Observational signatures of holographic models of 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 the fundamental physics underlying the big bang. Their form is surprisingly simple: Small amplitude: δt/t 10 5 Adiabatic Nearly Gaussian Nearly scale-invariant Any proposed cosmological model must be able to account for these basic features. Any predicted deviations (e.g. from Gaussianity) are likely to prove critical in distinguishing different models.
The power spectrum A Gaussian distribution is fully characterised by its 2-point function or power spectrum. From observations, the power spectrum takes the form: 2 S(q) = 2 S(q 0 ) (q/q 0 ) n S 1 The WMAP data yield (for q 0 = 0.002Mpc 1 ) 2 S(q 0 ) = (2.445 ± 0.096) 10 9, n S 1 = 0.040 ± 0.013, i.e., the scalar perturbations have small amplitude and are nearly scale invariant. These two small numbers should appear naturally in any theory that explains the data.
The bispectrum Non-Gaussianity implies non-zero higher-point correlation functions. The lowest order (hence easiest to measure) statistic is the 3-point function, or bispectrum, of curvature perturbations ζ: ζ(q 1 )ζ(q 2 )ζ(q 3 ) = (2π) 3 δ( q i )B(q i ) The amplitude of the bispectrum B(q i ) is parametrised by f NL : B(q i ) = f NL (shape function)
Non-Gaussianity Non-Gaussianity arises from nonlinearities in cosmological evolution. The three 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 CMB temperature fluctuations. (To linear order, T/T = (1/3)Φ). Primordial non-gaussianity is especially important as it allows us to constrain inflationary dynamics: Different models make different predictions for f NL and the shape function. e.g., single field slow-roll inflation f NL O(ɛ, η) 0.01.
Observational constraints From WMAP 7-yr data: Local form: f local NL = 32 ± 21, f equil NL = 26 ± 140 B local (q i ) = f local NL 6 5 A2 q 3 i q 3, i A = 2π 2 2 S(q). Equilateral form: B equil (q i ) = f equil NL 18 5 A2 1 q 3 i ( q 3 i 2q 1 q 2 q 3 +(q 1 q 2 2 +perms) ). The Planck data (expected next year) should be sensitive to f NL 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 inflationary universes in terms of a 3d quantum field theory without gravity. For conventional inflation, this dual QFT is strongly coupled. When the dual QFT is instead weakly coupled, we can model a universe which is non-geometric at very early times. In particular: These latter models provide a new mechanism for obtaining a nearly scale-invariant power spectrum. They are compatible with current observations, yet have a distinct phenomenology 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 in terms of a quantum field theory (QFT), without gravity, living in one 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 standard gauge/gravity duality plus specific analytic continuations:
Holographic formulae Via the holographic framework, cosmological observables are related to specific analytic continuations of QFT correlators: 2 S(q) = q 3 4π 2 Im[ T ( iq)t (iq) ] B(q i ) = 1 1 [ 4 i Im[ T ( iq i)t (iq i ) ] Im T ( iq 1 )T ( iq 2 )T ( iq 3 ) + i T (iq i )T ( iq i ) 2 ( T ( iq 1 )Υ( iq 2, iq 3 ) + cyclic perms) )] where T is the trace of the QFT stress tensor and Υ = δ ij δ kl δt ij /δg kl.
Holographic phenomenology Prototype dual QFT: 3d SU(N) Yang-Mills theory + scalars + fermions. Parameters: gym 2 ; the number of colours, N; QFT field content. Theory simplifies in t Hooft limit: N 1 but gym 2 N 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 To compute the cosmological power spectrum we need to evaluate the 2-pt function of T ij. The leading contribution is at 1-loop order: T (q)t ( q) N 2 q 3 Recalling the holographic formula, 2 S q3 T T 1 N 2
Power spectrum Spectrum scale invariant to leading order, independent of details of holographic theory. Moreover, The amplitude of the power spectrum 2 S (q 0) 1/N 2. Small observed amplitude 2 S (q 0) 10 9 N 10 4, justifying the large N limit. A complete calculation gives 2 S (q 0) = 16/π 2 N 2 (N A + N φ ), where N A = # gauge fields, N φ = # minimal scalars.
Spectral index 2-loop corrections give rise to a small deviation from scale invariance: n S 1 g 2 eff = g 2 YMN/q. The observed value n S 1 10 2 is then consistent with QFT being weakly interacting. To determine whether n S < 1 (red-tilted) or n S > 1 (blue-tilted) requires summing all 2-loop graphs, and will in general depend on the field content of the dual QFT. [Work in progress]
Running Irrespective of the details of the theory, the spectral index runs: α S = dn S /d ln q = (n S 1) + O(g 4 eff). This prediction is qualitatively different from slow-roll inflation, for which α s /(n S 1) is of first-order in slow roll. Running of this form is consistent with current data, and should be either confirmed or excluded by Planck. Observational signature #1
Constraints on running Solid line: α = (n s 1)
Tensor-to-scalar ratio Holographic model predicts r = 2 T 2 S ( ) NA + N φ + N χ + N ψ = 32 N A + N φ where N A = # gauge fields, N φ = # minimally coupled scalars, N χ = # conformally coupled scalars, N ψ = # fermions. An upper bound on r translates into a constraint on the field content of the dual QFT. r is not parametrically suppressed as in slow-roll inflation, nor does it satisfy the slow-roll consistency condition r = 8n T.
Non-Gaussianity Evaluating the QFT 3-pt function, our holographic formula predicts a bispectrum of exactly the equilateral form: B(q i ) = B equil (q i ), f equil NL = 5/36. Observational signature #2 This result is independent of all details of the theory. Result probably too small for direct detection by Planck, but the observation of larger f NL values would exclude our models.
Conclusions 4d inflationary universes may be described holographically in terms of dual non-gravitational 3d QFT. When dual QFT is weakly coupled obtain new holographic models with the following universal features: 1. Near scale-invariant spectrum of small amplitude perturbations. 2. The spectral index runs as α s = (n s 1). 3. The bispectrum is of the equilateral form with f equil NL = 5/36. Holographic models are testable: both predictions 2 and 3 may be excluded by the Planck data released next year.
Outlook
The scenario