G. Rossi

Morphology of Dark Matter Halosin the Cosmic Web

Graziano Rossi

Korea Institute for Advanced Study (KIAS)

Workshop on “Cosmic Web Morphology and Topology”

Copernicus Astronomical Center – Warsaw, Poland

July 15, 2011

G. RossiOUTLINE

1. Motivation and Goals

2. The Model

Shape parametersEllipsoidal evolutionWhite & Silk approximationInitial conditions

3. Halo Shapes and Simulations

Model summaryComparison with simulationsUniversal shape distribution

4. Basic Highlights & Cosmic Web Connection

MAIN REFERENCE

G. Rossi, R. Sheth & G. Tormen (2011), “Modelling the shapesof the largest gravitationally bound objects”, MNRAS, 1032

G. Rossi

INTRODUCTION

INTRODUCTION

MOTIVATION AND GOALS

MOTIVATION

Halos are not spherical!

Ellipsoidal descriptionimproves density profiles

Implications for weak andstrong lensing

GOALS

Use nonlinear dynamics to explain halo shapes

Statistical study of DM halo morphology

Link between ellipsoidal shape and primeval principles(gravitational instability)

Provide scheme for DM halo evolution (ai, bi, ci → af, bf, cf)

More general → understand virialization process and structureformation as a statistical + gravitational process

G. Rossi

INTRODUCTION

INTRODUCTION

WHY ANALYTIC TOOLS?

WHY BOTHER?Need control over simulations

Deeper physical understanding

Connection between simulations and actual survey data

A NICE REFEREE ...

“If we know that author’s analytic model would not work in matching the observational data, then whyshould we bother to consider it?”

OUR REPLY

Progress in science also (always?) comes from understanding why models fail. Highlightingshortcomings, and explaining their source, so that others do not have to follow the same dead-end, is animportant part of this process. We did not misrepresent our results - we were honest about theshort-comings of our model - and we pointed out ways in which future analyses (by ourselves or others)may result in improved agreement and understanding. That is the nature of science, and perhaps that iswhy other scientists (if not the referee) responded favourably to our posting on astro-ph – they clearly‘bothered to consider it’, even if it did not work.

G. Rossi

INTRODUCTION

INTRODUCTION

WHY TRIAXIALITY?

Life is not spherical (i.e. there are no spherical cows)

Statistics of Gaussian random fields disfavor spherical shapes

Model too idealized: at least a step further ...

Intrinsic shapes of galaxies are triaxial

Note that spherical overdensity criteria are currently used innumerical studies ...

Nonlinear clustering of halos and dark matter

Formation and evolution of galaxies

Galaxies and their relation with cosmic web → future surveys(i.e. EUCLID, WFIRST)

Structural properties of halos ⇐⇒ properties of galaxies

Nonspherical modelling of intracluster gas

Gravitational lensing observations (i.e. cosmic shear, flexion, ...)

G. Rossi

INTRODUCTION

INTRODUCTION

MODELLING: PHILOSOPY

TWOFOLD PROCEDURE

Two independent parts + various improvements

More general formalism

(1) Initial conditions: statistics

(2) Nonlinear dynamics: virialization process → gravity

ANALYTIC TOOLS

(1) Excursion sets formalism

(2) Ellipsoidal model

G. Rossi

THE MODEL

THE MODEL

SHAPE PARAMETERS

Ellipticity

e =λ1 − λ3

2δ

Prolateness

p =λ1 + λ3 − 2λ2

2δ

“Prolate” object → 0 ≥ p ≥ −e“Oblate” object → 0 ≤ p ≤ eSphere → e = 0, p = 0

a ≃ b ≫ c: Oblate object(disk-shaped) → pancake

b ≃ c ≪ a: Prolate object(cigar-shaped) → filament

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THE MODEL

THE MODEL

ELLIPSOIDAL EVOLUTION (1)

EQUATIONS OF MOTION: ELLIPSOID

d2Ak

dt2 = −4πGρ̄Ak

[1 + δ

3+

b′

k

2δ + λ′

ext,k

]

= −GA2

k

4πA3k ρ̄(1 + δ)

3− 4πGρ̄Ak

[b′

k

2δ + λ′

ext,k

]

= −GMA2

k

− 4πGρ̄Ak

[b′

k

2δ + λ′

ext,k

]

INITIAL CONDITIONS

Ak(ti), vk(ti) set by Zeldovich approximation. A3 ≥ A2 ≥ A1

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THE MODEL

THE MODEL

ELLIPSOIDAL EVOLUTION (2)

The complication arises from extra terms due to the ellipsoidalpotential in a known external tidal field:

Φ = −πG3

∑

k=1

[

bkρ+(2

3− bk

)

ρ̄]

A2k

DETAILS

b′

k = bk − 2/3

bk = A1(t)A2(t)A3(t)∫

∞

0dτ

[A2i (t)+τ ]

∏3j=1[A

2j (t)+τ ]1/2

λ′

ext,k = D(t)Di (t)

[λext,k (ti )− δi/3]

D is the linear theory growth mode of perturbations

λext,k (ti ) are the initial eigenvalues of the strain tensor

Extra term needed for Λ-cosmologies

G. Rossi

THE MODEL

THE MODEL

ELLIPSOIDAL EVOLUTION (3)

p=0, e=0.1 p=0, e=0.2 p=0, e=0.3

p=e/2, e=0.1 p=e/2, e=0.2 p=e/2, e=0.3

p=-e/2, e=0.1 p=-e/2, e=0.2 p=-e/2, e=0.3

Prolate objects → 2short, 1 long → pnegative (filaments)Oblate objects → 1short, 2 long → ppositive (pancakes)

At p fixed, if e ↑ thenobjects are moreelongated (filament-like)

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THE MODEL

THE MODEL

WHITE & SILK APPROXIMATION: EXTENSION (1)

Nelect the time dependence of the b′

k s in the equation ofmotion and replace the quantity with a “spherical average”

AN ANALYTIC APPROXIMATION

Ak (t) ≃a(t)a(ti )

{

Ak (ti )[

1−D(t)D(ti )

λk (ti )]

−Ah(ti)[

1−DDi

δi

3−

ae(t)a(t)

]}

where

Ah(ti) = 3/∑

k Ak (ti)−1

ae(t) = expansion factor of a universe with initial densitycontrast δi =

∑

k λk (ti)

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THE MODEL

THE MODEL

WHITE & SILK APPROXIMATION: EXTENSION (2)

Three critical times → collapse along each of the three axes

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THE MODEL

THE MODEL

INITIAL CONDITIONS

EXCURSION SETS: BASIC IDEAPick a particle at random and smooth the linear density field over ever smaller spheres around it, until thecriterion for collapse at some redshift z is satisfied. The mass in the sphere is then identified as that ofthe collapsed object to which the particle belongs

EXCURSION SET FORMALISM

Statistics of the four-dimensional field F (r,Rf), i.e. trajectories ofthe field as a function of the filter radius at a fixed position (withF the linear overdensity)

Rate at which random trajectories meet an absorbing barrier →mass function

Bound structures forming at t are regions above some initialcritical overdensity Fcr – for a given time, this is a plane ofconstant elevation

A diffusion-like prolem

Several nontrivial subtle points

More rigorous treatment → see Maggiore & Riotto (2010)

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THE MODEL

THE MODEL

EXCURSION SET ALGORITHM

DEFORMATION TENSOR MATRIX D

d11 = (−y1 − 3y2/√

15 − y3/√

5)/3

d22 = (−y1 − 2y3/√

5)/3

d33 = (−y1 −+3y2/√

15 − y3/√

5)/3

d12 ≡ d21 = y4/√

15

d13 ≡ d31 = y5/√

15

d23 ≡ d32 = y6/√

15

Eigenvalues of D are λ1, λ2, λ3

Determine δ, e,p which are simple combinations of theeigenvalues

Check if (δ, e,p) cross the ellipsoidal barrier B(δ, e,p) at themass-scale σ

If so exit – if not, continue the loop

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SIMULATIONCOMPARISONS

SIMULATION COMPARISONS

MODEL IN A NUTSHELL

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SIMULATIONCOMPARISONS

SIMULATION COMPARISONS

COMPARISON WITH SIMULATIONS

z=0 z=0

z=0 z=0

z=0 z=0

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SIMULATIONCOMPARISONS

SIMULATION COMPARISONS

COMPARISON WITH SIMULATIONS

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SIMULATIONCOMPARISONS

SIMULATION COMPARISONS

UNIVERSAL SHAPE FUNCTION

AXIS-RATIO DISTRIBUTIONS AT LATER TIMES

p(A12|A13, δ, σ) = 32(1−A13)

[

1 −(2A12−1−A13)

2

(1−A13)2

]

× exp{

− 58σ2 (2A12 − 1 − A13)

2}

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BASICHIGHLIGHTS

BASIC HIGHLIGHTS

RESULTS: SUMMARY

MAIN FINDINGS

In our model, more massive halos are more spherical

Trend is consistent with perturbation theory

Trend is consistent with observations, i.e. mass dependence ofthe shapes of early-type galaxies

In many numerical opposite trend: more massive halos are lessspherical, but contradictory results (see Park et al. 2010)

Nonlinear evolution does not change the conditional distributiongc(p|e, δ) → analytic explanation of Jing & Suto’s numerical fit

Universal conditional axis ratio distributions at late times

Critical dependence on halo finder

Isolated systems are more spherical

Fixed numerous flaws in previous literature

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BASICHIGHLIGHTS

BASIC HIGHLIGHTS

CAVIATS

A DIFFICULT TASK ...Direct comparison between theory and simulations challengingand subtle

How do you define a gravitationally bound object? What is anon-spherical dark matter halo?

Spherical overdensity, tree algorithms, two-step procedures withpost-processing, FoF, ...

Results in simulations are a function of the halo finding algorithm

When is a halo virialized or ‘relaxed’?

Ellipsoidal collapse is a significant oversimplification

Many details of nonlinear collapse are neglected

G. Rossi

BASICHIGHLIGHTS

BASIC HIGHLIGHTS

COSMIC WEB CONNECTION

IMPROVEMENTS: A FEW POSSIBILITIES ...Include violent relaxation effects

Test different collapse criteria

Account for correlations between halo properties andenvironment

Account for correlations between formation times and haloshapes

Account for correlations between halo shapes

Include correlated steps in the excursion sets algorithm

Include effects of baryonic physics

Include subhalos in the overall picture

G. Rossi

BASICHIGHLIGHTS

BASIC HIGHLIGHTS

HALO SHAPES: RATIONALE

MOTIVATION

Theoretical model for dark halo shapes

ACHIEVEMENTS/RESULTS

Satisfactory match with simulations only around M∗

Good agreement for conditional distributions

Analytic insights (i.e. WS approximation, explanations of fittingformulae, etc. ...)

RELEVANCE & ONGOING WORK

Investigate discrepancies model vs simulations → HORIZONRUN simulation @ KIAS

Numerous theoretical studies up-to-date (i.e. Robertson et al.2009; Maggiore & Riotto 2010 ...)

Relevance for EUCLID and WL → accurate shapes of galaxies