Transcript Red 3000

AY202a
Galaxies & Dynamics
Lecture 20:
Large Scale Structure
& Large Scale Flows
Cosmology from LSS
Compare the observed distribution of galaxies with
those predicted by models:
Tools:
Correlation functions
Topology
Power Spectrum
Count Statistics (counts-in-cells, …)
Void Probability Function
Genus GS Wavelets Fractals Filling Factor etc.
Correlation Functions
Angular correlation function ω(θ) is defined by
δPθ = N [1 + ω(θ)] δΩ
where N is the number of objects per steradian
and δPθ is the probability of finding an
object with solid angle δΩ at an angular
distance θ from a randomly chosen object.
(draw rings around each galaxy and count its
neighbors as a function of angular radius)
The Spatial Correlation Function ξ(r) or ξ(s)
is defined by
δPr = n [1 + ξ(r)] δV
where n is the volume number density of
objects and δPr is the probability of finding
an object within volume element δV at a
distance r from a randomly chosen object.
Peebles (and everyone since) found roughly
-γ
-γ
ξ(r) ~ B r
= (r/r0)
and observationally γ ~ 1.8
Correlation Function Estimation
Hamilton
ξ =
or
<DD> <RR>
<DR>2
= <DD>/<DR>
Landy & Szalay
ξ = (<DD> - 2<DR> + <RR>)/<RR>
Separation measures
How do we measure scales and separations in 3D?
Simplest way is just projected
r = tan(θ) D = tan(θ) (v1+v2)/2H0
But should velocity separation be included?
If so define the separation
s = (v12 + v22 – 2v1v2 cos θ)½ /H0
which works well outside clusters (a little messy with F.o.G.)
Not all galaxies are created equal…
What is the proper way to weight the galaxies in the
CF estimators?
Simple is Unit weight w(r) = 1 for each galaxy
But, if one only counts pairs, you will weight
galaxies in clusters more than those outside.
So use minimum variance weighting (Davis et al)
w(r,x) = [1 + 4 n(r) J3(x)]-1
where
J3(x) =  ξ(y) y2 dy = volume integral of the CF
x
0
2dFGRS
SDSS ‘05
R0=6.8 h-1 Mpc
a = -1.2
for K < -22
M. Westover
M. Westover
Angular and Spatial Correlation functions are
related by Limber’s Equation
(r1,r2)2 dr1 dr Ψ1 Ψ2 ξ(r12/r0)
ω(θ) =
[  r2 dr Ψ ]2
for a homogeneous model
[Limber ApJ 117, 134 (1953)]
Velocity Space Correlations
1960’s + 1970’s Layzer & Irvine, Geller & Peebles
made the connection between random galaxy
motions and gravitational clustering a.k.a.
The Cosmic Virial Theorem
<v2(r)> ~ G <ρ> ξ(r) r2
Roughly, the field velocity dispersion is related to
the mean mass density through the spatial
correlation function
Pairwise Velocity Dispersion
Calculate the bivariate CF ξ (rp,)
 is the l.o.s. separation of a pair in Mpc
= |(v1 – v2)| /H0
rp is the projected separation, also in Mpc
= tan θ (v1 + v2) /2H0
Predictions Marzke et al. 1995
ξ(rp,)
no clusters
One of many
connundrums
in the 1990’s
--- the data
indicated a
low 
2dF
σ vs 
results
Clusters Cluster, Too
Bahcall& Soniera, ‘83
Klypin & Kopylov ‘83
PGH 86, etc.
r0 ~ 15-25 h-1 Mpc
depending on richness
Counts in Cells Analysis
nocc = # of galaxies per occupied cell
nexp = expected # per cell
Filaments
Data
Sheets &
Intersecting Sheets
Filaments versus Surfaces
deLapparent, Geller & Huchra
The Power Spectrum
Suppose the Universe is periodic on a volume
VU. Consider the Simplest case, a volume
limited sample with equal weight galaxies, N
galaxies. Measure fluctuations on different
scales in volume V:
P(k) = (<|δk|2> - 1/N) (Σ |wk|2)-1 (1-|wk|2)-1
k
where δk = 1/N Σ e
j
And
ik.xj
- wk
w(x) is the window function for the survey
= 1 inside and = 0 outside the boundaries
So w(x) = V/VU
w
k
Σ wk e-ik.x
is the Fourier Transform of w(x)
Kronecker delta
This derives from
<|δk|2>
= δk0D + 1/N
variance
sample
mean
+ P(k)
Poisson Fluctuations Real Power
due to finite sampling
per mode
(see Peacock & Dodds; Park et al 1994)
LCRS vs CfA2+SSRS2
SDSS
Tegmark et al. 2004
SDSS vs and plus other measures
Red line are from a Monte Carlo Markov chain analysis of the WMAP for simple flat scalar
adiabatic models parameterized by the densities of dark energy, dark matter, and baryronic
matter, the spectral index and amplitude, and the reionization optical depth.
2dF
PS constraints
on 
Simulations
Industry started by S. Aarseth followed by
Efstathiou, White, Frenk & Davis and now
many others. (c.f. Virgo Consortium)
Big groups at MPI, NCSA, Chicago.
N-Body codes or N-body Hydro codes
(PP, PPM, Grid, SPH)
 =1 α = 3.2
 =0.09 α = 2.4
 = 1 α = 1.8
 = 1 α = 0.0
 = 0.2 α = 1.8
 = 1 α = 4.5
DEFW ‘85
kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk
Constrained Model (V. Springel)
LCDM
simulation
Filaments are
warm
Hydrogen
(~10^5 K)
250 Mpc Cube
Hernquist 2003
SCDM
VIRGO
Consortium
OCDM
LCDM
By the middle 1990’s it was clear- at least to the observers
– that SCDM was dead.
Baryon Acoustic Oscillations
Eisenstein et al ’05
noted that LRG
sample had large
effective volume
SDSS LRG Correlation Function
Simulations from
D. Eisenstein
•
Evolution of
Fluctuations of
different Stuff
based on Seljak & Zaldarriaga
(CMBfast code)
•
•
Today
Large Scale Motions
Rubin 1952 Distortions
deVaucouleurs 1956 Local Supergalaxy  Supercluster
Rubin, Ford, Thonnard, Roberts 1976 + answering papers
Peebles – Silk – Gunn early ’70’s Mass and Light
CMB dipole 1976-79 Wilkinson++, Melchiori++ (balloons)
Virgo Infall
Schechter ’80, TD ’80, DH ’82, AHMST ’82
Great Attractor --- 1985 Seven Samurai (BFDDL-BTW)
Kaiser 1985 Caustics
IRAS Surveys 1985  Davis, Strauss, Fisher, H, ++
ORS 1992 Santiago et al.
COBE Dipole ‘97
Flows and Dipoles
(Silk; Peebles; Gunn)
Gravity
g ~ M/R
Light
2
2
f ~ L/R
So, if <M/L> ~ constant
Gravity Vector
=
Flux Vector
Velocity Perturbations from TF fit to a Virgo Infall Model
z residuals, no infall
AHMST
1982
The Hunt
for the
Dipole
ORS
(Santiago et al.
including Marc)
What is the ideal All-Sky Survey?
Go to the near IR!
---- Beat extinction, the bane
of optical surveys
--- Select for the stars that
trace the baryonic mass
(not star formation)
2MASS Telescope at FLWO
CTIO 1.5-meter
6dF Fiber Positioner, SRC Schmidt, Coonabarabran
•
•
•
Magenta
V < 1000 km/s
Blue
1000 < V < 2000 km/s
Green
2000 < V < 3000 km/s
Red 3000 < v < 4000
Blue 4000< v < 5000
Green 5000 < v < 6000
Red 6000 < v < 7000
Blue 7000 < v < 8000
Green 8000 < v < 9000
Great Wall
LSC
We are here
Pisces-Perseus
KS < 11.25
2MRS Dipole
blue tri = FW - M81, Maffei’s & friends
(Erdogdu et al 2006)
red tri = FW - only LG
Density vs Flow Fields
Don’t do this!
Much Better!
CMB versus LG Reference Frames
Remember,
This Is A
Sphere!
Hectospec Positioner on MMT
300 Fibers
covering a
1 degree field
of view
D. Fabricant
Large
Synoptic
Survey
Telescope
8.4-m
7 degree FOV