Observational signatures of holographic models of inflation

Observational signaturesof holographic modelsof inflation

Paul McFadden

Universiteit van Amsterdam

First String Meeting 5/11/10

This talk

I. Cosmological observables & non-Gaussianity

II. Holographic models of inflation

III. Observational signatures

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.

From quantum fluctuations to galaxies

Imaging the primordial perturbations

COBE (1989)

Imaging the primordial perturbations

WMAP (2001)

Imaging the primordial perturbations

Planck (2009)

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.

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.

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)

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.

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.

II. Holographic models of inflation

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.

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)

Holographic framework

Our holographic framework for cosmology is based on standardgauge/gravity duality plus specific analytic continuations:

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.

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.

III. Observational signatures

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

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.

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]

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

Constraints on running

Solid line:

α = −(ns−1)

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 .

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.

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.

Outlook

The scenario