Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://hea-www.harvard.edu/astrostat/talks_1011/haiku_ksm_20100927.pdf Äàòà èçìåíåíèÿ: Mon Sep 27 20:12:22 2010 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 05:54:36 2012 Êîäèðîâêà:

Hierarchical Bayesian Models for Type Ia SN Light Curves, Dust and Cosmic Distances

Kaisey S. Mandel Harvard University 27 September 2010
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Hierarchical Brings deep knowledge from data Distant star glows, fades. -Bob Kirshner

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Cosmological Energy Content

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Supernova Cosmology: Constraining Cosmological Parameters using Distance vs.Velocity

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AAS 215

AAS 215

*+,+-./01+E+./.2+34
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Standard Candle Principle
1. Know or Estimate Luminosity L of a Class of Astronomical Objects 2. Measure the apparent brightness or flux F 3. Derive the distance D to Object using Inverse Square Law: F = L / (4 D)

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Type Ia Supernovae are Nearly Standard Candles
· · · ·
Progenitor : C/O White Dwarf Star accreting mass leads to instability Thermonuclear Explosion: Deflagration/Detonation Nickel to Cobalt to Iron Decay + radiative transfer powers the light cur ve SNe Ia progenitors have nearly same mass, therefore energy
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Credit: FLASH Center


Type Ia Supernova Apparent Light Curve
6 8

H!9
Obs. Mag. ! kc ! mwx

10 12 14 16 18 20 22 SN2005eq (CfA3+PTEL) !10 0 10 20 30 40 Obs. Days Since Bmax

J!7 I!4 R!2 V B+2
50 60

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Reading the Wattage of a SN Ia: Empirical Correlations

· ·

Width-Luminosity Relation: an obser ved correlation (Phillips) Obser ve optical SN Ia Light Cur ve Shape to estimate the peak luminosity of SN Ia more precisely: ~0.5 mag to ~0.2 mag error Color-Luminosity Relation

·

Intrinsically Brighter SN Ia have broader light curves and are slow decliners

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I will show you fear in a handful of Dust
1.5 1 0.5 0 0.6 0.4 0.2 0 !0.2 !0.4 1 0.5 0 !0.5 !17 Apparent Intrinsic !17.5 !18 RV = 3.1 RV = 2.4 RV = 1.7

B!V

V!R

V!I

Random Dust Effects: 1. Redder 2. Dimmer

MV or V0!µ

!18.5

!19

!19.5

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Obser ve in NIR to see through dust

· · · ·

Host Galaxy Dust presents a major systematic uncertainty in supernova cosmology inference Dust extinction has significantly reduced effect in NIR bands NIR SN Ia are good standard candles Observe in NIR!: PAIRITEL /CfA
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Statistical inference with SN Ia

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SN Ia cosmology inference based on empirical relations Statistical models for SN Ia are learned from the data Several Sources of Randomness & Uncertainty 1. Photometric errors 2. Intrinsic Variation and Correlations between L, Light Cur ve Shape, Color = Population Distribution of SN Ia 3. Random Peculiar Velocities in Nearby Hubble Flow 4. Host Galaxy Dust: extinction and reddening.

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How to incorporate this all into a coherent statistical model? Hierarchical Bayesian Model!
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Directed Acyclic Graph for SN Ia Inference with Hierarchical Modeling
· · · ·
Intrinsic Randomness Dust Extinction & Reddening Peculiar Velocities Measurement Error
Dust Pop

"Training" - Learn about Populations
A
s V s ,RV

µs AppLCs

z

s

Generative Model Global Joint Posterior Probability Density Conditional on all SN Data
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AbsLC

s

D

s

Training SN Ia AbsLC Pop
p Ap ,RV V

s = 1, . . . , NSN µp Prediction

AbsLC
12

p

AppLCp

D

p

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Statistical Computation with Hierarchical SN Ia Models: The BayeSN Algorithm
BayeSN MCMC Convergence
3

·

V

Strategy: Generate a
Markov Chain to sample global parameter space (populations & all individuals) => seek a global solution

2.5 2

SN2001az

SN2001ba

A

1.5 1 0.5 0 3 2.5 2

SN2006cp

SN2007bz

V

A

1.5 1 0.5 0 10
0 1 2 3 0 1 2 3

·

Chain explores/samples trade-offs/degeneracies in global parameter space for populations and individuals

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10

10 10

10

10

10

MCMC Sample

MCMC Sample

Multiple chains globally converge from random initial values
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BayeSN MCMC strategy
· · · · ·
Gibbs Sampling Metropolis-Hastings Parameter Expansion Generalized Conditional Sampling Parallel chains to diagnose convergence
SN Ia AbsLC Pop Dust Pop
s As ,RV V

µs AppLCs

z

s

AbsLC

s

D

s

Training
p Ap ,RV V

s = 1, . . . , NSN µp Prediction

AbsLC

p

AppLCp

D

p

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Results: Optical+NIR Hierarchical Inference
PTEL+CfA3 Light-curves
6 8
Apparent Magnitude

Marginal Posterior of Dust
0.6

AV

10 12 14 16 18 20 22 SN2005eq (CfA3+PTEL) !10 0 10 20 30 Phase 40

H!9 J!7 I!4 R!2 V
Extinction (mag)

0.5 0.4 0.3 0.2 0.1

SN2005eq

AH

B+2
0

50

60
0.35 0.3

0.3 0.4 0.5 Dust Law Slope R!1 V

0.3

0.4 R!1 V

0.5

6 8
Apparent Magnitude

H!9 J!7 I!4 R!2 V
SN2006ax (CfA3+PTEL) !10 0 10 20 30 Phase 40
Extinction (mag)

A

V

SN2006ax

10 12 14 16 18 20 22

0.25 0.2 0.15 0.1 0.05

AH

B+2
50 60

0

0.2

0.3 0.4 0.5 Dust Law Slope R!1 V

0.60.2

0.3

0.4 R!1 V

0.5

0.6
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Improved Constraints from Combining Optical with Infrared Light Cur ves

3.5 3 2.5

P( P( P( E(

µ µ µ µ

| BV) | BVRI) | BVRIJH) |z) ± (300 km/s)

1.5 1 0.5 1.5 1 0.5 1.5

BV SN2002bo BVRI

PDF

2

SN2002bo
1.5 1

Extinction AV

1

BVRIJH

0.5 0 30.5

0.5 31.2 31.4 31.6 31.8 32 32.2 32.4 32.6

31

31.5

32

32.5

33

33.5

Distance Modulus µ

Distance Modulus µ

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Improved Constraints from Combining Optical with Infrared Light Cur ves
4 3.5 3 P( P( P( E( µ µ µ µ | BV) | BVRI) | BVRIJH) |z) ± (300 km/s)
0.6 0.4 0.2

BV

SN2005ki:CSP

Extinction AV

2.5

0.6 0.4 0.2

BVRI

PDF

2 1.5 1 0.5 0 33.6

SN2005ki:CSP

0.6 0.4 0.2

BVRIJH

33.8

34

34.2

34.4

34.6

34.8

35

35.2

35.4

34

34.1

34.2

34.3

34.4

34.5

34.6

34.7

34.8

34.9

Distance Modulus µ

Distance Modulus µ

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Nearby Optical+NIR Hubble Diagram
38 37 36

h = 0.72 !pec = 150 km/s

µ(pred)

35 34 33 32 31 1 0.5 0

110 BVRI(JH) SN Ia (CfA3+PTEL+lit)
4

Optical Optical+NIR

10 CV Pred Err (All, cz > 3000 km/s) = 0.14 mag (0.139 ± 0.011 intr.)

Difference

!0.5 !1

CV Pred Err (Opt only & cz > 3000 km/s) = 0.15 mag (0.148 ± 0.014 intr.) CV Pred Err (Opt+NIR & cz > 3000 km/s) = 0.11 mag (0.102 ± 0.019 intr.) 3000 5000 7000 10000 Velocity [CMB+Virgo] (km/s) 15000

Cross-Validated Distance Predictions (Opt+NIR) RMS Distance Prediction Error = 0.11 mag (5.5% in distance)
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Summary
· · ·
Hierarchical models for SN Ia Light Cur ves, Dust,Distance BayeSN: MCMC for fitting hierarchical models for SN Ia SN Ia Optical+NIR: Constrain dust, predict distances better

References Mandel, K. , W.M. Wood-Vasey, A.S. Friedman, & R.P. Kirshner. Type Ia Supernova Light Curve Inference: Hierarchical Bayesian Analysis in the Near Infrared. 2009, ApJ, 704:629-651 Mandel, K., G. Narayan, & R.P. Kirshner. Type Ia Supernova Light Curve Inference: Hierarchical Modeling in the Optical and Near Infrared. 2010, in prep.
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