Non-Gaussianity in the CMB. Kendrick Smith (Princeton) Whistler, April 2012

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1 Non-Gaussianity in the CMB Kendrick Smith (Princeton) Whistler, April 2012

2 Why primordial non-gaussianity? Our best observational windows on the unknown physics of inflation are: The gravity wave amplitude Deviations from Gaussian statistics Gravity wave amplitude: smoking gun of inflation (versus alternatives e.g. ekpyrosis), signature of GUT-scale physics Primordial non-gaussianity: multi-parameter space which probes different aspects of the physics of inflation : signature of multifield inflation : signature of inflaton self-interactions...

3 Single-field slow-roll inflation Consider minimally coupled scalar field φ potential: V (φ) V (φ) M 1 Pl S = d 4 x 1 g 2 ( φ)2 V (φ) with flat (slow-roll) Assume simple reheating: inflaton perturbation to adiabatic mixture of standard model particles (δφ) decays ds 2 = dt 2 + a(t) 2 e 2ζ(x,t) dx 2 adiabatic curvature In this model, the initial curvature perturbation ζ(x) is a Gaussian field within (futuristic) observational limits (Maldacena 2002)

4 Non-Gaussianity (ca. 2003) non-gaussian Occurs generically in multifield models; natural range: (e.g. curvaton model, variable reheating, New Ekpyrosis) Conversely, observation of rule out single-field inflation would 21 (WMAP7) Observational errors : 5 (Planck) f loc 1 (LSST) Main observational signature: 3-point function in squeezed triangles

5 Non-Gaussianity (ca. 2012) Three-point shapes Four-point shapes Higher derivative shapes SUSY shapes SUSY shapes Blue = WMAP constraint exists in literature Green = no WMAP constraint exists but shape has been constructed Red = shape has not been constructed yet

6 Non-Gaussianity via cubic interactions Add interaction terms to inflationary Lagrangian Toy example: massless test scalar with σ 3 interaction S = 1 dσ 2 dη d 3 xa(η) 2 ( i σ) 2 + fa(η) 2 dη small coupling constant dσ dη 3 To first order in f, non-gaussianity shows up in the 3-point function k 2 k σ σ σ f 1 k1 k2 k3 σ 3 k 3 = 0 dη η2 e (k 1+k 2 +k 3 )η k 1 k 2 k 3 2f k 1 k 2 k 3 (k 1 + k 2 + k 3 ) 3 Signal-to-noise peaks on equilateral triangles:

7 Non-Gaussianity via QFT interactions General picture: each possible interaction term in the action corresponds to a different observable signal e.g. σ 3 interaction generates equilateral three-point function Which interactions should we look for? Assuming single-field inflation and no fine-tuning or special symmetries, the most general cubic interaction is f 1 σ 3 + f 2 σ( i σ) 2 For data analysis, it s convenient to orthogonalize by defining parameters f equil,forthog f equil =1.21f f 2 f orthog =0.108f f 2 Senatore, Smith & Zaldarriaga 2009

Shapes of non-gaussianity k 1 Curvature 3-point

depends only on shape of triangle k 3 k 2 To visualize,

1 ) k1 k2 k3 equilateral k 3 k 1 flattened k 3 k 1 f

8 Shapes of non-gaussianity k 1 Curvature 3-point function ζ ζ ζ k1 k2 k3 defined for closed triangles, depends only on shape of triangle k 3 k 2 To visualize, plot k2k 2 3ζ 2 ζ ζ vs side ratios (k 2 /k 1 ), (k 3 /k 1 ) k1 k2 k3 equilateral k 3 k 1 flattened k 3 k 1 f local f equilateral f orthogonal k 2 /k 1 squeezed k 2 /k 1 k 2 /k 1

9 Data analysis k 1 k 2 Diagram π 3 Curvature bispectrum ζ k1 ζ k2 ζ k3 k 3 dη η2 e (k 1+k 2 +k 3 )η k 1 k 2 k 3 CMB bispectrum a 1 m 1 a 2 m 2 a 3 m 3 (2i + 1) dη dr η 2 r 2 µ i (η, r) 4π m 1 m 2 m 3 Optimal estimator f η 2 dη 3 r 2 dr µ (η, r)(c 1 a) m m WMAP constraint

10 General case is computationally intractible Calculating CMB bispectrum from curvature bispectrum: 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) 4D oscillatory integral for each : too slow ( 1, 2, 3 ) Computing optimal estimator from data ( a m ): f = imi a 1 m 1 a 2 m 2 a 3 m 3 (C 1 a) 1 m 1 (C 1 a) 2 m 2 (C 1 a) 3 m 3 Number of terms in sum is 5 max : too slow Smith & Zaldarriaga (2007)

11 Example: Physical shapes are tractable! interaction ζ k1 ζ k2 ζ k3 k 1 k 2 Specialized to this case, estimator can be written in tractable form: ˆf = σ 3 µ (τ,r)= τ 2 dτ 2kdk π 0 r 2 dr dτ τ 2 e (k 1+k 2 +k 3 )τ k 1 k 2 k 3 d 2 n Generalizes to any tree diagram, e.g. 4-point estimators: m j (kr)e τk (k) π 3 k 3 µ (τ,r)(c 1 a) m Y m (n) 3 Smith, Senatore & Zaldarriaga, to appear

12 Computational problem 1: large number of terms ˆf = i η 2 i ( η) j r 2 j ( r) d 2 n µ (η i,r j )(C 1 a) m 3 m Sum of many terms, corresponding to points in (η, r) plane Proposed optimization algorithm to reduce computational cost General form: N Given estimator ˆX = ˆX i and covariance matrix Cov( ˆX i, ˆX j ) i=1 Find minimal (in sense defined by Cov) subset { ˆX i1, ˆX i2,, ˆX im } Original ˆX can be written w/fewer terms: w 1 ˆXi1 + + w M ˆXiM Specialized to ˆf : Start with many points in the (η, r) plane Optimization algorithm gives small number of points, weights

13 Optimization algorithm: example ˆf = η 2 dη r 2 dr d 2 n µ (η, r)(c 1 a) m m Unoptimized estimator: terms corresponding to dense sampling in the (r, η) plane Applying optimization algorithm reduces number of integration points (or terms in the estimator) to 86 η r Smith & Zaldarriaga

Computational problem 2: C 1 CMB map = vector a C 1 a =(S + N) 1 a N = Noise covariance matrix (diagonal in pixel space) S = Signal covariance matrix (diagonal in harmonic space) Appears to require

14 Computational problem 2: C 1 CMB map = vector a C 1 a =(S + N) 1 a N = Noise covariance matrix (diagonal in pixel space) S = Signal covariance matrix (diagonal in harmonic space) Appears to require inverting by-10 6 matrix! Proposed fast multigrid algorithm for solving (S + N)x = a iteratively (similar to elliptic PDE such as 2 x = a ) WMAP: ~15 core-min Planck: ~4 core-hours Smith, Zahn, Dore & Nolta 2008

15 WMAP results: local shape First optimal analysis: f loc = 38 ± 21 (1σ) At the time, results in the literature were difficult to interpret... f loc = 32 ± 34 f loc = 87 ± 30 f loc = 55 ± 30 (Creminelli et al) (Yadav & Wandelt) (Komatsu et al) (Smith, Senatore & Zaldarriaga 2009) Optimal estimator achieves smallest error bars and ensures uniqueness of result by removing choices in data analysis (f loc Current data is consistent with single-field inflation = 0) ; Planck will reduce error bar by factor ~few

16 WMAP results: single-field shapes 400 First optimal constraints: f equil = 155 ± 140 f orthog = 149 ± 110 f orthog 0 Planck: errors smaller by ~few f equil Master result which can be compared to all single-field models Senatore, Smith & Zaldarriaga 2009

17 Case study: DBI inflation String-motivated model of inflation (Alishahiha, Silverstein & Tong) L = 1 1+f(φ)( φ) 2 g s f(φ) + V (φ) Single field model, non-g parameterized by f equil,forthog 400 f equil = 0.35 c 2 s f orthog = c 2 s f orthog 0 DBI WMAP7: c s (95% CL) Senatore, Smith & Zaldarriaga f equil

(exaggerated) Generates squeezed 3-point function where ρ = projected

18 WMAP results: gravitational lensing Apparent locations of CMB hot and cold spots are deflected by intervening large scale structure (exaggerated) Generates squeezed 3-point function where ρ = projected matter density ρ l1 T l2 T l3 T l2 ρ l1 T l3 Smith, Zahn, Doré & Nolta 2008

19 WMAP results: gravitational lensing Consider a large (~10 deg) overdense region CMB appears slightly magnified; acoustic peaks move to lower l 2 C TT overdense region underdense region Correlation between long-wavelength density CMB power spectrum C TT and small-scale is equivalent to a three-point correlation T l2 ρ l1 ρ T l3

20 WMAP results: gravitational lensing NVSS: 1.4 GHz all-sky radio survey Use galaxy counts as tracer for projected matter density CMB CMB NVSS galaxy counts (Galactic coordinates) Three-point signal detected at First detection of CMB lensing! 3.4σ Planck: few percent CMB-only measurement (4-point function) Smith, Zahn, Doré & Nolta 2008

21 Conclusions and future outlook Theory: Primordial non-gaussianity is a powerful, multifaceted probe of early universe physics. Can map QFT interactions to observable signals Do we have a complete set of signals to look for? No; new examples are still emerging... single-field: higher derivative interactions Behbahani, Mirbabayi, Senatore & Smith in prep. multi-field: SUSY shapes Chen & Wang Baumann & Green Baumann, Green, Ferraro & Smith in prep.

22 Conclusions and future outlook CMB phenomenology: Can measure N-point correlation function T l1 T l2 T ln with full shape discrimination. One estimator per diagram Optimal data analysis requires solving several algorithmic and computational problems; when dust settled, WMAP data is consistent with Gaussian statistics Many shapes remain to be analyzed! higher derivative cubic interactions four-point statistics / quartic interactions SUSY shapes Planck will dramatically improve existing constraints (Jan 2013!)

The primordial CMB 4-point function

The primordial CMB 4-point function Kendrick Smith (Perimeter) Minnesota, January 2015 Main references: Smith, Senatore & Zaldarriaga (to appear in a few days) Planck 2014 NG paper (to appear last week