Modeling the Universe Interfacing Theory, Simulations, Statistical - - PowerPoint PPT Presentation

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Modeling the Universe

Tim Eifler (JPL/Caltech, University of Arizona) Interfacing Theory, Simulations, Statistical Methods, and Observations

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11

The Challenge

reduced data and catalogs summary statistics

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11

reduced data and catalogs

The Challenge

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Introducing CosmoLike

Weak Lensing, Galaxy Clustering, Clusters, CMB, CMB-LSS correlations Multi-Probe Covariances/Hybrid Estimators Galaxy bias models (linear, quadratic, HOD) Explore fundamental physics (cosmic acceleration, neutrinos, tests of gravity) Systematics (photo-z, shape uncertainties) Likelihood free inference Gaussianization of summary statistics Numerical Simulations/ Emulators Astrophysics (Intrinsic alignment, Baryonic Physics) Idea: consistent, multi-probe likelihood analysis software framework including

• Realistic statistical error bars (cross-probe covariances)
• Cross-correlations of observables/systematics
• Efficient treatment of nuisance parameters

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Project 1: Simulate a Multi-Probe Likelihood Analysis for LSST

cosmolike - cosmological likelihood analyses for photometric galaxy surveys CosmoLike release paper (www.cosmolike.info) Krause & TE 2017

Theory+Sims+Stats -> Obs

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Weak Lensing (cosmic shear) 10 tomography bins 25 l bins, 25 < l < 5000 Galaxy clustering 4 redshift bins (0.2-0.4,0.4-0.6,0.6-0.8,0.8-1.0) compare two samples: σz <0.04, redMaGiC linear + quadratic bias only : l bins restricted to R> 10 Mpc/h HOD modeling going to R>0.1 MPC/h Galaxy-galaxy lensing galaxies from clustering (as lenses) with shear sources Clusters - number counts + shear profile so far, 8 richness, 4 z-bins (same as clustering) tomographic cluster lensing (500 < l < 10000)

Example Data Vector and Systematics

shear calibration, photo-z (sources) IA, baryons b1, b2,… photo-z (lenses) N-M relation c-M relation

• ff-centering

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CosmoLike - “Inner Workings”

Krause & Eifler 2017 halo.c cosmo3d.c

growth factor

D(k,z) Plin(k,z) distances Pnl(k,z) Coyote U. Emulator

collapse density

𝜀c(z)

peak height

𝜉 (M,z) halo properties HOD, bias model

N(Mobs;zi) CXY(l;zi,zj)

scaling relation

Mobs(M)

cluster selection fuction

c(M,z) b(M,z) n(M,z)

z-distr.

n(z)

clusters.c

photo-z model

redshift.c

p r

• j

e c t i

• n

f u n c t i

• n

s L i m b e r a p p r

• x

.

cosmo2d.c

transfer function

T(k,z)

systematics.c

non-linear regime galaxy formation cluster finding intrinsic alignments baryons non-Gaussian photo-zs shear calibration ... .... ....

P(k,zj) Cov(zi,zj,zk,zl,l1,l2) Likelihood cosmological parameters

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Multi-Probes Forecasts: Covariance

details: Krause&TE ‘17

Cosmic Shear Galaxy- Galaxy Lensing Galaxy Clustering Cluster Lensing Clusters 7+ million elements

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The Power of Combining Probes

7 cosmological parameters 49 nuisance parameters

• Shear Calibration,
• Lens+Source photo-z,
• Linear galaxy bias
• Cluster Mass

Calibration

• Intrinsic Alignments

clustering cosmic shear clusterN 3x2pt 3x2pt+clusterN+clusterWL

Ωm σ8 h w0 wa wa w0 h σ8 Prob

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Zoom into w0-wa plane

clustering cosmic shear clusterN 3x2pt 3x2pt+clusterN+clusterWL

w0 wa

• Very non-linear gain in

constraining power

• Most stringent

requirements on numerical simulations, photo-z, shear calibration, etc flow from Multi-Probe statistical limits

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Project 2: Exploring WFIRST survey strategies

Project within the WFIRST Cosmology with the High Latitude Survey Science Investigation Team TE et al in prep

Theory+Sims+Stats -> Obs

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Individual vs multi- probe WFIRST analysis Modified Gravity All-In Systematics

76 dimensions (7 cosmology, 69 systematics)

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WFIRST - LSST synergies

Possible WFIRST extension of 1.6 years overlapping with LSST

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Project 3: New statistical methods to reduce Super-Computing needs

Precision matrix expansion - efficient use of numerical simulations in estimating errors on cosmological parameters Friedrich & TE 2018

Theory+Stats -> Sims

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The Problem: Inverse Covariance Estimation

Analytical covariance model relies on approximations that might be too imprecise for an LSST Y10 data set Estimation the covariance from numerical simulations (brute force), requires 10^5-10^6 realizations of an LSST Year 10 like survey to shield against noise in the estimator Why? The estimated inverse covariance is not the inverse of the estimated covariance High-dimensionality of the data vector -> many elements in the covariance

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ˆ Ψ = ν Nd 1 ν ˆ C1

p(π π π|ˆ ξ ξ ξ) ⇠ exp 1 2χ2 h π π π | ˆ ξ ξ ξ,C i! p(π π π)

χ2 h π π π | ˆ ξ ξ ξ,C i = ⇣ˆ ξ ξ ξ ξ ξ ξ[π π π] ⌘T C1 ⇣ˆ ξ ξ ξ ξ ξ ξ[π π π] ⌘

Standard Estimator

New idea: Include theory information into estimator

C = A+B

C = M+(B−Bm)

C = (1 +X) M ,

X := (B−Bm) M−1

where M = A+Bm is

Ψ = M−1 B B B B B B @

∞

X

k=0

(−1)kXk 1 C C C C C C A = M−1 ⇣ 1 −X+X2 +O h X3i⌘

ˆ Ψ2nd = M1 +M1BmM1BmM1 M1 ⇣ ˆ BBm ⌘ M1 M1 ˆ BM1BmM1 M1BmM1 ˆ BM1 +M1 ν2 ˆ BM1 ˆ Bν ˆ B tr ⇣ M1 ˆ B ⌘ ν2 +ν2 M1

Invert and expand as power series Build Estimator Only matrix multiplication, no inversion of estimated quantities

Idea: Estimate the inverse directly

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Idea: Estimate the inverse directly

ˆ Ψ = ν Nd 1 ν ˆ C1

p(π π π|ˆ ξ ξ ξ) ⇠ exp 1 2χ2 h π π π | ˆ ξ ξ ξ,C i! p(π π π)

χ2 h π π π | ˆ ξ ξ ξ,C i = ⇣ˆ ξ ξ ξ ξ ξ ξ[π π π] ⌘T C1 ⇣ˆ ξ ξ ξ ξ ξ ξ[π π π] ⌘

Standard Estimator

New idea: Include theory information into estimator

C = A+B

C = M+(B−Bm)

C = (1 +X) M ,

X := (B−Bm) M−1

where M = A+Bm is

Ψ = M−1 B B B B B B @

∞

X

k=0

(−1)kXk 1 C C C C C C A = M−1 ⇣ 1 −X+X2 +O h X3i⌘

ˆ Ψ2nd = M1 +M1BmM1BmM1 M1 ⇣ ˆ BBm ⌘ M1 M1 ˆ BM1BmM1 M1BmM1 ˆ BM1 +M1 ν2 ˆ BM1 ˆ Bν ˆ B tr ⇣ M1 ˆ B ⌘ ν2 +ν2 M1

Invert and expand as power series Build Estimator Only matrix multiplication, no inversion of estimated quantities

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ˆ Ψ = ν Nd 1 ν ˆ C1

Inverting quantities with “hats” is dangerous

Standard estimator

ˆ C := 1 ν

Ns

• i=1
• ˆ

ξ i − ¯ ξ ˆ ξ i − ¯ ξ T

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Idea: Estimate the inverse directly

ˆ Ψ = ν Nd 1 ν ˆ C1

p(π π π|ˆ ξ ξ ξ) ⇠ exp 1 2χ2 h π π π | ˆ ξ ξ ξ,C i! p(π π π)

χ2 h π π π | ˆ ξ ξ ξ,C i = ⇣ˆ ξ ξ ξ ξ ξ ξ[π π π] ⌘T C1 ⇣ˆ ξ ξ ξ ξ ξ ξ[π π π] ⌘

Standard Estimator

New idea: Include theory information into estimator

C = A+B

C = M+(B−Bm)

C = (1 +X) M ,

X := (B−Bm) M−1

where M = A+Bm is

Ψ = M−1 B B B B B B @

∞

X

k=0

(−1)kXk 1 C C C C C C A = M−1 ⇣ 1 −X+X2 +O h X3i⌘

ˆ Ψ2nd = M1 +M1BmM1BmM1 M1 ⇣ ˆ BBm ⌘ M1 M1 ˆ BM1BmM1 M1BmM1 ˆ BM1 +M1 ν2 ˆ BM1 ˆ Bν ˆ B tr ⇣ M1 ˆ B ⌘ ν2 +ν2 M1

Invert and expand as power series Build Estimator Only matrix multiplication, no inversion of estimated quantities

SLIDE 20

ˆ Ψ = ν Nd 1 ν ˆ C1

Standard Estimator

New idea: Include theory information into estimator

C = A+B

C = M+(B−Bm)

C = (1 +X) M ,

X := (B−Bm) M−1

where M = A+Bm is

Ψ = M−1 B B B B B B @

∞

X

k=0

(−1)kXk 1 C C C C C C A = M−1 ⇣ 1 −X+X2 +O h X3i⌘

ˆ Ψ2nd = M1 +M1BmM1BmM1 M1 ⇣ ˆ BBm ⌘ M1 M1 ˆ BM1BmM1 M1BmM1 ˆ BM1 +M1 ν2 ˆ BM1 ˆ Bν ˆ B tr ⇣ M1 ˆ B ⌘ ν2 +ν2 M1

Invert and expand as power series Build Estimator Only matrix multiplication, no inversion of estimated quantities

Idea: Estimate the inverse directly

p(π π π|ˆ ξ ξ ξ) ⇠ exp 1 2χ2 h π π π | ˆ ξ ξ ξ,C i! p(π π π)

χ2 h π π π | ˆ ξ ξ ξ,C i = ⇣ˆ ξ ξ ξ ξ ξ ξ[π π π] ⌘T C1 ⇣ˆ ξ ξ ξ ξ ξ ξ[π π π] ⌘

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No more inversion of “hat quantities”…

ˆ Ψ2nd = M1 +M1BmM1BmM1 M1 ⇣ ˆ BBm ⌘ M1 M1 ˆ BM1BmM1 M1BmM1 ˆ BM1 +M1 ν2 ˆ BM1 ˆ Bν ˆ B tr ⇣ M1 ˆ B ⌘ ν2 +ν2 M1

New estimator

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New estimator performance

Instead of >10^5 our new estimator only requires ~2000 numerical simulations (LSST case) Given that 1 sim is 1M CPUh, at 1c/CPUh New method reduces cost $1B to $20M (-> fund theorists!) Next step: data compression

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Project 4: Synergies of CMB- S4 and LSST

Looking through the same lens: Shear calibration for LSST, Euclid, and WFIRST with stage 4 CMB lensing Schaan, Krause, TE et al 2017

Obs -> Theory/Sims

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Ω0

m = 0.315+0.0097 −0.0093

0.800 0.825 0.850 0.875

σ8

σ8 = 0.831+0.0089

−0.0088

0.925 0.950 0.975 1.000

nS

nS = 0.965+0.0052

−0.0052

−1.25 −1.00 −0.75 −0.50

w0

w0 = −1.0+0.11

−0.11

−0.5 0.0 0.5 1.0

wa

wa = −0.0+0.33

−0.34

0.040 0.045 0.050 0.055

Ω0

b

Ω0

b = 0.049+0.0038 −0.0044

0.275 0.300 0.325 0.350

Ω0

m

0.60 0.65 0.70 0.75

h0

0.800 0.825 0.850 0.875

σ8

0.925 0.950 0.975 1.000

nS

−1.25 −1.00 −0.75 −0.50

w0

−0.5 0.0 0.5 1.0

wa

0.040 0.045 0.050 0.055

Ω0

b

0.60 0.65 0.70 0.75

h0

h0 = 0.67+0.021

−0.021

LSST multi-probe+CMB-S4 Lensing

This project is currently evolving into full LSST+CMB-S4 forecasts (cosmology+inflation models)

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m0 = −0.0+0.023

−0.022 −0.015 0.000 0.015 0.030

m1

m1 = −0.0+0.011

−0.011 −0.015 0.000 0.015 0.030

m2

m2 = −0.0+0.0075

−0.0075 −0.015 0.000 0.015 0.030

m3

m3 = 0.0+0.0061

−0.0061 −0.015 0.000 0.015 0.030

m4

m4 = 0.0+0.005

−0.005 −0.015 0.000 0.015 0.030

m5

m5 = −0.0+0.0044

−0.0044 −0.015 0.000 0.015 0.030

m6

m6 = −0.0+0.0039

−0.0039 −0.015 0.000 0.015 0.030

m7

m7 = −0.0+0.0036

−0.0036 −0.015 0.000 0.015 0.030

m8

m8 = −0.0+0.0032

−0.0032 −0.015 0.000 0.015 0.030

m0

−0.015 0.000 0.015 0.030

m9

−0.015 0.000 0.015 0.030

m1

−0.015 0.000 0.015 0.030

m2

−0.015 0.000 0.015 0.030

m3

−0.015 0.000 0.015 0.030

m4

−0.015 0.000 0.015 0.030

m5

−0.015 0.000 0.015 0.030

m6

−0.015 0.000 0.015 0.030

m7

−0.015 0.000 0.015 0.030

m8

−0.015 0.000 0.015 0.030

m9

m9 = −0.0+0.0029

−0.0028

Calibrating galaxy shape measurements is a major systematics for LSST

• Usually calibration is

achieved through costly simulations, which we hope are unbiased

• CMB data can be used to

self-calibrate LSST shape measurements

• Independent consistency

check or additional information

Use CMB information to offset one

• f the largest systematics in LSST

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Allows for independent LSST shear calibration at level of LSST requirements in highest z-bins (hard to achieve otherwise)

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Project 5: Test Accuracy in Numerical Simulations

Project in some shelf… might never see daylight…

Theory -> Sims

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Numerical simulations have uncertainties

Joint data vector details (LSST Y1, 18000 deg^2)

• WL Source Sample:
• 5 tomographic bins [0:2.5]
• 25 l-bins [30:5000]
• n_gal=13 gal/arcmin^2
• Clustering Lens Sample:
• 4 tomographic bins [0:1.0]
• 25 l-bins [30:5000]
• red sequence sample
• k_max cut-off, R=[2,5,10]

Mpc/h to justify linear galaxy bias models

• Galaxy galaxy lensing using

lens and source sample

Matter Power Spectrum Error Models

2 4 6 8 10 2 4 6 8 10

n=1 n=0.8 n=0.5

(A k/kNL)ˆn

∆P(k,z=0) in % k

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black: no error, Rmin=10 Mpc/h green: n=0.5, Rmin=10 Mpc/h blue: n=0.8, Rmin=10 Mpc/h red: n=1.0, Rmin=10 Mpc/h

LSST Y1

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Conclusions

Exciting because of the enormous amount of cosmological data from a variety of surveys Complex because smart+precise multi-probe and multi-data set analyses are hard We need creative research on systematics mitigation, precise error calculation, model building, data inference Critical to interface expertise in simulations, observations, analytical modeling, statistical methods

Future of cosmology is very exciting and very complex