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Cosmological Nonlinear Perturbations
J. Hwang, H. Noh, D. Jeong,
J. Gong
Post Newtonian approximation
General
Relativistic
Fully
Nonlinear
Weakly
Relativistic
Weakly
Nonlinear
?
Linear
Perturbation
Newtonian
Gravity
Studies of Large-scale Structure
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
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
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
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?
?
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
Perturbation Theory
Noh, JH, PRD (2004)
temporal comoving (v=0) gauge
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!
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)
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)
pure Einstein
corrections
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.
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
Dashed line: negative
P13,Einstein
P22
P11
P13,Newton
PTotal
P22 + P
13,Newton
Phys. Rev. Lett. 103, 121301, arXiv:0902.4285
D. Jeong, J. Gong, H. Noh, JH, arXiv:1010.3489
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
Minimally coupled scalar field
To third-order, assuming large-scale and slow-roll:
PN Approximation
Complementary!
Minkowski background
Robertson-Walker background
Newtonian gravitational potential
JH, Noh, Puetzfeld, JCAP03 (2008) 010
Newtonian, indeed!
JH, Noh, Puetzfeld, JCAP03 (2008) 010
determined by
Einstein’s equation
1PN order
JH, Noh, Puetzfeld, JCAP03 (2008) 010
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
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
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
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