscous pressure, usually
taken to have the following form:
Π = −3ζH = −3ζoραH, ζo = const, α = const
8
(2)
where ζ is the coefficient of bulk viscosity.
Eckart theory is NOT relativistic:
• Allows superluminal signals. From the equation
Π = −3ζH = −3ζoραH, ζo = const, α = const (3)
if H is switched off, then the viscous term 3H = 0 instantly (compare with usual wave equation). Need term
to gradually become zero.
• Equilibrium states unstable, perturbations grow
• Cannot describe non-stationary effects
Theory is first order, equations of algebraic type. Neglects
second order terms. Need differential evolution equations
which place equilibrium & dissipative variables on same footing.
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2.2
TRUNCATED ISRAEL-STEWART THEORY (TIS)
In this case, the viscous pressure Π obeys the following differential evolution equation:
Π + τ Π̇ = −3ζ H
(4)
where τ is the relaxation time for the irreversible process.
We notice that for τ = 0, the theory reduces to Eckart
theory.
• Causal, stable
• Problem: Admits pathological behaviour at late times
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2.3
FULL LINEAR ISRAEL-STEWART THEORY (LIS)
In this case, the viscous pressure Π obeys the following differential evolution equation:
[
]
1
τ̇ ζ̇ Ṫ
Π + τ Π̇ = −3ζH − Π τ 3H + − −
(5)
2
τ ζ T
where T is the temperature.
• Problem: Relies on assumption of small deviations from
equilibrium
• May not hold during early inflation and when dealing
with dark energy
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2.4
NONLINEAR ISRAEL-STEWART THEORY (NIS)
• FIS is linear, but causal
• Relies on assumption of small deviations from equilibrium
|Π| < p
• Is this justified?
• May not hold during early inflation and when dealing
with dark energy
• Hence theory extended [Maartens & Mendez (1997) PRD
55, 1937] to include nonlinear effects
12
In this case, the viscous pressure Π obeys the following differential evolution equation:
(
)−1
τ∗
τ Π̇ = −3ζ H − Π 1 + Π
[ ζ
]
1
τ̇ ζ̇ Ṫ
− Π τ 3H + − −
(6)
2
τ ζ T
where τ∗ is the characteristic time for nonlinear effects (the
linear Israel-Stewart theory is recovered for τ∗ = 0).
13
Since eqns difficult to solve, do a phase plane analysis. We
would like an ideal cosmological model to have the following
properties:
1. early inflation, corresponding to a source or unstable node
2. radiation dominated phase, corresponding to an unstable
critical point
3. matter dominated era, corresponding to a transient saddle point
4. current accelerating phase, which may or may not last
forever, possibly a stable attractor or sink
Can we achieve this with viscosity?
14
Ref: G Acquaviva & A Beesham (2014) PRD, 90, 023503
• Consider non-interacting mixture of dust and a viscous
fluid
• Equations difficult to solve in general
• Carry out a dynamical systems analysis
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To reduce the dynamical equations to an autonomous system, define the expansion-normalized variables:
3ρv
3Π
Ω = 2 , Π̃ = 2
(7)
θ
θ
and we get the dynamical system:
[
]
′
Ω = 3 (Ω − 1) (γ − 1) Ω + Π̃
(8)

(
)−1
2 Π̃
Π̃
k

Π̃′ = −3γ v 2 Ω 1 +
1+
3 ζ0
γ v2 Ω
[
]
2
Π̃ 2γ − 1
− Ω + −3 Π̃(γ − 1) (1 − Ω)
(9)
−3
Ω
2γ
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Figure 5 shows a Picture
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This model has some very desirable features. There exist
a nonzero set of trajectories which:
• Start off at a source Pd+ , which represents the radiation
dominated era
• Pass through a transient saddle point Pd0, the matter
dominated era, when structure formation can take place,
and
• Finally end up at a possible sink Pd−, which represents the
currently observed expanding stage (either polynomial or
exponential)
• Primordial inflation not included
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2.5
DISCONZI THEORY
A new first order formulation of bulk viscosity has been recently given by Disconzi et al [1, 2]. It is claimed that this
is causal and that the equilibrium states are stable.
The field equations are
1
Rab − Rgab + Λgab = Tab
(10)
2
where Rab is the Ricci tensor, R the Ricci scalar, gab the
metric tensor, Λ the cosmological term (which need not necessarily be constant) and the energy momentum tensor Tab
is given by
Tab = (ρ + p)uaub + pgab − ζ(gab + uaub)∇dC d
(11)
where ρ is the energy density, p the pressure, ua the 420
velocity of the fluid and ζ the coefficient of bulk viscosity.
The quantity C a is the dynamic velocity of the fluid defined
by
Ca = F ua
(12)
and F is the specific enthalpy of the fluid given by
ρ+p
F =
(13)
µ
where µ is the rest mass density of the fluid, conserved along
the fluid flow lines:
∇a(µua) = 0
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(14)
3
FIELD EQUATIONS
The FLRW metric is given by
[
]
2
dr
2(dθ 2 + sin2θdϕ2 (15)
ds2 = −dt2 + a2(t)
+
r
1 − kr2
where a is the scale factor, and k = 0, +1, −1 corresponding
to flat, closed, open models, respectively. Now we get
ȧ
a
∇au = 3
(16)
a
from which we find that
ȧ
a
(17)
∇aC = Ḟ + 3F
a
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From the previous equations, we get the following Raychaudhuri type equation:
(
)
ä
1
ȧ
2
Ḣ + H = = − ρ + 3p − 3ζ Ḟ + 9ζF − 2Λ (18)
a
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a
where H = ȧ/a is the Hubble parameter. In addition, we
also get the modified energy conservation equation
ρ̇ + 3(ρ + p)H − 3ζ(Ḟ + 3F H)H + Λ̇ = 0
(19)
Note that we have allowed for the possibility of a variable
cosmological parameter. For comparison with other theories,
we note that the modified energy conservation equation (19)
can also be written, for Λ = 0 as
ρ̇ + 3(ρ + p̄)H = 0
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(20)
where the total pressure p̄ is given by
p̄ = p + Π
(21)
where the viscous pressure Π is:
Π = −3ζ(Ḟ + 3F H)H
(22)
From equations (18) and (19), we can derive a Friedmanntype equation
3k
2
(23)
3H + 2 = ρ + Λ
a
For the most part, we shall assume a linear equation of state
p = ωρ
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(24)
where ω is constant. Equations (18) and (19) then become,
respectively:
)
ä
1(
2
Ḣ + H = = − (3ω + 1)ρ − 3ζ Ḟ + 9ζF H − 2Λ
a
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(25)
ρ̇ + 3(ω + 1)ρH − 3ζ(Ḟ + 3F H)H + Λ̇ = 0
(26)
Equations (23), (25) and (26) are the basic equations that
we will use for our analysis.
The function F is given by
(1 + ω)ρa3
F =
µ0
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(27)
4
SOLUTIONS
4.1
Power law solution
• k = 0, Λ = 0 (We shall consider Λ ̸= 0 later)
• ω = const
The equations admit the following solution:
a = ao
2
t 3(ω+1)
ρo
ρ= 2
t
(28)
(29)
F =
2
Fot ω+1
(30)
Π=
−2ω
Πot ω+1
(31)
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4.2
Exponential Solution
In Eckart theory, it is well-known that a simple exponential
solution of the type:
a = aoeHot
(32)
exists for all values of ω ̸= −1. In fact, such a simple solution
exists in the TIS, FIS and NFIS as well.
Let us investigate if such a solution exists in this formulation. We consider:
• k = 0, Λ = 0 (We shall consider Λ ̸= 0 later)
• α = 0 =⇒ ζ = const = ζo
• ω = const
From equation (23), the density ρ will also be constant.
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From the modified energy conservation equation (19):
ρ̇ + 3(ρ + p)H − 3ζ(Ḟ + 3F H)H + Λ̇ = 0
(33)
we get the following equation:
µo(ω + 1) − 6(ω + 1)ζoHoa3 = 0
(34)
from which we conclude that a = const unless ω = −1.
However, if ω = −1, then we see from the equation (27) for
F that there is no viscosity.
Hence, we conclude that a simple exponential solution of
the type a = aoeHot does not exist in this theory in contrast
to Eckart, TIS, LIS and NIS theories.
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4.3
Variable Λ
Consider
• ω = const ̸= −1
• Λ = H2
• ζ = ζoρα, α ≥ 0
Then we find the following solution:
a(t) = (−2kt2 + k1t + k2)1/2
(35)
We can easily find the other parameters from our equations.
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We list them for the case k = 0:
[
]
2n(ω − 1)µ0
α
+
ρ =
1/2
(6α − 1)k1(1 + ω)ζo(k1t + k2)
[
]
k3
(k1t + k2)3α
[
]
2n(ω − 1)µ0
ζ = ζo
+
(6α − 1)k1(1 + ω)ζo(k1t + k2)1/2
]
[
k3
ζo
(k1t + k2)3α
k12
Λ=
4(k1t + k2)2
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(36)
(37)
(38)
4.4
Variable Λ
Consider
• Λ = H2
•k=0
• ω = 1, p = ρ, stiff equation of state
From equations (18) and (19), we get, for ω = 1, k = 0,
( )2
3
3
ζ ρ̇a
ρa H
ä
ȧ
2
+ 12
=2 +2
(39)
µo
µo
a
a
Then we can find the following power law solution for a:
a(t) = tm, ȧ = mtm−1, ä = m(m − 1)tm−2
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(40)
The energy density is:
[
]1/(α+1)
m(2m − 1)(α + 1)µo
c2
ρ=
+
(41)
3m+1
6m(α+1)
ζo(6mα + 3m − 1)t
t
The cosmological parameter Λ has the form:
2
m
Λ = H2 = 2
t
and the viscous pressure Π is given by:
−2m(2m − 1)
Π=
t2
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(42)
(43)
4.5
Variable Λ
Consider:
•k=0
• ω = 1, p = ρ, stiff equation of state
We start from equations (18) and (19) again:
( )2
3
3
ζ ρ̇a
ρa H
ä
ȧ
2
+ 12
=2 +2
(44)
µo
µo
a
a
and obtain an exponential solution
a = emt
(45)
The energy density ρ is given by
[
]1/(α+1)
2m(α + 1)µo −3mt
ρ=
e
+ c1e−6m(α+1)t
(46)
3ζo(2α + 1)
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where c1 is a constant of integration. The cosmological constant is:
Λ = H 2 = m2
(47)
whereas the viscous pressure is also constant and given by
Π = −4m2
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(48)
5
CONCLUSION
• Further study required
• Temperature law - may also have pathological behaviour
• Dynamical systems analysis - need to define new variables
• More interesting solutions
• Relationship to other theories
• Looks simple but is it?
• So far, appears causal
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References
[1] Disconzi, MM, Kephart, TW & Scherrer, RJ, Phys. Rev.
D, 91:043532 (2015).
[2] Disconzi, MM, Kephert,
arXiv:1510.07187 (2015).
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TW & Scherrer,
RJ,