Environmental Environmental dependence of halo dependence of halo

formation timesformation times

Geraint HarkerGeraint Harker

MotivationMotivation

Assumptions about the environmental Assumptions about the environmental dependence of halo formation times are dependence of halo formation times are implicit in semi-analytic models:implicit in semi-analytic models: In merger trees based on extended Press-In merger trees based on extended Press-

Schechter theory the distribution of formation Schechter theory the distribution of formation redshifts is a function only of the mass of a halo, redshifts is a function only of the mass of a halo, and not of its environment.and not of its environment.

Even if not manifested in spatially-averaged Even if not manifested in spatially-averaged quantities, the dependence could become quantities, the dependence could become important when considering e.g. clustering important when considering e.g. clustering statistics.statistics.

Calculation of formation Calculation of formation redshiftredshift

Consider parent halos with 64 or more particles in the Consider parent halos with 64 or more particles in the merger trees calculated by John Helly using the merger trees calculated by John Helly using the SubFind groups.SubFind groups.

Starting at the final snapshot, move back slice by Starting at the final snapshot, move back slice by slice, and at each stage:slice, and at each stage: Identify the ‘main progenitor’ as the most massive identified Identify the ‘main progenitor’ as the most massive identified

in the merger tree.in the merger tree. If this has mass less than half the mass of the parent halo, If this has mass less than half the mass of the parent halo,

use the redshift of the previous snapshot as the formation use the redshift of the previous snapshot as the formation redshift.redshift.

Otherwise, move to the next redshift slice, identifying the Otherwise, move to the next redshift slice, identifying the main progenitor there, and so on.main progenitor there, and so on.

This corresponds with the formation time as defined This corresponds with the formation time as defined in e.g. the theoretical studies of Lacey & Cole (1993).in e.g. the theoretical studies of Lacey & Cole (1993).

The Marked Correlation The Marked Correlation FunctionFunction

Described in Sheth & Tormen (2004), this is a generalisation of Described in Sheth & Tormen (2004), this is a generalisation of the two-point correlation function.the two-point correlation function.

For each halo, For each halo, ii, we define a ‘mark’ , we define a ‘mark’ mmii. The mark may be, for . The mark may be, for example, the formation redshift. Then the marked correlation example, the formation redshift. Then the marked correlation function, function, M(r)M(r), is defined by, is defined by

where the sum is taken over all pairs of objects {where the sum is taken over all pairs of objects {ii,,jj} with } with separation separation rr==rrijij but where the mean is taken over all objects in but where the mean is taken over all objects in the sample.the sample.

Therefore if Therefore if M(r)M(r)>1 pairs of objects with separation >1 pairs of objects with separation rr tend have tend have a greater value of the mark than average.a greater value of the mark than average.

We do not need to choose a spatial scale to define environment We do not need to choose a spatial scale to define environment – the marked correlation function picks this out for us.– the marked correlation function picks this out for us.

},{

2

},{)(ji

ji ji

m

mmrM

Scaled formation redshiftScaled formation redshift

Instead of raw formation redshift, we use the Instead of raw formation redshift, we use the quantity ω̃B defined in Lacey & Cole (1993) by:quantity ω̃B defined in Lacey & Cole (1993) by:

This precisely scales out the mass This precisely scales out the mass dependence of formation redshift in dependence of formation redshift in extended Press-Schechter theory for scale-extended Press-Schechter theory for scale-free initial power spectra (and very nearly free initial power spectra (and very nearly does otherwise).does otherwise).

)(2

)0()(~22

form

MM

z

Scaling testsScaling tests

An example of a marked An example of a marked correlation functioncorrelation function

‘‘Scrambled’ means we Scrambled’ means we randomly permute the randomly permute the marks between the halos marks between the halos in the sample.in the sample.

Scrambling within mass Scrambling within mass bins implies we only bins implies we only permute the marks permute the marks between halos of similar between halos of similar mass.mass.

These processes would be These processes would be equivalent if the scaled equivalent if the scaled formation redshift formation redshift precisely scales out the precisely scales out the mass dependence.mass dependence.

Np=68768for M* halo

Different mass rangesDifferent mass ranges

Other tests of environmental Other tests of environmental dependencedependence

Lemson & Kauffmann Lemson & Kauffmann (1999) used as a (1999) used as a measure of environment measure of environment the overdensity within the overdensity within an annulus ofan annulus of2 < r / 2 < r / hh-1-1Mpc < 5 Mpc < 5 centred on the halo.centred on the halo.

They found no signal, They found no signal, but the better statistics but the better statistics available from the available from the Millennium Simulation Millennium Simulation do suggest a do suggest a dependence on dependence on environment.environment.

Further workFurther work

More experimentation with different mass ranges More experimentation with different mass ranges and comparison to traditional measures of and comparison to traditional measures of environment.environment.

A marked cross-correlation function?A marked cross-correlation function? Alternative marks: for example, the number of Alternative marks: for example, the number of

progenitors above a certain mass or different progenitors above a certain mass or different definitions of formation time.definitions of formation time.

Effects of the above on clustering statistics, either Effects of the above on clustering statistics, either of (sub)halos or semi-analytic galaxies.of (sub)halos or semi-analytic galaxies.

Further interpretation. Further interpretation. Different redshifts.Different redshifts.

Traditional tests of Traditional tests of environment suggested by environment suggested by marked correlation function marked correlation function

resultsresults

Further scaling testsFurther scaling tests

Compare the marked correlation function Compare the marked correlation function with ω̃B as the mark with the function when with ω̃B as the mark with the function when we: we: Use raw formation redshifts;Use raw formation redshifts; Shuffle marks within small mass bins;Shuffle marks within small mass bins; Scale formation redshifts by dividing through Scale formation redshifts by dividing through

by the mean formation redshift of halos of the by the mean formation redshift of halos of the same mass in the simulation.same mass in the simulation.

Rank the scaled formation redshifts then force Rank the scaled formation redshifts then force them to follow the analytic distribution.them to follow the analytic distribution.

The latter three make little difference to The latter three make little difference to the final result.the final result.

Lemson & Kauffmann figure Lemson & Kauffmann figure using scaled formation redshiftusing scaled formation redshift