1 1 Eric Linder University of California, Berkeley Lawrence Berkeley National Lab Course on Dark Energy Cosmology at the Beach 2009 JDEM constraints

2 2 Outline Lecture 1: Dark Energy in Space The panoply of observations Lecture 2: Dark Energy in Theory The garden of models Lecture 3: Dark Energy in your Computer The array of tools – Don’t try this at home!

3 3 Nature of Acceleration How much dark energy is there? energy density   How springy/stretchy is it? equation of state w, w Is dark energy static? Einstein’s cosmological constant . Is dark energy dynamic? A new, time- and space- varying field. How do we learn what it is, not just that it is?

4 4 What’s the Matter with Energy? They are off by a factor of 1,000,000,000,000, 000,000,000,000, 000,000,000,000, 000,000,000,000, 000,000,000,000, 000,000,000,000, 000,000,000,000, 000,000,000,000, 000,000,000,000, 000,000,000,000. This is modestly called the fine tuning problem. Why not just bring back the cosmological constant (  )? When physicists calculate how big  should be, they don’t quite get it right.

5 5 Matter Dark energy Today Size=2 Size=4 Size=1/2Size=1/4 We cannot calculate the vacuum energy to within But it gets worse: Think of the energy in  as the level of the quantum “sea”. At most times in history, matter is either drowned or dry. Cosmic Coincidence Why not just settle for a cosmological constant  ?  For 90 years we have tried to understand why  is at least times smaller than we would expect – and failed.  We know there was an epoch of time varying vacuum once – inflation.

6 6 On Beyond  ! We need to explore further frontiers in high energy physics, gravitation, and cosmology. New quantum physics? Does nothing weigh something? Einstein’s cosmological constant, Quintessence, String theory New gravitational physics? Is nowhere somewhere? Quantum gravity, supergravity, extra dimensions? We need new, highly precise data

7 7 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..

8 8 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)

9 9 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

10 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)] . But, ~  [(1+w)  ] ~  (1+w) HM p So if 1+w << 1, then  ~ /H << M p. It is very hard to directly reconstruct the potential. Goldilocks problem: Dark energy is unlike Inflation! . .

11 Dynamics of Quintessence Equation of motion of scalar field driven by steepness of potential slowed by Hubble friction Broad categorization -- which term dominates: field rolls but decelerates as dominates energy field starts frozen by Hubble drag and then rolls Freezers vs. Thawers  + 3H  = -dV(  )/d  ¨˙

12 Limits of Quintessence Distinct, narrow regions of w-w Entire “thawing” region looks like = -1 ± Need w experiments with  (w) ≈ 2(1+w). Caldwell & Linder 2005 PRL 95,  2 /2 - V(  )  2 /2 + V(  ).. w =

13 Calibrating Dark Energy Models have a diversity of behavior, within thawing and freezing. But we can calibrate w by “stretching” it: w  w(a  )/ a . Calibrated parameters w 0, w a. de Putter & Linder JCAP The two parameters w 0, w a achieve level accuracy on observables d(z), H(z). w(a)=w 0 +w a (1-a) This is from physics (Linder 2003). It has nothing to do with a Taylor expansion.

14 Latest Results for w Systematics already dominate error budget We do not know w(z) = -1 or what dark energy was doing at z>1. Kowalski et al. 2008, ApJ [arXiv: ]

15 Beyond Lambda Choose “motivated” models widely covering Beyond  physics. Includes thawing, freezing, phase transition, modGR, geometric. Rubin et al. 2008, ApJ [arXiv: ] Compare current data (SN+CMB+BAO) vs. 10 dark energy models. Most models have limit approaching  but two don’t.

16 Doomsday Model First dark energy model - Linde 1986 V(  ) = V 0 + V 0 (  -  0 ) Linear potential 2 parameters -  m and w 0 or t doom or V 0 Rolls down potential to negative density and universe collapses in finite time. (also see Weinber g 2008)

17 DGP Braneworld 2 parameters -  m and  k or  bw or r c H 2 =(8  G/3)  m +H/r c

18 Beyond Lambda While  is consistent with data, many varieties of physics are also. (2 models do better than .) Improvements in systematics will have large impact - e.g. Braneworld disfavored at  2 =+15 if statistical errors only. Uniform data set / analysis key, as is next generation ability to see w. Apart from testing “exotic cosmologies”, such comparisons are useful because model variety includes sensitivity to systematics that don’t “look like” . No indication of any such systematics. Diversity highlights need for physical priors before model selection useful.

19 Beyond Scalar Fields 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

20 Gravity Beyond 4D z=1 z=2 z=3  =1/2  =1 (BW) Can reproduce expansion or growth with quintessence, but not both. DGP Braneworld, and H  mods, obey freezer dynamics in w-w

21 Physics of Growth Perturb the acceleration equation by Peebles 1980 (pre-DM!) Generalization Growth index  = [1+w(z=1)] Accurate to level for dark energy and can describe deviations from Einstein gravity growth (as long as usual matter domination at high z). [Linder 2005, Linder & Cahn 2007] which conserves mass This determines growth of density inhomogeneities  =  /  Fitting function

22 Physics of Growth Growth g(a)=(  /  )/a depends purely on the expansion history H(z) -- and gravity theory. Expansion effects via w(z), but separate effects of gravity on growth. g(a) = exp {  0 a d ln a [  m (a)  -1] } Growth index  is valid parameter to describe modified gravity. Accurate to 0.1% in numerics. Similar to Peebles 1980 (  =0.6) and Wang & Steinhardt 1998 (constant w). 0 Linder 2005

23 Growth Beyond Growth Beyond  Gravitational growth index  is nearly constant, i.e. single parameter (not function) to describe growth separately from expansion effects. Derivable from 1st principles, even for modified gravity, accurate to 0.1% in growth. Minimal Modified Gravity (aka Beyond the Standard Model 2) uses simultaneous fit to expansion and growth {  m,w 0,w a,  }, as a benchmark model to explore the accelerating universe (cf. mSUGRA for dark matter).

24 The Nature of Gravity To test Einstein gravity, we need growth and expansion. Tension between distance and LSS mass growth reveals deviations from GR. Keep expansion history as w(z), growth deviation from expansion (modGR) by . Fit both simultaneously. Huterer & Linder 2006  gives deviations in growth from GR Bias:

25 Violating Matter Domination Gravitational growth index  depended on early matter domination. Need calibration parameter for growth, just like for SN (low z) and BAO (high z) distances. Beyond the Standard Model 3 simultaneous fit to {  m,w 0,w a, ,g * }. Next generation data can test  (  e )=0.005,  G early /G=1.4%,  ln a=1.7%. g(a) = g * exp {  0 a d ln a [  m (a)  -1] } g * is nearly constant, single parameter, handles early time deviations: modGR, early DE, early acceleration. Separate from ,w; accurate to 0.1%. Linder

26 Paths to Testing Gravity Alternate approaches: Solve for metric potentials ,  [e.g. Hu & Sawicki 2007] or parametrize  /  -1 (PPN) [e.g. Caldwell, Cooray, Melchiorri 2007; Jain & Zhang 2007; Zhang, Bean, Liguori, Dodelson 2007; Amendola, Kunz, Sapone 2007]. Test by spectroscopic vs. imaging surveys.  Galaxies Galaxy ClustersLinear regime LSS  Blue (dynamics)  Red (lensing/ISW)  -  Jain & Zhang 2007

27 Dark Energy Surprises There is still much theoretical research needed! Dark energy is… Dark Smooth on cluster scales Accelerating Maybe not completely! Clumpy in horizon? Maybe not forever! It’s not quite so simple! Research is what I'm doing when I don't know what I'm doing. - Wernher von Braun

28 Dark energy is a completely unknown animal. Not completely dark? [coupling to (dark) matter, to itself] Not energy? [modified gravity -- physics, not physical] Track record: Inner solar system motions  General Relativity Outer solar system motions  Neptune Galaxy rotation curves  Dark Matter Finding Our Way in the Dark Moral: Given the vast uncertainties, go for the most unambiguous insight.

29 What could go wrong? Potentials ,  ; anisotropic stress  s ; gravitational strength G(k,t) ; sound speed c s ; coupling . SN distances come from the FRW metric. Period. Lensing distances depend on deflection law (gravity) even if separate mass (gravity) -- (  -  ), c s,  s,G(k,t) BAO depends on standard CDM (matter perturbations being blind to DE). -- (  +  ),c s, ,  s,G(k,t) Clean Physics “Yesterday’s sensation is Today’s calibration and Tomorrow’s background.” --Feynman What could go right? Ditto.

30 END Lecture 2 For more dark energy theory resources, see Dynamics of Dark Energy (Copeland, Sami, Tsujikawa 2006) Dynamics of Quintessence, Quintessence of Dynamics (Linder 2007) and the references cited therein. Lecture 1: Dark Energy in Space The panoply of observations Lecture 2: Dark Energy in Theory The garden of models Lecture 3: Dark Energy in your Computer The array of tools – Don’t try this at home!