1 1 The Darkness of the Universe: The Darkness of the Universe: Acceleration and Deceleration Eric Linder Lawrence Berkeley National Laboratory 2 2 Discovery! Acceleration 3 3 accelerating decelerating accelerating decelerating cf. Tonry et al. (2003) Cosmic Concordance Supernovae alone Accelerating expansion > 0 CMB (plus LSS) Flat universe > 0 Any two of SN, CMB, LSS Dark energy ~75% 4 4 Acceleration and Particle Physics Key element is whether (aH) -1 = -1 is increasing or decreasing. I.e. is there acceleration: >0. Also, ~aH~H/T~T/M p for classical radiation, but during inflation this redshifts away and quantum particle creation enters. a.. Comoving scale -1 Time horizon scale Inflation The conformal horizon scale (aH) -1 tells us when a comoving scale (e.g. perturbation mode) leaves or enters the horizon. 5 5 Acceleration = Curvature The Principle of Equivalence teaches that Acceleration = Gravity = Curvature Acceleration over time will get v=gh, so z = v = gh (gravitational redshift). But, t t 0 parallel lines not parallel (curvature)! t0t0 t Height Time 6 6 Equations of Motion Expansion rate of the universe a(t) ds 2 = dt 2 +a 2 (t)[dr 2 /(1-kr 2 )+r 2 d 2 ] Friedmann equations (/a) 2 = H 2 = (8 /3M p 2 ) [ m + ] /a = -(4 /3M p 2 ) [ m + +3p ] Einstein-Hilbert action S = d 4 x -g [ R/2 + L + L m ] a.. 7 7 Spacetime Curvature Ricci scalar curvature R = R = 6 [ a/a + (/a) 2 ] = 6 ( a/a + H 2 ) Define reduced scalar curvature R = R/(12H 2 ) = (1/2) [1 + aa/ 2 ] = (1/2)(1-q) Note that division between acceleration and deceleration occurs for R =1/2 (q=0). Superacceleration (phantom models) is not (a) > 0, but (a/a) > 0, i.e. R > 8 8 Todays Inflation To learn about the physics behind dark energy we need to map the expansion history. Subscripts label acceleration: R = (1-q)/2 q = -a / 2 R =1/4 EdS R =1/2 acc R =1 superacc a.. 9 9 Equations of Motion Expansion rate of the universe a(t) ds 2 = dt 2 +a 2 (t)[dr 2 /(1-kr 2 )+r 2 d 2 ] Friedmann equations (/a) 2 = H 2 = (8 /3M p 2 ) [ m + ] /a = -(4 /3M p 2 ) [ m + +3p ] Einstein-Hilbert action S = d 4 x -g [ R/2 + L + L m ] a.. 10 Scalar Field Theory Scalar field Lagrangian - canonical, minimally coupled L = (1/2)( ) 2 - V( ) Noether prescription Energy-momentum tensor T =(2/ -g) [ ( -g L )/ g ] Perfect fluid form (from RW metric) Energy density = (1/2) 2 + V( ) + (1/2)( ) 2 Pressure p = (1/2) 2 - V( ) - (1/6)( ) 2.. 11 Scalar Field Equation of State Continuity equation follows KG equation [(1/2) 2 ] + 6H [(1/2) 2 ] = -V - V + 3H ( +p) = -V d /dln a = -3( +p) = -3 (1+w) + 3H = -dV( )/d Equation of state ratio w = p/ Klein-Gordon equation (Lagrange equation of motion) 12 Equation of State Reconstruction from EOS: (a) = c exp{ 3 dln a [1+w(z)] } (a) = dln a H -1 sqrt{ (a) [1+w(z)] } V(a) = (1/2) (a) [1-w(z)] K(a) = (1/2) 2 = (1/2) (a) [1+w(z)]. 13 Equation of State Limits of (canonical) Equations of State: w = (K-V) / (K+V) Potential energy dominates (slow roll) V >> K w = -1 Kinetic energy dominates (fast roll) K >> V w = +1 Oscillation about potential minimum (or coherent field, e.g. axion) V = K w = 0 14 Equation of State Examples of (canonical) Equations of State: d /dln a = -3( +p) = -3 (1+w) = (Energy per particle)(Number of particles) / Volume = E N a -3 Constant w implies ~ a -3(1+w) Matter: E~m~a 0, N~a 0 w=0 Radiation: E~1/ ~a -1, N~a 0 w=1/3 Curvature energy: E~1/R 2 ~a -2, N~a 0 w=-1/3 Cosmological constant: E~V, N ~a 0 w=-1 Anisotropic shear: w=+1 Cosmic String network: w=-1/3 ; Domain walls: w=-2/3 15 Expansion History Suppose we admit our ignorance: H 2 = (8 /3) m + H 2 (a) Effective equation of state: w(a) = -1 - (1/3) dln ( H 2 ) / dln a Modifications of the expansion history are equivalent to time variation w(a). Period. Observations that map out expansion history a(t), or w(a), tell us about the fundamental physics of dark energy. Alterations to Friedmann framework w(a) gravitational extensions or high energy physics 16 Expansion History For modifications H 2, define an effective scalar field with V = (3M P 2 /8 ) H 2 + (M P 2 H 0 2 /16 ) [ d H 2 /d ln a] K = - (M P 2 H 0 2 /16 ) [ d H 2 /d ln a] Example: H 2 = A( m ) n w = -1+n Example: H 2 = (8 /3) [g( m ) - m ] w= -1 + (g-1)/[ g/ m - 1 ] 17 Weighing Dark Energy SN Target 18 Exploring Dark Energy 19 Dark Energy Models Scalar fields can roll: 1)fast -- kination [Tracking models] 2)slow -- acceleration [Quintessence] 3)steadily -- acceleration deceleration [Linear potential] 4)oscillate -- potential minimum, pseudoscalar, PNGB [V~ n ] 20 Power law potential Normal potentials dont work: V( ) ~ n have minima (n even), and field just oscillates, leading to EOS w = (n-2)/(n+2) n024 w-101/31 21 Oscillations Oscillating field w = (n-2)/(n+2) Take osc. time 1, d ln ( -1)/dt