1. The Oddly QuietUniverse: How theCMB challengescosmology’sstandard model Glenn D. Starkman Craig Copi, Dragan Huterer & Dominik Schwarz Francesc Ferrer, Amanda Yoho Neil Cornish, David Spergel, & Eiichiro Komatsu Pascal Vaudrevange,, Dan Cumberbatch; Jean-Philippe Uzan, Alain Riazuelo, Jeff Weeks, R. Trotta, Pascal Vaudrevange, Bob Nichols, Peter Freeman Joe Silk, Andrew Jaffe, Anastasia Anarchiou
2. COBE - DMR Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level
3. The WMAP Sky NASA/WMAP Science team
4. COBE vs. WMAP
5. OutlineThe largest scaleproperties of the universe:Thethe angular problem low-l / large-angle power l from Cl to C(θ) spectrum ClBeyond Cl and C(θ) • seeing the solar system in the microwave backgroundAnd back: troubles in cosmological paradise
6. The WMAP Sky NASA/WMAP Science team
7. Angular Power Spectrum  T =  lmalm Ylm( , )  T =  lmalm= (2l+1)-1 m |alm|2 Ylm( , ) Standardmodel for the origin of fluctuations (inflation): alm are independent Gaussian random6 parameter fit to ~38 points variables, with < alm al’m’> = Cl  l l ‘mm’ ⇒ Sky is statistically isotropic and Gaussian random ALL interesting information in the sky is contained in Cl Cl = (2l+1)-1 m |alm| NASA/WMAP Science team
8. Measuring the shape of space Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level NASA/WMAP Science team
9. Click to edit Master text styles Cosmic Triangle Second level ● Third level ● Fourth level ● Fifth level
10. Angular Power Spectrum Ω= 1 ± .0x
11. Is there anythinginteresting left to learn about the Universe on large scales?
12. Motivation: “The Low-l Anomaly”
13. Beyond Cl: Searches for Departures from Gaussianity/Statistical Isotropy• angular momentum dispersion axes (da Oliveira-Costa, et al.)• Genus curves (Park)• Spherical Mexican-hat wavelets (Vielva et al.)• Bispectrum (Souradeep et al.)• North-South asymmetries in multipoint functions(Eriksen et al., Hansen et al.)• Cold hot spots, hot cold spots (Larson and Wandelt)• Land & Magueijo scalars/vectors• multipole vectors (Copi, Huterer & GDS; Schwarz, SCH; CHSS; also Weeks; Seljak and Slosar; Dennis)
14. Multipole VectorsQ: What are the directions associated with the l th multipole: ∆Tl (θ,φ)  m almYlm (θ,φ) ?Dipole (l =1) :m a1mY1m (θ,φ) = Α(1) (ûx(1,1), ûy(1,1), ûz(1,1)) (sinθ cosφ , sinθ sinφ , cosθ )Advantages: 1) û (1,1) is a vector, Α(1) is a scalar 2) Only Α(1) depends on C1
15. Shape and Alignment of the Quadrupole and OctopoleA. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga, A. Hamilton. Phys.Rev.D69:063516,2004astro-ph/0307282For each l, find the axis nl around which the angularmomentum dispersion : (L)2  ∑m m2 |alm (nl)|2 is maximizedResults: Probability• octopole is unusually “planar” 1/20?? (dominated by |m| = 3 if z  n3 ).• n2 .n3=0.9838 1/60
16. Multipole Vectors General l, write: m almYlm (θ,φ) ≈ Α (l) [(û (l,1)⋅ê)…(û (l, l) ê ) traces] - all{{alm, m=- l,…, l}, l =(0,1,)2,…}  {Α(l) ,{û (l,i),l =1,… l}, l = (0,1,)2,…} Advantages: 1) û(l,i) are vectors, Α (l) is a scalar 2) Only Α (l) depends on C l
17. Maxwell Multipole Vectors m almYlm (θ,φ) = [ (u (l,1)⋅)… (u (l, l) ⋅)r -1]r=1 manifestly symmetric AND trace free: 2 (1/r)  (r)J.C. Maxwell, A Treatise on Electricity and Magnetism, v.1, 1873 (1st ed.)
18. Notice: Area Vectors• Quadrupole has 2 vectors, i.e. quadrupole is a plane• Octopole has 3 vectors, i.e. octopole is 3 planes Suggests defining: w(l,i,j)  (û (l,i) x û (l,j)) “area vectors” Carry some, but not all, of the information For the experts: ● w(2,2,2) || n2 • octopole is perfectly planar if w(3,1,2) || w(3,2,3) || w(3,3,1) and then: n3 || w(3,I,j)
19. l=2&3 Area Vectors l=3 normal l=3 normal equinox l=2 normal l=2 dipole normal l=3 normal tic ip ecl P SGl=3 normal dipole equinox l=3 normal l=3 normal
20. Alignment probabilitiesProbability of the quadrupole and octopole planesbeing so aligned: (0.1-0.6)%Conditional probability of the aligned planes being soperpendicular to the plane of the solar system: (0.2-1.7)%Conditional probability of aligned planes perpendicularto the solar system pointing at the dipole/ecliptic: few% Net : < 10-5
21. Area vectors tell about the orientations of the normals of the multipole planesDON’T include all the information (multipole vectors do)Can rotate the aligned planes about their common axis!
22. l=2&3 : The Map
23. Quadrupole+Octopole Correlations -- Explanations: Cosmology?• Cosmology -- you’ve got to be kidding? you choose: the dipole or the ecliptic ?
24. Quadrupole+Octopole Correlations -- Explanations: Systematics?• Cosmology• Systematics -- but … how do you get such an effect? esp., how do you get a N-S ecliptic asymmetry? (dipole mis-subtraction?) how do you avoid oscillations in the time-ordered data?
25. Angular Power Spectrum At least 3 other major deviations in the Cl in 1st year data
26. Power spectrum: ecliptic plane vs. poles “First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: The Angular Power Spectrum” G. Hinshaw, et.al., 2003, ApJS, 148, 135 -- only v.1 on archive All 3 other major deviations are in the ecliptic polar Cl only!!
27. No Dip? © Boomerang Collaboration
28. The case against a systematic (cont.) ArcheopsFerrer, Starkman & Yoho (in progress): the anomaly in the first peak is (largely orentirely) localized to a region around the north ecliptic pole representing < 10% ofthe sky. Statistical significance tbd.
29. Dip? What dip?
30. Different weighting schemes: in the CMB:dip to a small patch of the north ecliptic sky.Amanda Yoho, Francesc Ferrer, Glenn D.389
31. No first peak at the N pole?
32. No first peak at the N pole?
33. Quadrupole+Octopole Correlations -- Explanations: more galaxy?• Cosmology• Systematics• The Galaxy: • has the wrong multipole structure (shape) • is likely to lead to GALACTIC not ECLIPTIC/DIPOLE/EQUINOX correlations
34. Quadrupole+Octopole Correlations -- Explanations: Foregrounds?• Systematics• Cosmology• The Galaxy• Foregrounds -- difficult: 1. Changing a patch of the sky typically gives you: Yl0 2. Sky has 5x more octopole than quadrupole 3. How do you get a physical ring perpendicular to the ecliptic 4. Caution: can add essentially arbitrary dipole, which can entirely distort the ring! (Silk & Inoue) 5. How do you hide the foreground from detection? T≈TCMB
35. Future data Dvorkin, Peiris & Hu: Planck TE cross-correlation will test models “where asuperhorizon scale modulation of the gravitational potential field causes the CMBtemperature field to locally look statistically anisotropic, even though globally themodel preserves statistical homogeneity and isotropy. Φ(x) = g1(x) [1 + h(x)] + g2(x)Where g1(x) and g2(x) are Gaussian random fields andh(x) is the modulating long-wavelength fieldFor dipolar h(x), predictions for the correlation between the first 10 multipoles of thetemperature and polarization fields can typically be tested at better than the 98% CL.For quadrupolar h, the polarization quadrupole and octopole should be moderatelyaligned. Predicted correlations between temperature and polarization multipolesout to ℓ = 5 provide testsat the ∼ 99% CL or stronger for quadrupolar models that makethe temperature alignment more than a few percent likely.
36. Future dataPlanck: • different systematics (direct T measurement) ⇒ Confirmation of low-l alignments significant
37. Future dataPlanck: • different systematics (direct T measurement) ⇒ Confirmation of low-l alignments significant • TE correlations – test particular models (Dvorkin, Peiris & Hu) of the quadrupole-octopole alignment
40. The Angular Correlation Function, C( )C() = < T(Ω1)T(Ω2)>Ω1.Ω2=cos But (established lore): Click to edit Master text styles Second level C() = l Cl Pl(cos ()) ● Third level ● Fourth level  Same information as Cl, just ● Fifth level differently organized IF • C() is obtained by a full sky average or • the sky is statistically isotropic, i.e. if <alm a*l’m’>=δll’ δmm’Cl NASA/WMAP science team
41. Is the Large-Angle Anomaly Significant? One measure (WMAP1): S1/2 = -11/2 [C()]2 d cos Only 0.15% of realizations ofinflationary CDM universewith the best-fit parametershave lower S!
42. Is the Large-Angle Anomaly Significant?One measure (WMAP1): S1/2 = -11/2 [C()]2 d cos
44. Two point angular correlation function -- WMAP1
45. Two point angular correlation function -- WMAP3
46. Statistics of C(θ)S1/2 = -1 1/2 [C()]2 dcos 
47. Origin of C(θ)
48. Is it an accident?Only 2% of rotated and cut full skies have this low a cut-sky S1/2
49. Statistics of C(θ)• 0.03% of realizations of the concordance modelof inflationary ΛCDM have so little cut sky large-angle correlation !• Either: this reflects a .03% probable full skyC(θ), or a 5% probable C(θ) and a 2%probably alignment with the galaxy
50. Statistics of C(θ)Efstathiou, Ma and Hanson:Direct calculation of C(θ) on the cut sky is asuboptimal estimator of C(θ) on the full sky.The estimator which is optimal if the sky is GRSI(C(θ) calculated from reconstructed full sky Cl) yieldsa larger S1/2, which is less significant in itssmallness.
51. Statistics of C(θ)Response: • So what ? This is completely consistent with what we said. What is odd is the CUT-SKY C(θ) ! • Not true anyway (CHSS in press): ● Reconstructed Cl are biased upward ● “Weighted” reconstructed Cl are not biased, but require smoothing when calculated at low Nside – smoothing causes contamination ● When reconstructing full sky Cl from smoothed cut skies – everything depends on how you fill the cut
52. Origin of C(θ)
53. The Conspiracy theory: minimizing S1/2To obtain S1/2 < 971 with the WMAP Clrequires varying C2, C3, C4 & C5!
54. Reproducing C(θ) To obtain S1/2<971 with the WMAP Cl, requires varying C2, C3, C4 and C5.The low-l Cl are therefore not measuringlarge angle (θ~π/l ), but smaller anglecorrelations
55. Low l = large angle?The low-l Cl are therefore not measuringlarge angle (θ~π/l ), but smaller anglecorrelations
56. “The Low-l Anomaly”
57. Violation of GRSIEven if we replaced all the theoretical Cl by their measured values up to l=20,Translation: The observed absence cosmic variance would give only a 3% of large-angle correlationchance of recovering this low an S1/2 in is inconsistentrealization with the a particular (>>97%) and most of thost of those are much most fits with the theory than is the of poorer fundamental prediction inflationary data current cosmology! (Copi, GDS in preparation)
58. SUMMARY If you believe the observed full-sky CMB:There are signs that the sky is NOTstatistically isotropic • These are VERY statistically significant (>>99.9%) • The observed low-l fluctuations appear correlated to the solar system (but not to other directions)
59. This lack of correlations/power could be due to: • Statistical fluctuation -- incredibly unlikely • features in the inflaton potential -- contrived, and << 3% chance of such low S1/2 • Topology
60. The microwave background in a “small” universeAbsence of long wavelength modes Absence of large angle correlations
61. Measuring the shape of space Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level
62. Three TorusClick to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level Same idea works in three space dimensions
63. Infinite number of tiling patterns Click to edit Master text stylesClick to edit Master text styles Second level Second level ● Third level ● Third level ● Fourth level ● Fourth level ● Fifth level ● Fifth level This one only works in hyperbolic space
64. Spherical Topologies This example only works in spherical space
65. The microwave background in a multi-connected universe figure: J. Shapiro-KeyN. Cornish, D. Spergel, GDS gr-qc/9602039, astro-ph/9708083, astro-ph/9708225, astro-ph/9801212
66. Matched circles in a three torus universe Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level
67. Search costThe WMAP sky contains over three million pixels (0.1 degree resolution)Full search naively takes ~n7 operations, where n2 is the number of sky pixels Total number of operations ~ 1023 (~10 million years on my laptop) Solution: Hierachical Search: n  n/4 Computational “tricks” (Fourier methods) n7  n6 log(n) (Reduced parameter space searches)
68. Statistics for matched circlesSpatial comparisons: Use a RES r Healpix grid (3 x 22r+2 pixels) Draw a circle radius  around center, linearly interpolate values at 2r+1 points around circle S12 = 2<T1() T2 () >  /(< T1 () 2>  + < T2 () 2> ) Perfect match S12 = 1 Random circles <S12> = 0Fourier space comparisons: Ti () = ∑m Tim eim Sij () = 2∑m mTimTjm e-im / ∑m m(|Tim|2 + |Tjm|2 )  is relative phase We write as: Sij () = ∑m sm e-im and calculate Sij () as an FFT of sm for a n / logn speed-up (to n6 log(n))
69. Matched Circles in Simulations
70. Blind test (simulated sky supplied by A. Riazuelo):Manifold (S3/Z2) with 98304 visible circle pairs at eachradius, αParameters chosen to maximize ISW, Doppler de-coherence--“worst case”. α missed made missed made false-negative 1st cut 1st cut 2nd cut2nd cut rate24 334 97970 1642 96328 2%30 154 98150 118 98032 0.4%36 55 98249 11 98238 0.07%42 19 98285 3 98282 0.02%48 13 98291 2 98289 0.02%54 8 98296 0 98296 <0.001%
71. Searching the WMAP Sky: antipodal circles Single α, 95% C.L threshold Circle statistic on WMAP sky
72. Implications• No antipodal matched circles larger than 25o at > 99% confidence • now extended to 20o by pre-filtering.• Unpublished: no matched circles > 25o. • Universe is >90% of the LSS diameter (~78Gyr) across (Vcell>75% VLSS) • Search is being complete on 5-year data • Sensitivity should improve to 10o-15oIf there is topology, it’s beyond(or nearly beyond) the horizon
73. SUMMARYIf you don’t believe the CMB inside the Galaxyis reliable then: CMB lacks large angle correlations • first seen by COBE (~5% probable), • now statistically much less likely (~.03% probable)
74. This lack of correlations/power could be due to: • Statistical fluctuation -- incredibly unlikely • features in the inflaton potential -- contrived, and << 3% chance of such low S1/2 • Topology -- not (yet?) seen
75. Conclusions We can’t trust the low-l microwave to be cosmic => inferred parameters may be suspect (, A, 8,…)Removal of a foreground will likely mean even lower C() at large angles expect P(Swmap| Standard model)<<0.03% Contradict predictions of generic inflationary models at >99.97% C.L.
76. While the cosmic orchestra may beplaying the inflation symphony,somebody gave the bass and the tubathe wrong score. They’re trying veryhard to hush it up.There is no good explanation for any of this. Yet.
77. Planck may teach us thatthis has all been a wildgoose chase,or show us that the Universestill has some importantmysteries for us to decipher
78. • SUMMARYangle correlations CMB shows signs of distinct lack of large -- the low-l Cl are measuring SMALL ANGLE not largeangle correlations -- this was first seen by COBE, but nowstatistically much stronger• This lack of correlations/power could be due to: • features in the inflaton potential -- contrived • Topology -- not (yet?) seen • Statistical fluctuation -- incredibly unlikely • There also signs of the failure of statistical isotropy • These are VERY statistically significant • 99.9%-99.995% • The observed fluctuations seem to be correlated to the solar system (but not to other directions with great statistical significance)
79. Conclusions • We can’t trust the low-l microwave to be cosmic => • inferred parameters may be suspect (, A, 8,…) • The low-l multipoles are measuring structure at smaller than expected scales.Removal of a foreground will mean even lower C() at large angles expect P(Swmap| Standard model)<<0.03% Contradict predictions of generic inflationary models at >99.97% C.L.
80. While the cosmic orchestra may beplaying the inflation symphony,somebody gave the bass and the tubathe wrong score. They’re trying veryhard to hush it up.There is no good explanation for any of this. Yet.
81. ©Scientific American
82. Click to edit Master text styles Second level ● Third level ● Fourth level ● Fifth level
83. Wilkinson Microwave Anisotropy Probe(WMAP) NASA/WMAP Science team
84. Explaining the Low-l Anomaly1. “Didn’t that go away?”2. “I never believe a posteori statistics.”3. Cosmic variance -- “I never believe anything less than a (choose one:) 5 10 20 result.”4. “Inflation can do that”5. Other new physics
85. CAUTION!Absence of modes on scale of topology Absence of large angle correlationsObservation of topology scale no theory of amplitudes of largest modes!including large scale metric distortionsDO NOT BELIEVE ANYPREDICTIONS/POSTDICTIONS/LIMITS THATDEPEND ON PRECISE MODE AMPLITUDES
86. Dodecahedral Space Tiling of the three-sphere by 120 regular dodecahedronsLuminet, Weeks, Lehoucq, Riazuelo, UzanNature Nov. 03 Evidence: low C2 and C2 : C3 : C4 Predictions: 1. Ω = 1.013 2. “Circles in the sky”
87. The search for matched circlesGeneral 6 parameter search:Location of first circle center (2)Location of second circle center (2)Radius of the circle (1)Relative phase of the two circles (1)Reduced 4 parameter search (back-to-back circles):Ø Location of first circle center (2)Ø Radius of the circle (1)Ø Relative phase of the two circles (1)
88. Directed searches for individual topologiesDirected reduced parameter searches (for specific geometries):Ø Location of first circle center (2)Ø Radii of first circle pair (0a or 1 )Ø Relative location of subsequent circle centers (1b or more)Ø Radii of subsequent circle pairs (0b or more)Ø Relative phasings of the circles (0b or more)Cautions (!!):Ø a : Depends on assumptions about expected power per mode, which are inflationary, and hence suspect on scales where the scale-free spectrum may be broken by existing scales (e.g. topology scale)Ø b : Depends on rigidity of manifold.Ø A posteori vs. a priori: Must be careful that if we search over enough “special cases” with sufficiently reduced parameter spaces we drastically increase our probability
89. Hierarchical“Old Code” Search full parameter space • Start at RES7 (0.45° pixels) -- search • Identify 1000 best circles at each circle radius () • Progress to RES9 (0.1° pixels) -- search 1000 neighbourhoods • Keep 1000 best matches at each radius: Smax () Completed unpublished negative 7 parameter search ( >25o)! Problem: Old code exhibits high false-negative rate when there are multiple true circle-pairs due to saturation in neighbourhood of best matched pair(s). Question: Would the “old code” have been saturated by false (eg. osculating) circles? Answer: Probably not -- we found true simulation circles at every true radius, just not all of the circles at each radius. Why unpublished? This was a good discovery code, but we can’t use it to
90. Hierarchical Search -- Version 2Current Code • Start at RES7 (0.45° pixels) -- search full parameter space • Identify 5000 best “neighbourhoods” at each circle radius () • Progress to RES9 (0.1° pixels) -- search the 5000 neighbourhoods • Keep 5000 best matches at each radius: Smax () is max value of thes Completed negative 5 parameter search astro-ph/0310233 7 parameter search -- had intended to run on Goddard computers, but WMAP monopolized them for ~ 2 years longer than planned -- all in 2-4 week intervals! The code is being ported to a new supercomputing cluster. Expect results thissummer. (Finally!)
91. Implications• No antipodal matched circles larger than 25o at > 99% confidence • now extended to 20o by pre-filtering.• Specific search for Poincare Dodecahedron down to 6o • Universe is not a “soccer ball” (of the claimed size)• Unpublished: no matched circles > 25o. • Universe is >90% of the LSS diameter (24 Gpc) across • Search is being repeated on new data • Sensitivity should improve to 10o-15oIf there is topology, it’s beyond(or nearly beyond) the horizon.
92. Shape and Alignment of the Quadrupole and OctopoleA. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga, A. Hamilton. Phys.Rev.D69:063516,2004astro-ph/0307282For each l, find the axis nl around which the angularmomentum dispersion : (L)2  ∑m m2 |alm (nl)|2 is maximizedResults: Probability• octopole is unusually “planar” 1/20?? (dominated by |m| = 3 if z  n3 ).• n2 .n3=0.9838 1/60
93. Alignment probabilitiesAll values in %, in a sample of 100,000 MCrealisations of Gaussian-random alm
94. Additional alignment of the observedquadrupole+octopole with physical directions
95. Percentile of additional alignmentwith physical directions • S(4,4) percentiles given the observed “shape” of l=2&3
96. Percentile of additional alignmentwith physical directions • S(4,4) percentiles given the observed “shape” of l=2&3
97. Percentile of additional alignmentwith physical directions x0.05 • S(4,4) percentiles given the observed “shape” of l=2&3
98. Did WMAP123 change the (large angle) story?Mildly changed quadrupole: • time dependence of satellite temperature • “galaxy bias correction” -- add power inside “galaxy cut”Reported something different for Cl: • maximum likelihood estimate of coefficients of Legendre polynomial expansion of C( ) instead of (2l+1)-1 m |alm|2Quadrupole and Octopole still strange: • planar (octopole) • aligned with each other • perpendicular to ecliptic • normal points toward equinox/dipole • oriented so that ecliptic separates extrema NOT Statistically Isotropic
99. Is the Large-Angle Anomaly Significant? WMAP1: S = -1 1/2 [C()]2 d cos  0.03% Only 0.15% of realizations ofinflationary CDM universewith the best-fit parametershave lower S!
100. Statistics of C(θ)S1/2 =  1/2 [C( d cos -1 )]2
101. Reproducing C(θ)To obtain S1/2<971 with the WMAP Cl, requires varying C2, C3, C4 and C5.
102. Two point angular correlation function statisticsS1/2 = -1 1/2 [C()]2 d cos 
103. Two point angular correlation function statisticsS1/2 = -1 1/2 [C()]2 d cos 
104. The Conspiracy theory: minimizing S1/2To obtain S1/2 < 971 with the WMAP Clrequires varying C2, C3, C4 & C5!
105. Low-l Cl measures small angle correlations
106. Low-l Cl measures small angle correlations
107. Correlations of higher multipoles Rankings of alignments with single best aligned direction.
108. “Angular momentum dispersions” of higher multipoles
109. Conclusions• We can’t trust the measured low-lmicrowave background to be cosmic
110. Conclusions • We can’t trust the measured low-l microwave background to be cosmicg foregroundably meaner at large scales& Raanen suggests at least 5-10x less quadrup
111. Evidence of less low-l power?NASA/WMAP science team
112. For the futureNeed models that: a) explain the observations b) make testable predictions: 1. polarization 2. spectrum
113. quadrupole map and vectors
114. octopole map and vectors
115. quadrupole+octopole map and vectors
116. quadrupole and octopolemultipole and area vectors
117. S(n,m) histograms
118. Correlations of higher multipoles Rankings of alignments with single best aligned direction.
119. “Angular momentum dispersions” of higher multipoles
120. “Angular momentum dispersions” of higher multipoles
121. Quadrupole vectors -- signal vs. foreground
122. Octopole vectors -- signal vs. foreground
123. Comparison with COBE
124. When SH happens: SH N N total pixels: Nc= N2/4 pattern pixels: Np = N background pixels: Nb = Nc-Np Assume: * binary pixels * pattern fidelity: fpWMAP Science Team * background fidelity: fb Probability of particular combination of Nc pixels: Number of combinations with fidelities fp and fb: Number of orientations:  N Possible sizes: Number of symbol pairs: ((Nletters*Nalphabets+ Ndigits) *Nfonts +
125. example: SH N=10 total pixels: [ (N/2)2] pattern pixels: Np = 3 N minimum pattern fidelity: fp=0.8 N=10 minimum background fidelity: fb=0.8Probability of all realizations of one particular pattern with suitable fidelity at this fixed size, all orientations: 0.07Increase pattern fidelity to 90%: 0.00005 Expected number of 2-character patterns: 4
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