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, R. Trotta, Pascal Vaudrevange, Bob Nichols, Peter Freeman Joe Silk, Andrew Jaffe, Anastasia Anarchiou How the CMB challenges cosmology’s standard model

COBE - DMR

The WMAP Sky NASA/WMAP Science team

COBE vs. WMAP

Outline The largest scale properties of the universe: C l the angular power spectrum C l The low- l / large-angle problem from C l to C  Beyond C l and C  seeing the solar system in the microwave background And back: troubles in cosmological paradise

The WMAP Sky NASA/WMAP Science team

Angular Power Spectrum C l = (2 l +1) -1  m |a l m | 2  T =  l m a l m Y l m ( ,  ) NASA/WMAP Science team C l = (2 l +1) -1  m |a l m | 2  T =  l m a l m Y l m ( ,  ) Standard model for the origin of fluctuations (inflation): a l m are independent Gaussian random variables, with = C l  l l ‘  mm’  Sky is statistically isotropic and Gaussian random ALL interesting information in the sky is contained in C l 6 parameter fit to ~38 points

Measuring the shape of space NASA/WMAP Science team

Cosmic Triangle

Angular Power Spectrum  x

Is there anything interesting left to learn about the Universe on large scales?

Motivation : “The Low-l Anomaly”

The microwave background in a “small” universe Absence of long wavelength modes  Absence of large angle correlations

Measuring the shape of space

Three Torus Same idea works in three space dimensions

Infinite number of tiling patterns This one only works in hyperbolic space

This example only works in spherical space Spherical Topologies

The microwave background in a multi-connected universe N. Cornish, D. Spergel, GDS gr-qc/ , astro-ph/ , astro-ph/ , astro-ph/ figure: J. Shapiro-Key

Matched circles in a three torus universe

Matched Circles in Simulations In a blind test >99% of circles found in a “deliberately difficult” universe

Searching the WMAP Sky: antipodal circles Circle statistic on WMAP sky Single  95% C.L threshold

Implications No antipodal matched circles larger than 25 o at > 99% confidence now extended to 20 o by pre-filtering. Unpublished: no matched circles > 25 o. Universe is >90% of the LSS diameter (~78Gyr) across (V cell >75% V LSS ) Search is being repeated on 5-year data Sensitivity should improve to 10 o -15 o If there is topology, it’s beyond (or nearly beyond) the horizon

Beyond C l : 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 Vectors Q: What are the directions associated with the l th multipole:  T l   m a l m Y l m  Dipole ( l =1) :  m a 1m Y 1m       û x (1,1), û y (1,1), û z (1,1) )  (sin   cos , sin   sin , cos   Advantages: 1) û (1,1) is a vector,    is a scalar 2) Only    depends on C 1

Multipole Vectors {{a l m, m=- l,…, l }, l =(0,1,)2,…}  {   l )  û ( l,i), l =1,… l }, l = (0,1,)2,…} - all traces ]  m a l m Y l m   (l)    û (l,1)  ê)…  û (l, l)  ê ) General l, write: Advantages: 1) û ( l,i) are vectors,  ( l ) is a scalar 2) Only  ( l ) depends on C l

Maxwell Multipole Vectors  m a l m Y l m    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 (1 st ed.)

Area Vectors Notice: 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) || n 2 octopole is perfectly planar if w (3,1,2) || w (3,2,3) || w (3,3,1) and then: n 3 || w (3,I,j)

l=2&3 Area Vectors equinox dipole l=2 normal l=2 normal l=3 normal ecliptic SGP

Alignment probabilities Probability of the quadrupole and octopole planes being so aligned: ( )% Conditional probability of the aligned planes being so perpendicular to the plane of the solar system: ( )% 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 planes about 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? possibilities -- correlation of beam asymmetry with observing pattern (S. Myer) Cosmology

Angular Power Spectrum At least 3 other major deviations in the C l 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, only v.1 on archive All 3 other major deviations are in the ecliptic polar C l 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.

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: Y l0 2. Sky has 5x more octopole than quadrupole 3. How do you get a physical ring perpendicular to the ecliptic Caution: can add essentially arbitrary dipole, which can entirely distort the ring! (Silk & Inoue) 4. How do you hide the foreground from detection? T≈T CMB 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.  (x) = g 1 (x) [1 + h(x)] + g 2 (x) Where g 1 (x) and g 2 (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.

“The Low-l Anomaly” The low quadrupole

“The Large-Angle Anomaly”

The Angular Correlation Function, C(  ) But (established lore): C(  ) =  l C l P l (cos (  )) IF C(  ) is obtained by a full sky average or the sky is statistically isotropic, i.e. if =  ll’  mm’ C l C(  ) =      cos   Same information as C l, just differently organized NASA/WMAP science team

Is the Large-Angle Anomaly Significant? One measure (WMAP1): S 1/2 =  -1 1/2 [C(  )] 2 d cos  Only 0.15% of realizations of inflationary  CDM universe with the best-fit parameters have lower S!

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

Two point angular correlation function -- WMAP1

Two point angular correlation function -- WMAP3

Statistics of C(θ) S 1/2 =  -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 S 1/2

Origin of C(θ)

The Conspiracy theory: minimizing The Conspiracy theory: minimizing S 1/2 To obtain S 1/2 < 971 with the WMAP C l requires varying C 2, C 3, C 4 & C 5 !

Reproducing C(θ) To obtain S 1/2 <971 with the WMAP C l, requires varying C 2, C 3, C 4 and C 5. The low- l C l are therefore not measuring large angle ( θ~π/ l ), but smaller angle correlations

Low l = large angle? The low- l C l 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 C l by their measured values up to l=20, cosmic variance would give only a 3% chance of recovering this low an S 1/2 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, Cumberbatch, 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)

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 Topology -- not (yet?) seen features in the inflaton potential -- contrived, and << 3% chance of such low S 1/2

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(S wmap | 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.

SUMMARY CMB shows signs of distinct lack of large angle correlations -- the low- l C l 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% % 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(S wmap | 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

Wilkinson Microwave Anisotropy Probe (WMAP) NASA/WMAP Science team

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 C 2 and C 2 : C 3 : C 4 Predictions: 1. Ω   “Circles in the sky” Luminet, Weeks, Lehoucq, Riazuelo, Uzan Nature 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):  Location of first circle center (2)  Radius of the circle (1)  Relative phase of the two circles (1)

Directed searches for individual topologies Directed reduced parameter searches (for specific geometries):  Location of first circle center (2)  Radii of first circle pair (0 a or 1 )  Relative location of subsequent circle centers (1 b or more)  Radii of subsequent circle pairs (0 b or more  Radii of subsequent circle pairs (0 b or more)  Relative phasings of the circles (0 b or more) Cautions (!!):  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)  b : Depends on rigidity of manifold.  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.

Search cost The WMAP sky contains over three million pixels (0.1 degree resolution) Full search naively takes ~n 7 operations, where n 2 is the number of sky pixels Total number of operations ~ (~10 million years on my laptop)  Hierachical Search: n  n/4  Computational “tricks” (Fourier methods)   Computational “tricks” (Fourier methods) n 7  n 6 log(n)  (Reduced parameter space searches) Solution:

Statistics for matched circles Spatial comparisons: Perfect matchRandom circles S 12 = 2  /(  +  ) Fourier space comparisons: T i (  ) = ∑ m T im e im  Use a RES r Healpix grid (3 x 2 2r+2 pixels) Draw a circle radius  around center, linearly interpolate values at 2 r+1 points around circle S ij (  ) = 2∑ m mT im T jm e -im  / ∑ m m(|T im | 2 + |T jm | 2 )  is relative phase We write as: S ij (  ) = ∑ m s m e -im  and calculate S ij (  ) as an FFT of s m for a n / logn speed-up (to n 4 log(n)) S 12 = 1 = 0

Blind test (simulated sky supplied by A. Riazuelo): Manifold (S 3 /Z 2 ) with visible circle pairs at each radius,  Parameters chosen to maximize ISW, Doppler de-coherence --“worst case”.  missed made missed madefalse-negative 1st cut 1st cut2nd cut2nd cutrate % % % % % <0.01% <0.01% <0.01% <0.01% <0.01% <0.01% % %

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: S max (  ) Completed unpublished negative 7 parameter search (  >25 o )! 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 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: S max (  ) is max value of these Completed negative 5 parameter search astro-ph/ 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 25 o at > 99% confidence now extended to 20 o by pre-filtering. Specific search for Poincare Dodecahedron down to 6 o Universe is not a “soccer ball” (of the claimed size) Unpublished: no matched circles > 25 o. Universe is >90% of the LSS diameter (24 Gpc) across Search is being repeated on new data Sensitivity should improve to 10 o -15 o 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 n l around which the angular momentum dispersion : (  L) 2  ∑ m m 2 |a l m (n l )| 2 is maximized Results: Probability octopole is unusually “planar” 1/20?? (dominated by |m| = 3 if z  n 3 ). n 2. n 3 = /60 A. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga, A. Hamilton. Phys.Rev.D69:063516,2004 astro-ph/

Alignment probabilities All values in %, in a sample of 100,000 MC realisations of Gaussian-random a l m

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

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 C l : maximum likelihood estimate of coefficients of Legendre polynomial expansion of C(  ) instead of (2 l +1) -1  m |a l m | 2 Quadrupole 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

Is the Large-Angle Anomaly Significant? 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) S 1/2 =  -1 1/2 [C(  )] 2 d cos 

Reproducing C(θ) To obtain S 1/2 <971 with the WMAP C l, requires varying C 2, C 3, C 4 and C 5.

Two point angular correlation function statistics S 1/2 =  -1 1/2 [C(  )] 2 d cos 

Two point angular correlation function statistics S 1/2 =  -1 1/2 [C(  )] 2 d cos 

The Conspiracy theory: minimizing The Conspiracy theory: minimizing S 1/2 To obtain S 1/2 < 971 with the WMAP C l requires varying C 2, C 3, C 4 & C 5 !

Low- l C l 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 scales work 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

“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: N c =  N 2 /4 pattern pixels: N p =  N background pixels: N b = N c -N p Assume: * binary pixels * pattern fidelity: f p * background fidelity: f b Probability of particular combination of N c pixels: Number of combinations with fidelities  f p and  f b : Number of orientations:  N Possible sizes: Number of symbol pairs: ((N letters *N alphabets + N digits ) *N fonts + N symbols) ) 2 ≈ 10 5

example: SH N=10 total pixels: [  (N/2) 2 ] pattern pixels: N p = 3 N minimum pattern fidelity: f p =0.8 minimum background fidelity: f b =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%: 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

Two point angular correlation function -- WMAP1

Two point angular correlation function -- WMAP3

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

CMB post-COBE © Boomerang Collaboration

CMB Post-COBE © Boomerang Collaboration

Wilkinson Microwave Anisotropy Probe (WMAP) NASA/WMAP Science team

CMB Post-COBE © Boomerang Collaboration