non-linear boltzmann equationmctp/sciprgpgs/events/2013/cap13/... · non-linear boltzmann su, lim,...

Non-LinearBoltzmann

Su, Lim,Shellard

IntroductionMotivation

Background

Part I:2nd-OrderBoltzmannEquationSolve Boltzmann Eq.

Calculate Bispectrum

Part II:Non-linearBoltzmannEqua-tion(PublishSoon)Why High Orders?

Lensing in Boltzmann

Generic Formalism

Summary

Non-linear Boltzmann Equationfor

the Cosmic Microwave Background

Simon S.-C. Su1 Eugene A. Lim2 E.P.S. Shellard1

1Centre for Theoretical CosmologyDAMTP

University of Cambridge

2Theoretical Particle Physics and Cosmology GroupPhysics Department

Kings College London

Cosmology After Planck Workshop, 2013

This trip is supported by

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Contents

1 IntroductionMotivationBackground

2 Part I: 2nd-Order Boltzmann EquationSolve Boltzmann Eq.Calculate Bispectrum

3 Part II: Non-linear Boltzmann Equation(Publish Soon)Why High Orders?Lensing in BoltzmannGeneric Formalism

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Why Non-linear Boltzmann Equation?

High `’s regimes in current observationsE.g. ` ≤ 2500 for Planck; even higher for ACT and SPTNon-linear effects can be important

Bispectrum probes non-GaussianitiesContamination from nonlinearity of GR

Planck Paper I Planck Paper XXIV

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

What is Boltzmann Equation?

Boltzmann Equation

L[I(xA, p0, ni)] = C(xA, p0, ni)

L→ Liouville Operator, free-streaming

C→ Collision Operator, e.g. Compton scattering

Liouville Operator

∂I∂η

+

(dp0

dη

)∂I∂p0 +

(dxI

dη

)∂I∂xI +

(dni

dη

)∂I∂ni

Redshift, e.g. SW and ISW

Time-delay, including Born correction of lensing

Lensing, including lens-lens coupling

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Non-linear Effects in Different Regimes

Part I: Early-time effects

E.g. intrinsic 2nd-order photon density,photon-SW couplings, ...

Dominated by 2nd-order

Part II: Late-time effects

E.g. Sunyaev-Zeldovich effect,weak-lensing effect,time-delay effect,redshift effect,foreground contaminations,CIB, ...

Higher-order effects can beimportant

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Non-linear Effects in Different Regimes

Part I: Early-time effects


Dominated by 2nd-order

Part II: Late-time effects


Higher-order effects can beimportant

Only consider free-streaming effects at late-time

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Contents




Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

A Glimpse of Complexity

Only Einstein Field Equations here ...

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Verify Numerical Solutions in 2nd Order

Among the 10 Einstein Field Equations, 4 are redundantAs constraint equations to check the numerical consistency

Equilateral limit: k0 = 0.6; k1 = 0.6; k2 = 0.6 (Mpc−1)

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Line of Sight Approach in 2nd-Order

2nd-order photon intensity

I[II](η0, k, n) =

∫ η0

0dr e−ik·nr−τ(η)

∫dk1dk2

(2π)32

δ(k− k1 − k2)S[II]T (η, k1, k1, n)

2nd-order source function: S [II]T (η, x, n, p0)

C[II]

+ τ′I[II]−2

(dp0

dη

)[I]∂I[I]

∂p0−(

dp0

dη

)[II]∂I∂p0−2

(dxI

dη

)[I]∂I[I]

∂xI−2

(dni

dη

)[I]∂I[I]

∂ni

Compton scattering

Photon-redshift coupling

2nd-order redshifts

Time-delay

Weak lensing

How to separate early-time and late-time effects of reshifts?

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Line of Sight Approach in 2nd-Order

2nd-order photon intensity

I[II](η0, k, n) =

∫ η0

0dr e−ik·nr−τ(η)

∫dk1dk2

(2π)32

δ(k− k1 − k2)S[II]T (η, k1, k1, n)


C[II]

+ τ′I[II]−2

(dp0

dη

)[I]∂I[I]

∂p0−(

dp0

dη

)[II]∂I∂p0 ((((((((((((hhhhhhhhhhhh−2

(dxI

dη

)[I]∂I[I]

∂xI−2

(dni

dη

)[I]∂I[I]

∂ni

Compton scattering


2nd-order redshifts

Time-delay Consider in Part II

Weak lensing Consider in Part II

How to separate early-time and late-time effects of reshifts?

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Understand Physics in 2nd-Order

Source function for photon-redshift coupling

S [II]T = −2

(dp0

dη

)[I]∂I[I]

∂p0 = −2 p0(∂ηΨ[I] − ni∂IΦ[I] ) ∂I

[I]

∂p0

Numerically unstable (hierarchy problem)

Does not converge at low `’s: S[II]T (k1, k2, n, r) =

∑`m S

[II]`m (k1, k2, r)Y`m(n)

Solution: Use integrating by parts, photon intensity

∼∫ η0

0 dηe−ik1·nr S[I]T (η, k1, n)

{e−ik2·nr Φ(k2, η) +

∫ η0η dη1e−ik2·nr1 [ Φ′(k2, η1) + Ψ′(k2, η1) ]}

Identify Photon-SW and Photon-ISW couplings

Separate early-time and late-time effects

LSS Time η Now

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Convergence Check for Hierarchy Problem

Converge at ` ∼ 10

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

2nd-Order Bispectrum Around Recombination


C[II]

+ τ′I[II]−2

(dp0

dη

)[I]∂I[I]

∂p0−(

dp0

dη

)[II]∂I∂p0 ((((((((((((hhhhhhhhhhhh−2

(dxI

dη

)[I]∂I[I]

∂xI−2

(dni

dη

)[I]∂I[I]

∂ni

Compton scattering

Photon-redshift coupling Only early-time effects

2nd-order redshifts Only early-time effects

CMB Bispectrum with flat-sky and thin-shell approximations

b`1`2`3 ≈r−4

LSS

(2π)2

∫ ∞−∞

dkz1dkz

2P(k1)P(k2)

∫ 0

rLSS

dr1dr2dr3e−i(kz1r1+kz

2r2+kz3r3)

S [I]T (k1, r1)S [I]

T (k2, r2)S [II]T (k1, k2, n, r3) + 1↔ 3 + 2↔ 3

S[II]T (k1, k2, n, r) =

∑`m S

[II]`m (k1, k2, r)Y`m(n)

Include m 6= 0 modes up to `max, need to be considered outside squeezed limit

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Checking with Analytic Solution

Analytic solution in squeezed limit

b`S`L`L = 2C`S C`L + C`L C`L − CTζ`S

C`L

dln(`2LC`L )

dln`L

For `S = 10

Flat-sky is good!

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Full 2nd-Order Bispectrum Around Recombination

For ` ≤ 2000fNL ∼ 0.88 (local)fNL ∼ 5.1 (equilateral)FNL ∼ 3.19S/N ∼ 0.69 (Noiseless)Systematic bias to Planck!

S.-C. Su, E. A. Lim, and E. P. S. Shellard (2012)

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Contents




Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Importance of High-Order Effects

SW, ISW, Doppler effects in Boltzmann eq. VS lensing, time delay in geodesic eq.Study these effects in the same framework?As a consistency check for the geodesic approach, any corrections?


C[II]

+ τ′I[II]−2

(dp0

dη

)[I]∂I[I]

∂p0−(

dp0

dη

)[II]∂I∂p0−2

(dxI

dη

)[I]∂I[I]

∂xI−2

(dni

dη

)[I]∂I[I]

∂ni

Compton scattering


2nd-order redshifts

Time-delay

Lensing

Derive lensing, redshifts and time-delay from high-order Boltzmann eq.

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Lensing in Boltzmann Equation

Lensed temperature anisotropies in Dyson series

Θ(n) =∫ η0

0 dη U(η0, η, n)S[I]T (−nr, η)

where U(η0, η, n) = T[

e∫ η0η dηV(η,n)

]

LSS Time η Now

Lens-source coupling

Lens-source coupling

Lens-lens coupling

Arbitrarily high order. . .

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Extension to Time-Delay and Reshifts

Liouville operator

∂I∂η

+

(dp0

dη

)∂I∂p0 +

(dxI

dη

)∂I∂xI +

(dni

dη

)∂I∂ni

Add the potential V of time-delay and redshift in the Dyson series

Θ(n) =

∫ η0

0dη U(η0, η, n)S[I]

T (−nr, η)

U(η0, η, n) = T[


]Exhaust all couplings between lensing, time-delay and redshift systematically

Generic formalism for couplings betweenfree-streaming photons and gravitational potentials

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Summary

Solved 2nd-order Boltzmann equation numerically

Calculated 2nd-order bispectrum around recombinationS/N∼ 0.69 systematic bias to Planck

Established formalism for free-streaming photonsInclude lensing, redshift and time-delay in Boltzmann equationExhaust all couplings with gravitational potential

DONE Early-time effects


ALMOST Late-time effects


Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

If we want to extract more information fromCMB, we have to study non-linear effects!

Thank you!

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Weak Lensing in Geodesic Equation (Review)

Remapping approach

Θ(n) = Θ(n + α)

α(n) = 2∫ rLSS

0dr

r − rLSS

r rLSS∇nΨNL

W (η, x)

Known assumptions in the remapping approachBorn approximation, i.e. unperturbed light pathsLens-lens couplings ignored1st-order unlensed temperature anisotropies,i.e. Θ ≈ Θ[I]

Fixed source at the last scattering surfaceNewtonian approximation in non-linear regime,

i.e. TΨ(k, η) −→ TΨ(k, η)

√PNL

Ψ (k,η)

PΨ(k,η)

How do these two approaches relate?(

dni

dη

)∂Pcd

∂ni←→ Θ(n) = Θ(n + α)

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Weak Lensing in Boltzmann Equation

With assumptionsBorn approximation, i.e.

(dxI

dη

)∂I∂xI ≈ 0

1st-order temperature anisotropies, i.e. Θ ≈ Θ[I]

Newtonian approximation:dnidη =

∑∞N=1

1N!

(dnidη

)[N]≈ −

∑∞N=1

SijN!∂J(Ψ[N] + Φ[N]) ≡ −Sij∂J(ΨNL + ΦNL)

NEW: screen projector: ∂Sij

∂nj ≈ 0 (corrections in 3rd/higher-order only)Perform the line of sight approach iteratively

The lensed temperature anisotropies

Θ(n) =I(η0, n)

4=

∫ η0


T (−nr, η) Dyson series!

where U(η0, η, n) = T[


]V(η, n) ≡

2r∇i

n[ΨNL

W (−nr, η)]

(�n,r)i

(�n,r)i X (−nr′, η′) ≡r − r′

r′∂

∂niX (−nr′, η′)

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Checkpoint for the New Approach

(dni

dη

)∂Pcd

∂ni −→ Θ(n) =

∫ η0


T (−nr, η)

Born approximation, i.e.(

dxI

dη

)∂I∂xI ≈ 0

1st-order temperature anisotropies, i.e. Θ ≈ Θ[I]

Newtonian approximation:dnidη =

∑∞N=1

1N!

(dnidη

)[N]≈ −

∑∞N=1

SijN!∂J(Ψ[N] + Φ[N]) ≡ −Sij∂J(ΨNL + ΦNL)

NEW: screen projector: ∂Sij

∂nj ≈ 0 (corrections in 3rd/higher-order only)

Θ(n) =

∫ η0


T (−nr, η) −→ Θ(n) = Θ(n +α)

Lens-lens couplings ignored

Fixed source at the last scattering surface

How good are these approximations?

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Physical Effects in 2nd-Order Bispectrum

Main effects: intrinsic 2nd-order photon density, photon-SWcouplings and quadratic collisions

Non-LinearBoltzmann

Su, Lim,Shellard


Background





Generic Formalism

Summary

Linear VS Second-Order Boltzmann Equation

Linear 2nd-order

Gaussian Non-Gaussian

Due to primordial fluctuations Sourced by quadratic 1st-order terms

Fourier transformation: x→ k

Modes decouple

k→ Convolution on k1 and k2

Modes coupleBlack-body spectrum Spectral distortion: y-typeSVT decomposition holds SVT decomposition breaks down

More fruitful physics in non-linear orders!

non-linear boltzmann equationmctp/sciprgpgs/events/2013/cap13/... · non-linear boltzmann su, lim,...

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