The Shape of the CMB Lensing Bispectrum

Transcription

1 The Shape of the CMB Lensing Bispectrum Duncan Hanson Berkeey Lensing Workshop, Apri 22, 2011 The shape of the CMB ensing bispectrum Lewis, Chainor, Hanson (2011) arxiv: CMB ensing and primordia non-gaussianity Hanson, Smith, Chainor, Liguori (2009) arxiv: () Apri 22, / 17

2 OUTLINE I INTRODUCTION Lensing Bispectra Overap with fnl oca II BISPECTRUM GUIDE: Significance in the high S/N imit Effect of ensing on the shape of other bispectra (particuary fnl oca) III PROSPECTS: Current status, utimate imits. Panck () Apri 22, / 17

3 THE T- CORRELATION The CMB temperature is given by T(ˆn) = (ˆn)+ ISW (ˆn)+[ ](ˆn)+..., where the ISW and ensing effects are χ ISW (ˆn) = 2 dχ Ψ(χˆn; η 0 χ) (ˆn) = 2 0 χ 0 dχ χ χ χ χ Ψ(χˆn; η 0 χ) This eads to a non-gaussian bispectrum: TTT = ISW( ) () Apri 22, / 17

4 THE E- CORRELATION There are aso ensing bispectra in poarization. No onger from cross-correation with ISW, but from overap between potentias which source and quadrupoes which source reionization E-modes (Lewis, Chainor, Hanson 2011). Recenty impemented in CAMB. z 1100 z 11 z 2 () Apri 22, / 17

5 THE LENSING BISPECTRUM The T- and E- correations are significant O(30%) on arge scaes, but fade quicky. The -induced T and E covariances are mosty at high-, whie the cross-correation is at ow-, so get a squeezed shape. T T T C Tψ /(C TT C ψψ ) 1/2 CEψ /(C EE C ψψ ) 1/ τ =0.09 τ = Lewis, Chainor, Hanson (2011) () Apri 22, / 17

6 USE OF LENSING-ISW The ensing-isw bispectrum is a (reativey) direct probe of dark energy (Sejak and Zadarriaga 1998, Godberg and Sperge 1998). Particuary good at breaking the anguar diamaeter distance degeneracy (Hu 2001). There is a significant overap with the fnl oca -type bispectrum Ψ NG ( x) = Ψ + f NL (Ψ 2 Ψ 2 ) (Smith and Zadarriaga 2006). dd (+1)C /2 E d (+1)C / Panck Idea Panck Idea EE (+1)C /2 BB (+1)C / Hu (2001) () Apri 22, / 17

7 LENSING-ISW AND f oca NL Why does the ensing-isw bispectrum project onto the f oca NL bispectrum? Lensing convergence resuts in a oca change of scae oca change of variance. corresponds to a oca change in the ampitude of the power spectrum oca fnl oca change of variance. So there is an overap between the oca and ISW bispectra (athough phases differ). The arge ampitude of the ISW-ensing bispectrum resuts in significant contamination for f oca NL. 4 b 4(+4)/µK x 108 () Apri 22, / 17

8 THINGS TO WORRY ABOUT Panck has the abiity to detect this signa (pots to come ater), but there are a few fundamenta things to worry about: In the event of a significant detection, what is the effect of signa variance? What is the effect of ensing on other bispectra (caims of arge effects from Cooray, Sarkar and Serra 2008), and is there a way to cacuate the ensing of the bispectrum non-perturbativey? Higher order ensing terms in C are known to be important for Panck, SPT (e.g. Aex s tak yesterday). ˆ ˆ () Apri 22, / 17

9 BIRDWATCHERS GUIDE TO BISPECTRUM CALCULATIONS We write the ensed temperature graphicay: T(ˆn) = (ˆn) + ISW (ˆn) + i i (ˆn) + i j ij (ˆn) +... Using the foowing gossary: Observabe fied: T Product of fieds: Gaussian fied:, ISW () Apri 22, / 17

10 SIGNAL VARIANCE In bispectrum-anguage, often think of the minimum-variance estimator for the ensing-isw ampitude as  = 1 F m 1 m 2 m 3 T 1 m 1 T 2 m 2 T 3 m 3 C TT 1 C TT C TT 3 B m 1m 2m 3 T 1 m 1 T 2 m 2 It s more natura to write  = 1 F m C T N ˆ m Tm. C TT T 3 m 3 () Apri 22, / 17

11 SIGNAL VARIANCE In bispectrum-anguage, often think of the minimum-variance estimator for the ensing-isw ampitude as  = 1 F m 1 m 2 m 3 T 1 m 1 T 2 m 2 T 3 m 3 C TT 1 C TT 2 It s more natura to write C TT 3 B m 1m 2m 3  = 1 F m C T N ˆ m Tm. C TT m ISW m () Apri 22, / 17

12 SIGNAL VARIANCE Treating ˆ as Gaussian, suggests the prescription  = 1 F m C T ˆ m Tm N C TT  = 1 F m (N C T ˆ m Tm + C )C TT From simuations, describes we the increase in variance due to signa. + (C T ) 2. m ISW m () Apri 22, / 17

13 SIGNAL VARIANCE FOR f oca NL The same picture appies to the squeezed trianges of fnl oca as we (sort of ike Munshi and Heavens 2009). Can picture as correation of two sma-scae modes, moduated by arge-scae mode. Potentia improvement on the CSZ approach (Creminei et. a. 2007, Smith et. a. 2011). Ψ Ψ Ψ () Apri 22, / 17 Ψ

14 BISPECTRUM LENSING For a squeezed bispectrum, safe to use the unensed short-eg approximation. Lensing terms generate couping between the two ong-waveength modes, e.g. = = ISW () Apri 22, / 17

15 BISPECTRUM LENSING For a squeezed bispectrum, safe to use the unensed short-eg approximation. Lensing terms generate couping between the two ong-waveength modes, e.g. = = ISW () Apri 22, / 17

16 BISPECTRUM LENSING We want to cacuate i i 1 2 i j ij 1 6 i j k ijk T + T + T +... Aways keeping one free (grayed out) to correate with ISW. Note that in rea space δ T(ˆn) = ( )[ˆn + (ˆn)] = T(ˆn), δ (ˆn) where T is the ensed gradient of the CMB. () Apri 22, / 17

17 BISPECTRUM LENSING We want to cacuate i i 1 2 i j ij 1 6 i j k ijk T + T + T +... Aways keeping one free (grayed out) to correate with ISW. In Fourier space we have (again with T( ˆn) = [ˆn + (ˆn)]) T( δ 1 ) δ( 2 ) T( 3 ) = i 2π δ( ) 2 T( 1 ) T( 1 ) 1 2π δ( )( 1 2 )C TT 1. Where the vaidity of is determined by the extent to which gradients and ensing commute. () Apri 22, / 17

18 BISPECTRUM LENSING So ensing of squeezed bispectra may be approximated simpy by accuratey ensing the ong egs, e.g. using CAMB. Works we for f oca NL. Corrects an O(10%) suppression at ow-. Aso expains the N (2) bias for ˆ m power spectrum at ow-. Sice through f oca NL Hanson et. a. (2009) () Apri 22, / 17

19 DETECTION ON DATA Currenty nu resuts on WMAP (e.g. Caabrese et. a. 2009). Utimate detection significance with T and E imited by signa variance to 9σ, athough non-inear effects coud make this higher (Mangii and Verde 2009). 1/σ TE no noise TE Panck T no noise T Panck max σ fnl σ ens Corr. f NL Bias σ marge f NL T T+E Panck T Panck T+E Lewis, Chainor, Hanson (2011) () Apri 22, / 17

20 PLANCK PROSPECTS Good match for Panck, with fu-sky coverage. Predict 4σ from temperature aone for inear ISW. Difficuty in that ow- in is aso where many scan-strategy associated systematics ive. Aso need to worry about things ike pecuiar veocity dipoe. C T /(C TT C ) 1/ ISW-ensing bispectrum (50µKarcmin, 7 FWHM) A chaenge, but exciting! () Apri 22, / 17

21 CONCLUSIONS ISW- directy traces dark energy / matter at z 2. E- is significant as we, and shoud be sensitive to even higher redshifts. Signa variance can be treated by approximating ˆ as Gaussian. non-perturbative ensed bispectra can be cacuated in the squeezed imit, using unensed short-egs. Aim for detection with Panck, which has sky-coverage and S/N for 4σ. () Apri 22, / 17