Transcript The Phenomenology of Large Scale Structure • Motivation: A biased view of dark matters • Gravitational Instability – The spherical collapse model – Tri-axial (ellipsoidal)

The Phenomenology of
Large Scale Structure
• Motivation: A biased view of dark matters
• Gravitational Instability
– The spherical collapse model
– Tri-axial (ellipsoidal) collapse
• The random-walk description
–
–
–
–
The halo mass function
Halo progenitors, formation, and merger trees
Environmental effects in hierarchical models
(SDSS galaxies and their environment)
• The Halo Model
Galaxy clustering depends on type
Large samples (SDSS, 2dF)
now available to quantify this
You can observe a lot just by watching. -Yogi Berra
Light is a biased tracer
Not all galaxies are fair tracers of dark matter;
To use galaxies as probes of underlying dark matter
distribution, must understand ‘bias’
How to describe different point
processes which are all built from
the same underlying density field?
THE HALO MODEL
Review in Physics Reports (Cooray & Sheth 2002)
The Cosmic
Background
Radiation
Cold: 2.725 K
Smooth: 10-5
Simple physics
Gaussian fluctuations
= seeds of subsequent
structure formation
= simple(r) math
Logic which follows
is general
N-body
simulations
of
gravitational
clustering
in an
expanding
universe
Cold
Dark
Matter
• Simulations
include gravity
only (no gas)
• Late-time field
retains memory of
initial conditions
• Cosmic
capitalism
Co-moving volume ~ 100 Mpc/h
Cold Dark Matter
• Cold: speeds are non-relativistic
• To illustrate, 1000 km/s ×10Gyr ≈ 10Mpc;
from z~1000 to present, nothing (except
photons!) travels more than ~ 10Mpc
• Dark: no idea (yet) when/where the stars
light-up
• Matter: gravity the dominant interaction
Models of halo abundances
and clustering:
Gravity in an expanding universe
Goal:
Use knowledge of initial conditions
(CMB) to make inferences about
late-time, nonlinear structures
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Hierarchical
models
Dark matter ‘haloes’ are
basic building blocks of
‘nonlinear’structure
Springel et al. 2005
MODELS OF
NONLINEAR
COLLAPSE
Assume a spherical cow….
Spherical evolution model
• Initially, Ei = – GM/Ri + (HiRi)2/2
• Shells remain concentric as object evolves; if
denser than background, object pulls itself
together as background expands around it
• At ‘turnaround’: E = – GM/rmax = Ei
• So – GM/rmax = – GM/Ri + (HiRi)2/2
• Hence (Ri/r) = 1 – Hi2Ri3/2GM
= 1 – (3Hi2 /8pG) (4pRi3/3)/M
= 1 – 1/(1+Di) = Di/(1+Di) ≈ Di
Virialization
• Final object virializes: −W = 2K
• Evir = W+K = W/2 = −GM/2rvir= −GM/rmax
– so rvir = rmax/2:
• Ratio of initial to final size = (density)⅓
– final density determined by initial overdensity
• To form an object at present time, must have
had a critical overdensity initially
• Critical density ↔ Critical link-length!
• To form objects at high redshift, must have
been even more overdense initially
Spherical collapse
Turnaround: E = -GM/rmax
size
Virialize: -W=2K
E = W+K = W/2
rvir = rmax/2
time
Modify gravity → modify collapse model
Exact Parametric Solution
(Ri/R) vs. q and (t/ti) vs. q
very well approximated by…
3
(Rinitial/R)
= Mass/(rcomVolume)
=1+d
−dsc
~ (1 – dLinear/dsc)
Virial Motions
• (Ri/rvir) ~ f(Di): ratio of initial and final
sizes depends on initial overdensity
• Mass M ~ Ri3 (since initial overdensity « 1)
• So final virial density ~ M/rvir3 ~ (Ri/rvir)3 ~
function of critical density: hence, all
virialized objects have the same density,
whatever their mass
• V2 ~ GM/rvir ~ M2/3: massive objects have
larger internal velocities/temperatures
Halos and Fingers-of-God
• Virial equilibrium:
• V2 = GM/r = GM/(3M/4p200r)1/3
• Since halos have same density, massive halos have
larger random internal velocities: V2 ~ M2/3
• V2 = GM/r = (G/H2) (M/r3) (Hr)2
= (8pG/3H2) (3M/4pr3) (Hr)2/2
= 200 r/rc (Hr)2/2 = W (10 Hr)2
• Halos should appear ~ ten times longer along line
of sight than perpendicular to it: ‘Fingers-of-God’
• Think of V2 as Temperature; then Pressure ~ V2r
HALO
ABUNDANCES
Assume a spherical cow….
(Gunn & Gott 1972; Press & Schechter 1974;
Bond et al. 1991; Fosalba & Gaztanaga 1998)
The Random Walk Model
Higher
Redshift
Critical
overdensity
This
patch
forms
halo of
mass M
smaller mass
patch within
more massive
region
MASS
From Walks to Halos: Ansätze
• f(dc,s)ds = fraction of walks which first cross
dc(z) at s
≈ fraction of initial volume in patches of
comoving volume V(s) which were just dense
enough to collapse at z
≈ fraction of initial mass in regions which
each initially contained m =rV(1+dc) ≈ rV(s)
and which were just dense enough to collapse
at z (r is comoving density of background)
≈ dm m n(m,dc)/r
The Random Walk Model
Higher
Redshift
Critical
overdensity
Typical mass smaller
at early times:
hierarchical clustering
MASS
Scaling laws
• Recall characteristic scale V(s) defined by
dc2(z) ~ s ~ s2(R) ~ ∫dk/k k3 P(k) W2(kR)
• If P(k) ~ kn with n>−3, then s ~ R–3–n
• Since M~R3, characteristic mass scale at z is
M*(z)~ [dc(z)] –6/(3+n)
• Since dc(z) decreases with time,
characteristic mass increases with time
→ Hierarchical Clustering
Random walk with absorbing barrier
s
• f(first cross d1 at s) = 0∫ dS f(first cross d0 at S)
× f(first cross d1 at s | first cross d0 at S)
• (where d1 >d0 and s>S )
• But second term is function of d1 −d0 and s − S
• (because subsequent steps independent of previous
ones, so statistics of subsequent steps are simply a shift
of origin –– a key assumption we will return to later)
•
s
f(d1,s) = 0∫ dS f(d0,S) f(d1−d0|s−S)
• To solve….
• …take Laplace Transform of both sides:
∞
• L(d1,t) = 0∫ ds f(d1,s) exp(–ts)
∞
s
= 0∫ ds exp(–ts) 0∫ dS f(d0,S) f(d1–d0,s–S)
∞
∞
-tS
= 0∫ dS f(d0,S) e s-S∫ ds f(d1–d0,s–S) e-t(s-S)
= L(d0,t) L(d1–d0,t)
• Solution must have form: L(d1,t) = exp(–Cd1)
• After some algebra (see notes): L(d1,t) = exp(–d1√2t)
• Inverting this transform yields:
• f(d1,s) ds = (d12/2πs)½ exp(–d12/2s) ds/s
• Notice: few walks cross before d12=2s
The Mass Function
• f(dc,s) ds = (dc2/2πs)½ exp(–dc2/2s) ds/s
• For power-law P(k): dc2/s = (M/M*)(n+3)/3
• n(m,dc) dm = (r/m)/√2p (n+3)/3 dm/m
(M/M*)(n+3)/6 exp[–(M/M*)(n+3)/3/2]
• (Press & Schechter 1974; Bond et al. 1991)
Simplification because…
• Everything local
• Evolution determined by cosmology
(competition between gravity and
expansion)
• Statistics determined by initial fluctuation
field: since Gaussian, statistics specified by
initial power-spectrum P(k)
• Fact that only very fat cows are spherical is
a detail (crucial for precision cosmology)
Only very fat cows are spherical….
(Sheth, Mo & Tormen 2001; Rossi, Sheth & Tormen 2007)
The Halo
Mass
Function
(Reed et al. 2003)
•Small halos
collapse/virialize
first
•Can also model
halo spatial
distribution
•Massive halos
more strongly
clustered
(current parametrizations by Sheth & Tormen 1999; Jenkins etal. 2001)
Theory predicts…
• Can rescale halo abundances to ‘universal’
form, independent of P(k), z, cosmology
– Greatly simplifies likelihood analyses
• Intimate connection between abundance and
clustering of dark halos
– Can use cluster clustering as check that cluster
mass-observable relation correctly calibrated
• Important to test if these fortunate
simplifications also hold at 1% precision
(Sheth & Tormen 1999)
Non-Maxwellian Velocities?
• v = vvir + vhalo
• Maxwellian/Gaussian velocity within halo
(dispersion depends on parent halo mass)
+ Gaussian velocity of parent halo (from
linear theory ≈ independent of m)
• Hence, at fixed m, distribution of v is
convolution of two Gaussians, i.e.,
p(v|m) is Gaussian, with dispersion
svir2(m) + sLin2 = (m/m*)2/3svir2(m*) + sLin2
Two contributions to velocities
~ mass1/3
• Virial motions
(i.e., nonlinear
theory terms)
dominate for
particles in
massive halos
• Halo motions
(linear theory)
dominate for
particles in low
mass halos
Growth rate of halo motions ~ consistent with linear theory
Exponential tails are generic
• p(v) = ∫dm mn(m) G(v|m)
F(t) = ∫dv
eivt
p(v) = ∫dm n(m)m
2s 2(m)/2
-t
e vir
2s 2/2
-t
e Lin
• For P(k) ~ k−1, mass function n(m) ~ power-law times
exp[−(m/m*)2/3/2], so integral is:
F(t) =
2s 2/2
-t
e Lin [1
+ t2svir2(m*)]−1/2
• Fourier transform is product of Gaussian and FT of
K0 Bessel function, so p(v) is convolution of G(v)
with K0(v)
• Since svir(m*)~ sLin, p(v) ~ Gaussian at |v|<sLin but
exponential-like tails extend to large v (Sheth 1996)
Comparison with simulations
Sheth & Diaferio 2001
• Gaussian core with exponential tails as expected!
EVOLUTION
AND
ENVIRONMENT
Spherical evolution model
• ‘Collapse’ depends
on initial over-density
Di; same for all initial
sizes
• Critical density
depends on cosmology
• Final objects all have
same density, whatever
their initial sizes
•Collapsed objects
called halos;
~ 200× denser than
background, whatever
their mass
(Figure shows particles at z~2 which, at z~0, are in a cluster)
Assume a spherical herd of spherical cows…
Initial spatial distribution within patch (at z~1000)...
…stochastic (initial
conditions Gaussian
random field); study
`forest’ of merger
history ‘trees’.
…encodes information about
subsequent ‘merger history’
of object
(Mo & White 1996; Sheth 1996)
Random Walk = Accretion history
High-z
Major merger
Low-z
overdensity
larger
mass at
low z
small mass
at high-z
MASS
Other features of the model
• Quantify forest of merger histories as
function of halo mass (formation times,
mass accretion, etc.)
• Model spatial distribution of halos: (halo
clustering/biasing)
– Abundance + clustering calibrates Mass
• Halos and their environment:
– Nature vs. nurture—key to simplifying models
of galaxy formation
Merger trees
• (Bond et al. 1991; Lacey
& Cole 1993)
• Fraction of M (halo which
virialized at T) which was
in m<M at t<T:
f[s,dc(t)|S,dc(T)] ds=
f[s−S,dc(t)−dc(T)] ds=
f(m,t|M,T) dm =
(m/M) N(m,t|M,T) dm
• N(m|M) is mean number
of smaller halos at earlier
time
• (see Sheth 1996 and Sheth &
Lemson 1999 for higher order
moments)
(from Wechsler et al. 2002)
Correlations with environment
Critical
over-dense
overdensity
under-dense
Easier to get here
from over-dense
environment
This
patch
‘Top-heavy’
forms
halo of mass function in
mass M dense regions
MASS
The Peak-Background Split
• Consider random walks centered on cells
which have overdensity d when smoothed
on some large scale V: M=rV(1+d) » M*
• On large scales (M » M*, so S(M) «1),
fluctuations are small (i.e., d «1),
so walks start from close to origin:
• f(m,t|M,T) dm = f[s−S,dc(t)−d] ds
≈ f[s,dc−d] ds ≈ f[s,dc] ds −d (df/ddc)
≈ f(s,dc) ds [1 −(d/dc) (dlnf/dlndc)]
≈ f(s,dc) ds [1 −(d/dc) (1 – dc2/s)]
Halo Bias on Large Scales
• Ratio of mean number density in dense regions to
mean number density in Universe:
N(m,t|M,T)/n(m,t)V = (M/m) f(m,t|M,T)/(rV/m)f(m,t)
[recall dense region had mass M = rV(1+d)]
• But from peak-background split:
f(m,t|M,T) ≈ f(m,dc) [1 −(d/dc)(1 – dc2/s)]
• N(m,t|M,T)/n(m,t)V ≈ (1+d) [1 −(d/dc) (1 – dc2/s)]
≈ 1 − (d/dc) (1 – dc2/s) + d = 1 + b(m)d
• Large-scale bias factor: b(m) ≡ 1 + (dc2/s – 1)/dc
– Increases rapidly with m at m»m*
(Cole & Kaiser 1989; Mo & White 1996; Sheth & Tormen 1999)
Halos and their environment
• Easier to get to here
from here
than from here
• Dense regions host
more massive halos
n(m,t|d) = [1 + b(m,t)d] n(m,t)
b(m,t) increases with m, so n(m,t|d) ≠ (1+d) n(m,t)
Fundamental basis for models of halo bias (and hence
of galaxy bias)
Most
massive
halos
populate
densest
regions
over-dense
underdense
Key to understand
galaxy biasing
(Mo & White 1996;
Sheth & Tormen 2002)
n(m|d) = [1 + b(m)d] n(m)
Correlations with environment
PAST
Critical
overdensity
over-dense
FUTURE
under-dense
This
patch
forms
halo of
mass M
At fixed mass,
formation history
independent of
future/environment
MASS
Environmental effects
• In hierarchical models, close
connection between evolution and
environment (dense region ~ dense
universe ~ more evolved)
• Observed correlations with
environment test hierarchical galaxy
formation models
Gastrophysics determined by
formation history of parent halo
Environmental effects
• Gastrophysics determined by
formation history of parent halo
• All environmental trends come from
fact that massive halos populate
densest regions
THE
HALO MODEL
Light is a biased tracer
Not all galaxies are fair tracers of dark matter;
To use galaxies as probes of underlying dark matter
distribution, must understand ‘bias’
How to describe different point
processes which are all built from
the same underlying distribution?
THE HALO MODEL
Center-satellite process requires knowledge of how
1) halo abundance;
2) halo clustering;
3) halo profiles;
4) number of galaxies per halo;
all depend on halo mass.
(Revived, then discarded in 1970s by Peebles, McClelland & Silk)
Halo
Profiles
• Not quite
isothermal
• Depend on halo
mass, formation
time; massive
halos less
concentrated
• Distribution of
shapes (axisratios) known
(Jing & Suto 2001)
Navarro,
Frenk &
White
(1996)
The halo-model of clustering
• Two types of pairs: both particles in same halo, or
particles in different halos
• ξdm(r) = ξ1h(r) + ξ2h(r)
• All physics can be decomposed similarly: ‘nonlinear’
effects from within halo, ‘linear’ from outside
The dark-matter correlation function
ξdm(r) = ξ1h(r) + ξ2h(r)
The 1-halo piece
• ξ1h(r) ~ ∫dm n(m) m2 ξdm(m|r)/r2
• n(m): number density of halos
• ξdm(m|r): fraction of total pairs, m2, in an mhalo which have separation r; depends on
density profile within m-halos
• Need not know spatial distribution of halos!
• This term only matters on scales smaller than
the virial radius of a typical M* halo (~ Mpc)
ξdm(r) = ξ1h(r) + ξ2h(r)
• ξ2h(r) = ∫dm1 m1n(m1) ∫dm2 m2n(m2) ξ2h(r|m1,m2)
r
r
• Two-halo term dominates on large scales, where
peak-background split estimate of halo clustering
should be accurate: dh ~ b(m)ddm
• ξ2h(r|m1,m2) ~ ‹dh2› ~ b(m1)b(m2) ‹ddm2›
• ξ2h(r) ≈ [∫dm mn(m) b(m)/r]2 ξdm(r)
• On large scales, linear theory is accurate:
ξdm(r) ≈ ξLin(r) so ξ2h(r) ≈ beff2 ξLin(r)
Halo-model of galaxy clustering
• Two types of pairs: only difference from dark matter
is that number of pairs in m-halo is not m2
• ξdm(r) = ξ1h(r) + ξ2h(r)
• Distribution within halos is small scale detail
Halo-model of galaxy clustering
• Halo abundances and clustering matter on large scales
• Spatial distribution within halos (halo density
profiles) only matters on small scales
• Different galaxy types populate different halo masses
The halo-model of galaxy clustering
• Write two components as
– ξ1gal(r) ~ ∫dm n(m) g2(m) ξdm(m|r)/rgal2
– ξ2gal(r) ≈ [∫dm n(m)g1(m)b(m)/rgal]2 ξdm(r)
– rgal = ∫dm n(m) g1(m): number density of galaxies
– ξdm(m|r): fraction of pairs in m-halos at separation r
• g2(m) is mean number of galaxy pairs in m-halos
(= m2 for dark matter)
• g1(m) is mean number of galaxies in m-halos
(= m for dark matter)
• Think of g1(m) as ‘weight’ applied to each dark matter
halo - galaxies ‘biased’ if g1(m) not proportional to m
Halo-model of un-weighted correlations
Write 1+ξ = DD/RR as sum of two components:
ξ1gal(r) ~ ∫dm n(m) g2(m) ξdm(m|r)/rgal2
ξ2gal(r) ≈ [∫dm n(m) g1(m) b(m)/rgal]2 ξdm(r)
≈ bgal2 ξdm(r)
g2(m) is mean number of galaxy pairs in m-halos
(= m2 for dark matter)
g1(m) is mean number of galaxies in m-halos
(= m for dark matter)
Halo-model of galaxy clustering
• Two types of pairs: only difference from dark matter
is that number of pairs in m-halo is not m2
• ξdm(r) = ξ1h(r) + ξ2h(r)
• Spatial distribution within halos is small-scale detail
Type-dependent clustering: Why?
populate
lower mass
halos = less
strongly
clustered
populate massive halos
= strongly clustered
Sheth & Diaferio 2001
Spatial distribution within halos second order effect (on >100 kpc)
Comparison with
simulations
• Halo model
calculation of x(r)
• Red galaxies
• Dark matter
• Blue galaxies
• Note inflection at scale
of transition from
1halo term to 2-halo
term
• Bias constant at large r
Sheth et al. 2001
x1h›x2h
x1h‹x2h →
Two approaches
• Halo Occupation Distribution
(Jing et al., Benson et al.; Seljak; Scoccimarro et al.)
– Model Ngal(>L|Mhalo) for range of L (Zehavi et al.;
Zheng et al.; Berlind et al.; Kravtsov et al.; Conroy et al.; Porciani,
Magliochetti; Collister, Lahav)
– Differentiating gives LF as function of Mhalo
(Tinker et al., Skibba et al.):
• Conditional Luminosity Function (Peacock, Smith):
– Model LF as function of Mhalo , and infer HOD
(Yang, Mo, van den Bosch; Cooray)
Higher-order moments
• n-th order correlation function depends on n-th
order moment of p(Ngal|Mhalo)
• In centre + Poisson satellite model, these are all
completely specified
• On large scales, higher order moments come from
suitably weighting perturbation theory results
• Incorporating halo shapes matters on small scales
(Smith, Watts & Sheth 2006)
Satellite galaxy counts ~ Poisson
• Write g1(m) ≡ ‹g(m)› = 1 + ‹gs(m)›
• Think of ‹gs(m)› as mean number of satellite
galaxies per m halo
• Minimal model sets number of satellites as
simple as possible ~ Poisson:
• So g2(m) ≡ ‹g(g-1)› = ‹gs (1+gs)› = ‹gs› +
‹gs2› = 2‹gs› + ‹gs›2 = (1+‹gs›)2 - 1
• Simulations show this ‘sub-Poisson’ model
works well (Kravtsov et al. 2004)
Halo Substructure
• Halo substructure = galaxies is good model
(Klypin et al. 1999; Kravtsov et al. 2005)
• Agrees with semi-analytic models and SPH
(Berlind et al. 2004; Zheng et al. 2005; Croton et al. 2006)
• Setting n(>L) = n(>Vcirc) works well for
all clustering analyses to date, including z~3
(Conroy et al. 2006)
Halo-model of mark correlations
Write WW as sum of two components (WD similar):
W1gal(r) ~ ∫dm n(m) g2(m) ‹W|m›2 ξdm(m|r)/rgal2
W2gal(r) ≈ [∫dm n(m) g1(m) ‹W|m› b(m)/rgal]2 ξdm(r)
On large scales, expect
WW = M(r) = 1+W(r)
DD
1+ξ(r)
=
1 + BW ξdm(r)
1 + bgal ξdm(r)
Gradients can be included (only matter for 1h term)
Note assumption!
• Whereas mark may correlate with halo
mass, there is no additional correlation
between mark and environment
• Greatly simplifies galaxy formation models
and interpretation of galaxy clustering:
– Some semi-analytic galaxy formation models
assume this explicitly (when use semi-analytic
merger trees rather than trees from simulation)
Assumptions (to test)
• Halo profiles depend on mass, not
environment
• Galaxy properties, so p(Ngal|L,m), and so
g1(m) and g2(m), depend on halo mass, not
environment
• All environmental dependence comes from
correlation between halo mass and
environment:
n(m|d) = [1+b(m)d] n(m)
– Mass function ‘top-heavy’ in dense regions
• 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
M
r<−19.5
SDSS
Abbas & Sheth 2007
• Environment
= neighbours
within 8 Mpc
• Clustering
stronger in
dense regions
• Dependence
on density
NOT
monotonic in
less dense
regions!
• Same seen in
mock catalogs
 Choice of scale not important
 Mass function ‘top-heavy’ in dense
regions
 Massive halos have smaller radii
(halos have same density whatever
their mass)
 Gaussian initial conditions?
 Void galaxies, though low mass,
should be strongly clustered
SDSS
 Little room for additional (e.g.
assembly bias) environmental effects
Halo Model is simplistic …
• Nonlinear physics on small scales from
virial theorem
• Linear perturbation theory on scales larger
than virial radius (exploits 20 years of hard
work between 1970-1990)
…but quite accurate!
Thus, one can …
• Model both real and redshift space observations
• Model clustering of thermal SZ effect as a weight
proportional to pressure, applied to halos/clusters
• Model clustering of kinetic SZ signal as a weight,
proportional to halo/cluster momentum
• Model weak gravitational galaxy-galaxy lensing as
cross-correlation between galaxies and mass in halos
• (see review article Cooray & Sheth 2002)
• Number density and clustering as function
of luminosity now measured in 2dF,SDSS
• Assuming there are NO large scale
environmental effects, halo model provides
estimates of luminosity distribution as
function of halo mass (interesting, relatively
unexplored connection to cluster LFs)
• Suggests BCGs are special population
(another interesting, unexplored connection
to clusters!)
The Halo Grail
Halo model
provides natural
framework
within which to
discuss, interpret
most measures
of clustering; it
is the natural
language of
galaxy ‘bias’
The Holy Grail
The Cup!
India Cricket
World Champions
Cracks in the standard model
• Sheth &Tormen (2004) measure correlation
between formation time and environment:
– At fixed mass, close pairs form earlier
– Point out relevance to halo model description
– Measurement repeated and confirmed by Gao et al.
(2005), Harker et al. (2006), Wechsler et al. (2006)
• Early formation  more clustered (even at fixed
mass) at low masses
• Does this matter for surveys which use clustering
of (primarily) luminous galaxies for cosmology
(Abbas & Sheth 2005, 2006; Croton et al. 2006)?
Close pairs
form at
higher
redshifts
Sheth & Tormen
2004
A direct test of the importance of
this effect using the SDSS
Based on
– Abbas & Sheth (2005): Clustering as
function of environment (theory)
– Abbas & Sheth (2006): Environmental
dependence of clustering in the SDSS
– Abbas & Sheth (2007): Strong clustering
of under-dense regions
Correlations between 3 variables
• c2 = ∑(zi – axi – byi)2
z|x,y
=
x
(c
–
c
c
)
+
y
(c
–
c
c
)
zx
zy
yx
zy
zx
xy
_____ __ _________ __ _________
szz
sxx (1 – cxy2) syy (1 – cxy2)
• z=formation time, x=mass, y=environment
– Hierarchical clustering: czx < 0
– Massive halos in dense regions: cxy > 0
– No correlation between formation time and
environment: czy = 0.
z|x,y
=
x
c
+
y
(–
c
c
)
zx
zx
xy
_____ __ _________ __ _________
szz
sxx (1 – cxy2) syy (1 – cxy2)