theoretical explanations for the accelerating...
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Theoretical Explanations for the accelerating Universe
JGRG16, Niigata, Japan 27 November 2006
Éanna Flanagan, Cornell
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• Observations show that the expansion of the Universe is accelerating, which according to general relativity implies the existence of a form of matter with negative pressure (dark energy).
The Universe today
a
a= −4πG
3[ρ + 3p]
• Acceleration if p < −13ρ
Dark matter: well-motivated candidate particles, recent evidence against modified gravity
Dark energy: nature, properties unknown
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Evidence for acceleration
• Friedmann equation:
1H2
0
a2
a2=
ΩM
a3+
ΩΛ
a3(1+w)+
ΩR
a4+
Ωk
a2,
1 = ΩM + ΩΛ + ΩR + Ωk
• Cosmic microwave background (standard yardstick)
ΩR ≈ 10−4, |Ωk| ≤ 0.05
• Infer a(t) from brightness and redshifts of standard candles (Type Ia supernova)
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Evidence for acceleration (cont.)
No Big Bang
1 2 0 1 2 3
expands forever
-1
0
1
2
3
2
3
closed
recollapses eventually
Supernovae
CMB
Clusters
open
flat
Knop et al. (2003)Spergel et al. (2003)Allen et al. (2002)
Supernova Cosmology Project
Ω
ΩΛ
M
(Spergel et. al. 2006)
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• Cosmological constant
- simplest model, agrees with data, theoretical problems
• Backreaction of perturbations
- no new physics required, seems unlikely, not yet ruled out
• Dynamical dark energy models
• Modifications of general relativity
Explanations for acceleration: categories
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• Cosmological constant
- simplest model, agrees with data, theoretical problems
• Backreaction of perturbations
- no new physics required, seems unlikely, not yet ruled out
• Dynamical dark energy models
• Modifications of general relativity
Explanations for acceleration: categories
Mixed models (generic)
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1. Partial classical models
- for example, , or
2. Complete classical models,
3. Weakly coupled effective field theories
4. “Natural”, weakly coupled field theories
5. UV complete theories, eg string theory models
Explanations for acceleration: frameworks
H2 = f(ρ) cs = cs(z), w = w(z)
S =∫
d4x . . .
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Simplest model: Cosmological constant
• The case , constant energy density • Puzzle: expect quantum loops to generate a much larger
energy density, • Unknown higher energy physics can in principle cancel out this
contribution, but it requires exquisite fine tuning.• String theory predicts a multiverse with huge number of
different vacua, each with its own . Anthropic principle could explain smallness of our
• Problem: life might still evolve if were 1000 times larger.
w = −1 ρΛ ∼ (10−3 eV)4
ρΛ ∼ E4cutoff
ρΛ
ρΛ
ρΛ
• Puzzle: The Universe has expanded by 35 orders of magnitude. Why are dark energy and matter comparable right now? Seems to require fine tuning of initial conditions.
(Loeb 2006)
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Dark energy versus time
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Cosmological constant (cont.) • One idea for solving the cosmological constant problem is the fat
graviton (Zee 2003, Sundrum 2003)
• The graviton is a composite particle with size• This suppresses loop diagrams that contribute to Λ
!grav
• Room for a fat graviton: 20 µm ! !grav ! 100 µm
Short distance tests of Newton’s law
Observed value of Λ
• Hard to construct explicit model
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• Cosmological constant
- simplest model, agrees with data, theoretical problems
• Backreaction of perturbations
- no new physics required, seems unlikely, not yet ruled out
• Dynamical dark energy models
• Modifications of general relativity
Explanations for acceleration: categories
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Backreaction of perturbations
Gab = 8πρuaub
gab = g(0)ab + εg(1)
ab + ε2g(2)ab
ρ = ρ(0) + ερ(1) + ε2ρ(2)
ua = u(0)a + εu(1)
a + ε2u(2)a
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Backreaction of perturbations
• Approximate momentum conservation forbids contributions to low spatial frequency observables (Hubble expansion rate) from interaction between subhorizon and superhorizon modes
k ∼ k1 + k2
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Backreaction of perturbations (cont.)
• Can backreaction drive the observed acceleration?
Super Horizon
Sub Horizon
Yes NoKolb et. al. 2004a,b; 2005,
Wiltshire 2005, Moffat 2005, Brandenberger 2005
Rasanen 2003, 2006; Notari 2005; Kolb et. al. 2005; Nambu & Tanimoto 2005; Alnes et. al.
2005; Kai et. al. 2006
Hirata & Seljack 2005; EF 2005; Geshnizjani et. al.
2005
Fry & Siegel 2005; Ishibashi & Wald 2005;
Linder 2005; Gruzinov et. al. 2006; Kasai et. al. 2006
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Backreaction of perturbations (cont.)
• Can backreaction drive the observed acceleration?
Super Horizon
Sub Horizon
Yes NoKolb et. al. 2004a,b; 2005,
Wiltshire 2005, Moffat 2005, Brandenberger 2005
Rasanen 2003, 2006; Notari 2005; Kolb et. al. 2005; Nambu & Tanimoto 2005; Alnes et. al.
2005; Kai et. al. 2006
Hirata & Seljack 2005; EF 2005; Geshnizjani et. al.
2005
Fry & Siegel 2005; Ishibashi & Wald 2005;
Linder 2005; Gruzinov et. al. 2006; Kasai et. al. 2006 ?
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Backreaction of perturbations (cont.)
• Can backreaction drive the observed acceleration?
Super Horizon
Sub Horizon
Yes NoKolb et. al. 2004a,b; 2005,
Wiltshire 2005, Moffat 2005, Brandenberger 2005
Rasanen 2003, 2006; Notari 2005; Kolb et. al. 2005; Nambu & Tanimoto 2005; Alnes et. al.
2005; Kai et. al. 2006
Hirata & Seljack 2005; EF 2005; Geshnizjani et. al.
2005
Fry & Siegel 2005; Ishibashi & Wald 2005;
Linder 2005; Gruzinov et. al. 2006; Kasai et. al. 2006 ?
• Objection: if , expect
- Other dimensionless numbers present, e.g.
• Objection: data requires , so pert. theory fails
- Already interesting if pert. theory predicts large effects
g(1)ab ∼ 10−5 g(2)
ab ∼ 10−10
knonlinear/kHubble
g(2)ab ∼ 1
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Superhorizon perturbations
• Original work used 2nd order pert. theory, complex, subtle errors• There is a simple counter-argument. Consider a Universe with
no subhorizon perturbations: local Taylor series expansions of general solutions of the Einstein-dust equations can be used.
• Observer measures luminosity distance , infers via DL
〈DL(z, θ,ϕ)〉θ,ϕ =z
H0+
12H0
(1 − q0)z2 + O(z3)
• We obtain
q0 =4π
3H20
ρ +1
3H20
[75σabσ
ab − ωabωab
]
• Negative would require large velocity gradients at horizon scale, excluded by observations
q0
q0 = −aa/a2
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Subhorizon perturbations
• Averaged Einstein equations: Gab[gcd] = 8πTab −⟨G(2)[δgcd; gcd]
⟩
gab ≡ 〈gab〉 , δgab ≡ gab − gab, Tab ≡ 〈Tab〉
• Can effective stress tensor of fluctuations drive acceleration?• Order of magnitude estimate for perturbation of size R and mass M
in Newtonian regime
δg ∼ M
R, Tab ∼ (∇δg)2 ∼ M2
R4
∆M
M∼ TabR3
M∼ M
R# 1
• Smallness confirmed by Newtonian pert. theory computations of Siegel and Fry (2005)
• Post-Newtonian corrections? Unresolved
(Ellis, 1984; Buchert 2003, 2005)
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Subhorizon perturbations (cont.)
• Observer measures and infers deceleration using flat FRW relation
DL(z) ≡ 〈DL(z, θ,ϕ)〉
q(z) =1− (1 + z)2D′′
L(z)(1 + z)D′
L(z)−DL(z)
• Is is possible to measure in any inhomogeneous dust Universe?
• Is it possible to reproduce observed ? • Do the spectrum of perturbations in our Universe give rise
to ?
q(z) < 0
q(z) < 0
q(z)
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Subhorizon perturbations (cont.)
• Observer measures and infers deceleration using flat FRW relation
DL(z) ≡ 〈DL(z, θ,ϕ)〉
q(z) =1− (1 + z)2D′′
L(z)(1 + z)D′
L(z)−DL(z)
• Is is possible to measure in any inhomogeneous dust Universe?
• Is it possible to reproduce observed ? • Do the spectrum of perturbations in our Universe give rise
to ?
q(z) < 0
q(z) < 0
q(z)YES
YES
• Analyzed using Bondi-Tolman models, spherically symmetric dust Universes (Nambu & Tanimoto 2005; Alnes et. al.
2005; Chuang et. al. 2005; Vanderveld et. al. 2006; Apostolopoulos et. al. 2006; Kai
et. al. 2006; Garfinkle 2006)
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Subhorizon perturbations (cont.)
• Observer measures and infers deceleration using flat FRW relation
DL(z) ≡ 〈DL(z, θ,ϕ)〉
q(z) =1− (1 + z)2D′′
L(z)(1 + z)D′
L(z)−DL(z)
• Is is possible to measure in any inhomogeneous dust Universe?
• Is it possible to reproduce observed ? • Do the spectrum of perturbations in our Universe give rise
to ? Unresolved
q(z) < 0
q(z) < 0
q(z)YES
YES
• Analyzed using Bondi-Tolman models, spherically symmetric dust Universes (Nambu & Tanimoto 2005; Alnes et. al.
2005; Chuang et. al. 2005; Vanderveld et. al. 2006; Apostolopoulos et. al. 2006; Kai
et. al. 2006; Garfinkle 2006)
• Requires detailed computations to confirm or refute claim of Kolb et. al. (2005)
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Subhorizon perturbations (cont.)
• Bondi-Tolman metric
ds2 = −dt2 +(
∂R(r, t)∂r
)2
dr2 + R(r, t)2dΩ2
R(r, t) ∝ r[t− t0(r)]2/3, t0(r) = −λr2/(r2 + D2)
(Vanderveld et. al. 2006)
q(z) =12
+32w(z)ΩΛ(z)
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• Cosmological constant
- simplest model, agrees with data, theoretical problems
• Backreaction of perturbations
- no new physics required, seems unlikely, not yet ruled out
• Dynamical dark energy models
• Modifications of general relativity
Explanations for acceleration: categories
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Dynamical models of dark energy
• Typically do not address cosmological constant problem• Can address the cosmic coincidence problem• Typically invoke new fundamental or effective fields
S = −∫
d4x√−g
[12(∇Φ)2 + V (Φ)
]
ρ =12Φ2 + V (Φ), p =
12Φ2 − V (Φ)
Φ2 ! V (Φ) =⇒ p ≈ −ρ
• Problem: expect loop corrections to spoil flatness of potential and small mass of scalar field
• One solution: scalar field is the size of compact extra dimensions (radion), protected by diffeomorphism invariance
Quintessence:
(e.g., Burgess et. al. 2006)
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Modified Gravity vs. Dark Energy
• Evidence for dark energy presumes validity of general relativity. Perhaps, instead, general relativity is modified on large scales.
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Modified Gravity vs. Dark Energy
• Evidence for dark energy presumes validity of general relativity. Perhaps, instead, general relativity is modified on large scales.
• How do we decide if a given a modification of the laws of physics involves a modification of gravity? Perhaps ask which side of the Einstein equation is modified, or which term in action is modified:
Gµν = 8πGTµν
S =∫
d4x√−g
R
16πG+ Smatter[gµν ,Ψmatter]
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Modified Gravity vs. Dark Energy
• Evidence for dark energy presumes validity of general relativity. Perhaps, instead, general relativity is modified on large scales.
• How do we decide if a given a modification of the laws of physics involves a modification of gravity? Perhaps ask which side of the Einstein equation is modified, or which term in action is modified:
Gµν = 8πGTµν
S =∫
d4x√−g
R
16πG+ Smatter[gµν ,Ψmatter]
• This criterion is in fact ambiguous. Instead, we should ask if there are universal, long-range, 5th forces between macroscopic bodies.
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Modified Gravity vs. Dark Energy
Example:
S =∫
d4x√−g
R
16πG+ Smatter[eα(Φ)gµν ,Ψmatter]−
∫d4x√−g
[12(∇Φ)2 − V (Φ)
]
gµν ≡ eα(Φ)gµν , Φ ≡ f(Φ),
S =∫
d4x√−g
[A(Φ)R16πG
− 12(∇Φ)2 − V (Φ)
]+ Smatter[gµν ,Ψmatter]
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Modified Gravity vs. Dark Energy
Example:
• In this theory, scalar field both acts like quintessence and mediates 5th forces.
• This mixed character is generic, since loop corrections generate matter couplings.
• Solar system tests of gravity (light bending, perihelion precession) require
• These theories can arise as effective description of extra dimensions• Other possible observational signatures: time evolution of effective
Newton’s constant (fine structure constant for generalized models)
S =∫
d4x√−g
R
16πG+ Smatter[eα(Φ)gµν ,Ψmatter]−
∫d4x√−g
[12(∇Φ)2 − V (Φ)
]
gµν ≡ eα(Φ)gµν , Φ ≡ f(Φ),
S =∫
d4x√−g
[A(Φ)R16πG
− 12(∇Φ)2 − V (Φ)
]+ Smatter[gµν ,Ψmatter]
|α′(Φ)| ! 10−2 if V ′′(Φ)! (A.U.)−2
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Modifying gravitational action: a catalog
• Are there successful models that are not “mostly quintessence”?
S[gµν ,Ψm] =∫
d4x√−g
f(R)16πG
+ Sm[gµν ,Ψm]
- Equivalent to last model with , ruled out by Solar System (Chiba 2003)
- Loopholes: (i) Can make with fine tuning (Nojiri & Odintsov 2003) (ii) In special cases nonlinearities in potential can be important, acts like chameleon field models
α(Φ) = Φ/√
6
S[gµν ,Ψm] =∫
d4x√−g
f(R,RµνRµν , RµνλσRµνλσ)16πG
+ Sm[gµν ,Ψm]
- Problems with ghosts/acausality (Chiba 2005, Navarro et. al. 2005, de Felice et. al. 2005)
V (Φ0) ∼ m2pH
20 , V ′′(Φ0)" (A.U.)−2
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Modifying gravitational action: a catalog
- Ruled out; predicts modifications of particle physics at energy scale
S[gµν ,∇µ,Ψm] =∫
d4x√−g
f(R)16πG
+ Sm[gµν ,Ψm]
∼√
H0Mp ∼ 10−3 eV
S[gµν ,∇µ,Ψm] =∫
d4x√−g
f(R, R)16πG
+ Sm[gµν ,Ψm]
- Some successful models. Equivalent to tensor bi-scalar model similar to the mixed models discussed earlier.
• Some interesting models based on extra dimensions do not have a simple effective 4-dimensional description, eg DGP model
(EF 2005)
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Modifying gravitational action: a catalog
- The case , k-essence theories, solve coincidence problem
S[gµν ,Φ,Ψm] =∫
d4x√−g
R
16πG+ Sm[eα(Φ)gµν ,Ψm]−
∫d4x√−g F (K, Φ), K ≡ (∇Φ)2
α(Φ) = 0
F (K, Φ) = F (K)
- Well posed initial value formulation and avoiding ghosts require and (Armendiariz-Picon 2005)
- When in addition have with local mininum, get ghost condensate model (Arkani-Hamed et. al. 2003). Cosmological attractor. String theory version found (Mukohyama 2006)
F,K > 0 F,K + 2KF,KK > 0
- Superluminal propagation for . Not a problem classically (Bruneton 2005) but may be in QFT (Arkani-Hamed et al. 2006)
F,KK > 0
- Novel gravitational effects for . Corrections to Newton’s law, violations of equivalence principle (EF & Vanderveld 2006)
α(Φ) != 0
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• Probes of the expansion history of the Universe
• Probes of the growth of perturbations
• Precision tests of general relativity
• Measurements of time evolution in fundamental constants of nature
• Specific observational windows (i) Supernovae (ii) Measurements of numbers of clusters using CMBR (iii) Weak gravitational lensing (iv) Baryon acoustic oscillations
Observational probes of dark energy
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Image credit: Max Tegmark
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• The backreaction explanation for the acceleration of the Universe has not been ruled out, but seems unlikely.
• If backreaction does not work, explaining the acceleration requires new fundamental physics.
• The dark energy might be a cosmological constant. We may never be able to explain its tiny size.
• The dark energy may be dynamical. There are a variety of theoretical scenarios and a variety of observational windows.
Conclusions