Signatures of primordial NG in CMB and LSS. Kendrick Smith (Princeton/Perimeter) Munich, November 2012

Transcription

1 Signatures of primordial NG in CMB and LSS Kendrick Smith (Princeton/Perimeter) Munich, November 2012

2 Primordial non-g + observations 1. CMB: can we independently constrain every interesting non-gaussian signal? 2. Large-scale structure: what non-gaussian signals can be constrained, and what are the degeneracies?

3 EFT of inflation π = Goldstone boson of spontaneously broken time translations 1-1 correspondence between operators in and -like parameters (Degree-N operator shows up in N-point CMB correlation function) S π = d 4 x g( ḢM2 pl) π 2 c 2 s ( iπ) 2 a 2 S π f NL + A c 2 s π 3 t + 1 c2 s c 2 s π t ( i π) 2 a 2 Equilateral+orthogonal 3-point shapes (Senatore, KMS & Zaldarriaga 2009) +Bπ 3 ttt + Cπ ttt π 2 ijk + Higher-derivative 3-point shapes (Behbahani, Mirbabayi, Senatore & KMS to appear) +D π 4 + E π 2 ( i π) 2 + F ( i π) 2 ( j π) point shapes (Senatore & Zaldarriaga 2009) +ρ πσ + Gσ 3 + Quasi single-field inflation (Chen & Wang 2009)

4 CMB data analysis Degree-N operator O (e.g. O = π 3 or O = π 4 ) Curvature N-point function ζ k1 ζ k2 ζ kn CMB N-point function a 1 m 1 a 2 m 2 a N m N CMB estimator E = i m i a 1 m 1 a 2 m 2 a N m N N i=1 ã i m i +

5 Computational difficulties Example: π 3 interaction Computing the curvature 3-point function is straightforward... ζ k1 ζ k2 ζ k3 = 0 dτ τ 2 e (k 1+k 2 +k 3 )τ k 1 k 2 k 3 2 k 1 k 2 k 3 (k 1 + k 2 + k 3 ) 3 k 1 k 2 π 3 k 3

6 Computational difficulties...but subsequent steps look intractable in full generality: CMB three-point function: 4D oscillatory integral for each ( i,m i ) a 1 m 1 a 2 m 2 a 3 m 3 = (21 + 1)( )( ) 4π dr dk 1 dk 2 dk 3 3 i= k 2 i π j i (k i r) i (k i ) m 1 m 2 m 3 ζ k1 ζ k2 ζ k3 CMB transfer function (computed numerically) CMB estimator: number of terms in sum is O( 5 max) E = i m i a 1 m 1 a 2 m 2 a 3 m 3 ã 1 m 1 ã 2 m 2 ã 3 m 3 + observed CMB multiples (appropriately filtered)

7 Factorizability = computability Suppose the curvature 3-point function is factorizable ζ k1 ζ k2 ζ k3 = f 1 (k 1 )f 2 (k 2 )f 3 (k 3 )+5perm. Define (and precompute) α (i) 2k 2 (r) = dk π CMB three-point function is fast to compute: f i (k)j (kr) a 1 m 1 a 2 m 2 a 3 m 3 = (21 + 1)( )( ) 4π dr r 2 α (1) 1 (r)α (2) 2 (r)α (3) (r)+5perm m 1 m 2 m 3 CMB estimator is fast to evaluate: E = r 2 dr d 2 n 3 i=1 m α (i) ãmy m (n) Komatsu, Spergel & Wandelt 2003 Creminelli, Nicolis, Senatore, Tegmark & Zaldarriaga 2005 KMS & Zaldarriaga 2006

8 Making shapes factorizable Two possibilities for making shape factorizable e.g. π 3 2 shape: ζ ζ ζ = k1 k2 k3 k 1 k 2 k 3 (k 1 + k 2 + k 3 ) 3 1. approximate by a factorizable shape ( template shape ) ζ k1 ζ k2 ζ k3 i k3 i + i=j k ik 2 j 2k 1k 2 k 3 k 3 1 k3 2 k3 3 equilateral template 2. perform an algebraic magic trick, e.g. find integral representation 0 e tk 1 e tk 2 e tk 3 ζ k1 ζ k2 ζ k3 = t 2 dt k 1 k 2 k 3 KMS & Zaldarriaga 2006

9 Factorizability + Feynman diagrams Observation: for π 3 shape, the integral representation is just undoing the last step of the Feynman diagram calculation ζ k1 ζ k2 ζ k3 0 dτ τ 2 e (k 1+k 2 +k 3 )τ k 1 k 2 k 3 k 1 π 3 k 2 = 2 k 1 k 2 k 3 (k 1 + k 2 + k 3 ) 3 k 3 Generalizes to any tree diagram, e.g. 4-point estimators: Smith, Senatore & Zaldarriaga, to appear

10 Factorizability + Feynman diagrams Ultimate generalization of KSW construction: Estimator Feynman rules which go directly from the diagram to the CMB estimator external line = CMB + harmonic-space factor α (r, t)ã m vertex = r 2 dr dt N-way real-space product internal line = harmonic-space factor A (r, t, r,t ) k 1 k 2 3 e.g. π 3 E = r 2 dr dt m α (r, t)ã m Y m (n) k 3

11 Example: resonant NG Just an example to illustrate the power of this method in finding a factorizable representation... ζ k1 ζ k2 ζ k3 1 k 2 1 k2 2 k2 3 sin A log k 1 + k 2 + k 3 k + A 1 i=j k i cos A log k 1 + k 2 + k 3 k j k Hard to see how this could ever be made factorizable, but going back to the physics gives the following factorizable representation! ζ k1 ζ k2 ζ k3 Re e (1+iA)log(i+iA)+iA log k AΓ(1 + ia) 1+ ia 2 1 k 1 k 2 2 k2 3 dx e (1+iA)x g(k 1,x)g(k 2,x)g(k 3,x) + 1 k 2 2 k k 1 k 2 k perm. g(k, x) =exp[ (1 + ia)ke x ]

12 Example: quasi single field inflation S π = d 4 x g 1 2 ( π) ( σ)2 M 2 2 σ2 + ρ πσ g 3! σ3 k 1 k 2 k 1 k 2 k 3 k 4 k 3 ζ k1 ζ k2 ζ k3 gρ 3 ζ k1 ζ k2 ζ k3 ζ k4 g 2 ρ 4 Because 3-point function and 4-point function depend on different combinations of parameters, either one can have larger signal-to-noise in different parts of the QSFI parameter space

13 Data analysis to do list... For any physical shape, current machinery seems to be sufficient to do the analysis! A (possibly incomplete) to do list: Higher derivative shapes (... π 3,... π πijk, 2 ) Quartic interactions ( π 4, π 2 i π 2, i π 2 j π 2, ) Quasi single-field inflation Solid inflation Anything else...?

14 Large-scale structure Local model: ζ(x) =ζ G (x)+ 3 5 f NLζ G (x) 2 Non-Gaussian contribution to halo bias on large scales: b 1 b(k) b 0 + f NL (k/ah) 2 as k 0 P mh (k) b(k)p mm (k) P hh (k) b(k) 2 P mm (k) P hh (k) 1/n P mh (k) P mm (k) Constraints are ultimately better than the CMB Planck: LSST: σ(f NL )=5 σ(f NL ) 1

15 NG halo bias: interpretation Correlation between long-wavelength mode and small-scale power Φ > 0 Φ < 0 Three-point function is large in squeezed triangles ζ kl ζ ks ζ ks f NL 1 k 3 L k3 S k S k S k L Locally measured fluctuation amplitude depends on value of Newtonian potential σ loc 8 = σ 8 (1 + 2f NL Φ) σ loc 8 near a point x Φ(x)

16 NG halo bias: interpretation This picture naturally leads to enhanced large-scale clustering σ loc 8 = σ 8 (1 + 2f NL Φ) δn h n h = log n h log ρ m b 0 δρ m ρ m + log n h log σ 8 b 1 /2 δσ 8 σ 8 δρ m = b 0 ρ m = b 0 + b 1 + b 1 f NL Φ f NL (k/ah) 2 δρm ρ m

17 General expression for non-gaussian clustering Schematic form: P mh (k) = b 0 + b N f N+2 (k) N=1 f N (k) = ζ k ζ ζ k1 kn 1 k i P mm (k) k squeezed limit P hh (k) = g MN (k) = Baumann, Ferraro, Green & KMS 2012 b N=1 b 0 b N f N+2 (k)+ MN ζ ζ ζ k1 km k ζ 1 k N P P ki =k k j = k b M b N g M+1,N+1 (k) k P mm (k) collapsed limit

18 Example 1: bias from squeezed 4-point function g NL model: ζ = ζ G + g NL ζ 3 G k 1 k 2 k 3 Simple example of non-gaussian model whose 4-point function ζ ζ ζ ζ k1 k2 k3 k4 is large in the squeezed limit k 1 0. k 4 Scale dependence of bias is same as f loc NL model Mass and redshift dependence are different, but hard in practice to discriminate f loc NL and g NL Smith, Ferraro & LoVerde 2011

19 Example 2: stochastic bias from collapsed 4-pt τ NL model: ζ ζ ζ = 6 k1 k2 k3 5 f NL(P ζ (k 1 )P ζ (k 2 )+2perm.) P ζ ζ ζ ζ =2τ k1 k2 k3 k4 NL ζ (k 1 )P ζ (k 2 )P ζ ( k 1 + k 3 ) + 23 perm. In simple local model (ζ = ζ G f NLζ 2 G ) one has τ NL =( 6 5 f NL) 2 but in general f NL,τ NL can be independent f NL k S k S k L k 1 τ NL k 2 k 3 k 4

20 Example 2: stochastic bias from collapsed 4-pt Our general expression predicts the following: f NL P mh (k) = b 0 + b 1 (k/ah) 2 P mm (k) 25 P hh (k) = b 2 f NL 0 +2b 0 b 1 (k/ah) 2 + b2 36 τ NL 1 (k/ah) 4 P mm (k) Qualitative prediction of τ NL model: stochastic halo bias Matter and halo fields are not proportional on large scales Gives some scope for distinguishing f NL,τ NL : Different bias values inferred from P mh (k), P hh (k) Different tracer populations are not 100% correlated Even with a single population, can separate k 2,k 4 terms Smith & LoVerde 2010

21 S π = Example: quasi-single field inflation d 4 x g 1 2 ( π) ( σ)2 M 2 2 σ2 + ρ πσ g 3! σ3 α = M 2 Squeezed/collapsed limits (where ) lim ζ k ζ ζ gρ 3 L ks ks k L 0 lim ζ k ζ ζ ζ g 2 ρ 4 1 k2 k3 k4 k 1 +k k 3 α L k3+α S 1 H 2 k 1 + k 2 3 2α k1 3+α k3 3+α Prediction: non-gaussian bias has spectral index given by b(k) =b 0 + b 1 gρ 3 (k/ah) 2 α τnl = g 2 ρ 4 Prediction: bias is mostly stochastic ( relative to the square of f NL = gρ 3 ) is enhanced

22 Large-scale structure: general picture Large-scale structure constraints are best understood as precise tests of statistical homogeneity of the universe on large scales Non-Gaussian models with large squeezed limits can be interpreted as large-scale inhomogeneity in statistics of small-scale modes, e.g: 10 3 Mpc 1 large-scale correlation between density and small-scale skewness 10 3 Mpc 1 large-scale inhomogeneity in small-scale power, uncorrelated to density ( stochastic )

23 Conclusions and future outlook CMB: Can measure N-point correlation function T T T l1 l2 ln with full shape discrimination. One estimator per diagram Statistical machinery is mature but many shapes unanalyzed! Large-scale structure: Future constraints on some models (e.g. fnl loc ) better than CMB Models without squeezed limits (e.g. single field) unconstrained Difficult to separate different N-point shapes (or different values of N) but some scope for discriminating models based on spectral index of the halo bias and stochastic vs non-stochastic bias