[PPT] - EFFECT OF BINARIES ON DARK MATTER ESTIMATES IN DWARF GALAXIES PowerPoint Presentation

SLIDE 1

EFFECT OF BINARIES ON DARK MATTER ESTIMATES IN DWARF GALAXIES

Caveats to Dwarf Galaxy Indirect Detection Limits

L A U R A J . C H A N G P R I N C E T O N U N I V E R S I T Y S m a l l G a l a x i e s , C o s m i c Q u e s t i o n s J u l y 3 1 , 2 0 1 9 I N C O L L A B O R AT I O N W I T H : L I N A N E C I B ( C A LT E C H )

SLIDE 2

Photon energy Counts

enhanced emission from dark matter-rich regions Weakly Interacting Massive Particles → gamma rays DM DM

W, Z, b...

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W, Z, b...

<latexit sha1_base64="3+bGj0ZTbcgrBnhQFim8WozTu6I=">AB8XicdVDLSgNBEOz1GeMr6tHLYBA8hGVXAnoMePEYwTxIsoTZySQZMju7zPQKYclfePGgiFf/xpt/42wSxGdBQ1HVTXdXmEh0PenZXVtfWNzcJWcXtnd2+/dHDYNHGqGW+wWMa6HVLDpVC8gQIlbyea0yiUvBVOrnK/dce1EbG6xWnCg4iOlBgKRtFKnVaFdCokdF23Xyp7btXLQXzXW5BfShmWqPdLb71BzNKIK2SGtP1vQSDjGoUTPJZsZcanlA2oSPetVTRiJsgm18I6dWGZBhrG0pJHP160RGI2OmUWg7I4pj89PLxb+8borDyATKkmRK7ZYNEwlwZjk75OB0JyhnFpCmRb2VsLGVFOGNqSiDeHz9/9J89z1Pde/qZrtWUcBTiGEzgDHy6gBtdQhwYwUHAPj/DkGOfBeXZeFq0rznLmCL7Bef0A7DGPGg=</latexit><latexit sha1_base64="3+bGj0ZTbcgrBnhQFim8WozTu6I=">AB8XicdVDLSgNBEOz1GeMr6tHLYBA8hGVXAnoMePEYwTxIsoTZySQZMju7zPQKYclfePGgiFf/xpt/42wSxGdBQ1HVTXdXmEh0PenZXVtfWNzcJWcXtnd2+/dHDYNHGqGW+wWMa6HVLDpVC8gQIlbyea0yiUvBVOrnK/dce1EbG6xWnCg4iOlBgKRtFKnVaFdCokdF23Xyp7btXLQXzXW5BfShmWqPdLb71BzNKIK2SGtP1vQSDjGoUTPJZsZcanlA2oSPetVTRiJsgm18I6dWGZBhrG0pJHP160RGI2OmUWg7I4pj89PLxb+8borDyATKkmRK7ZYNEwlwZjk75OB0JyhnFpCmRb2VsLGVFOGNqSiDeHz9/9J89z1Pde/qZrtWUcBTiGEzgDHy6gBtdQhwYwUHAPj/DkGOfBeXZeFq0rznLmCL7Bef0A7DGPGg=</latexit><latexit sha1_base64="3+bGj0ZTbcgrBnhQFim8WozTu6I=">AB8XicdVDLSgNBEOz1GeMr6tHLYBA8hGVXAnoMePEYwTxIsoTZySQZMju7zPQKYclfePGgiFf/xpt/42wSxGdBQ1HVTXdXmEh0PenZXVtfWNzcJWcXtnd2+/dHDYNHGqGW+wWMa6HVLDpVC8gQIlbyea0yiUvBVOrnK/dce1EbG6xWnCg4iOlBgKRtFKnVaFdCokdF23Xyp7btXLQXzXW5BfShmWqPdLb71BzNKIK2SGtP1vQSDjGoUTPJZsZcanlA2oSPetVTRiJsgm18I6dWGZBhrG0pJHP160RGI2OmUWg7I4pj89PLxb+8borDyATKkmRK7ZYNEwlwZjk75OB0JyhnFpCmRb2VsLGVFOGNqSiDeHz9/9J89z1Pde/qZrtWUcBTiGEzgDHy6gBtdQhwYwUHAPj/DkGOfBeXZeFq0rznLmCL7Bef0A7DGPGg=</latexit><latexit sha1_base64="3+bGj0ZTbcgrBnhQFim8WozTu6I=">AB8XicdVDLSgNBEOz1GeMr6tHLYBA8hGVXAnoMePEYwTxIsoTZySQZMju7zPQKYclfePGgiFf/xpt/42wSxGdBQ1HVTXdXmEh0PenZXVtfWNzcJWcXtnd2+/dHDYNHGqGW+wWMa6HVLDpVC8gQIlbyea0yiUvBVOrnK/dce1EbG6xWnCg4iOlBgKRtFKnVaFdCokdF23Xyp7btXLQXzXW5BfShmWqPdLb71BzNKIK2SGtP1vQSDjGoUTPJZsZcanlA2oSPetVTRiJsgm18I6dWGZBhrG0pJHP160RGI2OmUWg7I4pj89PLxb+8borDyATKkmRK7ZYNEwlwZjk75OB0JyhnFpCmRb2VsLGVFOGNqSiDeHz9/9J89z1Pde/qZrtWUcBTiGEzgDHy6gBtdQhwYwUHAPj/DkGOfBeXZeFq0rznLmCL7Bef0A7DGPGg=</latexit>

→ γ

<latexit sha1_base64="yhMrpue2AQlofr4U4bYshNm18BA=">AB+3icdVDLSgNBEJyNrxhfMR69DAbB0zIbF5O9Bbx4jGAekITQO5kQ2Z2l5lZNYT8ihcPinj1R7z5N04egoWNBRV3XR3hYng2hDy4WTW1jc2t7LbuZ3dvf2D/GhoeNUVansYhVKwTNBI9Y3XAjWCtRDGQoWDMcX8795i1TmsfRjZkrCthGPEBp2Cs1MsXOoPRwaUiu9wZwhSQi9fJG5QqQTEx8T1CfHLJUvOgyC4INhzyQJFtEKtl3/v9GOaShYZKkDrtkcS052CMpwKNst1Us0SoGMYsralEUimu9PF7TN8apU+HsTKVmTwQv0+MQWp9USGtlOCGenf3lz8y2unZlDpTnmUpIZFdLlokApsYjwPAve5YtSIiSVAFbe3YjoCBdTYuHI2hK9P8f+kUXI94nrXfrFaXcWRcfoBJ0hD5VRFV2hGqojiu7RA3pCz87MeXRenNdla8ZzRyhH3DePgGw8pTc</latexit><latexit sha1_base64="yhMrpue2AQlofr4U4bYshNm18BA=">AB+3icdVDLSgNBEJyNrxhfMR69DAbB0zIbF5O9Bbx4jGAekITQO5kQ2Z2l5lZNYT8ihcPinj1R7z5N04egoWNBRV3XR3hYng2hDy4WTW1jc2t7LbuZ3dvf2D/GhoeNUVansYhVKwTNBI9Y3XAjWCtRDGQoWDMcX8795i1TmsfRjZkrCthGPEBp2Cs1MsXOoPRwaUiu9wZwhSQi9fJG5QqQTEx8T1CfHLJUvOgyC4INhzyQJFtEKtl3/v9GOaShYZKkDrtkcS052CMpwKNst1Us0SoGMYsralEUimu9PF7TN8apU+HsTKVmTwQv0+MQWp9USGtlOCGenf3lz8y2unZlDpTnmUpIZFdLlokApsYjwPAve5YtSIiSVAFbe3YjoCBdTYuHI2hK9P8f+kUXI94nrXfrFaXcWRcfoBJ0hD5VRFV2hGqojiu7RA3pCz87MeXRenNdla8ZzRyhH3DePgGw8pTc</latexit><latexit sha1_base64="yhMrpue2AQlofr4U4bYshNm18BA=">AB+3icdVDLSgNBEJyNrxhfMR69DAbB0zIbF5O9Bbx4jGAekITQO5kQ2Z2l5lZNYT8ihcPinj1R7z5N04egoWNBRV3XR3hYng2hDy4WTW1jc2t7LbuZ3dvf2D/GhoeNUVansYhVKwTNBI9Y3XAjWCtRDGQoWDMcX8795i1TmsfRjZkrCthGPEBp2Cs1MsXOoPRwaUiu9wZwhSQi9fJG5QqQTEx8T1CfHLJUvOgyC4INhzyQJFtEKtl3/v9GOaShYZKkDrtkcS052CMpwKNst1Us0SoGMYsralEUimu9PF7TN8apU+HsTKVmTwQv0+MQWp9USGtlOCGenf3lz8y2unZlDpTnmUpIZFdLlokApsYjwPAve5YtSIiSVAFbe3YjoCBdTYuHI2hK9P8f+kUXI94nrXfrFaXcWRcfoBJ0hD5VRFV2hGqojiu7RA3pCz87MeXRenNdla8ZzRyhH3DePgGw8pTc</latexit><latexit sha1_base64="yhMrpue2AQlofr4U4bYshNm18BA=">AB+3icdVDLSgNBEJyNrxhfMR69DAbB0zIbF5O9Bbx4jGAekITQO5kQ2Z2l5lZNYT8ihcPinj1R7z5N04egoWNBRV3XR3hYng2hDy4WTW1jc2t7LbuZ3dvf2D/GhoeNUVansYhVKwTNBI9Y3XAjWCtRDGQoWDMcX8795i1TmsfRjZkrCthGPEBp2Cs1MsXOoPRwaUiu9wZwhSQi9fJG5QqQTEx8T1CfHLJUvOgyC4INhzyQJFtEKtl3/v9GOaShYZKkDrtkcS052CMpwKNst1Us0SoGMYsralEUimu9PF7TN8apU+HsTKVmTwQv0+MQWp9USGtlOCGenf3lz8y2unZlDpTnmUpIZFdLlokApsYjwPAve5YtSIiSVAFbe3YjoCBdTYuHI2hK9P8f+kUXI94nrXfrFaXcWRcfoBJ0hD5VRFV2hGqojiu7RA3pCz87MeXRenNdla8ZzRyhH3DePgGw8pTc</latexit>

→ γ

<latexit sha1_base64="yhMrpue2AQlofr4U4bYshNm18BA=">AB+3icdVDLSgNBEJyNrxhfMR69DAbB0zIbF5O9Bbx4jGAekITQO5kQ2Z2l5lZNYT8ihcPinj1R7z5N04egoWNBRV3XR3hYng2hDy4WTW1jc2t7LbuZ3dvf2D/GhoeNUVansYhVKwTNBI9Y3XAjWCtRDGQoWDMcX8795i1TmsfRjZkrCthGPEBp2Cs1MsXOoPRwaUiu9wZwhSQi9fJG5QqQTEx8T1CfHLJUvOgyC4INhzyQJFtEKtl3/v9GOaShYZKkDrtkcS052CMpwKNst1Us0SoGMYsralEUimu9PF7TN8apU+HsTKVmTwQv0+MQWp9USGtlOCGenf3lz8y2unZlDpTnmUpIZFdLlokApsYjwPAve5YtSIiSVAFbe3YjoCBdTYuHI2hK9P8f+kUXI94nrXfrFaXcWRcfoBJ0hD5VRFV2hGqojiu7RA3pCz87MeXRenNdla8ZzRyhH3DePgGw8pTc</latexit><latexit sha1_base64="yhMrpue2AQlofr4U4bYshNm18BA=">AB+3icdVDLSgNBEJyNrxhfMR69DAbB0zIbF5O9Bbx4jGAekITQO5kQ2Z2l5lZNYT8ihcPinj1R7z5N04egoWNBRV3XR3hYng2hDy4WTW1jc2t7LbuZ3dvf2D/GhoeNUVansYhVKwTNBI9Y3XAjWCtRDGQoWDMcX8795i1TmsfRjZkrCthGPEBp2Cs1MsXOoPRwaUiu9wZwhSQi9fJG5QqQTEx8T1CfHLJUvOgyC4INhzyQJFtEKtl3/v9GOaShYZKkDrtkcS052CMpwKNst1Us0SoGMYsralEUimu9PF7TN8apU+HsTKVmTwQv0+MQWp9USGtlOCGenf3lz8y2unZlDpTnmUpIZFdLlokApsYjwPAve5YtSIiSVAFbe3YjoCBdTYuHI2hK9P8f+kUXI94nrXfrFaXcWRcfoBJ0hD5VRFV2hGqojiu7RA3pCz87MeXRenNdla8ZzRyhH3DePgGw8pTc</latexit><latexit sha1_base64="yhMrpue2AQlofr4U4bYshNm18BA=">AB+3icdVDLSgNBEJyNrxhfMR69DAbB0zIbF5O9Bbx4jGAekITQO5kQ2Z2l5lZNYT8ihcPinj1R7z5N04egoWNBRV3XR3hYng2hDy4WTW1jc2t7LbuZ3dvf2D/GhoeNUVansYhVKwTNBI9Y3XAjWCtRDGQoWDMcX8795i1TmsfRjZkrCthGPEBp2Cs1MsXOoPRwaUiu9wZwhSQi9fJG5QqQTEx8T1CfHLJUvOgyC4INhzyQJFtEKtl3/v9GOaShYZKkDrtkcS052CMpwKNst1Us0SoGMYsralEUimu9PF7TN8apU+HsTKVmTwQv0+MQWp9USGtlOCGenf3lz8y2unZlDpTnmUpIZFdLlokApsYjwPAve5YtSIiSVAFbe3YjoCBdTYuHI2hK9P8f+kUXI94nrXfrFaXcWRcfoBJ0hD5VRFV2hGqojiu7RA3pCz87MeXRenNdla8ZzRyhH3DePgGw8pTc</latexit><latexit sha1_base64="yhMrpue2AQlofr4U4bYshNm18BA=">AB+3icdVDLSgNBEJyNrxhfMR69DAbB0zIbF5O9Bbx4jGAekITQO5kQ2Z2l5lZNYT8ihcPinj1R7z5N04egoWNBRV3XR3hYng2hDy4WTW1jc2t7LbuZ3dvf2D/GhoeNUVansYhVKwTNBI9Y3XAjWCtRDGQoWDMcX8795i1TmsfRjZkrCthGPEBp2Cs1MsXOoPRwaUiu9wZwhSQi9fJG5QqQTEx8T1CfHLJUvOgyC4INhzyQJFtEKtl3/v9GOaShYZKkDrtkcS052CMpwKNst1Us0SoGMYsralEUimu9PF7TN8apU+HsTKVmTwQv0+MQWp9USGtlOCGenf3lz8y2unZlDpTnmUpIZFdLlokApsYjwPAve5YtSIiSVAFbe3YjoCBdTYuHI2hK9P8f+kUXI94nrXfrFaXcWRcfoBJ0hD5VRFV2hGqojiu7RA3pCz87MeXRenNdla8ZzRyhH3DePgGw8pTc</latexit>

Image: NASA, DOE, Fermi-LAT collaboration

Fermi Large Area Telescope (Fermi-LAT)

THERMAL WIMP DARK MATTER (DM)

Indirect detection

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

2

SLIDE 3

3

Inner Galaxy: Dark matter halo density sharply peaked toward Galactic Center

DARK MATTER ANNIHILATION IN THE SKY

L . P i e r i e t a l . [ 0 9 0 8 . 0 1 9 5 ]

Dwarf galaxies: dark matter-dominated satellites of the main galaxy

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 4

3

Inner Galaxy: Dark matter halo density sharply peaked toward Galactic Center

DARK MATTER ANNIHILATION IN THE SKY

L . P i e r i e t a l . [ 0 9 0 8 . 0 1 9 5 ]

Dwarf galaxies: dark matter-dominated satellites of the main galaxy

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 5

INDIRECT DETECTION BENCHMARK: DWARF GALAXIES

4

F e r m i - L AT c o l l a b o r a t i o n a n d D E S c o l l a b o r a t i o n [ 1 6 1 1 . 0 3 1 8 4 ]

Galactic Center Excess (GCE), dark matter interpretation Analyze blank regions of the sky (no systematics)

Low astrophysical backgrounds (dust/gas) compared to other indirect detection targets

→ some of the most stringent and robust constraints

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 6

5

F e r m i - L AT c o l l a b o r a t i o n a n d D E S c o l l a b o r a t i o n [ 1 6 1 1 . 0 3 1 8 4 ]

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

DWARF GALAXY INDIRECT DETECTION CAVEATS

SLIDE 7

5

F e r m i - L AT c o l l a b o r a t i o n a n d D E S c o l l a b o r a t i o n [ 1 6 1 1 . 0 3 1 8 4 ]

Constraints rely on accurate J-factors

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

DWARF GALAXY INDIRECT DETECTION CAVEATS

SLIDE 8

5

F e r m i - L AT c o l l a b o r a t i o n a n d D E S c o l l a b o r a t i o n [ 1 6 1 1 . 0 3 1 8 4 ]

Constraints rely on accurate J-factors

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

DWARF GALAXY INDIRECT DETECTION CAVEATS

Use with caution: “Galaxies for which Published Kinematics May Not Reliably Translate to Masses”

J . D . S I m o n [ 1 9 0 1 . 0 5 4 6 5 ]

SLIDE 9

5

F e r m i - L AT c o l l a b o r a t i o n a n d D E S c o l l a b o r a t i o n [ 1 6 1 1 . 0 3 1 8 4 ]

Constraints rely on accurate J-factors

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

DWARF GALAXY INDIRECT DETECTION CAVEATS

Use with caution: “Galaxies for which Published Kinematics May Not Reliably Translate to Masses”

J . D . S I m o n [ 1 9 0 1 . 0 5 4 6 5 ]

SLIDE 10

INFERRING DWARF GALAXY J-FACTORS

6

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 11

INFERRING DWARF GALAXY J-FACTORS

6

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Important assumptions:

SLIDE 12

INFERRING DWARF GALAXY J-FACTORS

6

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Equilibrium

K . E l - B a d r y e t a l . [ 1 6 1 0 . 0 4 2 3 2 ]

Important assumptions:

SLIDE 13

INFERRING DWARF GALAXY J-FACTORS

6

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Equilibrium

K . E l - B a d r y e t a l . [ 1 6 1 0 . 0 4 2 3 2 ]

Important assumptions:
Spherical system
V. B o n n i v a rd e t a l . [ 1 4 0 7 . 7 8 2 2 ]

SLIDE 14

INFERRING DWARF GALAXY J-FACTORS

6

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Equilibrium

K . E l - B a d r y e t a l . [ 1 6 1 0 . 0 4 2 3 2 ]

Important assumptions:
Spherical system
V. B o n n i v a rd e t a l . [ 1 4 0 7 . 7 8 2 2 ]
Non-rotating system

SLIDE 15

INFERRING DWARF GALAXY J-FACTORS

6

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Equilibrium

K . E l - B a d r y e t a l . [ 1 6 1 0 . 0 4 2 3 2 ]

Important assumptions:
Spherical system
V. B o n n i v a rd e t a l . [ 1 4 0 7 . 7 8 2 2 ]
Non-rotating system
No accounting for binaries

A . W. M c C o n n a c h i e a n d P. C o t e [ 1 0 0 9 . 4 2 0 5 ] M . E . S p e n c e r e t a l . [ 1 8 1 1 . 0 6 5 9 7 ] M . E . S p e n c e r e t a l . [ 1 7 0 6 . 0 4 1 8 4 ] M . E . S p e n c e r e t a l . [ 1 8 1 1 . 0 6 5 9 7 ]

SLIDE 16

INFERRING DWARF GALAXY J-FACTORS

6

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Equilibrium

K . E l - B a d r y e t a l . [ 1 6 1 0 . 0 4 2 3 2 ]

Important assumptions:
Spherical system
V. B o n n i v a rd e t a l . [ 1 4 0 7 . 7 8 2 2 ]
Non-rotating system
No accounting for binaries

A . W. M c C o n n a c h i e a n d P. C o t e [ 1 0 0 9 . 4 2 0 5 ] M . E . S p e n c e r e t a l . [ 1 8 1 1 . 0 6 5 9 7 ] M . E . S p e n c e r e t a l . [ 1 7 0 6 . 0 4 1 8 4 ] M . E . S p e n c e r e t a l . [ 1 8 1 1 . 0 6 5 9 7 ]

How does this affect dark matter annihilation (decay) constraints?

SLIDE 17

INFERRING DWARF GALAXY J-FACTORS

6

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Equilibrium

K . E l - B a d r y e t a l . [ 1 6 1 0 . 0 4 2 3 2 ]

Important assumptions:
Spherical system
V. B o n n i v a rd e t a l . [ 1 4 0 7 . 7 8 2 2 ]
Non-rotating system
No accounting for binaries

A . W. M c C o n n a c h i e a n d P. C o t e [ 1 0 0 9 . 4 2 0 5 ] M . E . S p e n c e r e t a l . [ 1 8 1 1 . 0 6 5 9 7 ] M . E . S p e n c e r e t a l . [ 1 7 0 6 . 0 4 1 8 4 ] M . E . S p e n c e r e t a l . [ 1 8 1 1 . 0 6 5 9 7 ]

How does this affect dark matter annihilation (decay) constraints? Spherical Jeans equation 3d radial velocity dispersion, stellar density profile Halo mass (DM density profile)

SLIDE 18

INFERRING DWARF GALAXY J-FACTORS

6

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Equilibrium

K . E l - B a d r y e t a l . [ 1 6 1 0 . 0 4 2 3 2 ]

Important assumptions:
Spherical system
V. B o n n i v a rd e t a l . [ 1 4 0 7 . 7 8 2 2 ]
Non-rotating system
No accounting for binaries

A . W. M c C o n n a c h i e a n d P. C o t e [ 1 0 0 9 . 4 2 0 5 ] M . E . S p e n c e r e t a l . [ 1 8 1 1 . 0 6 5 9 7 ] M . E . S p e n c e r e t a l . [ 1 7 0 6 . 0 4 1 8 4 ] M . E . S p e n c e r e t a l . [ 1 8 1 1 . 0 6 5 9 7 ]

How does this affect dark matter annihilation (decay) constraints? In practice, observe line-of-sight projected quantities Line-of-sight projected velocity dispersion, stellar density profile Halo mass (DM density profile) J-factor ( + particle physics) Constraint on DM annihilation ( ) degeneracy M − β ρ − β

J . I . R e a d a n d P. S t e g e r [ 1 7 0 1 . 0 4 8 3 3 ]

SLIDE 19

OUTLINE OF METHODOLOGY

7

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 20

OUTLINE OF METHODOLOGY

7

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

The question

SLIDE 21

OUTLINE OF METHODOLOGY

7

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

The question In the cleanest scenario, what is the effect of binaries on dwarf galaxy J-factors?

SLIDE 22

OUTLINE OF METHODOLOGY

7

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

The question In the cleanest scenario, what is the effect of binaries on dwarf galaxy J-factors? The (simulated) data

SLIDE 23

OUTLINE OF METHODOLOGY

7

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Gaia Challenge spherical mocks:

Plummer light profile
Cusped DM profile
No velocity anisotropy

The question In the cleanest scenario, what is the effect of binaries on dwarf galaxy J-factors? The (simulated) data

G a i a C h a l l e n g e h t t p : / / a s t ro w i k i . p h . s u r re y. a c . u k / d o k u w i k i M . G . Wa l k e r a n d J . P e ñ a r r u b i a [ 1 1 0 8 . 2 4 0 4 ]

SLIDE 24

OUTLINE OF METHODOLOGY

7

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Gaia Challenge spherical mocks:

Plummer light profile
Cusped DM profile
No velocity anisotropy

The question In the cleanest scenario, what is the effect of binaries on dwarf galaxy J-factors? The (simulated) data

G a i a C h a l l e n g e h t t p : / / a s t ro w i k i . p h . s u r re y. a c . u k / d o k u w i k i M . G . Wa l k e r a n d J . P e ñ a r r u b i a [ 1 1 0 8 . 2 4 0 4 ]

The model

SLIDE 25

OUTLINE OF METHODOLOGY

7

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Nested Plummer light profile: sum of 2 Plummer, independent norm & scale
Broken power law DM profile: spans cusped ↔ cored

Gaia Challenge spherical mocks:

Plummer light profile
Cusped DM profile
No velocity anisotropy

The question In the cleanest scenario, what is the effect of binaries on dwarf galaxy J-factors? The (simulated) data

G a i a C h a l l e n g e h t t p : / / a s t ro w i k i . p h . s u r re y. a c . u k / d o k u w i k i M . G . Wa l k e r a n d J . P e ñ a r r u b i a [ 1 1 0 8 . 2 4 0 4 ]

The model

J . I . R e a d a n d P. S t e g e r [ 1 7 0 1 . 0 4 8 3 3 ]

SLIDE 26

OUTLINE OF METHODOLOGY

7

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Nested Plummer light profile: sum of 2 Plummer, independent norm & scale
Broken power law DM profile: spans cusped ↔ cored

Gaia Challenge spherical mocks:

Plummer light profile
Cusped DM profile
No velocity anisotropy

The question In the cleanest scenario, what is the effect of binaries on dwarf galaxy J-factors? The (simulated) data

G a i a C h a l l e n g e h t t p : / / a s t ro w i k i . p h . s u r re y. a c . u k / d o k u w i k i M . G . Wa l k e r a n d J . P e ñ a r r u b i a [ 1 1 0 8 . 2 4 0 4 ]

The model The method

J . I . R e a d a n d P. S t e g e r [ 1 7 0 1 . 0 4 8 3 3 ]

SLIDE 27

OUTLINE OF METHODOLOGY

7

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Nested Plummer light profile: sum of 2 Plummer, independent norm & scale
Broken power law DM profile: spans cusped ↔ cored

Gaia Challenge spherical mocks:

Plummer light profile
Cusped DM profile
No velocity anisotropy

The question In the cleanest scenario, what is the effect of binaries on dwarf galaxy J-factors? The (simulated) data

G a i a C h a l l e n g e h t t p : / / a s t ro w i k i . p h . s u r re y. a c . u k / d o k u w i k i M . G . Wa l k e r a n d J . P e ñ a r r u b i a [ 1 1 0 8 . 2 4 0 4 ]

The model The method Run through pipeline to get J-factors for Gaia Challenge

J . I . R e a d a n d P. S t e g e r [ 1 7 0 1 . 0 4 8 3 3 ]

SLIDE 28

OUTLINE OF METHODOLOGY

7

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Nested Plummer light profile: sum of 2 Plummer, independent norm & scale
Broken power law DM profile: spans cusped ↔ cored

Gaia Challenge spherical mocks:

Plummer light profile
Cusped DM profile
No velocity anisotropy

The question In the cleanest scenario, what is the effect of binaries on dwarf galaxy J-factors? The (simulated) data

G a i a C h a l l e n g e h t t p : / / a s t ro w i k i . p h . s u r re y. a c . u k / d o k u w i k i M . G . Wa l k e r a n d J . P e ñ a r r u b i a [ 1 1 0 8 . 2 4 0 4 ]

The model The method Run through pipeline to get J-factors for Gaia Challenge → Repeat analysis with injected binary motions

J . I . R e a d a n d P. S t e g e r [ 1 7 0 1 . 0 4 8 3 3 ]

SLIDE 29

8

STEP 1: LIGHT PROFILE FIT

10−2 10−1 100 101 R [kpc] 10−2 100 102 104 106 ΣPlumSphere(R) [counts · kpc−2]

1 component 2 components 3 components

log10(M0) = 3.27+0.00

−0.00

3 . 2 5 6 3 . 2 6 4 3 . 2 7 2 3 . 2 8

log10(M0)

− . 8 7 − . 8 6 5 − . 8 6

log10(a0)

− . 8 7 − . 8 6 5 − . 8 6

log10(a0)

log10(a0) = −0.87+0.00

−0.00

log10(M0) = 3.41+0.00

−0.00

− . 9 6 − . 9 − . 8 4 − . 7 8

log10(M1

M0)

log10(M1

M0) = −0.89+0.02 −0.02

− . 6 2 − . 6 − . 5 8 − . 5 6

log10(a0)

log10(a0) = −0.60+0.01

−0.01

3 . 3 9 3 . 4 5 3 . 4 2

log10(M0)

− . 5 7 − . 5 5 5 − . 5 4

log10(a1

a0)

− . 9 6 − . 9 − . 8 4 − . 7 8

log10(M1

M0)

− . 6 2 − . 6 − . 5 8 − . 5 6

log10(a0)

− . 5 7 − . 5 5 5 − . 5 4

log10(a1

a0)

log10(a1

a0) = −0.56+0.00 −0.00

Binned likelihood fit to stellar positions
Fits tend to be well-constrained

1 Plummer component 2 Plummer components

3000 tracers

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Preliminary Preliminary Preliminary

SLIDE 30

9

STEP 1: LIGHT PROFILE FIT

10−1 100 R [kpc] 10−2 10−1 100 101 102 103 104 ΣPlumSphere(R) [counts · kpc−2]

1 component 2 components 3 components

log10(M0) = 2.48+0.01

−0.01

2 . 4 5 2 . 4 7 5 2 . 5

log10(M0)

− . 9 − . 8 8 − . 8 6

log10(a0)

− . 9 − . 8 8 − . 8 6

log10(a0)

log10(a0) = −0.88+0.01

−0.01

log10(M0) = 2.64+0.01

−0.01

− 1 . 2 − 1 . 5 − . 9

log10(M1

M0)

log10(M1

M0) = −1.08+0.04 −0.04

− . 6 8 − . 6 4 − . 6 − . 5 6

log10(a0)

log10(a0) = −0.62+0.01

−0.01

2 . 6 2 . 6 4 2 . 6 8

log10(M0)

− . 8 − . 7 5 − . 7 − . 6 5

log10(a1

a0)

− 1 . 2 − 1 . 5 − . 9

log10(M1

M0)

− . 6 8 − . 6 4 − . 6 − . 5 6

log10(a0)

− . 8 − . 7 5 − . 7 − . 6 5

log10(a1

a0)

log10(a1

a0) = −0.74+0.02 −0.02

Binned likelihood fit to stellar positions
Fits tend to be well-constrained

1 Plummer component 2 Plummer components

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

500 tracers

Preliminary Preliminary Preliminary

SLIDE 31

10

STEP 2: VELOCITY DISPERSION FIT

Use Step 1 to constrain light profile parameters: float over middle 95% posterior

parameter ranges from light profile fit

Extract posterior distributions for DM parameters

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

3000 tracers

Optimistically assume velocity error of 0.2 km/s

103 102 101 100 101 R [kpc] 104 106 108 1010 ρDM(R) [M/kpc3]

Posteriors, 3 break(s) Theory (Zhao)

Preliminary

SLIDE 32

11

STEP 2: VELOCITY DISPERSION FIT

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

101 100 R [kpc] 105 106 107 108 109 1010 ρDM(R) [M/kpc3]

Posteriors, 3 break(s) Theory (Zhao)

500 tracers

Use Step 1 to constrain light profile parameters: float over middle 95% posterior

parameter ranges from light profile fit

Extract posterior distributions for DM parameters
Optimistically assume velocity error of 0.2 km/s

Preliminary

SLIDE 33

12

INJECTING BINARIES

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

As a starting point: Model and code for modeling binary motion from Spencer+ 2018 (If you have a favorite binary model we should try, please let us know!)

M . E . S p e n c e r e t a l . [ 1 8 1 1 . 0 6 5 9 7 ]

SLIDE 34

13

EFFECT OF UNMODELED BINARIES ON J-FACTORS

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

7.32 +0.02

−0.02

9.14 +0.25

−0.26

19.07 +0.07

−0.05

Ntracers log10 (M(<Rmax)/M⊙) log10 (M(<R1/2)/M⊙) log10 (J/(GeV2 cm−5)) fbinary 3000 3000 500 500 1 1 7.38 +0.17

−0.19

19.16 +0.09

−0.06

9.19 +0.23

−0.20

7.25 +0.04

−0.05

18.99 +0.19

−0.10

8.81 +0.18

−0.17

7.30 +0.04

−0.04

19.06 +0.16

−0.10

8.76 +0.21

−0.18 1 fbinary 18.8 18.9 19.0 19.1 19.2 19.3 19.4 log10(J [GeV2 cm−5]) 3000 tracers 1 fbinary 500 tracers

Preliminary Preliminary

SLIDE 35

13

EFFECT OF UNMODELED BINARIES ON J-FACTORS

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

7.32 +0.02

−0.02

9.14 +0.25

−0.26

19.07 +0.07

−0.05

Ntracers log10 (M(<Rmax)/M⊙) log10 (M(<R1/2)/M⊙) log10 (J/(GeV2 cm−5)) fbinary 3000 3000 500 500 1 1 7.38 +0.17

−0.19

19.16 +0.09

−0.06

9.19 +0.23

−0.20

7.25 +0.04

−0.05

18.99 +0.19

−0.10

8.81 +0.18

−0.17

7.30 +0.04

−0.04

19.06 +0.16

−0.10

8.76 +0.21

−0.18 1 fbinary 18.8 18.9 19.0 19.1 19.2 19.3 19.4 log10(J [GeV2 cm−5]) 3000 tracers 1 fbinary 500 tracers

Preliminary Preliminary

In progress:

Even more statistics? (Computationally expensive)
Injecting binary motion only in certain regions
More flexible DM profile? (Currently 3 breaks)
Other systematics?

SLIDE 36

14

CONCLUSIONS & EXTENSIONS

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

In very simple examples on mock data, the presence of unmodeled binaries can bias

estimates of dwarf galaxy J-factors ⇒ bias dark matter constraints derived

Effect of binaries becomes more drastic with increased statistics
With more stars measured and more accurate measurements, will this become a

more measurable effect?

With future multi-epoch binary measurements, could exclude confirmed binaries

from analysis

Statistical uncertainties in dwarf galaxy dark matter constraints need to be better

understood and characterized

Other important systematics: tidal disruption, deviations from equilibrium, non-

sphericity, …

J . D . S i m o n e t a l . [ 1 9 0 3 . 0 4 7 4 3 ]

SLIDE 37

BACKUP SLIDES

15

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 38

INDIRECT DETECTION BENCHMARK: DWARF GALAXIES

16

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 39

INFERRING DWARF GALAXY J-FACTORS

17

1 ν d dr(ν ¯ vr

2) + 2 βani ¯

vr

2

r = − GM(r) r2

M(r) ≈ 4π∫

r

ρDM(r′) r′2dr′

3d radial velocity dispersion velocity anisotropy

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 40

INFERRING DWARF GALAXY J-FACTORS

17

σ2

p(R) =

2 Σ(R) ∫

∞ R (1 − βani(r)R2

r2 ) ν(r)¯ v2

r(r)

rdr r2 − R2

Spherical Jeans equation

1 ν d dr(ν ¯ vr

2) + 2 βani ¯

vr

2

r = − GM(r) r2

M(r) ≈ 4π∫

r

ρDM(r′) r′2dr′

3d radial velocity dispersion velocity anisotropy

In practice, observe line-of-sight projected quantities

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 41

INFERRING DWARF GALAXY J-FACTORS

17

σ2

p(R) =

2 Σ(R) ∫

∞ R (1 − βani(r)R2

r2 ) ν(r)¯ v2

r(r)

rdr r2 − R2

Spherical Jeans equation

1 ν d dr(ν ¯ vr

2) + 2 βani ¯

vr

2

r = − GM(r) r2

M(r) ≈ 4π∫

r

ρDM(r′) r′2dr′

3d radial velocity dispersion velocity anisotropy

In practice, observe line-of-sight projected quantities

projected velocity dispersion

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 42

INFERRING DWARF GALAXY J-FACTORS

17

σ2

p(R) =

2 Σ(R) ∫

∞ R (1 − βani(r)R2

r2 ) ν(r)¯ v2

r(r)

rdr r2 − R2

Spherical Jeans equation

2d stellar density (“light profile”)

1 ν d dr(ν ¯ vr

2) + 2 βani ¯

vr

2

r = − GM(r) r2

M(r) ≈ 4π∫

r

ρDM(r′) r′2dr′

3d radial velocity dispersion velocity anisotropy

In practice, observe line-of-sight projected quantities

projected velocity dispersion

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 43

Nested Plummer light profile
18

FIDUCIAL SETUP

J . I . R e a d a n d P. S t e g e r [ 1 7 0 1 . 0 4 8 3 3 ]

Gaia challenge spherical mocks:
Plummer light profile
Cusped DM profile
Isotropic

Σ(R) =

Np

∑

i=1

Mia2

i

π(a2

i + R2)2

ν(r) =

Np

∑

i=1

3Mi 4πa3

i

× (1 + r2 a2

i ) −5/2

Abel transform

DM profile can span cusped ↔ cored

Broken power law DM profile: analytic formula for enclosed DM mass

ρ(r) = ρ0 ( r r0)

−γ0

ρ0 ( r rj)

−γj+1 j

∏

n=1 (

rn rn−1 )

−γn

r < r0 rj < r < rj+1

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 44

Nested Plummer light profile
18

FIDUCIAL SETUP

J . I . R e a d a n d P. S t e g e r [ 1 7 0 1 . 0 4 8 3 3 ]

Gaia challenge spherical mocks:
Plummer light profile
Cusped DM profile
Isotropic

Σ(R) =

Np

∑

i=1

Mia2

i

π(a2

i + R2)2

ν(r) =

Np

∑

i=1

3Mi 4πa3

i

× (1 + r2 a2

i ) −5/2

Abel transform

DM profile can span cusped ↔ cored

Broken power law DM profile: analytic formula for enclosed DM mass

ρ(r) = ρ0 ( r r0)

−γ0

ρ0 ( r rj)

−γj+1 j

∏

n=1 (

rn rn−1 )

−γn

r < r0 rj < r < rj+1

Analytic formulas for 2d/3d stellar density and enclosed DM mass profiles → more computationally tractable

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 45

Nested Plummer light profile
18

FIDUCIAL SETUP

J . I . R e a d a n d P. S t e g e r [ 1 7 0 1 . 0 4 8 3 3 ]

Gaia challenge spherical mocks:
Plummer light profile
Cusped DM profile
Isotropic

Σ(R) =

Np

∑

i=1

Mia2

i

π(a2

i + R2)2

ν(r) =

Np

∑

i=1

3Mi 4πa3

i

× (1 + r2 a2

i ) −5/2

Abel transform

DM profile can span cusped ↔ cored

Broken power law DM profile: analytic formula for enclosed DM mass

ρ(r) = ρ0 ( r r0)

−γ0

ρ0 ( r rj)

−γj+1 j

∏

n=1 (

rn rn−1 )

−γn

r < r0 rj < r < rj+1

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

SLIDE 46

19

STEP 1: LIGHT PROFILE FIT

10−1 100 r [kpc] 10−2 10−1 100 101 102 103 ΣPlumSphere(r) [counts · kpc−2]

1 component 2 components 3 components

log10(M0) = 1.88+0.03

−0.03

1 . 7 6 1 . 8 4 1 . 9 2 2 .

log10(M0)

− . 8 8 − . 8 − . 7 2

log10(a0)

− . 8 8 − . 8 − . 7 2

log10(a0)

log10(a0) = −0.77+0.02

−0.02

log10(M0) = 1.87+0.05

−0.10

− 1 . 5 − 1 . − . 5 .

log10(M1

M0)

log10(M1

M0) = −0.48+0.30 −0.29

− . 6 − . 4 − . 2 .

log10(a0)

log10(a0) = −0.57+0.11

−0.08

1 . 7 1 . 8 1 . 9 2 .

log10(M0)

− 1 . 5 − 1 . − . 5 .

log10(a1

a0)

− 1 . 5 − 1 . − . 5 .

log10(M1

M0)

− . 6 − . 4 − . 2 .

log10(a0)

− 1 . 5 − 1 . − . 5 .

log10(a1

a0)

log10(a1

a0) = −0.53+0.10 −0.35

Binned likelihood fit to stellar positions
Fits tend to be well-constrained

1 Plummer component 2 Plummer components

100 stars

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

Preliminary Preliminary Preliminary

SLIDE 47

20

STEP 2: VELOCITY DISPERSION FIT

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

3000 stars

ln(ρ0) = 23.34+1.37

−2.41

− 6 . 5 − 6 . − 5 . 5 ln(r0)

ln(r0) = −6.34+0.74

−0.50

− . 8 − . 4 . . 4 γ0

γ0 = −0.50+0.75

−0.39

. 5 1 . 1 . 5 γ1 − γ0

γ1 − γ0 = 0.69+0.69

−0.50

2 . 8 3 . 2 3 . 6 ln(r1

r0)

ln(r1

r0) = 3.39+0.36 −0.54

. 5 1 . 1 . 5 2 . γ2 − γ1

γ2 − γ1 = 1.20+0.76

−0.69

2 . 8 3 . 2 3 . 6 ln(r2)

ln(r2) = 3.71+0.23

−0.61

2 1 . 2 2 . 5 2 4 . ln(ρ0) 2 4 6 γ3 − γ2 − 6 . 5 − 6 . − 5 . 5 ln(r0) − . 8 − . 4 . . 4 γ0 . 5 1 . 1 . 5 γ1 − γ0 2 . 8 3 . 2 3 . 6 ln(r1

r0)

. 5 1 . 1 . 5 2 . γ2 − γ1 2 . 8 3 . 2 3 . 6 ln(r2) 2 4 6 γ3 − γ2

γ3 − γ2 = 1.39+4.28

−1.10

log10(M0) = 3.89+0.00

−0.00

− . 5 4 4 − . 5 4 − . 5 3 6 − . 5 3 2

log10(a0)

log10(a0) = −0.54+0.01

−0.00

− . 5 7 − . 5 5 5 − . 5 4

log10(M1)

log10(M1) = −0.55+0.02

−0.02

3 . 8 8 3 3 . 8 8 6 3 . 8 8 9 3 . 8 9 2

log10(M0)

− . 4 7 7 5 − . 4 7 5 − . 4 7 2 5 − . 4 7

log10(a1)

− . 5 4 4 − . 5 4 − . 5 3 6 − . 5 3 2

log10(a0)

− . 5 7 − . 5 5 5 − . 5 4

log10(M1)

− . 4 7 7 5 − . 4 7 5 − . 4 7 2 5 − . 4 7

log10(a1)

log10(a1) = −0.47+0.00

−0.00

Preliminary Preliminary

SLIDE 48

21

STEP 2: VELOCITY DISPERSION FIT

L . J . C H A N G | S M A L L G A L A X I E S , C O S M I C Q U E S T I O N S 2 0 1 9

500 stars

ln(ρ0) = 20.27+1.17

−2.82

− 3 . − 1 . 5 . ln(r0)

ln(r0) = −2.56+2.27

−1.06

− . 6 . . 6 1 . 2 γ0

γ0 = 0.54+0.71

−1.12

. 8 1 . 6 2 . 4 γ1 − γ0

γ1 − γ0 = 0.91+1.07

−0.64

1 2 3 ln(r1

r0)

ln(r1

r0) = 1.72+1.33 −1.26

2 4 6 γ2 − γ1

γ2 − γ1 = 1.32+3.60

−1.03

. 8 1 . 6 2 . 4 ln(r2)

ln(r2) = 0.79+1.37

−0.62

1 8 . 1 9 . 5 2 1 . ln(ρ0) 2 . 5 5 . 7 . 5 γ3 − γ2 − 3 . − 1 . 5 . ln(r0) − . 6 . . 6 1 . 2 γ0 . 8 1 . 6 2 . 4 γ1 − γ0 1 2 3 ln(r1

r0)

2 4 6 γ2 − γ1 . 8 1 . 6 2 . 4 ln(r2) 2 . 5 5 . 7 . 5 γ3 − γ2

γ3 − γ2 = 3.70+4.20

−2.75

log10(M0) = 2.93+0.01

−0.01

− . 6 3 − . 6 2 − . 6 1 − . 6

log10(a0)

log10(a0) = −0.62+0.02

−0.01

− 1 . − . 9 5 − . 9

log10(M1)

log10(M1) = −0.97+0.07

−0.06

2 . 9 2 2 . 9 2 8 2 . 9 3 6 2 . 9 4 4

log10(M0)

− . 6 − . 5 8 5 − . 5 7 − . 5 5 5

log10(a1)

− . 6 3 − . 6 2 − . 6 1 − . 6

log10(a0)

− 1 . − . 9 5 − . 9

log10(M1)

− . 6 − . 5 8 5 − . 5 7 − . 5 5 5

log10(a1)

log10(a1) = −0.57+0.02

−0.02

Preliminary Preliminary