observational signatures of holographic models of inflation
TRANSCRIPT
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Observational signaturesof holographic modelsof inflation
Paul McFadden
Universiteit van Amsterdam
First String Meeting 5/11/10
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This talk
I. Cosmological observables & non-Gaussianity
II. Holographic models of inflation
III. Observational signatures
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References
This talk is based on work with Kostas Skenderis:
• Observational signatures of holographic models of inflation,arxiv:1010.0244.
• The holographic universe, arxiv:1001.2007.
• Holography for cosmology, arxiv:0907.5542.
• Holographic non-Gaussianity, to appear shortly.
... and on-going work also with Adam Bzowski.
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From quantum fluctuations to galaxies
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Imaging the primordial perturbations
COBE (1989)
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Imaging the primordial perturbations
WMAP (2001)
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Imaging the primordial perturbations
Planck (2009)
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Primordial perturbations
The primordial perturbations offer some of our best clues as to thefundamental physics underlying the big bang. Their form is surprisinglysimple:
• Small amplitude: δT/T ∼ 10−5
• Adiabatic
• Nearly Gaussian
• Nearly scale-invariant
I Any proposed cosmological model must be able to account for thesebasic features.
I Any predicted deviations (e.g. from Gaussianity) are likely to provecritical in distinguishing different models.
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The power spectrum
A Gaussian distribution is fully characterised by its 2-point function orpower spectrum. From observations, the power spectrum takes the form:
∆2S(q) = ∆2
S(q0) (q/q0)nS−1
The WMAP data yield (for q0 = 0.002Mpc−1)
∆2S(q0) = (2.445± 0.096)× 10−9, nS−1 = −0.040± 0.013,
i.e., the scalar perturbations have small amplitude and are nearly scaleinvariant.
I These two small numbers should appear naturally in any theory thatexplains the data.
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The bispectrum
Non-Gaussianity implies non-zero higher-point correlation functions.
The lowest order (hence easiest to measure) statistic is the 3-pointfunction, or bispectrum, of curvature perturbations ζ:
〈ζ(q1)ζ(q2)ζ(q3)〉 = (2π)3δ(∑
qi)B(qi)
The amplitude of the bispectrum B(qi) is parametrised by fNL:
B(qi) = fNL × (shape function)
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Non-Gaussianity
Non-Gaussianity arises from nonlinearities in cosmological evolution. Thethree primary sources are:
1. Nonlinearities (interactions) in inflationary dynamics.
2. Nonlinear evolution of perturbations in radiation/matter era.
3. Nonlinearities in relationship between metric perturbations and CMBtemperature fluctuations. (To linear order, ∆T/T = (1/3)Φ).
Primordial non-Gaussianity is especially important as it allows us toconstrain inflationary dynamics:
I Different models make different predictions for fNL and the shapefunction. e.g., single field slow-roll inflation fNL ∼ O(ε, η) ∼ 0.01.
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Observational constraints
From WMAP 7-yr data: f localNL = 32± 21, fequilNL = 26± 140
I ‘Local’ form:
Blocal(qi) = f localNL
6
5A2
∑q3i∏q3i
, A = 2π2∆2S(q).
I ‘Equilateral’ form:
Bequil(qi) = fequilNL
18
5A2 1∏
q3i
(−∑
q3i −2q1q2q3 +(q1q
22 +perms)
).
The Planck data (expected next year) should be sensitive to fNL ∼ 5.
Non-Gaussianity potentially provides a strong test of inflationary models.
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II. Holographic models of inflation
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A holographic universe
Recently, we proposed a holographic description of 4d inflationaryuniverses in terms of a 3d quantum field theory without gravity.
I For conventional inflation, this dual QFT is strongly coupled.
I When the dual QFT is instead weakly coupled, we can model auniverse which is non-geometric at very early times.
In particular:
• These latter models provide a new mechanism for obtaining a nearlyscale-invariant power spectrum.
• They are compatible with current observations, yet have a distinctphenomenology from conventional (slow-roll) inflation.
• The Planck data has the power to confirm or exclude these models.
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Holography
Any quantum theory of gravity should have a dual description interms of a quantum field theory (QFT), without gravity, living inone dimension less.
Any holographic proposal for cosmology should specify:
1. The nature of the dual QFT
2. How to compute cosmological observables
(e.g. the primordial power spectrum & bispectrum)
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Holographic framework
Our holographic framework for cosmology is based on standardgauge/gravity duality plus specific analytic continuations:
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Holographic formulae
Via the holographic framework, cosmological observables are related tospecific analytic continuations of QFT correlators:
∆2S(q) =
−q3
4π2Im[〈T (−iq)T (iq)〉]
B(qi) = −1
4
1∏i Im[〈T (−iqi)T (iqi)〉]
Im[〈T (−iq1)T (−iq2)T (−iq3)〉
+∑i
〈T (iqi)T (−iqi)〉 − 2(〈T (−iq1)Υ(−iq2,−iq3)〉+ cyclic perms)
)]
where T is the trace of the QFT stress tensor and Υ = δijδklδTij/δgkl.
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Holographic phenomenology
Prototype dual QFT:
I 3d SU(N) Yang-Mills theory + scalars + fermions.
I Parameters: g2YM; the number of colours, N ; QFT field content.
Theory simplifies in ’t Hooft limit: N � 1 but g2YMN fixed.
To make predictions:
1. Compute correlation functions using perturbative QFT.
2. Apply holographic formulae to find cosmological observables.
3. Compare with observational data.
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III. Observational signatures
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Power spectrum
I To compute the cosmological power spectrum we need to evaluatethe 2-pt function of Tij .
I The leading contribution is at 1-loop order:
〈T (q)T (−q)〉 ∼ N2q3
I Recalling the holographic formula,
∆2S ∼
q3
〈TT 〉∼ 1
N2
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Power spectrum
I Spectrum scale invariant to leading order, independent of details ofholographic theory.
Moreover,
I The amplitude of the power spectrum ∆2S(q0) ∼ 1/N2.
I Small observed amplitude ∆2S(q0) ∼ 10−9 ⇒ N ∼ 104, justifying
the large N limit.
A complete calculation gives ∆2S(q0) = 16/π2N2(NA +Nφ), where
NA = # gauge fields, Nφ = # minimal scalars.
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Spectral index
2-loop corrections give rise to a small deviation from scale invariance:
nS − 1 ∼ g2eff = g2
YMN/q.
The observed value nS − 1 ∼ 10−2 is thenconsistent with QFT being weakly interacting.
I To determine whether nS < 1 (red-tilted) or nS > 1 (blue-tilted)requires summing all 2-loop graphs, and will in general depend onthe field content of the dual QFT.
[Work in progress]
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Running
I Irrespective of the details of the theory, the spectral index runs:
αS = dnS/d ln q = −(nS−1) +O(g4eff).
I This prediction is qualitatively different from slow-roll inflation, forwhich αs/(nS − 1) is of first-order in slow roll.
I Running of this form is consistent with current data, and should beeither confirmed or excluded by Planck.
⇒ Observational signature #1
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Constraints on running
Solid line:
α = −(ns−1)
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Tensor-to-scalar ratio
I Holographic model predicts
r =∆2T
∆2S
= 32
(NA +Nφ +Nχ +Nψ
NA +Nφ
)
where NA = # gauge fields, Nφ = # minimally coupled scalars,Nχ = # conformally coupled scalars, Nψ = # fermions.
I An upper bound on r translates into a constraint on the fieldcontent of the dual QFT.
I r is not parametrically suppressed as in slow-roll inflation, nor does itsatisfy the slow-roll consistency condition r = −8nT .
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Non-Gaussianity
Evaluating the QFT 3-pt function, ourholographic formula predicts a bispectrumof exactly the equilateral form:
B(qi) = Bequil(qi), fequilNL = 5/36.
⇒ Observational signature #2
I This result is independent of all details of the theory.
I Result probably too small for direct detection by Planck, but theobservation of larger fNL values would exclude our models.
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Conclusions
I 4d inflationary universes may be described holographically in termsof dual non-gravitational 3d QFT.
I When dual QFT is weakly coupled obtain new holographic modelswith the following universal features:
1. Near scale-invariant spectrum of small amplitude perturbations.
2. The spectral index runs as αs = −(ns − 1).
3. The bispectrum is of the equilateral form with fequilNL = 5/36.
I Holographic models are testable: both predictions 2 and 3 may beexcluded by the Planck data released next year.
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Outlook
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The scenario