1. 1 1 Dark Energy: Extending Einstein Eric Linder University of California, Berkeley Lawrence Berkeley National Lab

2. 2 2 From Data to Theory (and back) To compare observations and theory we need a statistical measure of goodness of fit. We need to compare the theory value, e.g. for distance-redshift, d lum = (1+z)  0 z dz’ / H(z’;  m,w(z’) ) to the data D lum i. For example  2 or likelihood  2 =  i,j [D lum i - d lum (z i )] COV -1 (i,j) [D lum j - d lum (z j )] t L = exp(-  2 /2) [Gaussian near max likelihood] We need 1) theory or robust parametrization w(z), 2) efficient method for estimating parameter errors given data characteristics.

3. 3 3 Fisher Matrix Fisher matrix gives lower limit for Gaussian likelihoods, quick and easy. F ij = d 2 (- ln L) / dp i dp j =  O (dO/dp i ) COV -1 (dO/dp j )  (p i )  1/(F ii ) 1/2 Example: O=d lum ( z=0.1,0.2,…1), p=(  m,w), COV=(  d/d)d  ij F  w =  k (dO k /d  )(dO k /dw)  k -2  2 (  )COV( ,w) COV( ,w)  2 (w) C = F -1 = ( ) F  F  w F w  F ww F = ( ) Also called information matrix. Add independent data sets, or priors, by adding matrices. e.g. Gaussian prior on  m= 0.28  0.03 via  2 = (  m -0.28) 2 /0.03 2 See: Tegmark et al. astro-ph/9805117 Dodelson, “Modern Cosmology”

4. 4 4 Survival of the Fittest Fisher estimates give a N-dimension ellipsoid. Marginalize (integrate over the probability distribution) over parameters not of immediate interest by crossing out their row/column in F -1. Fix a parameter by crossing out row/column in F. 1  (68.3% probability enclosed) joint contours have d  2 =2.30 in 2-D (not d  2 =1). Read off 1  errors by projecting to axis and dividing by 1.52=  2.30. Orientation of ellipse shows degree of covariance (degeneracy). Different types of observations can have different degeneracies (complementarity) and combine to give tight constraints.

5. 5 5 Model Independence We could check each theoretical model one by one against the data -- but there are 10 x of them, each with their own parameters. We’d also like to predict / design results of different experiments. Want model independent approach. Remember H(z)=[  m (1+z) 3 +  w exp{3  0 z d ln(1+z) [1+w(z)]} ] 1/2 Parametrize w(z). Keep close to the physics: both energy density and pressure enter the dynamics; directly related to kinetic/potential energy of scalar field.

6. 6 6 Model Independence Simplest parametrization, with physical dynamics, w(a)=w 0 +w a (1-a) Recall a=(1+z) -1. Virtues: Model independent Excellent approximation to exact field equation solutions Robust against bias Well behaved at high z Problems: Cannot handle rapid transitions or oscillations. N.B.: constant w lacks important physics; w(z)=w 0 +w 1 z is Taylor expansion about low z only - pathological at high z.

7. 7 7 Eigenmodes w 0, w a makes for easiest, robust comparison. But sometimes want nonparametric form. Eigenmodes of w(z) give independent principal components (but depend on model, experiment, and probe). Start with parameters of w i in z bins. Diagonalize Fisher matrix F=E T DE: D is diagonal, rows of E give eigenvectors. w(z) =  b i e i (z) Localized eigenmodes L=E T D 1/2 E Huterer & Cooray 2005 Huterer & Starkman 2003

8. 8 8 Design an Experiment Precision in measurement is not enough - one must beware degeneracies and systematics. p2p2 p1p1 *. Degeneracy: e.g. Aw 0 +Bw a =const Degeneracy: hypersurface, e.g. covariance with  m Systematic: offset error in data or model, e.g. evolution or Systematic: floor to precision, e.g. calibration

9. 9 9 Mapping History Data over a range of redshifts can be effective at breaking degeneracies. Plus one gets leverage from a long baseline in expansion history.

10. 10 Controlling Systematics Controlling systematics is the name of the game. Finding more objects is not. Must understand the sources, instruments, and the theory interpretation. Forthcoming experiments may deliver 100,000s of objects. But uncertainties do not reduce by 1/  N. Must choose cleanest probe, mature method, with multiple crosschecks.

11. 11 Complementarity Complementarity of techniques (e.g. SN,WL,CMB,…) improves precision breaks degeneracies immunizes against systematics

12. 12 Design an Experiment How to design an experiment to explore dark energy? Choose clear, robust, mature techniques Rotate the contours thru choice of redshift span Narrow the contours thru systematics control Break degeneracies thru multiple probes

13. 13 Optimize an Experiment Optimization depends on the question asked. Recall that physics divided into 2 classes: thawing and freezing.

14. 14 Design an Experiment How to design an experiment to explore dark energy? Choose clear, robust, mature techniques Rotate the contours thru choice of redshift span Narrow the contours thru systematics control Break degeneracies thru multiple probes With a strong experiment, we can even test the framework of physics.

15. 15 Acceleration = Curvature The Principle of Equivalence teaches that Acceleration = Gravity = Curvature Acceleration  over time will get v=gh/c, so z = v/c = gh/c 2 (gravitational redshift). But, t  t 0  parallel lines not parallel (curvature)! t0t0 t´ Height Time

16. 16 Dark energy is a completely unknown animal. A new theory or a new component? Track record: Inner solar system motions  General Relativity Outer solar system motions  Neptune Galaxy rotation curves  Dark Matter Finding Our Way in the Dark

17. 17 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

18. 18 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 ]

19. 19 The world is w(z) Don’t care if it’s braneworld, cardassian, vacuum metamorphosis, chaplygin, etc. Simple, robust parametrization w(a)=w 0 +w a (1-a) Braneworld [DDG] vs. (w 0,w a )=(-0.78,0.32) Vacuum metamorph vs. (w 0,w a )=(-1,-3) Also agree on m(z) to 0.01 mag out to z=2

20. 20 Hidden Dimensions Extra dimensions have been used for unification in physics since the 1920s. Large extra dimensions -- braneworlds -- can be tested astronomically. Spacetime is warped by e -y as one moves a distance y off a brane. Think of the spacetime properties as an index of refraction: such a spatial gradient  n localizes light (and the rest of physics). Gravity? Electro- magnetism?

21. 21 Warped Gravity On large (cosmological distances) there may be leaking gravity. The cosmic expansion would appear slower over these distances, i.e. accelerating today! Like localized light in a fiber optic, gravity will eventually leak off into hidden dimensions. Or think of a tuning fork: it radiates sound in all directions, but the waves are stronger if localized.

22. 22 DGP Braneworld Dynamics More than 3  from flatness r c = M Pl 2 /(2M 5 3 )  rc = (2H 0 r c ) -2 SNAP could determine  rc to  (  rc )=0.003 Fairbairn & Goobar astro-ph/0511029

23. 23 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

24. 24 Revealing Physics Time variation w(z) is a critical clue to fundamental physics. Modifications of the expansion history = w(z). But need an underlying theory -  ? beyond Einstein gravity? Growth history and expansion history work together. Linder 2004, Phys. Rev. D 70, 023511 cf. Lue, Scoccimarro, Starkman Phys. Rev. D 69 (2004) 124015 for braneworld perturbations

25. 25 Growth History Growth rate of density fluctuations g(a) = (  m /  m )/a

26. 26 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  = 0.55+0.05[1+w(z=1)] Works to 0.05 – 0.2%! 0

27. 27 Growth and Expansion Keep expansion history as w(z), growth deviation from expansion by . With  as free fit parameter, we can test framework, and the origin of dark energy. Paradigm: To reveal the origin of dark energy, measure w, w, and . e.g. use SN+WL.

28. 28 Going Nonlinear Efficient generation of grid of dark energy cosmologies. Linder & White 2005 PRD 72, 061304(R) Previous fit functions were only good to ~10% -- for . New technique is good to 1.5%, for general dark energy.

29. 29 Gravity’s Bias Neglecting modified gravity will bias the cosmology unless gravity is properly accounted for (e.g.  ). Huterer & Linder 2006

30. 30 Going Beyond Einstein To test Einstein gravity, we need growth and expansion. To test dark energy and GR, we need superb data. 9 parameter cosmology fit. Testing GR via growth index  degrades w 0, w a by 15-25%.

31. 31 Fitting Beyond Einstein How well can we fit gravity? N.B. it’s important to include other effects on large scale structure such as m. WL+SN+CMB can determine  to 8%.

32. 32 Dark Energy Surprises There is still much theoretical work 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!

33. 33 Heart of Darkness Is dark energy dark – only interacts gravitationally? Self interaction: pseudoscalar quintessence Coupling to matter: Chaplygin gas Leads to 5 th force: limited by lab tests Unify dark energy with dark matter?  Distorts matter power spectrum: ruled out unless within 10 -5 of  Coupling to gravitation: Scalar-tensor theories = Extended quintessence Can clump on subhorizon scales Can “turn on” from nonlinear structure formation?! Higher dimension gravity: Scalaron quintessence Can be written in terms of scalar-tensor and w eff Sandvik et al. 2003 The horror!

34. 34 Theory and Data Pinpointing Physics Is it  ? dynamics via w Checking Geometry allowing curvature Testing GR new gravity Thanks to Gary Bernstein, Dragan Huterer, Masahiro Takada for key contributions

35. 35 Complementarity Cosmic acceleration is so revolutionary we need the crosschecks, synergy, reduced influence of systematics, robust answers of complementary probes. SNAP space mission gives infrared and high redshift measurements, high resolution and lower systematics. SNAP wide field telescope gives multiple probes (e.g. SN Ia, Weak Lensing, Clusters, Strong Lensing, SN II) and rich astronomical resources. When you have a mystery ailment, you want a diagnosis with blood tests, EKG, MRI,...

36. 36 Frontiers of the Universe Breakthrough of the Year 1919 Cosmology holds the key to new physics in the next decade. 1998 2003 Let’s find out!