dissipation of dark matter - uni-bielefeld.de · 2013. 4. 26. · bulk viscous ﬂuid and the...

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


1 Motivation for non-standard pressureless CDM

2 The dynamics of the ΛvCDM model and observational constraintsBackground expansionPerturbationsComparison with observationsStructure Formation

3 Final Remarks


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


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


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)


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 ).


Integrated Sachs-Wolfe effect


Structure of the CMB power spectrum


Background expansion

A ΛvCDM universe?



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)



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.


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)


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.



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



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



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).


Structure Formation

Hierarchical Structure Formation: Merger Tree


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).


Structure Formation


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.


Structure Formation


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)


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)


Structure Formation


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)

]


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.


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


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.


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).


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


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


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


Structure Formation

Some words about numerical simulations -

http://ooo.aip.de/groups/cosmology/


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)


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)


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


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


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!


Final Remarks

We model CDM as a bulk viscous fluid: negative pressure


Final Remarks


We allow the existence of a bulk viscosity contributing to theobserved acceleration without being the major cause.


Final Remarks



Background data is able to place an upper bound to theviscosity parameter ξ . 0.2 (ξ0 . 107 Pa.s ), at 2σ.


Final Remarks




The dynamics of the linear sub horizon perturbations placesstronger constraints on the dark matter viscosityξ . 0.2× 10−10 (ξ0 . 10−3 Pa.s).


Final Remarks





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?


Final Remarks





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 ...

dissipation of dark matter - uni-bielefeld.de · 2013. 4. 26. · bulk viscous ﬂuid and the...

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