3D Matter and Halo density fields with Standard Perturbation Theory and local bias Nina Roth BCTP Workshop Bad Honnef October 4 th 2010

Introduction Tegmark et al. (2006) Galaxy power spectrum ● Statistics from galaxy density distribution ● Data from observations is better on small scales ● Contains information about the underlying dark matter (DM) distribution → Structure formation → Cosmological parameters ● DM distribution can be modeled by fluid equations → analytic solution exists only for large scales/ small densities: linear theory → full non-linear evolution: numerical simulations → simple extension to linear theory: SPT ● Final ingredient: a bias model to map DM to galaxies → very popular model: local bias

Standard perturbation theory D n (t) ~ D n (t) (Growth factor) Perturbative expansion: linear solution (P 11 ) Power spectrum: one-loop correction (P P 13 ) ● Before: one-loop corrections calculated from P 11 (e.g. ) → only power spectrum available ● Now: calculate density contrast field point-by-point on a grid → full 3D information contained in the field is available → input needed: initial density field from an N-body simulation

Goals ● Use 3rd-order SPT to calculate a DM density field ● Determine bias from DM halos in an N-body simulation ● Reconstruct halo field from SPT + bias parameters ● Compare with simulated halo density field and power spectra Provides test for both SPT and the bias model used Simulations: (Pillepich et al. 2010)

SPT matter field L=150 Mpc/h, R=12 Mpc/h, z=0L=1200 Mpc/h, R=28 Mpc/h, z=0 Comparing different expansion orders

SPT matter field L=150 Mpc/h, R=12 Mpc/h, z=0L=1200 Mpc/h, R=28 Mpc/h, z=0

Matter power spectra L=1200 Mpc/h, R=0, z=0L=150 Mpc/h, R=0, z=0

Local bias model L=150 Mpc/h, R=12 Mpc/h M h = x M סּ /h ● Fit to both non-linear matter and SPT ● Scatter: shot noise & intrinsic dispersion ● Choice of smoothing scale important ● Truncate after b 3 → to be consistent with SPT order → gives best fit (AIC) → requires smoothing

Halo density field L=150 Mpc/h, R=12 Mpc/h, z=0, M h = x M סּ /h

Halo power spectra The shot noise-term P h,22 ● P h,22 supposedly stays constant on very large scales (Heavens et al. 1998), while P h should behave like P m and fall off SPT halo field ● not really visible even in our large box ● dominated by b 2 δ 1 2

Halo power spectra Linear matter & b 1 Non-linear matter & b 1, b 2, b 3 SPT & b 1, b 2, b 3 L=150 Mpc/h, R=12 Mpc/h, z=0 The curse of local bias The local bias model is a problem! ● smoothing in k- & multiplication in x-space → halo field no longer smoothed on the same scale → not an effect of using SPT → limits usable scales to k < 0.2 h/Mpc → using smaller R allows larger k, but violates SPT assumption ( δ « 1 )

Summary ● For the matter density field, third-order SPT provides a better description than linear theory ● Linear biasing is not able to accurately reconstruct the halo field, the best result is obtained using 3 bias parameters. ● The bias parameters from the fits depend on the fitting process, but lead to comparable results in the halo power spectra ● Linear theory can be used to describe the matter power spectrum up to scales of k ~ 0.15 h/Mpc (within 1 σ of the simulations) ● Third-order SPT can be used up to k ~ 0.3 h/Mpc (within 1 σ) ● Halo power spectra from SPT and local bias are only accurate up to scales of k ~ 0.2 h/Mpc (just like linear theory + linear bias) ● Using the full non-linear matter field does not improve the halo power spectra → bias model is a problem Conclusion: One needs to be careful when using SPT and the local bias model for precision cosmology!

(Halo density field)

Method Problem: double 3D-integral for third order (t~N 3 ) Solution: third order as function of second order density and velocity divergence (Makino, Sasaki & Suto 1992)

Simulation parameters

Standard perturbation theory D n (t) ~ D n (t) (Growth factor) Perturbative expansion: linear solution Power spectrum: one-loop correction

Insert the SPT expansion: Halo density contrast field: Bias Halo density contrast: Third order expansion: Local Bias Model

Future Work Theoretical work by Heavens et. al (1998): ● large-scale/effective bias is not b 1 ● → contribution from P h,13 ● → depends on smoothing scale ● for even larger scales, P h,22 dominates ● → constant, shot-noise-like contribution ● → from peaks and troughs of density field ● I can now specifically look at the density field to understand where this behaviour may come from: ● → is there a physical origin? ● → or a relic from the SPT assumptions? b 1 =b 2 =b 3 =1, R=12 Mpc/h

Mass dependence Smith, Scoccimarro & Sheth (2007) (using ST mass function) R=12 Mpc/h b1b2b3b1b2b3

Linear bias is only valid for large scales, but our data is best for small scales. Linear theory

L=600 Mpc/h, R=50 Mpc/h contours: ±0.05, 0.15 (0.2) Halo density field

Problem: double 3D-integral for third order (t~N 3 ) Solution: third order as function of second order density and velocity divergence (Makino, Sasaki & Suto 1992) ● hierarchical method → higher orders in principle possible in similar time ● numerically more robust ● velocity divergence field to (n-1) th order as by-product