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Cosmological post-Newtonian Approximation compared with Perturbation Theory

J. Hwang KNU/KIAS 2012.02.17. Cosmological post-Newtonian Approximation compared with Perturbation Theory. Question. Action at a distance. Compared with Einstein’s gravity, is Newton's gravity reliable in near horizon scale simulation?. Linear deviation from

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Cosmological post-Newtonian Approximation compared with Perturbation Theory

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  1. J. Hwang KNU/KIAS 2012.02.17 Cosmological post-Newtonian Approximation compared with Perturbation Theory

  2. Question Action at a distance Compared with Einstein’s gravity, is Newton's gravity reliable in near horizon scale simulation? Linear deviation from homogeneous-isotropic background

  3. Newton’s theory: • Non-relativistic (no c) • Action at a distance, violate causality • c=∞ limit of Einstein’s gravity: 0th post-Newtonian limit • No horizon • Static nature • No strong pressure • No strong gravity • No gravitational waves • Incomplete and inconsistent • Einstein’s gravity: • Relativistic • Strong gravity, dynamic • Simplest

  4. 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 strength of gravity • Fully nonlinear • No strong gravity situation; weakly relativistic • Valid far inside horizon

  5. Studies of Large-scale Structure Fully Relativistic ? Weakly Relativistic Newtonian Gravity axis Weakly Nonlinear Fully Nonlinear Background World Model axis Linear Perturbation

  6. PT vs. PN Fully Relativistic “Terra Incognita” Numerical Relativity Perturbation Theory (PT) Weakly Relativistic Post-Newtonian (PN) Approximation Newtonian Gravity axis Weakly Nonlinear Fully Nonlinear Background World Model axis

  7. Linear Perturbation vs. 1PN Cosmological Nonlinear Perturbation (2nd and 3rd order) Fully Relativistic “Terra Incognita” Numerical Relativity Cosmological 1st order Post-Newtonian (1PN) Weakly Relativistic Newtonian Gravity axis Weakly Nonlinear Fully Nonlinear Background World Model axis Linear Perturbation

  8. Newtonian Theory

  9. Newtonian perturbation equations: Newtonian (0PN) metric: Mass conservation: Momentum conservation: Poisson’s equation:

  10. By combining: To linear order:

  11. Perturbation Theory

  12. Metric convention:(Bardeen 1988) Spatial gauge: Bardeen, J.M. in “Particle Physics and Cosmology” edited by Fang, L., & Zee, A. (Gordon and Breach, London, 1988) p1

  13. To linear order: Perturbed Lapse, Acceleration Curvature perturbation Perturbed expansion Shear

  14. Gauge-invariant combinations: : A gauge-invariant density perturbation based on the comoving gauge

  15. Relativistic/Newtonian correspondences: Uniform-expansion-gauge Uniform-curvature gauge Comoving gauge Zero-shear gauge Perturbed density, Perturbed velocity Perturbed gravitational potential Perturbed curvature JH, Noh, Gong (2012)

  16. 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!

  17. Linear order:Lifshitz (1946)/Bonnor(1957) (comoving-synchronous gauge) Second order:Peebles (1980)/Noh-JH (2004) (K=0, comoving gauge) Third order:JH-Noh (2005) Curvature perturbation in the comoving gauge ~10-5 Pure General Relativistic corrections Physical Review D 69 10411 (2004); 72 044012 (2005)

  18. The unreasonable effectiveness of Newtonian gravity in cosmology! Vishniac MN 1983 Pure Einstein Jeong et al 2011 Jeong, Gong, Noh, JH, ApJ 722, 1(2011)

  19. Post-Newtonian Approximation

  20. Newtonian gravitational potential Minkowski background Robertson-Walker background JH, Noh, Puetzfeld, JCAP 03 010 (2008)

  21. Zero-pressure 1PN equations: Nonlinear E-conservation: Mom-conservation: Raychaudhury-eq: G00-Gii Mom-constraint: G0i

  22. 1PN compared with Newtonian: 0PN: 1PN 1PN: v=u

  23. PN vs. PT

  24. Comparison (flat background): 1PN: Linear PT:

  25. Comparison: PT PN PN: gauge-invariant PT: depends on the gauge condition

  26. Comoving gauge:

  27. Zero-shear gauge:

  28. Uniform-expansion gauge:

  29. Noh, JH, Bertschinger (2012)

  30. For growing solution: (Takada & Futamase, MN 1999) Spurious mode Physical density fluctuations

  31. Newtonian interpretation: Newtonian: Einstein: Correspondence with mixed gauges: To second-order

  32. Question Compared with Einstein’s gravity, is Newton's gravity reliable in near horizon scale simulation?

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