Dark Energy: current theoretical issues and progress toward future experiments. A. Albrecht. KITPC Colloquium November

Transcription

1 Dark Energy: current theoretical issues and progress toward future experiments A. Albrecht UC Davis KITPC Colloquium November 4 29

2 95% of the cosmic matter/energy is a mystery. It has never been observed even in our best laboratories Ordinary Matter (observed in labs) Dark Energy (accelerating) Dark Matter (Gravitating)

3 American Association for the Advancement of Science

4 at the moment, the nature of dark energy is arguably the murkiest question in physics--and the one that, when answered, may shed the most light. American Association for the Advancement of Science

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6 Oct 23 29

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8 Cosmic acceleration Accelerating matter is required to fit current data Ordinary non accelerating matter Amount of w=- matter ( Dark energy ) Amount Supernova of gravitating matter Preferred by data c. 23

9 Cosmic acceleration Accelerating matter is required to fit current data Ordinary non accelerating matter Amount of w=- matter ( Dark energy ) BAO Amount Supernova of gravitating matter Kowalski, et al., Ap.J.. (28) Preferred by data c. 28

10 Cosmic acceleration Accelerating matter is required to fit current data Ordinary non accelerating matter Amount of w=- matter ( Dark energy ) BAO Amount Supernova of gravitating matter Kowalski, et al., Ap.J.. (28) Preferred by data c. 28 (Includes dark matter)

11 Dark energy appears to be the dominant component of the physical Universe, yet there is no persuasive theoretical explanation. The acceleration of the Universe is, along with dark matter, the observed phenomenon which most directly demonstrates that our fundamental theories of particles and gravity are either incorrect or incomplete. Most experts believe that nothing short of a revolution in our understanding of fundamental physics* will be required to achieve a full understanding of the cosmic acceleration. For these reasons, the nature of dark energy ranks among the very most compelling of all outstanding problems in physical science. These circumstances demand an ambitious observational program to determine the dark energy properties as well as possible. From the Dark Energy Task Force report (26) astro-ph/6959 *My emphasis

12 Dark energy appears to be the dominant component of the physical Universe, yet there is no persuasive theoretical explanation. The acceleration of the Universe is, along with dark matter, the observed phenomenon which most directly demonstrates that our fundamental theories of particles and gravity are either incorrect or incomplete. Most experts believe that nothing short of a revolution in our understanding of fundamental physics* will be required to achieve a full understanding of the cosmic acceleration. For these reasons, the nature of dark energy ranks among the very most compelling of all outstanding problems in physical science. These circumstances DETF = a HEPAP/AAAC demand an ambitious observational subpanel program to guide to determine planning the of dark energy properties as well as possible. future dark energy experiments From the Dark Energy Task Force report (26) astro-ph/6959 More info here *My emphasis

13 This talk Part : Part 2 A few attempts to explain dark energy Motivations, problems and other comments Theme: We may not know where this revolution is taking us, but it is already underway: Planning new experiments - DETF - Next questions

14 Some general issues: Properties: Solve GR for the scale factor a of the Universe (a= today): a 4π G Λ = ( ρ + 3p) + a 3 3 Positive acceleration clearly requires w p/ ρ < /3 (unlike any known constituent of the Universe) or a non-zero cosmological constant or an alteration to General Relativity.

15 Some general issues: Properties: Solve GR for the scale factor a of the Universe (a= today): a a 4π G Λ = ( ρ + 3p) Positive acceleration clearly requires w p/ ρ < /3 (unlike any known constituent of the Universe) or a non-zero cosmological constant or an alteration to General Relativity. See work by Pengjie Zhang (SHAO)

16 Some general issues: Numbers: Today, ρ DE M ( ev ) P Many field models require a particle mass of 3 mq ev H from mm ρ 2 2 Q P DE

17 Some general issues: Numbers: Today, ρ DE M ( ev ) P Many field models require a particle mass of 3 mq ev H from mm ρ 2 2 Q P DE Where do these come from and how are they protected from quantum corrections?

18 Some general issues: Two familiar ways to achieve acceleration: Properties: ) Einstein s cosmological constant Solve GR for the scale and factor relatives a of the ( w Universe = ) (a= today): a a 4π G Λ = ( ρ + 3p) ) Whatever drove inflation: Dynamical, Scalar field? Positive acceleration clearly requires w p / ρ < /3 (unlike any known constituent of the Universe) or a non-zero cosmological constant or an alteration to General Relativity.

19 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics Vacuum energy problem (we ve gotten nowhere with this) Λ = 2 Λ? Vacuum Fluctuations

20 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics The string theory landscape (a radically different idea of what we mean by a fundamental theory)

21 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics The string theory landscape (a radically different idea of what we mean by a fundamental theory) Theory of Everything? Theory of Anything

22 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics The string theory landscape (a radically different idea of what we mean by a fundamental theory) Not exactly a cosmological constant

23 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics De Sitter limit: Horizon Finite Entropy Banks, Fischler, Susskind, AA & Sorbo etc

24 De Sitter Space: The ultimate equilibrium for the universe? Horizon 2 S A= H =Λ Quantum effects: Hawking Temperature T = H = 8π G ε 3 DE

25 De Sitter Space: The ultimate equilibrium for the universe? Horizon 2 S A= H =Λ Quantum effects: Hawking Temperature T = H = Does this imply (via ) a finite Hilbert space for physics? S = ln N 8π G ε 3 DE Banks, Fischler

26 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics De Sitter limit: Horizon Finite Entropy Equilibrium Cosmology Rare Fluctuation Dyson, Kleban & Susskind; AA & Sorbo etc

27 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics De Sitter limit: Horizon Finite Entropy Equilibrium Cosmology Rare Fluctuation Dyson, Kleban & Susskind; AA & Sorbo etc Boltzmann s Brain?

28 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics De Sitter limit: Horizon Finite Entropy Equilibrium Cosmology Rare Fluctuation Dyson, Kleban & Susskind; AA & Sorbo etc This picture is in deep conflict with observation

29 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics De Sitter limit: Horizon Finite Entropy Equilibrium Cosmology Rare Fluctuation Dyson, Kleban & Susskind; AA & Sorbo etc This picture is in deep conflict with observation (resolved by landscape?)

30 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics De Sitter limit: Horizon Finite Entropy Equilibrium Cosmology Dyson, Kleban & Susskind; AA & Sorbo etc Rare Fluctuation This picture forms a nice foundation for inflationary cosmology

31 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics De Sitter limit: Horizon Finite Entropy Equilibrium Cosmology Contrast with eternal inflation! Dyson, Kleban & Susskind; AA & Sorbo etc Rare Fluctuation This picture forms a nice foundation for inflationary cosmology

32 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics De Sitter limit: Horizon Finite Entropy Equilibrium Cosmology Contrast with eternal inflation! Dyson, Kleban & Susskind; AA & Sorbo etc Rare Fluctuation This picture forms a nice foundation for inflationary cosmology AA 29

33 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics De Sitter limit: Horizon Finite Entropy Equilibrium Cosmology Dyson, Kleban & Susskind; AA & Sorbo etc Rare Fluctuation Perhaps saved from this discussion by instability of De Sitter space (i.e. Woodard et al)

34 Specific ideas: i) A cosmological constant Λ Nice textbook solutions BUT Deep problems/impacts re fundamental physics Λ is not the simple option

35 Some general issues: Alternative Explanations?: Is there a less dramatic explanation of the data?

36 Some general issues: Alternative Explanations?: Is there a less dramatic explanation of the data? For example is supernova dimming due to dust? (Aguirre) γ-axion interactions? (Csaki et al) Evolution of SN properties? (Drell et al) Many of these are under increasing pressure from data, but such skepticism is critically important.

37 Some general issues: Alternative Explanations?: Is there a less dramatic explanation of the data? Or perhaps Nonlocal gravity from loop corrections (Woodard & Deser) Misinterpretation of a genuinely inhomogeneous universe (ie. Kolb and collaborators)

38 Specific ideas: ii) A scalar field ( Quintessence ) Recycle inflation ideas (resurrect Serious unresolved problems Explaining/ protecting 5 th force problem dream?) 3 mq ev H Vacuum energy problem What is the Q field? (inherited from inflation) Why now? (Often not a separate problem) Λ=

39 Specific ideas: ii) A scalar field ( Quintessence ) Inspired Recycle by inflation ideas (resurrect Serious unresolved problems Explaining/ protecting 5 th force problem dream?) 3 mq ev H Vacuum energy problem What is the Q field? (inherited from inflation) Why now? (Often not a separate problem) Λ=

40 Specific ideas: ii) A scalar field ( Quintessence ) Recycle inflation ideas (resurrect Λ= dream?) Result? Serious unresolved problems Explaining/ protecting 5 th force problem 3 mq ev H Vacuum energy problem What is the Q field? (inherited from inflation) Why now? (Often not a separate problem)

41 Learned from inflation: A slowly rolling (nearly) homogeneous scalar field can accelerate the universe φ + 3H φ = V w 2 p φ + ρ V V ϕ

42 Learned from inflation: A slowly rolling (nearly) homogeneous scalar field can accelerate the universe φ + 3H φ = V Dynamical w 2 p φ + ρ V V ϕ

43 Learned from inflation: A slowly rolling (nearly) homogeneous scalar field can accelerate the universe φ + 3H φ = V Dynamical w 2 p φ = + ρ V V ϕ Rolling scalar field dark energy is called quintessence

44 Some quintessence potentials Exponential (Wetterich, Peebles & Ratra) PNGB aka Axion (Frieman et al) Exponential with prefactor (AA & Skordis) Inverse Power Law (Ratra & Peebles, Steinhardt et al)

45 Some quintessence potentials Exponential (Wetterich, Peebles & Ratra) V( ϕ) = Ve λϕ PNGB aka Axion (Frieman et al) V( ϕ) = V (cos( ϕ/ λ) + ) Exponential with prefactor (AA & Skordis) ( 2 ( ) ) V( ϕ) = V χ ϕ β + δ e λϕ Inverse Power Law (Ratra & Peebles, Steinhardt et al) V( ϕ) V m ϕ = α

46 Some quintessence potentials Exponential (Wetterich, Peebles & Ratra) V( ϕ) = Ve λϕ PNGB aka Axion (Frieman et al) V( ϕ) = V (cos( ϕ/ λ) + ) Exponential with prefactor (AA & Skordis) ( 2 ( ) ) V( ϕ) = V χ ϕ β + δ e λϕ Inverse Power Law (Ratra & Peebles, Steinhardt et al) V( ϕ) V m ϕ = α

47 Some quintessence potentials Exponential (Wetterich, Peebles & Ratra) V( ϕ) = Ve λϕ PNGB aka Axion (Frieman et al) V( ϕ) = V (cos( ϕ/ λ) + ) Exponential with prefactor (AA & Skordis) ( 2 ( ) ) V( ϕ) = V χ ϕ β + δ e λϕ Inverse Power Law (Ratra & Peebles, Steinhardt et al) V( ϕ) V m ϕ = α

48 Some quintessence potentials Exponential (Wetterich, Peebles & Ratra) V( ϕ) = Ve λϕ PNGB aka Axion (Frieman et al) V( ϕ) = V (cos( ϕ/ λ) + ) Exponential with prefactor (AA & Skordis) ( 2 ( ) ) V( ϕ) = V χ ϕ β + δ e λϕ Inverse Power Law (Ratra & Peebles, Steinhardt et al) V( ϕ) V m ϕ = α

49 Some quintessence potentials Exponential (Wetterich, Peebles & Ratra) V( ϕ) = Ve λϕ PNGB aka Axion (Frieman et al) V( ϕ) = V (cos( ϕ/ λ) + ) Exponential with prefactor (AA & Skordis) ( 2 ( ) ) V( ϕ) = V χ ϕ β + δ e λϕ Inverse Power Law (Ratra & Peebles, Steinhardt et al) V( ϕ) V m ϕ = α

50 The potentials Exponential (Wetterich, Peebles & Ratra) V( ϕ) = Ve λϕ PNGB aka Axion (Frieman et al) V( ϕ) = V (cos( ϕ/ λ) + ) Exponential with prefactor (AA & Skordis) ( 2 ( ) ) V( ϕ) = V χ ϕ β + δ e λϕ Stronger than average motivations & interest Inverse Power Law (Ratra & Peebles, Steinhardt et al) V( ϕ) V m ϕ = α

51 The potentials Exponential (Wetterich, Peebles & Ratra) V( ϕ) = Ve λϕ PNGB aka Axion (Frieman et al) V( ϕ) = V (cos( ϕ/ λ) + ) Exponential with prefactor (AA & Skordis) ( 2 ( ) ) V( ϕ) = V χ ϕ β + δ e λϕ Feng et al Inverse Power Law (Ratra & Peebles, Steinhardt et al) V( ϕ) V m ϕ = α Stronger than average motivations & interest See important work on this topic by Mingzhi Li et al and Bo

52 The potentials Exponential (Wetterich, Peebles & Ratra) V( ϕ) = Ve λϕ PNGB aka Axion (Frieman et al) V( ϕ) = V (cos( ϕ/ λ) + ) Exponential with prefactor (AA & Skordis) ( 2 ( ) ) V( ϕ) = V χ ϕ β + δ e λϕ Feng et al Inverse Power Law (Ratra & Peebles, Steinhardt et al) V( ϕ) V m ϕ = α Stronger than average motivations & interest See important work on this topic by Mingzhi Li et al and Bo

53 w p ρ they cover a variety of behavior. PNGB EXP IT AS w(a) a a = cosmic scale factor time

54 Dark energy and the ego test

55 Specific ideas: ii) A scalar field ( Quintessence ) Illustration: Exponential with prefactor (EwP) models: (( ) 2 ) ( ) V( ϕ) = V ϕ B A exp ϕλ All parameters O() in Planck units, AA & Skordis 999 motivations/protections from extra dimensions & quantum gravity Burgess & collaborators (e.g. B = 34 A =.5 λ = 8 V = )

56 Specific ideas: ii) A scalar field ( Quintessence ) V ( ϕ ) Illustration: Exponential with prefactor (EwP) models: (( ) 2 ) ( ) V( ϕ) = V ϕ B A exp ϕλ AA & Skordis 999 All parameters O() in Planck units, ϕ = β motivations/protections from extra dimensions ϕ & quantum gravity Burgess & collaborators (e.g. B = 34 A =.5 λ = 8 V = ) AA & Skordis 999

57 Specific ideas: ii) A scalar field ( Quintessence ) V ( ϕ ) Illustration: Exponential with prefactor (EwP) models: (( ) 2 ) ( ) V( ϕ) = V ϕ B A exp ϕλ AA & Skordis 999 All parameters O() in Planck units, ϕ = β motivations/protections from extra dimensions ϕ & quantum gravity Burgess & collaborators (e.g. B = 34 A =.5 λ = 8 V = ) AA & Skordis 999

58 Specific ideas: ii) A scalar field ( Quintessence ) Illustration: Exponential with prefactor (EwP) models: Ω, w Ω r Ω m Ω D w a AA & Skordis 999

59 Specific ideas: iii) A mass varying neutrinos ( MaVaNs ) Exploit m ρ ν /4 3 DE Issues Origin of acceleron (varies neutrino mass, accelerates the universe) gravitational collapse ev Faradon, Nelson & Weiner Afshordi et al 25 Spitzer 26

60 Specific ideas: iii) A mass varying neutrinos ( MaVaNs ) Exploit m ρ ν /4 3 DE Issues Origin of acceleron (varies neutrino mass, accelerates the universe) gravitational collapse ev Faradon, Nelson & Weiner Afshordi et al 25 Spitzer 26

61 Specific ideas: iii) A mass varying neutrinos ( MaVaNs ) Exploit m ρ ν /4 3 DE Issues Origin of acceleron (varies neutrino mass, accelerates the universe) gravitational collapse ev Faradon, Nelson & Weiner Afshordi et al 25 Spitzer 26

62 Specific ideas: iv) Modify Gravity Not something to be done lightly, but given our confusion about cosmic acceleration, very much worth considering. Many deep technical issues e.g. DGP (Dvali, Gabadadze and Porrati) Ghosts Charmousis et al

63 Specific ideas: iv) Modify Gravity Not something to be done lightly, but given our confusion about cosmic acceleration, very much worth considering. Many deep technical issues e.g. DGP (Dvali, Gabadadze and Porrati) Ghosts Charmousis et al Important work by Pengjie Zhang

64 This talk Part : Part 2 A few attempts to explain dark energy - Motivations, Problems and other comments Theme: We may not know where this revolution is taking us, but it is already underway: Planning new experiments - DETF - Next questions

65 This talk Part : Part 2 A few attempts to explain dark energy - Motivations, Problems and other comments Theme: We may not know where this revolution is taking us, but it is already underway: Planning new experiments - DETF - Next questions

66 This talk Part : Part 2 A few attempts to explain dark energy - Motivations, Problems and other comments Theme: We may not know where this revolution is taking us, but it is already underway: Planning new experiments - DETF - Next questions

67 This talk Part : Part 2 A few attempts to explain dark energy - Motivations, Problems and other comments Theme: We may not know where this revolution is taking us, but it is already underway: Planning new experiments - DETF - Next questions

68 Astronomy Primer for Dark Energy Solve GR for the scale factor a of the Universe (a= today): From DETF Positive acceleration clearly requires w p/ ρ < /3 unlike any known constituent of the Universe, or a non-zero cosmological constant - or an alteration to General Relativity. The second basic equation is 2 a 8πG N ρ Λ = + a 3 3 k a 2 Today we have H 2 2 a 8 G N π ρ Λ = = + k a 3 3

69 Astronomy Primer for Dark Energy Solve GR for the scale factor a of the Universe (a= today): From DETF Positive acceleration clearly requires w p/ ρ < /3 unlike any known constituent of the Universe, or a non-zero cosmological constant - or an alteration to General Relativity. The second basic equation is 2 a 8πG N ρ Λ = + a 3 3 k a 2 Today we have H See also Frieman Huterer and Turner for a review 2 2 a 8 G N π ρ Λ = = + k a 3 3

70 Hubble Parameter We can rewrite this as = 8πG Nρ 3H 2 + Λ 3H 2 k H 2 Ω ρ + Ω Λ + Ω k To get the generalization that applies not just now (a=), we need to distinguish between non-relativistic matter and relativistic matter. We also generalize Λ to dark energy with a constant w, not necessarily equal to -: non-rel. matter curvature rel. matter Dark Energy

71 What are the observable quantities? Expansion factor a is directly observed by redshifting of emitted photons: a=/(+z), z is redshift. Time is not a direct observable (for present discussion). A measure of elapsed time is the distance traversed by an emitted photon: This distance-redshift relation is one of the diagnostics of dark energy. Given a value for curvature, there is - map between D(z) and w(a). Distance is manifested by changes in flux, subtended angle, and sky densities of objects at fixed luminosity, proper size, and space density. These are one class of observable quantities for dark-energy study.

72 Another observable quantity: The progress of gravitational collapse is damped by expansion of the Universe. Density fluctuations arising from inflation-era quantum fluctuations increase their amplitude with time. Quantify this by the growth factor g of density fluctuations in linear perturbation theory. GR gives: This growth-redshift relation is the second diagnostic of dark energy. If GR is correct, there is - map between D(z) and g(z). If GR is incorrect, observed quantities may fail to obey this relation. Growth factor is determined by measuring the density fluctuations in nearby dark matter (!), comparing to those seen at z=88 by WMAP.

73 Another observable quantity: The progress of gravitational collapse is damped by expansion of the Universe. Density fluctuations arising from inflation-era quantum fluctuations increase their amplitude with time. Quantify this by the growth factor g of density fluctuations in linear perturbation theory. GR gives: This growth-redshift relation is the second diagnostic of dark energy. If GR is correct, there is - map between D(z) and g(z). If GR is incorrect, observed quantities may fail to obey this relation. Growth factor is determined by measuring the density fluctuations in nearby dark matter (!), comparing to those seen at z=88 by WMAP.

74 What are the observable quantities? Future dark-energy experiments will require percent-level precision on the primary observables D(z) and g(z).

75 Supernovae (exploding stars) can have well determined luminosities and can be (briefly) as bright as an entire galaxy. As very bright standard candles they can be used to probe great cosmic distances. Supernova

76 Dark Energy with Type Ia Supernovae Exploding white dwarf stars: mass exceeds Chandrasekhar limit. If luminosity is fixed, received flux gives relative distance via Qf=L/4πD 2. SNIa are not homogeneous events. Are all luminosityaffecting variables manifested in observed properties of the explosion (light curves, spectra)? Supernovae Detected in HST GOODS Survey (Riess et al)

77 Dark Energy with Type Ia Supernovae Example of SN data: HST GOODS Survey (Riess et al) Clear evidence of acceleration!

78 Riess et al astro-ph/6572

79 Dark Energy with Baryon Acoustic Oscillations Acoustic waves propagate in the baryonphoton plasma starting at end of inflation. When plasma combines to neutral hydrogen, sound propagation ends. Cosmic expansion sets up a predictable standing wave pattern on scales of the Hubble length. The Hubble length (~sound horizon r s) ~4 Mpc is imprinted on the matter density pattern. BAO seen in CMB (WMAP) Identify the angular scale subtending r s then use θ s =r s /D(z) WMAP/Planck determine r s and the distance to z=88. Survey of galaxies (as signposts for dark matter) recover D(z), H(z) at <z<5. Galaxy survey can be visible/nir or 2- cm emission BAO seen in SDSS Galaxy correlations (Eisenstein et al)

80 Dark Energy with Galaxy Clusters Galaxy clusters are the largest structures in Universe to undergo gravitational collapse. Markers for locations with density contrast above a critical value. Theory predicts the mass function dn/dmdv. We observe dn/dzdω. Dark energy sensitivity: Optical View (Lupton/SDSS) Mass function is very sensitive to M; very sensitive to g(z). Also very sensitive to misestimation of mass, which is not directly observed. Cluster method probes both D(z) and g(z)

81 Dark Energy with Galaxy Clusters Optical View (Lupton/SDSS) X-ray View (Chandra) 3 GHz View (Carlstrom et al) Sunyaev-Zeldovich effect

82 Galaxy Clusters from ROSAT X-ray surveys From Rosati et al, 999: ROSAT cluster surveys yielded ~few clusters in controlled samples. Future X-ray, SZ, lensing surveys project few x, detections.

83 Gravitational Lensing

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87 Dark Energy with Weak Gravitational Lensing Mass concentrations in the Universe deflect photons from distant sources. Displacement of background images is unobservable, but their distortion (shear) is measurable. Extent of distortion depends upon size of mass concentrations and relative distances. Depth information from redshifts. Obtaining 8 redshifts from optical spectroscopy is infeasible. photometric redshifts instead. Lensing method probes both D(z) and g(z)

88 Dark Energy with Weak Gravitational Lensing In weak lensing, shapes of galaxies are measured. Dominant noise source is the (random) intrinsic shape of galaxies. Large- N statistics extract lensing influence from intrinsic noise.

89 Choose your background photon source: Faint background galaxies: Use visible/nir imaging to determine shapes. Photometric redshifts. Hoekstra et al 26: QuickTime and a TIFF (Uncompressed) decompressor are needed to see this picture. Photons from the CMB: Use mm-wave highresolution imaging of CMB. All sources at z=88. (lensing not yet detected) 2-cm photons: Use the proposed Square Kilometer Array (SKA). Sources are neutral H in regular galaxies at z<2, or the neutral Universe at z>6. (lensing not yet detected)

90 Q: Given that we know so little about the cosmic acceleration, how do we represent source of this acceleration when we forecast the impact of future experiments? One Answer: (DETF, Joint Dark Energy Mission Science Definition Team JDEM STD) Model dark energy as homogeneous fluid all information contained in wa pa/ ρ a ( ) ( ) ( ) Model possible breakdown of GR by inconsistent determination of w(a) by different methods.

91 Q: Given that we know so little about the cosmic acceleration, how do we represent source of this acceleration when we forecast the impact of future experiments? One Answer: (DETF, Joint Dark Energy Mission Science Definition Team JDEM STD) Model dark energy as homogeneous fluid all information contained in wa pa/ ρ a ( ) ( ) ( ) Model possible breakdown of GR by inconsistent determination of w(a) by different methods. Also: Std cosmological parameters including curvature

92 Q: Given that we know so little about the cosmic acceleration, how do we represent source of this acceleration when we forecast the impact of future experiments? One Answer: (DETF, Joint Dark Energy Mission Science Definition Team JDEM STD) Model dark energy as homogeneous fluid all information contained in wa pa/ ρ a ( ) ( ) ( ) Model possible breakdown of GR by inconsistent determination of w(a) by different methods. Also: Std cosmological parameters including curvature We know very little now

93 Recall: Some general issues: Two familiar ways to achieve acceleration: Properties: ) Einstein s cosmological constant Solve GR for the scale and factor relatives a of the ( w Universe = ) (a= today): a a 4π G Λ = ( ρ + 3p) ) Whatever drove inflation: Dynamical, Scalar field? Positive acceleration clearly requires w p / ρ < /3 (unlike any known constituent of the Universe) or a non-zero cosmological constant or an alteration to General Relativity.

94 w a 95% CL contour ( ) wa ( ) = w + w a a (DETF parameterization Linder) DETF figure of merit: = / Area w

95 The DETF stages (data models constructed for each one) Stage 2: Underway Stage 3: Medium size/term projects Stage 4: Large longer term projects (ie JDEM, LST) DETF modeled SN Weak Lensing Baryon Oscillation Cluster data

96 DETF Projections Figure of merit Improvement over Stage 2 Stage 3

97 DETF Projections Figure of merit Improvement over Stage 2 Ground

98 DETF Projections Figure of merit Improvement over Stage 2 Space

99 DETF Projections Figure of merit Improvement over Stage 2 Ground + Space

100 A technical point: The role of correlations Combination Technique #2 Technique #

101 From the DETF Executive Summary One of our main findings is that no single technique can answer the outstanding questions about dark energy: combinations of at least two of these techniques must be used to fully realize the promise of future observations. Already there are proposals for major, long-term (Stage IV) projects incorporating these techniques that have the promise of increasing our figure of merit by a factor of ten beyond the level it will reach with the conclusion of current experiments. What is urgently needed is a commitment to fund a program comprised of a selection of these projects. The selection should be made on the basis of critical evaluations of their costs, benefits, and risks.

102 The Dark Energy Task Force (DETF) Created specific simulated data sets (Stage 2, Stage 3, Stage 4) Assessed their impact on our knowledge of dark energy as modeled with the w-wa parameters ( ) = + ( ) wa w w a a

103 The Dark Energy Task Force (DETF) Created specific simulated data sets (Stage 2, Stage 3, Stage 4) Assessed their impact on our knowledge of dark energy as modeled with the w-wa parameters Followup questions: In what ways might the choice of DE parameters biased the DETF results? What impact can these data sets have on specific DE models (vs abstract parameters)? To what extent can these data sets deliver discriminating power between specific DE models? How is the DoE/ESA/NASA Science Working Group looking at these questions?

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105 The Dark Energy Task Force (DETF) NB: To make concrete comparisons this work ignores various possible improvements to the DETF data models. Created specific simulated data sets (Stage 2, Stage 3, Stage 4) Assessed their impact on our knowledge of dark energy as modeled with the w-wa parameters (see for example J Newman, H Zhan et al & Schneider et al, H Zhan et al to appear, Followup questions: NIR synergies) ALSO In what ways might the choice of DE Ground/Space parameters synergies biased the DETF results? What impact can these data sets have on specific DE models (vs abstract parameters)? DETF To what extent can these data sets deliver discriminating power between specific DE models? How is the DoE/ESA/NASA Science Working Group looking at these questions?

106 The Dark Energy Task Force (DETF) Created specific simulated data sets (Stage 2, Stage 3, Stage 4) Assessed their impact on our knowledge of dark energy as modeled with the w-wa parameters Followup questions: NB: To make concrete comparisons this work ignores various possible improvements to the DETF data models. (see for example J Newman, H Zhan et al & Schneider et al, H Zhan et al to appear, NIR synergies i.e Dome A & LSST synergies ) In what ways might the choice of DE parameters ALSObiased the DETF results? Ground/Space synergies What impact can these data sets have on specific DE models (vs abstract parameters)? DETF To what extent can these data sets deliver discriminating power between specific DE models? How is the DoE/ESA/NASA Science Working Group looking at these questions?

107 The Dark Energy Task Force (DETF) Created specific simulated data sets (Stage 2, Stage 3, Stage 4) Assessed their impact on our knowledge of dark energy as modeled with the w-wa parameters Followup questions: In what ways might the choice of DE parameters biased the DETF results? What impact can these data sets have on specific DE models (vs abstract parameters)? To what extent can these data sets deliver discriminating power between specific DE models? How is the DoE/ESA/NASA Science Working Group looking at these questions?

108 Summary In what ways might the choice of DE parameters have skewed the DETF results? A: Only by an overall (possibly important) rescaling What impact can these data sets have on specific DE models (vs abstract parameters)? A: Very similar to DETF results in w-wa space To what extent can these data sets deliver discriminating power between specific DE models? A: DETF Stage 3: Poor DETF Stage 4: Marginal Excellent within reach (AA)

109 How good is the w(a) ansatz? w -2 Sample w(z) curves in w -w a space Sample w(z) curves for the PNGB models wa ( ) = w + w a w-wa can only do these a ( ) w w w Sample w(z) curves for the EwP models z z DE models can do this (and much more)

110 How good is the w(a) ansatz? w w w w -2 Sample w(z) curves in w -w a space Sample w(z) curves for the PNGB models Sample w(z) curves for the EwP models z z NB: Better than wa ( ) = w + w a ( ) w-wa can only do these wa ( ) = & flat DE models can do this (and much more) a w

111 w( a) Try N-D stepwise constant w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + N i= z N parameters are coefficients of the top hat functions T ai, a i + ( ) AA & G Bernstein 26 (astro-ph/68269 ). More detailed info can be found at

112 w( a) Try N-D stepwise constant w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + i= N parameters are coefficients of the top hat functions N AA & G Bernstein 26 (astro-ph/68269 ). More detailed info can be found at z T ai, a i + ( ) Used by Huterer & Turner; Huterer & Starkman; Knox et al; Crittenden & Pogosian Linder; Reiss et al; Krauss et al de Putter & Linder; Sullivan et al

113 w( a) Try N-D stepwise constant w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + i= N parameters are coefficients of the top hat functions AA & G Bernstein 26 N z T ai, a i + ( ) Allows greater variety of w(a) behavior Allows each experiment to put its best foot forward Any signal rejects Λ

114 w( a) Try N-D stepwise constant w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + i= N parameters are coefficients of the top hat functions N z T ai, a i + ( ) Any signal Convergence rejects Λ AA & G Bernstein 26 Allows greater variety of w(a) behavior Allows each experiment to put its best foot forward

115 Illustration of stepwise parameterization of w(a). discrete paramterization model w a Z=4 Z= Each bin height is a free parameter 5

116 Illustration of stepwise parameterization of w(a) Refine bins as much as needed discrete paramterization model. w a Z=4 Z= Each bin height is a free parameter 6

117 Illustration of stepwise parameterization of w(a) Refine bins as much as needed discrete paramterization model. w a Z=4 Z= Each bin height is a free parameter 7

118 Illustration of stepwise parameterization of w(a) Refine bins as much as needed discrete paramterization model. w a Z=4 Z= Each bin height is a free parameter 8

119 Illustration of stepwise parameterization of w(a) Refine bins as much as needed. discrete paramterization model FOMSWG settled on 36 bins w a Z=4 Z= Each bin height is a free parameter 9

120 Q: How do you describe error ellipsis in ND space? A: In terms of N principle axes f i and corresponding N errors : σ i 2D illustration: f = Axis σ f 2 = Axis 2 σ 2

121 Q: How do you describe error ellipsis in ND space? A: In terms of N principle axes f i and corresponding N errors σ i : Principle component analysis 2D illustration: f = Axis σ f 2 = Axis 2 σ 2

122 Q: How do you describe error ellipsis in ND space? A: In terms of N principle axes f i and corresponding N errors : σ i 2D illustration: NB: in general the a complete basis: w cf = i i i f is form f = Axis f 2 = Axis 2 σ σ 2 c i The are independently measured qualities with errors σ i

123 Q: How do you describe error ellipsis in ND space? A: In terms of N principle axes f i and corresponding N errors : σ i 2D illustration: NB: in general the a complete basis: w cf = i i i f is form f = Axis f 2 = Axis 2 σ σ 2 c i The are independently measured qualities with errors σ i

124 DETF stage 2 2 Characterizing 9D ellipses by principle axes and Stage 2 ; lin-a corresponding N Grid = 9, z max = 4, Tag = 443 errors σ i σ i Principle Axes f's f f's a i a i f's a a z-=4 z =.5 z =.25 z = 7 8 9

125 WL Stage 4 Opt 2 Characterizing 9D ellipses by principle axes and Stage 4 Space WL corresponding Opt; lin-a N Grid = 9, z max = 4, Tag errors = 443 σ i σ i Principle Axes f's f f's a i a i f's a a z-=4 z =.5 z =.25 z = 7 8 9

126 WL Stage 4 Opt 2 Characterizing 9D ellipses by principle axes and Stage 4 Space WL corresponding Opt; lin-a N Grid = 6, z max = 4, errors Tag = 543 σ i σ i Principle Axes f's f f's i a a i f's a Convergence a z-=4 z =.5 z =.25 z = 7 8 9

127 DETF(-CL) 9D (-CL) e4 e3 F DETF/9D Grid Linear in a zmax = 4 scale: Stage 3 Stage 4 Ground e4 e3 BAOp BAOs SNp SNs WLp ALLp Bska Blst Slst Wska Wlst Aska Alst Stage 4 Space Stage 4 Ground+Space e4 e4 e3 e3 BAO SN WL S+W S+W+B [S S B lst W lst ] [B S S lst W lst ] All lst [S S W S B IIIs ] S s W lst

128 DETF(-CL) 9D (-CL) e4 e3 F DETF/9D Grid Linear in a zmax = 4 scale: Stage 3 Stage 4 Ground e4 e3 BAOp BAOs SNp SNs WLp ALLp Bska Blst Slst Wska Wlst Aska Alst Stage 2 Stage 3 = order of magnitude (vs.5 for DETF) Stage 4 Space Stage 4 Ground+Space e4 e3 e4 e3 BAO SN WL S+W S+W+B [S S B lst W lst ] [B S S lst W lst ] All lst [S S W S B IIIs ] S s W lst Stage 2 Stage 4 = 3 orders of magnitude (vs for DETF)

129 Upshot of N-D FoM: ) DETF underestimates impact of expts 2) DETF underestimates relative value of Stage 4 Inverts vs Stage 3 cost/fom 3) The above can be understood approximately in Estimates terms of a simple rescaling (related to higher S3 vs S4 dimensional parameter space). 4) DETF FoM is fine for most purposes (ranking, value of combinations etc).

130 Summary In what ways might the choice of DE parameters have skewed the DETF results? A: Only by an overall (possibly important) rescaling What impact can these data sets have on specific DE models (vs abstract parameters)? A: Very similar to DETF results in w-wa space To what extent can these data sets deliver discriminating power between specific DE models? A: DETF Stage 3: Poor DETF Stage 4: Marginal Excellent within reach (AA)

131 Summary In what ways might the choice of DE parameters have skewed the DETF results? A: Only by an overall (possibly important) rescaling What impact can these data sets have on specific DE models (vs abstract parameters)? A: Very similar to DETF results in w-wa space To what extent can these data sets deliver discriminating power between specific DE models? A: DETF Stage 3: Poor DETF Stage 4: Marginal Excellent within reach (AA)

132 DETF stage 2 [ Abrahamse, AA, Barnard, Bozek & Yashar PRD 28] DETF stage 3 DETF stage 4

133 Summary In what ways might the choice of DE parameters have skewed the DETF results? A: Only by an overall (possibly important) rescaling What impact can these data sets have on specific DE models (vs abstract parameters)? A: Very similar to DETF results in w-wa space To what extent can these data sets deliver discriminating power between specific DE models? A: DETF Stage 3: Poor DETF Stage 4: Marginal Excellent within reach (AA)

134 Summary In what ways might the choice of DE parameters have skewed the DETF results? A: Only by an overall (possibly important) rescaling What impact can these data sets have on specific DE models (vs abstract parameters)? A: Very similar to DETF results in w-wa space To what extent can these data sets deliver discriminating power between specific DE models? A: DETF Stage 3: Poor DETF Stage 4: Marginal Excellent within reach (AA)

135 Summary In what ways might the choice of DE parameters have skewed the DETF results? A: Only by an overall (possibly important) rescaling What impact can these data sets have on specific DE models (vs abstract parameters)? A: Very similar to DETF results in w-wa space To what extent can these data sets deliver discriminating power between specific DE models? A: DETF Stage 3: Poor DETF Stage 4: Marginal Excellent within reach (AA)

136 WL Stage 4 Opt Principle Axes σ i σ i f's f f's a i a Characterizing 9D ellipses by principle axes and Stage 4 Space WL corresponding Opt; lin-a N Grid = 9, z max = 4, Tag errors = 443Express models & data in space of coefficients of w(a) modes i f's a a z-=4 z =.5 z =.25 z = 7 8 9

137 w = cf i i i DETF Stage 4 ground [Opt] c / σ 2 2 c / σ

138 w = cf i i i DETF Stage 4 ground [Opt] c / σ 4 4 c / σ 3 3

139 The different kinds of curves correspond to different trajectories in mode space (similar to FT s) PNGB EXP IT AS w(a) a

140 DETF Stage 4 ground Data that reveals a universe with dark energy given by will have finite minimum 2 distances χ to other quintessence models powerful discrimination is possible.

141 Summary In what ways might the choice of DE parameters have skewed the DETF results? A: Only by an overall (possibly important) rescaling What impact can these data sets have on specific DE models (vs abstract parameters)? A: Very similar to DETF results in w-wa space To what extent can these data sets deliver discriminating power between specific DE models? A: DETF Stage 3: Poor DETF Stage 4: Marginal Excellent within reach (AA)

142 Summary In what ways might the choice of DE parameters have skewed the DETF results? A: Only by an overall (possibly important) rescaling What impact can these data sets have on specific DE models (vs abstract parameters)? A: Very similar to DETF results in w-wa space To what extent can these data sets deliver discriminating power between specific DE models? Interesting contribution A: to discussion of Stage 4 DETF Stage 3: Poor DETF Stage 4: Marginal Excellent within reach (AA)

143 How is the DoE/ESA/NASA Science Working Group looking at these questions? i) Using w(a) eigenmodes ii) Revealing value of higher modes

144 DoE/ESA/NASA JDEM Science Working Group Update agencies on figures of merit issues formed Summer 8 finished Dec 8 (report on arxiv Jan 9, moved on to SCG) Use w-eigenmodes to get more complete picture also quantify deviations from Einstein gravity

145 How is the DoE/ESA/NASA Science Working Group looking at these questions? i) Using w(a) eigenmodes ii) Revealing value of higher modes

146 Principle Axes σ i f i DETF= Stage 4 Space Opt All f k=6 =, Pr = z a Mode Mode 2 Mode 3 Mode z

147 This talk Part : Part 2 A few attempts to explain dark energy - Motivations, problems and other comments Theme: We may not know where this revolution is taking us, but it is already underway: Planning new experiments - DETF - Next questions

148 This talk Part : Part 2 A few attempts to explain dark energy - Motivations, problems and other comments Theme: We may not know where this revolution is taking us, but it is already underway: Planning new experiments - DETF - Next questions Deeply exciting physics

149 This talk Part : Part 2 A few attempts to explain dark energy - Motivations, problems and other comments Theme: We may not know where this revolution is taking us, but it is already underway: Rigorous quantitative case for Planning new Stage experiments 4 (i.e. LSST, JDEM, Euclid) - DETF Advances in combining techniques - Next questions Insights into ground & space synergies

150 This talk Part : Part 2 A few attempts to explain dark energy - Motivations, problems and other comments Theme: We may not know where this revolution is taking us, but it is already underway: Rigorous quantitative case for Planning new Stage experiments 4 (i.e. LSST, JDEM, Euclid) - DETF Advances in combining techniques - Next questions Insights into ground & space synergies

151 This talk Part : Part 2 A few attempts to explain dark energy - Motivations, problems and other comments Theme: We may not know where this revolution is taking us, but it is already underway: Rigorous quantitative case for Planning new Stage experiments 4 (i.e. LSST, JDEM, Euclid) - DETF Advances in combining techniques - Next questions Insights into NIR synergies w Photo z-s

152 This talk Part : Part 2 Deeply exciting physics A few attempts to explain dark energy - Motivations, problems and other comments Theme: We may not know where this revolution is taking us, but it is already underway: Rigorous quantitative case for Planning new Stage experiments 4 (i.e. LSST, JDEM, Euclid) - DETF Advances in combining techniques Insights into - Next questions NIR synergies w Photo z-s (i.e Dome A & LSST synergies)

153 END

154 Kowalski, et al., Ap.J.. (28) [SCP] (Flat universe assumed)

155 Kowalski, et al., Ap.J.. (28) [SCP] (Flat universe assumed)

156 Additional Slides

157 Focus on wa ( ) 57

158 Focus on wa ( ) Cosmic scale factor = time 58

159 Focus on wa ( ) Cosmic scale factor = time wa = w+ w a DETF: ( ) ( ) a 59

160 Focus on wa ( ) (Free parameters = ) Cosmic scale factor = time wa = w+ w a DETF: ( ) ( ) a 6

161 Focus on wa ( ) (Free parameters = ) Cosmic scale factor = time wa = w+ w a DETF: ( ) ( ) a (Free parameters = 2) 6

162 Illustration of FOMSWG parameterization of w(a) w a Z=4 Z= 62

163 Illustration of FOMSWG parameterization of w(a). w.5 Measure w w ( ) a Z=4 Z= 63

164 Illustration of FOMSWG parameterization of w(a). discrete paramterization model w a Each bin height is a free parameter 64

165 Illustration of FOMSWG parameterization of w(a) Refine bins as much as needed discrete paramterization model. w a Each bin height is a free parameter 65

166 Illustration of FOMSWG parameterization of w(a) Refine bins as much as needed discrete paramterization model. w a Z=4 Z= Each bin height is a free parameter 66

167 Illustration of FOMSWG parameterization of w(a) Refine bins as much as needed discrete paramterization model. w a Each bin height is a free parameter 67

168 Illustration of FOMSWG parameterization of w(a) Refine bins as much as needed. discrete paramterization model FOMSWG settled on 36 bins w a Each bin height is a free parameter 68

169 w( a) N-D stepwise w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + N i= z N parameters are coefficients of the top hat functions T ai, a i + ( ) AA & G Bernstein 26 (astro-ph/68269 ). More detailed info can be found at

170 w( a) N-D stepwise w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + i= N parameters are coefficients of the top hat functions N AA & G Bernstein 26 (astro-ph/68269 ). More detailed info can be found at z T ai, a i + ( ) Used by Huterer & Turner; Huterer & Starkman; Knox et al; Crittenden & Pogosian Linder; Reiss et al; Krauss et al de Putter & Linder; Sullivan et al

171 w( a) N-D stepwise w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + i= N parameters are coefficients of the top hat functions N z T ai, a i + ( ) Allows greater variety of w(a) behavior Allows each experiment to put its best foot forward Any signal rejects Λ

172 w( a) N-D stepwise w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + i= N parameters are coefficients of the top hat functions N z T ai, a i + ( ) Any signal Convergence rejects Λ AA & G Bernstein 26 Allows greater variety of w(a) behavior Allows each experiment to put its best foot forward

173 w( a) N-D stepwise w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + i= N parameters are coefficients of the top hat functions N z T ai, a i + ( ) Any signal Convergence rejects Λ AA & G Bernstein 26 Allows greater variety of w(a) behavior Allows each experiment to put its best foot forward

174 DETF FoM w Marginalize over all other parameters and find uncertainties in w and w a ΛCDM value DETF FoM = (area of ellipse) errors in w and w a are correlated w a 74

175 From DETF: The figure of merit is a quantitative guide; since the nature of dark energy is poorly understood, no single figure of merit is appropriate for every eventuality. FOMSWG emphasis 75

176 How good is the w(a) ansatz? w -2 Sample w(z) curves in w -w a space Sample w(z) curves for the PNGB models wa ( ) = w + w a w-wa can only do these a ( ) w w w Sample w(z) curves for the EwP models z z DE models can do this (and much more)

177 How good is the w(a) ansatz? w w w w -2 Sample w(z) curves in w -w a space Sample w(z) curves for the PNGB models Sample w(z) curves for the EwP models z z NB: Better than wa ( ) = w + w a ( ) w-wa can only do these wa ( ) = & flat DE models can do this (and much more) a w

178 w( a) Try N-D stepwise constant w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + N i= z N parameters are coefficients of the top hat functions T ai, a i + ( ) AA & G Bernstein 26 (astro-ph/68269 ). More detailed info can be found at

179 w( a) Try N-D stepwise constant w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + i= N parameters are coefficients of the top hat functions N AA & G Bernstein 26 (astro-ph/68269 ). More detailed info can be found at z T ai, a i + ( ) Used by Huterer & Turner; Huterer & Starkman; Knox et al; Crittenden & Pogosian Linder; Reiss et al; Krauss et al de Putter & Linder; Sullivan et al

180 w( a) Try N-D stepwise constant w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + i= N parameters are coefficients of the top hat functions AA & G Bernstein 26 N z T ai, a i + ( ) Allows greater variety of w(a) behavior Allows each experiment to put its best foot forward Any signal rejects Λ

181 w( a) Try N-D stepwise constant w(a) ( ) ( ) w( a) = + w a = + wt i ai, a i + i= N parameters are coefficients of the top hat functions N z T ai, a i + ( ) Any signal Convergence rejects Λ AA & G Bernstein 26 Allows greater variety of w(a) behavior Allows each experiment to put its best foot forward

182 Q: How do you describe error ellipsis in ND space? A: In terms of N principle axes f i and corresponding N errors : σ i 2D illustration: f = Axis σ f 2 = Axis 2 σ 2

183 Q: How do you describe error ellipsis in ND space? A: In terms of N principle axes f i and corresponding N errors σ i : Principle component analysis 2D illustration: f = Axis σ f 2 = Axis 2 σ 2

184 Q: How do you describe error ellipsis in ND space? A: In terms of N principle axes f i and corresponding N errors : σ i 2D illustration: NB: in general the a complete basis: w cf = i i i f is form f = Axis f 2 = Axis 2 σ σ 2 c i The are independently measured qualities with errors σ i

185 Q: How do you describe error ellipsis in ND space? A: In terms of N principle axes f i and corresponding N errors : σ i 2D illustration: NB: in general the a complete basis: w cf = i i i f is form f = Axis f 2 = Axis 2 σ σ 2 c i The are independently measured qualities with errors σ i

186 DETF stage 2 2 Characterizing 9D ellipses by principle axes and Stage 2 ; lin-a corresponding N Grid = 9, z max = 4, Tag = 443 errors σ i σ i Principle Axes f's f f's a i a i f's a a z-=4 z =.5 z =.25 z = 7 8 9

187 WL Stage 4 Opt 2 Characterizing 9D ellipses by principle axes and Stage 4 Space WL corresponding Opt; lin-a N Grid = 9, z max = 4, Tag errors = 443 σ i σ i Principle Axes f's f f's a i a i f's a a z-=4 z =.5 z =.25 z = 7 8 9

188 WL Stage 4 Opt 2 Characterizing 9D ellipses by principle axes and Stage 4 Space WL corresponding Opt; lin-a N Grid = 6, z max = 4, errors Tag = 543 σ i σ i Principle Axes f's f f's i a a i f's a Convergence a z-=4 z =.5 z =.25 z = 7 8 9

189 DETF(-CL) 9D (-CL) e4 e3 F DETF/9D Grid Linear in a zmax = 4 scale: Stage 3 Stage 4 Ground e4 e3 BAOp BAOs SNp SNs WLp ALLp Bska Blst Slst Wska Wlst Aska Alst Stage 4 Space Stage 4 Ground+Space e4 e4 e3 e3 BAO SN WL S+W S+W+B [S S B lst W lst ] [B S S lst W lst ] All lst [S S W S B IIIs ] S s W lst