粒⼦軌道に基づく𝑁体ハロー構造の解析
京都⼤学 天体核研究室 M2

杉浦 宏夢with
樽家 篤史 (YITP), Yann Rasera (Paris7U)
2017-10-23 観測的宇宙論ワークショップ@弘前⼤

00/19

Introduction: What is Splashback Radius?
DM particle
Splashback radius 𝑟#$
Dark Halo

Splashback = first apocenter passage of particles, accreting to a halo.
01/19

ideal particle trajectory w/o angular momentum
Accretion of DM into a halo𝑑& 𝒓
𝐺𝑀 𝑡, 𝑟
=−𝒓..
&&𝑑𝑡𝑟

radiussp

Cf. Fillmore & Goldreich (1984),
Adhikari, Dalal, Chamberlain (2014)𝑟#$radial velocitytime

The first apocenter isthe “splashback point”.
sp
𝑟#$radius

It is not so easy to analyze particle motion
after shell-crossing in analytic way.
02/19

𝑅#$ as Halo Boundary
This radius can be regarded as a physically-motivated halo boundary.
Splashback mass 𝑀#$ is a physicallyreasonable definition of halo mass.
accretion

radial velocity

multi-steamsp
𝑟#$radius

1. boundary between multi-stream
region and accreting region
2. It does not exhibit “pseudo-evolution.”(More, Diemer & Kravtsov 2015)
3. detectable signs in density profile

03/19

𝑅#$ and Density ProfileWe can detect splashback radius
density [log]2 45 6by density slope 𝛾 =.
2 45 7

realistic profilesp

radius [log]radial velocity

sp

𝑟#$density slope of stacked N-body haloradius(cited from Diemer & Kravtsov 2014)04/19

Previous Works• SPARTA algorithm: how to calculate
𝑅#$ in 𝑁-body simulation
Diemer (2017)• Application of SPARTA to investigate
halo properties
Diemer et al. (2017)↑do not quantify asphericity↓do not use particle data

Other works:• 𝑅#$ depends on accretion rate and ΩCD .
Adhikari, Dalal & Chamberlain (2014),Shi (2016)• Weak lensing
More et al. (2016), Chang et al. (2017)• Relation to Halo Assembly Bias
More et al. (2016),
P. Busch & S. White (2017)

Cf. Jing & Suto (2002)• “Splashback Shell”
Mansfield, Kravtsov & Diemer (2017)05/19

Main idea of our work
1. It is possible to use not only the first apocenter, but also other apocenters.
2. It is possible to formulate asphericity of dark halos.
In order to do it, we define “period” of a particle
as the number of apocenter-passages.
This method gives characterization of halos, based on
purely dynamical structure without assumptions suchthat spherical overdensity or FoF.
06/19

Definition of Period
We propose a new particle attribute “period.”𝑝=0

For each particles, we define period 𝑝 as the number of
apocenter-passages.
𝑝=1
radius𝑝=2

𝑝=3

The point where 𝑝 = 0 → 1
is the splashback point.
-> “generalized splashback”

time

07/19

Meaning of Period
Clasiffication by period = decomposition of a halo into “phase shells.”

spParticles whose 𝑝 is a given value
consist of:
1. Accreting component
2. Ascending component

08/19

𝑁-body Simulation
𝑁-body simulation data is provided by Yann Rasera.
EdS universe (ΩC = 1), 𝐿 = 318 Mpc/ℎ, 𝑁 = 512 O , 𝐻D = 72 km/s/Mpc
Enough snapshots are stored:
60 snapshots between 𝑧 = 0
and 𝑧 = 1.43.

09/19

Halo Motion
We define “halo motion” by the following algorithm:
1. Choose 𝑁U particles closest to the center of the halo.
2. Read the previous snapshot, and determine
positions of these particles.
3. Set (density-weighted) center of mass of theseparticles as the “center” of the halo in this snapshot.
4. Repeat this procedure.
This reflects the motion of densest region of the halo.
10/19

Calculation of Period
1. Set 𝑝 = 0 for every particles in the initialsnapshot.
2. Read following data, and if
𝑣7 𝑧$WXY ≥ 0 and 𝑣7 𝑧 ≤ 0,record 𝑝 = 𝑝$WXY + 1, else 𝑝$WXY .
3. Repeat this procedure until 𝑧 = 0.
-> We get lists of 𝑝-values for each particles.
11/19

Difference from Diemer’s SPARTA
The motion of particles in a subhalo
1. Large motion in the main halo
2. Small motion in the subhalo.
-> we require that a particle must experience𝒓 𝑧` ⋅ 𝒓 𝑧& < 0
between adjacent apocenters.

This is an important difference from Diemer’s SPARTA, which deals withsubhalos as ONE object and particles belonging to them are NOT decomposed.
12/19

Result: Period Distribution

This region may not be correct values.
∵ limited number of snapshots

13/19

Result: Phase-shell Decomposition
𝑝=5
𝑝=3

𝑝=1

𝑝=6
𝑝=4

𝑝=2

𝑝=0

14/19

Inertial tensor & Halo shape
Inertial tensor of phase-shell 𝑝 is given by1Uhh𝐼de =f 𝑥d 𝑥e .
𝑁UUj kU(h)𝑥d : position of the particle 𝑎 relative to the C.M.
𝑝h : period of the particle 𝑎
Jing & Y. Suto (2002), D. Suto et al. (2016), etc.

Average radius 𝑅U is given by𝑅U =1
tr 𝐼de =3`/&&&&
𝑎 +𝑏 +𝑐

3

Eigenvalues of 𝐼de gives axis lengths 𝑎, 𝑏, 𝑐.
𝑎&𝐼de ∼𝑏&𝑐&
15/19

Result: Halo Shape

Jing & Suto (2002): isodensity sphere

Our approach can quantify
Distribution of axis ratio for 139 halos, more outer region.
`tpreliminary

Universal profile

periodOur result is• Larger axis ratio(=more spherical)• Larger variance

Axis ratio

Average radius

whose 𝑀&DD is 2.3 − 3.2×10 𝑀⊙

period16/19

Discussion: Axis Ratio

preliminary

Peculiar tail: This may
be due to major merger.
17/19

To-do list
• mass accretion history• Relation of 𝑅#$ to accretion history is interesting. (Diemer & Kravtsov
2014)• How does it effect on other quantities?

• Splashback mass(#$)

f 𝐼de 𝑥d 𝑥e ≤ 1d,e

• (Non-spherical) density profile & obervability of inner shells• “Second splashback” shell may be destroyed by spherical average.
18/19

Summary• Splashback radius of a halo
• My work: classify particles by the number of apocenter-passages.
• 𝑝 = 1 particles define a “splashback shell”.
• Higher-𝑝 particles define “inner shells.”• Halo structures:• Shell radius, axis ratio, density profile, …

• Future work: application to the observations.
19/19

Calculation of periodif ( flag== 0 ) and ( np.dot(r0,r1) >= 0 ):period.append(period[-1])else:flag = 1vr0 = np.dot( r0, v0 )vr1 = np.dot( r1, v1 )if ( vr0 > 0 ) and ( vr1 <= 0 ):period.append(period[-1]+1)
flag = 0
direction[id] = x1
else:
period.append(period[-1])Halo Mass
Non-spherical splashback mass:(#$)
f 𝐼de 𝑥d 𝑥e ≤ 1d,epreliminary

Result: Density Profile