15th May 2019
Resonant instabilities in cosmology
and their observational consequences

Y TP
YUKAWA INSTITUTE FOR
THEORETICAL PHYSICS

Probing dark matter from nonlinear
structure formation of the Universe
Atsushi Taruya (YITP)Papers related to parametric instabilities
A. TARUYAPHYSICAL REVIEW D, VOLUME 59, 103505

Parametric amplification of density perturbations in the oscillating inflation model
A. Taruya*
Department of Fundamental Sciences, FIHS, Kyoto
Kyoto 606-8501, Japan
28 May University,
1998~Received 11 December 1998; published 20 April 1999!
We study the adiabatic density perturbation in the oscillating inflation model, proposed by Damour and
Mukhanov. The recent study of cosmological perturbations during reheating shows that the adiabatic fluctuaPhysics
Letters B 428 Žoccurs
1998. 37–43
tion behaves like a perfect fluid and no significant
amplification
on superhorizon scales. In the oscillating inflation model, however, the accelerated expansion takes place during the oscillating stage and there
might be a possibility that the parametric amplification on small scales affects the adiabatic long-wavelength
perturbation. We analytically show that the density perturbation neglecting the metric perturbation can beamplified by the parametric resonance and the instability band becomes very broad during oscillating inflation.
Cosmologicalperturbation
withnumerically.
two scalar
fields
We examine this issue by solving
the evolution equation
for perturbation
We findthat theparametric resonance is strongly suppressed forin
the reheating
long-wave modes
comparable
to the Hubble horizon. The
after
inflation
result indicates that the metric perturbation plays a crucial role in the evolution of scalar field perturbation.
1
2
Therefore, in the single field case, there would be noA.
significant
of oscillating
inflation on the primorTaruyaimprint, Y. Nambu
dial spectrum of the adiabatic perturbation.
However, it could be expected that the oscillating inflation model
Department of Physics, Nagoya UniÕersity, Chikusa-ku, Nagoya 464-01, Japanin the multifield system gives an enormous amplification on large scales, which may lead to the production of
Received 29 August 1997; revised 20 February 1998
primordial black holes. @S0556-2821~99!04010-2#
Editor: M. Dine
PACS number~s!: 98.80.Cq
Abstract

We investigate the cosmological perturbation of two-scalar field model during the reheating phase after inflation. Using
the exact solution of the perturbation in long-wavelength
limit,
expressed
in termsof the background
quantities, we theI. INTRODUCTION
fore, asfarwhich
as istheadiabatic
perturbationis concerned,
analyze the behavior of the metric perturbation. The oscillating inflaton field gives rise to the parametric resonance of theprimordial power spectrum on large scales does not suffermassless scalar field and this leads to the amplification of the iso-curvature mode of the metric perturbations. q 1998
e dynamics of the coherently
oscillating
scalarfield
from the significant amplification by the parametric resoElsevier Science
B.V. All rights
reserved.

an important role in the early stage of the universe andnance. However, there remains a possibility that the parametPACS: 98.80.-k; 98.80.Bp
een studied by many Keywords:
authors.CosmologicalSpecifically,
the signifiric resonance can appear well inside the Hubble horizon. For
perturbation; Reheating; Scalar fields; Parametric amplificationof the parametric resonance is found in the recent inexample, consider the massless self-interacting inflaton. Negation of the reheating process @1#. Because of the coglecting the metric perturbation, it is known that the fluctuat oscillation of the background inflaton field, the lighttion of the inflaton field has the instability mode whosen field or the fluctuation1.of
the inflaton itself is amplified
Introduction
the Hubble
large-scale
inhomowavelength during
is muchreheatingsmalleraffects
than thehorizon@1,3#.

FIG. 3. Evolution of Q in the case of the parameters q50.1 andf c 51025 M̃ pl . For each figure, we start to calculate the background equations ~1! and ~2! from the value f 5q/ A6 after theslow-rolling regime. The initial conditions for the fluctuation Q are
set by Q51.0, Q 8 50 and Eq. ~8! is solved numerically. The horizontal axis denotes the cosmic time normalized by AA/M̃ pl : ~a! thefluctuation Q for the mode k510a e H e ; ~b! the fluctuation Q for themode k52a e H e ; ~c! the gauge-invariant quantity Q whose wave
length reaches the Hubble horizon size just after the oscillating
inflation ends, i.e., k5a e H e .

Contents
Nonlinear structure formation as a probe of dark matter :phase-space properties of cold dark matter (CDM) halos•Introduction•Phase-space structures of CDM halos•Summary and prospects
In collaboration withYann Rasera

Hiromu Sugiura(Observatoire de Paris)(Dept. Phys. Kyoto U)

Takahiro Nishimichi(YITP)

Dark matter & structure formation
Dark matter (DM)• Hypothetical invisible massive particles• ~30 % of the energy density of the Universe• Unknown microscopic origin (though many candidates)
Observational evidences:
Flat rotation curves
Weak lensing observations (e.g., Bullet clusters)
CMB & large-scale structure
DM is an important building block in cosmic structure formation

Nature of dark matterIn structure formation,of particular importance is cold nature of DM
velocity distribution was virtually null at an earlystage of structure formation

Cold dark matter (CDM)
e.g., Peebles (’82), Blumenthal et al. (’82), Bond et al. (’82), …

• Early growth of CDM fluctuations

Baryon “catch up”• Hierarchical clustering of structure formation
Irrespective of microscopic origin,(cuspy halos / substructure)

Such a system is macroscopically described by Vlasov-Poisson
equation starting with cold initial conditionulerian)
Perturbation
theorytzmannequation(Vlasov-Poisson
system)erian) Perturbation theory#$
Cosmological
Vlasov-Poisson
systems Boltzmann(Vlasov-Poisson
system)∂pequation∂∂Ψ ∂+ equation
−
m
f
(x,
p)
=
0,
(4.23
Boltzmann(Vlasov-Poisson
system)
2 ∂x
∂t
ma
∂x
∂p
Vlasov-Poisson
system in a cosmologicalbackground:#$∂equation:
p ∂
∂Ψ $∂
#
the Poisson
Distributionfunction
+
−
m
f
(x,
p)
=
0,
(4.2
2
Collisionless∂ ∂t p ma∂ #
∂Ψ ∂x
∂ ∂p
$
%
∂x
+
−
m
f
(x,
p)
=
0,
(4.23)m2
2
2
3
Boltzmann
eq.
∂t
ma
∂x
∂x
∂p
∇
Ψ(x)
=
4π
G
a
d
p
f
(x,
p)
−
ρ
.
(4.24
m
d with the Poisson equation:
3a$ith the Poisson equation: # m %2
2 %3$ρm . a(t) : scale factor (4.2ofΨ(x) = 4π particle.
G#am
d
p
f
(x,
p)
−
s of CDM∇(+baryon)
3 3
2
2
a
the Universe
potential∇ NewtonΨ(x) =
4π G a
d
p
f
(x,
p)
−
ρ
.
(4.24)m3a
he mass =ofLarge-N
CDM (+baryon)
particle.
limit (N→∞)
of N-body simulation
mass of CDM (+baryon) particle.

proximation
Cold initial flow (or single-stream flow):
Dirac’s delta function
am approximation&'
3
approximationnsatz
f (x, p) = n a {1 + δm (x)} δD p − m a v(x) .
(4.25
&

'

Velocity field
3Mass density field
Ansatz f (x, p) = n a {1 + δm (x)}&δD p − m a v(x).
'3Ansatz f (x, p) = n a {1 + δm (x)} δD p − m a v(x) .

aking the zeroth and first velocity moments of Eq. (4.23) yields (4.2
(4.25)
System
at
an
early
phase
is
reducedtopressureless
fluid
system∂δmzeroth
1 and first velocity moments of Eq. (4.23) yields
satz, taking the
+ and
∇→
[(1
+
δ
)v]
=
0,
(4.26
m
z, taking the zeroth
firstvelocity
momentsofEq.
(4.23)
yieldsfoundation of (Eulerian) perturbation theory

Fate of cold initial condition
velocity

In 1D cosmology (example)

Phase spacediverge !

densityDensity profile

position

Single-stream

position

Shell crossing !

position

position

position

Multi-stream flow(= formation of dark halo)

Shell-crossing & multi-stream flows are natural outcome of nonlinear
structure formation in CDM cosmology → Test for CDM paradigm

Boundary of CDM halos
16

Phase spase5

vr / v c

0.5
4

0
-0.5

3

-1

2

JCAP11(2d log ρ / d log r

(r)

1
r→∞
−3
⟶
r
(r/rs )(1 + r/rs )2(Navarro et al. ’97)

-1
-2

-4

Outskirt of density profile is found to
significantly deviate from NFW profile:
halo

0

-3

Diemer & Kravtsov (’14)
Adhikari et al. (’14)

This exactly happens at the boundary of
single-/multi-stream flow (≠viral radius)

Densityslope→ splashback radius

-5
0.1

0.5 1
r (Mpc h-1)

2 34

rofile of similar mass at rsp (as1.0
l below). This is consistent with
12
lashback feature.
than an NFW profile of similar mass at rsp (asooksteeper
atthelogarithmic
slopeof
Chang et al. (’18)
1.0
we discuss in more detail below). This is consistent with
0.6
•
SDSS
DR8
phot-zgals
the also
expectationa splashback taken
feature. by
ich isthe forapproach
An alternative is to look at the logarithmic slope of
Density
0.6
More
et
al.
(’16),
Baxter
et
al.
(’17)
udes
model
of taken by
theourcollapsed
profile, forwhichtheis alsoprofile
the approachprofileB17. This approach includes our model for the profile ofGalaxieswhich
isassumed
to
be
a
power
0.2
Galaxies
the infalling material, which is assumed to be a power
0.2
Lensing
Lensing•
DES
Y1
photo-zgals&
weak
lensing
In the bottomof Fig.
4 welogashow the loganel law.
of Fig.
4 we panel
showtherithmic slope of the collapsed profile inferred
from
the
Chang
et
al. (’18)
lapsedprofileinferredfromthegalaxy density and lensing measurements. We find that2
at rsp , the inferred collapsed
profilefromboth galaxy
ng measurements.
We
find
that
and lensing profiles Clusters
exhibit rapid steepening,achievingidentified
withvaluesprofile
much steeper
than theslope ofgalaxy
an NFW profile at
2
apsed
from
both
3
algorithm
scales around rsp andredMaPPer
beyond. This again
is consistent
NFW
ibitwith
rapid
steepening,achieving
the picturethat a splashback
feature exists at the
NFW
Densityslope
outskirtsoftheseclusters.
4n •the
slope
of
an
NFW
profileat
PlanckSZ distributions
clustersof rsp and the slope of the
The posterior
32
total profile
and again
the collapsed
in Fig. 4 are shown
eyond.
This
is profile
consistent+ Pan-STARRSin Fig.
5. Here we clearly see photo-z
that the galaxygalaxies
and lensNFW
4splashback
featureexistsattheing measurements of rsp and the slopes of the profiles
NFW
Zurcher
&
More
(’18)
rs. are consistent with each other, with the lensing mea46
surements having larger uncertainties. The measured
tionsof rspslope
and
the
slope
ofrspthelogarithmicof thetotal
profile at
is 3.6 ± 0.3
Densityslope
2
Detection
significance,8
and 3.5 ± 0.4 isfor atthe high-stat.
galaxy density andlensing pro0.2
0.5
1
2
5
10
apsed
profile
in
Fig.
4
are
shown
files, respectively. The measured logarithmic slope ofr [h Mpc]

but the results are still controversial

dlogr(r)/ (r)dlog (r)/dlogr

dlog coll (r)/dlogr

dlog (r)/dlogr

coll

coll

(r)/ (r)Detection of splashback signature1

Splashback radius: theoretical aspects
The Astrophysical Journal, 789:1 (18pp), 2014 July 1

Dependence of outer environment is important

The Astrophysical Journal, 789:1 (18pp), 2014 July 1• Mass accretion rate :Δ log(Mvir)Γ≡
Δ log(a)(Diemer & Kravstov ’14)

• Dark energy & modified gravity
d log ρ/d log r

d log ρ/d log r

-1.5
-2

-2.5

Adhikari et al. ’18

Figure 7. Dependenceof the slope[62]profilesthe mass
accretion rate
GR 1e11
andonCMB-S4[63]).
0Observatory
occurrence of a recent major merger. In both panels, F5
the1e11
red line shows
8e121.5 < ν <median density profile of all halos in the peak heightGR
range
-0.5
F5 8e12z = 0, previously shown in Figure 5. In the top panel,
theGRsample is fur
particlesIII.
SPLASHBACK
INchange
MODIFIED
split by accretion rate,measuredas the logarithmic
particles
F5 in halo m-1
MODELSΓ ≡ ∆ log(Mvir )/∆ log(a), with differences evaluatedfor the main progenand descendant halo at z = 0.5 and z = 0. Halos with high mass accre
rates exhibit very-1.5
different median profiles compared to their slowly accre
Modifiedgravitybeen
invo
counterparts. The bottom
panel shows
the samemodels
samples,have
but with
the additio
ternatives
dark energy
understandcondition that the -2
halos
have notto
undergonea majorto
mergersince z = t
The profiles are veryaccelerated
similar to thosein the top panel,whichdemonstrates
expansionof theuniverse.
Insystematic deviationsin the shape
of
the
outer
profilecorrelatewiththe-2.5gravity
is modified on large scales but onoves
mass accretion rate rather than a sharp increase of mass due to a recent mhigher density environments, general relat
merger.
-3restored to be consistent with the stringent(A color version of this figure is available in the online journal.)

Sub-halos-1.0
w =ww ==-1.0-2.0
w = -0.5-1.5 w = -2.0
w = -0.5

-2
-2.5
-3
-3.5

[60] anObserv[60] and WFIRST [61]) and CMB survey-1dlog ρ / dlogr

-1

Γ=1.5

-4
1
r/r200m

III.

GR
Mod
-3.5tests of GR in the solar system. Most the
est therefore invoke screening ternati
mechanismf(R)-4these modifications in high density regions.

accelergravity

The figure shows
strikingly
clear
betweenm
1 fcorrelation
on atheHu-Sawicki(R) [36] andthe Dvaaccretion rate andthe steepness
themedianoutertwo
profi
Porrati(DGP) model[64]
that utilize
d
r of
(Mpc
h-1)

Beyond splashback radius
Splashback radius is just one of the rich CDM characteristics
Beyond splashback radius,

Multi-stream structure is supposed to be developed
Alternative probe of dark matter
3

Radialvelocity2

Radiusv/V200

1
0
-1
-2
-3
0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

r/R200How can we look at multi-stream ?

Tracing multi-stream flow with particle
trajectories in N-body simulation
H. Sugiura, AT,Yann & Nishimichi (in prep.)

Distance from halo center

Keeping track of apocenter passage(s) for particle trajectories,
number of apocenter passages, p, is stored for each particle= SPARTA algorithm + α
p=0

p=1

N-body simulation
by Y. Rasera

p=2(Diemer’17; Diemer et al.’17)

p=3

Tiling phase-spacestreams with p
PresentTime• L=316Mpc/h, N=512^3

11,000 halos• 60 snapshots at 0<z<1.43(M200

1013 M )(Observatoire de Paris) • Einstein-de Sitter universe (⌦m = 1, ⌦⇤ = 0)

Multi-stream flow in CDM halo
H. Sugiura, AT,Yann & Nishimichi (in prep.)3

Radialvelocity2

Radiusv/V200

1
0
-1
-2
-3
0.00

0.25

0.50

0.75

Halo center

1.00

r/R200

1.25

1.50

1.75

2.00

Multi-stream flow in CDM halo
H. Sugiura, AT,Yann & Nishimichi (in prep.)3

Radialvelocity2

Radiusv/V200

1
0

p=1

-1
-2
-3
0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

r/R200Phase-space distribution of particles
classified by # of apocenter passage, p
Are these really multi-stream flow ?

p=2

p=3

p=4

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0.0 131✏ = 0.2✏ = 0.2= 0.6✏ =✏0.6✏ =✏1.0= 1.0
24
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non-dimensional
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46
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non-dimensional
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68

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2• Scaling parameters ofvelocity & radius
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4
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0.6
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Comparison with self-similar solution
Sugiura et al. (in prep.)

vr /v200

Using particles with p=1~5 (# of apocenter passage) to fit:
An example
to show a good agreement
2.0

Self-similar

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

-1.0

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p=1 -2.0

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Self-similar

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0.5

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i
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p=2 -2.0

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y
r
na

2.0

1.5

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0.5

2.0

Self-similar

p=3 -2.0

p=4 -2.0

r/r200Self-similar

p=5

2.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Comparison with self-similar solution
Sugiura et al. (in prep.)

Using particles with p=1~5 (# of apocenter passage) to fit:
vr /v200

C = 8.65+4.52−3.20

An example
to show a good agreement
2.0
1.5

1.0

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

-1.0

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y
r
a
n
i

Self-similar

y
m
r
i
l
a
e
n
r
i
P
m
i
l
e
r
P
1.5
1.0

U/v200

1.5

-2.0
0.0

Self-similar

1.6

0.8

0.5

0.5

0.0

0.0

-0.5

-0.5

10

0.0

0.5

1.0

8
6

1.0

-1.0

4

-1.0

-1.0

-1.5

2

-1.5

-1.5

p=2 -2.0

1.5

1.5

1.0

0.5

-0.5

1.5

Self-similar

2.0

1.2

s

2.0

Self-similar

r/r200

U = 0.99+0.20−0.19

p=4 -2.02.4
5 p=3
10 15
-2.0 20 25 0.8 1.2 1.6 2.0

s = 4.50+1.55
−1.53

2

4

6

p=5
8 10

0.8 200
1.0 0.0 0.2 0.4 0.6
C/r0.0
2.0 0.0 0.2 0.4 0.6 0.8 1.0
s 0.8 1.0
200 0.2 0.4 0.6 U/v

Comparison with self-similar solution
Example of good fit

Density map
p=0

p=1

p=2

p=3

p=4

y/R200All

x/R200

Radial velocity vr/V200

Radial phase space

p=1

p=2

p=3
radius r/R200

p=4

p=5

Comparison with self-similar solution
Example of good fit

Density map
p=0

p=1

p=2

p=3

p=4

y/R200All

x/R200

Radial velocity vr/V200

Radial phase space

p=1

p=2

p=3
radius r/R200

p=4

p=5

Comparison with self-similar solution
Example of bad fit

Density map
p=0

p=1

p=2

p=3

p=4

y/R200All

x/R200

Radial velocity vr/V200

Radial phase space

p=1

p=2

p=3
radius r/R200

p=4

p=5

Statistical properties~40% of halos are found to be better fitted to self-similar solution
図 Ry

s のエラーバーX パラメータ決定精度が良いかは微妙だが- sbest と k8W- d8W 区間持っているので- sbest がハローの自己相似解フィットを特徴づける指標として意味があるこ
best =
ると思われるX

1~3)• Massive halos tend to give a better fit to self-similar solution• A large scatter between fitting parameter s and 200
0.30

200 X

(Best-fit) (Measured)

M200 < 1014 M
M200

0.20

1014 M

Light
0.10
0.09
0.08
0.07
0.06

Heavy

0.05
0.04
0.03
0

#miBQM Q7 `2/m+2/

1

2

3

sbest

4

5

6

Γ
200
Measured accretion
rate
200

図 3 .Bbi`B#miBQM Q7 ?HQ ++`2iBQM `i2

2

reducedreduced 2 χ(with s

M200 < 1014)M
Δ log(M20014
Γ200 ≡
M200 10 M
Δ log(a)3.5
3.0(using z=0 & 0.5 data)

2.5
2.0

Heavy

1.5
1.0
0.5

Light

0.0
0

1

2

3

sbest

4

5

6

s

Best-fit accretion rate in self-similar solution (M ∝ a )2
RR B7 (7Q`
議論用, 論文として整形するときにこのグラフを消す
X 図 Rk で十分だしそっちのが
HBM2
?HQb
HB;?i2`
best M/ sbest 7Q` r2HH@}ii2/ ?HQbX h?2 #Hm2 図Summary
Dark matter halo as cosmological probe of dark matter& fresh look at its phase-space properties• Distinctive feature of cold dark matter in phase space{• Multi-stream structure• Sharp divergence in density（shell-crossing）
Outskirts of halo → Splashback radius• Tracing multis-dream flow with particle trajectories··· comparison with self-similar solution~40% of halos are better fitted to self-similar solution
Investigation of phase-space properties of dark matter are
fun, and would help clarifying the origin of dark matter