G. Starkman - The Oddly Quiet Universe

Glenn D. StarkmanCraig Copi, Dragan Huterer & Dominik Schwarz

Francesc Ferrer, Amanda YohoNeil 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

The Oddly Quiet Universe: How the CMB challenges cosmology’s standard model

Click to edit Master text stylesSecond level

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Fifth level

COBE - DMR

The WMAP Sky

NASA/WMAP Science team

COBE vs. WMAP

OutlineThe largest scale properties of the universe:

the angular power spectrum Cl

The low-l / large-angle probleml from Cl to C(θ)

Beyond Cl and C(θ)• seeing the solar system in the microwave background

And back:troubles in cosmological paradise

The WMAP Sky


Angular Power Spectrum

Cl = (2l+1)-1 m |a lm|2

T = lmalm Ylm( , )


Cl = (2l+1)-1 m |a lm|2

T = lmalm Ylm( , ) Standard model for the origin of fluctuations (inflation):

alm are independent Gaussian random 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

6 parameter fit to ~38 points

Measuring the shape of spaceClick to edit Master text styles

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Cosmic Triangle


Ω= 1 ± .0x

Is there anything interesting left to learn about the Universe on

large scales?

Motivation: “The Low-l Anomaly”

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)

Multipole VectorsQ: What are the directions associated

with the l th multipole:

∆Tl (θ,φ) m a lmYlm (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

Shape and Alignment of the Quadrupole and Octopole

For each l, find the axis nl around which the angular momentum dispersion :

( L)2 ∑m m2 |a lm (nl)|2is maximized

Results: Probability

•octopole is unusually “planar” 1/20?? (dominated by |m| = 3 if z n3 ).• n2 .n3=0.9838 1/60

A. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga, A. Hamilton. Phys.Rev.D69:063516,2004 astro-ph/0307282

Multipole Vectors

alm, m=- l,…, l, l =(0,1,)2,…

Α(l) ,û (l,i),l =1,… l, l = (0,1,)2,…

- all traces]m almYlm (θ,φ) ≈

Α (l) [(û (l,1)⋅ê)…(û (l, l) ê )

General l, write:

Advantages: 1) û(l,i) are vectors, Α(l) is a scalar

2) Only Α(l) depends on Cl

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.)

Area VectorsNotice:

• 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)

l=2&3 Area Vectorsequinox

equinox

dipole

dipolel=2 normal

l=2 normall=3 normal

l=3 normal

l=3 normal

l=3 normal

l=3 normal

l=3 normal

eclip

tic

SG

P

Alignment probabilities

Probability of the quadrupole and octopole planes being so aligned:

(0.1-0.6)%

Conditional probability of the aligned planes being so perpendicular to the plane of the solar system:

(0.2-1.7)%

Conditional probability of aligned planes perpendicular to the solar system pointing at the dipole/ecliptic:

few%Net : < 10-5

Area vectors tell about the orientations of the normals of the multipole planes

DON’T include all the information (multipole vectors do)

Can rotate the aligned planesabout their common axis!

l=2&3 : The Map

Quadrupole+Octopole Correlations -- Explanations: Cosmology?

• Cosmology -- you’ve got to be kidding? you choose: the dipole or the ecliptic ?

Quadrupole+Octopole Correlations -- Explanations: Systematics?

• 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?

• Cosmology


At least 3 other major deviations in the Cl in 1st year data

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!!

No Dip?

© Boomerang Collaboration

The case against a systematic (cont.)

Archeops

Ferrer, Starkman & Yoho (in progress): the anomaly in the first peak is (largely or entirely) localized to a region around the north ecliptic pole representing < 10% of the sky. Statistical significance tbd.

Dip? What dip?

Different weighting schemes:

Degree-scale anomalies in the CMB: localizing the first peak dip to a small patch of the north ecliptic sky.Amanda Yoho, Francesc Ferrer, Glenn D. Starkmanhttp://arXiv.org/pdf/1005.5389

http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Yoho,%20Amanda%22

http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Yoho,%20Amanda%22

http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Ferrer,%20Francesc%22

http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Starkman,%20Glenn%20D.%22

No first peak at the N pole?

Quadrupole+Octopole Correlations -- Explanations: more galaxy?

• Systematics

• The Galaxy:• has the wrong multipole structure (shape)• is likely to lead to GALACTIC not ECLIPTIC/DIPOLE/EQUINOX

correlations

• Cosmology

Quadrupole+Octopole Correlations -- Explanations: Foregrounds?

• Systematics

• Foregrounds -- difficult:1. Changing a patch of the sky typically gives you: Yl0

2. Sky has 5x more octopole than quadrupole3. 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

• Cosmology

• The Galaxy

Future data

Dvorkin, Peiris & Hu: Planck TE cross-correlation will test models “where a superhorizon scale modulation of the gravitational potential field causes the CMB temperature field to locally look statistically anisotropic, even though globally the model preserves statistical homogeneity and isotropy.

Φ(x) = g1(x) [1 + h(x)] + g2(x)

Where g1(x) and g2(x) are Gaussian random fields and h(x) is the modulating long-wavelength field

For dipolar h(x), predictions for the correlation between the first 10 multipoles of the temperature and polarization fields can typically be tested at better than the 98% CL. For quadrupolar h, the polarization quadrupole and octopole should be moderately aligned. Predicted correlations between temperature and polarization multipoles out to ℓ = 5 provide testsat the 99% CL or stronger for quadrupolar models that make ∼the temperature alignment more than a few percent likely.

Future data

Planck:

• different systematics (direct T measurement)⇒ Confirmation of low-l alignments significant

Future data

Planck:

• different systematics (direct T measurement)⇒ Confirmation of low-l alignments significant

• TE correlations – test particular models(Dvorkin, Peiris & Hu) of the quadrupole-octopole alignment

“The Low-l Anomaly”

The low

quadrupole

“The Large-Angle Anomaly”

The Angular Correlation Function, C( )



Fifth level

But (established lore):C( ) = l Cl Pl(cos ( ))

IF• C( ) is obtained by a full sky averageor• the sky is statistically isotropic,i.e. if <alm a*l’m’>=δll’ δmm’Cl

C( ) = < T( Ω1)T(Ω2)>Ω1.Ω2=cos

Same information as Cl, just differently organized

NASA/WMAP science team

Is the Large-Angle Anomaly Significant?

One measure (WMAP1): S1/2 = -11/2 [C( )]2 d cos

Only 0.15% of realizations of inflationary CDM universe with the best-fit parameters have lower S!


One measure (WMAP1): S1/2 = -11/2 [C( )]2 d cos

“The Low-l Anomaly? What Low-l Anomaly?”

Two point angular correlation function -- WMAP1

Statistics of C(θ)

S1/2 = -1 1/2 [C( )]2 d cos

Origin of C(θ)

Is it an accident?

Only 2% of rotated and cut full skies have this low a cut-sky S1/2

Statistics of C(θ)

• 0.03% of realizations of the concordance model of inflationary ΛCDM have so little cut sky large-angle correlation !

• Either: this reflects a .03% probable full sky C(θ), or a 5% probable C(θ) and a 2% probably alignment with the galaxy

Statistics of C(θ)

Efstathiou, Ma and Hanson: Direct calculation of C(θ) on the cut sky is a suboptimal 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) yields a larger S1/2, which is less significant in its smallness.

Statistics of C(θ)

Response:

• 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

• So what ? This is completely consistent with what we said. What is odd is the CUT-SKY C(θ) !

Origin of C(θ)

The Conspiracy theory: minimizing S1/2

To obtain S1/2 < 971 with the WMAP Cl requires varying C2, C3, C4 & C5!

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 measuring

large angle (θ~π/l ), but smaller angle correlations

Low l = large angle?

The low-l Cl are therefore not measuring

large angle (θ~π/l ), but smaller angle correlations

“The Low-l Anomaly”

Violation of GRSI

Even if we replaced all the theoretical Cl by their measured values up to l=20,

cosmic variance would give only a 3% chance of recovering this low an S1/2 in

a particular realization

Translation: The observed absence of large-angle correlation

is inconsistent (>>97%) with the most fundamental prediction of

inflationary cosmology!

and most of thost of those are much poorer fits with the theory than is the

current data (Copi, GDS in preparation)

SUMMARY

If you believe the observed full-sky CMB:

There are signs that the sky is NOT statistically isotropic

• These are VERY statistically significant (>>99.9%)

• The observed low-l fluctuations appear

correlated to the solar system (but not to other directions)

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

The microwave background in a “small” universe

Absence of long wavelength modes

Absence of large angle correlations

Measuring the shape of spaceClick to edit Master text styles

Second level Third level

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Three Torus



Fifth level

Same idea works in three space dimensions

Infinite number of tiling patterns



Fifth level

This one only works in hyperbolic space



Fifth level

This example only works in spherical space

Spherical Topologies

The microwave background in a multi-connected universe

N. Cornish, D. Spergel, GDS gr-qc/9602039, astro-ph/9708083, astro-ph/9708225, astro-ph/9801212

figure: J. Shapiro-Key

Matched circles in a three torus universe



Fifth level

Search cost

The 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)

Hierachical Search: n n/4 Computational “tricks” (Fourier methods) n7 n6 log(n)(Reduced parameter space searches)

Solution:

Statistics for matched circles

Spatial comparisons:

Perfect match Random circles

S12 = 2<T1( ) T2 ( ) > /(< T1 ( ) 2> + < T2 ( ) 2> )

Fourier space comparisons:

Ti ( ) = ∑m Tim eim

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

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 S ij ( ) as an FFT of sm

for a n / logn speed-up (to n6 log(n))

S12 = 1 <S12> = 0

Matched Circles in Simulations

Blind test (simulated sky supplied by A. Riazuelo):

Manifold (S3/Z2) with 98304 visible circle pairs at each radius, αParameters chosen to maximize ISW, Doppler de-coherence --“worst case”.

α missed made missed made false-negative

1st cut 1st cut 2nd cut2nd cut rate

24 334 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% 60 1 98303 0 98303 <0.001% 65 2 98302 0 98302 <0.001% 71 5 98299 0 98299 <0.001% 76 1 98303 0 98303 <0.001% 80 2 98302 0 98302 <0.001% 85 0 98304 0 98304 <0.001% 90 0 98304 0 98304 <0.001%

Searching the WMAP Sky: antipodal circles

Circle statistic on WMAP sky

Single α, 95% C.L threshold

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-15o

If there is topology, it’s beyond (or nearly beyond) the horizon

SUMMARY

CMB lacks large angle correlations• first seen by COBE (~5% probable), • now statistically much less likely (~.03% probable)

If you don’t believe the CMB inside the Galaxy is reliable then:

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

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.

While the cosmic orchestra may be playing the inflation symphony, somebody gave the bass and the tuba the wrong score. They’re trying very hard to hush it up.

There is no good explanation for any of this.

Yet.

Planck may teach us that this has all been a wild goose chase,

or show us that the Universe still has some important mysteries for us to decipher

SUMMARY• CMB shows signs of distinct lack of large angle correlations

-- the low-l Cl are measuring SMALL ANGLE not large angle correlations

-- this was first seen by COBE, but now statistically 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)

Conclusions• We can’t trust the low-l microwave to be cosmic =>

• inferred parameters may be suspect ( , A, 8,…)

• 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.

• The low-l multipoles are measuring structure at smaller than expected scales.

While the cosmic orchestra may be playing the inflation symphony, somebody gave the bass and the tuba the wrong score. They’re trying very hard to hush it up.

There is no good explanation for any of this.

Yet.

©Scientific American



Fifth level

Wilkinson Microwave Anisotropy Probe(WMAP)


Explaining the Low-l Anomaly

1. “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

CAUTION!

Absence of modes on scale of topology

Absence of large angle correlations

Observation of topology scale

no theory of amplitudes of largest modes!

including large scale metric distortions

DO NOT BELIEVE ANY PREDICTIONS/POSTDICTIONS/LIMITS THAT DEPEND ON PRECISE MODE AMPLITUDES

Dodecahedral Space

Tiling of the three-sphere by 120 regular dodecahedrons

Evidence: low C2 and C2 : C3 : C4

Predictions: 1. Ω = 1.013 2. “Circles in the sky”

Luminet, Weeks, Lehoucq, Riazuelo, UzanNature Nov. 03

The search for matched circles

General 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)

Directed searches for individual topologies

Directed 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 of false positives.

Hierarchical Search“Old Code”

• Start at RES7 (0.45° pixels) -- search full parameter space• 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-pairsdue 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 quote confidence levels on a limit.

Hierarchical Search -- Version 2

Current 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 these

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!)

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-15o

If there is topology, it’s beyond (or nearly beyond) the horizon.

Shape and Alignment of the Quadrupole and Octopole

For each l, find the axis nl around which the angular momentum dispersion :

( L)2 ∑m m2 |a lm (nl)|2is maximized

Results: Probability

•octopole is unusually “planar” 1/20?? (dominated by |m| = 3 if z n3 ).• n2 .n3=0.9838 1/60

A. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga, A. Hamilton. Phys.Rev.D69:063516,2004 astro-ph/0307282

Alignment probabilities

All values in %, in a sample of 100,000 MC realisations of Gaussian-random alm

Additional alignment of the observed quadrupole+octopole with physical directions

Percentile of additional alignment with physical directions

• S(4,4) percentiles given the observed “shape” of l=2&3

Percentile of additional alignment with physical directions

• S(4,4) percentiles given the observed “shape” of l=2&3

x0.05

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 |a lm|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


WMAP1: S = -1 1/2 [C( )]2 d cos

Only 0.15% of realizations of inflationary CDM universe with the best-fit parameters have lower S!

0.03%

Statistics of C(θ)

S1/2 = -1 1/2 [C( )]2 d cos

Reproducing C(θ)

To obtain S1/2<971 with the WMAP Cl, requires varying C2, C3, C4 and C5.

Two point angular correlation function statistics

S1/2 = -1 1/2 [C( )]2 d cos

The Conspiracy theory: minimizing S1/2

To obtain S1/2 < 971 with the WMAP Cl requires varying C2, C3, C4 & C5!

Low-l Cl measures small angle correlations

Correlations of higher multipoles

Rankings of alignments with single best aligned direction.

“Angular momentum dispersions” of higher multipoles

Conclusions• We can’t trust the measured low-l microwave background to be cosmic

Conclusions

• Removal of a contaminating foreground (or systematic) will probably mean

even LESS power at large scaleswork by Schwarz & Raanen suggests at least 5-10x less quadrupole or octopole.

• We can’t trust the measured low-l microwave background to be cosmic

Evidence of less low-l power?

NASA/WMAP science team

For the future

Need models that:

a) explain the observations

b) make testable predictions:

1. polarization

2. spectrum

quadrupole map and vectors

octopole map and vectors

quadrupole+octopole map and vectors

quadrupole and octopole multipole and area vectors

S(n,m) histograms

Correlations of higher multipoles

Rankings of alignments with single best aligned direction.

“Angular momentum dispersions” of higher multipoles

Quadrupole vectors -- signal vs. foreground

Octopole vectors -- signal vs. foreground

Comparison with COBE

When SH happens:

WMAP Science Team

SH

N

N

total pixels: Nc= N2/4 pattern pixels: Np = Nbackground pixels:

Nb = Nc-Np

Assume: * binary pixels * pattern fidelity: fp * 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 + Nsymbols)) 2 ≈ 105

example:

SH

N=10

N=10 total pixels: [ (N/2)2] pattern pixels: Np = 3 Nminimum pattern fidelity: fp=0.8minimum background fidelity: fb=0.8

Probability of all realizations of one particular pattern with suitable fidelity at this fixed size, all orientations:

0.07

Increase pattern fidelity to 90%: 0.00005

Expected number of 2-character patterns: 4

quadrupole multipole & area vectors - WMAP123

octopole multipole & area vectors - WMAP123

quadrupole &octopole multipole & area vectors - WMAP123

ILC123-ILC1 quadrupole multipole & area vectors

Quadrupole-Octopole correlations -- WMAP1 vs. WMAP123

Correlations with physical directions -- WMAP1 vs. WMAP123

Additional Correlations with physical directions -- WMAP123

Additional Correlations with physical directions -- WMAP1 vs. WMAP123



Fifth level

CMB post-COBE


CMB Post-COBE


Wilkinson Microwave Anisotropy Probe(WMAP)


CMB Post-COBE


G. Starkman - The Oddly Quiet Universe

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