Effects of Survey Geometry

on Power Spectrum Covariance

for Cosmic Shear Survey

Ryuichi Takahashi (Hirosaki U)

with Shunji Soma, Masahiro Takada

and Issha Kayo

RT+, in preparation

What survey shape is the BEST （or WORST） to extract the cosmological information in weak-lensing cosmic shear survey ?

Abstract

Quiz

Which observational survey geometry is better to extract the cosmological information in weak-lensing cosmic shear survey ?

“Square” “Belt” “Elongated Rectangular”

the surface areas are same

VS

“Belt” “Elongated Rectangular”

Quiz

better

the surface areas are same

Which observational survey geometry is better to extract the cosmological information in weak-lensing cosmic shear survey ?

“Square” VS

Quiz2

“Square” “Sparse Sampling”

the surface areas are same

Which observational survey geometry is better to extract the cosmological information in weak-lensing cosmic shear survey ?

VS

Quiz2

“Square” “Sparse Sampling”

the surface areas are same

Which observational survey geometry is better to extract the cosmological information in weak-lensing cosmic shear survey ?

VS better

・ larger-scale fluctuations can be measured @large scales

The above observational shapes have advantages at both large and small scales:

“Sparse Sampling”

“Belt” “Elongated Rectangular”

Summary

・ the non-Gaussian error can be suppressed @small scales

（e.g. Kaiser 1986, 1998; Kilbinger & Schneider 2004; Chiang+ 2013）

・ larger-scale fluctuations can be measured @large scales

The above observational shapes have advantages at both large and small scales:

“Sparse Sampling”

“Belt” “Elongated Rectangular”

Summary

（e.g. Kaiser 1986, 1998; Kilbinger & Schneider 2004; Chiang+ 2013）

・ the non-Gaussian error can be suppressed @small scales

“Sparse Sampling”

“Belt” “Elongated Rectangular”

Summary

Signal-to-Noise in power spectrum measurement can be 3 times higher than the S/N in the square observational shape (→ 9 times larger survey area)

Introduction

Weak-lensing cosmic shear is a powerful observational tool to constrain the cosmological models (including dark energy).

previous surveys : SDSS, CHFTLS, COSMOS, …

on-going & future surveys : Subaru HSC, DES, LSST, …

from LSST homepage

Convergence field 𝜅(𝜃 ) : projected surface mass density at angular position

Convergence power spectrum 𝐶(ℓ)

𝐶 ℓ =1

𝑁ℓ 𝜅 (ℓ)

2

ℓ

: convergence in Fourier domain

： # of mode

𝜃

: multipole ℓ

𝜅 (ℓ)

𝑁ℓ

（=shear power spectrum）

∆𝐶(ℓ)

𝐶(ℓ)=

2

𝑁ℓ

: # of mode

Gaussian error ∝

Measurement error @ large scales

Measurement errors of cosmological parameters

𝑁ℓ

∝ observational area

Gaussian error

（observational area） 1/2

1

（observational area） 1/2

1 ∝

Measurement error = Gaussian error + non-Gaussian error

depends on non-linearity of fluctuations （scale, redshift）

&“survey shape”

“What observational shape is the best to reduce the non-Gaussian error?”

Measurement error @ small scales

（Meiksin & White 1999; Scoccimarro+ 1999）

Small-scale fluctuations are correlated due to non-linear mode coupling

Log-normal Convergence field

Convergence field is assumed to follow log-normal distribution

: minimum convergence ← empty beam

: variance

the mean is zero

𝜅0

𝜎𝜅2

𝜅 0

𝑑𝑃

𝑑𝜅

𝜅0

which is supported by gravitational-lensing ray-tracing simulations （e.g. Jain+ 2000; Taruya+ 2002; Das & Ostriker 2006; RT+ 2011）

Log-normal Convergence Maps

・ 1000 maps prepared

・ each map is a square shape 203x203 deg (= 4π stradian)

・ input power spectrum at redshift 0.9 ← halo-fit nonlinear model

𝜃1

𝜃2

203deg

20

3d

eg

0

cut a region & calculate the power spectrum

・ angular resolution = 1arcmin

2

Log-normal Convergence Maps

We calculate the power spectra & covariance from the 1000 maps

Cumulative SN up to ℓmax

𝑆

𝑁

2= 𝐶𝑖𝑗

−1ℓ𝑖,ℓ𝑗<ℓmax

𝐶𝑊 ℓ𝑖 𝐶𝑊(ℓ𝑖)

SN（Signal-to-Noise ratio） in power spectrum measurement

𝐶𝑊(ℓ)

𝐶𝑖𝑗

: power spectrum

: covariance

Results

rectangular observational shape

𝜃𝐴

𝜃𝐵

𝜃𝐴 × 𝜃𝐵 = 100 deg2

with an area of 100deg^2

Cumulative SN for rectangular observational shapes

Survey area = 100deg^2

𝑧𝑠 = 0.9

Cumulative SN up to l_max

Maximum multipole

Cumulative SN for rectangular observational shapes

Survey area = 100deg^2

𝑧𝑠 = 0.9

Cumulative SN up to l_max

Maximum multipole

2 times better

3 times better

“What observational survey shape is the best (or worst) to suppress the non-Gaussian error in the power spectrum measurement”

Problem setting ：

1. prepare 100 small patches, each patch has an area of 1x1 deg^2

2. place these 100 patches on the all sky (203x203 deg^2)

→ survey area is 100deg^2 in total

100 deg^2 area in total

Cumulative SN up to l_max

3 times better

3 times better

Maximum multipole

𝜎𝑊2 =

𝑑2𝑞

2𝜋 2𝑊 𝑞

2𝐶(𝑞)

=1

𝑆𝑊2 𝑑2𝜃1𝑑

2𝜃2𝑊 𝜃 1 𝑊(𝜃 2)𝜉( 𝜃 1 − 𝜃 2 )

Two-point correlation function

The non-Gaussian error depends on variance of convergence in the survey region:

Small non-Gaussian error Small variance 𝜎𝑊2

（Hilbert+ 2011; Takada & Hu 2013）

Two

-po

int

corr

ela

tio

n f

un

ctio

n o

f co

nve

rge

nce

ξ has a minimum at θ=15deg

Best separation among the patches is ～15deg

Separation angle

100 deg^2 area in total

Each patch is separated by ～15deg separation

Sparse sampling

1x1 deg^2 patches are placed in a configuration 10x10 with a separation θsep

Cumulative SN up to l_max

Maximum multipole

Cumulative SN up to l_max

Maximum multipole

more sparse distribution is better

・ larger-scale fluctuations can be measured @large scales

The above observational shapes have advantages at both large and small scales:

“Sparse Sampling”

“Belt” “Elongated Rectangular”

Summary

・ the non-Gaussian error can be suppressed @small scales

（e.g. Kaiser 1986, 1998; Kilbinger & Schneider 2004; Chiang+ 2013）

Power Spectrum Covariance for Log-normal Field

𝑊 𝜃 = 1 0

𝑆𝑊 = 𝑑2𝜃 𝑊(𝜃 )

Total survey area

𝐶𝑊(ℓ) =1

𝑆𝑊 𝑑2𝑞 𝑊 ℓ − 𝑞

2𝐶(𝑞)

Survey window function

Circular-averaged band power spectrum

𝐶 𝑊 ℓ =1

𝑆𝑊

𝑑2ℓ

𝑁ℓℓ ∈ℓ

𝑑2𝑞

2𝜋 2𝑊 ℓ − 𝑞

2𝐶(𝑞)

𝑁ℓ = 2𝜋ℓ∆ℓ : # of modes in (ℓ − ∆ℓ 𝟐 , ℓ + ∆ℓ 𝟐 )

inside the survey region otherwise

Power spectrum convolved with the window function

Power spectrum covariance

𝐶𝑖𝑗 = 𝐶 𝑊 ℓ𝑖 , 𝐶 𝑊 ℓ𝑗 =1

𝑆𝑊

2

𝑁ℓ𝐶 𝑊(ℓ𝑖)

2𝛿𝑖𝑗 + 𝑇𝑊(ℓ𝑖 , ℓ𝑗)

𝐶𝐺𝑖𝑗 =

1

𝑆𝑊

2

𝑁ℓ𝐶 𝑊(ℓ𝑖)

2𝛿𝑖𝑗

𝐶𝑇0𝑖𝑗 =

1

𝑆𝑊

𝑑2ℓ

𝑁ℓ𝑖ℓ ∈ℓ𝑖

𝑑2ℓ′

𝑁ℓ𝑗ℓ ∈ℓ𝑗

𝑇(ℓ, −ℓ, ℓ′, −ℓ′)

（Takada & Hu 2013）

𝐶𝜎𝑊𝑖𝑗 =

4

𝜅02 𝜎𝑊

2 𝐶 ℓ𝑖 𝐶(ℓ𝑗) 𝜎𝑊2 =

𝑑2𝑞

2𝜋 2 𝑊 𝑞 2𝐶(𝑞)

Variance of convergence in the survey region

= 𝐶𝐺𝑖𝑗+ 𝐶𝑇0

𝑖𝑗 + 𝐶𝜎𝑊𝑖𝑗

Gaussian non-Gaussian terms

∝ 1 / (survey area) depends on the survey geometry

（beat-coupling term）

The trispectrum can be written in term of power spectra for the log-normal field

Hilbert+ (2011) showed that a four point correlation function of log-normal field can be written in terms of its two point function

→ Power spectrum covariance can be written in terms of its power spectrum

Gaussian

→ three independent data

→ single data

Non-Gaussian Cls are fully correlated at different k

Non-Gaussian error depends on survey geometry

“Optimal” survey geometry can reduce the non-Gaussian errors

シミュレーションマップを使った解析

: # of realizations

𝐶 𝑊,𝑟 ℓ =1

𝑆𝑊

1

𝑁ℓ 𝜅 𝑊,𝑟(ℓ

′)2

ℓ′ ∈ℓ

Cylindrically-averaged power spectrum estimator

Mean power spectrum

𝐶𝑊 ℓ =1

𝑁𝑟 𝐶 𝑊,𝑟(ℓ)

𝑁𝑟

𝑟=1

𝐶𝑖𝑗 = 𝐶 𝑊 ℓ𝑖 , 𝐶 𝑊 ℓ𝑗 =1

𝑁𝑟 − 1 𝐶 𝑊,𝑟 ℓ𝑖 − 𝐶𝑊 ℓ𝑖 𝐶 𝑊,𝑟(ℓ𝑖) −𝐶𝑊 ℓ𝑗

𝑁𝑟

𝑟=1

Covariance matrix

𝑁𝑟 = 1000

Cumulative SN(Signal-to-Noise ratio) up to ℓmax

𝑆

𝑁

2= 𝐶𝑖𝑗

−1ℓ<ℓmax

𝐶𝑊 ℓ𝑖 𝐶𝑊(ℓ𝑖)

（Taruya+ 2002） Ray-tracing simulation

Covariance matrix (diagonal elements)

Multipole

Variance

Multipole

𝐶𝑖𝑗

𝐶𝑖𝑖𝐶𝑗𝑗

Covariance matrix (off-diagonal elements)