Non-Gaussianity From Inflation April 19, 2006

Eiichiro Komatsu

University of Texas at Austin

Non-Gaussianity From Inflation April 19, 2006

CMB High-z Cl

usters

LSS

Observational Constr

aints on Non-Gaussia

nity

Why Study NG? (Why Care?!)

• Who said that CMB should be Gaussian?

 Don’t let people take it for granted!

 It is remarkable that the observed CMB is (very close to bein g) Gaussian.

 The WMAP map, when smoothed to 1 degree, is entirely do minated by the CMB signal.

 If it were still noise dominated, no one would be surprised that the ma p is Gaussian.

 The WMAP data are telling us that primordial fluctuations are very close to being Gaussian.

 How common is it to have something so close to being Gaussian in as tronomy? E.g., Maxwellian velocity distribution, what else?

• It may not be so easy to explain that CMB is Gaussian, unless we have a compelling early universe model that predicts Gaussian primordial fluctuations: Inflation.

 “Gaussianity” should be taken as seriously as “Flatness”.

Gaussianity vs Flatness

• People are generally happy that geometry of our Universe is flat.

 1-total=-0.003 (+0.013, -0.017) (68% CL) (WMAP03+H

ST)

 Geometry of our Universe is consistent with being flat t o ~3% accuracy at 95% CL.

• What do we know about Gaussianity?

 For GfNLG2, -54<f_NL<114 (95% CL) (WMAP03)

 Primordial fluctuations are consistent with being Gaussi an to ~0.001% accuracy at 95% CL.

• In a way, inflation is supported more by Gaussiani

ty of primordial fluctuations than by flatness. (Just

kidding.)

Let’s Hunt Some NG!

• The existing CMB data already suggest that primordial fluc tuations are very close to being Gaussian; however, this d oes not imply, by any means, that they are perfectly G aussian.

 In fact, we would be in a big trouble if f_NL turned out to be too clos e to zero. Second-order GR perturbations in the standard cosmol ogical model must produce f_NL~5 or so. (See Sabino Matarrese’s and Nicola Bartolo’s Talks, and Michelle Liguori’s Poster)

• Some inflationary models produce much larger f

_NL

. We ma y be able to distinguish candidate inflationary models by N G. (Not to mention that the simplest, single-field, slow-roll models produce tiny NG.)

• The data keep getting better.

• If we could find it, it would lead us to something huge.

How Do We Test Gaussianity

of CMB?

Finding NG?

•

Two approaches to

•

I. Null (Blind) Tests / “Discovery” Mode

 This approach has been most widely used in the literature.

 One may apply one’s favorite statistical tools (higher-order correlations, topolog y, isotropy, etc) to the data, and show that the data are (in)consistent with Gauss ianity at xx% CL.

 PROS: This approach is model-independent.

 CONS: We don’t know how to interpret the results.

 “The data are consistent with Gaussianity” --- what physics do we learn from that? It is not clear what could be ruled out on the basis of this kind of test.

•

II. “Model-testing” Mode

 Somewhat more recent approaches.

 Try to constrain “NG parameter(s)” (e.g., fNL)

 PROS: We know what we are testing, we can quantify our constraints, and we c an compare different data sets.

 CONS: Highly model-dependent. We may well be missing other important NG si gnatures.

Recent Tendency

• I. Null (Blind) Tests / “Discovery” Mode

 This approach is being applied mostly to the “large-sc ale anomaly” of the WMAP data.

 North-south asymmetry

 Quadrupole-octopole alignment

 Some pixels are too cold (Marcos Cruz’s Poster)

 “Axis of Evil” (Joao Magueijo’s Talk)

 Large-scale modulation

• II. “Model-testing” Mode

 A few versions of fNL have been constrained using the bispectrum, Minkowski functionals and other statistic al methods.

App. II: What Do We Need?

• We need to know the predicted form of stati stical tools as a function of model paramet ers to fit the data.

• ^{For }

^G

^f

^NL

^

^G2

, there are only three statistic al tools for which the analytical predictions are known:



The angular bispectrum

 Komatsu & Spergel (2001); Babich & Zaldarriaga (2004)



The angular trispectrum

 Okamoto & Hu (2002); Kogo & Komatsu (2006)



Minkowski functionals

 Hikage, Komatsu & Matsubara (2006)

Simplified Model:

• Working assumption: f

_NL

is independent of scales

 Clearly an oversimplification! (Note, however, that this form i s actually predicted from curvaton models and the non-linear Sachs-Wolfe effect in the large-scale limit.)

• Why use this ansatz? Current observations are not yet sensitive to scale-dependence of f

_NL

, but are only sens itive to the overall amplitude.

 See Creminelli et al. (2005) for an alternative ansatz.

• Sensitivity Goal: f

_NL

~1

 Why f_NL~1? NG “floor”: the ubiquitous signal from the secon d-order GR produces something like f_NL~5, which would set t he lower limit to which one may hope to detect the primordial non-Gaussianity.

 It may be possible to achieve f_NL~3 using the angular bispect rum from the Planck and CMBPol data. How do we go from t here to f_NL~1?





 

x ^{ }G

 

x ^{ f}NL_G²

 

x

Bartolo, Komatsu, Matarrese & Riotto (20

04)

R

5

 3



How Do They Look?



   x ^{ }

G

  x ^{ f}

NL



_G²

  x

Simulated temperature maps from

f

_NL

=0 f

_NL

=100

f

_NL

=1000 f

_NL

=5000

Is One-point PDF Useful?

Conclusion: 1-point PDF is not very u seful. (As far as CMB is concerned.)

A positive

f

_NL yields nega tively skewed temperatur e anisotropy.

One-point PDF from WMAP

• The one-point distribution of CMB temperature anisotropy looks pretty Gaussian.



Galaxy has been masked.



Left to right: Q (41GHz), V (61GHz), W (94GHz).

Spergel et al. (2006)

Angular Bispectrum, B _lmn

• A simple statistic that captures all of the information contained in the third-order moment of CMB anisotropy.

• Theoretical predictions exist.

• Statistical properties well understood.

How Does It Look?

• ^Primordial



Inflation



Second-order PT

• ^Secondary



Gravitational lensing



Sunyaev-Zel’dovich effe

• ^Nuisance ct



Radio point sources



Our Galaxy

Komatsu & Spergel (2001)

?Angular Bispectrum, B _lmn ?

•

Physical non-Gaussian signals should be generated in real space (via e.g., non-linear coupling), and thus should be mo re apparent in real space.

 The Central Limit Theorem makes alm coefficients more Gaussi an!

 Non-Gaussianity localized in real space is obscured and spread ove r many l’s and m’s.

•

Challenges in the analysis

 Sky cut complicates the analysis in harmonic space in many ways.

 Computationally expensive (but not impossible).

 8 hours on 16 procs of an SGI Origin 300 for measuring all configur ations up to l=512.

Optimal “Cubic” Statistics

•

^Motivation

 We know the shape of the angular bispectrum.

 Find the “best statistic” in real space that is most sensitive to th e kind of non-Gaussianity we are looking for.

•

^Results

 The statistics already combine all configurations of the bispectr um optimally.

 Optimized just for fNL: Maximum adaptation of the approac h II.

 1000 times faster for l_max=512 (30 sec vs 8 hours)

 4000 times faster for l_max=1024 (4 minutes vs 11 days)

•

One can also find an optimized statistics just for point s ources.

Komatsu, Spergel & Wandelt (2005)

Cubic Estimator = Skewness of Filtered Maps

•B(x) is a Wiener reconstructed primordial potential field.

•A(x) picks out relevant configurations of the bispectrum.



S

_prim

  dVA x   ^B

²

^{  x  f}

^NL



A r, ˆ   n ^  ^

l

  r ^b

l

C

_l

 ^a

^lm

^Y

^lm

^{  n ˆ}

lm



B r, ˆ   n ^  ^

l

  r ^b

l

C

_l

 ^a

^lm

^Y

^lm

^{  n ˆ}

lm





_l

  r ^  ^d

³

^kg

Tl

  k ^j

l

  kr



_l

  r ^  ^d

³

^{kP k}   ^g

Tl

  k ^j

l

  kr

Wiener-reconstructed Primordial Curvature

Reconstruction can be made even better by including the polarization data.

(See Ben Wandelt’s Talk)





 

3 1 T

T

On the largest scale,

It Works Very Well.

•The statistics tested against simulations

UNBIASED UNCORRELATE

D UNBIASED

Komatsu et al. (2006)

Bispectrum Constraints

•

Still far, far away from f

NL~1, but it could put so me interesting limits on parameters of curvaton models, ghost inflation, and DBI inflation model s.



58  f_NL 134(95%)

Komatsu et al. (2003); Spergel et al. (2006)



54  f_NL 114(95%)

(1yr) (3yr)

Angular Trispectrum, T _lmpq (L)

• Why care?! Two reasons.



The trispectrum could be non-zero even when the bispectrum is exactly zero.



We may increase our sensitivity to primordial NG

by including the trispectrum in the analysis.

Not For WMAP, But Perhaps For Planck…

• Trispectrum (~ ^f

NL 2

) is more sensitiv e than the bispect rum (~ ^f

NL

) when ^f

N L

is large.

 At Planck resolution, the trispectrum woul d be detected more s ignificantly than the b ispectrum, if f_NL > 50.

Kogo & Komatsu (2006)

Minkowski Functionals

• ^Morphology

 Area

 Contour length

 Euler characteristics (or “Genus”)

 The number of hot spots minus cold spots.

• All quantities are evaluated as a function of the peak height relative to r.m.

s.

MFs from WMAP



f

_NL

 137(95%)



125  f

_NL

 139(95%)

(1yr)

Komatsu et al. (2003); Spergel et al. (2006)

(3yr)

Area Contour Length Genus

Analytical Calculations

•

The analytical formulae s hould be very useful: we do not need to run NG si mulations for doing MFs any more.

•

The formulae indicate th at MFs are actually as se nsitive to fNL as the bispe ctrum; however, MFs do not contain any more info rmation than the bispectr um does.

Hikage, Komatsu & Matsubara (2006)

Using Galaxies

• Not only CMB, but also the large-scale str ucture of the universe does contain inform ation about primordial fluctuations on larg e scales. (See Peter Coles’s Talk.)

• One example: Galaxy Bispectrum

Cosmic Inflation Probe (CIP), a galaxy survey measuring 10 mil lion galaxies at 3<z<6, would offer an opportunity to use this for mula to constrain f_NL~5 (note that the scale measured by CIP is smaller than that measured by CMB by a factor of ~10!)

CIP

Using High-z Objects

• Massive objects forming at high-z are e xtremely rare: they form at high peaks o f (nearly) Gaussian random field.

• Even a slight distortion of a Gaussian tai

l can enhance (or reduce) the number of

high-z object dramatically. The higher th

e mass is, or the higher the redshift is, t

he bigger the effect becomes.

•

The current WMAP limits still permit large changes in the number of objects at high z.

•

A golden object, like a few times 10¹⁴ solar masses at z=3, would be a smoking gun.

Implications for high- z objects

Komatsu et al. (2003)

Summary

•

There have been a lot of development in this field, and there is still a lot more to do. It is timely to have a workshop that focus es on non-Gaussianity from Inflation!

 We need more accurate predictions for the form of observables, such a s the CMB bispectrum, trispectrum, Minkowski functionals, and others, from

 Various models of inflation

 Second-order PT, including second-order effects in Boltzmann equation

•

Not just CMB!

 Galaxies and high-z objects might give us some surprises.

•

Toward the sensitivity goal, f_NL~1.

 What would be the best way to achieve this sensitivity?

 Currently, the CMB bispectrum seems to be ahead of everything else.

 The temperature plus polarization bispectrum would allow us to get down t o f_NL~3. How do we break this barrier?

•

Yet, the real surprise might come from the Approach I.