Nonlinear Power Spectrum Emulator

Christian Wagner - September 24 2009 - Potsdam

Nonlinear Power Spectrum Nonlinear Power Spectrum EmulatorEmulator

Christian WagnerChristian Wagner

in collaboration with Katrin Heitmann, Salman Habib, in collaboration with Katrin Heitmann, Salman Habib, David Higdon, Brian Williams, Earl Lawrence (Los David Higdon, Brian Williams, Earl Lawrence (Los

Alamos), Alamos),

and Martin White (Berkeley)and Martin White (Berkeley)


(Tegmark & Zaldarriaga 2002)


MotivationMotivation

Power spectrum Power spectrum is a is a key statistic in cosmologykey statistic in cosmology

derived from the density fieldderived from the density field Cosmology dependent, including Cosmology dependent, including Dark EnergyDark Energy and and Theory of Theory of

GravityGravity Measured by various probes: Galaxy clustering (BAO), Lyman Measured by various probes: Galaxy clustering (BAO), Lyman

Alpha Forest, Cosmological Alpha Forest, Cosmological Weak lensingWeak lensing, …, … Precise theoretical predictions needed to derive unbiased Precise theoretical predictions needed to derive unbiased

cosmological parameter estimates from observational datacosmological parameter estimates from observational data Huterer & Takada 2005: Huterer & Takada 2005: 1% accuracy1% accuracy needed for near-term WL needed for near-term WL

experimentsexperiments Currently used fitting-formulas accurate to 5-10% (e.g. HaloFit by Currently used fitting-formulas accurate to 5-10% (e.g. HaloFit by

Smith et al. 2003)Smith et al. 2003) Precision N-body simulations very expensivePrecision N-body simulations very expensive MCMC needs to evaluate about 10,000 – 100,000 trial cosmologies MCMC needs to evaluate about 10,000 – 100,000 trial cosmologies

=> More than 30 years on current supercomputers=> More than 30 years on current supercomputers

)(

)(x

x

2)()( kkP


IdeaIdea

Build an Build an emulatoremulator from a “small” number of very accurate from a “small” number of very accurate N-body simulationsN-body simulations1)1) Demonstrate Demonstrate 1% accuracy1% accuracy for a single cosmology for a single cosmology

(arxiv:0812.1052)(arxiv:0812.1052)2)2) Develop Develop framework of the emulatorframework of the emulator: simulation design, : simulation design,

interpolation scheme, … (arxiv:0902:0429)interpolation scheme, … (arxiv:0902:0429)3)3) Build emulator from Build emulator from simulation suitesimulation suite and make it publicly and make it publicly

available (almost done)available (almost done) Problems:Problems:

At smaller scales (k>1 h/Mpc) At smaller scales (k>1 h/Mpc) baryonic physicsbaryonic physics becomes becomes important (White 2004, Zhang & Knox 2004, Jing et al. 2006, important (White 2004, Zhang & Knox 2004, Jing et al. 2006, Rudd et al. 2008) Rudd et al. 2008)

High-dimensional High-dimensional parameter spaceparameter space => Choice of cosmological parameters and priors=> Choice of cosmological parameters and priors

Aim:Aim: Prediction of the nonlinear matter power spectrum Prediction of the nonlinear matter power spectrum out to k ~ 1 h/Mpc with 1% accuracy between z=0 and z=1 out to k ~ 1 h/Mpc with 1% accuracy between z=0 and z=1 for for flat wCDMflat wCDM cosmologies cosmologies


Convergence tests to assure 1% Convergence tests to assure 1% accuracyaccuracy

Code comparisonCode comparison Box sizeBox size Starting redshiftStarting redshift ICs (ZA or 2LPT)ICs (ZA or 2LPT) Mass resolutionMass resolution Time steppingTime stepping Force resolutionForce resolution

~1 Gpc/h box~1 Gpc/h boxwith 1024with 102433 particlesparticleszzstartstart~200 with ZA~200 with ZA


Cosmic Calibration Cosmic Calibration FrameworkFramework

Flat wCDM cosmologies: w, Flat wCDM cosmologies: w, mm, , bb, n, nss, and , and 88 Priors from CMB and other probesPriors from CMB and other probes Hubble constant determined by CMB constraint: Hubble constant determined by CMB constraint: llAA==ddlsslss/r/rss=302.4 (WMAP5)=302.4 (WMAP5)

Sampling the parameter spaceSampling the parameter space Grid: e.g. 3Grid: e.g. 355=243 (not small), only 3 values per dimension=243 (not small), only 3 values per dimension Random sampling produces clusters and voids in the Random sampling produces clusters and voids in the

parameter spaceparameter space Orthogonal Array – Latin Hypercube sampling: Orthogonal Array – Latin Hypercube sampling:

space filling and good sampling in projected dimensionsspace filling and good sampling in projected dimensions Interpolation scheme: PC decomposition, Gaussian Interpolation scheme: PC decomposition, Gaussian

Process modelingProcess modeling


37 cosmological models37 cosmological models


Performance of the interpolation Performance of the interpolation schemescheme

HaloFit used as a proxy for the simulationsHaloFit used as a proxy for the simulations


Coyote UniverseCoyote Universe

37 cosmological models37 cosmological models 16 low + 4 medium + 1 high-resolution 16 low + 4 medium + 1 high-resolution

simulation per model + perturbation simulation per model + perturbation theory for the largest scalestheory for the largest scales

11 outputs between z=4 and z=011 outputs between z=4 and z=0 ~ 800 simulations~ 800 simulations ~ 60 Terabyte data~ 60 Terabyte data ~ 2 million CPU-hours~ 2 million CPU-hours ~ six months on the Coyote cluster~ six months on the Coyote cluster


Holdout Test for 6 ModelsHoldout Test for 6 Models


Out-of-Sample Test (Out-of-Sample Test (CDM)CDM)


Conclusion & OutlookConclusion & Outlook

Nonlinear matter power spectrum prediction Nonlinear matter power spectrum prediction accurate to 1% out to k~1 h/Mpcaccurate to 1% out to k~1 h/Mpc

Small number (~40) of cosmological models Small number (~40) of cosmological models sufficient to cover the range of interest (5 sufficient to cover the range of interest (5 parameters)parameters)

Use Coyote Emulator instead of HaloFitUse Coyote Emulator instead of HaloFit LRG mock catalogs for BOSSLRG mock catalogs for BOSS Emulator for the mass function instead of Emulator for the mass function instead of

fitting formula?fitting formula? Extend the parameter space to non-constant Extend the parameter space to non-constant

w?w? Go beyond k = 1 h/Mpc?Go beyond k = 1 h/Mpc?

Nonlinear Power Spectrum Emulator

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