1. Feasibility of detecting dark energy using bispectrum Yipeng Jing Shanghai Astronomical Observatory Hong Guo and YPJ, in preparation

2. Exploring Dark Energy ----Physical Principles Measuring the luminosity distance---standard candles Measuring the angular distance---standard rulers Measuring the shape of a known object Measuring the dynamical evolution of the structures----linear growth factor D(z) Dynamical DE or w(z): measuring the geometry or DM dynamics at z=0—2

4. Power spectrum Bispectrum Reduced Bispectrum Density Fluctuation Definition of the bispectrum Basics about the bispectrum method to measure the linear growth factor

5. General properties of bispectrum The quantity measures the correlation of the densities at three points in space; It is vanished for Gaussian density fluctuation field; But it is generated by gravitational clustering of matter; It can be also induced by selecting the density field in a biased way (e.g. the galaxy density field)

6. Bias Relation 2nd order Perturbation Theory Q_m depends on the shape of P(k) only Can measure D(z) through measuring b_1 On sufficiently large scale

7. Why Bispectrum In principle, one can measure the growth factor by measuring the power spectrum and the bispectrum since D(z) =1/b, without relying on the assumptions on bias and dynamics etc; measure sigma_8 and DE; Bispectrum is of great use in its own right: non- Gaussian features (inflation), bias factor (galaxy formation), nonlinear evolution

8. The key problems when measuring the growth factor Nonlinear evolution of dark matter clustering; Nonlinear coupling of galaxies to dark matter; Is there any systematic bias in measuring D(z)? On which scales ? Feasibility to measure with next generation of galaxy surveys (especially for those at high redshift) ? Simulation requirement ： Large volume and high resolution

9. Cosmological N-body simulations at SHAO with 1024 3 particles (PP-PM, Jing et al. 2007) Box size (Mpc/h) M_p (M_sun/h) realizations LCDM11502.2E73 LCDM23001.8 E94 LCDM36001.5 E104 LCDM412001.2 E 114 LCDM518004.0 E 114

10. Distribution of dark matter and galaxies ---simulations Density of dark matter Galaxy distribution based on a semi-analytical model (Kang et al. 2005). Red for E and blue for S galaxies

11. Test of the 2nd order Perturbation Theory Valid on scales larger than that of k=0.1 h/Mpc (less than 10%)

12. Halo model ： not perfect but helpful

13. Halo model ： understanding the nonlinear evolution (but two-halo term sensitive to upper limit in the integral)

14. Test of the bias model Using Semi-Analytic Model of Millennium Simulation (Croton et al. 2006) to build Mock sample of “galaxies”. mock galaxies: 600 Mpc/h (3 realizations) and 1200 Mpc/h (4 realizations) 500 Mpc/h 1200 Mpc/h Millennium Simulation

15. Probability of galaxies in halos

16. Systematics: a few percent level; Non-linear Q_m used; Valid on slightly smaller scales (k<0.2 h/Mpc) Error bars need to be estimated carefully

17. b2: may tell about galaxy formation Positive for brightest galaxies (M_r<-22.5), negative for bright and faint galaxies

18. Error bars of bispectrumare comparable to the Gaussian fluctuation on large scales k<0.1 h/Mpc (Dark Matter)

19. Error bars of B_g comparable to the Gaussian case Mock galaxies

20. Preliminary conclusions 2nd perturbation theory for the bispectrum of dark matter is valid for k<0.1 Mpc/h at redshift 0 Also valid for variance Delta^2(k)<0.3 at high redshift; The bias expansion valid on slightly larger scales (about <0.1 Mpc/h) The error is close to the Gaussian one Unbiased measurement of b1 and b2, therefore, dark energy and galaxy formation, promising Feasibility study with ongoing redshift surveys, especially at high redshifts, is being undertaken; Accurate prediction for Q_m needs to be done (cf. loop-corrections, Sccocimarro et al.)