New Ekpyrotic Cosmology and Non-Gaussianity

Transcription

1 New Ekpyrotic Cosmology and Non-Gaussianity Justin Khoury (Perimeter) with Evgeny Buchbinder (PI) Burt Ovrut (UPenn) hep-th/ , hep-th/ , hep-th/ Related work: Lehners, McFadden, Turok & Steinhardt, hep-th/ Creminelli & Senatore, hep-th/ Koyama & Wands, hep-th/ Finelli, hep-th/ , Brandenberger & Finelli, Tolley & Wesley, hep-th/ Koyama, Mizuno, Vernizzi and Wands, hep-th/ Lehners & Steinhardt, to appear

2 Wandelt Effect From Date: Mon, 12 Nov :02: (EST) From: Justin Khoury To: Subject: Re: Very Early Universe workshop - talk title? dear prof shellard, the title of my talk will be "New Ekpyrotic Cosmology and Non-Gaussianity". looking forward to this exciting meeting! all best wishes, justin Detection of primordial non-gaussianity (fnl) in the WMAP 3-year data at above 99.5% confidence Authors: Amit P. S. Yadav, Benjamin D. Wandelt (Submitted on 7 Dec 2007 (v1), last revised 9 Dec 2007 (this version, v2))

3 Ekpyrotic Cosmology Before big bang, universe underwent a long phase of slow contraction (Spiritually the opposite of inflation) This is driven by a scalar field with steep, negative potential: V (φ) = V 0 e φ/λ V (φ) φ where Λ ɛm Pl and ɛ 1 Leads to scaling solution, corresponding to slow contraction ( ) a(t) ( t) 2ɛ V 0 ; φ(t) = Λ log 2Λ 2 (1 6ɛ) t2 w = 1 3ɛ 1 1

4 Origin of scale invariance Since ɛ 1, scale factor is nearly constant => ignore gravity φ = 2V with solution t = φ d φ e φ/2λ 2V0 = 2Λ eφ/2λ 2V0 = 2 V, φφ Moving on to perturbations: δφ k + ( k 2 + V,φφ ) δφk = 0 factor of 2 => scale-inv which reduces to δφ k + (k 2 2t 2 ) δφ k = 0

5 However this time-delay mode exactly projects out the curvature perturbation => not scale invariant ζ Lyth, hep-ph/ JK, Ovrut, Steinhardt and Turok, hep-th/ Brandenberger and Finelli, hep-th/ ζ Problematic because is constant on super-horizon scales (in the absence of entropy perturbations). This is true independent of GR Lyth, Malik and Sasaki, astro-ph/ Caveat: New (stringy) effects at the singularity could save the day by mixing the time-delay and curvature modes Tolley, Steinhardt and Turok, hep-th/ McFadden, Turok and Steinhardt, hep-th/ Battefeld, Patil and Brandenberger, hep-th/

6 Idea: Consider scale invariant entropy perturbations, convert it onto curvature pertn, so that invariant well before the bounce ζ is scale Lehners, McFadden, Turok and Steinhardt, hep-th/ Buchbinder, JK and Ovrut, hep-th/ Creminelli and Senatore, hep-th/

7 New Ekpyrotic Cosmology Consider 2 scalar fields, each with its own ekpyrotic potential: V (φ 1, φ 2 ) = V 1 e φ 1/Λ 1 V 2 e φ 2/Λ 2 Again this leads to scaling solution a(t) ( t) 2(ɛ 1+ɛ 2 ) ; φ i (t) = Λ i log ( ) V i 2Λ 2 i (1 6(ɛ 1 + ɛ 2 )) t2 However not an attractor because of tachyonic instability Lehners, McFadden, Turok and Steinhardt, hep-th/ Buchbinder, JK and Ovrut, hep-th/ Koyama and Wands, hep-th/

8 Tachyon most easily seen by using new field variables φ = Λ 1φ 1 + Λ 2 φ 2 Λ Λ 2 2 ; χ = Λ 2φ 1 Λ 1 φ 2 Λ Λ 2 2 in terms of which the potential becomes ( ) V (φ, χ) = V 0 e φ/λ 1 + χ2 2Λ Scaling soln corresponds to rolling along φ, while χ remains fixed.

9 Tachyon is good Since χ is fixed, its fluctuations are entropy pertns. Key feature of potential: V,χχ = V,φφ = 2 t 2 from form of potential from scaling solution Substitute in pertn eqn: δχ k + (k 2 2t 2 ) δχ k = 0 which implies a scale invariant spectrum Tachyonic direction is therefore essential in generating scale-invariant entropy pertn spectrum.

10 Converting to curvature pertn Entropy pertn sources ζ on large scales ζ 2Ḣ φ θ δχ where tan θ(t) = χ/ φ describes the angle in field space Sharp turn in field trajectory imprints scale-invariant δχ onto curvature perturbation Alternative conversion mechanisms: Koyama and Wands, hep-th/ Battefeld, hep-th/ Lehners et al., hep-th/

11 Final Amplitude In sharp turn approx, k 3/2 ζ k θ H ɛmpl where and H θ is Hubble at the end of ekpyrotic phase, is overall change in field direction. Nearly identical to inflationary expression k 3/2 ζ k H ɛinf M Pl

12 What about the bounce? Using ghost condensate mechanism, can generate a non-singular bounce, without introducing ghosts or other pathologies Creminelli, Luty, Nicolis and Senatore, JHEP 12, 080 (2006) Can successfully merge the ekpyrotic phase with this subsequent ghost condensation phase Buchbinder, JK, Ovrut, hep-th/ Find that ζ goes through the bounce unscathed and emerges in the hot big bang phase with a scale-invariant spectrum.

13 Non-Gaussianity ( V (φ, χ) = V 0 e φ/λ 1 + χ2 2Λ 2 + α 3 3! χ 3 ) Λ where α i are expected to be O(1) 2 sources of non-gaussianity: -- Intrinsic NG because of δχ self-interactions: δχ k1 δχ k2 δχ k3 (Since ζ δχ, curvature pertn inherits this NG.) -- Non-linear relation between ζ and δχ : ζ(x) = a 1 δχ(x) + a 2 δχ 2 (x) + a 3 δχ 3 (x)... (Local form)

14 Both contributions give NG of local form: ζ k1 ζ k2 ζ k3 = (2π) 3 δ 3 ( k 1 + k 2 + k 3 ) i.e. encoded in a single parameter f NL ( ) 6 5 f NL [P ζ (k 1 )P ζ (k 2 ) +...] Result is generically large: f NL = f int NL + f conv NL = 5 24( θ) 2 ɛ Buchbinder, JK, Ovrut, hep-th/ ( 1 4 ) 3 α 3 θ WMAP 3-year: f NL ɛ 1 36 < f NL < 100 See also: Creminelli & Senatore, hep-th/ Koyama et al, hep-th/ Lehners & Steinhardt, to appear Yadav & Wandelt (2007): 26.9 < f NL < at 95%CL

15 NG offers distinguishing prediction from singlefield, slow-roll inflation Large inflationary NG can be obtained in DBI inflation, modulon/dgz mechanism & curvaton Different shape for 3-pt function * f NL 1 O(1) generically Ω σ * Identical shape for 3-pt * Distinguishable with 4-pt function

16 Four-point amplitude: ζ k1 ζ k2 ζ k3 ζ k4 = (2π) 3 δ 3 ( i ki ) [T (k 1, k 2, k 3, k 4 ) + T (k 1, k 2, k 3, k 4 )] where T (k 1, k 2, k 3, k 4 ) = 1 2 τ NL {P ζ (k 1 )P ζ (k 2 )P ζ (k 14 ) + 23 perm.} ; T (k 1, k 2, k 3, k 4 ) = κ NL {P ζ (k 1 )P ζ (k 2 )P ζ (k 3 ) + 3 perm.} Therefore have two shape parameters: τ NL & κ NL Both are generically large: τ NL, κ NL f 2 NL 10 4 WMAP (estimated): Planck: τ NL < 10 4 τ NL < 600 Kogo & Komatsu (2006)

17 Explicitly: τ NL = 1 16( θ) 4 ɛ 2 ( α23( θ) 2 ) κ NL = α 3(2α 3 θ 1) + θα 4 12( θ) 3 ɛ 2. ; α i where recall that arise from self-interactions ( V (φ, χ) = V 0 e φ/λ 1 + χ2 2Λ 2 + α 3 χ 3 ) 3! Λ By contrast, simplest curvaton models rely on a (free) massive scalar field to generate entropy perturbations α i are zero τ NL f 2 NL (as in our case) κ NL f NL (much smaller) Byrnes, Sasaki & Wands (2006) Hence, shape of 4-pt is generically different

18 New Ekpyrotic Ghost: Naively there is a ghost: Kallosh, Kang, Linde & Mukhanov, hep-th/ L = 1 2 ( φ) Λ 2 ( φ) L = 1 2 ( φ)2 µ χ µ φ 1 2 Λ2 χ L ψ φ + χ which reproduces original upon integrating out. Can diagonalize kinetic term by shifting : L = 1 2 ( ψ) ( χ)2 1 2 Λ2 χ Creminelli et al, hep-th/ However, mass of ghost is at the cutoff, so can t trust it. New physics at scale Λ can remove it. χ e.g. L = 1 2 ( φ)2 µ χ µ φ ( χ) Λ2 χ 2 L = 1 2 ( ψ)2 3 2 ( χ)2 1 2 Λ2 χ 2 ( ( /Λ Integrate out χ 2, then reproduce original L with correction O ) n )

19 Our model: L = M 4 P (X) + V (φ) ( φ)2 Λ where X = ( φ)2 2m 4 Near the bounce, dispersion relation is Ignore Ḣ ω 2 = 2ḢM 2 Pl M 4 k 2 + (ω2 k 2 ) 2 Λ term for a moment, this has two poles: ω 2 k4 2Λ 2 k2 ω 2 > Λ 2 Hence, in regime of validity of EFT, dispersion relation reduces to ω 2 2ḢM 2 Pl M 4 k 2 + k4 Λ 2 where term is important within regime and k 4 ω 2 Λ 2 k 2 Λ 2

20 Summary: New Ekpyrotic Cosmology is a candidate alternative theory of the early universe. Predictions: * Amplitude and spectral tilt: identical to inflation * NG generically much larger than slow-roll inflation * Negligible primordial gravity wave amplitude on large scales

21

22

23

24

25