Gaussianity, Flatness, Tilt, Running, and Gravity Waves: WMAP and Some Future Prosp

ects

Eiichiro Komatsu

University of Texas at Austin

July 11, 2007

Cosmology and Strings: 6 Numbers

• Successful early-universe models must produce:– The universe that is nearly flat, |K|<O(0.02)

– The primordial fluctuations that are • Nearly Gaussian, |fNL|<O(100)

• Nearly scale invariant, |ns-1|<O(0.05), |dns/dlnk|<O(0.05)

• Nearly adiabatic, |S/R|<O(0.2)

Cosmology and Strings: 6 Numbers

• A “generous” theory would make cosmologists very happy by producing detectable primordial gravity waves (r>0.01)…– But, this is not a requirement yet. – Currently, r<O(0.5)

How Flat is our Universe?• The peak positions of CMB aniso

tropy measured by WMAP 3yr data gave:K = 1m

= 0.3040 + 0.4067

CMB alone cannot constrain curvature.

Flatness: WMAP+

• The current best combination is WMAP+BAO extracted from LRGs in SDSS.– CMB and BAO a

re absolute distance indicators.

– SNs measure only relative distances.

+H0 from HST

+BAO from LRGs

+SNs from SNLS

+SNs from “Gold”

+P(k) from 2dFGRS

+P(k) from SDSS

We Need Absolute Distance Indicators to Constrain K

• The angular diameter distance DA(z) is related to the luminosity distance DL(z) by– DL(z)=(1+z)2 DA(z)

• When curvature is smaller than the comoving distance (z), DA(z) may be expanded in Taylor series as– DA(z) ~ (z)+ (z)/6, where (z)=c*int(dz/Hubble)

• Therefore, nearly cancels in a relative distance between objects that are not greatly separated.– Relative distances between supernovae are less sensitive to curvat

ure.– Relative distances between CMB (at z=1090) and BAO (at z=0.35 f

or LRGs) are more sensitive to curvature.

WMAP 3yr Results: 2% Flatness

• WMAP3+HST: 0.048<K<0.020 (95%)

– Note that the HST result, H0=728 km/s/Mpc, is also an absolute distance measurement, so it can be as good as BAO in terms of constraining curvature. However, the distance error from HST (8%) is bigger than the error from BAO (3%). A room for improvement.

• WMAP3+SNs: 0.035<K<0.013

• WMAP3+BAO: 0.032<K<0.008

– This is, in a way, a test of inflation to 2% level.

0.1% Flatness in Future?

• Planck CMB data + future BAO data at z=3 (from the planned galaxy surveys e.g., HETDEX, WFMOS, BOSS) would yield ~0.1% determination of flatness.– Knox, PRD 73, 023503 (2006)

• Therefore, flatness would offer 0.1% test of inflation in 5-10 years.

Gaussianity vs Flatness• So, we are generally happy that geometry of our

observable Universe is flat.– Geometry of our Universe is consistent with being flat

to ~2% accuracy at 95% CL.

• What do we know about Gaussianity?– Let’s take a usual model, LfNLL

2

L~10-5 is the linear curvature perturbation in the matter era• WMAP 3yr: WMAP 3yr: 54<f54<fNLNL<114<114 (95% CL) • One can improve on this by ~15%, see Creminelli et al.

– Therefore, is consistent with being Gaussian to ~100(10-5)2/(10-5)=0.1% accuracy at 95% CL.

• The Truth: Inflation is supported more by Gaussianity than by flatness.

How Would fNL Modify PDF? One-point PDF is not very useful for measuring primordial NG. We need something better:

•Bispectrum:

54<f54<fNLNL<114<114

•Trispectrum:

•N/A (yet)

•Minkowski functionals:

70<f70<fNLNL<91<91

Gaussianity vs Flatness: Future

• Flatness will never beat Gaussianity.– In 5-10 years, we will know flatness to 0.1% level.– In 5-10 years, we will know Gaussianity to 0.01%

level (fNL~10), or even to 0.005% level (fNL~5), at 95% CL.

• However, a real potential in Gaussianity test is that we might detect something at this level (multi-field, curvaton, DBI, ghost cond., new ekpyrotic…)– Or, we might detect curvature first?– Is 0.1% curvature interesting/motivated?

Confusion about fNL (1): Sign• What is fNL that is actually measured by

WMAP?• When we expand as =L+fNLL

2, is Bardeen’s curvature perturbation (metric space-space), H, in the matter dominated era.– Let’s get this stright: is not Newtonian potential (metric time-time)– Newtonian potential is . (There is a minus sign!)– In the SW limit, temperature anisotropy is T/T=(1/3).– A positive fNL resultes in a negative skewness of T.

• It is useful to remember that fNL positive = Temperature skewed negative (more cold spots)= Matter density skewed positive (more objects)

Confusion about fNL (2): Primordial vs Matter Era

• In terms of the primordialprimordial curvature perturbation in the comoving gauge, R, Bardeen’s curvature perturbation in the matter era is given by LL=+=+(3/5)(3/5)RRLL at the linear level (notice the plus sign).

• Therefore, R=RL+(3/5)fNLRL2

• There is another popular quantity, =+R. (Bardeen, Steinhardt & Turner (1983); Notice the plus sign.) =L+(3/5)fNLL

R=RL(3/5)fNLRL2

R=RLfNLRL2

=L(3/5)fNLL2

FAQ: Why Care About Matter Era?• A. Because that’s what we measure.• It is important to remember that fNL receives thre

e contributions:1. Non-linearity in inflaton fluctuations,

– Falk, Rangarajan & Srendnicki (1993)– Maldacena (2003)

– Non-linearity in - relation– Salopek & Bond (1990; 1991)– Matarrese et al. (2nd order PT)– N papers; gradient-expansion papers

1. Non-linearity in T/T- relation– Pyne & Carroll (1996); Mollerach & Matarrese (1997)

g(+mpl-1f)

mpl-1g(

+mpl

-1f)

T/T~ gT(+f)

T/T~gT[L+(f+gf+ggf)L2]

Komatsu, astro-ph/0206039

fNL ~ f+ gf+ ggfin slow-roll

•g=1•fin slow-roll

•g~O(1/)•fin slow-roll

•g=1/3•ffor Sachs-Wolfe

3 Ways to Get Larger Non-Gaussianity from Early Universe

1. Break slow-roll: ff• Features in V()

• Kofman, Blumenthal, Hodges & Primack (1991); Wang & Kamionkowski (2000); Komatsu et al. (2003); Chen, Easther & Lim (2007)

• DBI inflation• Chen, Huang, Kachru & Shiu (2004)

• Ekpyrotic model, old and new

fNL ~ f+ gf+ ggf

3 Ways to Get Larger Non-Gaussianity from Early Universe

2. Amplify field interactions (without breaking scale invariance): f

• Often done by non-canonical kinetic terms• Ghost inflation

• Arkani-Hamed, Creminelli, Mukohyama & Zaldarriaga (2004)

• DBI Inflation• Chen, Huang, Kachru & Shiu (2004)

fNL ~ f+ gf+ ggf

3 Ways to Get Larger Non-Gaussianity from Early Universe

3. Suppress the perturbation conversion factor, gg

• Generate curvature perturbations from isocurvature (entropy) fluctuations with an efficiency given by g.• Linde & Mukhanov (1997); Lyth & Wands (2

002)• Curvaton predicts gcurvaton which can be a

rbitrarily small• Lyth, Ungarelli & Wands (2002)

• New Ekpyrotic model?

fNL ~ f+ gf+ ggf

Confusion about fNL (3): Maldacena Effect

• Maldacena’s celebrated non-Gaussianity paper (Maldacena 2003) uses the sign convention that is minus of that in Komatsu & Spergel (2001):– fNL(Maldacena) = fNL(Komatsu&Spergel)

• The result: cosmologists and high-energy physicists have often been using different sign conventions.

• It is always useful to ask ourselves, “do we get more cold spots in CMB for fNL>0?” – If yes, it’s Komatsu&Spergel convention. – If no, it’s Maldacena convention.

Positive fNL = More Cold Spots

€

x( ) = ΦG x( ) + fNLΦG2 x( )Simulated temperature maps from

fNL=0 fNL=100

fNL=1000 fNL=5000

Gaussianity Tests: Future Prospects

• The current status: fNL<102 (95%) from WMAP– Komatsu et al. (2003); Spergel et al. (2006); Creminelli et al.

(2006)

• Planck (temperature + polarization): fNL<6 (95%)– Yadav, Komatsu & Wandelt (2007)

• High-z galaxy survey (e.g., ADEPT): fNL<7 (95%)– Sefusatti & Komatsu (2007)

• CMB and LSS are independent. By combining these two constraints, we get fNL<4.5 (95%). This is currently the best constraint that we can possibly achieve in the foreseeable future (~10 years)

Tilt, ns

• A constraint from WMAP 3yr results: 0.074 < ns1 < 0.010 (95%)– Or, ns=0.9580.016 (68%)– 2.6 hint of ns<1.

• However, it must be always remembered that this constraint assumes zero gravity waves, r=0.– “tensor-to-scalar ratio”, r = hGW

2/2 – CMB power spectrum from GWs is propo

rtional to r.

nsr Degeneracy: ns=0.9580.120r

Let’s Understand ns-r Degeneracy

Model Power SpectraScalar T

Tensor T (prop to r)

Scalar E

Tensor E (prop to r)

Tensor B (prop to r)

ns: Tilting Spectrum

nnss>1: Need more power on large an>1: Need more power on large angular scales, more gravity wavesgular scales, more gravity waves

ns: Tilting Spectrum

nnss<1: Need less power on large ang<1: Need less power on large angular scales, less gravity wavesular scales, less gravity waves

ns: 3yr Highlight

• When no gravity waves exist…

• 1yr Result (WMAP only fit)– ns=0.990.04

• 3yr Result (WMAP only fit)– ns=0.9580.016

• Where did this dramatic improvement come from?

ns- Degeneracy: Finally Broken

Parameter Degeneracy Line from Temperature Data Alone

Polarization Data Nailed Tau

1 Year WMAP

3 Year WMAP

1 Year WMAP + other CMB

Polarization From Reionization• CMB was emitted at z~1100.• Some fraction of CMB was re-scattered in a reionized unive

rse.• Photons scatterd away from our line of sight -> temperatur

e damping• Photons scattered into our line of sight -> polarized light

z=1100, ～ 1

z～ 11, ～0.1

First-star formation

z=0

IONIZED

REIONIZED

NEUTRAL

e-

e-e-

e-

e-e- e-

e-e-e-

e-

e-

e- e- e-

Temperature Damping, and Polarization Generation

“Reionization Bump”

2

e-2

Running Index, dns/dlnk

• Constraints from WMAP 3yr:– dns/dlnk = 0.0550.031 (w/o GWs)– dns/dlnk = 0.0850.043 (w GWs)– No strong evidence for scale dependent ns, yet. Note that sl

ow-roll inflation models usually predict dns/dlnk~O(0.001).

• We need to go beyond CMB to constrain this better: we need good measurements of P(k) at small scales.– E.g., when clustering data of Lyman-alpha clouds are includ

ed, dns/dlnk = 0.0150.012 (Slosar, McDonald & Seljak 2007; w/o GWs)

Tilt and Running: Future• The future lies in the large-scale structure data.

– CMB becomes unusable for precision cosmology at small spatial scales due to the Silk damping and secondary anisotropy.

– CMB is limited to k~0.2 Mpc1

• The large-scale structure data can extend the spatial dynamic range by an order of magnitude, up to k~2 Mpc1. (10x better than CMB)– ns1 = O(0.001) and dns/dlnk = O(0.001) are within o

ur reach (Takada, Komatsu & Futamase 2006)

Primordial Gravity Waves

• 3yr limits from WMAP only (95%):– r<0.65 (no running ns)

– r<1.1 (with running ns)

• The constraint is still dominated entirely by the temperature data.– With polarization data only, the constraint i

s r<2.2 (95%; with or without running)

Primordial Gravity Waves: Future

• B-mode polarization is the only way to go.– If CMB polarization can’t see it, no direct detection

experiments (even BBO) will see it.

• WMAP 3 years (2006): r < 2.2 (B-mode 95%)• WMAP 8 years (2009): r ~ 0.3• WMAP 12 years (2013): r ~ 0.2• Planck 1.5 years (launch 2008; results >201

0): r ~ 0.05• CMBPol/EPIC (20xx): r~0.01

GW(k)

k

~k-2

RDMD

CMB anisotropy

Pulsar timing

LISA LIGO

Entered the horizon during

Speaking of Primordial Gravity Waves…Usual Cartoon Picture

Numerical Solution: Traditional

Flat?

€

GW 0 10−10 E inf

1016GeV

⎛

⎝ ⎜

⎞

⎠ ⎟4

Primordial Gravity Waves as a “Time Machine”

€

ds2 = a2(τ )[−dτ 2 + (δ ij + hij )dx idx j ]

hij = 0 ⇒ ˙ ̇ h ij + 2˙ a

a˙ h ij + k 2hij = 0 in FRW spacetime

0 0

)( 2

22

=+⇒=

++−=

ijijij

jiijij

hkhh

dxdxhdtds&&

δ

in Minkowski spacetime

Cosmological Redshift

Therefore, the gravity wave spectrum is sensitive to the entire history of cosmic expansion after inflation.

Improving CalculationsImproving Calculations• Change in the background expansion law Relativistic Degrees of Freedom: g*(T)

Radiation Content of the Early Universe

• Neutrino physics Neutrino Damping (J. Stewart 1972, Rebhan & Schwarz 1994, Weinberg 2004, Dicus & Repko 2005 ) Collisionless Damping due to Anisotropic Stress

€

˙ ̇ h ij + 2˙ a

a˙ h ij + k 2hij =16πGπ ij

4

0

3/1

0*

*20

2

3

8−−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛==

aa

gg

HG

H Rρπ

Watanabe & Komatsu (2006)

Relativistic Degrees of Freedom: g*(T)

€

ρrad = ρ i =i

∑ π 2

30g*(T)T 4 / ∝ a−4,

but ∝ g*−1/ 3a−4

In the early universe, L↔↔↔ −+ ννγγ ee

?4−∝ aradρGW

RDMD

kRD

g*(T)

T, k

€

GW =ρGW ,0

ρ rad ,0

≠ const., but ∝g*(Thc )

g*0

⎛

⎝ ⎜

⎞

⎠ ⎟

−1/ 3

Relativistic Degrees of Freedom: g*(T)

Particle Contents: rest massphoton 0neutrinos 0e-, e+ .51 MeVmuon 106 MeVpions 140 MeVgluon 0u quark 5 MeVd quark 9 MeVs quark 110 MeVc quark 1.3 GeVtauon 1.8 GeVb quark 4.4 GeVW bosons 80 GeVZ boson 91 GeVHiggs boson 114 GeVt quark 174 GeV

SUSY ?~1TeV

QGP P.T.~180MeV

e-,e+ ann.~510keV

Collisionless damping of tensor modes by anisotropic stress due to neutrino free-streaming

tot

, |||| , ||||ρρν

ν ≡Π=Π= ×+×+ fhh

Asymptoticsolution:

645.0||

0 ,8031.0 ,)sin(

)0()(

2=∝⇒

==−

→

A

Akk

Ahh

GW

kk

35.5% less!

Deviation from equilibrium distribution of ν couples with GWs though gravity.

kkkk Gahkha

ah ,

2,

2,, 162 λλλλ π Π=++ &&&&

The Most Accurate Spectrum of GW in the Standard Model of Particle Physics

Watanabe & Komatsu (2006)

Old Result

Features in the Spectrum

Matter-radiation equalitye+e- annihilationNeutrino decoupling QGP phase transition ElectroWeak P.T.SUSY breaking

Reheating (1014 GeV)GUT scale (1016 GeV)Planck scale (1019 GeV) Hz

Hz

Hz

Hz

Hz

Hz

Hz

Hz

Hz

11

9

7

3

4

7

10

10

16

10

10

10

10

10

10

10

10

10

−

−

−

−

−

−CMB ~10-18 Hz WMAP GW0 < 10-11

Planck GW0 < 10-13

Pulsar timing ~10-8 Hz GW0 < 10-8

LISA ~10-2 Hz GW0 < 10-11

DECIGO/BBO ~ 0.1 Hz GW0 < ?Adv. LIGO ~102 Hz GW0 < 10-10

Detector sensitivitiesCosmological events

Hz100

)(

GeV10

6/1

hc*hc60 ⎟

⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛≅ − TgTf

Cosmological Events and Sensitivities

Summary• WMAP is doing fine.• We continue to improve on the accuracy of 6 key param

eters for cosmology vs strings:– Flatness– Gaussianity– Tilt– Running– Gravity waves– Adiabaticity (which I have not talked about)

• FYI: Improved calculations of the primordial gravity wave spectrum.– A straight, featureless line is obsolete!