Forward Modelling in Cosmology Alexandre Refregier ICTP 14.5.2015 - PowerPoint PPT Presentation

Forward Modelling in Cosmology Alexandre Refregier ICTP 14.5.2015

Cosmological Probes Cosmic Microwave Background Gravitational Lensing Supernovae Galaxy Clustering

Wide-Field Instruments CMB Planck, SPT, ACT, Keck Imaging VST, DES, Pann-STARRS, LSST VIS/NIR Euclid, WFIRST, Subaru Boss, Wigglez, DESI, HETDEX Spectro LOFAR, GBT, Chimes, BINGO, GMRT, Radio BAORadio, ASKAP , MeerKAT, SKA

Impact on Cosmology Amara et al. 2008 Stage IV Stage IV+Planck Stage IV Surveys will challenge all sectors of the Stage IV+Planck Stage IV cosmological model: • Dark Energy: w p and w a with an error of 2% and 13% respectively (no prior) • Dark Matter: test of CDM paradigm, precision of 0.04eV on sum of neutrino masses (with Planck) • Initial Conditions: constrain shape of primordial power spectrum, primordial non-gaussianity • Gravity: test GR by reaching a precision of 2% on the growth exponent ( d ln m / d ln a m ) → Uncover new physics and map LSS at 0<z<2: Low redshift counterpart to CMB surveys

Challenges Current: Radiation-Matter transition High-precision Cosmology era with CMB Next stage: Matter-Dark Energy transition High-precision Cosmology with LSS surveys, different from CMB: ‣ 3D spherical geometry ‣ Multi-probe, Multi-experiments ‣ Non-gaussian, Non-Linear ‣ Systematics limited ‣ Large Data Volumes

Bayesian Parameter Estimation ‣ Bayesian inference: p( θ |y)=p(y| θ ) × p( θ )/P(y) ‣ In practice: Evaluation of p(y| θ ) is expensive, N θ is large ( ≥ 7) ‣ MCMC: produce a sample { θ i } distributed as p( θ |y) (e.g. CosmoMC Lewis & Bridle 2002, CosmoHammer, Akeret+ 2012 ) θ 2 p( θ i |D) θ 1

Forward Modelling ‣ Bayesian inference relies on the computation of the likelihood function p(y| θ ) ‣ In some situations the likelihood is unavailable or intractable (eg. non-gaussian errors, non-linear measurement processes, complex data formats such as maps or catalogues) ‣ Simulation of mock data sets may however be done through forward modelling mag r50 class ellip Angular scale 90 � 18 � 1 � 0.2 � 0.1 � 0.07 � 23.5 2.3 0.11 0.23 6000 5000 22.1 1.2 0.89 0.02 4000 D � [ µ K 2 ] 3000 24.1 3.2 0.76 0.54 2000 24.2 4.3 0.45 0.65 1000 0 2 10 50 500 1000 1500 2000 2500 22.7 3.1 0.91 0.32 Multipole moment, �

Approximate Bayesian Computation review: Turner & Zandt 2012, see also: Akeret et al. 2015 ‣ Consider reference data set y and simulation based model with parameters θ which can generate simulated data sets x ‣ Define: • Summary statistics S to compress information in the data • Distance measure ρ (S(x),S(y)) between data sets • Threshold ε for the distance measure ‣ Sample prior p( θ ) and accept sample θ * if ρ (S(x),S(y))< ε , where x is generated from model θ * ‣ ABC approximation to posterior: p( θ |y) ≃ p( θ | ρ (S(x),S(y))< ε ) ‣ Use Monte Carlo sampler with sequential ε to sample ABC posterior (eg. ABC Population Monte Carlo)

Gaussian Toy Model Iteration: ¡0, ¡ ² : ¡0.500 Iteration: ¡2, ¡ ² : ¡0.414 Iteration: ¡4, ¡ ² : ¡0.298 Iteration: ¡6, ¡ ² : ¡0.231 Akeret et al. 2015 8 8 8 8 6 6 6 6 4 4 4 4 2 2 2 2 0 0 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.6 0.8 1.0 1.2 1.4 Data set y: N samples drawn µ µ µ µ Iteration: ¡8, ¡ ² : ¡0.181 Iteration: ¡10, ¡ ² : ¡0.135 Iteration: ¡12, ¡ ² : ¡0.103 Iteration: ¡14, ¡ ² : ¡0.083 8 8 8 8 from gaussian distribution 6 6 6 6 4 4 4 4 with known σ and unknown 2 2 2 2 0 0 0 0 0.6 0.8 1.0 1.2 1.4 0.8 0.9 1.0 1.1 1.2 1.3 0.8 0.9 1.0 1.1 1.2 1.3 0.8 0.9 1.0 1.1 1.2 mean θ µ µ µ µ Iteration: ¡16, ¡ ² : ¡0.063 Iteration: ¡18, ¡ ² : ¡0.050 Iteration: ¡20, ¡ ² : ¡0.041 Iteration: ¡22, ¡ ² : ¡0.031 40 40 40 40 30 30 30 30 S ummary statistics : S(x)=<x> 20 20 20 20 10 10 10 10 0 0 0 0 0.8 0.9 1.0 1.1 1.2 0.90 0.95 1.00 1.05 1.10 0.90 0.95 1.00 1.05 1.10 0.96 0.98 1.00 1.02 1.04 D istance: ρ (x,y) = |<x>-<y>| µ µ µ µ Iteration: ¡24, ¡ ² : ¡0.024 Iteration: ¡26, ¡ ² : ¡0.020 Iteration: ¡28, ¡ ² : ¡0.016 Iteration: ¡30, ¡ ² : ¡0.012 40 40 40 40 30 30 30 30 20 20 20 20 10 10 10 10 0 0 0 0 0.96 0.98 1.00 1.02 1.04 0.96 0.98 1.00 1.02 1.04 0.96 0.98 1.00 1.02 1.04 0.96 0.98 1.00 1.02 1.04 µ µ µ µ Iteration: ¡32, ¡ ² : ¡0.010 Iteration: ¡34, ¡ ² : ¡0.010 50 50 ABC ¡PMC ABC ¡analytic 40 40 Bayesian 30 30 20 20 10 10 0 0 0.96 0.98 1.00 1.02 1.04 0.96 0.98 1.00 1.02 1.04 µ µ

Image Modelling UFig: Ultra Fast Image Generator Bergé et al. 2013, Bruderer et al. 2015 data y : SExtractor catalogue Bertin & Arnouts 1996 model: parametrised distribution of intrinsic galaxy properties UFig

ABCPMC Akeret et al. 2015 Mahalonobis distance: q ( y − µ y ) T Σ − 1 S ( y ) = y ( y − µ y ) q ( x − µ y ) T Σ − 1 S ( x ) = y ( x − µ y ) , ρ (S(x),S(y))= 1D KS distance

Monte-Carlo Control Loops Refregier & Amara 2013 'Δ'Inputs' Input' Image'Simula1ons' 0' Data' (UFig)' 2' Lensing'Measurements' Other'Diagnos1cs' Lensing' Lensing' Lensing' 3.2' Other' Other' 1' Other' 3.1'

UFig Ultra Fast Image Generator Bergé et al. 2013; Bruderer et al. 2015 DES SV UFig 7x10 6 ¡galaxies ¡(R<29) ¡ 3x10 4 ¡stars ¡ 2.5 ¡min ¡on ¡a ¡single ¡core

HOPE Akeret et al. 2014 • Just-In-Time compiler for astrophysical computations • Makes Python as fast as compiled languages • HOPE translates a Python function into C++ at runtime • Only a @jit decorator needs to be added @hope.jit def improved(x, y) : return x**2 + y**4 • Supports numerical features commonly used in astrophysical calculations For more information see: http://hope.phys.ethz.ch

MCCL: First Implementation Bruderer et al. 2015 mag 14 16 18 20 22 24 26 28 6 10 3 10 2 10 1 5 10 0 10 3 10 2 10 1 5 4 0 10 0 0 10 3 Size [pixels] 1 2 10 2 0 0 0 0 0 10 1 10000 3 10 0 10000 5000 1 0 0 10 3 10 2 1000 2000 2 0 0 5 0 10 1 0 0 2000 2 5 10 0 0 0 10 3 100 10 2 10 1 1 10 0 10 3 DES 10 2 UFig 10 1 0 10 0 6 14 16 18 20 22 24 26 28 5 mag Size [pixels] 4 3 2 1 0 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3

Tolerance Analysis Bruderer et al. 2015 0 . 10 PSF - uncorr PSF - uncorr PSF - uncorr PSF - corr PSF - corr PSF - corr DES SV DES SV DES SV DES 5y DES 5y DES 5y 0 . 05 95% CL 95% CL 95% CL m 1 0 . 00 − 0 . 05 − 0 . 10 30 . 5 30 . 6 5 . 0 5 . 5 0 . 36 0 . 42 mag 0 σ N e 1 ,rms 0 . 10 PSF - uncorr PSF - uncorr PSF - uncorr PSF - corr PSF - corr PSF - corr DES SV DES SV DES SV DES 5y DES 5y DES 5y 0 . 05 95% CL 95% CL 95% CL m 1 0 . 00 − 0 . 05 − 0 . 10 0 . 40 0 . 48 0 . 24 0 . 30 0 . 135 0 . 150 e 2 ,rms σ θ

UFIG/BCC Busha, Wechsler et al. 2015; Chang et al. 2015 Transfer Function Blind Cosmology Partial Ultra Fast Image Simulated Simulated Challenge (BCC) DESDM Generator (UFig) DES images DM catalogs Catalogs software ? Our Universe CTIO / DECam DES images DESDM DM catalogs software Nord et al. 2015, Nicola et al. 2015 + Integration of spectroscopy simulations

Conclusions ‣ Upcoming and future LSS surveys have great promise for cosmology but will require new data analysis approaches ‣ Forward modelling is a promising approach to analyse complex data sets ‣ ABC can provide an approximation to the posterior in cases when the likelihood is not available