Covariance Matrix of the Matter Power Spectrum

Ryuichi Takahashi （Nagoya U.）

with N. Yoshida, M. Takada, T. Matsubara, N. Sugiyama,

I. Kayo, A.J. Nishizawa, T. Nishimichi, S. Saito, A. Taruya

ref: RT et al. 2009, arXiv:0902.0371

Dark energy

Negative pressure（P<0）

Accelerates the expansion rate

of the Universe

Its nature still elude us

Recent Cosmological observations

（CMB, large scale structure, SNIa）

reveal that ～70% of the cosmic

energy density is dark energy.

Big problem not only for cosmology but also physics

Maybe it is related to vacuum energy？

cosmic energy density

Its nature is revealed by astronomical observations

SDSS data

Measuring large scale galaxy distribution provides

a strong constraint on the nature of dark energy

peak at 100Mpc/h can be used as “standard ruler”

gala

xy c

orr

ela

tio

n f

un

cti

on

distance（Mpc/h）

100Mpc/h

（Eisenstein et al. 2005）

100Mpc/h

distance D

θ angle

measuring galaxy distribution

Mpc/h100D

Standard ruler

100Mpc/h

distance D

θ angle

dark energy density

measuring galaxy distribution

Standard ruler

Mpc/h100D

distance D

angle θ

100Mpc/h

θ

distance D

angle

dark energy density

measuring galaxy distribution

Standard ruler

Mpc/h100D

distance D

angle θ

100Mpc/h

θ

measuring galaxy distribution

Mpc/h100D

Directly measure the distance to the galaxies

Standard ruler

distance D

angle

Equation of state w （=pressure/density）

1

Pw

Dark Energy

similar to the cosmological constant （w=-1）

D

D

w

w

4

relation to the distance measurement accuracy

1at z

（WFMOS feasibility study report）

WFMOS

dis

tan

ce m

easu

rem

en

t accu

racy （

%）

Future & On going plans

WFMOS（Subaru & Gemini telescope, photometric）

Lyman-break galaxies at z~3

Emission line galaxies at z~1

FMOS（Subaru telescope, Near Infrared, Photometric）

HSC（Subaru telescope, Narrow band filter） 2011+

Lyα emitter at z=3 ～2% error Matsuda＋

LRG（luminous red galaxies） at z=0.3-0.6

Hα emitter at z=1 Totani, Sumiyoshi＋

Eisenstein＋

SDSSⅢ（BOSS） 2009+

～1% error

xkierdVkP )()(

Galaxy power spectrum （Percival et al. 2007）

波数

100Mpc/h

Accurate theoretical model of P(k) （<1%） is necessary

1st peak at k=0.06h/Mpc

How accurately the cosmological parameters are

determined from the galaxy power spectrum ?

parameter estimation error

1.01w 05.0or

The likelihood analysis of power spectrum

・ analysis

)()()()(),(cov thobsthobs

,

12

jjii

ji

ji kPkPkPkPkk

2

02.0or

covariance matrix

three sources of statistical error

Volume of survey area （= number of Fourier modes）

Shot noise

Non-Gaussian error

error ∝ （volume）^-1/2

Correlation between different Fourier modes k via

non-linear gravitational evolution

Each mode k is not independent

Covariance matrix of P(k) between k1 and k2

)(ˆ)(ˆ)(ˆ)(ˆ),cov( 221121 kPkPkPkPkk

),(11

)(2

21,

2

21kkT

VnkP

Nkk

gk

)()()()(),( 22112

3

1

3

21

21

kkkkV

kd

V

kdkkT

kk

Nk : number of mode

T : trispectrum

（e.g. Scoccimarro, Zaldarriaga & Hui 1999）

kV1 non-Gaussian contribution arising

from mode-coupling shot noise

Gaussian term

volume

Non-Gaussian error

: power spectrum estimator in each realization

: mean power spectrum

)(ˆ kP

)(kP

5000 realizations

V=（1Gpc/h）^3 256^3 particles & mesh

PM（Particle-Mesh） simulations

)(ˆ1

)( kPN

kPi

i

r

i

ii

r

kPkPkPkPN

kk )()(ˆ)()(ˆ1

1),cov( 221121

Mean

Covariance

Largest realizations for BAO

Our data are publically available

64-bit PC cluster

Intel Xeon 3GHz

12 nodes×4 cpus

= 48 processes

Memory total 16GB×12

HDD ～20TB

At lab., Nagoya U.

12 nodes

code : Gadget2

)(ˆ kP

6 realizations

colored curves :

for each realization

thick black curve : mean P(k)

V=(1Gpc/h)^3

Variance of P（k）

solid curve:

perturbation theory

dashed curve:

halo model

Comparison with Analytical Models

symbols :

simulation outputs

（Scoccimarro et al. 1999）

（Cooray & Hu 2001）

correlation matrix

How many realizations we need ?

（Kayo et al. in preparation）

vertical axis：variance of covariance

solid line : theoretical prediction for Gaussian density fluctuation

How many realizations we need ?

200 realizations for 10% accuracy

800 realizations for 5% accuracy

rN

2

cov

covcov

2

2

2

cov

variance of variance

necessary number of galaxy mock catalogue

Nr : number of realizations

)()(),(cov 2121

1

2

max2,1

kPkPkkN

S

kk

Signal-to-noise ratio

non-Gaussian error degrades the S/N for k>0.2h/Mpc

（Hamilton, Rimes, Neyrinck, Szapudi, Lee, Pen, … ）

)()(),(cov 2121

1

2

max2,1

kPkPkkN

S

kk

Signal-to-noise ratio

non-Gaussian error degrades the S/N for k>0.2h/Mpc

（Hamilton, Rimes, Neyrinck, Szapudi, Lee, Pen, … ）

× factor 4

× factor 3

jikk

jixd

kdP

xd

kdPkkF

ln

)(

ln

)(),(cov 21

21

1

max2,1

2/11

ji

i

i Fx

x

),,,,,( 22

8 whhhnx cbSi

Parameter estimation

Fisher matrix

Gaussian error

assumption is OK

six cosmological parameters

solid curves : Gaussian error

symbols : with Non-Gaussian error

ratio = NG/G

・ parameter degeneracy

・ non-Gaussian degradation

This is because …

2/1

ii

i

i Fx

x

no parameter degeneracy

Gaussian error

assumption

underestimates

the errors

by a factor of 2-4

sub box effects

)()(~

)(2

3~ kPWkdVP

W kkk

1,2,3i

2

ii

ii2

2

2sin)(

~

xk

xkW k

1Gpc/h

500Mpc/h

1/8 size

blue & red symbols are results

for subbox

dashed curve：P(k) convolved with

window function

Beat-coupling （Takada & Kayo in preparation）

mode-coupling larger than subbox volume

)()()(21

1716

121111111

2

kPkPPV

),cov( 11 kk

L

L : size of subbox

S/N is further degraded by 30%

work in progress

L=1Gpc/h N=1024^3 TPM simulation

～ hundreds realizations

（M ～ 7×10^10 M_SUN/h）

Making halo catalogue

HSC Lyα emitter at z=3 （Matsuda +）

BOSS LRG（luminous red galaxies） at z=0.3-0.6

WFMOS Lyman-break galaxies at z~3

Emission line galaxies at z~1

FMOS Hα emitter at z=1 （Totani、Sumiyoshi+）

（Eisenstein +） （SDSSⅢ）

Summary

non-Gaussianity in the covariance matrix affects the S/N

and the parameter estimation for k>0.2h/Mpc

variance of angular averaged P(k) (monopole component)

in redshift space is larger than in real space by 16% (z=1),

20% (z=3) due to the redshift space distortion

probability distribution of is almost Gaussian even

in non-linear regime

variance of covariance decreases as where Nr

is the number of realizations

)(ˆ kP

2/1/2

rN

3rd order density perturbation

),(),()()(),( 321 akakkaDak

)()(),()( 112

3

2 kqqkqqFqdk

)()()(),,()( 1113

33

3 kpqpqkpqpqFpdqdk

*

31

2

2

2

1

2Re2

leading correction to the linear power spectrum

linear power spectrum leading correction term

(one-loop correction)

（Makino, Suto & Sasaki 1992; Jain & Bertschinger 1994）

)(lin kP

)()( 1322 kPkP

EdS 以外の宇宙モデル

近似：EdS モデルでの結果（上式）でスケールファクター a を

線形成長率 D（a） に変更

5.0

ln

lnM

ad

Ddf なら厳密に近似はＯＫ

（e.g. Bernardeau, et al. 2002, Physics Reports）

)()()(),( 1322

4

lin

2 kPkPakPaakP

実際は 6.0

ln

lnM

ad

Ddf

ダークエネルギー入り宇宙で３次摂動まで計算

),(),(),(),(),( 3

qkqqkqqdk

k

),(),(),(),(2

3),(

),( 32

qkqqkqqdkHkH

kM

2

)(),(

pqp

pqp

22

2

2

)()(),(

qpqp

qpqp

V

Bernardeau 1994; Matsubara 1995 for lambda CDM model

基礎方程式 （連続の式＆オイラー方程式） δ：密度場、 ｖ：速度場

aawwaw a 1)( 0

relative errors

P22 P13

EdS model との相対誤差は <2% at z=0

ixd

kPd

ln

)(ln

derivative of P(k) 8

Sn

h

b

w

w

fiducial parameter %5

jikk

jixd

kdP

xd

kdPkkF

ln

)(

ln

)(),(cov 21

21

1

max2,1

2/11

ji

i

i Fx

x

),,,,,( 8 whnx wbSi

Parameter estimation

Fisher matrix

Gaussian error

assumption is OK

six cosmological parameters

8

Sn

h

b

w

w

solid lines : Gaussian error symbols : with Non-Gaussian error

・ parameter degeneracy

・ non-Gaussian contribution

This is because …

2/1

ii

i

i Fx

x

no parameter degeneracy

Gaussian error

assumption

underestimates

the errors

by a factor of 2-4

8

Sn

h

b

w

w

Covariance matrix の分散

error = 1/（number of realizations）^1/2

全非対角成分＠初期条件（z=20）

100 400 4000

Number of realizations

diagonal off-diagonal

Non-Gaussian error in Weak lensing

（Cooray & Hu 2001; Takada & Jain 2008; Eifler, Schneider & Hartlap 2008）

Gaussian error の仮定は10-20％くらい誤差を過小評価

コバリアンスは誤差だけでなくベストフィット値にも影響（？）

Model A :

高田さんハローモデル

Model B :

simulation results in

Semboloni et al 2007

（Ichiki, Takada, Takahashi 2008）

CHFTLS WL data

64-bit PC cluster DE

Intel Xeon 3 GHz

10 nodes×4 cpus

HDD 150GB×10 + 480GB×4

+ 1TB×12

Memory total 16GB×10

At lab., Nagoya U.

code : Gadget2

),(2

),( ||

22

|| kkPN

kkP s

k

s

)(2

5

1

3

21

9

1

7

4

5

6

3

41

)(2

2

2

432

2kP

NkP s

k

s

2

2222

5

1

3

211

d

4324

2

9

1

7

4

5

6

3

411 d

)(1)(22 kPdkP rs

variance in redshift space

additional coefficient due to the redshift space distortion

LHS :

RHS :

in linear regime

angular

averaging

k

||k : line of sight direction

: perpendicular direction

Finger of god effect erases the variance

・in linear regime, simulation results reproduce the redshift

space distortion (red solid lines)

variance in real space & in redshift space

・in non-linear regime, the variances in redshift space do not

increase comparing to in real space

（Meksin & White 1999）

16% 20%

forecast for WFMOS survey

%1.3%80.0 wWFMOS1 z=0.5-1.3 V=4(Gpc/h)^3 b=1.3

WFMOS3 z=2.3-3.3 V=1(Gpc/h)^3 b=3 %0.6%1.1 w

stretch parameter α Measurement accuracy for

acoustic peak positions kk α=1 : fiducial

nPbP 12

TbT 4

z=0.5-1.3 V=4(Gpc/h)^3 b=1.7 n=5e-4(Mpc/h)^-3

z=2.3-3.3 V=1(Gpc/h)^3 b=3.2 n=5e-4(Mpc/h)^-3

S/N for WFMOS survey

nP=1 @ k=0.2h/Mpc

（Gaztanaga, Cabre & Hui 2008）

SDSS 75000 LRG @ z=0.15-0.47

lin

e o

f sig

ht （M

pc/h）

perpendicular （Mpc/h）

Galaxy correlation function in two dimension

distance D

θ angle

distance D(z) & Hubble rate H(z) are determined

)(/ zHz D

Galaxy correlation function in two dimension

Baryon Acoustic Oscillations

at z=1100

photon fluctuation baryon （proton & electron）

density fluctuation

CMB temperature

fluctuation

dark matter density fluctuation

Thomson scattering

gravitational interaction

Baryon Acoustic Oscillations

at z=1100

photon fluctuation baryon （proton & electron）

density fluctuation

CMB temperature

fluctuation

dark matter density fluctuation

Thomson scattering

gravitational interaction

Galaxy distribution

sound horizon at recombination 100Mpc/h

),cov(),cov(

),cov(),(

2211

2121

kkkk

kkkkr

off-diagonal elements

: correlation matrix

Probability distribution of the power spectrum estimator

The results are well described by the Gaussian distribution

Likelihood analysis with non-Gaussian error

)()()()(),(cov 2211

,

21

12

max21

kPkPkPkPkk thobsthobs

kkk

2exp 2L)()(ˆ)( kAPkPAkP ith

A=1 fiducial

)()( kPkP iobs

fitting parameter is only an amplitude A

（i-th realization）