A modal bispectrum estimator for the CMB bispectrum

Transcription

1 A modal bispectrum estimator for the CMB bispectrum Michele Liguori Institut d Astrophysique de Paris (IAP) Fergusson, Liguori and Shellard (2010)

2 Outline Summary of the technique 1. Polynomial modes 2. What we measure: f NL, mode spectrum, shape reconstruction Results from WMAP-5 1. Model independent: mode spectrum and bispectrum reconstr. 2. Scale independent shapes 3. Running shape: feature in the inflaton potential Extension to Planck Summary

3 Bispectrum domain l 3 squeezed equilateral l max l 2 l 2 l 3 flat flat l 1 l 1 l max B! 1! 2! 3 = "! 1! 2! 3 % ( m $ ' a 1 m # m 1 m 2 m! 3 & 1 a 2 m! 2 a 3! 3 ; b! 1! 2! 3 = h! 1! 2! 3 B! 1! 2! 3

4 Mode expansion We define the scalar product: f,g = " w 3 (! 1,! 2,! 3 ) f (! 1,! 2,! 3 )g! 1,! 2,! 3! 1! 2! 3 ( ) We expand the bispectrum in terms of separable orthonormal functions defined in the shaded domain (tetrapyd) with scalar product above.

5 Mode estimation = β 0 + β 1 + β 2 + Goal: for a given dataset, extract best-fit β i, i=1,,p The basis elements pictured on the right are by construction factorizable Apply position space cubic statistics by Komatsu, Spergel and Wandelt (2003) to each separable template on the right to estimate the amplitudes β i Orthonormal basis è β i uncorrelated (in first approx.)

6 f NL estimation Expand theoretical shape until a good level of correlation is achieved Extract mode amplitude from the data up to the highest mode in the shapes under study Correlate to get f NL ˆ f NL =! " shape #! $ obs. N

7 WMAP (5-year) mode reconstruction Fergusson, Liguori and Shellard 2010 l max = 500 V+W coadded map 31 modes Inverse variance pre-filtering

8 Local WMAP Equil. WMAP shape reconstruction using first 31 modes (distance ordering). Comparison with local and equilateral

9 WMAP f NL estimation Scale invariant shapes: ü Constant ü Local ü Warm ü Flat ü Equilateral family Separable ansatz DBI Ghost inflation Single field ü Orthogonal Running shapes: ü Sharp feature in the inflaton potential

10 Local (fnl = 39 ± 20) fnl = 54.4 ± 29.4 fnl = ± (sep.) Equilateral (fnl = 155 ± 140) Orthogonal (fnl = -214 ± 110) fnl = ± (DBI) fnl = ± (ghost) fnl = ± (single) fnl= ± 133.3

11 Flat f NL = 18.1 ± 14.9 Constant f NL = ± 116.8

12 Scale invariance breaking feature A step in the inflaton potential breaks slow-roll ü Glitches in the power spectrum ü A large NG can be generated ü Scale invariance is broken Sinusoidal running in the shape (parameters: amplitude, period, phase) # S ~ f feat. sin K & + " NL % ( $ k * ' (Chen et al. 2006)

13 Amplitude computed for all phases and several scales between l=150 and l= combinations of scale and phase

14 WMAP l*= 150 φ=0

15 Mode decomposition at Planck resolution Extend mode decomposition to much higher l max. That requires many more modes. l max : 500 è 2000 n side : 512 è 2048 p max : 7 è 18 n modes : 31 è 274 Nothing conceptually different, but numerical stability and optimization issues required several technical modifications to the original WMAP pipeline. Currently being used on Planck data.

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17 Summary ü We introduced an estimator of primordial NG based on a separable modal expansion of the bispectrum. Nice features of the modal estimator: 1. It allows to separate any shape in a general clear mathematical framework. 2. It allows model independent reconstruction of the 3-point function 3. It makes multi-shape studies faster and simpler 4. Through mode spectrum and shape reconstruction it allows a better monitoring of potential contaminants ü We applied our estimators to WMAP 5-yr data and constrained a large number of models, including first constraints on feature models ü We extended our pipeline to high angular resolutions and we are now applying it to Planck data (as well as WMAP7)

18 Bispectrum estimator A ML bispectrum estimator of f NL has been shown to be optimal ˆ f NL = 1 N "! i m i m 1 m B 1 m 2 m 3 a! 1! 1! 2! 3 m 2 m 3 a! 2 a! 3 C! 1 C! 2 C! 3 In presence of rotational invariance breaking terms f ˆ NL = 1 N " m B 1 m 2 m 3! 1! 2! 3 ( C #1 a) m 1! 1 ( C #1 a) m 2! 2 ( C #1 a) m 3! # #1 3C! 1 m 1! 2 m 2 3 ( C #1 a) m 3! 3

19 WMAP constraints WMAP 7-yrs WMAP 5-yrs Local -10 < f NL < 74-4 < f NL < 80 Equilateral -214 < f NL < < f NL < 435 Orthogonal -410 < f NL < < f NL < 71 (95% c.l)