Gaussianity, Flatness, Tilt, Running, and Gravity Waves: WMAP and Some Future Prospects
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Transcript of Gaussianity, Flatness, Tilt, Running, and Gravity Waves: WMAP and Some Future Prospects
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Gaussianity, Flatness, Tilt, Running, and Gravity Waves: WMAP and Some Future Prosp
ects
Eiichiro Komatsu
University of Texas at Austin
July 11, 2007
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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)
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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)
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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.
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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
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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.
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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.
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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.
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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.
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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
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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?
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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)
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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
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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)
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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
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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
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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
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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
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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.
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Positive fNL = More Cold Spots
€
x( ) = ΦG x( ) + fNLΦG2 x( )Simulated temperature maps from
fNL=0 fNL=100
fNL=1000 fNL=5000
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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)
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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.
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nsr Degeneracy: ns=0.9580.120r
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Let’s Understand ns-r Degeneracy
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Model Power SpectraScalar T
Tensor T (prop to r)
Scalar E
Tensor E (prop to r)
Tensor B (prop to r)
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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
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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
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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?
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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
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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-
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Temperature Damping, and Polarization Generation
“Reionization Bump”
2
e-2
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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)
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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)
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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)
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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
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GW(k)
k
~k-2
RDMD
CMB anisotropy
Pulsar timing
LISA LIGO
Entered the horizon during
Speaking of Primordial Gravity Waves…Usual Cartoon Picture
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Numerical Solution: Traditional
Flat?
€
GW 0 10−10 E inf
1016GeV
⎛
⎝ ⎜
⎞
⎠ ⎟4
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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.
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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)
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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
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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
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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=++ &&&&
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The Most Accurate Spectrum of GW in the Standard Model of Particle Physics
Watanabe & Komatsu (2006)
Old Result
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Features in the Spectrum
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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
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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!