Non-linear Matter Bispectrum in General Relativity SG Biern Seoul National Univ. with Dr. Jeong and Dr. Gong. The Newtonian Cosmology is enough for matter collapsing?? (New Perspective on Cosmology?) (Old Perspective on Cosmology? ) (weakly)

1. Motivation Jeong et al.(2009)

Hubble Flow (Gaussian) Initial Density Perturbation Peculiar Motion 2. Introduce Perturbation Cosmology FRW

3. Introduce Perturbation quantities Shift Non-spatial Orthogonality Newtonian Potential (Gravitational Time Dilation) Curvature perturbation Isotropic Pressure Energy FluxAnisotropic Stress Energy Density( ) 4-Velocity( )

Thanks to the Diffeomorphism Invariant, We can fix temporal and spatial Gauge. 4. Gauge Fixing Temporal Spatial Mix

World-line 5. Comoving Gauge: v=0, γ=0 Time bases are not orthogonal whereas spatial bases are orthogonal. ▶ Taking perpendicular hypersurface with respect to fluid world-line. ▶ The density solution is same with the SG for linear order, we can obtain the Linear Powerspectrum from CAMB. ▶ For second order CDM perturbation equations in the CG correspond to fully Newtonian cosmology in Eulerian Coordinate(Hwang et al. 2004) ▶ Easy to solve – We have up to 4 th order solutions.

6. How to Obtain Solutions in the Comoving Gauge Fully Newtonian Perturbation Pure Relativistic Contribution ● Continuity Equation ● Euler Equation

By definition If this vanished, the spatial metric parts become flat. -> Newtonian geometry. Where Negative ! This is suppressed for sub-horizon. Over-dense corresponds to α 0 Thus, This indicates under-dense for super-horizon?

Express Higher order solutions in terms of linear solutions. Up to 2 nd order, there are no terms whose coefficient is Therefore, there is No Relativistic correction Kernel. For 3 rd order, We can split the Kernel as, For 4 th order, We can split the Kernel as, Relativistic Parts contain the Curvature perturbation φ. Hence, the relativistic parts can be characterized by

Jeong et al.(2010 Apj) 6. How About 1-loop Power-Spectrum???

7. Matter Bispectra – Tree Level

Matter Bispectra – 1-loop

8. Vector Setting

9. Tree Level Result for Several Gauges Equilateral ZSG = Zero Shear Gauge(Conformal Newtonian) β=0=γ CG = Comoving Gauge γ=0=v UCG = Uniform Curvature Gauge(Flat Slicing) φ=0=γ SG = Synchornous Gauge β=0=α FoldedSqueezed(α=100)

10. Equilateral Limit Bispectrum for the Comoving Gauge

Total Sum Bispectrum in Equilateral Limit

11. Folded Limit Bispectrum for the Comoving Gauge

Total Sum Bispectrum in Folded Limit

12. N-Body Simulation Result Equilateral LimitFolded Limit E. Sefusatti at el.(2010),

13. Loosely Squeezed Limit Bispectrum for the Comoving Gauge(α=10)

Total Sum Bispectrum in Loosely Squeezed Limit

14. Tightly Squeezed Limit Bispectrum for the Comoving Gauge(α=100)

Total Sum Bispectrum in Tightly Squeezed Limit

15. Conclusion 5. Despite of this interesting result, they are much far from observations, hence, Newtonian cosmology and Newtonian N-body simulation are valid for sub-horizon. 1.General Relativity perturbation is a necessary tool to understand our Universe especially for Large scale structure. 2. There are Gauge Issues in GR by contrast to Newtonian Gravity. 3. Thanks to the Plank data, we can treat the initial random field as a Gaussian Random field. Also, We can obtain higher order perturbation solutions in terms of Linear power spectra. 4. The Relativistic Bispectra are suppressed by Newtonian for the equilateral and folded limit. However, they dominates for very large scale in the tightly squeezed limit.