Cosmological Nonlinear Density and Velocity Power Spectra. J. Hwang UFES Vitória November 11, 2015
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1 Cosmological Nonlinear Density and Velocity Power Spectra J. Hwang UFES Vitória November 11, 2015
2 Perturbation method: Perturbation expansion All perturbation variables are small Weakly nonlinear Strong gravity; fully relativistic Valid in all scales Fully nonlinear and Exact perturbations Post-Newtonian method: Abandon geometric spirit of GR: recover the good old absolute space and absolute time Newtonian equations of motion with GR corrections Expansion in strength of gravity Excluding TT perturbation Fully nonlinear No strong gravity; weakly relativistic Valid far inside horizon Case of the Fully nonlinear and Exact perturbations
3 Relativistic vs. Nonlinear Pert. Fully Relativistic Cosmological Nonlinear (NL) Perturbation (2 nd and 3 rd order) Fully NL and Exact Pert. Eqs. Terra Incognita Numerical Relativity Weakly Relativistic Newtonian Gravity axis Weakly Nonlinear Background World Model axis` Linear Perturbation Cosmological 1 st order Post-Newtonian (1PN) Fully Nonlinear
4 Fully NL & Exact Pert. Theory HJ & Noh, MNRAS 433 (2013) 3472 Noh, JCAP 07 (2014) 037
5 Covariant (1+3) approach The linear perturbations are so surprisingly simple that a perturbation analysis accurate to second order may be feasible using the methods of Hawking (1966). One could then judge the domain of validity of the linear treatment and, more important, gain some insight into the non-linear effects. Sachs and Wolfe (1967)
6 Convention: (Bardeen 1988) Decomposition, possible to NL order (York 1973) No TT-pert! Spatial gauge condition No anisotropic stress Spatial gauge: Temporal gauge still not taken yet! Complete spatial gauge fixing. Remaining variables are spatially gauge-invariant to fully NL order! Lose no generality! HJ & Noh, MNRAS 433 (2013) 3472; Noh, JCAP 07 (2014) 037
7 HJ & Noh, MNRAS 433 (2013) 3472; Noh, JCAP 07 (2014) 037 Metric: Energy-momentum tensor: Internal energy Four-vector: Coordinate three-velocity of fluid Fluid three-velocity measured by Eulerian observer : Lorentz factor Scalar- & vector-type decomposition:
8 Metric convention: Inverse metric: Exact! Using the ADM and the covariant formalisms the rest are simple algebra. We do not even need the connection! HJ & Noh, MNRAS 433 (2013) 3472; Noh, JCAP 07 (2014) 037
9 Fully Nonlinear Perturbation Equations without taking temporal gauge condition: Noh, JCAP 07 (2014) 037
10 Tensor-type to linear order Noh, JCAP 07 (2014) 037
11 with To Background order: ADM energy-constraint Trace of ADM propagation Covariant E-conservation Noh, JCAP 07 (2014) 037
12 To linear order without taking temporal gauge condition: (Bardeen 1988) Extrinsic curvature Definition of kappa, K i i ADM energy-conservation, G 0 0 ADM momentum-conservation, G 0 i ADM propagation, trace, G i i ADM propagation, tracefree, G i j - Energy-conservation, T c 0;c Momentum-conservation, T c i;c
13 Meanings of variables: Lapse function: Lapse Intrinsic curvature: Perturbed curvature Trace of extrinsic curvature: Trace-free extrinsic curvature: Shear Expansion scalar Acceleration vector Shear tensor Rotation tensor Perturbed expansion HJ & Noh, MNRAS 433 (2013) 3472; Noh, JCAP 07 (2014) 037
14 Temporal gauge (slicing, hypersurface): v = 0 from third order in the presence of vector type perturbation Applicable to NL orders! Fully NL formulation Not available Complete gauge fixing. Remaining variables are gauge-invariant to fully NL order!
15 Zero-pressure Irrotational Fluid HJ & Noh, MNRAS 433 (2013) 3472 Noh, JCAP 07 (2014) 037
16 Conservation equations: Energy-conservation to NL order: Covariant energy-conservation = 0 Energy-conservation ADM energy-conservation Covariant energy-conservation: HJ & Noh, MNRAS 433 (2013) 3472; Noh, JCAP 07 (2014) 037
17 Covariant energy-conservation: Comoving gauge + irrotational Zero-pressure Comoving gauge + irrotational (v i = 0) + zero-pressure: = 0, Background order Exact! To 12 th -order perturbation, say:
18 Zero-pressure fluid in the comoving gauge Exact equations: Covariant energy-conservation: Trace of ADM propagation: ADM momentum constraint: RHS = pure Einstein s gravity corrections, starting from the third order, all involving Definition of kappa + ADM momentum constraint: Identify:
19 Linear-order: Second-order: Relativistic/Newtonian correspondence to second order. This equation is valid to fully nonlinear order in Newtonian theory. Third-order: Pure relativistic correction appearing from third order. All involving φ.
20 Power spectra: JH, Jeong & Noh, arxiv:
21 Leading Nonlinear Density Power-spectrum in the Comoving gauge: Pure Einstein Vishniac MN 1983 Jeong et al 2011 Unreasonable effectiveness of Newton s gravity in cosmology! Jeong, et al., ApJ 722, 1 (2011)
22 General Relativistic Continuity and Euler equations to Third order in the Comoving Newtonian gauge: Vector Tensor Vector Tensor JH, Jeong & Noh, arxiv:
23 Nonlinear Density Power-spectrum with vector and tensor contributions: P P13 ES P E 13 P EV 13 P N 13 P N 22 P 11 P ET 13 JH, Jeong & Noh, arxiv:
24 Nonlinear Velocity Power-spectrum with vector and tensor contributions: P P13 ES P E 13 P EV 13 P N 13 P N 22 P 11 P ET 13 JH, Jeong & Noh, arxiv:
25 Newtonian Limit Chandrasekhar, ApJ (1965): 0PN, Minkowski JH, Noh & Puetzfeld, JCAP (2008): cosmological Here: as a limit of FNL&E PT JH & Noh, JCAP 04 (2013) 035
26 Infinite speed-of-light Limit in ZSG & UEG: Covariant energy-conservation: 1 ZSG 1 Subhorizon limit 1 ADM momentum-constraint: Tracefree ADM propagation:
27 Equations in Newtonian limit: Covariant energy-conservation: Covariant momentum-conservation: Trace of ADM propagation:
28 With Relativistic Pressure Infinite speed-of-light limit, except for pressure JH & Noh, JCAP 10 (2013) 054
29 Case with Relativistic Pressure: Covariant energy-conservation: Covariant momentum-conservation: Trace of ADM propagation: No pressure! Previous works were not successful in guessing the correct forms. (Whittaker 1935; McCrea 1951; Harrison 1965; Coles & Lucchin 1995; Lima et al. 1997; ; Harko 2011) JH & Noh, JCAP 04 (2013) 035
30 Post-Newtonian Approximation Chandrasekhar, ApJ (1965): 1PN, Minkowski JH, Noh & Puetzfeld, JCAP (2008): cosmological Here: as a limit of FNL&E PT Noh & JH, JCAP 08 (2013) 040
31 1PN convention: (Chandrasekhar 1965) ~Shear Identification: PT 1PN 1PN equations! (JH, Noh & Puetzfeld, JCAP 2008) Covariant E-conservation: JH, Noh & Puetzfeld, JCAP 03 (2008) 010
32 Basic 1PN Equations: Tracefree ADM propagation: Covariant energy-conservation: Covariant momentum-conservation: Fourth-order in perturbation! Trace of ADM propagation: ADM momentum-constraint: JH, Noh & Puetzfeld, JCAP 03 (2008) 010; Noh & JH, JCAP 08 (2013) 040
33 Gauge conditions: JH, Noh & Puetzfeld, JCAP 03 (2008) 010
34 Propagation speed issue Under the general gauge: Trace of ADM propagation: Propagation speed is gauge dependent! of the potential JH, Noh & Puetzfeld, JCAP 03 (2008) 010
35 Resolution using Weyl tensor:.. E ij, H ij equations: JH, Noh & Puetzfeld, JCAP 03 (2008) 010
36 Fully NL and exact cosmological pert.: 1. Formulation } MN 433 (2013) Multi-component fluids, fields arxiv: Minimally coupled scalar field JCAP 07 (2014) Newtonian limit JCAP 04 (2013) with relativistic pressure JCAP 10 (2013) PN equations JCAP 08 (2013) 040 Future extentions: 1. Anisotropic stress Relativistic magneto-hydrodynamics 2. Light propagation (geodesic, Boltzmann) 3. 2 and higher order PN equations 4. Gauge-invariant combinations 5. TT perturbation should be handled perturbatively Applications: 1. Relativistic NL perturbations 2. Fitting and Averaging 3. Backreaction 4. Relativistic cosmological numerical simulation
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