()Inflation, dark energy, dark matter, …

Shuichiro Yokoyama (KMI, Nagoya Univ.)   2019

contents• Overview• Inflation• Dark energy• Dark matter• SummaryOverview
success of standard cosmology

LSS
M. Betoule et al.: Joint cosmological analysis of the SNLS and SDSS SNe Ia.
46

coh

C

0.12
0.11
0.08
0.11

HST

44

Table 9. Values of coh used in the cosmological fits. Those values correspond to the weighted mean per survey of the values
shown in Figure 7, except for HST sample for which we use theaverage value of all samples. They do not depend on a specificchoice of cosmological model (see the discussion in §5.5).

M(G) + ↵X1

42

SNLS

40

SDSS

38

µ = m?B

samplelow-z
SDSS-II
SNLS
HST

36

Low-z
34

σcoh

CMB

µ

0.15

µ⇤CDM

0.2

10 2

0.1

0.05

0
0

0.4
0.2
0.0
0.2
0.4

0.5
redshift

1

10 1

z

100

Fig. 8. Top: Hubble diagram of the combined sample. The distance modulus redshift relation of the best-fit ⇤CDM cosmology for a fixed H0 = 70 km s 1 Mpc 1 is shown as the blackline. Bottom: Residuals from the best-fit ⇤CDM cosmology as
a function of redshift. The weighted average of the residuals in
logarithmic redshift bins of width z/z ⇠ 0.24 are shown as
black dots.

SNe
©ESA

Overview• Standard component in cosmology?
Any others?

inflation

r 0.002

0.6

Constraint on CMB
0.4
0.2
0.0

power lawhybridnew

WMAP + 2dF
0.8

r 0.002

Tensor-to-scalar ratio

1.0

50 60

WMAP + CBI + VSA
50 60

0.6
0.4
0.2
0.0
0.90

0.95

1.00

ns

Spectral index

1.05

0.90

WMAP 2007

0.95

1.0

ns

Figure 3.2: (From Ref. [117]) Joint two-dimensional marginalized contours (68% anlevels) for inflationary parameters (r0.002 , ns ). r0.002 = 16ϵ(= 8r) represents a tensor
k = 0.002/M pc. In this figure, they assume a power-law primordial power spectrum(Upper left) WMAP only. (Upper right) WMAP+SDSS. (Lower left) WMAP+2dFG

Constraint on CMB

Planck 2018

Beyond standard??

le End Road, London, E1 4NS, United Kingdom

Inflaton with a non-minimal coupling

mail: tomot@cc.saga-u.ac.jp, ttenkan1@jhu.edu

tract. We study models of inflation where the scalar field that drives inflation

minimally to gravitywith
via ⇠ 2 R, or where the gravity sector is enlarged by anonsider the
0.08 original Higgs inflation, Starobinsky inflation, and two di↵erent
metricPalatini
0.07
ξ= 2
ξ= 2
nario
wherearethe inflaton is aξ=scalarfield other than the Higgs, and discuss
MB parameters
ξ= 5predicted
5
0.06
ξ=10
ξ=10
stinguished
from
each
other
by
measuring the tensor-to-scalar ratio and runn
WMAP5
0.05
no coupling
tral index0.04of primordial curvature
perturbations, on top of the amplitude an
0.03
x of the perturbations.
We consider both metric and Palatini theories of gravit
0.02
detailed studies
of non-minimally coupled models can help to identify the in
0.01
0
how
they mayprovide
for
a way
to
also distinguish between di↵erent theories
-4
-2
0
2
4
4
χ/M
e present context.
Best model?
Higgs inflation – mechanism and predictions

Predictions: CMB parameters and Higgs boson massV/(λφ/4)

0.4

50 60

0.3

2

SM+ ξh R

0.2

0.1

0.0

0.94

0.96

0.98

1.00

1.02

pl

Takahashi,index
n frame≃ 1 −Tenkanen(2018)
the cases spectralwith theJordanpotential
assumed as
2 ≃ 0.97(4N+3)
192
4
tensor/scalarr ≃ (4N+3)≃ 0.0033cases are(right).
Those forratio
the metricandPalatini2
he metric and the Palatini cases are almost identical for8(4N+9)δT/T ∼ 10−5=⇒

62/Q`X"2x`mFQp!HKmX/2√ξ ≃ 47000
λHiggs inflation – consistency and possibilities

10 / 22(Higgs inflation, Bezrukov, Shaposhnikov (2008))

ore suitable for slow-roll inflation (see the right panel

çè Starobinsky model (R^2 inflation)

Inflaton with a spectator field- extention to multi-field case -tuations from the inflaton can also contribute to the primordial fluctuthe power spectrum can be given by the sum of those contribution4 :Ps (k) = Ps(φ) (k) + Ps(χ) (k),(A.9)

inflaton
spectator(k) are the primordial power spectra generated by the inflaton φ andd χ. Due to the fact that the energy density of χ is subdominant during(χ)(φ)
index and its runnings for Ps are diﬀerent from those for Ps which(χ)10)–(A.8). To write down the spectral index and its runnings for Ps ,e the slow-roll parameters for χ:χ)

JCAP09(2014)

U ′′U ′ U ′′′
(U ′ )2 U (4)
(2)
(3)
,
ξχ ≡
,
σχ ≡
,
3H∗2
(3H∗2 )2
(3H∗2 )3
(U ′ )3 U (5)(U ′ )4 U (6)
(5)
≡
,
ζχ ≡
,
2 )4
2 )5
(3H
(3H
∗
Figure 3.∗The accessible region of the quadratic curvaton
models in the
chaotic inflation. The
Fujita,Kawasaki,
SYmodels(2014)
pairs of circles represent the predictionof theoriginal chaotic
for N = 50 (left) N = 60
“Encyclopædia curvatonis”(right). The curvaton with appropriate R > 0 and |⌘ | < 0.1 can produce {n , r} in the region
Figure1.
Predictionsof a few inflationarymodels (Large Fiel
al below
fortheχdashedfield
a primerespect Vennin,to the!
lines.andSince thedotted lines indicates
denote the case ofthe⌘ = 0,derivative
only the curvaton with
with aKoyama,Wands(2015)
and6, Higgs Inflation V ∝ [1 − exp(− 2/3φ/M )]2 - the Star
negative mass squared generates the region to the left of the dotted lines. Blue and red shaded
regionst represent(i) denotesthederivative.
Hexample,the
Hubbleparameterat fthe∗ is
BICEP2and only
Planck i-th
constraints,
respectively [4]. For
the pointof theVredNottensor-to-scalarratio,butalsochange∝star,
1 + spectralcos(φ/f ) withindex= 3Mshould
and Power
Law Inflation V ∝ ex{n , r} = {0.945, 0.15}, cannot be realized by the p = 2 chaotic + curvaton model, but the p = 4
nflation.
using
these
slow-rollalongwith fieldthose
massiveσ is defined
added, and when the reheating scenario is of tmodel with a By
negativemass squarecurvatonand the p = 6, parameters
8 model with a positive
massa curvaton
èdegenerate?
can reproduce it. One can see that in the observationally favored region, the models areheavilythestandard“curvaton
scenario”) and 9th (right panel, correspond. (A.3),thespectral
index
and
its
runnings
for
χ
are
given
as
degenerated.
p

s

s

Pl

Pl

R^2 inflation and …

to be 0.5 and 0.6, and the range of f and N0 as
respectively. As one can see, for these paramete
and the R2 -inflation partially degenerate in the rthe prediction of these models in the ns –αs plane
two models using the expected constraint on the

Figure 6. The predictions for r, ns and αs for the R2 -inflation (red circles) and thecase(1971),(blue circles).
The leftpanel…shows the predictions in the r-ns plane, and the
Nariai and Tomita
Starobinsky(1980),
those in the αs -ns plane.

In Fig. 6, we plot the predictions for r, ns and αs for R2 -inflation (R - inﬂation
quartic-spectator case (blue circles). The left panel
shows the predictio
bove expressions, the spectral index and its running of the total primordial
plane, and the right one shows those in the αquartics − -nspectators plane. Here, w
rum are respectively given bythat r in the quartic-spectator case can be consistent with that in the!
"!
" N =1 50 − 60. Comparing this figure and Fig. 8 , it is quite diﬃcult to0(χ)= (1 − Qχ ) n(φ)−(p + 2 − 2Qχ ) ,s − 1 + Qχ ns − 1 =quartic-spectatorcase from R2 -inflation by using the constraint on α
2N0
$ we have also checked that even if
the minihalo
1 # observations. In fact,
2− Qχ ) αs(φ) + Qχ αs(χ) + (1 − Qχ ) Qχ (∆ns )2 = −
p
+
2
−
4Q
+
2Q
χ
χ , to diﬀerentiate these models evenorder running
2N02 βs , it is still diﬃcult
precision of βs obtained by observationsof 21cm fluctuations from min(4.12)
2natural inflation with a massless spectator• Natural-spectator case(φ)(χ) R^2 inflation (N = 50 - 60)= ns −ns and Qχ represents the fraction of the contribution from χ-field2(N = 50 – 60,,

2
Figure(see8. Thepredictions
for r,nInand
αs for the
RFigure-inflation(redcircles)the
andforther,quartic-spectator
7.
The
predictions
n
and
α
for
the
R
-inflatis bessasimilarfashion,letusconsidermulti-field
case
where
the
infl
power spectrum
AppendixA).
Q
canrelatedtothetensor-to-scalarχphi^4-inflation
withamassless
spectator
case (blue circles). The left panelgiven
showsbythepredictions
in
the
r-n
plane,
and
the
right
one
shows
model (purple
circles).
The leftthe predictions is inflation) paneltheslow-rollso-called“natural”
withthe showspotential:
(N
=
50
–
60)
ress the eﬀective
spectral
index
and
its
runnings
withthethose in the αs -ns plane.
those in the αs -ns plane.
%&
eed to define the fraction of the contribution to4p
the (total) powerφ
4r = (1 − Qχ )16ϵ =; tensor-to-scalar(1 − Qχ ) . ratio(4.13)
χ fields as
V
(φ)
=
Λ
1 − cos(0 ≤ φ ≤ πf ),N0fplane, and the right
one shows those in αs − ns plane. Here, we choose Qχ so that r(φ)(χ)
5 Conclusion
2 -inflation for N =Ps (k0 )Ps (k0 ) case can be consistentinthequartic-spectator
with
that
in
the
R
0
we
fix
the
e-foldingnumber,N,
fixingr
corresponds
tofixingQ.
,
Q≡,(A.16)
0
χ
χ
(φ)
(χ)
(φ)
(χ)
Ps (k0 ) + Ps (k050) − 60. Comparing
Ps (k0 ) + Pthis)s (k0figure
and Fig.
6 , it isTakahashi,quite diﬃcult
to SY
distinguishtheSekiguchi,
Tashiro,in preparation
quartic-spectator
caseWith
fromthese
R2 -inflationby using the constraint on αs obtained fromare to be evaluatedat the pivot scale.
variables, theHow to discriminate?
• Higher order ?

Local type non-Gaussianity inmulti-scalar inflation
Linear perturbation (free propagation) ßà Gaussianè through the non-linear interaction
è non-zero higher order perturbations
è non-Gaussianity!!
è observed by higher order correlation functions!
Standard single slow-roll case;slow-roll suppressed!!
Spectator case;transfer from spectator to curvature perturbations

Distinguishing models by Local type NG• Single inflation vs spectator (curvaton, …)
Critical value;
Spectator (curvaton) scenario è
due to the transfer from spectator to adiabatic curvature pert.
Single scalar inflation è
slow-roll suppression

How to discriminate?
• Higher order ?
• Smaller scales?

Primordial fluctuations from inflationa; scale factor expansion of the UniverseH; expansion ratetypical size of the Universe

Sub-horizon
Super-horizon

generation

Quantum fluctuations(causal) are stretched byinflationary expansion
è“classical” super-horizon(acausal) fluctuations(basically determined at
the horizon exit)è
primordial fluctuations
with different scales(wavenumber) would
bring us the informationof different stages ofinflation !!

⟨ζ(k)ζ(k′ )⟩ = (2π)3 δ (3) (k + k′ )Pζ (k),(4.

Below wecurvaturegive explicit perturbations.
expressions for theseHere,runnings
the ′slow-rollparameters for theDiraceing primordialδ (3)using(k +k) is a 3-dimensional
single-field and multi-field models.
Runningsof thespectralindexofasction and P
initial condition
thealinear matter power spectru
ζ (k) determines
Single-fieldcase equation. By taking into account the scale-dependen
by P (k) A.1
throughthe Poisson
perturbative
approach
Assuming
slow-roll
single-field inflation
with a canonical kinetic term, ns and theectral index,
we acanperturbatively
writemodelPs asSpectralrunning parameters
canindex
be explicitly written down with the slow-roll parameters, which are
1 as! potential"ns −1+
defined using the inflatonV (φ)αs ln(k/k0 )+ 1 βs ln2 (k/k0 )k!
"
Ps (k)
1 2 = VA′ s 2
ϵ ≡ Mpl
, k0

2

2 V
η ≡ Mpl3!

′′,

V ′ V ′′′
(2)4
ξ ≡ Mpl
,
2

(4.

,
V
V
V
(A.3)
Taylorseriesofscaledependenceofindex′ )2 V (4)′ )3 V (5)′ )4spectral
(6)
(V
(V
(V
V
(3) ≡respectively
68
10 and theareamplitude,the
Mpl, theτ (4)≡ Mpl, pivotζ (5) scale≡ Mpl. leading-ord
s , k0 and ns σ
34Below we giveV
V explicit expressions for these runnings
V 5 using the slow-roll para
single-fieldmodels.
index, and we have expanded Ps in terms
ofandlnmulti-field
k up tothe 2nd order. The expansio
give explicit expressions
for theserunnings
using
the slow-rollforForthetheseslow-rollinflation,theseparameters
can index
be theexpressed
slow-rollparameters
Using
slow-rollparameters,
the parametersspectraland thew.r.t.
runnings
are givenby: 2 asA.1 Single-fieldnts
the 1stand 2nd orders are denoted
as αs (≡case
dns /d ln k) and βs (≡ d ns /d ln2 kandinmulti-fieldmodels.
a slow-roll single-field inflation model with a canonicalkinetic ter
ns − 1 =and
−6ϵ the+ 2η,quadraticAssuming(A.4)
egle-fieldcall therunningofn,respectively.
Intheframewocase runnings2(2) running parameters can be explicitly written down with the slow-roll parame
α
=
−24ϵ
+
16ϵη
−
2ξ
,as
(A.5)
s the
a slow-roll single-field
inflation
modelwith a canonicalkinetic
term,
nand
definedusing
thetheinflaton
potential
V (φ) ass andow-rollinflation,
runnings
such
α
β
can
be
explicitly
writtendown bss322(2) are(2)(3)
arameters can be explicitlyβwritten
down
with
the
slow-rollparameters,
which
!
"
+V2ηξ
+ 2σ , V ′′(A.6)V ′ V ′′′s = −192ϵ + 192ϵ η − 32ϵη − 24ϵξ′ 212

engso-calledslow-rollin Appendix
the inflaton potential
V (φ) as parameters. We provide
2 those expressions
2
4 A. ,
ϵ2 ≡2 Mpl
,
η3
≡ Mpl
, (2)
ξ (2)
4
3
2 (2)
(3)≡ Mpl
γs = −2304ϵ + 3072ϵ η − 1024ϵ η 2− 384ϵV ξ + 64ϵη + 148ϵηξ+ 36ϵσV
V2! ′ "2
′′
′
′′′
′
2
(4)
′
3
(5)
,1 Mwearbitrarily
higherordersweAppendVcan expand P
Vs up toV V (2)(V ) V and hence(V) V also, in(V ′ )4 V
2
2
(2)
4
(3)
2
(2)
2
(4)
(3)
6
(4)
8
(5)
10
,
η ≡ Mpl −6ησ,ξ ξ≡ M−, )σ − ≡τ ≡M, potential
ζ (A.7)≡ Mpl− 2η
2τMpl , V 3 ,pl 2(ξ
V;pl inflaton’s
2 pl V
V
V2
V4
V5
(A.3)
expressionsforhigherorder
runnings
in
terms
of
the
slow-rollparameters
noton′ )3 V (5) 5
4 (V ′ )4 V (6)
3 2
3 (2)
2 3
2 (2)
(V ′ )2 V (4)(Vδ=−36864ϵ+61440ϵη−
30720ϵ
η
−
7680ϵ
ξ
+
4736ϵ
η
+
5448ϵ
ηξrunnings are giv
6
(4)
8
(5)
10
s
Using
these
slow-rollparameters,thespectral
index
and
the
≡ Mpl
,
τ ≡ Mpl,ζ ≡ Mpl
.
3
4
5
V
V
V
ngle-fieldcase but also
2 (3)the multi-field(2)+744ϵforσ − 128ϵη 4 − 652ϵη 2 ξcase.
− 164ϵ(ξ (2) )2 − 340ϵησ (3) − 52ϵτ (4)dependtheonhigherorder
parameters
not appear in “r” and “n_s”
ns −= −6ϵdo+ 2η,e slow-roll parameters,
spectralindex
and slow-roll
the runnings
are givenby:1which
3 (2)
2 (3)
(2) 2
(4)
(2) (3)+2η ξ + 14η σ + 8η(ξ α)s =+ −24ϵ12ητ2 + 16ϵη+ 10ξ+ 2ζ (5) .
(A.8)− 2ξ (2)σ,1 = −6ϵ + 2η,(A.4)
βs = −192ϵ3 + 192ϵ2 η − 32ϵη 2 − 24ϵξ (2) + 2ηξ (2) + 2σ (3) ,recasts based on the Fisher analysis

to be 0.5 and 0.6, and the range of f and N0 as
respectively. As one can see, for these paramete
and the R2 -inflation partially degenerate in the rthe prediction of these models in the ns –αs plane
two models using the expected constraint on the

As examples

Figure 6. The predictions for r, ns and αs for the R2 -inflation (red circles) and thecase (blue circles). The left panel shows the predictions in the r-ns plane, and thethose in the αs -ns plane.

r – n_s plane

In Fig. 6, we plot the predictions for r, ns and αs for R2 -inflation (R - inﬂation
quartic-spectator case (blue circles). The left panel
shows the predictio
bove expressions, the spectral index and its running of the total primordial
plane, and the right one shows those in the αquartics − -nspectators plane. Here, w
rum are respectively given bythat r in the quartic-spectator case can be consistent with that in the!
"!
" N =1 50 − 60. Comparing this figure and Fig. 8 , it is quite diﬃcult to0(χ)= (1 − Qχ ) n(φ)−(p + 2 − 2Qχ ) ,s − 1 + Qχ ns − 1 =quartic-spectatorcase from R2 -inflation by using the constraint on α
2N0
$ we have also checked that even if
the minihalo
1 # observations. In fact,
2− Qχ ) αs(φ) + Qχ αs(χ) + (1 − Qχ ) Qχ (∆ns )2 = −
p
+
2
−
4Q
+
2Q
χ
χ , to diﬀerentiate these models evenorder running
2N02 βs , it is still diﬃcult
precision of βs obtained by observationsof 21cm fluctuations from min(4.12)
2natural inflation with a massless spectator• Natural-spectator case(φ)(χ) R^2 inflation (N = 50 - 60)= ns −ns and Qχ represents the fraction of the contribution from χ-field2(N = 50 – 60,,

2
Figure(see8. Thepredictions
for r,nInand
αs for the
RFigure-inflation(redcircles)the
andforther,quartic-spectator
7.
The
predictions
n
and
α
for
the
R
-inflatis bessasimilarfashion,letusconsidermulti-field
case
where
the
infl
power spectrum
AppendixA).
Q
canrelatedtothetensor-to-scalarχphi^4-inflation
withamassless
spectator
case (blue circles). The left panelgiven
showsbythepredictions
in
the
r-n
plane,
and
the
right
one
shows
model (purple
circles).
The leftthe predictions is inflation) paneltheslow-rollso-called“natural”
withthe showspotential:
(N
=
50
–
60)
ress the eﬀective
spectral
index
and
its
runnings
withthethose in the αs -ns plane.
those in the αs -ns plane.
%&
eed to define the fraction of the contribution to4p
the (total) powerφ
4r = (1 − Qχ )16ϵ =; tensor-to-scalar(1 − Qχ ) . ratio(4.13)
χ fields as
V
(φ)
=
Λ
1 − cos(0 ≤ φ ≤ πf ),N0fplane, and the right
one shows those in αs − ns plane. Here, we choose Qχ so that r(φ)(χ)
5 Conclusion
2 -inflation for N =Ps (k0 )Ps (k0 ) case can be consistentinthequartic-spectator
with
that
in
the
R
0
we
fix
the
e-foldingnumber,N,
fixingr
corresponds
tofixingQ.
,
Q≡,(A.16)
0
χ
χ
(φ)
(χ)
(φ)
(χ)
Ps (k0 ) + Ps (k050) − 60. Comparing
Ps (k0 ) + Pthis)s (k0figure
and Fig.
6 , it isTakahashi,quite diﬃcult
to SY
distinguish
Sekiguchi,
Tashiro,(2018) thequartic-spectator
caseWith
fromthese
R2 -inflationby using the constraint on αs obtained fromare to be evaluatedat the pivot scale.
variables, theAs contribution
one can see, fromfor theseparameter values, the natural-spectator model
presents therespectively.
fraction of theχ-field
degenerate in the r-ns plane. However, when we compare
ppendix A).andQχ thecan R
be2 -inflation
related topartiallythe tensor-to-scalar
the prediction of these models in the ns –αs plane, it would be easy to distinguish these
two models using the expected constraint on the ns –αs plane shown in Fig. 8.
4pQχ )16ϵ = (1 − Qχ ) .
(4.13)
N0
Figure 6. The predictions for r, ns and αs for the R2 -inflation (red circles) and the

As examples

case (blue circles). The left panel shows the predictions in the r-ns plane, and theumber, N0 , fixing r corresponds tothosefixingQχα. s -ns plane.
in then_s – alpha_s plane

In Fig. 6, we plot the predictions for r, ns and αs for R2 -inflation (R - inﬂation
quartic-spectator case (blue circles). The left panel shows the predictio
bove expressions, the spectral index and its running of the total primordial
quartic- spectator
plane,and the right one shows those in the αs − ns plane. Here, w
rum are respectively given bythat r in the quartic-spectator case can be consistent with that in the!
"!
" N =1 50 − 60. Comparing this figure and Fig. 8 , it is quite diﬃcult to0(χ)= (1 − Qχ ) n(φ)−(p + 2 − 2Qχ ) ,R - inﬂations − 1 + Qχ ns − 1 =quartic-spectatorcase from R2 -inflation by using
the constraint on α
2N0natural- spectator that even if$ we have also
thechecked
1 # observations. In fact,(φ)(χ)
2 minihalo
2− Qχ ) αs + Qχ αs + (1 − Qχ ) Qχ (∆ns ) = −p + 2 − 4Qχ + 2Qχ ,order runningto diﬀerentiate these models even
2N02 βs , it is still diﬃcult
precision of βs obtained by observationsof 21cm fluctuations from min(4.12)
2

2

natural inflation with a massless spectator• Natural-spectator case(φ)(χ) R^2 inflation (N = 50 - 60)= ns −ns and Qχ represents the fraction of the contribution from χ-field(N = 50 – 60,,

Inbea related
similarfashion,let us consider the multi-field case where the infl2
power
spectrum(see
Appendix
A).
Q
can
to
the
tensor-to-scalar2χnquartic-spectatorphi^4-inflation
7.
The
predictions
forather,masslesss and αspectators for the R -inflation (red circles) and the natural-spectatoror theFigureR-inflation(redcircles)with
and)the the
given
bythe
the
so-called in“natural”
with(purplecircles).
The
left
panel
shows
predictions
the r-ns inflation
plane, andrightpotential:
one shows= 50–sindex60)
themodeleﬀectiveandandits runnings
withtheslow-roll
eress
predictionsin(Nspectralther-nplane,
the
right
one
shows
%
&
thefraction
αs -ns ofplane.
eed to thosedefineinthe
the contribution to4p
the (total) powerφ
4r = (1 − Qχ )16ϵ =; tensor-to-scalar(1 − Qχ ) . ratio(4.13)
χ fields as
V
(φ)
=
Λ
1 − cos(0 ≤ φ ≤ πf ),N0f2

for r,
ns andαs for R -inflation(φ)(χ) (red circles) andPs (kPs (k0 )0)
fix5 left
theConclusion
e-foldingnumber,N0 , fixing
r corresponds
to fixing Qχ .
Qχ the≡ (φ),(A.16)).weThe
panel, shows
predictions(φ)(χ)(χ) in the r − ns
Ps (k0 ) + Ps (k0 )Ps (k0 ) + Ps (k0 )hose in the αs − ns plane. Here, we choose Qχ so Sekiguchi, Takahashi, Tashiro, SY (2018)
are to be evaluated at the pivot scale. With2 these variables, theHow to discriminate?
• Higher order ?
• Smaller scales?

SKA ??

=accountrav

Γ (p, k2 − p)ΓX (−k1 , −p, −k2 + p)PL (p)PL (|k2 − p|) + 5 perms.,
Γ (k1 )PL (k1 )2 X(2π)3 X!
nd to bed3 p (3)
1
(1)
(1)
oop,1
tting
the ΓX (k1 )ΓX (k2 )Γ (−k1 , p, −k2 − p)PL (k1 )BL (k2 , p, −k2 − p) + 5 perms.,=
is
3 X
4 TEST FOR
NON-GAUSSIANITY
2(2π)
Mpc, we!
3
1dp non-Gaussianity
(1)
(2)
(2)
oop,2
The
presenceofprimordial
aﬀectsthe pri=
1.78.
d
=
Γ
(p,
k
−
p)P
(k
)B
(−k
Γ
(k
)Γ
(−k
,
−k
)
1
1
2
2
L
1
L
2 , p, k2 − p) + 5 perms.,
is
X
X
X
3
mordial gravitational (2π)
potential
perturbations (Eq. 1), whichs is neg-2!
will
inducenon-Gaussian (2)characteristics into the density
ured
lind3 p (2)
(1)
oop,3
k2 + p, k3 − p) + 2 perms..
(3
Γ
(p,
k
−
p)Γ
=
Γ
(k
)
2
3 + p)P
1 , peaks
X (−p, kThis
iswillL (p)B
aﬀectL (ktheound by X 1field through(2π)3 Xthe Poisson equation.
of the initial matter density distribution, where the darksar samctions
PL , BL and
TL respectively
denotebi- andpeaks
tri-spectra
matterhalos collapse
on thesuchpower-,
overdensity
aboveofathe linear density field, which arlustering
throughthreshold δc . Non-Gaussianities aﬀect the high mass tailranges.
(rare events) of the halo mass function,
where for a posiled to a3⟨δ)δL (k2 )⟩ =functions(2π) willδ(k+ kgenerated
local order
L (k1correlation
1be2 )PL (k…)
1 ), leadingØ higher(bispectrum,
tive
fNL
more
high-sigma
peaks
ole range
to a larger number of high-mass dark
matter halos. Besides
at these⟨δL (k1 )δL (k2 )δL (k3 )⟩ = (2π)3 δ(k1 + k2 + k3 )BL (k1 , k2 , k3 ),the eﬀect of non-Gaussianity in the mass function of hainties oflos, the existence of a gravitational potential bispectrum oft have a⟨δL (k1 )δL (k2 )δL (k3 )δL (k4 )⟩ = (2π)3 δ(k1 + k2 + k3 + k4 )TL (k1 , k2 , k3 , k4 ).
(4ØScale-dependentbias
the local type can introduce an extra scale dependent term
t coming
in thefielddarkis matterand hencein the perturbations
galaxy bias Φ through the function M(k)tions
atthe welinear density
related halo
to thebias
primordial
curvature(Dalal et al. 2008; Matarrese & Verde 2008), giving
he model2D(z)k 2 T (k)rested inδL (k) = M(k)Φ(k); M(k) =,(5
his range2∗ )(1 + z∗ ) H02 Ωm0
3 D(zH0 Ωm δc (z)
E
E
bE
(k,
z,
f
)
=
b
+
3f
(b
−
1)
(16)
NL
NL
ef f
G
G
2 T (k)k 2
c
D(z), H0 and Ωm0 are the transfer function, the linear growth factor, the Hubble parameter at presene(k),quasarthethematter density
Here zdependence
an!!arbitrary∗ denotes
nndseewhere parameter,bG is therespectively.
linear Gaussianbias measured
in redshift at the matter-dominated
1/k^2
threlation (5),linear power
giventransferbyh−1theMpc.
the theprevious
section,spectrum
T (k) isis thefunction andGalaxies,
minihalos !!!
0
ved thatδc (z) = δc (z)/D(z) ∼ 1.686/D(z) assuming
an Einstein-de
2
non-Gaussian
PLD(z)(k) = being{M(k)}(k), growth fac(6ddition a
Sitte cosmology, with
thePΦlinear

Primordial NG in LSS

10nter for the Early Universe, Graduate School of Science,
FIG.of
3: Tokyo,Theconstraint
on fmatteras (solid)
a functionof (dashed) at z = 0.4 (blue, top)
Figure7: marginalized
Left:Bunkyo-ku,
Power spectrumof dark
andJapan
HI
and
NL
10
10 z = 2.5101010ersityTokyo
113-0033,
n
[h
Mpc
]
the maximum
redshift,the redshift
0.2 on< zfNL< (top); HI Gaussian bias (middle); effective(red, bottom),with fassuming
Forecastrange
1s error
NL = 10. Right:
umamoto
University,
2-39-1
Kurokami,
Kumamoto
860-8555, Japanzmax IM
with
width
∆z =
0.5. Heretake etfivesurveyvolume(bottom).
FromweCameraal. tracers(2013). (mass
ics,bins)
TheforUniversity
Japaneach redshift ofbin.Tokyo, Bunkyo-ku, Tokyo 113-0033,FIG.
1: 1σ marginalized errors in as a function of comoving
eq
orth
local
(top), fNL(middle) and fNLtute for the Physics and Mathematics of the Universe, galaxy number density, fNL
One of the most important features on horizon scales is primordial
non-Gaussianity.
(bottom)for survey with Many
Vsurvey = 10 h−3 Gpc3 and z = 1.
versity of Tokyo, Kashiwa, Chiba, 277-8568, Japan
-5

-4

-3

-2

3

-1

-3

g

SKA as a galaxy surveyσ(fNL)σ(fNL)

models of inflation predict a small amount of non-Gaussianity in the statistical distribution of primordial
This produces
a signal in the bispectrum,
but also
the1000power
spectrum
1-tracer2fluctuations.
raints on
local-type
primordial
non-Gaussianity
parameter
fNLin witha Vsurvey-33
continuum
survey2-tracer=10h Gpc , z=2
3-tracer–sinceprimordial
non-Gaussianity
inducesascale-dependent
correction
totheGaussianbias:
by Euclid, a continuum galaxy survey by Square Kilometre Array (SKA),
b !a bgeneral+ Db. Thisexpression
correction grows
largecovariancescales as Db µmatrix
fNL k 2 forofprimordial
non-Gaussianity
We derive
foronthethe power
eq
fNL
100
1 type, where f
of
the
local
is
the
non-Gaussian
parameter.
NL
tiple tracers to show how the so-called multitracer technique improves
In Camera et al. (2013), an analysis is given of the constraining power of IM surveys over fNLorthicular we clarify
the role of the overlap fraction of multiple tracers andnon-Gaussianity; their results are summarised in Fig. 7. This shows that the forecast errors on local
tracers. Our0.5Fisher
matrix analysis indicates that stringent constraints
fNL
10
5-tracers
fNL can be taken down towards s fNL . 3 for a deep enough survey with sufficient dishes. We
Euclid : same
number density
ined even with a single
survey,assuming five mass bins. When Euclid
recast their analysis
accordingtothe updated specifics of Table 2, and adopt a SKA1-MID IM
SKAon: galaxytypeare improved to σ(fombined, constraints
fNL
NL ) = 0.61 (0.50).
-5
-4
-2
-1
10
10
10
survey0.25
operating for 10, 000 hours at a system temperature of 20 K. The chosen10bandwidth
is-33 10-3ng [h Mpc ]Euclid1050,
SKA1
Euclid+SKA1
SKA2
Euclid+SKA2
therefore 350
where
we keep
thelast, high-frequency bins
between1000 and 1050 MHz
SKA1
(2)
?
rveys
Mario G. Santos
Yamauchi,Takahashi (2016)
for foreground removal. The bandwidth is further subdivided into constantbins
of for
10SY,
FIG. 2: frequency
Same as Fig.
1 but
z = 2. The green dotted lines
Yamauchi, Takahashi,
Oguri (2014)
FIG. MHz
4: Theexpected
marginalized
fNLin forrepresent
limit from Planck [3, 4].
width,
collected
into ‘chunks’constraintsof 20 by 20onbins
order to construct
a 65theby current
65 tomographic
each survey and combinations.
è Othermatrix. (To deal with the large number of bins, we use a block diagonal tomographic matrix
wheretypes of NG ?
ON
allowsustomeasurethescale-dependentbias
accurately
we correct for the overlapping, as described
configuration
eventually
HI IMin Camera et al. 2013.) Suchby asummingover allpossible triangular configurations.
even
at
large
scales,leading
tostrongconstraints
fNL
factor2
improvement
The explicit expression
ison
given
by .[32]
yields a constraint on the primordial non-Gaussianity parameterk! compared with Plancksitytracers
fluctuations
drastically is
improve the constraint, simply
because
" −1
# ∂B
∂B
s fNL = 2.3,
(7.2)
F
=
·
C
(B,
B)
· β , (26)αβmultitracer
technique does not take eﬀect for the one∂θα∂θ
he theearly
Universe.
k ,k ,k =k
tracercase.
Furthermore,combining
multipleredshift
namely morethethan three times better than the current constraint from Planck (using the large-scale
n-Gaussianity,
bins improves substantiallythe constraint, as is shown
where B = {B0I (k1 , k2 , k3 )}, I runs over the mass bins
structure convention).
widely,be- that galaxy samples(g1 g2 g3 ), and θα are free parameters to be determined
in Fig. partly
3. We findas et.
faral.asz =Santos,(2015)
IM
surveys will
also allow us
testconstraint.
Einstein’s theory of generalrelativity The
for thefirst
by observations.
marginalized
expected 1σ error on
3.2 predict(sixthSKAbin)smallcontributesignificantly
to to
the
odels
α
m of dark matter (solid) and HI (dashed) at z = 0.4 (blue, top) and z = 2.5
max

1

2

3

min

SKA as a galaxy survey
Higher order non-Gaussianities?
3
3

gNL/104

2

SKA1
EuclidEuclid+SKA1
SKA2
Euclid+SKA2

1
0
-1
-2

τNL

trum have diﬀerent redshift-, scale- and halo-mass dewith
pendences.; aTheparametercorrection to therelated
bias roughly
scales as
22 2
42βf z/k for thefNL , βprimordialand βg z/k for gNL .
f z /k for τNL , bispectrum
Because the correction to the bias from fNL is identi(3-pt.
func.)
cal to the one from gNL = (βf /βg )fNL , fNL and gNL
are degenerate for a single tracer case. However, in thecase with multiple tracers, the diﬀerent halo-mass dependence of βf and βg would break their degeneracy [23]. On
the other hand, because the bias correction due to τNL; parameters
relatedwithhas larger dependenceson the scaleand redshift,
its detectability would
enhanced for a trispectrum
survey with larger
thebeprimordial
angular scale and higher redshift coverage. Thus, we ex(4-pt.
pect that, with the multitracer technique,widefunc.)
and deepsurveys will be powerful to probe higher-order PNG.

SKA:lmin=2(solid),lmin=3(dashed)

-3
80
60
40
20
0
-20
-40
-60
-80
-6

-4

-2

0

fNL

III.

RESULTS

To study the required survey level needed to test
the consistency relation, we proceed to the Fisher
analysis. The Fisher matrix is defined by Fαβ =# ∂ C!J (ℓ)!
!I (ℓ) "∂C−1, where the indices I, J
Covℓ,I,J ∂θ α
IJ ∂θ β
′

α

2

4

6 -3

-2

-1

0

1

2

3

4

gNL/10

Yamauchi,contours
Takahashi(2015)
FIG. 1: Forecast 1σ marginalizedin (fNL, τNL ),
(gNL , τNL ), and (fNL , gNL ) planes with the vanishing fiducial
values of the nonlinear parameters. To see the dependence on
the minimal multipole we also plot the results for SKA withℓSKA
min = 3 in the dashed line.

arity parameters assuming zmin = 6. Shaded region is excluded by current

-10

5

-6

-4

-2

0

fNL

2

4

6

10-1

-6

-4

-2

0

fNL

2

zmin fNL gNL /103 ⌧NL
4
1.0
0.93
26 FIG. 1: Expected 2-dim constraints on the nonlinearity paramete
observations (see text for details).
6
1.3
2.2
28
Planck+SKA
8
2.2
8.1
33
10
3.8
14
58
SKA SY
FFTT
Sekiguchi, Takahashi, Tashiro,(2018)
2
5
4
0.38
0.58
0.56
total e↵ective area Atot [m ] 10
107
…)
0.48
0.75 (lots0.58 of works, bandwidth⌫ [MHz]1
COrE+FFTT 6
8
0.63
0.98
0.61
beam width ✓ [arcmin]
9
10
0.83
integration time t [hour]
1000
virialized objects
with the1.2virial0.67
temperature

SKA as a probe of EoR

rvations

Minihalos as biased tracersü

too low to cause the star formation

TABLE
TABLE IV: zmin -dependence of the constraint on fNL, gNL II: The survey specifications for 21 cm observations
neutral gas
andü⌧NLfilled
.
10

4

103

10

modelsungaussiton

2

mixed curvaton
with

101

1

with

ing ande figure,
L | > 2.

10-1

mixed modulated
with

COrE
SKA
FFTT

100

-6

-4

-2

0

2

4

6

datasetfNL
CMB alone
Planck
4.0
COrE
2.0
SKA
1.4
+Planck 1.3
+COrE
1.1
FFTT
0.51
+Planck 0.50
+COrE 0.48

gNL /103

⌧NL

41
18
2.3
2.3
2.2
0.79
0.78
0.75

610
160
28
28
27
0.59
0.58
0.58

FIG. 2:
Regions where the consistency relation TABLE⌧NL = III: Constraints on fNL , gNL and ⌧NL at 12
(36/25)fNL Powerful
can be excluded
at 1 .
parameters
being marginalized
to discriminate
large NGothermodels(multi-inflation)!!!over.

T
a

with

Higher NGs in multi-scalar inf.

arity parameters assuming zmin = 6. Shaded region is excluded by current

-10

5

-6

-4

-2

0

fNL

2

4

6

10-1

-6

-4

-2

0

fNL

2

zmin fNL gNL /103 ⌧NL
4
1.0
0.93
26 FIG. 1: Expected 2-dim constraints on the nonlinearity paramete
observations (see text for details).
6
1.3
2.2
28
Planck+SKA
8
2.2
8.1
33
10
3.8
14
58
SKA SY
FFTT
Sekiguchi, Takahashi, Tashiro,(2018)
2
5
4
0.38
0.58
0.56
total e↵ective area Atot [m ] 10
107
…)
0.48
0.75 (lots0.58 of works, bandwidth⌫ [MHz]1
COrE+FFTT 6
8
0.63
0.98
0.61
beam width ✓ [arcmin]
9
10
0.83
integration time t [hour]
1000
virialized objects
with the1.2virial0.67
temperature

SKA as a probe of EoR

rvations

Minihalos as biased tracersü

too low to cause the star formation

TABLE
TABLE IV: zmin -dependence of the constraint on fNL, gNL II: The survey specifications for 21 cm observations
neutral gas
andü⌧NLfilled
.
10

4

103

10

modelsungaussiton

2

mixed curvaton
with

101

1

with

ing ande figure,
L | > 2.

10-1

mixed modulated
with

COrE
SKA
FFTT

100

-6

-4

-2

0

2

4

6

datasetfNL
CMB alone
Planck
4.0
COrE
2.0
SKA
1.4
+Planck 1.3
+COrE
1.1
FFTT
0.51
+Planck 0.50
+COrE 0.48

gNL /103

⌧NL

41
18
2.3
2.3
2.2
0.79
0.78
0.75

610
160
28
28
27
0.59
0.58
0.58

FIG. 2:
Regions where the consistency relation TABLE⌧NL = III: Constraints on fNL , gNL and ⌧NL at 12
(36/25)fNL Powerful
can be excluded
at 1 .
parameters
being marginalized
to discriminate
large NGothermodels(multi-inflation)!!!over.

T
a

with

How to discriminate?
• Higher order ?
• Smaller scales?

SKA ??

SKA as a probe of EoR
Kohri, et al. (2013)

Matter fluctuations in EoR(lots of works, …)

Ø High-z è “gravitational“ non-linearity is not so large (~1 Mpc?)

PlanckPlanck + SKA phase1
Planck + SKA phase2
Planck + Omniscope
CMBpolCMBpol + SKA phase1
CMBPol + SKA phase2
CMBpol + Omniscope
COrE
COrE + SKA phase1
COrE + SKA phase2
COrE + Omniscopeδns
4.11 × 10−3
2.03 × 10−3
1.73 × 10−3
6.04 × 10−4
2.10 × 10−3
1.46 × 10−3
1.33 × 10−3
5.53 × 10−4
2.13 × 10−3
1.47 × 10−3
1.34 × 10−3
5.54 × 10−4

δαs6.59 × 10−3
2.90 × 10−3
2.36 × 10−3
1.07 × 10−3
2.36 × 10−3
2.07 × 10−3
1.84 × 10−3
1.00 × 10−3
2.43 × 10−3
2.09 × 10−3
1.85 × 10−3
1.00 × 10−3δβs9.95 × 10−3
2.21 × 10−3
1.52 × 10−3
7.31 × 10−4
4.37 × 10−3
1.61 × 10−3
1.21 ×10−3
6.86 × 10−4
4.47 × 10−3
1.63 × 10−3
1.22 × 10−3
6.87 × 10−4

Table 4: 1σ uncertainties for ns , αs and βs from various data sets. We take kref =0.05 Mpc−1 to derive these constraints.

Tegmark et al. (2004)

quantities, we take kref = 0.05 Mpc−1 in the most analysis presented below. However,
we also discuss how this reference scale aﬀects the determinations of n , α and β at the3

10 ∆βs0-5

SKA as a probe of EoR-50103∆ns5-50103∆αs5

Sekiguchi, Takahashi, Tashiro, SY (2017)
Shimabukuro, Ichiki, Inoue, SY (2014), …

Minihalos as a probe of smaller scales

Iliev et al. (2002), …

ü virialized objects with the virial temperature

too low to cause the star formation

ü filled. Expected constraints
on the neutral
primordialgas
power spectrum from observations of 21 cm signals
ihalos in combination with the CMB. Contours depict 1 errors with other cosmologicalrs being marginalized. We assume 21 cm signals from minihalos can be measured down to

10-1

1

r

10

r

7.7
3.2
4.6
2.4
1.7
1.3
1.2
0.95
0.955

ns

0.96

10

3

↵s

10.7
2.9
2.9
1.6
2.0
1.3
1.6
1.1
0.965
ns

0.97

10

3

2

s

15.1
6.5
1.5
0.79
0.63
0.44
0.39
0.28 0.98
0.975

1.5
1

103 βsPlanck
COrE
SKA
10-2
FFTT
Planck+SKA
Planck+FFTT-3
COrE+SKA10
COrE+FFTT 0.95

10

0

-1
10-3

103 αs

-1

0.5

-0.5

0.95

10

103 αs-2

ü corresponding scales;
3

2
1.5

0.5
0
-0.5

0.955

0.96

0.965
ns

0.97

-1

-0.5

0
0.5
103 αs

1

0.975

0.98

1.5

2

-2
0.95

0

R2 modelBrane
Figuremodel
6. Predictions for ns , ↵s , s and r for the R2 -inflation (yellow), br
natural-spectator model (green) and the inverse monomial-spectator mode
Inverse monomial-spectator
panel shows the predictions in the ns – r plane, and the upper right one s
Natural-spectator
plane. The lower panel shows the predictions in the ↵s – s plane.

-1
-1.5
-2
0.95

-0.005-0.01-0.015-0.02-0.025-0.03-0.035-0.04-0.045-0.05-0.055
-1.5

-1.5

0.955

We can distinguish !

Constraints on parameters -0.005
for the primordial power spectrum. For 21 cm observations,

0.96

0.965
ns

0.97

0.975

0.98

As seen in the upper left in Fig. 6, models discussed here canpredictions for ns and r, however, when we compare the predictionns – ↵s and the ↵s – s planes, we can see that those models give dwould be helpful to distinguish the model. It should be noted herebe easily di↵erentiated by using the expected constraint from futur
on the ns – ↵s and ↵s – s planes shown in Fig. 4. On the other handifficult to be distinguished from the expected constraint investigate

0

103

University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japanhe Physics and Mathematics of the Universe,
Tokyo, Kashiwa, Chiba, 277-8568, -5Japan

Observational inflation cosmology

-5
0
0
5
local-type primordial non-Gaussianity parameterfNL 5with a-5
10 ∆n
10 ∆α
a continuum galaxy survey by Square Kilometre Array (SKA),
a general expression for the covariance matrix of the powerers to show how the so-called multitracer technique improvesclarify the role of the overlap fraction of multiple tracers and
Our Fisher matrix analysis indicates that stringent constraints
Higherorder
with a single• survey,
assuming
five constraints
mass bins.
Figure
4. Expectedon theWhen
primordialEuclid
power spectrum from observations of 21 cm signals
minihalos toin combination
the(0.50).
CMB. Contours depict 1 errors with other cosmologicalconstraints on fNL are from
improvedσ(fNL ) =with
0.61

uations is
Universe.
ianity, thepartly bedict smalle we quan-

3

3

s

s

SKA

parameters being marginalized. We assume 21 cm signals from minihalos can be measured down tozmin = 6.

• Smaller scales

10 3 ns 10 3 ↵s 10 3 s
Planck
7.7
10.7
15.1
COrE
3.2
2.9
6.5
allows us to measure thescale-dependentbias
accurately
SKA
4.6
2.9
1.5
even at large scales, leading
tostrongconstraints
on
fNL .
FFTT
2.4
1.6
0.79
Planck+SKA
1.7
2.0
0.63
Planck+FFTT
1.3
1.3
0.44
COrE+SKA
1.2
1.6
0.39
COrE+FFTT
0.95
1.1
0.28
Table 3. Constraints on parameters for the primordial power spectrum. For 21 cm observations,zmin = 6 is assumed.

On the other hand, as can be read o↵ by Eq. (3.2), the signal-to-noise ratio in observations of the 21cm signal from minihalos rapidly increases1 at low redshifts. Therefore, theFuture wide and deep surveys with Euclid in op-

Dark energy

Accelerated expansion of the Universe at present …
Supernova Cosmology Project3
Knop et al. (2003)
Spergel et al. (2003)
Allen et al. (2002)

No Big Bang

2
Supernovae

ΩΛ

1
CMB

ex pands forev er
llyrecollapses ev entu a0Clusters

clo
fla
t

-1

se
d

en

op

0

1

2

ΩM

3

energy density(log scale)radiation

Most probable?
• Cosmological constantdark mattercosmologicalconstantèshould be resolved: why so small?
why now?

time (log scale)resolved by string landscape picture?

Susskind, Weinberg, …

not allowed by swampland conjecture ??
Obied, Ooguri, Spodyneiko, Vafa (2018)

cosmological constant; w = -1
from https://www.kitp.ucsb.edu/activities/stringvacua20

Inflation like• Quintessence (potential driven)
Lots of models …
(like an inflation zoo …)

equation of state parameter isc.f. cosmological constant; w = -1
k-essence
would be smaller than -1 .. (violation of energy condition?)

Thawing and freezing• Quintessence (potential driven)
Lots of models …

motivated by SUGRA, …

(like an inflation zoo …)
See, e.g., Caldwell and Linder (2005), …

Basically, two types of quintessence modelThawing modelFreezing model (tracker)

e.g.,

e.g.,

starts slow-rolling (thawing)around the present time

fast rolling àslow rolling (freezing)Thawing and freezing• Quintessence (potential driven)
Evolution of equation of state
Thawing model

9

Freezing model
Chiba (2010)

Chiba (2009)

n=1

n=1/4

early timepresent

for (b) the logarithmic potential, V = M 4 log φ, and forearly timepresent

Eric V. Linder
Physics Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720(Dated: October 29, 2018)
We present evidence that the simplest particle-physics scalar-field models of dynamical dark energy can be separated into distinct behaviors based on the acceleration or deceleration of the field as
it evolves down its potential towards a zero minimum. We show that these models occupy narrow
regions in the phase-plane of w and w′ , the dark energy equation-of-state and its time-derivativein units of the Hubble time. Restricting an energy scale of the dark energy microphysics limits
how closely a scalar field can resemble a cosmological constant. These results, indicating a desired
measurement resolution of order σ(w′ ) ≈ (1 + w), define firm targets for observational tests of thephysics of dark energy.

Thawing and freezing

ions and experiments at the close of they have transformed our understanding of thehe Universe. A consistent picture has emergedthat nearly three-quarters of the cosmos is
ark energy” — some sort of gravitationallyaterial responsible for the accelerated expanUniverse (for reviews see [1, 2, 3]). Proposals
k energy include Einstein’s cosmological conr a dynamical field such as quintessence. Herew scalar field dynamics separates into distinct
which, through future cosmological measurereveal the nature of the new physics accelerniverse.
s cosmological constant (Λ) is attributed tom zero-point energy of the particle physics
ith a constant energy density ρ, pressure pation-of-state w ≡ p/ρ = −1. In contrast,
e is a proposed time-varying, inhomogeneous
spatially-averaged equation-of-state w > −1
8]. The simplest physical model consists ofd, slowly rolling in a potential characterized
mely low mass. (This is similar to inflation,of accelerated expansion in the early universe,
nergy scale many orders of magnitude lower.)lar field evolving in a very shallow potential
istinguishable from a Λ, the task of elucidatysics of dark energy becomes diﬃcult if obcontinue to find that w is close to −1, e.g.
In this letter, we examine the likely behavfields and characterize them into two distinct

tivated by particle physics beyond the standard model,
this treatment allows a broad, model-independent assessment of a quintessence scalar field slowly relaxing in a
potential.
The physics is straightforward: the field φ will seek toroll towards the minimum of its potential V , according
to the Klein-Gordon equation φ̈ + 3H φ̇ = −dV /dφ. The
rate of evolution is driven by the slope of the potential
and damped by the cosmic expansion through the Hubble• Quintessence (potential driven)
Evolution of equation of state

Freezing modelTime variation of w

Thawing model′

To discriminate these models,
It should be important to observe
not only equation of state but also
its time dependence !

Caldwell and Linder (2005)′

Modified gravity• Scalar tensor theory
è non-minimally coupled scalar field
could also include the derivative coupling
In general, we can consider the Lagrangianwhich has infinite terms with including higher order derivatives …

Infinite possible theories ?
Is there any guiding principles?
• from the first principle (string theory, or …?) (top-down)• Based on some philosophy (respect some symmetry, stability condition, …)
è free from ghost instabilities !!

Lagrangian
aequ
scag
most
generalscalar-tensor
theoryhavingsecond-order
field
equa
The
generalized
Galileon(5)[19]describes
is fielda mostfurther18] retaining
second-order
field
equations.
precisely,
firstTheone
determinesthemostgeneralscalar-tensor
theoryhavingsecond-orderthesecondderivativesof the field.More
One second-ordercan introducefield
gravityand conside18] retaining
equations.
Mo
general scalar-field
theoryonafixed
Minkowski
backgroundwhich
yieldsasecond-order
The
Lagrangian(5)describesascalar-field
theoryonafixed
Minkowski
background
However,
Horndeski
determinedthetheorystarting
from
However, Horndeski determinedthetheoryfromgµν and∂µtheoryto ∇starting. However,since tcoµongeneralscalar-fielda
fixedMinkowsOne
canintroduce
gravityandconsideracovariantversionof(5)
bypromotingηtofield equation, assuming that the Lagrangian contains atcovariantization
most second derivatives
of φ
µν
leads
to
higherderiv
field
equation,assuming
that
the Lagrangian
than
those
made
for
derivingthegeneralized
Galileon,
and
than
those
made
for
derivingthegeneralized
Galileon,
and
g
and
∂
to
∇
.
However,sincecovariantderivatives
donotcommute,
the
naive
µν
µ
and is polynomialin ∂µµ ∂ν φ. One
then promotes the theorydangerous.
to a covariant
one in thesameFor
example,
one
would
ha
and
is
polynomial
in
∂
∂
φ.
One
then
promoteµν
covariantizationleads
to higher
derivatives
interms
the field
equations,
which
would beLagrangian
looks
very
diﬀerent
from(8):
way as above
by adding [20]appropriate(unique)counter
so
that
the
field
equations
Lagrangian
[20]
looks
very
diﬀerentfrom(8):
as aboveappropriateµ R , (unique)ν
dangerous. For example, one would have way
derivatives
of bytheadding
Ricci ctensor
X∇
[∇
∇
∇
φ−∇
µν
4
µ
ν
are of second order both for φ#and# the metric. The
thus
are ofgeneralized
second orderGalileon
both for can
φ and
thebe
metr
see,
e.g.,
Kobayashi,1901.07183(review
paper)
2∇canµν
ν 2 be done
νobtained. It should be noted
that
this
procedure
inarbitraryspacetimeµtheµ µthisνσµ
νσ
αβγ
obtained.
Itshould
be
noted
that
proce
µνscalar-field
νµσ(R
σµν ∇
νσ
µ
νσ
αβγ
c
X∇
[∇
∇
∇
φ
−
∇
∇
φ]
=
−c
X∇
φ)
,
(6)
in
equationofmotion
4
µ
ν
ν
µ
4
κ
φ
φ
φ
+
κ
φ
φ
R
+
L
=
δ
κ
φ
R
+
κ
φ
φ
φ
+
κ
φ
φ
R
Ltheory
=
δ
κ
φ
R
+
•
Horndeski
1X
3α1
1Xα In3α
1 αβγ βγ dimensions.
β
γ
βγ
α
µνσ
α
β
γ
βγ
µνσ
fourcurvature-dependent
dimensions,
thegiven
Lagrang
dimensions. In four dimensions, the Lagrangian for
Galileon isaddingterms
3 3the bygeneralized
in the scalar-field equationof motion. Suchhigherderivativeterms
can be canceledby [19]
!
!
of
(5)
that
leads
to
second-orderfield"eby [19] Lagrangian;
µν µν
αβ αβ
µ µν ν
µ µν ν
by adding curvature-dependent
terms
appropriately
to2F
covariant
+ δ+
(F (F
+ 2W
)R
2F
φ
+
2κ
+
2W
)R
φφ
φThe
+
2κ
φφβφβ
given
byα(5).
[17]X
α version
XEq.
8 φ8αφφ
!
"
α
β
µνδµν
β
αβ αβ +is+
L = G2 (φ,− G3 (φ, X)✷φ + G4 (L=X)leads− Gto3 (φ,X)✷φ + GX)R + G(✷φ)−X)φµν
φµν and the metricof G(5)2 (φ,thatsecond-order
field
both
the2 scalarfield
4 (φ,equations
4X for
! +
Gc5X
µν
L
=
c
φ
+
c
X
−
X✷φ
+ !2W
− Xκ
✷φ
+
κ
.
− µν
6−(F6φ(F+G
2W
+
κ
.
1
2
3
is given by [17]
φ 5X
φ3Xκ
8)
9
"
φ−
8 ) ✷φ9(✷φ+ G5 (φ, X)G φµν −(✷φ) − 3✷φφµν φµν + 2φµν φνλ φµλ , 2 µν 6 (8)c5+ G5 (φ, X)G φµν −"# G φµν − X
c4 2
6+ c5 X2µναnR] +G2c,4 X
G3 ,(✷φ)
G4 , andare arbitrary func
c1 φn + c2 X[α−1c[α− φG5φµν
αnwhere
]X
3
α...α
α2=
3 X✷φ
α1 α
α12δ α2+...δ
1L
n...α
2
Here,
δ
:=
n!δ
is
the
generalized
Kronecker
2
Here,
δ
:=
n!δ
δ
...δ
is
the
generalized
Kroneckerµµ
ntheµ1 1µfunctions
where G2 , G3 , G4µ, 1andG2n5...µ
are
of
φ and
X. fHereand hereafter
wen arbitrary
1µµ2µ...µnotationandf
2c5µ2 "µnuse
X := ∂f /∂Xµ #φ := ∂f /3µν R is the Ricciνλ tensor, Gµν is t
2 µν
where
X
(✷φ)
−
3✷φφ
φ
+
2φ
φ
φλ(8)
, is now kno
(7)
+
c
X
G
φ
−
µν
µν
5
µν
use the notation
f
:=
∂f
/∂X
and
f
:=
∂f
/∂φ
for
a
function
f
of
φ
and
The
generalized
Galileon
arefunctions
arbitraryfunctions
φ and
and
X.
We
have
anotherfu
X arbitrary
φ functions
µν X.
κ9 κ
are
arbitrary
φof andX.
We
have
anotherfun
3of of9 areand now X := −g φµ φν /2. Here, thmostgeneralscalar-tensor
theoryhavingsecont
The generalized Galileon (8) is now known as
theHorndeski
theory[20],the
i.e.,
the
in
the
secondline
are“counterthis
mustsatisfy
F
=
2
(κ
+
2Xκ
−
κ
)
and
hence
isn
where
R
is
the
Ricci
tensor,G
isthe
Einsteintensor,
φ
:=
∇
φ,
φ
:=
∇
µν (κ3 +
µ )
µ
µν
µ∇
νφ
X 2
3However,
3X
1φ
this
must
satisfy
F
=
2Xκ
−
κ
and
hence
is
X
3X
1φ
Horndeski
determinedthetheoryThe most
scalar-tensor
theorymost generalandscalar-tensor
theoryhavingsecond-order field
four dimensions.
µνgeneralinequations
the field inequations.
The counter
now X := −gφµ φ
ν /2. Here, the fourth term in the first line and the first term
than
those
made
for
derivingthegeneralize
have
a determinedfunction
of
φ,
W
=
W
(φ),
which
can
beSinceabsorbedafunction
of
φ,
W
=
W
(φ),
which
can
be
absorbedincovariant
Galileon.
thefield equhavingsecond-order
field
equationsin
4D
However,haveHorndeski
thetheorystarting
fromthediﬀerent
assumptionsin the second line are the “counter terms” introduced to remove higher derivatives
Lagrangian[20]
looks very
diﬀerent
from (8):firstderivatives
of
φ,
the
Galileanshithan those
madeforderivingthegeneralized
Galileon,andtheoriginal
form
the
F
+
2W
→
F
.
Thus,
we
have
the
same
number
of
free
free
from
ghost
instabilities
associatedwiththehigherderivative
terms
in
the
field
equations.
The
counterterms
areunique.
ThistheoryiscalledtheoldnewFold + 2W → Fnew . Thus, we have thetheory.
same# number of free fu
Onlythe
secondof t
2 propertyµinvolveν σµLagrangian
αβγ thecould
haveνσextra d.o.f.
Lagrangian [20]
looks
very
diﬀerentfrom(8):
covariant
Galileon.
Since thefield
equations derivedfrom(7)
κ1X φinα φthe+Lfield= δµνσκ1 φα Risβγ+equivalenc
thegeneralized
Galileon
theory.
Nevertheless,theβ φγcoequations,maintained
thefirst
generalized
Galileon
theory.
Nevertheless,theequivalence
3$#
derivativesof φ, the Galilean shift symmetry is broken
in the covariant
Galileon!
2
theory (7)four spacetimµν
αβ is formulatedµµ ν)Rσinnatureis++δthe(Fφ+φ2W
propertyi.e.,µν second-order
αφ
φµ φνtrivial.
φσof+theκ φGalileon,
φµ R νσ2κ
φ φ αβ + 2FofX φtheL = theory.
δ αβγapparently
κ Onlyφµ Rtheνσsecond+farκfrom

Modified gravity

18] retaining second-order field equations. More precisely, first one determines the mostgeneral scalar-field theory on a fixed Minkowski background which yields a second-orderfield equation, assuming that the Lagrangian contains at most second derivatives of φand is polynomial in ∂µ ∂ν φ. One then promotes the theory to a covariant one in the same
way as above by adding appropriate (unique) counter terms so that the field equations
are of second order both for φ and the metric. The generalized Galileon can thus besee, e.g., Kobayashi, 1901.07183 (review paper)
obtained. It should be noted that this procedure can be done in arbitrary spacetime• Horndeski theory
dimensions. In four dimensions, the Lagrangian for the generalized Galileon is givenby [19] Lagrangian;!
"2µν
L = G2 (φ, X) − G3 (φ, X)✷φ + G4 (φ, X)R + G4X (✷φ) − φ φµν
"
G5X !
3
µν
νλ µ
µν
(✷φ) − 3✷φφ φµν + 2φµν φ φλ ,(8)
+ G5 (φ, X)G φµν −6where G2 , G3 , G4 , and G5 are arbitrary functions of φ and X. Here and hereafter we
Due to thefexistenceof non-minimal coupling between scalar d. o. f. and gravity,
use the notation
X := ∂f /∂X and fφ := ∂f /∂φ for a function f of φ and X.
The generalized Galileon (8) is now known as the Horndeski theory [20], i.e., themost generalscalar-tensor
theoryhaving second-orderfield equations
in four dimensions.
in thistheory, thegravitationallaw would
be changed!
However, Horndeski determined the theory starting from the diﬀerent assumptions
than those made for deriving the generalized Galileon, and the original form of theThat is, we can test not only by cosmological observations
Lagrangian [20]
from
(8):
butlooks
also byvery
local diﬀerent
gravity test,
GW experiments,and more..
$#2
L = δ αβγ κ φµ R νσ + κ φµ φν φσ + κ φ φµ R νσ + 2κ φ φµ φν φσModified gravity

Horndeski [150] was obtained by performing a disformal transformation [151example of degenerate higher-order scalar-tensor (DHOST) theories beyondgµν → g̃µν = C(φ, X)gµν + D(φ, X)φµφν .
[150] was obtained by performing a disformal transformation[151]

Modified gravity

This is a generalization of the familiar conformal transformation, g̃µν = C(φgµν → g̃µν = C(φ, X)gµν+ D(φ,transformationX)φµφν .
(108)
disformal(108) is invertible
if

generalization of the familiar conformal transformation, g̃µν =C(CC(φ)gµν .X The− XC
+ 2X 2 DX ) ̸= 0.
see, e.g., Kobayashi, 1901.07183 (review paper)transformation (108) is invertible if• Beyond ?

Since the disformal transformation contains derivatives of φ, the theory tr
2 Horndeski has higher-order field equations. Nevertheless, it is a degenerfromC(C − XCX + 2X
DX ) ̸
= 0.
(109)with (2 + 1) degrees of freedom because an invertible field redefinition does n
Horndeskicontains
theory:The
mostgeneralscalar-tensor
theorythenumberofof
freedom [152,153, 154]. This example i
disformal transformation
derivativesof physicalφ,the degrees
theorytransformed
existenceof asecond-orderwideritclasshealthyscalar-tensor
theories
than the Horndesk
havingfield
equationsin 4D
ndeski has higher-order field equations.
Nevertheless,is a ofdegeneratetheoryDegeneratehigher-orderscalar-tensor
theories have been constru
1) degrees of freedom because an invertible
field redefinitiondoes
notchangelittle bit instrong?
investigated systematically[147, 155, 156, 157, 158, 159]. Let us follow
er of physical degrees of freedom [152, 153, 154]. This example implies theand consider the extension of Horndeski’s G4 Lagrangian given byof a wider class of healthy scalar-tensor theories than the
Horndeskiclass.
Healthyextension
5!
nerate higher-order scalar-tensor theories have been constructed and
L = f (φ, X)R +AI (φ, X)LI ,ed systematically in [147, 155, 156, 157, 158, 159]. Let us follow Ref. [147]Degenerate Higher-Order Scalar-Tensor theories (DHOSTI=1
theories)der the extension of Horndeski’s G4 Lagrangian given bywhere

L = f (φ, X)R +

5
!
I=1

AI (φ, X)LI ,see, e.g., Langlois, 1811.06271 (review paper)

L1 = φµν φµν ,L2 =(110)(✷φ)2 ,

L4 = φµ φµα φαν φν ,L3 = ✷φφµ φν φµν

L5 = (φµ φν φµν )2 .

These five
constituents
exhaust allextra
the possible
quadratic
terms in second dWith so-called ``degeneracy
conditions”,
pathologicald. o. f. doesn’t
appear.
µνofφ, and the Horndeski
theory is the special case with A2 = −A1 =2µ νWhat is DE?
• Points for observations
ü Cosmological constantü Quintessence models – thawing type-- freezing type
ü Scalar-tensor theories – Horndeski, DHOST theories,…

measurement the gravitational law

Equation of statel Expansion of the Universe
In principle,
if we could measure the long time history of the expansion rate of the Universe,
we can also get the information about the evolution of the ``equation of state”.

Perturbative
parameterization

ç CMB, LSS and SNe

Planck (2018)

Cosmological gravitational law
l Evolution of matter inhomogeneities in the Universe
In cosmology, we treat the spatial inhomogeneities of matter distributions,(including galaxy distributions on large scales), as perturbations
on the background homogeneous and isotropic Universe (FLRW Universe).

Such a matter inhomogeneity evolves
through the gravitational interaction!
valuable information aboutthe ``cosmological” gravitational law!!
Univ. of Chicago

Linear growthsee, e.g., Kobayashi, 1901.07183 (review paper)l Evolution of matter inhomogeneities in the Universe
Measurethe ``linear” growth of matter (DM) density contrast (inohomogeneities),
to find the cosmological gravitational law
and test the scalar-tensor theories !
usually, parameterized as

In GR,ora is a scale factor (time coordinate)

0.4

−1

0.2

−1.2

0
0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

−1.4
0

0.1

Ωm

Linear growth14), Vol. 00, No. 0

0.1

21

0.2

0.3

0.4

0.5

0.6

Ωm

Figure 20. Left : Cosmological constraints in the ⌦⇤ vs ⌦m plane. Right:
Cosmological constraints in the w vs ⌦m plane. The inner and outer contours show the 68 and 95% confidence-level two-dimensional marginalised
constraints. All contours are showed assuming a flat ⇤CDM-model. The
blue contour represents the cosmological constraints using BOSS DR12
galaxies, the red contour shows the gain when adding the eBOSS quasar
sample and the green contour also includes the results from Ly-↵ measurements. All results are consistent with a ⇤CDM Universe.

Figure 18. Parameter contours for f 8 , DA and H for the predictions by
the 5 companion papers using the same DR14Q dataset for traditional RSD
analyses. Blue contours show the results presented in this work in configuration space, and red contours show the predictions by Hou et al. (2018) in
configuration space too using a second RSD modeling. The Fourier Space
based analyses are shown in green contours for the results by Gil-Marin
et al. (2018) using a third RSD modeling, in magenta contours for the results by Ruggeri et al. (2018) and in orange contours for Zhao et al. (2018),
both using redshift weighting techniques but with a different model.

arXiv:1809.09148v2 [astro-ph.CO] 27 Jan 2019
Cosmology frowith Subaru H

Chiaki H IKAGE1 , Masa
Rachel M ANDELBAUM
Hironao M IYATAKE7,8,9
Robert A RMSTRONG10Paul H O13 , Bau-Ching
Alexie L EAUTHAUD15 ,Sogo M INEO4 , Shoken
Hitoshi M URAYAMA1,17
Melanie S IMET19,9 , Jos
Michael A. S TRAUSS11
Yousuke U TSUMI23 , Sh1
Kavli Institute for the Physof Tokyo, Chiba 277-8582,
2
Research Center for the E3
Department of Physics, Un
4
National Astronomical Obs
5
The Inter-University CentePune, 411007, India
6
McWilliams Center for Cos
Pittsburgh, PA 15213, USA
7
Institute for Advanced Res
8
Division of Particle and As
University, Nagoya, Aichi 4
9Jet
Propulsion Laboratory,
10
Lawrence Livermore Nati
11
Department of Astrophys
08544, USA
12
Department of Astronomy
13
Academia Sinica InstituteTaiwan14
SOKENDAI (The Gradua
Japan
15
Department of Astronomy
St., Santa Cruz, CA 9506
16
Hiroshima University, Hig
17
Department of Physics an
CA 94720, USA
18
Theoretical Physics Grou
Berkeley, CA 94720, USA
University of California, R

19eft) and in the ⌦m -S8 (↵ = 0.45) plane (right), where S8 (↵) ⌘ 8 (⌦m /0.3)↵ , in the fiducial
n. For comparison, we plot cosmic shear results from KiDS-450 with correlation function (CF)mators (QE) (Köhlinger et al. 2017) and DES Y1 (Troxel et al. 2018b) with the same set ofas well as WMAP9 (Hinshaw et al. 2013) (yellow) and Planck 2015 CMB constraints without CMB

Figure 21. Measurements of f 8 (z) with redshift compared to the predicZarrouk et al. (2018)tion from the flat ⇤-CDM+GR model with Planck parameters. The f 8 (z)result presented in this work for the quasar sample is represented by the *marker and is obtained using 3-multipole fit. The error bar represents the total systematic error that includes the statistical precision and the systematicWhat is DE?
• Points for observations
ü Cosmological constant

SKA ??

ü Quintessence models – thawing type-- freezing type
ü Scalar-tensor theories – Horndeski, DHOST theories,…

measurement the gravitational law

(0)

+ 2cM

3 1 + w(0) + 2 (cMcT ) + 1

2 cM

3w(0)

cB

cH

cM

rms in Eq. (61) are the generalization of the previous results derived in the ca
the third term appears when at least either of cH and is nonvanishing, namely
Horndeski. Equation (61) is general in the sense that we have not yet imposed ↵T0), as discussed around Eq. (56), can be written in terms of (p, q, cH , ) as
⇢h
i
3
2
=
(p + 4q) ( 3 + 6p + 10q) 8pq
Bertolami, Gomes (2018)2( 3 + 6p + 10q)(3p + 11q)i
1h3q(1 + 2q)( 3 + 6p + 16q)(cH ++ ( 3 + 6p + 10q) + 8q (3p + 2q) cH +22pq + 3qcH + (3p + 5q)

SKA as a galaxy survey / HI intensity mapping

B.

Observational constraints

Santos et al. (2015)

Figure 1. Predicted SKA constraints at various stages on the equation of state parameter of dark energy (with BOSS and
Planck priors).3,instance,
webyinvestigate
constraints
onspectrum,
DHOST
based on current observational lmeasuring the BAO signaturein the radio of the galaxy power
SKA will allowtheories
for precise
measurements of thePlanck
equation of (2018)state of dark energy, w = p/⇢, of the form:
dex . For instance,measurementsfrom the BOSS DR12 give the lim
w = w clustering+ w (1 a) ,(1)
BOSS DR14(2018) in Ref. [50] (basewhere p on
and ⇢ arethe pressureand energy density
energy, w and w
are some real parameterssedtheanalysis
inof dark
Fourier
space)
and and = 0.609± 0.079
a is the scale factor. In Fig. 1 one can see that assuming a Gaussian distribution around the fiducial model
w = 1, which corresponds to the cosmological constant as dark energy, the predicted SKA results combinedace).
constraints
from
DR14
are
given as = 0.55 ± 0.19 in Ref. [51] awith Planck The
and Baryon
Oscillation Spectroscopic Survey(BOSS)BOSS
priors constrain
considerably theallowedregion of parameters. The expected combined precision is compared with the one from Euclid mission.
addinganalysis).
thespectrum
typical value of the deviation from th
Furthermore, bytomographic
studying weak gravitational lensing
the measurement of theSincedark matter power
will be possible, and bounds on the mass and the number of neutrinos families can be obtained.
bservations
as shown above can be roughly estimated as . O(0.1), let us emplo
2.2 Testing gravity to the limit
0aFigure 2. SKA predictions for the constraints on the growth of structures index (including BOSS and Planck priors).30aand spacial scalar perturbations of the metric, respectively:

4= µ̃(a, k)

5

,

(3)

6

and the third one which is the growth rate, f (a, k), (or its index ):✓ 2◆
a 8⇡GN ⇢f (a, k) =
,
3H 2

(4)

ds2 = ( 1 + 2 ) dt2 + (1 + 2 ) ij dxi dxj ,one finds a correction to the Newtonian potential

✓ ◆n
d
GN M
r
=
1 + 2↵2,drr
rv

Implication for ST theories

where the dimensionlessconstant↵ parameterises
theSY
coupling
Hirano,Kobayashi,
Yamauchi,(2019) of the sc
n > 0 is model dependent. A solar mass object has rv ⇠ O(0.1 kpc) [rection to GR is strongly suppressed in the solar system. In the case of
Vainshtein screening is fully e↵ective [21–23]. For beyond Horndeski theor
Modificationof gravity
works outside extended bodies
but breaks
down inside matter [24]. The e
astrophysical
Newtonian perturbations in
were
found to bebodies;of the form [24–27]
d
GN M (r) ⌥1 GN d2 M (r)
=
+
dr
r2
4
dr2
d
GN M (r) 5⌥2 GN dM (r)
=
,
dr
r2
4r
dr

Rr
where M (r) ⌘ 4⇡ 0 s2 ⇢(s)ds, and the parameters ⌥1 and ⌥2 are non v
theory contains beyond Horndeski terms in its Lagrangian.
This opens up the possibility of testing beyond Horndeski theories
objects such as stars [25, 26, 28–30] and galaxy clusters [27]. Currently
the range 0.22 < ⌥1 < 0.027 where the lower bound comes from the Cof white dwarf stars [30] and the upper bound comes from consistency offor hydrogen burning with the lowest mass hydrogen burning star [28, 29]
we note that prior to the white dwarf constraint, Ref. [26] was able to p
⌥1 > 2/3 by requiring a sensible stellar profile (with a mass density thatradius). The best constraint on ⌥2 = 0.22+1.22
1.19 comes from the agreem
and hydrostatic mass of galaxy clusters [27].
Constraining these parameters is important because they are dire11

Growth in higher order ?
0.15

11

Yamauchi, Tashiro, SY (2017)

expected constraint from0.1galaxy bispectrum
0.05
ξ

0
0.1

-0.05
0.05
ξ

Second order growth

0.15

-0.1
0

-0.15
-0.05
0.4

0.45

0.5

-0.1

0.55
γ

0.6

0.65

0.7

-0.15
0.4

0.45

0.5

0.55
γ

0.6

0.65

0.7

Linear growth

2: Forecast 1σ and 2σ marginalized contours in (γ,assumingξ) plane for SKA1MID(blue), SKA2(red) and Euclid(green), marginalover the equation-of-state parameter and bias parameters. For comparison, we also plot the kinetic gravity braiding model
le boxes), and the large ξ model (orange triangle).

G. 2: Forecast 1σ and 2σ marginalized contours in (γ, ξ) plane for SKA1MID(blue), SKA2(red) and Euclid(green), marginalng over the equation-of-state parameter and bias parameters. For comparison, we also plot the kinetic gravity braiding modelWhat is DE?

SKA??
• Points for observations
ü Cosmological constantü Quintessence models – thawing type-- freezing type
ü Scalar-tensor theories – Horndeski, DHOST theories,…

Dark matterDark matter• Existence of dark matterü Lots of astro/cosmological observations(rotation curve, LSS, CMB)

Basically, gravitational interaction©PlanckUnknown ``weakly” interacting particles?
SUSY particles (WIMP LSP)
QCD axion, sterile neutrino,
Axion Like Particles (string theory) …

from SDSSunknown particles?
• lots of particle experiments

axion - photon coupling©Majorovits (MADMAX)©Snowmass project (2013)

è No evidence …
è Beyond standard ``WIMP”?

Sterile neutrino©IceCube collabo.

BHs as DM• How about other candidates?
Basically, only gravitational interaction

Black holesHowever,

Success of standard big bang model
It is completely difficult to explain the dark matterby standard astrophysicalblackeraholes (steller BHs) …
Star formation

How aboutDark
if BHmattercould
be formed in the early Universe?
dominated era
è Primordial black hole!!

©ESAoCúċ?jJ`Ui(&Ǎ

ǐêġ

#



Ćġǡď

>4,<

ƀĶŎŦ*8,+Ľ1*ǐêġ'-ƴǏė×,ÔBasics of PBH

@?('8(8(,ġ,ƴǏC#

Mgto+*?('4Ǎaround 1015 g would have interesting observational consequences, especially for the EGB limits. However, this
discussion is rather technical, so it is relegated to an
Appendix.

4ƀ-ġ,ǌ'=@)+<#&ÝÞŵǔ+.=4@?,'ÝÞ,Ǉ¨((8+ǍII. DEFINITIONS AND DESCRIPTION OFBACKGROUND COSMOLOGY

Zeldovich and Novikov (1967)
In this section, Hawking
we present(1971)
some relevant definitions andǐêġ' ?(Ƌ=@Ä,Ǎ£Ņ-ÝÞĨ,ġúċ+Ǎ?(Ƌ& ?ŭŴƌ8 4
background equations.
Weassume(1974),
that the• Primordial Black Hole (PBH)
Carr andHawking… standardÏ&(Ƌ=@4$4>Ħǁ'(+*>uĝ'ÝÞ,Ĩ+'

ġ-Ǎ

ġ-8(8(ǍƀC2(D)½!CDM model applies, with the age of the Universe being
ü BHs formed in the early Universe
inflation)
t0 ¼ 13:7(afterGyr, theHubble parameter being h ¼ 0:72 andÂ­ÙcmYMgtoR_nI ,ƨ+ű>4à-¡
Ĩ+' ġ -ÝÞĨ(Ƥ#&8ƪŝ=ĚõƄ# Ý
the time of photon decoupling being tdec ¼ 380 kyr(Ƌ=@& 4 ­ÙcmYMgto -ÝÞƪü direct gravitational collapse[165,166].
of a overdense(horizonThroughoutregion
the paper
we put scale)c ¼ @ ¼ kB ¼"+úċ@ cmYMgto'
1. The Friedmann equation in the radiation era isƑø& ?(
(-)'ƍ+ ( ?(Ą

Mgto

&

?(

ü mass of formed BH ~ Hubble horizon mass at the formation

(-ĢǔCǋ@.ÝÞ-Ĩ+¼#&)D4<#&ƪŝýĚŰ,ÝÞ-ǤŊǤâöŘĊ+>ġśÉ,ÝÞ'*Ĵǅ-ÛÉ

nt@.^q^q,mass range. The code em# (Ƌ=*ĢĨ+s< similar
to the one used byimǨ >
dying
the effects of decaying+Ľ1&ŗ+ő,ǞÌǍ£'£ÝÞ,Ƒø

s+Ǎ£³ƊǧǍ
follows. Section II describes
4
nd
defines various quantities
cmYMgteviewsmYMgto black hole evaporation
,< +¡ǃonemission. Section IV theniving from cosmological nu-

8"G
4"3 G
2
especiallyfor the PBH
in theera
Hformed# radiation-dominated¼D 81, 104019
g$ T 4 ; (2010)¼(2.1)
PHYSICALREVIEW
3

45

! "
4"
t
&
3
5
where
g
countsthenumberofrelativisticof$
M ¼ $MPH ¼$#H % 2:03 ' 10 $M(degrees:
freedom.
3 This can be integrated to give1 s"& 2 (2.3)"& 1=2 !
!
gT
t % 0:738

$

s;
(2.2)
1
MeV
10:75
Here $ is a numerical factor which depends on the detailsof gravitational collapse. A simple
calculation
ﬃﬃﬃ 3 analyticalto their values at the start ofwhere g$ and T arepnormalized
suggests that ittheis aroundð1= Since
3Þ %we0:2
the radia-PBHs
BBN epoch.
areduring
only consideringtion era [4], although
the first
hydrodynamical
calculations
which form
duringthe radiation era (theones generated
beforeinflation
being
dilutednegligible
density),gave a somewhat
smallervalue[5].
The tofavoredvalue
has the``Conservative” constraints

HSC

Dark matter window !!

Niikura et al.; 1710.02151v3

10 BH-BH merger events and 1 NS-NS

Existence of about 30 M_sun BH !!!
©LIGO/VIRGO collaboration

OGLE-IV (data) results-exoplanet search OGLE (Optical Gravitational Lensing Experiment) IV;
Udalski, Szymanski, Szymanski, 1504.05966

5-years monitoring observations of stars in the Galactic bulge
26Vol. 65

Fig. 13. OGLE-IV sky coverage in the Galactic coordinates. Credit: Jan Skowron.

A. A.
29

Fig. 15. OGLE-IV Galactic center fields. Cadence of observations of the inner Galactic bulge microlensing fields is color-coded. On average: red fields – one
observation every 19 minutes throughout the visibility of the field ( z < 62 deg), yellow fields – one observation per hour, green fields – 2–3 observations per night,
dark blue fields – one observation per night, cyan fields – one observation per two nights. Silver color marks additional fields which were regularly observed in theyears 2010–2013. Transparent fields in the outer Galactic bulge and beige fields of the Galactic disk are monitored during the OGLE Galaxy Variability Survey(see also Fig. 21). Credit: Jan Skowron, background photograph: Krzysztof Ulaczyk.

Earth-mass PBHs???
1

Number of events per bin

100

0

log tE
1

2

3

100

1,000

Best-fitting model
95% conf. limit
Sumi et al. (2011)
10 Earths per star
5 Earths per star

10

1
0.1

1

10
tE (days)

6 microlensing events with short-time scale
Figure 1

Mroz et al. arXiv:1707.07634

Observed
distributionof timescalesof 2,617 high-quality microlensingè detectionof earth
mass PBHs?

events discovered by OGLE in 2010–2015. The purple line is the best-fitting model. The
dotted line constrains the 95% confidence limit on the number of wide-orbit or unboundEarth mass BHs, LIGO BHs and DM

Possible detection
by LIGO/VIRGO ?
(or upper limit)
Detected by OGLE?
PTA

Dark matter window !!

Sasaki, Suyama, Tanaka, SY (2016),
Niikura, Takada, SY, Sumi, Masaki (2019)

Can we probe PBH-DM by SKA?
• SKA as a probe of high redshift Universe
Ø Tashiro and Sugiyama (2012), and more.. (e.g., Poulin et al. (2017), …)
X-ray photons emitted by accretion of matter onto PBHs

The eﬀect of primordial black holes on 21cm fluctuations

1.0

1.0

z = 30

0.8

0.6

xH

xH

0.8

10 Mì
10 2 Mì
10 3 Mì

0.4

10 7 Mì
0.1

1

comoving radius [Mpc]

M = 10 3 Mìz = 50z = 40z = 30z = 200.2

10 6 Mì

0.01

0.6

0.2

10 4 Mì
10 5 M

0.2

0.0
0.001

3

10

0.0
0.001

0.01

0.1

1

10

comoving radius [Mpc]

Figure 1. The neutral fraction of hydrogen as a function of comoving distance from the source. The left panel shows the dependence
on mass at z = 30. The lines represents the neutral fraction with masses ranging from 10 M⊙ to 107 M⊙ from left to right. The rightpanel shows the dependence on the redshift for a PBH with M = 103 M⊙ . The dotted-dashed, dotted, solid and dashed lines represent
the neutral fraction at z = 50, z = 40, z = 30 and z = 20, respectively.

change the optical depthCan we probe PBH-DM by SKA?
• SKA as a probe of high redshift Universe
Ø Gong and Kitajima (2017)
Poisson like isocurvature fluctuations of PBHs

106
104
102
0
10
10-2
10-4
10-6
10-8
10-10
10-3 10-2 10-1 100 101 102
k [hMpc 1](a) MPBH = M

k3P(k)/2⇡2

k3P(k)/2⇡2

Matter power spectrum

103

104

106
104
102
0
10
10-2
10-4
10-6
10-8
10-10
10-3 10-2 10-1 100 101 102
k [hMpc 1](b) MPBH = 100M

103

104

Can we probe PBH-DM by SKA?
• SKA as a probe of high redshift Universe
From EDGES result …

depending on
the astrophysical parameters(accretion rate, …)..
Hektor et al. (2018)

Can we probe PBH-DM by SKA?
• SKA as a PTA

Pulsar Timing Array is known
to be an GW detection experiment.
SKA can find lots of radio Pulsars!

Barak, et al. (2018)

Can we probe PBH-DM by SKA?
See, e.g. Saito and J. Yokoyama (2009)

Density fluctuations with large amplitude è collapse è PBH formation

Based on the cosmological perturbation up to the second order,

Density fluctuations would be source of the tensor modes, that is, gravitational waves!!
frequency (wave number) of induced GWs  horizon scale at the reenter
mass of PBH  horizon mass at the reenter

Can we probe PBH-DM by SKA?
Primordial scalar power• SKA as a PTA

10-2

ζ

10-4
10-6
10-8
10-10 4
10

106

108
1010
k [Mpc-1 ]

1012

1014

FIG. 2. µ-dist. ⇣ ! R. k is artificially normalized by assuming an appropriate cosmic evolution after
phase-4.

SKA boundthe invariant norm of the potential tilt, and min(rI rJ V ) is the minimum eigenvalue of the
Hessian rI rJ V in an orthonormal frame.

In terms of inflation, this conjecture claims that any single continuous inflationary phasecannot last so long. That is, if it is true, one needs multiple phases of inflation to explain
sufficient e-folds ⇠ 50–60 for our observable universe in total, as we are proposing. In fact
our model always satisfies the second condition during any inflation phase as

Tada, SY in prep.

induced GWB

min(rI rJ V )' i > 1,V

for all phase-i,(16)while the first condition is satisfied apart from the inflationary trajectory. The existenceof large negative eigenvalue in the Hessian matrix generally causes a negligibly short (lessthan 1 e-fold) inflationary phase unless the initial value of the first slow-roll parameter✏H is significantly small (see Eq. (11)) and then the corresponding amplitude of the powerbyproduct• SKA as a PTA
Primordial magnetic field x primordial magnetic field è primoprrdial GWs

6

104
103

B [nG]

BBN

Magnetic
reheating

102
101

LIGO
LISA(forecast)

CMB dist.

Current PTA
SKA PTA(forecast)

0

10

100

105

1010
kp [Mpc-1]

1015

1020

Saga, Tashiro, SY (2018)

FIG. 3: Upper bounds on the amplitude of PMFs obtained from direct detection measurements of GWB; current PTAs (blacksummary0103

University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japanhe Physics and Mathematics of the Universe,
Tokyo, Kashiwa, Chiba, 277-8568, -5Japan

Observational inflation cosmology

-5
0
0
5
local-type primordial non-Gaussianity parameterfNL 5with a-5
10 ∆n
10 ∆α
a continuum galaxy survey by Square Kilometre Array (SKA),
a general expression for the covariance matrix of the powerers to show how the so-called multitracer technique improvesclarify the role of the overlap fraction of multiple tracers and
Our Fisher matrix analysis indicates that stringent constraints
Higherorder
ions
Gravitational
Waveswith a bysingle•Primordialsurvey,assuming
fivemass bins.
Figure
4. Expected
constraints
on theWhen
primordialEuclid
power
spectrum from observations of 21 cm signals
minihalos toin combination
the(0.50).
CMB. Contours depict 1 errors with other cosmologicalconstraints on fNL are from
improvedσ(fNL ) =with
0.61

Kamionkowski

3

3

s

s

SKA

1,2

parameters being marginalized. We assume 21 cm signals from minihalos can be measured down to1zmin = 6.

, and Fabian Schmidt• Smallerscaleslogy, Mail Code350-17,
Pasadena, CA 91125
10 3 andns 10 3 ↵s 10 3 s
Planck7.7
10.7 MD 15.1
s Hopkins University, 3400 N. Charles
St., Baltimore,
21210
COrE
3.2
2.9
6.5
allowsustomeasurethescale-dependentbias
accurately
ed: September 14, 2018)
SKA
4.6
2.9
1.5
even at large scales, leading
tostrongconstraints
on
fNL .
FFTT
2.4
1.6
0.79
Planck+SKA
1.7
suations
to theisintensity pattern of 21-cm
radiation fromthe2.0dark 0.63
Planck+FFTT
1.3
1.3
0.44
into
curl
and
curl-freecomponents.
Lensingbyprimordial
COrE+SKA
1.2
1.6
0.39
Universe.
COrE+FFTT
ponent,while the contribution fromlensing by0.95density1.1fluc- 0.28
ianity, therpartly
fluctuationsin the 21-cm background extend to very smallbe- And more;
Table 3. Constraints on parameters for the primordial power spectrum. For 21 cm observations,i↵erent
probe
di↵erentin è
redshiftspace.
dict smallfrequencies
Ø 21cm global
signal(small scaleshells
structuregas temp.
evo., …) Yoshiura, Takahashi^2 (2018)
zmin =
6 is assumed.
Ø 21cmlensing (detecting
Book, Kamionkowski,
Schmidtenwe
quanwith
which
to reconstruct
the PGWs)
curl componentof thelens-(2011)9 as can be read o↵ by Eq. (3.2), the signal-to-noise ratio in obserothers conceivably as small On
as ther⇠10hand,—far smaller than thosevations of the 21cm signal from minihalos rapidly increases at low redshifts. Therefore, theFuture wide and deep surveys with Euclid1 in op-

What is DE?

SKA• Points for observations
ü Cosmological constantü Quintessence models – thawing type-- freezing type
ü Scalar-tensor theories – Horndeski, DHOST theories,…
And more;Ø DE constraints from early Universe Kohri et al. (2016)
Ø growth index, or MG in astrophysical bodies, in early Universe?

Can we probe DM by SKA?
• PBH-DM
SKA bound

21cm from early Universe(evo. of gas temperature, ionization history, ..)Induced Primordial Gravitational Waves
And more;Ø small scale DM clumps with PTA Oguri, Kashiyama (2018)
Ø any other idea? microlensing, radio sources, …

dn/dlnM [h3 Mpc-3]

Can we probe other DM by SKA?
• Baryon – DM interaction

CDM
mWDM=10keV
mWDM=6keV
mWDM=2keV

104
102
100

z=10

10
evolution of gas temperature
10-4
Fialkov,where
Barkana,
Cohen(2018),
V is the volume
observed.
Important assumptions must be made about the astrophysical10-6
in order to apply these equations. The magnetic field must be inhomogeneouson
10-8
Cheungenvironment
et al. (2018),
scales that can be seen by the axion, and the plasma density must be less than that required
4 theresonance conversion. While there are great uncertainties in the small scale structure
104ofKovetzfor
etal.
(2018),…
10
Galaxy, the average electron density of order 1cm and the extremely high plasma density

2

-2

CDM

3

3
-3
dn/dlnM
dn/dlnM [h[h3 Mpc
Mpc-3]]

FIG. 1: Evolution of T21 with three di↵erent minimum halo masses. The black line is the EDGESmWDM=10keV

mWDM=6keV
mWDM=2keV

22
10required for resonance conversion of a non-relativistic axion, leads us to believe that 10
thereisbest-fit model.

zh = (19.5,
19.0, 18.6), andmerit in investigating the use of radio telescopes
further.

z=20= (1.54, 1.52, 1.48), respectively. These
three configurations
00

z=10

10
10

• Warm3 DM
Observing axion conversion in the interstellar medium

can be treated as the representative systematic uncertainties. In Fig. 1, we show their

-2
10-2

22

impacts on the plane (z, T21 ) for m = 200 GeV and h vi10
= 3 ⇥ 10 25 cm3 s 1 with the-4 black line is the EDGES-4
assumption of DM 100% annihilation to electron-positron pairs.
10The
10

with the SKA
SKA as a probeof small scale structure

best-fit model (EDGES BF). Clearly, the red solid line with the -6
resolution of the minimum
-6

10
10

halo mass Mmin = 10 3 M gives the weakest and the most conservative limit than the other-8-8Axion-photon conversion search
Modulation of grav. potential

22

21

13

FIG
int
var

2

• Axion DM (Ultra light particles, ..)

23

12

14

hc

Shimabukuro, Inoue, Ichiki, SY (2014),
Sekiguchi, Tashiro (2014),

dn/dlnM [h3 Mpc-3]

resolution Mmin = 10 3 M as
10
In order to determine the energy produced intwo. Hereafter, we will present our result only based on mass10
a given region in the Milky Way, we mustbeing conservative.
4
105 106 107 108 109 1010 1011 1012
410
we compute the theo- -1
make an assumption about both the den- In this paragraph, we would like to explicitly demonstrate10how
z=20
M [h Msun]sity of dark matter at a given point and theretical prediction of the DM modified T21 signal. First of all, after
2 all the DM ingredients10m eV⇥
strength and profile of the magnetic field.are included, we insert Eqs. (3) and (4) into the RECFAST code [32] to obtain the DM0 Halo mass functions
10
FIG.
9:
at
z 10= 10 (top) 10and z = 20
10
We use Equation 1 to determine the photonmodified matter temperature Tg (z). Secondly, we perform the RECFAST 10computation again(bottom)for CDM (red), and WDM with
mWDM = 10 keVT-2IMI
production rate at the Galactic centre andby switching o↵ the DM contribution in order to calculate mattertemperature without DM
TL
REN(blue)10and 2 10keV (green).
CUR
Athroughout the interstellar medium, makingannihilation. Then, comparing the matter temperature from both-4scenarios, one can obtain
PPT nss of0
5 yr rs ⇥ 10
10ulsa
the assumption that the related magneticthe change of Tg by DM annihilation Tg . This quantity is useful
20 p
in the10 next step. Before
SKA s
f
o
rs
0n
10 y ars ⇥ 5field does not change on timescales visible toperforming the third step, let us make a reasonable assumption
10-6that Ts is fully coupled to
puls
100
10
the axion. Kelley, Quinn (2017)-8 Therefore, in the third
Tg at z = 15 20 as indicated by the T21 signal probedotherby EDGES.
are possible and these observations
10interpretations
We calculate the density of dark matter
Scalar Field
10cannotyetbetaken
as Dark
definitive
evidence of WDM or
4
5
6
7 Matter 8
6
using an adjusted NFW profile with a central
10
10
10
10
10 109 1010 1011 1012
measurement
of
its
mass.
-1
10
core density of 10GeVcm 3 and a density at
10
10
10
M [h
Msun]Hz⇥10
observed frequency
3
Khmelnitsky,
Rubakov(2014)
Earth of 0.4GeVcm . The strength of the
Figure 1. Pulsar timing signal from the scalar field dark matter (3.9) for a range of scalarmagnetic field is assumed to have a strength
FIG. 9: Halo mass functions at z = 10 (top) and z = 20
15

16

17

9

8

7

6

FIG