1. Gauge WonderLand :Journey to another exotic gauges SG Biern Seoul National Univ.

2. Context 1. Introduce GR Perturbation 2. Higher-Order Gauge Transformation 3. Strategy 4. Comoving Gauge 5. Derive another gauge solutions 6. Gauge dependent linear power spectra 7. Non-linear Power spectra 8. Summary

3. 1. Introduce General Relativity Perturbations

4. (Sonego and Bruni, CMP, 193 (1998), 209.) 2. Higher-Order Gauge Transformation Flow1 Flow2 Flow1

5. 3. Strategy Gauge Transformation via 2. We have a Numerical Powerspectrum solution for the synchronous(comoving) gauge. (even for Non linear order solutions for matter dominant epoch) 4. We do not need to solve the partial differential equations but need some tedious algebras. 3. By using the gauge transformation, We are going to obtain non-linear powerspectra of another gauges. 1. To study physics, we need physical observbles (e.g. Power spectrum or Bispectrum)

6. 4. Comoving gauge The comoving Gauge 1. Power Spectrum in the CG is intuitionally reasonable. 2. For second order CDM perturbation equations in the CG have Coincidence with fully Newtonian cosmology.(Hwang et al. 2004) - Advantage of the Comoving Gauge(CG)- World-line 3. Easy to solve – We have up to 3 rd order solutions(Jeong et. al 2010)

7. 5. Derive another gauge solutions The ZeroShear Gauge(ZSG) (Conformal Newtonian) The Uniform Curvature Gauge(UCG) (Flat slicing) The Uniform Expansion Gauge(UEG) CGGT Comoving Horizon wave number occurs at linear order. All of these are Gauge Invariant!!

8. 6. Gauge Dependent Linear Power Spectrums We move to the Fourier Space from the real Space via Comoving Horizon wave number z=6(MD) CG ZSG UCG UEG

9. 7. Non-linear Power Spectrum 7.1 CG Power-spectra Jeong et al.(2010 Apj) Comoving Horizon occurs at 3 rd order.

10. 7.2 Higher-order density gauge transformation(1) -From CG to ZSG- -From CG to UCG-

11. 7.3 1-loop Powerspectra CG ZSG UCG UEG Comoving Horizon wave number z=6(MD) UEG is work in progress

12. 0 7.4 Higher-order density gauge transformation(2)

13. 7.5 1-loop Powerspectra -Work in progress- But there is no hope. For observable small scale we could not distinguish which gauge we are in.

14. 8. Summary & Confusions 5. Choosing Gauge in GR means choosing coordinate system. - Anti-intuitional situation happens in some gauges. - How can we interpret? 6. As we saw, sub-horizon limit there is no way to distinguish which gauges we are in. Hence, it would have nothing to do with observations. Discussions like these are meaningless? 1.Thanks to the perturbation method, we could obtain up to 3 rd order solutions in the commoving gauge. 2. Thanks to the gauge transformation method, we also could obtain up to non-linear order solutions in another gauges without solving differential equations. 3. Thanks to the gauge transformation, we could obtain Weakly non-linear power spectra in some gauges. 4. Thanks to above, we could see gauge dependency of gauge invariant variable power spectra which is a physical observable in principle. -> A tragedy starts.

15. Sorry For confusions