1. 1 Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture October 15 2010

2. 2 How can one accelerate the universe?

3. 3 1)A cosmological constant

4. 4 How can one accelerate the universe? 1)A cosmological constant AKA

5. 5 How can one accelerate the universe? 1)A cosmological constant AKA

6. 6 How can one accelerate the universe? 1)A cosmological constant AKA

7. How can one accelerate the universe? 2) Cosmic inflation

8. 8 How can one accelerate the universe? 2) Cosmic inflation V

9. 9 How can one accelerate the universe? 2) Cosmic inflation Dynamical V

10. How can one accelerate the universe? 1)A cosmological constant 2) Cosmic inflation

11. How can one accelerate the universe? 3) Modify Einstein Gravity

12. 12 Focus on

13. 13 Focus on Cosmic scale factor = “time”

14. 14 Focus on Cosmic scale factor = “time” DETF:

15. 15 Focus on Cosmic scale factor = “time” DETF: (Free parameters = ∞)

16. 16 Focus on Cosmic scale factor = “time” DETF: (Free parameters = ∞) (Free parameters = 2)

17. 17 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 w0-wa parameters

18. 18 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

19. 19 DETF Projections Stage 3 Figure of merit Improvement over Stage 2 

20. 20 DETF Projections Ground Figure of merit Improvement over Stage 2 

21. 21 DETF Projections Space Figure of merit Improvement over Stage 2 

22. 22 DETF Projections Ground + Space Figure of merit Improvement over Stage 2 

23. 23 Followup questions:  In what ways might the choice of DE parameters have skewed 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?

24. 24 Followup questions:  In what ways might the choice of DE parameters have skewed 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?

25. 25 w0-wa can only do these DE models can do this (and much more) w z How good is the w(a) ansatz?

26. 26 w0-wa can only do these DE models can do this (and much more) w z How good is the w(a) ansatz? NB: Better than & flat

27. 27 Illustration of stepwise parameterization of w(a) Z=4 Z=0

28. 28 Measure Z=4 Z=0 Illustration of stepwise parameterization of w(a)

29. 29 Each bin height is a free parameter Illustration of stepwise parameterization of w(a)

30. 30 Each bin height is a free parameter Refine bins as much as needed Illustration of stepwise parameterization of w(a)

31. 31 Each bin height is a free parameter Refine bins as much as needed Z=4 Z=0 Illustration of stepwise parameterization of w(a)

32. 32 Each bin height is a free parameter Refine bins as much as needed Illustration of stepwise parameterization of w(a)

33. 33 Try N-D stepwise constant w(a) AA & G Bernstein 2006 (astro-ph/0608269 ). More detailed info can be found at http://www.physics.ucdavis.edu/Cosmology/albrecht/MoreInfo0608269/astro-ph/0608269 http://www.physics.ucdavis.edu/Cosmology/albrecht/MoreInfo0608269/ N parameters are coefficients of the “top hat functions”

34. 34 Try N-D stepwise constant w(a) AA & G Bernstein 2006 (astro-ph/0608269 ). More detailed info can be found at http://www.physics.ucdavis.edu/Cosmology/albrecht/MoreInfo0608269/astro-ph/0608269 http://www.physics.ucdavis.edu/Cosmology/albrecht/MoreInfo0608269/ N parameters are coefficients of the “top hat functions” Used by Huterer & Turner; Huterer & Starkman; Knox et al; Crittenden & Pogosian Linder; Reiss et al; Krauss et al de Putter & Linder; Sullivan et al

35. 35 Try N-D stepwise constant w(a) AA & G Bernstein 2006 N parameters are coefficients of the “top hat functions”  Allows greater variety of w(a) behavior  Allows each experiment to “put its best foot forward”  Any signal rejects Λ

36. 36 Try N-D stepwise constant w(a) AA & G Bernstein 2006 N parameters are coefficients of the “top hat functions”  Allows greater variety of w(a) behavior  Allows each experiment to “put its best foot forward”  Any signal rejects Λ “Convergence”

37. 37 2D illustration: Axis 1 Axis 2 Q: How do you describe error ellipsis in N D space? A: In terms of N principle axes and corresponding N errors :

38. 38 Q: How do you describe error ellipsis in N D space? A: In terms of N principle axes and corresponding N errors : 2D illustration: Axis 1 Axis 2 Principle component analysis

39. 39 2D illustration: Axis 1 Axis 2 NB: in general the s form a complete basis: The are independently measured qualities with errors Q: How do you describe error ellipsis in N D space? A: In terms of N principle axes and corresponding N errors :

40. 40 2D illustration: Axis 1 Axis 2 NB: in general the s form a complete basis: The are independently measured qualities with errors Q: How do you describe error ellipsis in N D space? A: In terms of N principle axes and corresponding N errors :

41. 41 Principle Axes Characterizing ND ellipses by principle axes and corresponding errors DETF stage 2 z-=4z =1.5z =0.25z =0

42. 42 Principle Axes Characterizing ND ellipses by principle axes and corresponding errors WL Stage 4 Opt z-=4z =1.5z =0.25z =0

43. 43 Principle Axes Characterizing ND ellipses by principle axes and corresponding errors WL Stage 4 Opt “Convergence” z-=4z =1.5z =0.25z =0

44. 44 DETF(-CL) 9D (-CL)

45. 45 DETF(-CL) 9D (-CL) Stage 2  Stage 4 = 3 orders of magnitude (vs 1 for DETF) Stage 2  Stage 3 = 1 order of magnitude (vs 0.5 for DETF)

46. 46 Upshot of ND FoM: 1)DETF underestimates impact of expts 2)DETF underestimates relative value of Stage 4 vs Stage 3 3)The above can be understood approximately in terms of a simple rescaling (related to higher dimensional parameter space). 4)DETF FoM is fine for most purposes (ranking, value of combinations etc).

47. 47 Upshot of ND FoM: 1)DETF underestimates impact of expts 2)DETF underestimates relative value of Stage 4 vs Stage 3 3)The above can be understood approximately in terms of a simple rescaling (related to higher dimensional parameter space). 4)DETF FoM is fine for most purposes (ranking, value of combinations etc).

48. 48 Upshot of ND FoM: 1)DETF underestimates impact of expts 2)DETF underestimates relative value of Stage 4 vs Stage 3 3)The above can be understood approximately in terms of a simple rescaling (related to higher dimensional parameter space). 4)DETF FoM is fine for most purposes (ranking, value of combinations etc).

49. 49 Upshot of ND FoM: 1)DETF underestimates impact of expts 2)DETF underestimates relative value of Stage 4 vs Stage 3 3)The above can be understood approximately in terms of a simple rescaling (related to higher dimensional parameter space). 4)DETF FoM is fine for most purposes (ranking, value of combinations etc).

50. 50 An example of the power of the principle component analysis: Q: I’ve heard the claim that the DETF FoM is unfair to BAO, because w0-wa does not describe the high-z behavior to which BAO is particularly sensitive. Why does this not show up in the 9D analysis?

51. 51 DETF(-CL) 9D (-CL) Specific Case:

52. 52 BAO z-=4z =1.5z =0.25z =0

53. 53 SN z-=4z =1.5z =0.25z =0

54. 54 BAO DETF z-=4z =1.5z =0.25z =0

55. 55 SN DETF z-=4z =1.5z =0.25z =0

56. 56 z-=4z =1.5z =0.25z =0 SN w0-wa analysis shows two parameters measured on average as well as 3.5 of these DETF 9D

57. 57 Upshot of ND FoM: 1)DETF underestimates impact of expts 2)DETF underestimates relative value of Stage 4 vs Stage 3 3)The above can be understood approximately in terms of a simple rescaling 4)DETF FoM is fine for most purposes (ranking, value of combinations etc). Inverts cost/FoM Estimates S3 vs S4

58. 58 Upshot of ND FoM: 1)DETF underestimates impact of expts 2)DETF underestimates relative value of Stage 4 vs Stage 3 3)The above can be understood approximately in terms of a simple rescaling 4)DETF FoM is fine for most purposes (ranking, value of combinations etc).  A nice way to gain insights into data (real or imagined)

59. 59 Followup questions:  In what ways might the choice of DE parameters have skewed 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?

60. 60 A: Only by an overall (possibly important) rescaling Followup questions:  In what ways might the choice of DE parameters have skewed 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?

61. 61 Followup questions:  In what ways might the choice of DE parameters have skewed 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?

62. 62 How well do Dark Energy Task Force simulated data sets constrain specific scalar field quintessence models? Augusta Abrahamse Brandon Bozek Michael Barnard Mark Yashar +AA + DETF Simulated data + Quintessence potentials + MCMC See also Dutta & Sorbo 2006, Huterer and Turner 1999 & especially Huterer and Peiris 2006

63. 63 The potentials Exponential (Wetterich, Peebles & Ratra) PNGB aka Axion (Frieman et al) Exponential with prefactor (AA & Skordis) Inverse Power Law (Ratra & Peebles, Steinhardt et al)

64. 64 The potentials Exponential (Wetterich, Peebles & Ratra) PNGB aka Axion (Frieman et al) Exponential with prefactor (AA & Skordis) Inverse Power Law (Ratra & Peebles, Steinhardt et al)

65. 65 The potentials Exponential (Wetterich, Peebles & Ratra) PNGB aka Axion (Frieman et al) Exponential with prefactor (AA & Skordis) Inverse Power Law (Ratra & Peebles, Steinhardt et al) Stronger than average motivations & interest

66. 66 The potentials Exponential (Wetterich, Peebles & Ratra) PNGB aka Axion (Frieman et al) Exponential with prefactor (AA & Skordis) PRD 2008 Inverse Tracker (Ratra & Peebles, Steinhardt et al)

67. 67 …they cover a variety of behavior.

68. 68 Challenges: Potential parameters can have very complicated (degenerate) relationships to observables Resolved with good parameter choices (functional form and value range)

69. 69 DETF stage 2 DETF stage 3 DETF stage 4

70. 70 DETF stage 2 DETF stage 3 DETF stage 4 (S2/3) (S2/10) Upshot: Story in scalar field parameter space very similar to DETF story in w0-wa space.

71. 71 Followup questions:  In what ways might the choice of DE parameters have skewed 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?

72. 72 A: Very similar to DETF results in w0-wa space Followup questions:  In what ways might the choice of DE parameters have skewed 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?

73. 73 Followup questions:  In what ways might the choice of DE parameters have skewed 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?

74. 74 Michael Barnard PRD 2008 Followup questions:  In what ways might the choice of DE parameters have skewed 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?

75. 75 Problem: Each scalar field model is defined in its own parameter space. How should one quantify discriminating power among models? Our answer:  Form each set of scalar field model parameter values, map the solution into w(a) eigenmode space, the space of uncorrelated observables.  Make the comparison in the space of uncorrelated observables.

76. 76 Principle Axes Characterizing 9D ellipses by principle axes and corresponding errors WL Stage 4 Opt z-=4z =1.5z =0.25z =0 Axis 1 Axis 2

77. 77 ●●●● ● ■ ■ ■ ■ ■ ■■■■■ ● Data ■ Theory 1 ■ Theory 2 Concept: Uncorrelated data points (expressed in w(a) space) 0 1 2 0 51015 X Y

78. 78 Starting point: MCMC chains giving distributions for each model at Stage 2.

79. 79 DETF Stage 3 photo [Opt]

80. 80 DETF Stage 3 photo [Opt]

81. 81 DETF Stage 3 photo [Opt]  Distinct model locations  mode amplitude/σ i “physical”  Modes (and σ i ’s) reflect specific expts.

82. 82 DETF Stage 3 photo [Opt]

83. 83 DETF Stage 3 photo [Opt]

84. 84 Comments on model discrimination Principle component w(a) “modes” offer a space in which straightforward tests of discriminating power can be made. The DETF Stage 4 data is approaching the threshold of resolving the structure that our scalar field models form in the mode space.

85. 85 Comments on model discrimination Principle component w(a) “modes” offer a space in which straightforward tests of discriminating power can be made. The DETF Stage 4 data is approaching the threshold of resolving the structure that our scalar field models form in the mode space.

86. 86 Followup questions:  In what ways might the choice of DE parameters have skewed 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? A: DETF Stage 3: Poor DETF Stage 4: Marginal… Excellent within reach

87. 87 Followup questions:  In what ways might the choice of DE parameters have skewed 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? A: DETF Stage 3: Poor DETF Stage 4: Marginal… Excellent within reach Structure in mode space

88. 88 Followup questions:  In what ways might the choice of DE parameters have skewed 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? A: DETF Stage 3: Poor DETF Stage 4: Marginal… Excellent within reach

89. 89 Followup questions:  In what ways might the choice of DE parameters have skewed 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?

90. 90 DoE/ESA/NASA JDEM Science Working Group  Update agencies on figures of merit issues  formed Summer 08  finished ~now (moving on to SCG)  Use w-eigenmodes to get more complete picture  also quantify deviations from Einstein gravity  For today: Something we learned about normalizing modes

91. 91 NB: in general the s form a complete basis: The are independently measured qualities with errors Define which obey continuum normalization: then where

92. 92 Define which obey continuum normalization: then where Q: Why? A: For lower modes, has typical grid independent “height” O(1), so one can more directly relate values of to one’s thinking (priors) on

93. 93 Principle Axes

94. 94 Principle Axes

95. 95 Upshot: More modes are interesting (“well measured” in a grid invariant sense) than previously thought.

96. 96 A: DETF Stage 3: Poor DETF Stage 4: Marginal… Excellent within reach (AA)  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 w0-wa space Summary To what extent can these data sets deliver discriminating power between specific DE models?

97. 97 A: DETF Stage 3: Poor DETF Stage 4: Marginal… Excellent within reach (AA)  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 w0-wa space Summary To what extent can these data sets deliver discriminating power between specific DE models?

98. 98 A: DETF Stage 3: Poor DETF Stage 4: Marginal… Excellent within reach (AA)  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 w0-wa space Summary To what extent can these data sets deliver discriminating power between specific DE models?

99. 99 A: DETF Stage 3: Poor DETF Stage 4: Marginal… Excellent within reach (AA)  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 w0-wa space Summary To what extent can these data sets deliver discriminating power between specific DE models?

100. 100 A: DETF Stage 3: Poor DETF Stage 4: Marginal… Excellent within reach (AA)  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 w0-wa space Summary To what extent can these data sets deliver discriminating power between specific DE models?

101. 101 A: DETF Stage 3: Poor DETF Stage 4: Marginal… Excellent within reach (AA)  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 w0-wa space Summary To what extent can these data sets deliver discriminating power between specific DE models?

102. 102 A: DETF Stage 3: Poor DETF Stage 4: Marginal… Excellent within reach (AA)  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 w0-wa space Summary To what extent can these data sets deliver discriminating power between specific DE models?

103. 103  How is the DoE/ESA/NASA Science Working Group looking at these questions? i)Using w(a) eigenmodes ii)Revealing value of higher modes

104. 104 Principle Axes

105. 105 END

106. 106 Additional Slides

107. 107

108. 108

109. 109 Stage 2

110. 110 Stage 2

111. 111 Stage 2

112. 112 Stage 2

113. 113 Stage 2

114. 114 Stage 2

115. 115 Stage 2

116. 116 DETF Stage 3 photo [Opt]

117. 117 Eigenmodes: Stage 3 Stage 4 g Stage 4 s z=4z=2z=1z=0.5z=0

118. 118 Eigenmodes: Stage 3 Stage 4 g Stage 4 s z=4z=2z=1z=0.5z=0 N.B. σ i change too

119. 119 DETF Stage 4 ground [Opt]

120. 120 DETF Stage 4 ground [Opt]

121. 121 DETF Stage 4 space [Opt]

122. 122 DETF Stage 4 space [Opt]

123. 123 The different kinds of curves correspond to different “trajectories” in mode space (similar to FT’s)

124. 124 DETF Stage 4 ground  Data that reveals a universe with dark energy given by “ “ will have finite minimum “distances” to other quintessence models  powerful discrimination is possible.

125. 125 Consider discriminating power of each experiment (  look at units on axes)

126. 126 DETF Stage 3 photo [Opt]

127. 127 DETF Stage 3 photo [Opt]

128. 128 DETF Stage 4 ground [Opt]

129. 129 DETF Stage 4 ground [Opt]

130. 130 DETF Stage 4 space [Opt]

131. 131 DETF Stage 4 space [Opt]

132. 132 Quantify discriminating power:

133. 133 Stage 4 space Test Points Characterize each model distribution by four “test points”

134. 134 Stage 4 space Test Points Characterize each model distribution by four “test points” (Priors: Type 1 optimized for conservative results re discriminating power.)

135. 135 Stage 4 space Test Points

136. 136 Measured the χ 2 from each one of the test points (from the “test model”) to all other chain points (in the “comparison model”). Only the first three modes were used in the calculation. Ordered said χ 2 ‘s by value, which allows us to plot them as a function of what fraction of the points have a given value or lower. Looked for the smallest values for a given model to model comparison.

137. 137 Model Separation in Mode Space Fraction of compared model within given χ 2 of test model’s test point Test point 4 Test point 1 Where the curve meets the axis, the compared model is ruled out by that χ 2 by an observation of the test point. This is the separation seen in the mode plots. 99% confidence at 11.36

138. 138 Model Separation in Mode Space Fraction of compared model within given χ 2 of test model’s test point Test point 4 Test point 1 Where the curve meets the axis, the compared model is ruled out by that χ 2 by an observation of the test point. This is the separation seen in the mode plots. 99% confidence at 11.36 …is this gap This gap…

139. 139 DETF Stage 3 photo Test Point Model Comparison Model [4 models] X [4 models] X [4 test points]

140. 140 DETF Stage 3 photo Test Point Model Comparison Model

141. 141 DETF Stage 4 ground Test Point Model Comparison Model

142. 142 DETF Stage 4 space Test Point Model Comparison Model

143. 143 PNGB ExpITAS Point 10.001 0.10.2 Point 20.0020.010.51.8 Point 30.0040.041.26.2 Point 40.010.041.610.0 Exp Point 10.0040.0010.10.4 Point 20.010.0010.41.8 Point 30.030.0010.74.3 Point 40.10.011.19.1 IT Point 10.20.10.0010.2 Point 20.50.40.00040.7 Point 31.00.70.0013.3 Point 42.71.80.0116.4 AS Point 10.1 0.0001 Point 20.20.1 0.0001 Point 30.2 0.10.0002 Point 40.60.50.20.001 DETF Stage 3 photo A tabulation of χ 2 for each graph where the curve crosses the x-axis (= gap) For the three parameters used here, 95% confidence  χ 2 = 7.82, 99%  χ 2 = 11.36. Light orange > 95% rejection Dark orange > 99% rejection Blue: Ignore these because PNGB & Exp hopelessly similar, plus self-comparisons

144. 144 PNGB ExpITAS Point 10.0010.0050.30.9 Point 20.0020.042.47.6 Point 30.0040.26.018.8 Point 40.010.28.026.5 Exp Point 10.010.0010.41.6 Point 20.040.0022.17.8 Point 30.010.0033.814.5 Point 40.030.016.024.4 IT Point 11.10.90.0021.2 Point 23.22.60.0013.6 Point 36.75.20.0028.3 Point 418.713.60.0430.1 AS Point 12.41.40.50.001 Point 22.32.10.80.001 Point 33.33.11.20.001 Point 47.47.02.60.001 DETF Stage 4 ground A tabulation of χ 2 for each graph where the curve crosses the x-axis (= gap). For the three parameters used here, 95% confidence  χ 2 = 7.82, 99%  χ 2 = 11.36. Light orange > 95% rejection Dark orange > 99% rejection Blue: Ignore these because PNGB & Exp hopelessly similar, plus self-comparisons

145. 145 PNGB ExpITAS Point 10.01 0.41.6 Point 20.010.053.213.0 Point 30.020.28.230.0 Point 40.040.210.937.4 Exp Point 10.020.0020.62.8 Point 20.050.0032.913.6 Point 30.10.015.224.5 Point 40.30.028.433.2 IT Point 11.51.30.0052.2 Point 24.63.80.0028.2 Point 39.77.70.0039.4 Point 427.820.80.157.3 AS Point 13.23.01.10.002 Point 24.94.61.80.003 Point 310.910.44.30.01 Point 426.525.110.60.01 DETF Stage 4 space A tabulation of χ 2 for each graph where the curve crosses the x-axis (= gap) For the three parameters used here, 95% confidence  χ 2 = 7.82, 99%  χ 2 = 11.36. Light orange > 95% rejection Dark orange > 99% rejection Blue: Ignore these because PNGB & Exp hopelessly similar, plus self-comparisons

146. 146 PNGB ExpITAS Point 10.01.093.6 Point 20.010.17.329.1 Point 30.040.418.467.5 Point 40.090.424.184.1 Exp Point 10.040.011.46.4 Point 20.10.016.630.7 Point 30.30.0111.855.1 Point 40.70.0518.874.6 IT Point 13.52.80.014.9 Point 210.48.50.0118.4 Point 321.917.40.0121.1 Point 462.446.90.2129.0 AS Point 17.26.82.50.004 Point 210.910.34.00.01 Point 324.623.39.80.01 Point 459.756.623.90.01 DETF Stage 4 space 2/3 Error/mode Many believe it is realistic for Stage 4 ground and/or space to do this well or even considerably better. (see slide 5) A tabulation of χ 2 for each graph where the curve crosses the x-axis (= gap). For the three parameters used here, 95% confidence  χ 2 = 7.82, 99%  χ 2 = 11.36. Light orange > 95% rejection Dark orange > 99% rejection