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 (SDSS)

The halo-model of clustering Two types of particles: central + ‘satellite’ ξ obs (r) = ξ 1h (r) + ξ 2h (r) ξ 1h (r) = ξ cs (r) + ξ ss (r)

Luminosity dependent clustering Zehavi et al SDSS Centre plus Poisson satellite model (two → five free parameters) provides good description Think of as how galaxies ‘weight’ halos (~ T X,Y X, Y SZ ) central satellites

Zehavi et al SDSS = f cen (m) [1 + ]  1 + m/15m L

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 ) –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

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  w 2 = (Hamaus et al. 2010) Minimize E 2 = / (Cai et al. 2010)

Mass is mass-weighted halos Write ‘Wiener filter’ of model in which some halos are seen, others are not Stochasticity E 2 = 1 – C wm 2 /C ww /C mm Wiener ‘filter’ is that weight which minimizes stochasticity: w(m) = m/  + f dust b dust b(m) P h /[1 + ∑nb 2 P h ]

Note … w(m) = m/  + f dust b dust b(m) P h /[1 + ∑nb 2 P h ]  m/m min  m min  + f dust b 2 P h /[1 + n h b 2 P h ] ~ 1 + m/m min  m min f dust b 2 P h /[1 + n h b 2 P h ]) ~ 1 + m/m min  dust  h n h b 2 P h /[1 + n h b 2 P h ]) ~ 1 + m/5m min

E 2 opt = P 1h dust /P m + (f dust b dust ) 2 /[1 + ∑nb 2 P h ] (P h /P m ) → 0 when f dust = 0 → P 1h dust /P m when ∑nb 2 P h » 1 if massive halos missing, E cannot be made arbitrarily small → (f dust b dust ) 2 /[1 + ∑nb 2 P h ] when P h ~P m

Considerable gains at low masses E 2 = N/(S + N) = 1/(S/N + 1) = 1/(nb 2 P + 1) Optimal weighting yields same precision with fewer objects

(nb 2 ) eff P = 1/E 2 – 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 k 2 dependence Then weight galaxies/halos by other parameters (e.g., mass, luminosity, color) to check that k 2 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 n gal and ξ gal (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 M r <−19.5

Aside: Stochastic Nonlinear Bias Environmental dependence of halo mass function provides accurate framework for describing bias (curvature = ‘nonlinear’; scatter = ‘stochastic’) G 1 (M,V) = ∫dm N(m|M,V) g 1 (m)

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