Observational test of modified gravity modelswith future imaging surveys

Kazuhiro Yamamoto (Hiroshima U.)

Edinburgh Oct. 24-26

K.Y. ， Bassett, Nichol, Suto, Yahata, (PRD 2006)K.Y. ， Parkinson, Hamana, Nichol, Suto, (PRD 2007)Discussion by HSC Weak Lensing Working Group

INTRODUCTION

Modified Gravity models as alternative to the dark energy

f(R) gravity model, TeVeS theory, DGP model, etc. ・・・

ambitious challenges to the fundamental physics necessary step to go beyond the standard model ?

All these models may not be complete, but are

A lot of observational projects of the dark energy are proposed,

These results might be useful to test modified gravity theory.

WFMOS, HSC, DES, DUNE, LSST, JDEM, BOSS, ・・・

Future feasibility of testing gravity models ?Optimized strategy of future survey of HSC ?

2

Investigation of the observational consequences of typical model is thought-provoking, because we can learn what can be possible signatures of such generalized gravity models.

The DGP model as an example (Dvali, Gabadadze, Porrati, 00)

Brane world scenario, (3+1)-dim brane in (4+1)-dim. bulk

It is possible to construct a self-accelerating universe, without introducing dark energy, by choosing a scale parameter, rc=M4

2/2M52,

defined by the ratio of the Planck scales, properly.

(Deffayet, 01)

Modified Friedmann equation (flat universe)

mc

2 ρ3

8π=

r

H-H

G

Modification of expansion history changes the distance redshift relation

z

zH

dzz

0 )'(

')(

３

can be tested using SNe, BAO, CMBModified relation of the background expansion, distance-redshift relation

(K.Y., Bassett, Nichol, Suto, Yahata)

(e.g., Maartens, Majerroto 06)

Constraint using Baryon Oscillation

4

dlnP(k)/dlnk　 The Λ-model and the DGP model the same cosmological parameter 　　　　　　　 and the same data analysis,

Area 2000 deg2

n = 5×10-4 (h-1Mpc)-3

0.5 ＜ z ＜ 1.3

WFMOS-like sample

One can distinguish between the Lambda model and the DGP model clearly

difference of H(z) and r(z), 　　　　　 the peaks shift.

The background expansion is parameterized in general (flat universe)

z1

z3w)ww3(1

m3m2

02 ez))(1Ω-(1

z)(1Ω

HH(z) 0a

a

The expansion history of the DGP model is reproduced by the dark energy model of the equation of state

z1z

0.32-0.78w(z)

Ωm ～ 0.3

The expansion history of M.G. can be described equivalently by the parameterization of the dark energy model.

z1z

www(z) 0 a

(Linder, 04)

5

Perturbation is important as an independent information

(Maartens, Koyama 06)Perturbation of the cosmological DGP model

)sin(21)(21 22222222 dddtadtds

022

2

2

2

a

k

dt

dH

dt

d

mGak

3

114 22

maG

k 22

3

8

23121

HHHrc

mGak 22 8

Evolution of Growth factor

Modified Poisson equation

Anisotropic stress

(sub-horizon sclale)

6

Evolution of Growth factor

ΛCDM

DGP Dark Energy

Same expansion H(z)

a

aD )(1

a

growth factor is important to distinguish between the gravity models.

7

Phenomenological description of the perturbation for generalized gravity models

(Amin, Wagoner, Blandford 07; Jain, Zhang 07; Hu, Sawicki 07; Caldwell, Cooray, Melchiotti 07…)

(Amendola, Kunz, Sapone 07)

δρm

22 a4π=Φk

0=Φ+Ψ

G

General relativity

δm

22 ρaQ4πΦk

ηΦΦΨ

G

Generalized model

1Q

0η

)sin()(21)(21 22222222 ddfdtadtds K

0 Ψak

dtdδ

2Hdt

δd2

2

2

2

0δρη)Q(14dtdδ

2Hdt

δd2

2

G

8

Parameterization of the growth factor

a

m aa

da

a

aD0

1 1-)'('

'exp

)(

2

30

20

m aH

aΩH=aΩ

)()(

-

(Lahav 91, Wang, Steinhardt 98, Percival 05, Linder 05)

680=γ .

　 γ characterizes the modification of gravity

the dark energy model in general relativity

the DGP model

56.0~55.0_

))1(1(05.055.0

zwγ

The difference of the growth of density perturbation is described by γ.

Other way of description of modified gravity9

The fitting formula works

ΛCDM

DGP Dark Energy γ ＝ 0.56γ ＝ 0.68

Parameterization by γ reproduces the evolution

Evolution of growth factor

10

Observational constraints on γ

20.017.067.0

4.03.06.0

Porto & Amendola (07)

Nesseris & Perivolaropoulos (07)

3.02.00.0

a

0 m1 1-)(a'Ω

a'da'

expa(a)D η)(1γ

Lyman-α forest clustering

Galaxy clustering and redshift space distortion

Measurement of γ 　11

Importance of measuring γ as a consistency test of the growth of density perturbation and gravity model

The weak shear is useful to measure the evolution of the density perturbation, and to test modified gravity models

( Ishak, Upadhye, Spergel 06; Huteter, Linder 07; Amendola, Kunz, Sapone 07; Jain, Zhang 07 Heavens, Kitching, Verde 07; etc…)

)( )(,)(

))(())(()()( χzχl→knonlineaer

22

20

20

jiij massP

a2

ΩH3χzWχzWχd=lP ∫

)'(

(z)-)'(

'

)'('

1)(

1

, ]max[ z

z

dz

zdNdz

NzW

i

i

z

zzi

i

Weak shear power spectrum

')'(

'dz

zdNdz=N

1+i

i

z

zj ∫

dz

zdN )(Number count per unit solid angle

mGak 22 8

12

Feasibility of measuring γ with the HSC Weak Lens survey?

Fisher Matrix Analysis

a

0 m1 1-)(a'Ω

a'da'

expa(a)D γGrowth of density perturbation

Background expansion

1)-(a3w)ww-3(1m

-3m

20

2 e)aΩ-(1aΩHH(a) 0 aa

a)-(1www(a) 0 a

Analysis in the 7 parameters space

sma nhww ,,,,, 80 γ

marginalized

flat universe, Assumption;

13

β0

0

)(z/z-α1)/β)+Γ((αz

βg ezN=

dz

dN(z)1+α

3=β50=α ,.

dz

dNdzz

N

1=z ∫

gm

dz

dNdz=N ∫g

)/( 38Γ

z=z m

0

067.0

.min30exp9.0t

meanz

( )440

30

t

g 30=N.

.minexp

Amara & Refregier (06)

arcmin-2

(SNAP simulation)

(number density arcmin-2)

(mean redshift)

expt exposure time

Modeling of Galaxy distribution

/ 1 Field of View / 1 passband filter

The Validity of this relation for the HSC is now investigated by WLWG.

14

operationtnt

1.1

n timeobservatio Total

passbandnumberexp

2

2ViewofFieldArea

Total observation time = 100 nights (fixed)

Field of view = 1.5 degree

Overheard time = 10% of exposure time + operation time (toperation=5 minutes)

expt (one band exposure time for one FoV)

Total survey area

Assumption of the HSC WL survey

4passbandnumber n

15

Total survey area=1700 deg.2 texp=10mins./1FoV/passband

Total survey area as a function of texp

Total observation time = 100 nights4 passband filters

16

1σ error as a function of texp

Constraint on γ from the WL shear power spectrum

photo-z error

)1(05.0 zz

5.105.0 z only used the sample

The observation of 100 nights will be difficult to achieve

1.0

17

Marginalized Fisher matrix

Constraint on γ from the WL shear power spectrum + galaxy power spectrum (BAO) from WFMOS like spectroscopy survey

1σ error as a function of texp

34 Mpch10×4=n ]/[-

4.18.0 z

Assumed additional spectroscopy survey of the same survey area as the WL survey

of the number density

in the redshift range

Combination with the WFMOS improves the constraint 07.0

WL + BAO

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Summary & Conclusion

Dark energy survey is useful to test modified gravity models

Simple consistency test is to measure γ parameter

The weak lensing method is useful to constrain γ

The HSC alone would not provide a strong constraint, but the combination with the WFMOS improve it, 　　　 and Δγ≦0.07 might be possible. (2σ level for differentiating between the DE and the DGP) Slightly depends on the modeling of the galaxy count, dN/dz

Synergy with the cluster count ?

finding the optimized survey strategy of HSC

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HSC Weak Lens Working Group is investigating it