Theoretical Explanations for the accelerating Universe

JGRG16, Niigata, Japan 27 November 2006

Éanna Flanagan, Cornell

• 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

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)

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)

• 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

• 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)

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 . . .

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)

Dark energy versus time

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

• 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

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

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

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

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 ?

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

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

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)

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)

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)

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)

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)

• 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

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)

Modified Gravity vs. Dark Energy

• Evidence for dark energy presumes validity of general relativity. Perhaps, instead, general relativity is modified on large scales.

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]

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.

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]

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

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

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)

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

• 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

Image credit: Max Tegmark

• 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