1. TBA
TBA

2. SG Biern Seoul National Univ.
Where we live in ?
Road to Dark energy model Discrimination.
SG Biern
Seoul National Univ.

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

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

5. 3. Introduce GR perturbation Gauge Mode

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

7. 5. Comoving Gauge: v=0, γ=0 World-line
Time base and spatial bases are not orthogonal whereas spatial bases are orthogonal each other.
▶ 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 4th order solutions.

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

9. Where
By definition
If this vanished, the spatial metric parts become flat.
-> Newtonian geometry.

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

11. 5. Power-Spectrum Correlation function
: It measures the excess over random probability that two galaxy
Bernardeau.et.al(2002)

12. Bad News!!

13. 6. EdS VS ΛCDM(Newtonian)
R. Takahashi arXiv:

14. Good News!!

15. 6. EdS VS ΛCDM(Relativistic)
Biern and Gong in preparation

16. Bad News!!

18. News??

19. Mystery of General Relativity
UCG
ZSG
UEG
CG or SG
K_H
Density Solution has a gauge dependence > Powerspectra depend on gauge choices
All of these are NOT observable > Need to consider bias, SW effect, Lensing effect, etc. (J. Yoo, et. al , J. Yoo, D. Jeong, )

20. How about Bispectrum?

21. Bi-Spectra
Bernardeau.et.al(2002)

23. Bad News!!

24. Total Sum Bispectrum in Equilateral Limit
Biern et al

25. Total Sum Bispectrum in Folded Limit
Biern et al

26. Good News!!

27. Total Sum Bispectrum in Loosely Squeezed Limit
Biern et al

28. Total Sum Bispectrum in Tightly Squeezed Limit
Biern et al

29. News??

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

31. There is still the gauge dependence
Of course, these are not observable.
Sub-horizon, density solutions of many popular gauge converge to the Newtonian cosmology solution. (Except for Synchronous and uniform density gauge)

32. Higher-order density gauge transformation(2)

33. Future Work

34. The leading order bispectrum contains second order solution and relativistic second order solution is very effective for large scale.
Since the relativistic effect is sensitive to the dark energy effect, the relativistic effect considered observable bispectrum should be interesting.
Bertacca et al

35. Summary
Powerspectrum and Bispectrum in Newtonian cosmology are ineffective to distinguish dark energy model.
Relativistic Powerspectrum is sensitive to Dark energy model.
Relativistic Powerspectrum has a gauge dependence. However, with considering relativistic effect these become gauge independent. Unfortunately, they are still effective on large scale.
Due to the early splitting of Newtonian and relativistic bispectrum, we need to study the gauge invariant observable bisepctrum.

36. Time to come back home