Cosmological post-Newtonian Approximation compared with

Perturbation Theory

J. HwangKNU/KIAS2012.02.17

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

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

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

FullyRelativistic

FullyNonlinear

WeaklyRelativistic

WeaklyNonlinear

?

Studies of Large-scale Structure

NewtonianGravity axis

Linear Perturbation

Background World Model axis

FullyRelativistic

FullyNonlinear

WeaklyRelativistic

Post-Newtonian (PN)Approximation

Pert

urb

ati

on

Th

eory

(P

T)

“Terra Incognita”Numerical Relativity

PT vs. PN

WeaklyNonlinear

NewtonianGravity axis

Background World Model axis

FullyRelativistic

FullyNonlinear

WeaklyRelativistic

“Terra Incognita”Numerical Relativity

Cosmological 1st order Post-Newtonian (1PN)

Cosmological Nonlinear Perturbation (2nd and 3rd order)

Linear Perturbation vs. 1PN

WeaklyNonlinear

NewtonianGravity axis

Linear Perturbation

Background World Model axis

Newtonian Theory

Mass conservation:

Momentum conservation:

Poisson’s equation:

Newtonian perturbation equations:

Newtonian (0PN) metric:

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:

Comoving gauge Zero-shear gauge

Uniform-expansion-gauge Uniform-curvature gauge

Perturbed density, Perturbed velocity

Perturbed gravitational potential Perturbed curvature

JH, Noh, Gong (2012)

Relativistic/Newtonian correspondence includes Λ, but assumes:

1. Flat Friedmann background2. Zero-pressure3. Irrotational4. Single component fluid5. No gravitational waves6. Second order in perturbations

Relaxing any of these assumptions could lead to pure general relativistic effects!

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)

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

Post-Newtonian Approximation

Minkowski background

Robertson-Walker background

Newtonian gravitational potential

JH, Noh, Puetzfeld, JCAP 03 010 (2008)

Zero-pressure 1PN equations:

Nonlinear

E-conservation:

Mom-conservation:

Raychaudhury-eq:G0

0-Gii

Mom-constraint:G0

i

1PN compared with Newtonian:

0PN:

1PN: 1PN

v=u

PN vs. PT

Comparison (flat background):

1PN:

Linear PT:

Comparison: PT PN

PN: gauge-invariantPT: 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?