G. Starkman - The Oddly Quiet Universe

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

COBE - DMR
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The WMAP Sky

NASA/WMAP Science team

Outline
The largest scale
properties of the universe:
Thethe angular problem
low-l / large-angle power
l from Cl to C(θ)
spectrum Cl
Beyond Cl and C(θ)
• seeing the solar system in the microwave background

And back:
troubles in cosmological paradise

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 random
6 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|

Measuring the shape of space
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Cosmic Triangle
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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 Vectors
Q: 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

Shape and Alignment of the
Quadrupole and Octopole
A. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga, A. Hamilton. Phys.Rev.D69:063516,2004
astro-ph/0307282

For each l, find the axis nl around which the angular
momentum dispersion :
(L)2  ∑m m2 |alm (nl)|2
is maximized
Results:
Probability

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

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

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.)

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)

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
SG
l=3 normal

dipole

equinox l=3 normal l=3 normal

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 planes
about their common axis!

Quadrupole+Octopole
Correlations -- Explanations:
Cosmology?

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

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?

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.

Different weighting schemes:

in the CMB:
dip to a small patch of the north ecliptic sky.Amanda Yoho, Francesc Ferrer, Glenn D.
389

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

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

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 Angular Correlation
Function, C( )
C() = < T(Ω1)T(Ω2)>Ω1.Ω2=cos
But (established lore): Click to edit Master text styles
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C() = l Cl Pl(cos ()) ●
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 Same information as Cl, just ●
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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

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!

Significant?

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

function -- WMAP3

Statistics of C(θ)
S1/2 = -1 1/2 [C()]2 d
cos 

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:

•
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

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

Violation of GRSI
Even 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 correlation
chance 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)

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 space
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Three Torus
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Same idea works in three space dimensions

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

Spherical Topologies
This example only works in
spherical space

The microwave background in
a multi-connected universe

figure: J. Shapiro-Key

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

Matched circles in a three
torus universe
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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)

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

Statistics for matched circles
Spatial 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> = 0
Fourier 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))

Matched Circles in
Simulations

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

Manifold (S3/Z2) with 98304 visible circle pairs at each
radius, α
Parameters 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%

Searching the WMAP Sky:
antipodal circles

Single α, 95% C.L threshold

Circle statistic on WMAP sky

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
If you don’t believe the CMB inside the Galaxy
is reliable then:
CMB lacks large angle correlations
•
first seen by COBE (~5% probable),
•
now statistically much less likely
(~.03% probable)


•

•
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

•
SUMMARYangle correlations
CMB shows signs of distinct lack of large
-- the low-l Cl are measuring SMALL ANGLE not large
angle correlations
-- this was first seen by COBE, but now
statistically much stronger

•
•
features in the inflaton potential -- contrived
•
Topology -- not (yet?) seen
•

•
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,…)
•
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.

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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

Luminet, Weeks, Lehoucq, Riazuelo, Uzan
Nature Nov. 03

Evidence: low C2 and C2 : C3 : C4

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

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

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

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 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!)

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.

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

x0.05

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

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|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

Significant?

WMAP1:
S = -1 1/2 [C()]2 d cos 

0.03%

Only 0.15% of realizations of
inflationary CDM universe
with the best-fit parameters
have lower S!

Statistics of C(θ)
S1/2 =  1/2 [C( d cos
-1 )]2


Reproducing C(θ)

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

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

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

g foreground
ably mean
er at large scales
& Raanen suggests at least 5-10x less quadrup

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+octopole map and vectors

quadrupole and octopole
multipole and area vectors

Quadrupole vectors --
signal vs. foreground

Octopole vectors --
signal vs. foreground

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: fp
WMAP 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 +

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=0.8

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

CMB post-COBE


CMB Post-COBE


Wilkinson Microwave Anisotropy Probe
WMAP)


G. Starkman - The Oddly Quiet Universe

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