Cosmological post-Newtonian Approximation compared with Perturbation Theory

J. Hwang KNU/KIAS 2012.02.17

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

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

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

Fully Relativistic Weakly Relativistic Newtonian Gravity axis Background World Model axis Weakly Nonlinear Linear Perturbation ?

Fully Nonlinear

PT vs. PN

Fully Relativistic

“Terra Incognita”

Numerical Relativity Weakly Relativistic Newtonian Gravity axis Background World Model axis Weakly Nonlinear Post-Newtonian (PN) Approximation Fully Nonlinear

Linear Perturbation vs. 1PN

Fully Relativistic Cosmological Nonlinear Perturbation (2 nd and 3 rd order)

“Terra Incognita”

Numerical Relativity Weakly Relativistic Newtonian Gravity axis Background World Model axis Weakly Nonlinear Linear Perturbation Cosmological 1 st order Post-Newtonian (1PN) Fully Nonlinear

Newtonian Theory

Newtonian perturbation equations:

Newtonian (0PN) metric: Mass conservation: Momentum conservation: Poisson’s equation:

By combining: To linear order:

Perturbation Theory

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

To linear order:

Perturbed Lapse, Acceleration Curvature perturbation Perturbed expansion Shear

Gauge-invariant combinations:

: A gauge-invariant density perturbation based on the comoving gauge

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)

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!

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)

The unreasonable effectiveness of Newtonian gravity in cosmology!

Pure Einstein Vishniac MN 1983 Jeong et al 2011

Jeong, Gong, Noh, JH, ApJ 722, 1(2011)

Post-Newtonian Approximation

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

Zero-pressure 1PN equations:

E-conservation: Mom-conservation: Nonlinear Raychaudhury-eq: G 0 0 -G i i Mom-constraint: G 0 i

1PN compared with Newtonian: 0PN: 1PN: 1PN

v=u

PN vs. PT

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

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

Comoving gauge:

Zero-shear gauge:

Uniform-expansion gauge:

Noh, JH, Bertschinger (2012)

For growing solution:

(Takada & Futamase, MN 1999)

Spurious mode Physical density fluctuations

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

Question

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