S E.H. +S.F. = + 1 2! M 2(t) 4 (g ) ! M 3(t) 4 (g ) 3 + M 1 (t) 3. (g )δK µ µ M 2 (t) 2. δk µ νδk ν µ +... δk µ µ 2 M 3 (t) 2

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1 S E.H. +S.F. = d 4 x [ 1 g 2 M PlR 2 + MPlḢg 2 00 MPl(3H Ḣ) ! M 2(t) 4 (g ) ! M 3(t) 4 (g ) 3 + M 1 (t) 3 2 (g )δK µ µ M 2 (t) 2 δk µ µ 2 M 3 (t) ] δk µ νδk ν µ +... g π π a 2 ( iπ) 2 S π = d 4 x g [ M 2 PlḢ ( π 2 ( iπ) 2 a 2 ) +2M 4 2 ( π 2 + π 3 π ( ) iπ) 2 43 ] M 43 π 3 a 2 Parameters: H, Ḣ, Ḧ, M 2, M 3 Structure is set by the symmetries, the requirement that everything can be incorporated into the metric by a suitable choice of coordinates. Specific signs and coefficients in front of various terms, requirement of certain interactions, difference between time 1 and space derivatives.

2 S π = ) ( d 4 x g [ M 2 PlḢ ( π 2 ( iπ) 2 a 2 ) +2M 4 2 ( π 2 + π 3 π ( ) iπ) 2 43 ] M 43 π 3 a 2 M2 changes the dispersion relation of modes, introducing a sound speed. Same term that changes the propagation speed generates interactions. M 4 2 = 1 c2 s c 2 s M 2 Pl Ḣ 2 ω 2 = c 2 sk 2 S π = d 4 x g [ M 2 Pl Ḣ c 2 s ( ) π 2 c 2 ( i π) 2 s a 2 M 2 Pl Ḣ c 2 s π ( iπ) 2 a ] c 3 MPlḢ 2 π 3 c 4 s 2

3 Observational Consequences: 3-pt function k 1 k 3 k 2 k 6 1 ζ k1 ζ k2 ζ k3 = F ( k 2 k 1, k 3 k1 ) Higher order moments, departure from Gaussianity are sensitive to the interactions. Even after requiring scale invariance and translation invariance the three point function is still an arbitrary function of two continuous variable. 3

4 S π = d 4 x g [ M 2 PlḢ ( π 2 ( iπ) 2 a 2 ) +2M 4 2 ( π 2 + π 3 π ( ) iπ) 2 43 ] M 43 π 3 a 2 Just two specific shapes for this two shapes are predicted from each of the two possible interactions: π 3 π( i π) 2 4

5 Estimating the 3-pt function S π = d 4 x g [ M 2 Pl Ḣ c 2 s ( ) π 2 c 2 ( i π) 2 s a 2 M 2 Pl Ḣ c 2 s π ( iπ) 2 a ] c 3 MPlḢ 2 π 3 c 4 s ζ 3 ζ 2 3/2 L 3 L 2 1 ζ, E H c 2 s f NL = ζ3 ζ 2 2, f NL ζ L 3 L 2 1 ζ f E H c 2 NL 1 s c 2 s.

6 ) Two shapes are possible x ( x = x/c s ) π c =( 2MPlḢc 2 s ) 1/2 π S π = dt d 3 x g 1 2 ( ) π c 2 ( i π c ) 2 a 2 1 ( 8 Ḣ M Pl 2 c5 s ) 1/2 π c ( i π c ) 2 a c 3 ( 8 Ḣ M Pl 2 c5 s ) 1/2 π c 3 Λ 4 c 5 s Ḣ M 2 Pl c 7 sm 4 2 ζ 3 ζ 2 3/2 (H Λ )2

7 The Squeezed limit k 3 << k 1 k 2 ζ(x) =ζ g (x)+f NL ζ 2 g (x)+ Maldacena 2002 Non-Gaussianities In single clock models there is a direct connection between the departures from scale invariance and the three point function.

8 Consistency relation lim k 1 0 ζ κ 1 ζ κ 2 ζ κ 3 = (2π) 3 δ 3 ( i ki )(n s 1)P k1 P k3 ζ ki ζ kj = (2π) 3 δ 3 ( k i + k j )P ki while n s 1 is th ζζ k 3+(n s 1). k 1 k 2 k 3 ds 2 = dt 2 + e 2ζ(x) a(t) 2 dx i dx i For modes outside the horizon ζ(x)ζ(0) ζ 1 = x d ζ 1 dx ζ(x)ζ(0) ζ 1

9 The shapes in pictures 9

10 Observational Consequences: 4-pt function k 1 k 4 k 2 k 3 k 9 1 ζ k1 ζ k2 ζ k3 ζ k4 = F ( k 2 k 1, k 3 k 1, k 4 k 1, θ, φ) Higher order moments have even more freedom. Even after requiring scale invariance and translation invariance the four point function is still an arbitrary function of five continuous variable. Senatore & MZ in preparation. 10

11 Could one have an observable 4-pt function without a much larger 3-pt? S E.H. +S.F. = d 4 x [ 1 g 2 M PlR 2 + MPlḢg 2 00 MPl(3H Ḣ) ! M 2(t) 4 (g ) ! M 3(t) 4 (g ) 3 + M 1 (t) 3 2 g π π a 2 ( iπ) 2 (g )δK µ µ M 2 (t) 2 δk µ µ 2 M 3 (t) ] δk µ νδk ν µ +... Although there are quartic interactions inside (g 00 +1) 2 and (g 00 +1) 3 quartic terms, those are very small given current constraints on the three point function. 11

12 Could one have a large 4-pt function g π π a 2 ( iπ) 2 (g ) 2 contains quartic terms M2 4 ( i π) 4. τ NL = ζ 4 / ζ 2 3 τ NL ζ 2 L 4 L 2 ζ 4 ζ 2 2 L 4 L 2 1 ζ 2 τ E H c 4 NL 1 s c 4 s 1 E H c 4 s ζ 2 [f NL ζ] 1 N 1/2 pix, [τ NL ζ 2 ] 1 N 1/2 pix, Errors on the level of non-g are similar for 3 and 4 pt functions. The signal generated by these operators is very small.

13 Could one have a large 4-pt function What about starting with an operator that does not have a cubic interaction? M4 4 (δg 00 ) 4 M4 4 ( ( 16 π 4 32 π 3 ( µ π) π 2 ( µ π) 4 8 π( µ π) 6 +( µ π) 8) ζ 4 ζ 2 τ NLζ 2 L 4 2 M 4 4 E H L 2 ḢM 2 Pl ζ 2 τ NL M 4 4 ḢM 2 Pl Λ 4 U (ḢM2 Pl )2 M 4 4 (ḢM2 Pl ) τ NL τ NL ζ 2 L 4 L 2 H4 E H Λ 4 U

14 Could one have an observable 4-pt function without a much larger 3-pt? Could one have a (g 00 +1) 4 terms without (g 00 +1) 2 and (g 00 +1) 3 ones? i π π i π ( M4 4 (δg 00 ) 4 M π 4 32 π 3 ( µ π) π 2 ( µ π) 4 8 π( µ π) 6 +( µ π) 8) Although loop corrections generate 3-pt interactions they can be consistently small. M4 4 (δg 00 ) 4 ḢMPl(δg 2 00 ) 2, ḢM2 Pl(δg 00 ) 3 f NL 14 1

15 k 9 1 ζ k1 ζ k2 ζ k3 ζ k4 = F ( k 2 k 1, k 3 k 1, k 4 k 1, θ, φ) Although an arbitrary function of five continuous variable only one possibly large shape from single clock models, created by the interaction π 4 15

16 Searching for the signal The best data set is the one with the largest number of high signal to noise measurements (pixels, Fourier modes). Constraints go as WMAP and future all sky CMB experiments are the most promising. Surveys of hydrogen at high redshift using its 21 cm line could potentially do better.

17 The basics of CMB Anisotropies All 3 effects have the same origin 14 Gpc

18 Analysis Optimal weight Three point function Amplitude of primordial fluctuations Primordial 3 point function Computationally very difficult unless F is factorizable:

19 ( )( ) F π( i π) 2(k 1,k 2,k 3 )= π( iπ)2 fnl 2 Φ (24K 3 6 8K 2 2 K 3 3 K 1 8K 4 2 K K 3 3 K 3 1 6K 2 2 K K 6 1 ) K 39 K 3 1 F π 3(k 1,k 2,k 3 ) = 162 fnl π3 2 1 Φ K 33 K. 3 1 f π( iπ) 2 NL = f π3 NL = ( 1 1 ) c 2 s ( 1 1 c 2 s, )( c c2 s ) F (k, k, k) =f NL 6 2 Φ k 6 ( ) K 1 = k 1 + k 2 + k 3, K 2 = (k 1 k 2 + k 2 k 3 + k 3 k 1 ) 1/2 K 3 = (k 1 k 2 k 3 ) 1/3.

20 F (1) F (2) = k physical i F (1) (k 1,k 2,k 3 )F (2) (k 1,k 2,k 3 )/ (P k1 P k2 P k3 ) cos(f (1),F (2) )= F (1) F (2) (F (1) F (1) ) 1/2 (F (2) F (2) ) 1/2. Figure 2: Left: Cosine of single-field shape with the equilateral shape as we vary c 3 with c s 1, the regime in which it is independent of c s. The two horizontal lines represent when the scalar product is equal to ±0.7, to give a rough measure of when the cosine becomes small. Right: Cosine with the local shape.

21 Current constraints on single clock models From WMAP 5 yr data release 125 f equil NL % CL 369 f orthog NL 71 95% CL c s % CL Power spectrum Φ k1 Φ k2 Φ k3 = (2π) 3 δ (3) ( i f π( iπ) 2 NL = f π3 NL = ki )F (k 1,k 2,k 3 ) ( 1 1 ) c 2 s ( 1 1 c 2 s F (k, k, k) =f NL 6 2 Φ k 6,, )( c c2 s ) ( ) Smith, Senatore, MZ

22 One cannot put a bound on cs if only one shape is measured. 22

23 Current constraints on the local shape 4 f local NL 80 95% CL Our error bars are roughly 40% smaller than previous analysis. We outperform the previous analysis on both the large and small scales. Our results are robust to doing different cuts on the data (3 yr vs 5yr, details of the mask, range of l used). All the differences we see are consistent with being statistical. We see no evidence of foreground contamination and we are robust to the procedure used to subtract point sources. Smith, Senatore & MZ

24 Figure 1: Current constraints on f local NL. Errors in this figure and throughout the paper are 2-σ. Panel (a) best results from WMAP 5 years from the WMAP team [10] and WMAP 3 years from Yadav & Wandelt [11] together with the large scale structure results from Slosar et al [15] and the results from this paper using our optimal method (OPT). Panel (b) comparison of [10] and [11] for the same choice of analysis parameters (l max = 500, raw maps and the Kp0 mask). Panels (c) and (d) show the effect of the mask for cleaned and raw maps respectively (from [10]).

25 Ê = 1 N B l 1 l 2 l 3 ( l1 l 2 l 3 m 1 m 2 m 3 ) [ (C 1 â) l1m1(c 1 â) l2m2(c 1 â) l3m3 3C 1 l 1 m 1,l 2 m 2 (C 1 â) l3 m 3 ]

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27 Could one extract more information using other statistics? 4 Point function Other statistics have been suggested such as Minkowski functionals, various wavelets, etc. It has also been suggested that higher order terms in the expansion can be detected. Kogo & Komatsu, Hu & Okamoto One can compute the full likelihood for fnl and calculate the best constraints that can be obtained by any statistic, the Cramer-Rao bound. Basically all the information is contained in the three point function. Creminelli, Senatore and MZ

28 Construction of the Action S = d 4 x g[ 1 2 M 2 Pl R c(t)g00 Λ(t)+ 1 2! M 2(t) 4 (g ) ! M 3(t) 4 (g ) 3 + M 1 (t) 3 2 (g )δK µ µ M 2 (t) 2 δk µ µ 2 M 3 (t) ] δk µ νδk ν µ +..., H 2 = 1 3M 2 Pl ä a = Ḣ + H2 = 1 3M 2 Pl Exercises: 1. Show that the various terms are [ ] invariant under time dependent spatial c(t)+λ(t) changes in coordinates 2. Vary the action with respect to the [ ] 2c(t) Λ(t). [ metric ] to obtain background equations. Plug solution back into the [ action ] for the fluctuations. Convince yourself that the action starts quadratic. S = d 4 x g[ 1 2 M 2 Pl R + M 2 PlḢg00 M 2 Pl (3H2 + Ḣ)+ 1 2! M 2(t) 4 (g ) ! M 3(t) 4 (g ) 3 + M 1 (t) 3 2 (g )δK µ µ M 2 (t) 2 δk µ µ 2 M 3 (t) ] δk µ νδk ν µ (10)

29 Connection to scalar field examples ] [ [ ] ] Simple Examples: d 4 x g [ 1 ] 2 ( φ)2 V (φ) d 4 x g [ φ 0 (t) 2 2 g 00 V (φ 0 (t)) φ 0 (t) 2 = 2M 2 P Ḣ and V (φ(t)) = M 2 Pl (3H2 + Ḣ) ] K-Inflation L = P (X, φ), with X = g µν µ φ ν φ. S = d 4 x gp( φ 0 (t) 2 g 00, φ(t)) M 4 n(t) = φ 0 (t) 2n n P/ X n

30 Introducing the π: An analogy L = D µ φ 2 V ( φ ) 1 4 F µνf µν φ e iθ(x) φ D µ φ = µ φ + iqa µ φ A µ A µ 1 q µθ a radius of the minimum reparametrize φ φ = ρe iα/a What is the Lagrangian for the fluctuations? Call M the curvature in the radial direction.

31 L = 1 2 ( ρ)2 1 2 M 2 ρ F 2 q 2 a 2 A µ A µ ( α) aq µαa µ Massive radial excitations Vector modes Goldstone Boson φ e iθ(x) φ φ = ρe iα/a Can always choose gauge where α is 0. At low energies massive vector that has three polarization. m 2 A = q 2 a 2 However at energies above the mass of the vector the mixing terms are unimportant so talking about the Goldstone is useful. The gauge symmetry of the example is equivalent to the freedom in GR to change the time coordinate. The gauge where there is no Goldstone (the Unitary gauge) is equivalent to the gauge where there is no π α π E mix = m A ɛh

32 φ =1 φ =(π 1,π 2,,σ) σ = 1 π 2 1 π2 2 φ 2 ( π 1 ) 2 +( π 2 ) ( π π) 2 1 π 2 1 π2 2 m A < E < m A q The Goldstones interact but there is a range of energies for which they are both weakly coupled and are decoupled from the rest. In Inflation because H falls exactly in this range, the description with the Goldstones is quite simple.

33 Gravitational waves S E.H. +S.F. = d 4 x [ 1 g 2 M PlR 2 + MPlḢg 2 00 MPl(3H Ḣ) ! M 2(t) 4 (g ) ! M 3(t) 4 (g ) 3 + M 1 (t) 3 2 (g )δK µ µ M 2 (t) 2 δk µ µ 2 M 3 (t) ] δk µ νδk ν µ +... The action for gravitational wave perturbations in unchanged by the additional terms. Each polarization of the gravity waves, once properly normalized, has the same quadratic action as a massless scalar field so it will have fluctuations. M PL h µν φ h µν H M PL

34 The energy scale of inflation: gravitational waves Inflation predicts the presence of a stochastic Background of Gravitational Waves If Comparable to density perturbations Directly measure the expansion rate during Inflation. This measurement has taken a greater significance now that it appears that GW might not be observable in string-inflation.

35 Current constraints come from fitting the shape of the temperature power spectrum. Current constraints from WMAP 5 yrs r is ratio of contributions on large scales function[h, Ḣ] CMB probes a narrow range of energy scales function[ḣ,ḧ]

36 Effective field theory for Quintessence S = [ d 4 xa 3 1 ( ρq + p Q +4M 4) π (ρ Q + p Q ) ( π)2 a Ḣ(ρ Q + p Q )π (ρ Q + p Q )ḣπ M ( ) 2 ] 2 3H π 2 3Ḣπ + ḣ 2 2 π a 2. (3.1) ( ) 1/2 c 2 s H M Pl Creminelli, et al Figure 2: On the quintessential plane, we show the theoretical constraints on the equation of state and speed of sound of quintessence, in the presence of the operator M. Instability regions are dashed. Where 1+w Q and c 2 s have opposite sign we have a ghost-like instability corresponding to negative kinetic energy. For w Q < 1, the dashed regions in the left-lower panel is unstable by gradient (c 2 s H M/M 2 ) and Jeans ((1 + w Q )Ω Q 1) instabilities, while the strip close to the vertical axis corresponds to the stability window (2.29). Furthermore, the strip around the horizontal axis given in eq. (3.4) corresponds to the Ghost Condensate. Above this region, in the right-upper panel, we find standard k-essence.

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