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Cosmological Post-Newtonian Approximation with...
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Cosmological Post-Newtonian Approximation with Dark Energy
J. Hwang and H. Noh2008.05.05
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Studies of Large-scale Structure
NewtonianGravity
Linear Perturbation
?
GeneralRelativistic
WeaklyRelativistic
NewtonianLinear Weakly
NonlinearFullyNonlinear
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Perturbation Theory vs. Post-Newtonian
GeneralRelativistic
Post-NewtonianApproximation
PerturbationTheory
“Terra Incognita”Numerical Relativity
WeaklyRelativistic
NewtonianLinear Weakly
NonlinearFullyNonlinear
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Cosmology and Large-Scale StructureGeneralRelativistic
Cosmological Nonlinear Perturbation (2nd and 3rd order)
Likely to benot relevant in cosmology
Cosmological 1st order Post-Newtonian (1PN)
WeaklyRelativistic
NewtonianLinear Weakly
NonlinearFullyNonlinear
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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
2 <<cv
RcGM
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ΛCosmological constant, Λ, included.Λ appears only in the background equations!
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Relativistic/Newtonian correspondence:
Background order: Spatial curvature/Total energy
Cosmological constant
Friedmann (1922)/Milne and McCrea (1934)
Linear perturbation:Density perturbationDensity
Lifshitz (1946)/Bonnor (1957)
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Newtonian equations: Peebles (1980) Fully nonlinear!
Relativistic equations: Noh-Hwang (2004) To second order! K=0
Gravitational wavescomoving gauge
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Physical Review D, 72, 044012 (2005).
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Linear order:
Second order: K=0
Third order: K=0
Physical Review D, 72, 044012 (2005). Pure General Relativistic corrections
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ϕTo linear order: Curvature perturbation
In the comoving gauge, flat background (including Λ):
Curvature perturbation in the comoving gauge
CMB: Curvature perturbation in the zero-shear gauge
Sachs-Wolfe effect COBE, WMAP
Physical Review D, 72, 044012 (2005)
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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.
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Density power spectrum to second-order:K=0 =Λ:
Pure General Relativistic corrections (δ1·δ3)
Newtonian(δ2·δ2, δ1·δ3)
Vishniac (1983)/Noh-Hwang (2008)
Physical Review D, Submitted (2008)
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Background world model:Relativistic: Friedmann (1922)Newtonian: Milne-McCrea (1934)
Coincide for zero-pressureLinear structures:Relativistic: Lifshitz (1946)Newtonian: Bonnor (1957)
Coincide for zero-pressureSecond-order structures:Newtonian: Peebles (1980)Relativistic: Noh-Hwang (2004)
Coincide for zero-pressure, no-rotationThird-order structures: Relativistic: Hwang-Noh (2005)
Pure general relativistic correctionsδT/T ~ 10-5 factor smaller, independent of horizon
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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!
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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. Third-order perturbations→ Corrections → Small, independent of horizon
4. Background curvature→ Corrections
5. Pressure→ Relativistic even to the background and linear order
6. Rotation→ Corrections → Newtonian correspondence in the small-scale limit
7. Multi-component zero-pressure irrotational fluids→ Newtonian correspondence
8. Multi-component, third-order perturbations→ Corrections → Small, independent of horizon
Physical Review D, 76, 103527 (2007)
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Cosmological post-Newtonian metric (Chandrasekhar 1965):
Minkowski background
JCAP03 (2008) 010
Robertson-Walker background metric
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Newtonian limitNewtonian metric:
- Newtonian gravitational potentialEquations:
Newtonian, indeed!JCAP03 (2008) 010
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First Post-Newtonian equations (without taking temporal gauge):
JCAP03 (2008) 010
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Conclusion Perturbation method: Fully relativistic but weakly nonlinear.Pure relativistic third-order corrections:
PN approximation: Fully nonlinear but weakly relativistic.PN corrections:
Newtonian theory looks quite reliable in cosmological dynamics.Secular effects? Require numerical simulation.Equations are presented without taking the temporal gauge.Newtonian: action at a distance (Laplacian) → PN: propagation with speed of light (D’Alembertian)PN approximation including a scalar field as a Dark Energy? In progress.
462
2
22 1010~~~ −− −Φ
cv
cRcGM δ
52 10~~ −Φ
cvδϕ