dark energy with imperfect fluids - heidelberg universitycosmo/talks/... · 2012. 6. 18. ·...

IGNACY SAWICKI ITP, Heidelberg

Co-authors

Luca Amendola

Martin Kunz

Ippocratis Saltas

It’s still in preparation

13 June 2012 ITP, Heidelberg 2

Gen. Jack D. Ripper

“God willing, we will prevail, in peace and freedom from fear, and in true health, through the purity and k-essence of our natural... fluids. God bless you all.”


We have an understanding gap!

The ontological gap

• Dark energy and modified gravity treated differently

The universality gap

• Is DE a fluid?

• Does MG change the gravitational field?

The epistemological gap

• We only observe the motion of baryons


The background we know…


Supernova Cosmology Project Suzuki et al. (2011)

𝑤 ≃ −1

…the perturbations, not yet.

+𝑓𝜇𝜈 𝑔𝛼𝛽

𝐺eff

𝜂 ≡Φ +Ψ

Φ−Ψ

+𝑇𝜇𝜈DE

𝑤

𝑐s2

Mo

dified

Gravity

Dark En

ergy


𝐺𝜇𝜈 = 𝑇𝜇𝜈DM

Stuff Not Stuff

False Dichotomy?

• Equivalent to ℒ𝜙 = 𝑓′(𝜙)𝑅 + 𝑉(𝜙)

𝑓(𝑅)

• Just like 𝑓 𝑅 but not universally coupled Coupled DE

• Scalar must be coupled to gravity for consistency Galileons


All

an e

xtra

fo

rm o

f m

atte

r

The Purity of ?-Essence

Name Terms Extra Features

C.C. Λ No perturbations

Quintessence 𝑋 − 𝑉(𝜙) 𝑤 ≠ −1

k-essence 𝐾(𝜙, 𝑋) 𝑐s2 ≠ 1

KGB 𝐺(𝑋)□𝜙 𝑤 < −1 possible, coupling

ℒ4, ℒ5 𝐺𝑋 𝑋 𝐵𝜇𝜈𝐵

𝜇𝜈 − 𝐵2

+ 𝐺 𝑋 𝑅 𝜋, ?

𝑓 𝑅 𝑓′ 𝜙 𝑅 + 𝑉(𝜙) Coupling, 𝜋


𝑆 = d4𝑥 −𝑔 𝑅 + ℒ 𝜙, 𝛻𝜇𝜙,𝐵𝜇𝜈 , 𝑅𝜇𝜈 + ℒext

1 diff

2 diff

𝑋 = 𝜕𝜇𝜙2/2

𝑚 ≡ 2𝑋 𝐵𝜇𝜈 = 𝛻𝜈𝛻𝜇𝜙

Why not just solve EoM?

Too much info

Solution properties clear

Source of gravity

Relations not closed without EoM


EoM

EMT

EMT Decomposition


𝐺𝜇𝜈 = 𝑇𝜇𝜈 +𝑇𝜇𝜈ext

1 + 𝑓

𝑇𝜇𝜈 = ℰ𝑢𝜇𝑢𝜈 + 𝒫 ⊥𝜇𝜈 +2𝑞(𝜇𝑢𝜈) + 𝜏𝜇𝜈

𝑞𝜇 =⊥𝜇𝜆 𝛻𝜆𝑞

𝜏𝜇𝜈 = ⊥𝜇𝛼⊥𝜈

𝛽−1

3⊥𝜇𝜈⊥

𝛼𝛽 𝛻𝛼𝛻𝛽𝜋

𝑢𝜇 = 𝛻𝜇𝜙/𝑚 ⊥𝜇𝜈= 𝑔𝜇𝜈 + 𝑢𝜇𝑢𝜈

Perturbed EMT Conservation


𝛻𝜇𝑇𝜇𝜈 =𝑇𝜇𝜈ext 𝛻𝜇𝑓

1 + 𝑓 2

𝛿ℰ + 3𝐻 𝛿ℰ + 𝛿𝒫 + Ξ = 𝜌𝑆1

Ξ + 5𝐻Ξ − ℰ + 𝒫𝑘2Ψ

𝑎2−

𝑘2

𝑎2𝛿𝒫 +

2

3

𝑘2

𝑎2𝛿𝜋 = 𝜌𝑆2

d𝑠2 = − 1 + 2Ψ d𝑡2

+ 𝑎2 𝑡 1 + 2Φ d𝒙2

Ξ ≡ ℰ + 𝒫 Θ −𝑘2

𝑎2𝛿𝑞 + 𝑞 Θ

Closure Relations: Hydrodynamics


Yakov Zel’dovich (1914-1987) Major T. J. “King” Kong (1919-1964)

Hydro in CMB Cosmology

Lagrangian

• QED

Particles

Thermal

• 𝑓 𝜔, 𝑇, 𝜇 =2

exp𝜔

𝑇+1

EMT

Boltzmann

• 𝛻𝜇𝑇𝜇𝜈 = 0

No equations of motion for field values

Variables are 𝑇 and 𝜇

Evolution is conservation of perturbed EMT


Hydrodynamics or Not?


𝒫 = 𝒫(𝑇, 𝜇) ℰ = ℰ 𝑇, 𝜇

𝛿𝒫 = 𝒫,𝑇𝛿𝑇 + 𝒫,𝜇𝛿𝜇 𝛿ℰ = ℰ,𝑇𝛿𝑇 + ℰ,𝜇𝛿𝜇

𝛿𝒫 = 𝑐s2𝛿ℰ+ ℰ 𝑐a

2 − 𝑐s2 𝛿𝜇/𝜇

𝑐s2 ≡ 𝒫,𝑇/ℰ,𝑇

𝑐a2 ≡ 𝒫 /ℰ

Fluid

A Real Hydrodynamical Fluid


Hydrodynamics or Not?


𝒫 = 𝒫(𝑇, 𝜇) ℰ = ℰ 𝑇, 𝜇

𝛿𝒫 = 𝒫,𝑇𝛿𝑇 + 𝒫,𝜇𝛿𝜇 𝛿ℰ = ℰ,𝑇𝛿𝑇 + ℰ,𝜇𝛿𝜇


2 − 𝑐s2 𝛿𝜇/𝜇

𝑐s2 ≡ 𝒫,𝑇/ℰ,𝑇

𝑐a2 ≡ 𝒫 /ℰ

Fluid

𝒫 = 𝒫(𝜏, 𝑇) ℰ = ℰ 𝜏, 𝑇

𝛿𝒫rf = 𝒫,𝑇𝛿𝑇 𝛿ℰrf = ℰ,𝑇𝛿𝑇

𝛿𝒫rf = 𝑐s2𝛿ℰrf


2 − 𝑐s2 𝛿𝜏

Time-Evolving Substance

So what?

Take k-essence

The EMT is

So the pressure perturbation is

k-essence in general is not a fluid, since 𝜙 is the rest-frame clock

But: shift-symmetric k-essence does describe a superfluid



2 − 𝑐s2 𝛿𝜏

ℒ = 𝐾(𝜙,𝑚)

𝒫 𝜙,𝑚 = 𝐾 ℰ 𝜙,𝑚 = 𝑚𝐾𝑚 − 𝐾

𝛿𝒫 = 𝐶2𝛿ℰ + ℰ 𝑐a2 − 𝐶2

𝛿𝜙

𝑚

Putting it all together:


𝛿ℰ + 3𝐻 𝛿ℰ + 𝛿𝒫 + Ξ = 𝜌𝑆1

Ξ + 5𝐻Ξ − ℰ + 𝒫𝑘2Ψ

𝑎2−

𝑘2

𝑎2𝛿𝒫 +

2

3

𝑘2

𝑎2𝛿𝜋 = 𝜌𝑆2

𝛿𝒫 = 𝐶2𝛿ℰ − 3 𝑐a2 − 𝐶2

𝑎𝐻

𝑘

2 Ξ

𝐻 Ξ = ℰ + 𝒫 Θ

𝛿′′ +1

2− 3𝑤 −

3𝑤eff

2𝛿′ + 𝐶2

𝑘2

𝑎2𝐻2𝛿

−3 Ω𝑋 1+𝑤 +𝑤−3𝑤𝑤eff

2𝛿 =

3

21 + 𝑤 Ωm𝛿m

Subhorizon

So is solving EoM OK?

• BE/FD lowest state highly populated at 𝑇 = 0

• Axions

• Superfluid

Condensates

• Describe thermo potentials by scalars

• All shift-symmetric

Effective Thermo

• Always classical because?

• Scalar part of the gravity sector

Gravity


We treat the scalar completely differently to all other “stuff”

k-essence Brans-Dicke

𝑓 = const is just k-essence

𝐾 = 𝑉 𝜙 is ~𝑓 𝑅 gravity

Simplest class containing anisotropic stress and 𝑐s

2 ≠ 1

Stay in the Jordan frame


𝑆 = d4𝑥 −𝑔 1 + 𝑓(𝜙)𝑅

2+ 𝐾 𝜙, 𝑋 + ℒm[𝑔𝜇𝜈]

Is this “stuff” or “not-stuff”?

Speed of Sound

We can perturb this, collecting 2-derivative

𝒢𝜇𝜈𝛻𝜇𝛻𝜈𝛿𝜙 + 𝒪 𝛻𝜇𝜙𝛻𝜇𝛿𝜙… = 0

Contravariant acoustic metric


𝛻𝜇 𝐾𝑋𝛻𝜇𝜙 − 𝐾𝜙 =

1

2𝑓𝜙𝑅

𝒢𝜇𝜈 = −𝐷𝑢𝜇𝑢𝜈 + 𝐷𝑐s2 ⊥𝜇𝜈

𝐷 = 𝐸𝑚 + 3𝑓𝜙2/2 1 + 𝑓

𝑐s2 = 𝐷−1(𝑃𝑚 + 3𝑓𝜙

2 2 1 + 𝑓 )

k-essence: 𝑐s2 = 𝑃𝑚/𝐸𝑚

𝑓(𝑅): 𝑐s2 = 1

How many variables?


ℰ = 𝐸(𝜙,𝑚) − ϰ 𝜃

𝒫 = 𝑃 𝜙,𝑚 + ϰ 𝑚 𝑚+ 2

3𝜃

𝜋 = −ϰ

ϰ ≡ ln(1 + 𝑓)

𝑚~𝜙 𝜃 = 𝛻𝜇𝑢

𝜇

𝛽 = ϰ 2

𝑚2𝐷> 0

ϰ 𝑚

𝑚+ ϰ 𝑐s

2𝜃 +ϰ 𝐸𝜙

𝑚𝐷= 𝛽(𝐸 − 3𝑃 + 𝜌m)

ℰ = 𝐸(𝜙,𝑚) − ϰ 𝜃

𝒫 = 𝑃 𝜙,𝑚 − ϰ 𝑐s2 − 2

3𝜃 + 𝛽𝜌m

𝜋 = −ϰ(𝜙)

Two Types of MDE

Late Matter Domination Deep Matter Domination

k-essence terms dominate background

energy conserved

𝛽 ≪ ϰ /𝐻

𝑐s2 > 3𝛽

Ωm can be large

two-derivative terms dominate background

𝑤 irrelevant for evolution of ℰ

Physics just like 𝑓 𝑅 𝛽 = 1/3

𝑐s2 = 1


ℰ + 3𝐻 ℰ + 𝒫 =ϰ 𝜌m1 + 𝑓

ϰ ≪ 𝐻

Closure Relations

Parameter Perfect Regime

Imperfect Regime

𝐶2 𝑐s2 𝑐s

2 − 23

Π 0 1

Σ1 𝑐s2 − 𝑐a

2 𝑐s2 − 𝑐a

2 − 23

Σ2 small 0

𝜛1 0 1

𝛽 𝛽 𝛽


𝛿𝒫 ≃ 𝐶2𝛿ℰ + Σ1𝑎𝐻

𝑘

2 Ξ

𝐻+ Σ2

Ξ

𝐻+

𝛽𝛿𝜌

1 + 𝑓

𝑘2

𝑎2𝛿𝜋 ≃ Π𝛿ℰ + 𝜛1

𝑎𝐻

𝑘

2 Ξ

𝐻

𝑘T2

𝑎2𝐻2≡Ω𝑋 1 + 𝑤

𝛽

𝑘2

𝑎2𝛿𝒫 + 2

3𝑘2

𝑎2𝛿𝜋

∼ 𝑐s2𝑘2

𝑎2𝛿ℰ

Imperfect Regime

Lensing potential

• 𝑘2

𝑎2Φ−Ψ = 𝛿ℰ + 𝛿𝜌m

1+𝑓− 𝑘

2

𝑎2𝛿𝜋 ≃ 𝛿𝜌m

1+𝑓

Shear parameter

• 𝜂 ≡Φ+Ψ

Φ−Ψ=

𝑘2 𝑎2𝛿𝜋 𝛿𝜌m1+𝑓

≃𝛿ℰ 1+𝑓

𝛿𝜌m=

Ω𝑋

Ωm

𝛿

𝛿m


Evolution from Conservation


𝛿′′ +1

2− 3𝑤 −

3𝑤eff

2𝛿′ + 𝑐s

2 𝑘2

𝑎2𝐻2𝛿

−𝑀12(Ω𝑋, 𝑤)𝛿 =

3

21 + 𝑤 Ωm𝛿m

𝛿′′ −3

2

1

2+ 2𝑤 + 𝑤eff 𝛿

′ + 𝑐s2 𝑘

2

𝑎2𝐻2𝛿

−𝑀22(Ω𝑋, 𝑤)𝛿 = −𝛽

Ωm

Ω𝑋

𝑘2

𝑎2𝐻2𝛿m

Imperfect

Perfect

𝑘T2

𝑎2𝐻2= Ω𝑋(1 + 𝑤)/𝛽

Not the 𝑓(𝑅) solution

DE very clustered inside Jeans length

Particular solution 𝛿ℰ = −𝛽

𝑐s2𝛿𝜌m1+𝑓

∝ 𝑎−2

In deep MDE 2Φ +Ψ = 0

But the homogeneous mode decays slower

𝛿 = 𝑎(1+3𝑤)/2𝐴 cos2𝑐s

1+3𝑤

𝑘

𝑎𝐻+ 𝜑

Φ is dominated by DE and oscillates


𝛿′′ −3

2

1

2+ 2𝑤 𝛿′ + 𝑐s

2 𝑘2

𝑎2𝐻2𝛿 ≃ −𝛽

Ωm

Ω𝑋

𝑘2

𝑎2𝐻2𝛿m

The Takeaway

Conservation of the EMT contains all the useful and a minimum of useless information

Classical scalar fields do not in general obey hydrodynamical closure relations given understanding of the d.o.f. can calculate them: we have a prescription

More general scalar theories contain a new scale, separating the

perfect and imperfect

The real speed of sound (causality) determines the Jeans length

Gravity not really modified in MG fluid carries anisotropy need perturbations of DE to split potentials coupling gives large DE perturbations inside Jeans horizon


dark energy with imperfect fluids - heidelberg universitycosmo/talks/... · 2012. 6. 18. ·...

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