Nonlinear Power Spectrum Emulator

1. Nonlinear Power Spectrum Emulator Christian Wagner in collaboration with Katrin Heitmann, Salman Habib, David Higdon, Brian Williams, Earl Lawrence (Los Alamos), and Martin White (Berkeley)

2. (Tegmark & Zaldarriaga 2002)

3. Motivation • Power spectrum is a key statistic in cosmology derived from the density field • Cosmology dependent, including Dark Energy and Theory of Gravity • Measured by various probes: Galaxy clustering (BAO), Lyman Alpha Forest, Cosmological Weak lensing, … • Precise theoretical predictions needed to derive unbiased cosmological parameter estimates from observational data • Huterer & Takada 2005: 1% accuracy needed for near-term WL experiments • Currently used fitting-formulas accurate to 5-10% (e.g. HaloFit by Smith et al. 2003) • Precision N-body simulations very expensive • MCMC needs to evaluate about 10,000 – 100,000 trial cosmologies => More than 30 years on current supercomputers

4. Idea • Build an emulator from a “small” number of very accurate N-body simulations 1) Demonstrate 1% accuracy for a single cosmology (arxiv:0812.1052) 2) Develop framework of the emulator: simulation design, interpolation scheme, … (arxiv:0902:0429) 3) Build emulator from simulation suite and make it publicly available (almost done) • Problems: • At smaller scales (k>1 h/Mpc) baryonic physics becomes important (White 2004, Zhang & Knox 2004, Jing et al. 2006, Rudd et al. 2008) • High-dimensional parameter space => Choice of cosmological parameters and priors • Aim: Prediction of the nonlinear matter power spectrum out to k ~ 1 h/Mpc with 1% accuracy between z=0 and z=1 for flat wCDM cosmologies

5. Convergence tests to assure 1% accuracy • Code comparison • Box size • Starting redshift • ICs (ZA or 2LPT) • Mass resolution • Time stepping • Force resolution • ~1 Gpc/h box with 10243 particles zstart~200 with ZA

6. Cosmic Calibration Framework • Flat wCDM cosmologies: w, wm, wb, ns, and s8 • Priors from CMB and other probes • Hubble constant determined by CMB constraint: lA=pdlss/rs=302.4 (WMAP5) • Sampling the parameter space • Grid: e.g. 35=243 (not small), only 3 values per dimension • Random sampling produces clusters and voids in the parameter space • Orthogonal Array – Latin Hypercube sampling: space filling and good sampling in projected dimensions • Interpolation scheme: PC decomposition, Gaussian Process modeling

7. 37 cosmological models

8. Performance of the interpolation scheme HaloFit used as a proxy for the simulations

9. Coyote Universe • 37 cosmological models • 16 low + 4 medium + 1 high-resolution simulation per model + perturbation theory for the largest scales • 11 outputs between z=4 and z=0 • ~ 800 simulations • ~ 60 Terabyte data • ~ 2 million CPU-hours • ~ six months on the Coyote cluster

10. Holdout Test for 6 Models

11. Out-of-Sample Test (LCDM)

12. Conclusion & Outlook • Nonlinear matter power spectrum prediction accurate to 1% out to k~1 h/Mpc • Small number (~40) of cosmological models sufficient to cover the range of interest (5 parameters) • Use Coyote Emulator instead of HaloFit • LRG mock catalogs for BOSS • Emulator for the mass function instead of fitting formula? • Extend the parameter space to non-constant w? • Go beyond k = 1 h/Mpc?