The Shape of Things to Come?

Martin HendryDept of Physics and Astronomy

University of Glasgow, UK

• Some other examples of shape restriction / Wicksell problems

• Wider perspective on mapping dark matter

Are dark matter halos Are dark matter halos cuspycuspy??

ρ(r) = δs

(r /rs)(1+ r /rs)2

NFW profile (1996)

Density ‘cusp’ at smallfor CDM models

But rotation curve observations of low surface brightness galaxies in poor agreement with NFW profile.

A problem with CDM?...

1−∝ rρ r

(From Kuzio de Naray et al. 2006)

CDMΛ

From Lineweaver (1998)

The accelerating universeThe accelerating universe

In a flat, Friedmann model universe with dark matter and dark energy, the Universe is predicted to change from matter dominated to dark energy dominated fairly recently.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Ω

mΩ

ΛΩ

Present-day0/ RR

• When did this ‘turnaround’ occur?

• Related questions for dark energy equation of state w(z)

DeconvolvingDeconvolving projected brightness profilesprojected brightness profiles

Luminosity densityProjected SB profile

• Forward fitting of parametric model

• Regularised inversion

Formal solution:

Gebhardt et al. 1996

NonNon--thermal thermal bremsstrahlungbremsstrahlung spectra from solar flaresspectra from solar flares

In the past 10 years, e.g. SOHO and RHESSI have monitored the solar atmosphere with unprecedented spectral and imaging resolution.

Considerable recent interest in reconstructing the distribution of electron energies.

This problem can be cast as an Abel integral equation.

NonNon--thermal thermal bremsstrahlungbremsstrahlung spectra from solar flaresspectra from solar flares

( ) 1-1-2-2 keVsmphotons),(

4)( dEEQEF

rVnI ∫

∞

=ε

επ

ε

Bremsstrahlung cross-sectionElectron energyspectrum

Observed photonspectrum

Bethe-Heitler cross-section (e.g. Piana 1994)⎟⎟⎠

⎞⎜⎜⎝

⎛

−−−+

=E

Ee

EcmQEQ

ε

ε

εε

1111log),(

20

( ) dEE

EF∫∞

−=

ε εεψ )(Can be reformulated as (see Brown 1971)

where )()( εεεψ I ′−∝

Solution by regularised inversion (see e.g. Brown et al. 2006)

Non-thermal bremsstrahlung spectra from solar flares

Brown et al. (2006)

Comparison of different inversiontechniques on mock spectra.

Formulate as matrix equation

Solve by ‘data adaptive binning’or by minimising

To date, no inclusion of shape information.

Regularisation matrix

Mapping the dark matter distribution on larger scalesMapping the dark matter distribution on larger scales

One (erstwhile popular) method is to use z-independent galaxy distance indicators

If we estimate r we also get an estimate of the l.o.s. component

• Our ‘observed’ galaxy peculiar velocities are directly probing the distribution of matter.

• Spatial distribution of galaxies is a biased tracer of the mass

• Comparison of observed and predicted peculiar velocities can test models for the relationship between luminous and dark matter galaxy formation

Mapping the dark matter distribution on larger scales

One (erstwhile popular) method is to use z-independent galaxy distance indicators

If we estimate r we also get an estimate of the l.o.s. component

Line of sighto n

duest

n

cz

Mapping the dark matter distribution on larger scales

One (erstwhile popular) method is to use z-independent galaxy distance indicators

If we estimate r we also get an estimate of the l.o.s. component

Line of sighto n

duest

n

czLine of sighto n n

duest

n

cz rutrue

Mapping the dark matter distribution on larger scales

One (erstwhile popular) method is to use z-independent galaxy distance indicators

If we estimate r we also get an estimate of the l.o.s. component

See Tom Loredo’s talk for an overview of the various biases and systematics involved in reconstructing

Line of sighto n

duest

n

czLine of sighto n n

duest

n

cz rutrue

What ever happened to the peculiar velocity field?....

0

0.2

0.4

0.6

0.8

1

1989 1991 1993 1995 1997 1999 2001 2003 2005

Year

CDF

of #

pap

ers

on A

DS

Peculiar velocity (258)CMBR (2181)

What ever happened to the peculiar velocity field?....What ever happened to the peculiar velocity field?....

0

0.2

0.4

0.6

0.8

1

1989 1991 1993 1995 1997 1999 2001 2003 2005

Year

CDF

of #

pap

ers

on A

DS

Peculiar velocity (258)CMBR (2181)

But the potential advantages of peculiar velocities as a direct tracer of the dark matter distribution remain true…

2,000 ° 10,000°

17,000°

Comparing predicted and observed peculiar velocity fieldsComparing predicted and observed peculiar velocity fields

PSCz predictedvelocity field

(15,000 galaxies)

Comparing predicted and observed peculiar velocity fieldsComparing predicted and observed peculiar velocity fields

PSCz predictedvelocity field

(15,000 galaxies)

6dFGS observed

velocity field (15,000

galaxies)

Velocity Velocity –– velocity comparisons velocity comparisons

Archetype is VELMOD (Willick et al 1998)

‘Forward’ VELMOD

∏ Θ= );,|( iii czmpL ηMaximise likelihood of observing distance indicator data, given a velocity field model ‘Inverse’ VELMOD

∏ Θ= );,|( iii czmpL η

Θ = parameters of distance indicator and velocity model

VELMOD also requires a parametric model for)|(,LF),,,( rczprmS η

Velocity Velocity –– velocity comparisons velocity comparisons

Archetype is VELMOD (Willick et al 1998)

‘Forward’ VELMOD

∏ Θ= );,|( iii czmpL ηMaximise likelihood of observing distance indicator data, given a velocity field model ‘Inverse’ VELMOD

∏ Θ= );,|( iii czmpL η

Θ = parameters of distance indicator and velocity model

VELMOD also requires a parametric model for)|(,LF),,,( rczprmS η

See also Poster # 14

See also Poster # 14

Bulk Flow StatisticsBulk Flow Statistics

∫∞Ω

=0

22

2.1202 )(~)(2

)(v dkkRWkPHR m

π

Model

and fit by minimising parametersof distance indicator

Compare with theoretical predictions

nVB ˆ⋅−= ii czd

Constraining bulk flows with peculiar velocitiesConstraining bulk flows with peculiar velocities

102 peculiar velocities with infinite precision

Following Colless (2003)

Constraining bulk flows with peculiar velocitiesConstraining bulk flows with peculiar velocities

102 peculiar velocities with 20% precision…

Following Colless (2003)

Constraining bulk flows with peculiar velocitiesConstraining bulk flows with peculiar velocities

103 peculiar velocities with 10% precision…

Following Colless (2003)

Constraining bulk flows with peculiar velocitiesConstraining bulk flows with peculiar velocities

104 peculiar velocities with 20% precision

Following Colless (2003)

What can peculiar velocities do for us?What can peculiar velocities do for us?

Burkey & Taylor (2004) perform a Fisher matrix analysis to compare parameterconstraints for a 6dF z-only survey and z+v survey

Parameters:

amplitude of the galaxy power spectrum

power spectrum shape parameter

redshift-space distortion parameter

correlation between luminous and dark matter

mg AbA =

hmΩ=Γ

bm6.0Ω≈β

gr

What can peculiar velocities do for us?What can peculiar velocities do for us?

Burkey & Taylor (2004) perform a Fisher matrix analysis to compare parameterconstraints for a 6dF z-only survey and z+v survey

Parameters:

amplitude of the galaxy power spectrum

power spectrum shape parameter

redshift-space distortion parameter

correlation between luminous and dark matter

Inclusion of peculiar velocity information greatly improves estimation of andand reduces their correlation.

mg AbA =

hmΩ=Γ

bm6.0Ω≈β

gr

grβ

SummarySummary

• A number of astrophysical problems which may be amenable to shape restricted estimation, from solar physics to cosmology

• Many challenging problems in exploiting the potential of galaxy peculiar velocity for mapping dark matter and testing galaxy bias models (see also SPS summary talk).