Dissipation of Dark Matter - uni-bielefeld.de · 2013. 4. 26. · Bulk Viscous fluid and the...
Transcript of Dissipation of Dark Matter - uni-bielefeld.de · 2013. 4. 26. · Bulk Viscous fluid and the...
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Dissipation of Dark Matterbased on H. Velten and D. J. Schwarz, Physical Review D 86, 083501
(2012)
Hermano Velten
Bielefeld University
Bielefeld, 25.04.2013
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
1 Motivation for non-standard pressureless CDM
2 The dynamics of the ΛvCDM model and observational constraintsBackground expansionPerturbationsComparison with observationsStructure Formation
3 Final Remarks
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Real Universe presentsdissipative effects
Second viscosity (or bulkviscosity - ξ) appears in processwhich are accompanied by achange in volume (i.e. density)of the fluid.
Cosmological Bulk viscosity canbe tested against observationaldata.
⋆ wCDM in galaxy cluster: Serra A. and Romero, M., MNRAS
Letters 415, 1, L74, (2011)1111
Aims of this work:To assign such physical property to Dark Matter
To set an upper bound to the Dark Matter viscosity:ξallowed < ξmax
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Introducing dissipative phenomena in relativistic cosmology
Perfect fluid:Tαβ = pηαβ + (p + ρ)UαUβ and Nα = nUα
Imperfect fluid:Tαβ = pηαβ + (p + ρ)UαUβ +∆Tαβ and Nα = nUα +∆Nα
4-velocityVelocity of energy transport (Landau) → T i0 vanishesVelocity of particle transport (Eckart) → N i vanishes
We adopt Eckart’s theory∆Tαβ = −ηHαγHβδWγδ−χ(H
αγUβ+HβγUα)Qγ−ξHαβ ∂Uγ
∂xγ
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Bulk Viscous fluid and the (Unified) Dark Sector
Dark matter and Dark energy would be differentmanifestations of a single dark component
The Chaplygin gas(
p = −Aρ
)
and the Bulk Viscous fluid
realise this idea
Example: The density of the Chaplygin gas behaves as
ρ = (A+B
a6)1/2 (1)
a → 0 : ρ→ a−3
a → 1 : ρ→ cte
The Bulk Viscous pressure:
pv = −ξΘ; Θ = 3H; ξ = ξ0
(
ρvρv0
)ν
(2)
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Integrated Sachs-Wolfe effect challenges the Bulk Viscous fluid
Bulk viscous fluid shows acceptable results at backgroundlevel and even concerning the matter power spectrum data(W.S. Hipolito-Ricaldi, H.E.S. Velten, W. Zimdahl, JCAP 0906:016 (2009); W.S. Hiplito-Ricaldi, H.E.S.
Velten, W. Zimdahl, Phys.Rev.D 82, 063507 (2010); J. C. Fabris, P. L. C. de Oliveira, H. E. S. Velten,
Eur.Phys.J. C71, 1773 (2011))
Li & Barrow, PRD 79, 103521(2009) However, the choicesν = 0 and ν = −0.5alleviate such largeamplification fo theintegrated Sachs-Wolfeeffect (Hermano Velten and Dominik J.
Schwarz, JCAP 09 (2011) 016 ).
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Integrated Sachs-Wolfe effect
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure of the CMB power spectrum
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Background expansion
A ΛvCDM universe?
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Background expansion
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Background expansion
The dynamics of the ΛvCDM model
Our model has the same structure as the standard flat ΛCDM one.
H2 = H20
Ωb0(1 + z)3 +Ωr0(1 + z)4 +Ωv(z) + ΩΛ
. (3)
pv = −Θξ (4)
ξ = ξ0
(
ρvρv0
)ν
, (5)
where ξ0 and ν are constants and ρv0 is the density of the vCDMfluid at z = 0.Constraint from the assumption Ωk = 0
ΩΛ = 1− Ωb0 − Ωr0 − Ωv0 (6)
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Background expansion
The vCDM energy balance for such model reads,
(1 + z)dΩv(z)
dz− 3Ωv (z) + (7)
ξ
(
Ωv(z)
Ωv0
)ν[
Ωr0(1 + z)4 +Ωb0(1 + z)3 +Ωv (z) + ΩΛ
]1/2
= 0
We denote ν = 0 (ν = −1/2) as model A (B).
ξ = 9H0ξ0ρc0c2
= 24πGξ0c2H0
Standard CDM fluid is recovered if ξ = 0.
Fixing Ωb0 = 0.043 and Ωr0 = 8.32× 10−5 (WMAP-7) theremmaing free parameters of our viscous models are ξ and ΩΛ.
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Perturbations
The ISW effect (a net change in the energy of CMB photonsproduced by time evolving potentials wells) can be calculated by
(
∆T
T
)
ISW
= 2
∫ η0
ηr
ψ′dη. (8)
Scalar perturbations in the Newtonian gauge (with no shear)
ds2 = a2 (η)[
− (1 + 2ψ) dη2 + (1− 2ψ) δijdxidx j
]
. (9)
The perturbed Einstein equation reads
−k2ψ − 3Hψ′ − 3H2ψ =3H2
0a2
2Ωb∆b +Ωv∆v , (10)
−k(
ψ′ +Hψ)
=3H2
0a2
2ΩbΘb + (1 + wv)ΩvΘv (11)
ψ′′+3Hψ′+(2H′+H2)ψ =3a2H2
0Ωv
2
[
−wv
3H(kΘv + 3Hψ + 3ψ′) + νwv∆v
]
.
(12)
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Comparison with observations
Constraints on model parameters
The Supernovae (SN) data (Constitution sample).
Baryon Acoustic Oscillations BAO data from the recentWiggleZ Dark Energy Survey.
The position of the observed CMB peak l1, obtained by theWMAP project, that is related to the angular scale lA.
Information about the ISW effect: relative amplifications (Q)of the ISW effect calculated as (J.B. Dent, S. Dutta and T.J. Weiler, Phys. Rev.
D79, 023502 (2009).)
Q ≡
(
∆TT
)v
ISW(
∆TT
)ΛCDM
ISW
− 1. (13)
If Q > 0 (< 0) the ΛvCDM model produces more (less)temperature variation to the CMB photons via the ISW effectthan the fiducial ΛCDM model.
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Comparison with observations
Model A (ν = 0). Preferred parameters (2σ) between Q = 0% and Q = 40%!
0.45 0.50 0.55 0.60 0.65 0.70 0.75
0.0
0.1
0.2
0.3
0.4
0.5
0.6
WL
Ξ
13Gyrs
14Gyrs
Q=40%
Q=0%
2Σæ
CMB
SN BAO
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Comparison with observations
Model B (ν = −1/2).
0.45 0.50 0.55 0.60 0.65 0.70 0.75
0.0
0.1
0.2
0.3
0.4
0.5
0.6
WL
Ξ
13Gyrs
14Gyrs
2Σæ
Q=40%
Q=0%
CMB SN BAO
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Comparison with observations
Preliminary Conclusion
Considering the isotropic and homogeneous background only,viscous dark matter is allowed to have a bulk viscosity
ξ . 0.2 (ξ0 . 107 Pa.s ), at 2σ,
also consistent with the expected integrated Sachs-Wolfe effect(which plagues some models with bulk viscosity).
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Hierarchical Structure Formation: Merger Tree
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Meszaros equation for the vCDM matter
Assuming the confornal Newtonian gauge in the absence ofanisotropic stresses
ds2 = a(η)2[
− (1 + 2ψ) dη2 + (1− 2ψ) δijdxidx j
]
(14)
we can calculate the perturbed part of the energy-momentumbalances. These equations read
∆ = −(1 + w)
(
Θ
a− 3ψ
)
+ 3Hw∆− 3HδP
ρ; (15)
Θ = −H(1− 3w)Θ−w
1 + wΘ+
k2
a (1 + w)
δP
ρ+
k2
aψ, (16)
where Θ = ik jvj is the divergence of the perturbed fluid velocityand total pressure is P ≡ Peff = pk(kinetic) + Π(viscous).
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Meszaros equation for the vCDM matter
In order to find a single equation for the density contrast we stillneed the Poisson equation
k2
a2ψ + 3H
(
ψ + Hψ)
= −4πGρδ. (17)
For sub-horizon modes we take the large-k limit of the aboveequation as well as to neglect ψ in 15. Hence, the relativisticevolution of the density contrast is
∆ + (2H − 3Hwv) ∆ + (18)[
−4πGρ (1 + wv)− 6H2wv + 9H2w2v − 3Hwv − 3Hwv
]
∆ = (19)
−k2
a2δΠ
ρ− 3H
˙δΠ
ρ+δΠ
ρ
(
−15H2 − 3H)
(20)
where we assumed P = Π. Hence wv = Π/ρ.
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Meszaros equation for the vCDM matter
or, equivalently
a2∆′′ +
(
3 +aH ′
H− 3wv
)
a∆′ + (21)
[
−3H2
0Ω
2H2(1 + wv)− 6wv + 9w2
v − 3aH ′
Hwv − 3w ′
va
]
∆ = (22)
−k2
H2a2δΠ
9Ω− a
δΠ′
3Ω+δΠ
9Ω
(
−15− 3aH ′
H
)
(23)
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Eckart pressure in the relativistic theory
We identify the viscous pressure as Π = −ξuγ;γ which up to firstorder reads
δΠ
ρ= νwv∆−
wvΘ
3Ha+ wvψ +
wvψ
H(24)
or, equivalently
δΠ
Ω= 9
νwv(1 + wv)
1 + 2wv
∆+wv
1 + 2wv
(
3a∆′ + 9wv∆)
. (25)
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Meszaros equation for the vCDM matter
Sub horizon vCDM perturbations obey to the following Meszaros-like equation
a2 d2∆v
da2+
[
a
H
d H
da+ 3 + A(a) + B(a)k
2]
ad∆v
da+
[
+C(a) + D(a)k2−
3
2
]
∆v = P(a), (26)
A(a) = −6wv +a
1 + wv
dwv
da−
2a
1 + 2wv
dwv
da+
3wv
2(1 + wv)
B(a) = −
wv
3a2H2(1 + wv)
C(a) =3wv
2(1 + wv)− 3wv − 9w
2v−
3w2v
1 + wv
(
1 +a
H
dH
da
)
− 3a
(
1 + 2wv
1 + wv
)
dwv
da+
6awv
1 + 2wv
dwv
da
D(a) =w2v
a2H2(1 + wv)
P(a) = −3νwvad∆v
da+ 3νwv∆v
[
−
1
2+
9wv
2+
−1 − 4wv + 2w2v
wv(1 + wv)(1 + 2wv)ad wv
da−
k2(1 − wv)
3H2a2(1 + wv)
]
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Growth of sub horizon perturbations k = 0.2hMpc−1 (galaxy cluster scale)
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
1.2
a
D
Ξ= 2 x 10-4
Ξ= 2 x 10-5
Ξ= 2 x 10-6
k = 0.2 h Mpc-1
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
1.2
a
D
Ξ= 2 x 10-4
Ξ= 2 x 10-5
Ξ= 2 x 10-6
k = 0.2 h Mpc-1
Solid line is the standard CDM growth ∆CDM ∼ a.
Dashed lines correspond to the viscous models with differentvalues of the viscosity coefficient.
The initial conditions, i.e. the power spectrum at thematter-radiation equality, are set using the CAMB code.
Viscous dark halos at cluster scales (k = 0.2hMpc−1) are ableto follow the typical CDM growth only if ξ . 10−6.
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Same as before: Growth of sub horizon perturbations k = 0.2hMpc−1 (galaxy
cluster scale) - ν = 0
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
1.2
a
D
Ξ= 2 x 10-4
Ξ= 2 x 10-5
Ξ= 2 x 10-6
k = 0.2 h Mpc-1
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Growth of sub horizon perturbations k = 100hMpc−1
0.00 0.05 0.10 0.15 0.20 0.250.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
a
D
Ξ= 2 x 10-7
Ξ= 2 x 10-8
Ξ= 2 x 10-9
k = 100 h Mpc-1
0.00 0.05 0.10 0.15 0.20 0.250.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
aD
Ξ= 2 x 10-9
Ξ= 2 x 10-8
Ξ= 2 x 10-7
k = 100 h Mpc-1
Left: Model A. Right: Model B.
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Growth of sub horizon perturbations k = 1000hMpc−1 (dwarf galaxy scale)
0.00 0.05 0.10 0.15 0.20 0.250.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
a
D
Ξ= 2 x 10-9
Ξ= 2 x 10-10
Ξ= 2 x 10-11k = 1000 h Mpc-1
0.00 0.05 0.10 0.15 0.20 0.250.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
a
D
Ξ= 2 x 10-10
Ξ= 2 x 10-9
Ξ= 2 x 10-8
k = 1000 h Mpc-1
Smaller scales place stronger constraints on the viscosityvalue, e.g for dwarf galaxies ξ . 10−11 (for model B we findξ . 10−10).
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Same as before: Growth of sub horizon perturbationsk = 1000hMpc−1 (dwarf
galaxy scale) - ν = 0
0.00 0.05 0.10 0.15 0.20 0.250.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
a
D
Ξ= 2 x 10-9
Ξ= 2 x 10-10
Ξ= 2 x 10-11k = 1000 h Mpc-1
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Maximum viscosity allowed in order to obtain the same linear standard CDM
1 10 100 100010-13
10-12
10-11
10-10
10-9
10-8
k HhMpcL
Ξm
ax
Ν = 0
Ν = -12
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Maximum viscosity allowed in order to obtain the same linear standard CDM (SI units)
1 10 100 1000
10-5
10-4
0.001
0.01
0.1
1
k HhMpcL
Ξ0
maxHP
a.sL
Ν = 0
Ν = -12
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Some words about numerical simulations -
http://ooo.aip.de/groups/cosmology/
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
Newtonian Cosmology - E. A. Milne (1934); E. A. Milne, W. H.
McCrea (1934)
ρ+∇r . (ρv) = 0 , (27)
v + (v.∇r )v = −∇rΨ−∇rP
ρ, (28)
∇2rΨ = 4πGρ, (29)
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
neo-Newtonian Cosmology - McCrea (1951); E. R. Harrison (1965)
ρ+∇r . (ρv) + P∇r .v = 0 , (30)
(
∂v
∂t
)
r
+ (v.∇r )v = −∇r .Ψ−∇rP
ρ+ P, (31)
∇2rΨ = 4πG [ρ+ 3P] . (32)
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
!!PRELIMINARY!!Eckart theory
0.05 0.10 0.15 0.20 0.25 0.30 0.350.2
0.4
0.6
0.8
1.0
1.2
a
D
k = 1000 h Mpc-1 Ν = 0Ξ= 10-11
Ξ= 5 x 10-11
Ξ= 10-10
New
GR
neo-New
0.2 0.4 0.6 0.8 1.00.2
0.4
0.6
0.8
1.0
1.2
a
D
k = 0.2 h Mpc-1Ν = 0
Ξ = 5 x 10
-6
Ξ= 10-5
Ξ= 5 x 10-5
NewGR
neo-New
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
!!PRELIMINARY!!Eckart theory
0.05 0.10 0.15 0.20 0.25 0.30 0.350.2
0.4
0.6
0.8
1.0
1.2
1.4
a
D
Ν = 14 Ξ= 2 x 10-11
NewGRneo-New
k = 1000 h Mpc-1
0.4 0.5 0.6 0.7 0.8 0.9 1.00.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
a
D
Ν = 14 Ξ= 10-5
New
GR
neo-New
k = 0.2 h Mpc-1
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Structure Formation
!!PRELIMINARY!!Muller-Israel-Stewart theory
0.05 0.10 0.15 0.20 0.250.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
a
D
NewGRneo-New
Τ = H0-1
Τ = 0.5 x H0-1
Τ = 0.25 x H0-1
Τ = 0.1 x H0-1
Ξ= 10-10
Ν = 0
k = 1000 h Mpc-1
Viscous DM fluid with large relaxation time → Standard CDM!
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Final Remarks
We model CDM as a bulk viscous fluid: negative pressure
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Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Final Remarks
We model CDM as a bulk viscous fluid: negative pressure
We allow the existence of a bulk viscosity contributing to theobserved acceleration without being the major cause.
![Page 41: Dissipation of Dark Matter - uni-bielefeld.de · 2013. 4. 26. · Bulk Viscous fluid and the (Unified) Dark Sector Dark matter and Dark energy would be different manifestations](https://reader035.fdocument.pub/reader035/viewer/2022071408/6100b071295a564124547fa0/html5/thumbnails/41.jpg)
Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Final Remarks
We model CDM as a bulk viscous fluid: negative pressure
We allow the existence of a bulk viscosity contributing to theobserved acceleration without being the major cause.
Background data is able to place an upper bound to theviscosity parameter ξ . 0.2 (ξ0 . 107 Pa.s ), at 2σ.
![Page 42: Dissipation of Dark Matter - uni-bielefeld.de · 2013. 4. 26. · Bulk Viscous fluid and the (Unified) Dark Sector Dark matter and Dark energy would be different manifestations](https://reader035.fdocument.pub/reader035/viewer/2022071408/6100b071295a564124547fa0/html5/thumbnails/42.jpg)
Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Final Remarks
We model CDM as a bulk viscous fluid: negative pressure
We allow the existence of a bulk viscosity contributing to theobserved acceleration without being the major cause.
Background data is able to place an upper bound to theviscosity parameter ξ . 0.2 (ξ0 . 107 Pa.s ), at 2σ.
The dynamics of the linear sub horizon perturbations placesstronger constraints on the dark matter viscosityξ . 0.2× 10−10 (ξ0 . 10−3 Pa.s).
![Page 43: Dissipation of Dark Matter - uni-bielefeld.de · 2013. 4. 26. · Bulk Viscous fluid and the (Unified) Dark Sector Dark matter and Dark energy would be different manifestations](https://reader035.fdocument.pub/reader035/viewer/2022071408/6100b071295a564124547fa0/html5/thumbnails/43.jpg)
Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Final Remarks
We model CDM as a bulk viscous fluid: negative pressure
We allow the existence of a bulk viscosity contributing to theobserved acceleration without being the major cause.
Background data is able to place an upper bound to theviscosity parameter ξ . 0.2 (ξ0 . 107 Pa.s ), at 2σ.
The dynamics of the linear sub horizon perturbations placesstronger constraints on the dark matter viscosityξ . 0.2× 10−10 (ξ0 . 10−3 Pa.s).
A hint for the ”small scale problems“ of the CDM scenario:⋆ We do not observe so many sub-galactic structures as
predicted. (vCDM seems to provide such mechanism!)⋆ Does vCDM produce a cusped density profile (as predicted
by CDM) or a cusped (as observations indicate) one?
![Page 44: Dissipation of Dark Matter - uni-bielefeld.de · 2013. 4. 26. · Bulk Viscous fluid and the (Unified) Dark Sector Dark matter and Dark energy would be different manifestations](https://reader035.fdocument.pub/reader035/viewer/2022071408/6100b071295a564124547fa0/html5/thumbnails/44.jpg)
Outline Motivation for non-standard pressureless CDM The dynamics of the ΛvCDM model and observational constraints Final Rema
Final Remarks
We model CDM as a bulk viscous fluid: negative pressure
We allow the existence of a bulk viscosity contributing to theobserved acceleration without being the major cause.
Background data is able to place an upper bound to theviscosity parameter ξ . 0.2 (ξ0 . 107 Pa.s ), at 2σ.
The dynamics of the linear sub horizon perturbations placesstronger constraints on the dark matter viscosityξ . 0.2× 10−10 (ξ0 . 10−3 Pa.s).
A hint for the ”small scale problems“ of the CDM scenario:⋆ We do not observe so many sub-galactic structures as
predicted. (vCDM seems to provide such mechanism!)⋆ Does vCDM produce a cusped density profile (as predicted
by CDM) or a cusped (as observations indicate) one?
Only full numerical simulations could be able to predict thefinal clustering patterns in the case of Viscous CDM ...