Holographic Dark Energy Mathijs van de Mast Rijksuniversiteit - - PowerPoint PPT Presentation

Holographic Dark Energy

Mathijs van de Mast

Rijksuniversiteit Groningen

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Introduction

contents

The holographic principle The infrared cutoff Holographic dark energy Agegraphic dark energy Ricci dark energy

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Introduction

The holographic principle

Several restrictions on the number of degrees of freedom of a system Classical field theory + Ultraviolet cutoff Smax = ln N(V ) = V ln(2)l−3

p

Bekenstein bound: black hole entropy, scales with horizon area Holographic principle: All phenomena in a region of space can be fully described by a set of DoF’s on the bounding surface. Information density: 1 bit per planck area.

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Introduction

recording information

No distribution of matter will ever require more than 1 bit per Planck area

• n the screen.

Figure: A black hole projected onto the screen. Figure from Susskind (1994)

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Introduction

Fine tuning

In conventional quantum gravity, entropy scales with volume. S ∝ VC 3

UV .

Quantum corrections to vacuum energy are typically divergent. Limit the volume of a system: L3C 3

UV Sbh ≡ πL2M2 P

L acts as IR cutoff, and scales with C −3

UV

But then QFT fails on large scales.

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Introduction

Choosing an IR cutoff

Tighter constraint: L3C 4

UV LM2 P

So that L scales like C −2

UV

Largest IR cutoff, holographic dark energy density: ρhde = 3c2M2

PL−2

L ∼ H−1 implies CUV ∼ 10−2.5eV Problem: EoS parameter w = 0 Particle horizon size lph = a t

dt a

EoS parameter still more than −1/3: we need another number.

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Holographic dark energy

The future event horizon

The HDE model is independent of the vacuum energy bound. Future event horizon: Rh = a ∞

t dt a = a

∞

a da Ha2

HDE energy density: ρhde = 3c2M2

pR2 h = 3α2M2 pa−2(1−1/c)

w = − 1

3 − 2 3c

c is a free parameter. Equation of motion for Ωhde:

Ωhde dx 1 Ω2

hde = (1 − Ωhde)

• 1

Ωhde + 2 c√Ωhde

• .

Ωhde dx

is always positive.

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Holographic dark energy

The solution for c = 1

High a limit: 1

3 ln (1 − √Ωhde) ≃ ln a + x0

This results in √Ωhde = 1 − 3−823e−3x0a−3 Low a limit: Ωhde = ex0a Result: ρhde ≃= ex0ρm,0a−2 EoS for arbitary c: w = − 1

3 − 2 3c

Ωhde,0 + 1

6c

Ωhde,0(1 − Ωhde,0)

• 1 + 2

c

Ωhde,0

• z

c = 1 and Ωhde,0 = 0.73 implies w = −0.903 + 0.104z, which is in excellent agreement with observation

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Holographic dark energy

Problems with HDE

Locality. Causality. Circular dependency on acceleration.

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Agegraphic dark energy

The K´ arolyh´ azy relation

A distance t in Minkowski space-time cannot be measured to a better accuracy than δt = λt2/3

P t1/3

The minimal energy of a cell of this size inside a region of size t is ρq ∼ Eδt2

δt3 ∼ 1 t2

Pt2 ∼ m2 P

t2

The original agegraphic dark energy model takes the time scale t to be the age of the universe. This results in the density ρhde ∼ l−2

P l−2

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Agegraphic dark energy

Original agegraphic model

The age of the universe T = a

da Ha

Agegraphic energy density ρade = 3n2m2

P

T 2

Equation of motion: Ω′

ade ≡ dΩade dx

= Ωade(1 − Ωade)

• 3 − 2

n

√Ωade

• Equation of state parameter: wade = −1 + 2

3n

√Ωade In the matter-dominated epoch, the solution is Ωade ∝ a3. Problem: There is no dark energy-dominated epoch. Problem: In the matter dominated epoch, ρhde ∝ a−3 and ρhde ∝ a3

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Agegraphic dark energy

New agegraphic model

The time scale is taken to be the conformal age of the universe. ρade = 3n2m3

P

η2

, with η ≡ dt

a =

• da

a2H

Equation of motion: dΩade

da

= Ωade

a (1 − Ωade)

• 3 − 2

n √Ωade a

• Equation of state: wade = −1 + 2

3n √Ωade a

At late times, Ωhde → 1 In the matter dominated epoch, wade = −2/3 Solution: Ωade = n2a2/4. The conformal age is a logical choice in the FRW metric, and it is the causal time.

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Agegraphic dark energy

Interaction

What if dark energy and matter can exchange energy? The energy conservation becomes ˙ ρade + 3Hρade(1 + wade) = −Q ˙ ρm + 3Hρm(1 + wm) = Q New EoM: dΩade

da

= Ωade

a

• (1 − Ωade)
• 3(1 + wm) − 2

n √Ωade a

• −

Q 3m2

pH3

• New EoS: wade = −1 + 2

3n √Ωade a

−

Q 3Hρade

Note that interaction allows wq to ”cross the phantom divide”

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Agegraphic dark energy

Problems with the agegraphic model

The squared speed of sound v2

s,ade = dpade/dρade = − Ω′

ade

9n(1+wade)√Ωade + wade

Figure: Fractional energy densities, EoS and speed of sound as a function of e-folding time x. Image from Kim et al. (2007)

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Ricci dark energy

Ricci dark energy

The Ricci dark energy model takes the average radius of the Ricci scalar curvature R−1/2 as a length scale. The Ricci scalar is R = −6( ˙ H + 2H2) The dark energy density is ρrde =

3α 8πG ( ˙

H + 2H2) = −

α 16πG R = α 2−αΩm0e−3x + f0e−(4−2/α)x

At high redshift, the RDE behaves similar to dark matter.

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Ricci dark energy

Equation of state parameter

Figure: Evolution of the EoS parameter w for the Ricci dark energy as a function

• f redshift. Image from Gao et al. (2008)

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Ricci dark energy

Energy densities

At early times, the densities of dark energy and matter are comparable. Acceleration began at low redshift, solving the coincidence problem.

Figure: Evolutions of radiation density (crosses), non-relativistic matter density (solid line) and Ricci dark energy density (circles). Image from Gao et al. 2008

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Ricci dark energy

Age of the universe

The age t =

1 H0

• 1

1+z

dx h :

Figure: Age of the universe as a function of redshift. Observations of three old

• bjects are plotted as well. Image from Gao et al. 2008

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Ricci dark energy

Problems with the Ricci model

Dark energy density is proportional to the Ricci tensor. This means that it is relatively small during the radiation dominated epoch, which solves the coincidence problem. This model seems to work fine, but there is no clear physical motivation for it. Perhaps quantum gravity will offer one, some day.

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see ya!

Questions?

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