The Shape of Things to Come? - PDF Free Download

The Shape of Things to Come? Martin Hendry Dept 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 cuspy? NFW profile (1996) ρ(r) = δ s (r /r s )(1+ r /r s ) 2 1 Density cusp ρ r for CDM models at small r But rotation curve observations of low surface brightness galaxies in poor agreement with NFW profile. A problem with CDM?... (From Kuzio de Naray et al. 2006)

Λ CDM From Lineweaver (1998)

The 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. Ω Ω Λ 1 0.8 0.6 0.4 Ω m 0.2 0 0 1 2 3 4 5 Present-day R / R 0 When did this turnaround occur? Related questions for dark energy equation of state w(z)

Deconvolving projected brightness profiles Projected SB profile Luminosity density Formal solution: Forward fitting of parametric model Regularised inversion Gebhardt et al. 1996

Non-thermal bremsstrahlung spectra 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.

Non-thermal bremsstrahlung spectra from solar flares ε nv I ( ε ) = F de kev 2 4π r -2-1 -1 ( E) Q( ε, E) photons m s Observed photon spectrum Electron energy spectrum Bremsstrahlung cross-section 2 Q m ec + ε 0 1 1 E Bethe-Heitler cross-section Q( ε, E) = log (e.g. Piana 1994) ε E 1 1 ε E Can be reformulated as (see Brown 1971) where ( ε ) ε ( ε ) ψ I ψ ( ε ) = ε F ( E) E ε de 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 inversion techniques on mock spectra. Formulate as matrix equation Solve by data adaptive binning or by minimising Regularisation matrix To date, no inclusion of shape information.

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 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 n o Line of sight d u est cz u true n n n r See Tom Loredo s talk for an overview of the various biases and systematics involved in reconstructing

What ever happened to the peculiar velocity field?... 1 CDF of # papers on ADS 0.8 0.6 0.4 0.2 Peculiar velocity (258) CMBR (2181) 0 1989 1991 1993 1995 1997 1999 2001 2003 2005 Year

What ever happened to the peculiar velocity field?... 1 CDF of # papers on ADS 0.8 0.6 0.4 0.2 Peculiar velocity (258) CMBR (2181) 0 1989 1991 1993 1995 1997 1999 2001 2003 2005 Year 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 fields PSCz predicted velocity field (15,000 galaxies)

Comparing predicted and observed peculiar velocity fields PSCz predicted velocity field (15,000 galaxies) 6dFGS observed velocity field (15,000 galaxies)

Velocity velocity comparisons Archetype is VELMOD (Willick et al 1998) Maximise likelihood of observing distance indicator data, given a velocity field model L L Forward VELMOD = p m i η, cz ; Θ) ( i i Inverse VELMOD = p η m, cz ; Θ) ( i i i Θ = parameters of distance indicator and velocity model VELMOD also requires a parametric model for S( m, η, r), LF, p( cz r)

Bulk Flow Statistics v 2 ( R) = 2 H 0 Ω 2 2π 1.2 m 0 ~ P( k) W 2 ( kr) dk Model d i = cz i V B nˆ and fit by minimising parameters of distance indicator Compare with theoretical predictions

Constraining bulk flows with peculiar velocities 10 2 peculiar velocities with infinite precision Following Colless (2003)

Constraining bulk flows with peculiar velocities 10 2 peculiar velocities with 20% precision Following Colless (2003)

Constraining bulk flows with peculiar velocities 10 3 peculiar velocities with 10% precision Following Colless (2003)

Constraining bulk flows with peculiar velocities 10 4 peculiar velocities with 20% precision Following Colless (2003)

What can peculiar velocities do for us? Burkey & Taylor (2004) perform a Fisher matrix analysis to compare parameter constraints for a 6dF z-only survey and z+v survey Parameters: amplitude of the galaxy power spectrum A g = ba m power spectrum shape parameter redshift-space distortion parameter Γ = Ω m h 0.6 β Ωm b correlation between luminous and dark matter r g

Summary 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).