Dr Martin Hendry Dept of Physics and Astronomy University of Glasgow, UK

Dr Martin Hendry Dept of Physics and Astronomy University of Glasgow, UK martin@astroglaacuk

Dr Martin Hendry, Russell Johnston Dept of Physics and Astronomy University of Glasgow, UK

Who am I? William Thompson (Lord Kelvin) 1824-1907

Who am I? William Thompson (Lord Kelvin) 1824-1907 There is nothing new to be discovered in physics now All that remains is more and more precise measurement

My Research Interests: Cosmology: galaxy distance indicators galaxy redshift surveys cosmological parameters Gravitational wave data analysis: Bayesian inference methods LISA data analysis Multi-messenger astronomy See: wwwastroglaacuk/users/martin/supa-dahtml

Arthur S Eddington George McVitie Fred Menger

From Tom s lecture:

Outline of lecture Some background on surveys, population studies: Notation, nomenclature, limitations, inferential goals With thanks to Tom Loredo! A selective history of galaxy redshift surveys Non-parametric approaches the basic ideas Some examples of robust applications: Testing the completeness of galaxy catalogues Rauzy (2001) Extending the Rauzy method Johnston et al (2007) Constraining galaxy peculiar velocities Rauzy & Hendry (2000) Probing galaxy evolution Efron & Petrosian (1992, 1999) Johnston (2009) Future directions

Fundamental Equations

Edwin Hubble

Spectrum of a Distant Galaxy Spectrum of a nearby galaxy Energy Wavelength

The First Redshift Surveys CfA Survey #1 : 1977-1982 Surveyed ~1100 galaxies Redshift depth, z ~ 005

The First Redshift Surveys CfA Survey #2 : 1985-1995 Surveyed ~18000 galaxies Redshift depth, z ~ 005

1995 (LAS CAMPANAS) The largest structures in LCRS are much smaller than the survey size The size of the structures is similar in both samples LCRS

IRAS PSCz : 1992 1996, 15,000 galaxies Catalogued over 83% of the sky - Largest full sky survey

Surveys The Next Generation The Two Degree Field Galaxy Redshift Survey Ran from 1998 to 2003 Used the multifibre spectrograph on the Anglo Australian Telescope Measured redshifts for ~220,000 galaxies out to z ~ 02 Photometry from the APM galaxy catalogue, m b 1945

The Two Degree Field Galaxy Redshift Survey (2dFGRS)

= 100 Mpc diameter

Surveys The Next Generation The Sloan Digital Sky Survey (SDSS) Most ambitious ongoing survey to date Uses a dedicated 25m telescope on Apache Point NM Measures more than 600 galaxy spectra in a single observation DR7: >900000 galaxies (Abazajian et al 2008)

2dFGRS CfA SDSS SDSS

Following Loredo & Hendry (2009)

Example of the Parametric Approach: Calibrating the Tully-Fisher relation True TF relation M () = a + b Spectral line width parameter, measures galaxy rotation speed = Calibrate in a cluster assume all galaxies at the same distance:- P( m, r) = P( r, m, ) P( r, m, ) dm 2 [ m ( M ( ) μ( r) )] + S( m, )exp 2 2 + S( m, )exp 2 2 2 [ m ( M ( ) μ( r) )] dm Biases fitted parameters, but can be corrected iteratively From Strauss & Willick 1995

In what follows we will focus on the latter robust approach, reviewing recent progress and identifying some challenges for the future

Flux limit

Basic ideas: ( z i, L i ) For each object, at we can determine the maximum redshift at which the object would have been observable (Alternatively, we could consider the minimum luminosity at which the object would be observable at this redshift) This defines a rectangular region within which the sampled population should be complete

Basic ideas: This is the principle behind the V / Vmax test (Kafka, 1967; Schmidt, 1968) Assuming homogeneity, Significant departures from this value can be interpreted as evidence of incompleteness and/or evolution Schmidt outlines a variant used to estimate LF:

Further developments? Huchra & Sargent (1973) Investigated effects of clustering Felten (1976) Showed superiority to Classical approach (eghubble,1936) Avni & Bahcall (1980) Extended to multiple overlapping samples Hudson & Lynden Bell (1991) Recast for the galaxy diameter function Eales (1993) Investigated LF evolution using V / Vmax Qin & Xie (1997) Page & Carrera (2000) Generalised to other distributions: N / Nmax Improved modeling of systematics close to flux limit of the survey Sheth (2007) Extend to photometric redshifts (see later)

Further developments? For a good introduction to the latest in parametric approaches, see Kelly, Fan and Vestergaard (2008) A flexible method of estimating luminosity functions astro-ph/08052946

C- method The idea of a plateau region is also central to the C- (C minus) method of Lynden-Bell (1971) Model:

Further developments? Jackson (1974) Choloniewski (1987) Extended to multiple data sets Simultaneous estimation of the number density Caditz & Petrosian (1993) Introduced Gaussian smoothing Subbarao et al (1996) Extended to photometric redshifts (see later) Perceived weaknesses? Reconstructs CDF of Luminosity Function; errors? Influenced development of semi-parametric SWML method (Efstathiou et al 1988)

SWML method Model: Likelihood:

In what follows we will focus on the latter robust approach, reviewing recent progress and identifying some challenges for the future So, what can we do with product-limit estimators?

Robust Method: the basics Assumption: luminosity function is Universal 1 dp = S( m, r, l, b) ( r, l, b) ( M ) dldbdrdm A Selection effects Spatial distribution Luminosity function Null hypothesis (Rauzy 2001; Johnston et al 2007, Johnston 2009) S( m, z, l, b) ( mlim m) ( z, l, b) Step function Angular and radial Selection function

Distance modulus μ Example 1: robust completeness test M lim (μ i ) (M i, μ ) i S 1 S 2 Absolute magnitude M m lim

Example 1: robust completeness test m * > m lim

Example 1: robust completeness test Define:- = F( M ) F( ) M lim where Can show:- M F ( M ) = f ( x) dx P1: U[0,1], μ P2: uncorrelated

Example 1: robust completeness test Also:- ˆ i = ri n + 1 i n i r i = n( S 1 ) = n( S 1 S2) E i = 1 2 V i = 1 12 n n i i 1 + 1 but only for m* m lim

Define: SSRS2 (da Costa et al 1998) T C = N gal i= 1 (ˆ N i gal i= 1 E( ) = T C V i 0 1 2) var ( ) = 1 T C T C ~ N(0,1) for large N gal m lim =1535 Requires no model for the LF Independent of spatial clustering Independent of redshift selection

Extended to include faint and bright limits Johnson et al (2007)

Also adapted to essentially reverse the roles of M and Z, constructing a test statistic based on the CDF of the Z distribution T Should be equivalent for a universal LF (But what about evolution?)

The doubly-truncated case requires a choice of z (or M) The results are not sensitive to this choice, but can we optimize it eg to maintain constant SNR? (Work in progress)

In some cases the double truncation doesn t affect the faint limit Ignoring bright limit Including bright limit

But for 2dFGRS it clearly does! Impact on 2dFGRS as a probe of evolution, cosmology?

Example 2: Peculiar Velocity Field Modeling IRAS PSCz : 1992 1996, 15,000 galaxies Catalogued over 83% of the sky - Largest full sky survey

Example 2: Peculiar Velocity Field Modeling We can estimate galaxy distance and compare with their redshift to estimate their radial peculiar velocity Comparing observed peculiar velocities with the reconstructed density and velocity field from all-sky redshift surveys let s us constrain the amount of dark matter in the Universe Galaxy density field Peculiar velocity field v pec ( r) 4 r (r )(r - r) 3 g = d 3 r - r v pec = g Redshift distortion parameter

Density density comparisons Archetype is POTENT (Bertschinger & Dekel 1988; Dekel et al 1999) v pec = V ( r) = u( r,, ) dr V r 0 Need only radial components, but everywhere! Interpolate u(r) on a regular grid Smoothing window x v pec = g

Velocity velocity comparisons Archetype is VELMOD (Willick & Strauss 1997, Willick et al 1998) Maximise likelihood of observing Tully-Fisher data, given a velocity field and TF model L L Forward VELMOD = p m i, cz ; ) ( i i Inverse VELMOD = p m, cz ; ) ( i i i = parameters of TF relation and velocity model VELMOD also requires a parametric model for S( m,, r), LF, p( cz r)

Velocity velocity comparisons Triple-value regions VELMOD also requires a parametric model for S( m,, r), LF, p( cz r)

VELMOD Results Consistent picture of ~ 04-06 Significantly discrepant with results of density-density comparison What is the origin of this discrepancy? cf Berlind et al 2001, estimation still OK in non-linear local biasing schemes Rauzy & Hendry (2000) robust approach

Example 2: Peculiar Velocity Field Modeling Assuming define u = log 1 v / cz Can show:- v( r) v ( r) ( ) 5 10 P3: uncorrelated, u Estimate via ( i, u i ) = 0

Example 2: Peculiar Velocity Field Modeling I = 06 ± 01 Strength: Robust support for VELMOD analysis: validity of inhomogeneous Malmquist corrections Weakness: Completeness requirement may restrict sample size and depth From Rauzy & Hendry 2000

Example 3: Modeling Source Evolution The potential of V / Vmax as a diagnostic of source evolution is clear, but how do we test a specific evolutionary model or constrain its parameters? Ideally we would like to test a parametric model of evolution, while adopting a non-parametric approach for everything else! Efron & Petrosian (1992) suggests a possible route, again closely based on product-limit estimators and very similar to C- and robust Null hypothesis: LF and spatial distribution are independent Applied to galaxies and quasars

Example 3: Modeling Source Evolution Permutation test of the independence of y i x i = log( z i ) = m 5log( zi ) Then generalised to i y i = m i c log( z i ) Find value of the parameter for which (NH) c = 0 t c c 5 IF this is evidence for source evolution

Example 3: Modeling Source Evolution Galaxies results: Consistent with c = 5 although rather wide CIs: 90%: [480, 882] Quasars results: c = 0 Consistent with!! 90%: [-040, 011] Evidence for evolution of LF!

Example 3: Modeling Source Evolution Can we adapt EP92 (extended EP99 to double truncation; see also Schafer 2007) to test specific parametric LF evolution models? Evolving characteristic luminosity: which implies that ie Two approaches: (1) Test of independence via correlationcoefficient (cf Example 2) (2) Test of independence via relative entropy

(1) Test of independence via correlationcoefficient We assume (NH, ie correct ) where Define ˆ Then we determine such that = 0

(1) Test of independence via relative entropy ~ U[0,1] ~ U[0,1] Under NH (ie correct ) we expect We divide the unit square into equal cells and define 1 (minus) relative entropy, for trial value of ˆ We determine which minimises S

Example 3: Modeling Source Evolution Should this work? ~ U[0,1] ~ U[0,1] Will using the wrong also significantly affect?

Test using mock catalogs

Some results Mock MGC catalogs, using corr coeff 200 mocks

Some results Mock MGC catalogs, using corr coeff 200 mocks CIs determined from mock histogram

Some results Mock MGC catalogs, using corr coeff 200 mocks

Some results Mock MGC catalogs, using corr coeff 200 mocks CIs determined from mock histograms

Area and Size of Redshift Surveys 100E+09 SDSS photo-z 100E+08 100E+07 No of objects 100E+06 SDSS main SDSS red SDSS abs line 100E+05 CfA+ SSRS 2dF 2dFR 100E+04 SAPM QDOT LCRS 100E+03 100E+04 100E+05 100E+06 100E+07 100E+08 100E+09 100E+10 100E+11 Volume in Mpc 3 From Martinez (2006)

Future Directions: Photometric redshifts Probe strong spectral features (4000Å break) Difference in flux through filters as the galaxy is redshifted Following Lahav (2006)

Bayesian Photo-z likelihood prior Redshift z Benitez 2000 (BPZ)

Example: SDSS data (ugriz; r < 1777) ANNz (5:10:10:1) HYPERZ From Collister & Lahav 2004

Photo-z for SDSS: comparison From Collister & Lahav 2004

LSST Survey of 20,000 sq deg ~3 billion galaxies: Weak lensing maps Baryon acoustic oscillations ~1 million supernovae

Current status and future directions Following Loredo (2006) Important directions for future research!