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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).