testing physics of the early universe observationallykomatsu/... · inflation likes this result...
TRANSCRIPT
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Testing Physics of the Early Universe Observationally:
Are Primordial Fluctuations Gaussian, or Non-Gaussian?
Eiichiro Komatsu(Texas Cosmology Center, University of Texas at Austin)
Tufts/CfA/MIT Cosmology Seminar, Tufts UniversityApril 14, 2009
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How Do We Test Inflation?
• How can we answer a simple question like this:
• “How were primordial fluctuations generated?”
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Power Spectrum
• A very successful explanation (Guth & Pi; Hawking; Starobinsky; Bardeen, Steinhardt & Turner) is:
• Primordial fluctuations were generated by quantum fluctuations of the scalar field that drove inflation.
• The prediction: a nearly scale-invariant power spectrum in the curvature perturbation, ζ:
• Pζ(k) = A/k4–ns
• where ns~1 and A is a normalization. 3
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ns<1 Observed
• The latest results from the WMAP 5-year data:
• ns=0.960 ± 0.013 (68%CL; for tensor modes = zero)
• ns=0.970 ± 0.015 (68%CL; for tensor modes ≠ zero)
• tensor-to-scalar ratio < 0.22 (95%CL)
• Another evidence for inflation
• Detection of non-zero tensor modes is a next important step
Komatsu et al. (2009)
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Anything Else?• One can also look for other signatures of inflation. For
example:
• Isocurvature perturbations
• Proof of the existence of multiple fields
• Non-zero spatial curvature
• Evidence for the Land Scape, if curvature is negative. Rules it out if positive.
• Scale-dependent ns (running index)
• Complex dynamics of inflation 5
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Anything Else?• One can also look for other signatures of inflation. For
example:
• 95%CL limits on Isocurvature perturbations
• <8.9% (axion CDM); <2.1% (curvaton CDM)
• 95%CL limits on Non-zero spatial curvature
• <1.8% (positive curvature); <0.8% (negative curvature)
• 95%CL limits on Scale-dependent ns
• –0.068 < dns/dlnk < 0.0126
Komatsu et al. (2009)
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Beyond Power Spectrum
• All of these are based upon fitting the observed power spectrum.
• Is there any information one can obtain beyond the power spectrum?
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Bispectrum
• Three-point function!
• Bζ(k1,k2,k3) = (amplitude) x (2π)3δ(k1+k2+k3)b(k1,k2,k3)
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Why Study Bispectrum?
• It probes the interactions of fields - new piece of information that cannot be probed by the power spectrum
• But, above all, it provides us with a critical test of the simplest models of inflation: “are primordial fluctuations Gaussian, or non-Gaussian?”
• Bispectrum vanishes for Gaussian fluctuations.
• Detection of the bispectrum = detection of non-Gaussian fluctuations 10
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Gaussian? WMAP5
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Take One-point Distribution Function
•The one-point distribution of WMAP map looks pretty Gaussian.–Left to right: Q (41GHz), V (61GHz), W (94GHz).
•Deviation from Gaussianity is small, if any.12
Spergel et al. (2008)
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Inflation Likes This Result
• According to inflation (Guth & Yi; Hawking; Starobinsky; Bardeen, Steinhardt & Turner), CMB anisotropy was created from quantum fluctuations of a scalar field in Bunch-Davies vacuum during inflation
• Successful inflation (with the expansion factor more than e60) demands the scalar field be almost interaction-free
• The wave function of free fields in the ground state is a Gaussian!
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But, Not Exactly Gaussian
• Of course, there are always corrections to the simplest statement like this
• For one, inflaton field does have interactions. They are simply weak – of order the so-called slow-roll parameters, ε and η, which are O(0.01)
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Non-Gaussianity from Inflation•You need cubic interaction terms (or higher order) of fields.–V(φ)~φ3: Falk, Rangarajan & Srendnicki (1993) [gravity not included yet]
–Full expansion of the action, including gravity action, to cubic order was done a decade later by Maldacena (2003)
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Computing Primordial Bispectrum•Three-point function, using in-in formalism (Maldacena 2003; Weinberg 2005)
•HI(t): Hamiltonian in interaction picture–Model-dependent: this determines which triangle shapes will dominate the signal
•Φ(x): operator representing curvature perturbations in interaction picture
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Why Study Bispectrum?• Because a detection of the bispectrum has a best chance of
ruling out the largest class of inflation models.
• Namely, it will rule out inflation models based upon
• a single scalar field with
• the canonical kinetic term that
• rolled down a smooth scalar potential slowly, and
• was initially in the Bunch-Davies vacuum.
• Detection of the bispectrum would be a major breakthrough in cosmology. 17
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“fNL”
• fNL = the amplitude of bispectrum, which is
• Βζ(k1,k2,k3)=<ζ(k1)ζ(k2)ζ(k3)>
=fNL(2π)3δ(k1+k2+k3)b(k1,k2,k3)
• b(k1,k2,k3) is a model-dependent function that defines the shape of triangles predicted by various models.
k1k2
k3
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Forms of b(k1,k2,k3)• Local form [can be generated by multi-field
models]
• blocal(k1,k2,k3) = (6/5)[Pζ(k1)Pζ(k2)+cyc.]
• Equilateral form [can be generated by non-canonical kinetic terms, e.g., DBI]
• bequilateral(k1,k2,k3) = (18/5){–[Pζ(k1)Pζ(k2)+cyc.] – 2[Pζ(k1)Pζ(k2)Pζ(k3)]2/3
+ [Pζ(k1)1/3Pζ(k2)2/3Pζ(k3)+cyc.]}
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Squeezed Limit
Equilateral Limit
Komatsu & Spergel (2001)
Babich, Creminelli & Zaldarriaga (2004)
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Local Form Non-Gaussianity• The local form bispectrum, Βζ(k1,k2,k3)=(2π)3δ(k1+k2+k3)fNLlocal[(6/5)Pζ(k1)Pζ(k2)+cyc.]
• is equivalent to having the curvature perturbation in position space, in the form of:
• ζ(x)=ζgaussian(x) + (3/5)fNL[ζgaussian(x)]2
• This provides a useful model to parametrize non-Gaussianity, and generate initial conditions for, e.g., N-body simulations.
• This can be extended to higher-order:
• ζ(x)=ζgaussian(x) + (3/5)fNL[ζgaussian(x)]2 + (9/25)gNL[ζgaussian(x)]3
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What if fNL is detected?
• A single field, canonical kinetic term, slow-roll, and/or Bunch-Davies vacuum, must be modified.
• Multi-field (curvaton)
• Non-canonical kinetic term (k-inflation, DBI)
• Temporary fast roll (features in potential)
• Departures from the Bunch-Davies vacuum
• It will give us a lot of clues as to what the correct early universe models should look like.
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Local
Equil.
Bump+Osci.Folded
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Decoding Bispectrum• Hydrodynamics at z=1090
generates acoustic oscillations in the bispectrum
• Well understood at the linear level (Komatsu & Spergel 2001)
• Non-linear extension?
• Nitta, Komatsu, Bartolo, Matarrese & Riotto, arXiv:0903.0894
• fNLlocal~0.5
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Measurement
• Use everybody’s favorite: χ2 minimization.
• Minimize:
• with respect to Ai=(fNLlocal, fNLequilateral, bsrc)
• Bobs is the observed bispectrum
• B(i) is the theoretical template from various predictions23
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Journal on fNL (95%CL)• Local
• –3500 < fNLlocal < 2000 [COBE 4yr, lmax=20 ]
• –58 < fNLlocal < 134 [WMAP 1yr, lmax=265]
• –54 < fNLlocal < 114 [WMAP 3yr, lmax=350]
• –9 < fNLlocal < 111 [WMAP 5yr, lmax=500]
• Equilateral
• –366 < fNLequil < 238 [WMAP 1yr, lmax=405]
• –256 < fNLequil < 332 [WMAP 3yr, lmax=475]
• –151 < fNLequil < 253 [WMAP 5yr, lmax=700]
Komatsu et al. (2002)
Komatsu et al. (2003)
Spergel et al. (2007)
Komatsu et al. (2008)
Creminelli et al. (2006)
Creminelli et al. (2007)
Komatsu et al. (2008)24
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Latest on fNLlocal
• CMB (WMAP5 + most optimal bispectrum estimator)
• –4 < fNLlocal < 80 (95%CL)
• fNLlocal = 38 ± 21 (68%CL)
• Large-scale Structure (Using the SDSS power spectra)
• –29 < fNLlocal < 70 (95%CL)
• fNLlocal = 31 +16–27 (68%CL)
Smith et al. (2009)
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Slosar et al. (2009)
(Fast-moving field!)
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Exciting Future Prospects
• Planck satellite (to be launched on May 6, 2009)
• will see fNLlocal at 8σ, IF (big if) fNLlocal=40
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A Big Question
• Suppose that fNL was found in, e.g., WMAP 9-year or Planck. That would be a profound discovery. However:
• Q: How can we convince ourselves and other people that primordial non-Gaussianity was found, rather than some junk?
• A: (i) shape dependence of the signal, (ii) different statistical tools, and (iii) different tracers
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(i) Remember These Plots?
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(ii) Different Tools
• How about 4-point function (trispectrum)?
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Local Form Trispectrum• For ζ(x)=ζgaussian(x) + (3/5)fNL[ζgaussian(x)]2 +
(9/25)gNL[ζgaussian(x)]3, we obtain the trispectrum:
• Tζ(k1,k2,k3,k4)=(2π)3δ(k1+k2+k3+k4) {gNL[(54/25)Pζ(k1)Pζ(k2)Pζ(k3)+cyc.] +(fNL)2[(18/25)Pζ(k1)Pζ(k2)(Pζ(|k1+k3|)+Pζ(|k1+k4|))+cyc.]}
k3
k4
k2
k1
gNL
k2
k1
k3
k4
fNL2 30
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Trispectrum: if fNL is ~50, excellent cross-check for Planck
•Trispectrum (~fNL2)
•Bispectrum (~ fNL)
Kogo & Komatsu (2006)
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(Slightly) Generalized Trispectrum
• Tζ(k1,k2,k3,k4)=(2π)3δ(k1+k2+k3+k4) {gNL[(54/25)Pζ(k1)Pζ(k2)Pζ(k3)+cyc.] +τNL[(18/25)Pζ(k1)Pζ(k2)(Pζ(|k1+k3|)+Pζ(|k1+k4|))+cyc.]}
The local form consistency relation, τNL=(fNL)2, may not be respected –
additional test of multi-field inflation!
k3
k4
k2
k1
gNL
k2
k1
k3
k4
fNL2 32
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Trispectrum: Next Frontier
• A new phenomenon: many talks given at the IPMU non-Gaussianity workshop emphasized the importance of the trispectrum as a source of additional information on the physics of inflation.
• τNL ~ fNL2; τNL ~ fNL4/3; τNL ~ (isocurv.)*fNL2; gNL ~ fNL; gNL ~ fNL2; or they are completely independent
• Shape dependence? (Squares from ghost condensate, diamonds and rectangles from multi-field, etc)
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(ii) Different Tracers
• New frontier: large-scale structure of the universe as a probe of primordial non-Gaussianity
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New, Powerful Probe of fNL •fNL modifies the power spectrum of galaxies on very large scales–Dalal et al.; Matarrese & Verde–Mcdonald; Afshordi & Tolley
•The statistical power of this method is VERY promising–SDSS: –29 < fNL < 70 (95%CL); Slosar et al.
–Comparable to the WMAP 5-year limit already
–Expected to beat CMB, and reach a sacred region: fNL~1 35
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Effects of fNL on the statistics of PEAKS
• The effects of fNL on the power spectrum of peaks (i.e., galaxies) are profound.
• How about the bispectrum of galaxies?
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Previous Calculation• Scoccimarro, Sefusatti & Zaldarriaga (2004); Sefusatti &
Komatsu (2007)
• Treated the distribution of galaxies as a continuous distribution, biased relative to the matter distribution:
• δg = b1δm + (b2/2)(δm)2 + ...
• Then, the calculation is straightforward. Schematically:
• <δg3> = (b1)3<δm3> + (b12b2)<δm4> + ...Non-linear Bias BispectrumNon-linear Gravity
Primordial NG 37
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Previous Calculation
• We find that this formula captures only a part of the full contributions. In fact, this formula is sub-dominant in the squeezed configuration, and the new terms are dominant.
Non-linear Bias
Non-linear Gravity
Primordial NG
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Non-linear Gravity
• For a given k1, vary k2 and k3, with k3≤k2≤k1
• F2(k2,k3) vanishes in the squeezed limit, and peaks at the elongated triangles. 40
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Non-linear Galaxy Bias
• There is no F2: less suppression at the squeezed, and less enhancement along the elongated triangles.
• Still peaks at the equilateral or elongated forms. 41
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Primordial NG (SK07)
• Notice the factors of k2 in the denominator.
• This gives the peaks at the squeezed configurations. 42
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New Terms
• But, it turns our that Sefusatti & Komatsu’s calculation, which is valid only for the continuous field, misses the dominant terms that come from the statistics of PEAKS.
• Jeong & Komatsu, arXiv:0904.0497
Donghui Jeong 43
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MLB Formula
• N-point correlation function of peaks is the sum of M-point correlation functions, where M≥N.
Matarrese, Lucchin & Bonometto (1986)
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Bottom Line
• The bottom line is:
• The power spectrum (2-pt function) of peaks is sensitive to the power spectrum of the underlying mass distribution, and the bispectrum, and the trispectrum, etc.
• Truncate the sum at the bispectrum: sensitivity to fNL
• Dalal et al.; Matarrese&Verde; Slosar et al.; Afshordi&Tolley
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Bottom Line
• The bottom line is:
• The bispectrum (3-pt function) of peaks is sensitive to the bispectrum of the underlying mass distribution, and the trispectrum, and the quadspectrum, etc.
• Truncate the sum at the trispectrum: sensitivity to τNL (~fNL2) and gNL!
• This is the new effect that was missing in Sefusatti & Komatsu (2007).
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Real-space 3pt Function
• Plus 5-pt functions, etc...47
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New Bispectrum Formula
• First: bispectrum of the underlying mass distribution.
• Second: non-linear bias
• Third: trispectrum of the underlying mass distribution.48
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Local Form Trispectrum
• For general multi-field models, fNL2 can be more generic: often called τNL.
• Exciting possibility for testing more about inflation! 49
Φ=(3/5)ζ
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Local Form Trispectrum
k3
k4
k2
k1
gNL
k2
k1
k3
k4
fNL2 (or τNL)50
Φ=(3/5)ζ
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Trispectrum Term
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Trispectrum Term
Most Dominant in the Squeezed Limit
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Shape Results
• The primordial non-Gaussianity terms peak at the squeezed triangle.
• fNL and gNL terms have the same shape dependence:
• For k1=k2=αk3, (fNL term)~α and (gNL term)~α
• fNL2 (τNL) is more sharply peaked at the squeezed:
• (fNL2 term)~α3
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Key Question
• Are gNL or τNL terms important?
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1/k2
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Importance Ratios
• fNL2 dominates over fNL term easily for fNL>1!
kαkαk
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Redshift Dependence
• Primordial non-Gaussianity terms are more important at higher redshifts.
• The new trispectrum terms are even more important. 59
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Summary• Non-Gaussianity is a new, powerful probe of
physics of the early universe
• It has a best chance of ruling out the largest class of inflation models
• Various forms of fNL available today — 1.8σ at the moment, wait for WMAP 9-year (2011) and Planck (2012) for more σ’s (if it’s there!)
• To convince ourselves of detection, we need to see the acoustic oscillations, and the same signal in the bispectrum and trispectrum, of both CMB and the large-scale structure of the universe. 61
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Additional Remarks
• Unusually healthy interactions between observers and theorists: astronomers, cosmologists, phenomenologists, high-energy theorists
• The list of participants in workshops on non-Gaussianity speaks for its diversity
• Interdisciplinary efforts
• Lots of important contributions from young people
• New “industry” – active field, something new every month 62
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Now, let’s pray:
• May Planck succeed!
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Now, let’s pray:
•May the signal be there!
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