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Cosmological Nonlinear Perturbations J. Hwang, H. Noh, D. Jeong, J. Gong

Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

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Page 1: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Cosmological Nonlinear Perturbations

J. Hwang, H. Noh, D. Jeong,

J. Gong

Page 2: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Post Newtonian approximation

Page 3: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation
Page 4: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

General

Relativistic

Fully

Nonlinear

Weakly

Relativistic

Weakly

Nonlinear

?

Linear

Perturbation

Newtonian

Gravity

Studies of Large-scale Structure

Page 5: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

General

Relativistic

Fully

Nonlinear

Weakly

Relativistic

Weakly

Nonlinear

Post-Newtonian

Approximation

Per

turb

ati

on

Th

eory

“Terra Incognita”

Numerical Relativity

Linear

Perturbation

Newtonian

Gravity

Perturbation Theory vs. Post-Newtonian

Page 6: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

General

Relativistic

Fully

Nonlinear

Weakly

Relativistic

Weakly

Nonlinear

“Terra Incognita”

Numerical Relativity

Linear

Perturbation

Newtonian

Gravity

Cosmological 1st order

Post-Newtonian (1PN)

Cosmological Nonlinear

Perturbation (2nd and 3rd order)

Cosmology and Large-Scale Structure

Page 7: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Perturbation method: Perturbation expansion.

All perturbation variables are small.

Weakly nonlinear.

Strong gravity; fully relativistic!

Valid in all scales!

Post-Newtonian method: Abandon geometric spirit of GR: recover the good old

absolute space and absolute time.

Provide GR correction terms in the Newtonian equations of motion.

Expansion in

Fully nonlinear!

No strong gravity situation; weakly relativistic.

Valid far inside horizon

1~~2

2

22

c

v

Rc

GM

c

Page 8: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

time

scale Horizon ~ ct

(~3000Mpc)

Distance between

two galaxies

(~1Mpc)

Acceleration era Radiation era Matter era

Quantum

generation

Relativistic linear stage

conserved evolution

Newtonian

Nonlinear evolution

Recombination

(~380,000yr)

Microscopic (~10-30cm)

Macroscopic (~10cm)

(~10-35sec)

radiation=matter present (~13.7Gyr)

DE era?

?

Page 9: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Origin and evolution of LSS

Quantum origin - Space-time quantum fluctuations from uncertainty pr.

- Become macroscopic due to inflation.

- Nonlinear effect? NL perturbations

Linear evolution (Relativistic) - Linear evolution of the macroscopic seeds.

- Structures are described by conserved amplitudes.

Nonlinear evolution (Newtonian) - Nonlinear evolution inside the horizon.

- Newtonian numerical computer simulation.

- Relativistic effect?

NL perturbations, PN approximation

Page 10: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Perturbation Theory

Page 11: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Noh, JH, PRD (2004)

Page 12: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

temporal comoving (v=0) gauge

Page 13: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Assumptions:

Our relativistic/Newtonian correspondence includes Λ, but

assumes:

1. Flat Friedmann background

2. Zero-pressure

3. Irrotational

4. Single component fluid

5. No gravitational waves

6. Second order in perturbations

Relaxing any of these assumptions could lead to pure general

relativistic effects!

Page 14: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Einstein’s gravity corrections to Newtonian cosmology:

1. Relativistic/Newtonian correspondence for a zero-pressure,

irrotational fluid in flat background without gravitational waves.

2. Gravitational waves → Corrections

3. Background curvature → Corrections

4. Pressure → Relativistic even to the background and linear order

5. Rotation → Corrections

→ Newtonian correspondence in the small-scale limit

6. Multi-component zero-pressure irrotational fluids

→ Newtonian correspondence

7. Third-order perturbations → Corrections

→ Small, independent of horizon

8. Multi-component, third-order perturbations → Corrections

→ Small, independent of horizon

Physical Review D, 76, 103527 (2007)

Page 15: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Linear order: Lifshitz (1946)/Bonnor(1957)

Second order: Peebles (1980)/Noh-JH (2004)

Third order: JH-Noh (2005)

Physical Review D 72 044012 (2005) Pure General Relativistic corrections

(comoving-synchronous gauge)

Curvature perturbation in the comoving gauge

~10-5

(K=0, comoving gauge)

Page 16: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

pure Einstein

corrections

Page 17: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

1. Relativistic/Newtonian correspondence to the second order

2. Pure general relativistic third-order corrections are small ~5x10-5

3. Correction terms are independent of presence of the horizon.

Page 18: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Second-order power spectrum

Physical Review D, 77,123533 (2008)

General Relativistic contribution!

Relativistic/Newtonian

132211

31

2

2

2

1

2

321

)Re(2||||||

PPP

P

EinsteinNewton

EinsteinNewton

PPP ,13,1313

,3,33

Pure Einstein

Page 19: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation
Page 20: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Dashed line: negative

P13,Einstein

P22

P11

P13,Newton

PTotal

P22 + P

13,Newton

Phys. Rev. Lett. 103, 121301, arXiv:0902.4285

Page 21: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

D. Jeong, J. Gong, H. Noh, JH, arXiv:1010.3489

Page 22: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Power spectrum

General Relativistic contribution!

Relativistic/Newtonian

Pure Einstein

132211

3

321

)()()2()()(

PPPP

PDkqkqk

EinsteinNewton

EinsteinNewton

PPP ,13,1313

,3,33

D. Jeong, J. Gong, H. Noh, JH, arXiv:1010.3489

Page 23: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Minimally coupled scalar field

Page 24: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

To third-order, assuming large-scale and slow-roll:

Page 25: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation
Page 26: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

PN Approximation

Page 27: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Complementary!

Page 28: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Minkowski background

Robertson-Walker background

Newtonian gravitational potential

JH, Noh, Puetzfeld, JCAP03 (2008) 010

Page 29: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Newtonian, indeed!

JH, Noh, Puetzfeld, JCAP03 (2008) 010

Page 30: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

determined by

Einstein’s equation

1PN order

JH, Noh, Puetzfeld, JCAP03 (2008) 010

Page 31: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

46

2

2

221010~~~

c

v

cRc

GM PN corrections:

Secular effects? Require numerical study.

Newtonian: action at a distance (Laplacian)

PN: propagation with speed of light (D’Alembertian)

Propagation speed depends on the gauge choice!

Propagation speed of the (electric and magnetic parts of) Weyl

tensor is “c”. Gravity’s propagation speed is “c”!

Exists electromagnetic analogy

Propagation speed of potential depends on the gauge choice:

Coulomb gauge vs. Lorentz gauge

Propagation speed of the field is “c”. JH, Noh, Puetzfeld, JCAP03 (2008) 010

Page 32: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Conclusion Perturbation method: Fully relativistic but weakly nonlinear.

Relativistic/Newtonian correspondence for zero-pressure to

2nd .

General relativistic corrections appear, otherwise.

Pure relativistic third-order corrections:

Pure GR correction in matter power spectrum: negligible

Leading NL correction to inflation PS: negligible

PN approximation: Fully nonlinear but weakly relativistic.

PN corrections:

Newtonian theory looks quite reliable in cosmological

dynamics. (gravitational lensing requires PN effect!)

Quantitative effects require numerical study.

46

2

2

221010~~~

c

v

cRc

GM

5

210~~

cv

Page 33: Cosmological Nonlinear Perturbationsconf.kias.re.kr/cosmology10/ppt/1st_day/J. Hwang.pdf · Gravity Cosmological 1st order Post-Newtonian (1PN) Cosmological Nonlinear Perturbation

Likely to be

not relevant

in cosmology!

“Terra Incognita”

General

Relativistic

Fully

Nonlinear

Weakly

Relativistic

Weakly

Nonlinear Linear

Perturbation

Newtonian

Gravity

Cosmology and Large-Scale Structure