April 10, Degeneracies between canonical and. non-canonical inflation. Rhiannon Gwyn, AEI Potsdam. Introduction. Non-canonical.

Transcription

1 Degeneracies April 10, 2013 [ ] Gwyn, Rummel and Westphal, Resonant non-gaussianity with equilateral properties, [ ] Gwyn, Rummel and Westphal, Relations canonical and,

2 Outline

3 Physics in the early universe What can we learn about UV physics from cosmology? Can we differentiate models of?

4 Motivation 1 Planck: local f NL severely constrained, putting pressure on multifield models (NC kinetic terms, varying c s ) relatively unconstrained 2 f equil NL 3 NC kinetic terms are also fairly generic in string theory models of 4 However, there is degeneracy noncanonical models even at the 3pt function level (Non gaussianities) 5 in [ ] and [ ] we try to understand this degeneracy better...

5 UV sensitivity of The UV-complete theory in which operates is unknown, so we take an EFT approach: Effective field theory: corrections from higherdimensional operators should be suppressed by the cut-off Λ: L eff = L relevant + n O n c n Λ n 4 However, the EFT can be sensitive to the UV physics: Eta problem: Mass dimension 6 corrections can spoil the flatness of the potential: O 6 O Mp 2 4 φ 2 Mp 2 V eff = V m2 0 φ2 + O 4 Mp 2 φ 2 < O 4 > V 0 η = Mp 2 V V O(1).

6 UV sensitivity of kinetic terms In particular, kinetic terms arise when massive degrees of freedom are integrated out: L = 1 2 ( φ) ( ρ)2 + ρ M ( φ)2 1 2 M2 ρ 2 L eff = 1 2 ( φ)2 + ( φ)4 M 4 for H M at energy scales H << M. E.g., the DBI action [Silverstein and Tong, ] [ ] L DBI = Λ 4 1 ( φ)2 Λ 4 1 V (φ) 1 2 ( φ)2 + 1 ( φ) 4 8 Λ V (φ)

7 Lagrangian A single scalar field coupled minimally to gravity (X = 1 2 φ 2 ): For example, S = d 4 x g 4 [ M 2 p 2 R 4 + p(x, φ) p can = X V (φ) p DBI = 1 ( ) 1 2f (φ)x 1 V (φ) f (φ) p Tach = V (φ) 1 2 X Λ 4 ] p K = K (φ)x + X 2 Λ 4

8 Take separable action Inflation p(x, φ) = Λ 4 S(X) V (φ). The ary solution is given by X inf (A) satisfying 2X dp Λ 4 dx = A where A = V is the noncanonicalness parameter. 3Hλ 2 NCI is attractive overshoot/icftp reduced when the NC regime is relevant [Franche, RG, Underwood and Wissanji: & ]

9 Observational signatures of NCI NCI models P(X, φ) can lead to an observable amount of nongaussianity, of the equilateral type:[chen, Huang, Kachru, Shiu: ] where c 2 s = f equil NL cs 2 ( 1 + 2X p ) 1 XX p X Potentially clear observational signature of NCI! Not yet ruled out by data...

10 New Planck bounds: [ etc] Constraints on NCI fnl local = 2.7 ± 5.8 f equil NL = 42 ± 75 fnl ortho = 25 ± 39 c s 0.02

11 Degeneracy This looks like a clear signal should be possible, but we can still have degeneracy with canonical models...

12 Field redefinitions for simple Lagrangians p(x, φ) can transform to a canonical action via a field redef eg p(x, φ) = 1 2φ 2 ( µ φ) 2 V (φ) using ψ = ln φ. for more general p(x, φ) can always transform a canonical theory to a noncanonical one via canonical s [Bean et al, ]: p = F, p = F φ φ for a generating functional F (φ, φ). However only separable NC theories with quadratic potentials can be transformed to canonical theories this way (AFAIK...). [RG, Rummel and Westphal, ]

13 Onshell Can we construct a potential V can (φ) which gives rise (in a canonical theory) to the same trajectory X inf (φ) as in the noncanonical theory?

14 Noncanonical theory: Π inf (φ) p 1 φ 3H Π = 2X p X H 2 ρ = 3M 2 p = 2Xp X p 3M 2 p Onshell Canonical theory: φ = V can(φ) 3H(φ) H 2 (φ) = V can(φ) 3 Given some X(φ) = X inf, integrate 6Xdφ = dv can Vcan V can (φ) = ( ) 2 φ 3 Vcan + dφ φ 0 2 X inf (φ )

15 DBI + Inflection point potential V inf (φ) = V 0 + λ(φ = φ 0 ) + β(φ φ 0 ) 3

16 Canon vs Noncanon 1000 can. regime 1 can. regime A Ε V Η V can. regime 1 can. regime ΕDBI 10 8 ΗDBI 10 8 Εcan ΚDBI Ηcan

17 (Non)Canonical theory: 2 1 H 2 1 s(k) = 8π 2 Mp 2 c s ɛ 2 t (k) = 2 H 2 π 2 Observables Mp 2 k=ah csk=ah n s (k) 1 = 2ɛ η csk=ah n t (k) = 2 ɛ k=ah ɛ = Ḣ ; η = ɛ H 2 Hɛ so in the canonical limit ɛ ɛ V ; η 4ɛ V 2η V Recall that cs 2 = 1 + 2X p XX p X note that time of horizon crossing is different for scalar modes in NCI

18 Comparison of Observables N e 500 can. regime 2 s DBI can n s t

19 Analytic understanding? For theories with (1) a canonical limit where V = V can and (2) a speed limit st X inf = Λ 4 R when A is large (from finite convergence radius), 2 s(k), 2 t (k), N e match when. V can V ; c s = 2R A for A 1 can have V V can and V >> V can in some intermediate regime for A cs 2 (A) = A X inf A 2X inf 1 A for X n inf = X inf (A n ). Then we get the matching condition for DBI: X DBI inf = Λ4 2 A A 2 No other working examples... DBI special?

20 What about nongaussianities??

21 Resonant NG Axionic shift symmetry will receive small periodic modulations from NP effects [Chen, Easther, Lim & Flauger and Pajer ] V (φ) = V 0 (φ) + Λ 4 cos( φ f ) G(k [ 1, k 2.k 3 ) = f res 2ɛ sin( ln K ) + ] cos() +... k 1 k 2 k 3 f k where f res = 3b 2π 8 b = Λ 4 V 0 (φ )f K = k 1 + k 2 + k 3. NG comes from δ where δ = axion decay constant. Ḧ 2HḢ 2ɛ ( ) 3/2 f in interaction term. f is the

22 Resonant NG Less than 10 % overlap with the other shapes (local, equilateral, orthogonal)

23 V (φ) = V 0 (φ) + i δ 2ɛ Multiple sources A i cos( φ + c i f i ) H = i f i 3b i cos( φ 0 + c i f i ) G(k 1, k 2, k 3 ) k 1 k 2 k 3 = i 3 2πb i 8 ( ) 3/2 2ɛ 2ɛ sin( f i f i ln K k + c i f i ) Can choose b i, f i, c i to get an overlap with a periodic equilateral shape for N = O(10) terms!

24 One-dimensional limit x 2 = k 2 k 1, x 3 = k 3 k 1, x ± = x 2 ± x 3 Resonant NG is to 1st order in f i 2ɛ a fn of x +, k 1 but not x : ( 2ɛ sin ln K ) ( 2ɛ = sin (ln(1 + x + ) + ln k ) 1 ) f i k f i k Can only reproduce NG which is predominantly a function of x +, such as equil: S eq (k 1, k 2, k 3 ) = (k 1 + k 2 k 3 )(k 1 + k 3 k 2 )(k 3 + k 2 k 1 ) k 1 k 2 k 3 S x 0 eq (x + ) = 4(x + 1) x+ 2 C(S eq, S x 0 eq ) = 0.93

25 Degeneracies Periodic approximation Need to approximate a scale-inv shape by a scale-dep x 0 shape: make a periodic generalisation of Sequil. The overlap is still considerable: per C(Sequil, Sequil ) = 0.83 Can now fourier synthesize.

26 Fourier series Fourier expansion for N = 5(left) and N = 10(right)

27 Observational constraints f i < 1; b i < 1; b i f i < ɛ ɛ f eq NL < as is, the power spectrum constraint implies a resonantly generated f equil NL O(1) if shift symmetry is collectively broken [Behbahani & Green, ] (i.e. scale invariance is protected by several independent symmetries), N pt functions are no longer hierarchically suppressed with N. Then can have f NL up to 140 without implying a large oscillation in the power spectrum for small field models expect no NG (f too large to have fourier sum)

28 Canonical/Noncanonical /Future Work We suspect the description in terms of a canonical theory may be special to the DBI case We don t know why this works (asking for fluctuations around the bg to match...) Might be able to match 3pt observables Resonant NG Possible string theory/axion monodromy embedding? (series of instanton corrections expected) Applications elsewhere?