Effects of Survey Geometry on Power Spectrum Covariance for Cosmic Shear Survey

Transcription

1 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

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

3 Quiz Which observational survey geometry is better to extract the cosmological information in weak-lensing cosmic shear survey? Square VS Belt Elongated Rectangular the surface areas are same

4 Quiz Which observational survey geometry is better to extract the cosmological information in weak-lensing cosmic shear survey? Square VS Belt Elongated Rectangular better the surface areas are same

5 Quiz2 Which observational survey geometry is better to extract the cosmological information in weak-lensing cosmic shear survey? Square Sparse Sampling VS the surface areas are same

6 Quiz2 Which observational survey geometry is better to extract the cosmological information in weak-lensing cosmic shear survey? Square Sparse Sampling VS better the surface areas are same

7 Summary Sparse Sampling Belt Elongated Rectangular The above observational shapes have advantages at both large and small scales: larger-scale fluctuations can be scales (e.g. Kaiser 1986, 1998; Kilbinger & Schneider 2004; Chiang+ 2013) the non-gaussian error can be scales

8 Summary Sparse Sampling Belt Elongated Rectangular The above observational shapes have advantages at both large and small scales: larger-scale fluctuations can be scales (e.g. Kaiser 1986, 1998; Kilbinger & Schneider 2004; Chiang+ 2013) the non-gaussian error can be scales

9 Summary Sparse Sampling Belt Elongated Rectangular 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)

10 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

11 Convergence field κ(θ) : projected surface mass density at angular position θ Convergence power spectrum C(l) (=shear power spectrum) C l = 1 N l κ (l) 2 l l κ(l) N l : multipole : convergence in Fourier domain : # of mode

12 Measurement large scales Gaussian error C(l) C(l) = 2 N l N l : # of mode observational area Gaussian error 1 (observational area) 1/2 Measurement errors of cosmological parameters 1 (observational area) 1/2

13 Measurement small scales Small-scale fluctuations are correlated due to non-linear mode coupling (Meiksin & White 1999; Scoccimarro+ 1999) 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?

14 Log-normal Convergence field Convergence field is assumed to follow log-normal distribution which is supported by gravitational-lensing ray-tracing simulations (e.g. Jain+ 2000; Taruya+ 2002; Das & Ostriker 2006; RT+ 2011) κ 0 2 σ κ : minimum convergence empty beam : variance the mean is zero dp dκ κ 0 0 κ

15 203deg Log-normal Convergence Maps input power spectrum at redshift 0.9 halo-fit nonlinear model each map is a square shape 203x203 deg 2 (= 4π stradian) angular resolution = 1arcmin 1000 maps prepared θ 2 cut a region & calculate the power spectrum 0 203deg θ 1

16 Log-normal Convergence Maps We calculate the power spectra & covariance from the 1000 maps SN(Signal-to-Noise ratio) in power spectrum measurement Cumulative SN up to l max S N 2 = C ij 1 l i,l j <l max C W l i C W (l i ) C W (l) : power spectrum C ij : covariance

17 Results rectangular observational shape with an area of 100deg^2 θ A θ B θ A θ B = 100 deg 2

18 Cumulative SN for rectangular observational shapes Survey area = 100deg^2 Cumulative SN up to l_max z s = 0.9 Maximum multipole

19 Cumulative SN for rectangular observational shapes Survey area = 100deg^2 Cumulative SN up to l_max 2 times better 3 times better z s = 0.9 Maximum multipole

20 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

21 100 deg^2 area in total

22 Cumulative SN up to l_max 3 times better 3 times better Maximum multipole

23 The non-gaussian error depends on variance of convergence in the survey region: (Hilbert+ 2011; Takada & Hu 2013) σ W 2 = d 2 q 2π 2 W q 2 C(q) = 1 S W 2 d2 θ 1 d 2 θ 2 W θ 1 W(θ 2 )ξ( θ 1 θ 2 ) Two-point correlation function Small non-gaussian error Small variance σ W 2

24 Two-point correlation function of convergence ξ has a minimum at θ=15deg Best separation among the patches is ~15deg Separation angle

25 100 deg^2 area in total Each patch is separated by ~15deg separation

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

27 Cumulative SN up to l_max Maximum multipole

28 Cumulative SN up to l_max more sparse distribution is better Maximum multipole

29 Summary Sparse Sampling Belt Elongated Rectangular The above observational shapes have advantages at both large and small scales: larger-scale fluctuations can be scales (e.g. Kaiser 1986, 1998; Kilbinger & Schneider 2004; Chiang+ 2013) the non-gaussian error can be scales

30 Power Spectrum Covariance for Log-normal Field Survey window function Total survey area W θ = 1 0 inside the survey region otherwise S W = d 2 θ W(θ) Power spectrum convolved with the window function C W (l) = 1 S W d 2 q W l q 2 C(q) Circular-averaged band power spectrum CW l = 1 S W l l d 2 l N l d 2 q 2π 2 W l q 2 C(q) N l = 2πl l : # of modes in (l l 2, l + l 2)

31 Power spectrum covariance (Takada & Hu 2013) C ij = CW l i, C W l j = 1 S W 2 N l C W(l i ) 2 δ ij + T W (l i, l j ) Gaussian non-gaussian terms = C G ij+ C T0 ij + C σ W ij 1 / (survey area) C G ij = 1 S W 2 N l C W(l i ) 2 δ ij depends on the survey geometry (beat-coupling term) C T0 ij = 1 S W l l i d 2 l N li d2 l T(l, l, l, l ) l l j N lj C σ W ij = 4 κ 0 2 σ W 2 C l i C(l j ) σ W 2 = d 2 q 2π 2 W q 2 C(q) Variance of convergence in the survey region

32 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

33

34

35

36 Gaussian three independent data Non-Gaussian Cls are fully correlated at different k single data Non-Gaussian error depends on survey geometry Optimal survey geometry can reduce the non-gaussian errors

37 シミュレーションマップを使った解析 Cylindrically-averaged power spectrum estimator C W,r l = 1 S W 1 N l κ W,r(l ) 2 l l Mean power spectrum C W l = 1 N r C N W,r (l) r r=1 N r = 1000 : # of realizations Covariance matrix C ij = CW l i, C W l j = 1 N r 1 N r r=1 C W,r l i C W l i C W,r (l i ) C W l j Cumulative SN(Signal-to-Noise ratio) up to l max S N 2 = Cij 1 l<l max C W l i C W (l i )

38 Ray-tracing simulation (Taruya+ 2002)

39 Covariance matrix (diagonal elements) Variance Multipole

40 Covariance matrix (off-diagonal elements) C ij C ii C jj Multipole