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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.”
13 June 2012 ITP, Heidelberg 3
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
13 June 2012 ITP, Heidelberg 4
13 June 2012 ITP, Heidelberg 5
The background we know…
13 June 2012 ITP, Heidelberg 6
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
13 June 2012 ITP, Heidelberg 7
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
13 June 2012 ITP, Heidelberg 8
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
13 June 2012 ITP, Heidelberg 9
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
13 June 2012 ITP, Heidelberg 10
EoM
EMT
13 June 2012 ITP, Heidelberg 11
EMT Decomposition
13 June 2012 ITP, Heidelberg 12
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
13 June 2012 ITP, Heidelberg 13
𝛻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
13 June 2012 ITP, Heidelberg 14
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
13 June 2012 ITP, Heidelberg 15
Hydrodynamics or Not?
13 June 2012 ITP, Heidelberg 16
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
13 June 2012 ITP, Heidelberg 17
Hydrodynamics or Not?
13 June 2012 ITP, Heidelberg 18
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
𝒫 = 𝒫(𝜏, 𝑇) ℰ = ℰ 𝜏, 𝑇
𝛿𝒫rf = 𝒫,𝑇𝛿𝑇 𝛿ℰrf = ℰ,𝑇𝛿𝑇
𝛿𝒫rf = 𝑐s2𝛿ℰrf
𝛿𝒫 = 𝑐s2𝛿ℰ+ ℰ 𝑐a
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
13 June 2012 ITP, Heidelberg 19
𝛿𝒫 = 𝑐s2𝛿ℰ+ ℰ 𝑐a
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:
13 June 2012 ITP, Heidelberg 20
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
13 June 2012 ITP, Heidelberg 21
We treat the scalar completely differently to all other “stuff”
13 June 2012 ITP, Heidelberg 22
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
13 June 2012 ITP, Heidelberg 23
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
13 June 2012 ITP, Heidelberg 24
𝛻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?
13 June 2012 ITP, Heidelberg 25
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
13 June 2012 ITP, Heidelberg 26
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
13 June 2012 ITP, Heidelberg 27
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
13 June 2012 ITP, Heidelberg 28
Evolution from Conservation
13 June 2012 ITP, Heidelberg 29
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
13 June 2012 ITP, Heidelberg 30
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
13 June 2012 ITP, Heidelberg 31