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Galaxy Clustering Topology in the Sloan Digital Sky Survey

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Collaborators

Changbom Park (KIAS)

Juhan Kim (KIAS)

Rich Gott (Princeton U.)

Michael Vogeley (Drexel U.)

David Weinberg (Ohio State U.)

Topology of Large scale structure

Standard cosmological model:

1. LSS arises from primordial zero-point energy-density fluctuations (Bardeen, Steinhardt & Turner 1983)

2. The density fluctuations have random phases or a Gaussian density distribution; which has a known topology.

The gaussian random field has analytically calculable genus curve.

Departures of the genus curve from the random phase shape:

variation in the PS slope, skewness in the initial density field, biasing in the distribution of galaxies relative to mass, redshift space distortion, gravitational evolution, and so on.

Probe of the non-Gaussianity !

Topological Genus (Gott, Melott & Dickinson 1986)

Isodensity contour surfaces at a given density threshold level

G = (number of holes in the surface of constant density)-

(number of isolated regions surrounded by the surfaces)

Genus analysis of LSSAdvantages of Genus Measurement

Advantage for the detection of the non-gaussianity

Gaussian random field has analytically calculable genus curve; g = G/V

: threshold density in units of standard deviation

of

What do we expect the genus curve to look like?

Many holes

Multiply connected

Isolated clusters

& voids

(Weinberg, Gott & Melott 1987)

“Meat-ball Topology“

“Swiss-cheese Topology“

Genus-Related Statistics

Amplitude drop RA

RA = Aobs / ARP-PS

Shift parameter of the peak, Δν:

By fitting Gobs(ν) over –1<ν<1

Asymmetry parameters

AV & AC

A = ∫ Gobs(ν) d ν/∫ Gfit(ν) d ν

where intervals are

-1.2~-2.2 (AV), 1.2~2.2 (AC)

g

△ν

AC

AV

To Measure the departures of the observed

genus curve from the random phase

expectation

Final SDSS DR7 Main Galaxy Sample

(Choi et al. 2010)

The Cosmic Runner (Park et al. 2005)

The Sloan Great Wall (Gott et al. 2005)

The CfA Great Wall & the man

(de Lapparent et al. 1986)

△ν

AV

AC

G=373

±18 (4.7%)

SDSS Main

galaxies

Error estimates from 27

mock surveys

(20483p1433.6s,

20483p1024s LCDM sim.)

Choi et al. 2010

Fewer voids and fewer superclusters when compared

with the Gaussian genus curve: voids and superclusters

are more connected.

Distribution of early-type/red galaxies has smaller genus density, is more

meat-ball shifted, has

more isolated clusters, and fewer voids.

Test of Galaxy Formation Models

- Whether or not various models of galaxy formation are consistent
- with measurement of the SDSS galaxy clustering topology?

1. HGC, a Halo-Galaxy one-to-one Correspondence model [Kim, Park & Choi 2008]

- Each gravitationally self-bound, tidally stable dark halo (central or sub halo) above certain mass contains one galaxy above certain luminosity. Only galaxy number density is used to constrain the model.

2. HOD, Halo Occupation Distribution [Yang et al. 2007]

Galaxies populating in the dark halos with a halo occupation distribution model

3. SAM, Semi-Analytic Models of galaxy formation

Merger-tree + physical processes put in

Croton et al. (2006) & Bower et al. (2006)\'s of SAM (which differ mainly by AGN feedback and cooling); Bertone et al. (2007)\'s SAM (galactic wind)

Deviations due to combined effects of initial condition,

gravitational evolution and biasing depend on models.

Yang et al. 2007

Croton et al. 2006

Bower et al. 2006

Kim et al. 2008

SDSS DR7 Main

1. Amplitude agrees - PS (but Bower et al. !)

2. too positive : all models show sponge topology (too positive thresholds).

3. Strongly disagree with observed void and cluster abundances.

Overall, no model reproduces all features of the observed topology!

To estimate the statistical significance of the failure of each model

i: the four genus-related statistics

j: the two volume-limited samples

Curve: Vobs is replaced by the average

value over the 64 mock samples.

No existing galaxy formation models

reproduce the topology of the SDSS main galaxy sample near the smoothing scales, 6.1 and 7.1 h-1Mpc.

The probability for the HGC model to be consistent with the observation is only 0.4%. The HOD and three SAM models are absolutely ruled out by this test.

: 9.1 h-1Mpc scale

: 7.0 h-1Mpc scale

Color subsets of SAM mock galaxies completely fail to explain the observed topology.

Findings …

[ Observations ]

1. Topology of LSS measured from SDSS DR7

2. Dependence of LSS topology on scale, luminosity, morphology & color is measured.

Early-type/red galaxies has smaller genus, is more meat-ball shifted, has more clusters,

3. Topology bias of galaxy distribution with respect to matter is measured.

Topology bias is significantly large and scale-dependent.

[ Comparison with galaxy formation models ]

4. Topology at quasi- and non-linear scales can be used to constrain galaxy formation mechanism.

All models fail to explain the observed meat-ball shift of large-scale galaxy distribution.

SAM and HOD models fail to explain cluster and void abundances.

Color subsets of SAM models completely fail to explain the observed topology.

Galaxy formation models should be tuned to explain not only

the amplitude but also the topology of galaxy clustering!

2. SDSS DR7 Luminous Red Galaxies

LRGs (red dots) provide six times more

cosmological information than typical ones.

±11.1 (3.9%)

- 129 Mock LRGs:
- 43 independent Past light cone data:LCDM simulations (60003p7200s, 72103p10815s, 41203p6592s), Halo-galaxy assignment (HGC models), PSB halo finding
- For the most massive galaxies, the HGC model does seem to work well. initially Gaussian ΛCDM model successfully reproduces the observed topology of LRGs at large Scales.
- Comparison with the random phase
- expectation:
- Meat ball shifted and more connected
- voids and more isolated clusters.
- Comparison with Perturbation Theory expectation:
- Gravitational evolution effects on genus. Void part can not be explained by perturbation theory.

Analytic formula for the genus in weakly nonlinear regime due to gravitational evolution

Matsubara (1994)

Effect of finite sampling and redshift space distortionon genus curve

The redshift space distortion effects on the

genus curve are small in the weakly-linear scales.

DR 7 Luminous Red Galaxies

Findings..

For the most massive galaxies, the HGC model (Kim et al. 2008) does seem to work well.

Initially Gaussian ΛCDM model successfully reproduce the observed topology of LRGs at 21h-1Mpc scales.

LRG distribution has meat ball topology.

Voids are more connected and clusters are more isolated when compared

with the Gaussian genus curve.

The deviation from the random phase expectation can be explained by perturbation theory: Gravitational evolution effects on genus.

Still, void abundance in very low density regions ( < -2) are not explained.

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