Hierarchical Bayesian Models for Type Ia SN Light Curves ... · 9/27/2010 · Summary • Hierarchical models for SN Ia Light Curves, Dust,Distance • BayeSN: MCMC for ﬁtting

Kaisey S. MandelHarvard University27 September 2010

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

1Monday, September 27, 2010

Hierarchical

Brings deep knowledge from data

Distant star glows, fades.

-Bob Kirshner


Cosmological Energy Content


Supernova Cosmology:Constraining Cosmological Parameters

using Distance vs. Velocity

AAS 215

!"##$%&'%()*"+$(

! !!""!#!$%&!#!$%&!!

'!(%()'!(%()

*(%(+*(%(+

! !"#$%&'(#)*+,%!"#$%&'(#)*+,%('(-!.%*"#$%/01#%('(-!.%*"#$%/01#%12)/+%,+$,0)()(312)/+%,+$,0)()(3

! 45'/"$)'(%'6%45'/"$)'(%'6%77%%%%1)$2%,+*#2)6$81)$2%,+*#2)6$8

AAS 215

!"##$%&! !""#$%&#'()(#&*'+,*'#!""#$%&#'()(#&*'+,*'#! -*)&(././0#1234#5.)6#-*)&(././0#1234#5.)6#

"(&0*#"7589#:*)#)7#;*))*&#"(&0*#"7589#:*)#)7#;*))*&#:*<(&()*#,7"7&#(/'#'+:)#:*<(&()*#,7"7&#(/'#'+:)#(/'#&*'+,*#:%:)*=().,:(/'#&*'+,*#:%:)*=().,:

! >?(=././0#@#;(/'#>?(=././0#@#;(/'#(/7=("%(/7=("%

! A&7<(0()*#<&7;(;.".)%#A&7<(0()*#<&7;(;.".)%#:+&B(,*#B7&#*(,6#4C#)7#:+&B(,*#B7&#*(,6#4C#)7#,7:=7"70%#D#;*))*&#,7:=7"70%#D#;*))*&#*:).=()*:#7B#:%:)*=().,#*:).=()*:#7B#:%:)*=().,#

! !"#$%&%'(")##,7/:)&(./):#,7/:)&(./):#7/#,7/:)(/)#57/#,7/:)(/)#5

*+,+-./01+*+,+-./01+EE+./.2+345657++./.2+345657+EE+./89+34:47+./89+34:474Monday, September 27, 2010

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 / (4π D)


Type Ia Supernovae areNearly 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 curve

• SNe Ia progenitors have nearly same mass, therefore energy

Credit: FLASH Center

6


Type Ia Supernova ApparentLight Curve

!10 0 10 20 30 40 50 60

6

8

10

12

14

16

18

20

22B + 2SN2005eq (CfA3+PTEL)

V

R ! 2

I ! 4

J ! 7

H ! 9

Obs. Days Since Bmax

Ob

s.

Ma

g. !

kc !

mw

x


Reading the Wattage of a SN Ia:Empirical Correlations

• Width-Luminosity Relation: an observed correlation (Phillips)

• Observe optical SN Ia Light Curve 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

8


I will show you fear in a handful of Dust

0

0.5

1

1.5

B!V

!0.4

!0.2

0

0.2

0.4

0.6

V!R

!19.5!19!18.5!18!17.5!17

!0.5

0

0.5

1

V!I

MV or V

0!µ

RV = 3.1 R

V = 2.4 R

V = 1.7

Apparent

Intrinsic

Random Dust Effects:1. Redder 2. Dimmer


Observe 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

10

1989ApJ...345..245C


Statistical inference with SN Ia

• 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 Curve Shape, Color = Population Distribution of SN Ia

3. Random Peculiar Velocities in Nearby Hubble Flow

4. Host Galaxy Dust: extinction and reddening.

• How to incorporate this all into a coherent statistical model? Hierarchical Bayesian Model!

11


Directed Acyclic Graph for SN Ia Inferencewith Hierarchical Modeling

• Intrinsic Randomness• Dust Extinction & Reddening• Peculiar Velocities • Measurement Error

“Training” - Learn about Populations

12

Generative Model

Global Joint Posterior Probability

Density Conditional on all

SN Data

zs

Ds

µs

AppLCs

s = 1, . . . , NSN

AsV , Rs

V

AbsLCs

Training

PredictionApV , Rp

Vµp

DpAppLCpAbsLCp

DustPop

SN IaAbsLC

Pop


Statistical Computation with Hierarchical SN Ia Models: The BayeSN Algorithm

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

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

Multiple chains globally converge from random

initial values

0

0.5

1

1.5

2

2.5

3

AV

SN2001az

SN2001ba

100

101

102

103

0

0.5

1

1.5

2

2.5

3

MCMC Sample

AV

SN2006cp

BayeSN MCMC Convergence

100

101

102

103

MCMC Sample

SN2007bz


BayeSN MCMC strategy

• Gibbs Sampling

• Metropolis-Hastings

• Parameter Expansion

• Generalized Conditional Sampling

• Parallel chains to diagnose convergence

zs

Ds

µs

AppLCs

s = 1, . . . , NSN

AsV , Rs

V

AbsLCs

Training

PredictionApV , Rp

Vµp

DpAppLCpAbsLCp

DustPop

SN IaAbsLC

Pop


Results: Optical+NIR Hierarchical InferencePTEL+CfA3 Light-curves Marginal Posterior of Dust

!10 0 10 20 30 40 50 60

6

8

10

12

14

16

18

20

22B + 2SN2005eq (CfA3+PTEL)

V

R ! 2

I ! 4

J ! 7

H ! 9

Phase

Ap

pa

ren

t M

ag

nitu

de

!10 0 10 20 30 40 50 60

6

8

10

12

14

16

18

20

22

B + 2SN2006ax (CfA3+PTEL)

V

R ! 2

I ! 4

J ! 7

H ! 9

Phase

Ap

pa

ren

t M

ag

nitu

de

0.2 0.3 0.4 0.5 0.60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Dust Law Slope RV

!1

Extinction (

mag)

AV

0.2 0.3 0.4 0.5 0.6R

V

!1

SN2006ax

AH

0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

Dust Law Slope RV

!1

Extin

ctio

n (

ma

g)

AV

0.3 0.4 0.5R

V

!1

SN2005eq

AH


Improved Constraints from Combining Optical with Infrared Light Curves

0.5

1

1.5

BV

SN2002bo

0.5

1

1.5

Extinction A

V

BVRI

31.2 31.4 31.6 31.8 32 32.2 32.4 32.6

0.5

1

1.5

Distance Modulus µ

BVRIJH

30.5 31 31.5 32 32.5 33 33.50

0.5

1

1.5

2

2.5

3

3.5

SN2002bo

Distance Modulus µ

PD

F

P(µ | BV)

P(µ | BVRI)

P(µ | BVRIJH)

E(µ |z) ± (300 km/s)


Improved Constraints from Combining Optical with Infrared Light Curves

33.6 33.8 34 34.2 34.4 34.6 34.8 35 35.2 35.40

0.5

1

1.5

2

2.5

3

3.5

4

SN2005ki:CSP

Distance Modulus µ

PD

F

P(µ | BV)

P(µ | BVRI)

P(µ | BVRIJH)

E(µ |z) ± (300 km/s)

0.2

0.4

0.6 BVSN2005ki:CSP

0.2

0.4

0.6

Extinction A

V

BVRI

34 34.1 34.2 34.3 34.4 34.5 34.6 34.7 34.8 34.9

0.2

0.4

0.6

Distance Modulus µ

BVRIJH


Nearby Optical+NIR Hubble Diagram

Cross-Validated Distance Predictions

(Opt+NIR) RMS Distance Prediction Error = 0.11 mag (5.5% in distance)

104

31

32

33

34

35

36

37

38

µ(p

red)

h = 0.72

!pec

= 150 km/s

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

3000 5000 7000 10000 15000!1

!0.5

0

0.5

1

Velocity [CMB+Virgo] (km/s)

Diffe

rence

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

CV Pred Err (Opt+NIR & cz > 3000 km/s) = 0.11 mag (0.102 ± 0.019 intr.)CV Pred Err (Opt only & cz > 3000 km/s) = 0.15 mag (0.148 ± 0.014 intr.)

Optical

Optical+NIR


Summary• Hierarchical models for SN Ia Light Curves, Dust,Distance

• BayeSN: MCMC for fitting hierarchical models for SN Ia

• SN Ia Optical+NIR: Constrain dust, predict distances better

19

ReferencesMandel, 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.


Hierarchical Bayesian Models for Type Ia SN Light Curves ... · 9/27/2010 · Summary • Hierarchical models for SN Ia Light Curves, Dust,Distance • BayeSN: MCMC for ﬁtting

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