Cosmic Web: skeleton, connectivity, non-gaussianityhome.kias.re.kr/MKG/upload/cosmology2016/talk...

Skeleton of the Cosmic Web

Cosmic Web: skeleton, connectivity,non-gaussianity

Dmitry Pogosyan

1Physics Department, University of Alberta

November 2, 2016

with: S. Codis, C. Pichon, (IAP),T. Sousbie, C. Gay, Dick Bond, L. Kofman . . .


Large scale filamentary structure connecting clusters ofgalaxies is evident both in data and simulations

Sloan DS Survey Horizon (IAP) project


From Peaks to Filaments, Skeleton of the Cosmic Web

• The high rare peaks of thedensity field largely definehow large scale structurelooks like. Filaments aredense bridges between thepeaks

• The Skeleton of LSS tracesthe Cosmic Web of filamentsand provides the next level ofdetail to LSS understanding.

• At large scales, peaks andfilaments can be traced backto initial density field

Initial Final


Geometry of cosmological density field

Famous worksDoroshkevich, 1970Arnold, Zeldovich, Shandarin, 1982Bond, Bardeen, Kaiser, Szalay, 1986

Geometrical questions:

• abundance and shapes of peaksof different scale

• Length of filamentary bridges

• Connectivity of peaks byfilamentary ridges

• Relation between the properties offilaments and clusters

• Percolation characteristics offilamentary network

Sims: Horizon project + skeleton(DisPerSE, T. Sousbie 2011)Skeleton - geometrical descriptionof filaments


Skeleton as tracer of the filaments

Dark matter skeleton inHorizon 4π simulationT. Sousbie et al, 2011

Halo spin wrt the skeletonCodis et al, 2012, Noam Libeskind,Elmo Tempel, Stefan GottlöberTidal torgue near filamentsCodis et al, 2015


Fully connected Global Skeleton

• skeleton is a set of gradient linesconnecting peaks

• skeleton line between peakspasses through one saddle point

• Connectivity Nc is number ofpeaks the given peak isconnected to

• Nc as the function of peakproperties, and/or restriction onsaddles is the main object

• Properties of saddles are criticalfor percolation

• but not every segment of theskeleton is a dense filament


Basic info on Gaussian Peaks and Extrema Counts (BBKS)2D

- 4 -2 0 2 40.000

0.005

0.010

0.015

0.020

Ν

dn

dΝ

⟨npeak⟩=1

8p

3πR 2∗

, ⟨nsaddle⟩=1

4p

3πR 2∗

R peak−to−peak ≈ 6.6R∗ = 6.6σ1

σ2

⟨nsaddle⟩⟨npeak⟩

= 2 ⇒ ⟨Nc ⟩= 4

3D

- 4 -2 0 2 40.000

0.001

0.002

0.003

0.004

0.005

Ν

dn

dΝ

⟨n∓−−⟩=29p

15∓18p

10

1800π2R 3∗

R peak−to−peak ≈ 6.8R∗ = 6.8σ1

σ2

⟨nsaddle⟩⟨npeak⟩

≈ 3.055 ⇒ ⟨Nc ⟩= 6.1 ?


Connectivity of the Global Skeleton

Notable Result:2D ⟨Nc ⟩= 4 - average connections of a peak3D ⟨Nc ⟩= 6 - average connections of a peak

Matches well with2⟨ns a d ⟩/⟨nma x ⟩= 42⟨ns a d ⟩/⟨nma x ⟩ ≈ 6.1


ν dependence of the connectivity

−4 −2 0 2 4

0.1

1.0

10.0

0.2

2.

20.

0.05

0.5

5.

n =2

n =3

n =4

n =5

n =6

n =7

n =8

n =9

γ=0.58

Constrast ν

PD

F

PDF skeleton node density 2D

0 2 4 6 8 10

0.001

0.01

0.1

1.0

10.0

ν =−1.9

ν =−1.1

ν =−0.3

ν =0.5

ν =1.3

ν =2.1

ν =3.0

ν =3.8

γ=0.58

Degree nP

DF

PDF skeleton node density 2D

Notable Result:High peaks tend to have more connectionsPeaks with large number of connections are predominantly high


Towards connectivity theory, Nc

Idea: Count the number of saddles up to R . . . , conditional on the properties ofthe peak. But what is R ? Some characteristic distance to the next peak. Letus look at a peak with height ν=δ/σ= 2.5. How far is the next peak ?Compute conditional peak densities and define R as the most probabledistance to the next peak

ν=2

ν=3

ν=4

0 5 10 15 20�/�*0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

(�*)�<��(�)|��(�)>

if to any other peak, R ≥R peak−to−peak

and increases with ν

ν*=0

ν*=0.5

ν*=1

ν*=1.5

ν*=2

ν*=2.5

0 2 4 6 8 10 12 14�/�*0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

(�*)�<��(��ν>ν*)|��(��ν=�)>

if to other high peak, R <R peak−to−peak

(Kaiser bias)


Nc ≈ counting saddles to the next peak

Number of saddles to distance R conditional on the height ν of centralpeak

ν=2

ν=3

ν=4

2 3 4 5 6��/�*

2

4

6

8

��(<��)

• Lower density of saddles around higher peaks are compensatedby larger distances to the next peak. Nc remain relativelyinsensitive to ν ? Is there an inconsistency with measurements ?


Local Skeleton

• Near the peak real filamentsfollows ridges of the density

• Task is to count the number offilaments Nf that leave a peak.

• Filament structure reflects mergerhistory and flow pattern around inthe peak-patch.

• Depends on distance, (very)locally each peak is elliptical andhas two ridges leaving it.

• Thus (corse grained) skeletonline bifurcate, Nf =Nc −Nb i f u r .

• frequent close bifurcations - linkto catastrophy theory ?


Local number of filaments Nf near the peak

2D ⟨Nf ⟩= 3 3D ⟨Nf ⟩= 4

• Notable Results: average number of branches from a peak

• Interesting link: Adhesion model that enforces these values by construction

• Nbifurcations =Nc −Nf

• Issues: not all branches have high density and are physically important


Counting filaments as maxima on a sphere around the peak

2D, Nf (r )

random

pk, ν=1

pk, ν=2

pk, ν=3

pk, ν=4

0 2 4 6 8�/�*0

1

2

3

4

5

6

7<��(�)|��(�)>

3D, Nf (r )

Ν=2

Ν=3

Ν=4

0 2 4 6 8r�R*0

2

4

6

8

10

<NskelHrLÈnpkH0L>

Preliminary conclusions

• At r ∼ 0 Nf = 2 if we have peak at the center. Nf = 1 otherwise

• Nf (r ) does not depend on peak height, but distance to the next peakdoes, which explains dependence of connectivity on height


Not all filaments are equally prominent. Counting importantones

all Νf

Ν=3, Νf>1

Ν=3, Νf>2

Ν=4, Νf>1

Ν=4, Νf>2

0 5 10 15r�R*0

2

4

6

8

10<NskelHr,>Ν f LÈnpkH0,ΝL>

• Number of dense ν f > 2 filamentary bridges is increasing with the heightof the central peak

• Not very rare ν= 3 central peak has two (branches of) dense filaments,i.e it sits in one dominant filament on average

• Rare ν= 4 peak is at intersection of three prominant branches.


Connectivity of a non-Gaussian field differ from the Gaussian• In cosmological simulations, as density becomes more

non-Gaussian, connectivity of the Cosmic Web decreases• This leads to model dependent history of the connectivity at

different redshifts.

Nc

Y-axis to be divided by 2

0.0 0.2 0.4 0.6 0.8 1.0

8

9

10

11

12

Expansion factor

Me

an

de

gre

e

Expansion factor

Me

an

de

gre

eCDM

ΛCDM


Non-Gaussian 3D Extrema Counts (Gay et al, 2011)

⟨n∓−−⟩=29p

15∓18p

10

1800π2R 3∗

+5p

5

24π2p

6πR 3∗

�

q 2 J1

�

−8

21

J13�

+10

21⟨J1 J2⟩

�

- 4 -2 0 2 40.000

0.001

0.002

0.003

0.004

0.005

Ν

dn

dΝ

Measurement

Theory

-2 0 2 4Ν

-4

-2

2

4

106¶nextHΝL

¶ Ν

H1L

voidspeaks

At σ≈ 0.2⟨ns a d d l e ⟩⟨np e a k ⟩

≈ 2.5 ⇒Nc ≈ 5

( cubic moments evaluate, with some spectral dependence, to ≈ 0.1, see Gay,Pichon, Pogosyan, 2011)


Summary and what is left behind the scenes• Description of the filametary Cosmic Web poses many questions that can be

formulated in geometrical or topological language.

• This allows for novel computation and powerful analytical technique as well asmethods of analysis of simulations and data.

• Skeleton, in different version, is one such technique that has been provenextremely helpful ’marking’ large scale structure for study its effect, for example,on galaxy properties and formation (not in this talk).

• In this talk we focused on connectivity of the skeleton of the Cosmic Web, whichallowed as to start formulate analytical explanation of how it works.

• We found evidence of increased connectivity for rare peaks - i.e more massivegalaxy clusters.

• Evolving cosmic density field becomes non-Gaussian as non-linearity developes.This affects prominence of the filaments and connectivity of the Web. Differentcosmological models, as the result, has different history of connectivity whichmay be observed.

• Formalism of geometrical measures in non-gaussian regime that we developed(not in this talk) can provide the basis for analysing (mildly) non-linear skeleton.Simple example of direct extrema counts support simulation measurements ondecrease of connectivity with non-linearity.


Cosmic Web in 2D Adhesion Model

back


Towards the local description of the Skeleton


Towards the local description of the Skeleton

(∇∇ρ) ·∇ρ =λ∇ρ

Cosmic Web: skeleton, connectivity, non-gaussianityhome.kias.re.kr/MKG/upload/cosmology2016/talk...

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