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

Holographic Dark Energy Mathijs van de Mast Rijksuniversiteit Groningen Mathijs van de Mast (RUG) Holographic Dark Energy 1 / 20

Introduction contents The holographic principle The infrared cutoff Holographic dark energy Agegraphic dark energy Ricci dark energy Mathijs van de Mast (RUG) Holographic Dark Energy 2 / 20

Introduction The holographic principle Several restrictions on the number of degrees of freedom of a system Classical field theory + Ultraviolet cutoff S max = 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. Mathijs van de Mast (RUG) Holographic Dark Energy 3 / 20

Introduction recording information No distribution of matter will ever require more than 1 bit per Planck area on the screen. Figure: A black hole projected onto the screen. Figure from Susskind (1994) Mathijs van de Mast (RUG) Holographic Dark Energy 4 / 20

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: L 3 C 3 UV � S bh ≡ π L 2 M 2 P L acts as IR cutoff, and scales with C − 3 UV But then QFT fails on large scales. Mathijs van de Mast (RUG) Holographic Dark Energy 5 / 20

Introduction Choosing an IR cutoff Tighter constraint: L 3 C 4 UV � LM 2 P So that L scales like C − 2 UV Largest IR cutoff, holographic dark energy density: ρ hde = 3 c 2 M 2 P L − 2 L ∼ H − 1 implies C UV ∼ 10 − 2 . 5 eV Problem: EoS parameter w = 0 � t dt Particle horizon size l ph = a 0 a EoS parameter still more than − 1 / 3: we need another number. Mathijs van de Mast (RUG) Holographic Dark Energy 6 / 20

Holographic dark energy The future event horizon The HDE model is independent of the vacuum energy bound. Future event horizon: � ∞ � ∞ dt da R h = a a = a Ha 2 t a HDE energy density: ρ hde = 3 c 2 M 2 p R 2 h = 3 α 2 M 2 p a − 2(1 − 1 / c ) w = − 1 3 − 2 3 c c is a free parameter. Equation of motion for Ω hde : � � Ω hde 1 1 2 hde = (1 − Ω hde ) Ω hde + . c √ Ω hde Ω 2 dx Ω hde is always positive. dx Mathijs van de Mast (RUG) Holographic Dark Energy 7 / 20

Holographic dark energy The solution for c = 1 3 ln (1 − √ Ω hde ) ≃ ln a + x 0 High a limit: 1 This results in √ Ω hde = 1 − 3 − 8 2 3 e − 3 x 0 a − 3 Low a limit: Ω hde = e x 0 a Result: ρ hde ≃ = e x 0 ρ m , 0 a − 2 EoS for arbitary c : w = − 1 3 − 2 � Ω hde , 0 + 1 1 + 2 � � � Ω hde , 0 (1 − Ω hde , 0 ) � Ω hde , 0 z 3 c 6 c c c = 1 and Ω hde , 0 = 0 . 73 implies w = − 0 . 903 + 0 . 104 z , which is in excellent agreement with observation Mathijs van de Mast (RUG) Holographic Dark Energy 8 / 20

Holographic dark energy Problems with HDE Locality. Causality. Circular dependency on acceleration. Mathijs van de Mast (RUG) Holographic Dark Energy 9 / 20

Agegraphic dark energy The K´ arolyh´ azy relation A distance t in Minkowski space-time cannot be measured to a better accuracy than δ t = λ t 2 / 3 P t 1 / 3 The minimal energy of a cell of this size inside a region of size t is P t 2 ∼ m 2 ρ q ∼ E δ t 2 1 P δ t 3 ∼ t 2 t 2 The original agegraphic dark energy model takes the time scale t to be the age of the universe. This results in the density ρ hde ∼ l − 2 P l − 2 Mathijs van de Mast (RUG) Holographic Dark Energy 10 / 20

Agegraphic dark energy Original agegraphic model � a da The age of the universe T = 0 Ha Agegraphic energy density ρ ade = 3 n 2 m 2 P T 2 √ Ω ade ade ≡ d Ω ade 3 − 2 � � Equation of motion: Ω ′ = Ω ade (1 − Ω ade ) dx n √ Ω ade Equation of state parameter: w ade = − 1 + 2 3 n In the matter-dominated epoch, the solution is Ω ade ∝ a 3 . Problem: There is no dark energy-dominated epoch. Problem: In the matter dominated epoch, ρ hde ∝ a − 3 and ρ hde ∝ a 3 Mathijs van de Mast (RUG) Holographic Dark Energy 11 / 20

Agegraphic dark energy New agegraphic model The time scale is taken to be the conformal age of the universe. ρ ade = 3 n 2 m 3 P , with η 2 � dt da � η ≡ a = a 2 H √ Ω ade � � Equation of motion: d Ω ade = Ω ade 3 − 2 a (1 − Ω ade ) da n a √ Ω ade Equation of state: w ade = − 1 + 2 3 n a At late times, Ω hde → 1 In the matter dominated epoch, w ade = − 2 / 3 Solution: Ω ade = n 2 a 2 / 4. The conformal age is a logical choice in the FRW metric, and it is the causal time. Mathijs van de Mast (RUG) Holographic Dark Energy 12 / 20

Agegraphic dark energy Interaction What if dark energy and matter can exchange energy? The energy conservation becomes ρ ade + 3 H ρ ade (1 + w ade ) = − Q ˙ ρ m + 3 H ρ m (1 + w m ) = Q ˙ √ Ω ade � � � � New EoM: d Ω ade = Ω ade 3(1 + w m ) − 2 Q (1 − Ω ade ) − da a n a 3 m 2 p H 3 √ Ω ade New EoS: w ade = − 1 + 2 Q − 3 n a 3 H ρ ade Note that interaction allows w q to ”cross the phantom divide” Mathijs van de Mast (RUG) Holographic Dark Energy 13 / 20

Agegraphic dark energy Problems with the agegraphic model The squared speed of sound Ω ′ v 2 s , ade = dp ade / d ρ ade = − 9 n (1+ w ade ) √ Ω ade + w ade ade Figure: Fractional energy densities, EoS and speed of sound as a function of e-folding time x . Image from Kim et al. (2007) Mathijs van de Mast (RUG) Holographic Dark Energy 14 / 20

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 + 2 H 2 ) The dark energy density is 2 − α Ω m 0 e − 3 x + f 0 e − (4 − 2 /α ) x 8 π G ( ˙ 3 α H + 2 H 2 ) = − α α ρ rde = 16 π G R = At high redshift, the RDE behaves similar to dark matter. Mathijs van de Mast (RUG) Holographic Dark Energy 15 / 20

Ricci dark energy Equation of state parameter Figure: Evolution of the EoS parameter w for the Ricci dark energy as a function of redshift. Image from Gao et al. (2008) Mathijs van de Mast (RUG) Holographic Dark Energy 16 / 20

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 Mathijs van de Mast (RUG) Holographic Dark Energy 17 / 20

Ricci dark energy Age of the universe 1 1 dx � 1+ z The age t = h : 0 H 0 Figure: Age of the universe as a function of redshift. Observations of three old objects are plotted as well. Image from Gao et al. 2008 Mathijs van de Mast (RUG) Holographic Dark Energy 18 / 20

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. Mathijs van de Mast (RUG) Holographic Dark Energy 19 / 20

see ya! Questions? Mathijs van de Mast (RUG) Holographic Dark Energy 20 / 20