MARK CORRELATIONS

AND OPTIMAL WEIGHTS

(Cai, Bernstein & Sheth 2010)

Light is a biased tracer

Understanding bias important for understanding mass

Galaxy clustering depends on luminosity, color, type, ...

Zehavi et al. 2010 (SDSS)

The halo-model of clustering• Two types of particles: central + ‘satellite’

• ξobs(r) = ξ1h(r) + ξ2h(r)

• ξ1h(r) = ξcs(r) + ξss(r)

Luminosity dependent clustering

Zehavi et al. 2005 SDSS

• Centre plus Poisson satellite model (two → five free parameters) provides good description• Think of <N|m> as how galaxies ‘weight’ halos (~ TX,YX, YSZ)

cen

tral

sate

llite

s

Zehavi et al. 2010 SDSS

<Ngal|m> = fcen(m) [1 + <Nsat|m>] 1 + m/15mL

Halo model of full SED (colors, sizes ...)• Repeat HOD analysis for each discrete bin in

color and luminosity (and size, and ...)– Many covariant free parameters– Most current parameterizations are not self-

consistent (i.e. summing over colors in a luminosity bin does not give luminosity HOD)

• Use p(SED|L,density) from data– But what choice for density?

• Use bimodality + center-satellite split

L-dependence of clustering +

Bimodal SED-magnitude relation+

Assume p(SED|L) depends neither on mass of host halo, nor on being central

or satellite=

Accurate self-consistent model (Skibba & Sheth 2009)

Tool for understanding galaxy formation,

+ making mock catalogs for

cosmology, cluster finders,

photo-z methods

Mark Correlations• Weight galaxies when measuring clustering

signal; divide by unweighted counts– Simple to incorporate into Halo Model (Sheth 2005)

• WW(r)/DD(r) → no need for random catalog

• Error scales as scatter in weights times scatter in pair counts (Sheth et al. 2005) – If scatter in weights small, can do better than typical

cosmic variance estimate– Basis for recent excitement about constraining

primordial non-Gaussianity from LSS

Sheth, Jimenez, Panter, Heavens 2006

Close pairs (~ galaxies in clusters) more luminous, older than average

SDSS/MOPED +

Mark correlation analysis

Predicted inversion of SFR-density relation at z >1 (if densest regions

today were densest in the past)

Confirmed by zCosmos

• Radius of circle represents total mass in stars formed, in units of average stellar mass formed at same redshift

• Star formation only in less dense regions at low z?

Sheth, Jimenez, Panter, Heavens 2006

Sheth, Jimenez, Panter, Heavens 2006

What is the weight that must be applied to each halo so that the halo catalog best represents the underlying dark matter field?

Options

• Weight each halo equally (~standard)

• Weight each halo by its bias factor – correct if halos are Poisson sampling of mass,

a standard (and incorrect!) assumption

• Weight each halo by its mass – after all, we want the mass (rarely done!)

• Optimal weight must also account for missing mass (mass in ‘dust’)

• Minimize w2 = <(w h – bm)2>

(Hamaus et al. 2010)

• Minimize E2 = <(m – w h)2>/<m2> (Cai et al. 2010)

Mass is mass-weighted halos

• Write ‘Wiener filter’ of model in which some halos are seen, others are not

• Stochasticity E2 = 1 – Cwm2/Cww/Cmm

• Wiener ‘filter’ is that weight which minimizes stochasticity:

w(m) = m/+ fdustbdust b(m) Ph/[1 + ∑nb2 Ph]

Note …

• w(m) = m/+ fdustbdust b(m) Ph/[1 + ∑nb2 Ph]

m/mminmmin+ fdustb2Ph/[1 + nhb2 Ph]

~ 1 + m/mminmminfdustb2Ph/[1 + nhb2 Ph])

~ 1 + m/mmindusth nhb2Ph/[1 + nhb2 Ph])

~ 1 + m/5mmin

E2opt = P1h

dust /Pm + (fdustbdust)2 /[1 + ∑nb2 Ph] (Ph/Pm)

→ 0 when fdust = 0

→ P1hdust /Pm when ∑nb2 Ph» 1

if massive halos missing, E cannot be

made arbitrarily small

→ (fdustbdust)2 /[1 + ∑nb2 Ph] when Ph~Pm

Considerable gains at low masses

E2 = N/(S + N) = 1/(S/N + 1) = 1/(nb2 P + 1) Optimal weighting yields same precision

with fewer objects

(nb2)eff P = 1/E2 – 1

= 3 gives

‘volume limited’

estimate of power

spectrum

Not targeting massive

halos is a bad idea

Targeting galaxies which

prefer low mass halos

is inefficient (costly)

Luminosity (or stellar mass) thresholded samples are not far from optimal

On going• Easy to incorporate

– Mass-dependent selection function– Uncertainty in mass estimate (N.B. this affects

both m and b in optimal w)

• Determine optimal observable to use as weight (e.g., color? stellar mass?) for a given galaxy sample

• Redshift space effects/reconstructions – N.B. b/b = (E/2) (P/P)

• Effect of nonlinear bias, weight functions

Primordial non-Gaussianity

• Apply optimal weight to get clean measure of k2 dependence

• Then weight galaxies/halos by other parameters (e.g., mass, luminosity, color) to check that k2 piece scales as expected

• Can get large range of bias factors if weight is (large scale) environment

• Environment is number of neighbours within 8Mpc

30% densest

30% least dense

• Assume cosmology → halo profiles, halo abundance, halo clustering

• Calibrate g(m) by matching ngal and ξgal(r) of full sample

• Make mock catalog assuming same g(m) for all environments

• Measure clustering in sub-samples defined similarly to SDSS

SDSS

Abbas & Sheth 2007

Mr<−19.5

• Environment = neighbours within 8 Mpc

• Clustering stronger in dense regions

• Dependence on density NOT monotonic in less dense regions!

• Same seen in mock catalogs; little room for extra effects!

SDSS

Abbas & Sheth 2007

Will clustering data tell us if halos are 200× critical density?

Background density? Something else?

• Galaxy distribution remembers that, in Gaussian random fields, high peaks and low troughs cluster similarly

• N.B. ‘Assembly bias’ is commonly defined as the dependence of clustering on a parameter other than halo mass. This is not quite right – the effect here does indeed have clustering (at fixed halo mass) dependent on environment, yet it is perfectly consistent with the excursion set/peak background split approach.

There is much to be gained by thinking of different galaxy types

and properties as simply representing the effect of applying different weights to

the same underlying halo catalog