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

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

3. Newton’s theory: Non-relativistic (no c) Action at a distance, violate causality c=∞ limit of Einstein’s gravity: 0 th 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. Fully Relativistic Fully Nonlinear Weakly Relativistic Weakly Nonlinear ? Studies of Large-scale Structure Newtonian Gravity axis Linear Perturbation Background World Model axis

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

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

8. Newtonian Theory

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

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, AccelerationCurvature perturbation Perturbed expansionShear

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

15. Relativistic/Newtonian correspondences: Comoving gauge Zero-shear gauge Uniform-expansion-gauge Uniform-curvature 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) Second order: Peebles (1980)/Noh-JH (2004) Third order: JH-Noh (2005) Physical Review D 69 10411 (2004); 72 044012 (2005) Pure General Relativistic corrections (comoving-synchronous gauge) Curvature perturbation in the comoving gauge ~10 -5 (K=0, comoving gauge)

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

19. Post-Newtonian Approximation

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

21. Zero-pressure 1PN equations: Nonlinear E-conservation: Mom-conservation: Raychaudhury-eq: G 0 0 -G i i Mom-constraint: G 0 i

22. 1PN compared with Newtonian: 0PN: 1PN: 1PN v=uv=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?