CMB bispectrum

Rikkyo University

Takashi Hiramatsu

Collaboration with Ryo Saito (Yukawa Institute for Theoretical Physics, Kyoto) Atsushi Naruko (Tokyo Institute of Technology) Misao Sasaki (Yukawa Institute for Theoretical Physics, Kyoto)

Seminar, 07 Dec 2016 @ ICG

CMB bipectrum 2/28

Many kinds of information on inflation, for example,

Temperature fluctuations of

Planck Collaboration, arXiv://1502.01589

CMB bipectrum 3/28

Gaussian :

Non-Gaussian :

3-point function (Bispectrum)

Bispectrum gives the statistical properties beyond the power specttrum,

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... parameterised by

primordial generated by non-linearity

?inflation after inflation

Unfortunately dominant ...

How large ?

New inflationChaotic inflation

Power-law inflationDBI inflationK-inflation

Hybrid inflationMSSM inflationBrane inflation...

CMB bipectrum 5/28

Collision term ofThomson scattering(only for photons)

Matsubara, “Uchuron no Butsuri” (Tokyo Univ.)

Photon/Neutrino

CDM/Baryon

Photon's Thomson scatteringterm is derived from Boltzmann eq.of baryons.

Gravity

Boltzmann eqs.

Continuity/Euler eqs.

Perturbed Einstein eqs.

Photon polarisation Massless neutrino temperature

CMB bipectrum 7/28

Integral form of Boltzmann equation

Seljak, Zaldarriaga, APJ 469 (1996) 437

suppressed by tight-couplingbetween baryons-photons

directly solving

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Integrated Sachs-Wolfe (ISW) effect

Source at LSS

CMB bipectrum 9/28

CosmoLib : Huang, JCAP 1206 (2012) 012CMBFAST : Seljak, Zaldarriaga, APJ469 (1996) 437CAMB : Lewis, Challinor, APJ538 (2000) 473

CLASS II : Blas, Lesgourgues, Tram, JCAP 1107 (2011) 034existing codes

Rel

ativ

e er

ror

from

CA

MB

(%)

parameters

CMB bipectrum 10/28

Tensor TT Tensor TE Tensor EE, BB

All kinds of spectra are consistent to those computed by CAMB with ~1%

CMB bipectrum 11/28

LinearBoltzmann eqs.

2nd-order Boltzmann eqs.

Line-of-sight formula

CAMB, CMBfast, Class, CosmoLib,...

CMBquick, SONG, CosmoLib2

2nd-orderline-of-sight formula

cmb2nd

CMBquick : Pitrou, Uzan, Bernardeau, JCAP 07 (2010) 003]SONG : Petinarri et al., JCAP 1304 (2013) 003CosmoLib2 : Huang, Vernizzi, PRL 110 (2013) 101303

CMB bipectrum 12/28

(cf. Fidler, Koyama, Pettinari, JCAP 04 (2015) 037)

R.Saito, Naruko, Hiramatsu, Sasaki, JCAP10(2014)051 [arXiv:1409.2464]

Line-of-sight is bended by the gravity potential 'curve'-of-sight

CMB bipectrum 13/28

More presicely, ....

and expanding it up to 2nd-order of the solution of geodesic eq.,

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Source x ISWSource x LensingSource x Time-delaySource x Deflection

ISW x ISW

ISW x Time-delayISW x Lensing

We find totally 7 combinations that contribute to

[Source] x [gravitational] [ISW] x [gravitational]

TD

L

D

R.Saito, Naruko, Hiramatsu, Sasaki, JCAP10(2014)051 [arXiv:1409.2464]

ISW

Temp. fluc. on LSS = 'Source'

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e.g. source x lensing

[source] x [gravitational]

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Bispectrum templates

Verde et al., MNRAS 313 (2000) L141Gangui et al., APJ 430 (1994) 447

Komatsu, Spergel, PRD63 (2001) 063002

Quantify the magnitude of NG

templatessignals

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Komatsu, Spergel, PRD63 (2001) 063002

is minimised.

local-typeequilateral-typeorthogonal-type

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Local Equilateral Orthogonal Folded

Source x ISW 1.25(-3) 1.24(0) 4.11(-2) 3.93(-1)

Source x Lensing 8.86(0) -4.57(-1) -2.83(+1) 4.35(+1)

Source x Time-delay 2.82(-1) 4.35(-1) -3.45(-1) 6.93(-1)

Source x Deflection 1.82(-2) 1.76(-1) -3.00(-1) 5.27(-1)

ISW x ISW 1.31(-4) 5.19(-2) 1.13(-1) 1.64(-3)

ISW x Lensing 7.63(-2) 1.60(-1) -6.19(-1) 1.01(0)

ISW x Time-delay -1.84(-1) -1.48(-1) 1.33(-1) -2.59(-1)

(Single-template fitting)

- Lensing effect ([Src x Lens] + [ISW x Lens]) dominates as expected.- The whole lensing effect leads to

Note :this sum is referred to as “ISW-Lensing” in many literatures.

m309e

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Remapping approarch

Neglecting the thickness of LSS

Lensed photon is expanded in terms of lensing potential

Leading contribution to lensing bispectrum

Lensing potential Hu, PRD 62 (2000) 043007

Goldberg, Spergel, PRD 59 (1999) 103002

Zaldarriaga, PRD 62 (2000) 063510

Review : Lewis, Challinor, PR 429 (2006) 1

Hanson et al., PRD 80 (2009) 083004

5 perms.

Last

-sca

tter

ing

surf

ace

CMB bipectrum 20/28

Recovery of remapping approach

5 perms.

Remapping approach

Local Equilateral Orthogonal Folded

Remapping 8.94(0) -2.40(-1) -2.91(+1) 4.48(+1)m309e

Local Equilateral Orthogonal Folded

COS (Lensing) 8.93(0) -2.97(-1) -2.89(+1) 4.45(+1)m309e

We, for the first time, justify the remapping approach as a scheme to estimatethe lensing effect. In the other words, thickness of LSS doesn't affect so much.

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A : Source or ISWB : GravitationalBispectra by tensor Curve-of-sight formula

Leading contributions

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Source x ISWSource x LensingSource x Time-delaySource x Deflection

ISW x ISW

ISW x Time-delayISW x Lensing

Source x ISWSource x LensingSource x Time-delaySource x Deflection

Source x ISWSource x LensingSource x Time-delaySource x Deflection

ISW x ISW

ISW x Time-delayISW x Lensing

ISW x Deflection

ISW x ISW

ISW x Time-delayISW x Lensing

ISW x Deflection

Totally,we have 7+7+7+8=29 kinds of fNL.

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(Single-template fitting)

m320c

PRELIMINARY

Local Equilateral Orthogonal Folded

Source x ISW -2.14(-1) 1.42(-2) 3.69(-1) -5.64(-1)

Source x Lensing -6.65(-1) 1.23(-1) 1.66(0) -2.52(0)

Source x Time-delay -3.68(-2) -1.17(-3) 5.27(-2) -8.17(-2)

Source x Deflection -1.53(-2) -5.50(-2) 1.39(-1) -2.34(-1)

ISW x ISW -1.75(-3) 1.58(-3) 9.20(-3) -1.36(-2)

ISW x Lensing -5.17(-3) -6.34(-3) 3.47(-2) -5.58(-2)

ISW x Time-delay 4.89(-2) 2.24(-3) -5.00(-2) 7.79(-2)

scalar tensor

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(Single-template fitting)

m320c

PRELIMINARY

Local Equilateral Orthogonal Folded

Source x ISW 3.38(-7) -3.32(-5) -1.34(-5) 8.46(-6)

Source x Lensing 1.09(-5) 2.71(-4) 2.31(-5) 6.40(-5)

Source x Time-delay 6.05(-5) 4.03(-5) -3.95(-5) 7.57(-5)

Source x Deflection 4.34(-9) -3.40(-5) -6.80(-6) -1.98(-6)

ISW x Lensing 1.32(-3) -3.09(-2) -2.45(-2) 2.65(-2)

ISW x Time-delay -1.12(-4) -3.53(-4) 1.57(-4) -3.72(-4)

ISW x Deflection -9.25(-5) 3.62(-3) 2.96(-3) -3.24(-3)

tensor scalar

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(Single-template fitting)

m320c

PRELIMINARY

Local Equilateral Orthogonal Folded

Source x ISW 1.16(-7) 6.96(-7) -4.67(-6) 7.47(-6)

Source x Lensing -8.31(-7) -1.97(-5) -3.01(-6) -2.59(-6)

Source x Time-delay -1.60(-5) -3.91(-7) 1.43(-5) -2.22(-5)

Source x Deflection 3.00(-7) 4.66(-5) 7.52(-6) 5.52(-6)

ISW x ISW -6.39(-5) -6.39(-4) 7.63(-4) -1.41(-3)

ISW x Lensing -7.64(-5) 1.36(-3) 1.02(-3) -1.07(-3)

ISW x Time-delay 1.39(-5) 5.79(-5) -1.77(-4) 2.94(-4)

ISW x Deflection 1.80(-5) -3.59(-3) -1.82(-3) 1.50(-3)

tensor tensor

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(Single-template fitting)

m320c

PRELIMINARY

Local Equilateral Orthogonal Folded

Scalar x Scalar 9.05(0) 1.46(0) -2.94(+1) 4.59(+1)

Scalar x Tensor -8.89(-1) 7.83(-2) 2.21(0) -3.39(0)

Tensor x Scalar 1.18(-3) -2.74(-2) -2.14(-2) 2.30(-2)

Tensor x Tensor -1.25(-4) -2.78(-3) -2.04(-4) -7.07(-4)

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New CMB Boltzmann code implemeting 'curve'-of-sight formulas

* 1st-order scalar and tensor are completed. (TT, TE, EE, BB)

* Different schemes from CAMB, but consistent within O(1)%

* Implemented “curve”-of-sight formulas (2nd-order line-of-sight) for scalar and tensor temperature fluctuations.

* Implemented Komatsu-Spergel bispectrum estimator.

* Implemented remapping approximation.

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- Implement the curve-of-sight formulas for polarisation

- Implement the 2nd-order Boltzmann equations (cf. SONG, CosmoLib2)

- 2nd-order gravitational waves, magnetic field from [1st-order]2

- y-distortion to photon's distribution function ?

To-do

Applications ?

SONG : Petinarri et al., JCAP 1304 (2013) 003CosmoLib2 : Huang, Vernizzi, PRL 110 (2013) 101303

e.g. Saga et al., PRD 91 (2015) 024030 Saga et al., PRD 91 (2015) 123510