Measuring the CMASS galaxy bias using the bispectrum techniqueindico.ictp.it/event/a12214/session/12/contribution/8/... · 2014. 5. 5. · Measuring the CMASS galaxy bias using the

IntroductionMask of the Survey

Galaxy MocksNon-linear bias in real space

Non-linear bias in z-spaceConclusions

Measuring the CMASS galaxy bias using thebispectrum technique

Hector Gil Marın (ICG, Portsmouth)..

In collaboration with: W. Percival, L. Verde, C. Wagner, J. Norena

Non-linear bias Workshop. Trieste (Italia) 8th October 2013

Hector Gil-Marın Non-linear bias Workshop. Trieste Measuring the CMASS galaxy bias




Outline

i Introduction

ii Mask of the Survey

iii Galaxy Mocks

iv Non-linear bias (real space)

v Non-linear bias (z-space)

vi Conclusions





BOSSGalaxy BiasBispectrum of GalaxiesMocks

Introduction to BOSS

Apache Point Observatory (APO) 2.5-m telescope for five years from2009-2014.

Part of SDSS-III project. BOSS: Baryon Oscillation SpectroscopicSurvey

Map the spatial distribution of luminous red galaxies and quasars

Total coverage area 10,000 square degrees

Redshifts of 1.5 millionluminous galaxies to z ' 0.7

Wavelength: 360-1000 nm






Galaxy Bias

Galaxies are a biased tracers of dark matter.

Simple bias modelδg (x) = b1δ(x)

Non-linear local bias model

δg (x) =∑i

bii !

(δi (x)− σi )

Non-local bias model (see talks tomorrow)

δg = b1δ +b22δ2 + γ2G2






Bispectrum of Galaxies

The Bispectrum is sensitive tonon-Gaussianities, gravity and biasing.

Test of General Relativity

Detect potential primordial non-Gaussianities

Break degeneracies in the bias parameters






Mocks

Fast synthetic galaxy catalogues.

What has been included in the mocks is known, so one can work outwhat are the best estimators, and test systematics.

As pipelines become complex they are more difficult to capture bytheoretical modelling, and mocks a need.

Covariance matrix require a large number of realisations, theproduction of fast mock galaxy catalogues may provide them.





Angular & Radial maskHow to deal with it

Angular & Radial Mask

in a Box

Constant mean density

Box with periodic boundaryconditions

Real Survey

z-dependent mean density

Regions not observed →gaps






Angular & Radial Mask

1⋅10-4

2⋅10-4

3⋅10-4

4⋅10-4

5⋅10-4

6⋅10-4

7⋅10-4

1100 1200 1300 1400 1500 1600 1700 1800

– n (

r )

r [Mpc/h]






How to deal with it: Power Spectrum

Feldman et al. (1994)

F2(r) ≡ I−1/22 [n(r)− αns(r)] .

where α ≡ N/Ns and Ii ≡ αi∫d3r nis(r).

FKP estimator

〈|F2(k)|2〉 =

∫d3k′

(2π)3Pgal(k′)|G2(k− k′)|2 + (1 + α)

I1I2

with,

G2(k) ≡ I−1/22

∫d3r n(r)e+ik·r → δD(k)

Note the convolution between Pgal and |G2|2






How to deal with it: Power Spectrum

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0.01 0.1

Pow

er S

pect

rum

k [h/Mpc]

Plin Plin x W






How to deal with it: Bispectrum

and for the bispectrum,

F3(r) ≡ I−1/33 [n(r)− αns(r)] .

FKP estimator

〈F3(k1)F3(k2)F3(k3)〉−Bsn =

∫d3k′

(2π)3d3k′′

(2π)3Bgal(k′, k′′)Q3(k1−k′, k2−k′′)

where,

Q3(kA, kB) ≡ I−13

[I1/22 G2(kA)× I

1/22 G2(kB)× I

1/22 G∗2 (kA + kB)

],







and for the bispectrum,

F3(r) ≡ I−1/33 [n(r)− αns(r)] .

FKP estimator

〈F3(k1)F3(k2)F3(k3)〉 − Bsn ' Bgal(k1, k2)

where,

Q3(kA, kB) ≡ I−13

[I1/22 G2(kA)× I

1/22 G2(kB)× I

1/22 G∗2 (kA + kB)

],







We apply the mask geometry to a 2LPT halo periodic box.2LPT haloes: 20 realizations × 2.4 Gpc/h (unmasked).

2.0⋅109

3.0⋅109

4.0⋅109

5.0⋅109

6.0⋅109

7.0⋅109

8.0⋅109

0 0.2 0.4 0.6 0.8 1

0.063 0.060 0.052 0.041 0.028 0.021

B(θ

12)

[(M

pc/h

)6 ]

θ12/π

k3 [h/Mpc]

k1=0.021 h/Mpck2=2 k1

MaskNo Mask

5.0⋅1091.0⋅10101.5⋅10102.0⋅10102.5⋅10103.0⋅10103.5⋅10104.0⋅1010

0 0.2 0.4 0.6 0.8 1

0.042 0.040 0.034 0.025 0.013 0

θ12/π

k3 [h/Mpc]

k1=0.021 h/Mpck2= k1








2.0⋅1084.0⋅1086.0⋅1088.0⋅1081.0⋅1091.2⋅1091.4⋅1091.6⋅1091.8⋅109

0 0.2 0.4 0.6 0.8 1

0.153 0.146 0.127 0.099 0.068 0.051

B(θ

12)

[(M

pc/h

)6 ]

θ12/π

k3 [h/Mpc]

k1=0.051 h/Mpc

k2=2 k1

MaskNo Mask

1.0⋅109

2.0⋅109

3.0⋅109

4.0⋅109

5.0⋅109

6.0⋅109

0 0.2 0.4 0.6 0.8 1

0.102 0.097 0.083 0.060 0.032 0

θ12/π

k3 [h/Mpc]

k1=0.051 h/Mpck2= k1








5.0⋅107

1.0⋅108

1.5⋅108

2.0⋅108

2.5⋅108

3.0⋅108

3.5⋅108

0 0.2 0.4 0.6 0.8 1

0.300 0.287 0.250 0.194 0.133 0.100

B(θ

12)

[(M

pc/h

)6 ]

θ12/π

k3 [h/Mpc]

k1=0.1 h/Mpc

k2=2 k1

MaskNo Mask

3.0⋅108

6.0⋅108

9.0⋅108

1.2⋅109

1.5⋅109

1.8⋅109

0 0.2 0.4 0.6 0.8 1

0.200 0.190 0.162 0.118 0.062 0

θ12/π

k3 [h/Mpc]

k1=0.1 h/Mpc

k2= k1





2LPT haloesPower SpectrumBispectrum

Galaxy Mocks

There are 2 sets of mocks available for the BOSS collaboration.

2nd order Lagrangian Perturbation Theory Mocks (2LPT) byManera et al. 2012, arXiv.1203.6609

Quick Particle Mesh (QPM) Mocks by White et al. 2013,arXiv.1309.5532

In order to obtain the galaxy mocks,

Mock-Haloes are produced

They are populated with galaxies using some prescription, such asHOD

We can compare 2LPT haloes with N-body haloes predictions for thepower spectrum and bispectrum to check the validity of theapproximation.In order to have a similar biasing, we have to limit the mass o haloes to,

2LPT haloes: Mmin ' 5× 1012 M�/h

N-body haloes: Mmin ' 9× 1012 M�/h






2LPT Haloes: Power spectrum

Halo-halo power spectrum relative to the non-linear matter one.2LPT haloes: 20 realisations × 2.4 Gpc/hN-body haloes: 5 realisations × 1.5 Gpc/hSurvey volume ∼ 2 [Gpc/h]3

3.5

4

4.5

5

5.5

6

0.01 0.1

[P-P

nois

e] /

Pnl

k [h/Mpc]

2LPTHNbody






2LPT Haloes: Bispectrum

Halo-halo-halo bispectrum.2LPT haloes: 20 realisations × 2.4 Gpc/hN-body haloes: 5 realisations × 1.5 Gpc/hSurvey volume ∼ 2 [Gpc/h]3

2.0⋅109

3.0⋅109

4.0⋅109

5.0⋅109

6.0⋅109

7.0⋅109

8.0⋅109

0 0.2 0.4 0.6 0.8 1

0.063 0.060 0.052 0.041 0.028 0.021

B(θ

12)

[(M

pc/h

)6 ]

θ12/π

k3 [h/Mpc]

k1=0.021 h/Mpck2=2 k1

2LPT HaloesNbody Haloes

5.0⋅1091.0⋅10101.5⋅10102.0⋅10102.5⋅10103.0⋅10103.5⋅10104.0⋅1010

0 0.2 0.4 0.6 0.8 1

0.042 0.040 0.034 0.025 0.013 0

θ12/π

k3 [h/Mpc]

k1=0.021 h/Mpck2= k1








2.0⋅108

4.0⋅108

6.0⋅108

8.0⋅108

1.0⋅109

1.2⋅109

1.4⋅109

1.6⋅109

0 0.2 0.4 0.6 0.8 1

0.153 0.146 0.127 0.099 0.068 0.051

B(θ

12)

[(M

pc/h

)6 ]

θ12/π

k3 [h/Mpc]

k1=0.051 h/Mpc

k2=2 k1


1.0⋅109

2.0⋅109

3.0⋅109

4.0⋅109

5.0⋅109

6.0⋅109

0 0.2 0.4 0.6 0.8 1

0.102 0.097 0.083 0.060 0.032 0

θ12/π

k3 [h/Mpc]

k1=0.051 h/Mpck2= k1








5.0⋅107

1.0⋅108

1.5⋅108

2.0⋅108

2.5⋅108

3.0⋅108

3.5⋅108

0 0.2 0.4 0.6 0.8 1

0.300 0.287 0.250 0.194 0.133 0.100

B(θ

12)

[(M

pc/h

)6 ]

θ12/π

k3 [h/Mpc]

k1=0.1 h/Mpc

k2=2 k1


3.0⋅108

6.0⋅108

9.0⋅108

1.2⋅109

1.5⋅109

1.8⋅109

0 0.2 0.4 0.6 0.8 1

0.200 0.190 0.162 0.118 0.062 0

θ12/π

k3 [h/Mpc]

k1=0.1 h/Mpc

k2= k1





Bias FormalismKernelsMask EffectBest Triangle ConfigurationFitting FormulaBispectrum vs. Power spectrumGalaxy Mocks

Bias Formalism

We stay at the local bias model with two free parameters: b1 and b2.Bias model chosen

δh(x) = b1δ(x) +1

2b2[δ(x)2 − σ2].

δh(k) = b1δ(k) +1

2b2

∫dq

(2π)3δ(q)δ(k− q).

Power Spectrum,

Phh(k) ' b21P(k) + b1b2

∫dq

(2π)3B(k,q) + . . .

b2eff(k) ≡ Phh(k)

P(k)= b21 +

b1b2P(k)

∫dq

(2π)3B(k,q)






Bias Formalism

Bispectrum,

Bhhh(k1, k2, k3) = b31B(k1, k2, k3) +[b21b2P(k1)P(k2) + cyc.

]+ b22 terms

Tree level approximation,

Halo Bispectrum

Bhhh(k1, k2, k3) =[b31P(k1)P(k2)2F2(k1, k2) + b21b2P(k1)P(k2)

]+ cyc.

We write Bhhh as a function of Phh.

Bhhh(k1, k2, k3) =

[b31

b2eff(k1)b2eff(k2)2Phh(k1)Phh(k2)F2(k1, k2)+

+b21b2

b2eff(k1)b2eff(k2)Phh(k1)Phh(k2)

]+ cyc.

For simplicity we assume that beff is k-independent.Hector Gil-Marın Non-linear bias Workshop. Trieste Measuring the CMASS galaxy bias





Bias Formalism

Rewrite the mapping between Bhhh and Phh as a function of these newbias parameters: b′i .

Halo Bispectrum

Bhhh(k1, k2, k3) =

[2

b′1Phh(k1)Phh(k2)F2(k1, k2) +

b′2b′21

Phh(k1)Phh(k2)

]+cyc.

where the connection to the true bias parameters is,

b′1 ≡b4eff

b31, b′2 ≡ b2

b4eff

b41

if b2 very small, beff ' b1 and b′i ' bi . We could measure the biasparameter without any matter information.






Improved Kernel for dark matter

The behaviour of tree level approximation can be improved modifying the2-point kernel,

F2(ki , kj) =5

7+

1

2cos(θij)

(kikj

+kjki

)+

2

7cos2(θij),

F eff2 (ki , kj) =

5

7a(ni , ki )a(nj , kj) +

1

2cos(θij)

(kikj

+kjki

)b(ni , ki )b(nj , kj)

+2

7cos2(θij)c(ni , ki )c(nj , kj),

The functions a(n, k), b(n, k) and c(n, k) depend on 9 free parametersthat need to be calibrated from simulations. You can find their definitionand the values of these free parameters in HG-M et al. (2012)arXiv.1111.4477






Improved kernel for dark matter

N-body dark matter particles: 5 realisations × 1.5 Gpc/h.

F2(k1, k2) → SPT kernel

F eff2 (k1, k2) → Effective kernel

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.9 0.95 1 1.05 1.1

b 2

b1

k3<0.20

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.9 0.95 1 1.05 1.1

b 2

b1

k3<0.30






Mask Effect

2LPT Haloes. F eff2 used.

20 realizations × 2.4 Gpc/h (unmasked)Error-bars correspond to 1σ dispersion of 1 single realization.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1

b’2

b’1

k3<0.2 h/Mpc

no maskmask

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1

b’2

b’1

k3<0.3 h/Mpc






Best Triangle Configuration

2LPT Haloes. F eff2 used.

20 realizations × 2.4 Gpc/h.Error-bars correspond to 1σ dispersion of 1 single realization.

0.4

0.6

0.8

1

1.2

1.4

2.3 2.4 2.5 2.6 2.7 2.8

b’2

b’1

k3<0.2 h/Mpc

k2/k1=1

k2/k1=2

combined

0.4

0.6

0.8

1

1.2

1.4

2.3 2.4 2.5 2.6 2.7 2.8

b’2

b’1

k3<0.3 h/Mpc






Fitting Formula

F eff2 (ki , kj) =

5


1

2cos(θij)

(kikj

+kjki


+2


2.0⋅108

4.0⋅108

6.0⋅108

8.0⋅108

1.0⋅109

1.2⋅109

1.4⋅109

1.6⋅109

0 0.2 0.4 0.6 0.8 1

0.153 0.146 0.127 0.099 0.068 0.051

B(θ

12)

[(M

pc/h

)6 ]

θ12/π

k3 [h/Mpc]

k1=0.051 h/Mpc

k2=2 k1

Theoretical FormulaNbody Haloes

1.0⋅109

2.0⋅109

3.0⋅109

4.0⋅109

5.0⋅109

6.0⋅109

0 0.2 0.4 0.6 0.8 1

0.102 0.097 0.083 0.060 0.032 0

θ12/π

k3 [h/Mpc]

k1=0.051 h/Mpck2= k1






Fitting Formula

F eff2 (ki , kj) =

5


1

2cos(θij)

(kikj

+kjki


+2


5.0⋅107

1.0⋅108

1.5⋅108

2.0⋅108

2.5⋅108

3.0⋅108

0 0.2 0.4 0.6 0.8 1

0.300 0.287 0.250 0.194 0.133 0.100

B(θ

12)

[(M

pc/h

)6 ]

θ12/π

k3 [h/Mpc]

k1=0.1 h/Mpc

k2=2 k1

Theoretical FormulaNbody Haloes

3.0⋅108

6.0⋅108

9.0⋅108

1.2⋅109

1.5⋅109

1.8⋅109

0 0.2 0.4 0.6 0.8 1

0.200 0.190 0.162 0.118 0.062 0

θ12/π

k3 [h/Mpc]

k1=0.1 h/Mpc

k2= k1






Bispectrum vs. Power Spectrum

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.85 1.9 1.95 2 2.05 2.1

b 2

b1

k3<0.30

Eff. kernel NbodyH

SPT kernel NbodyH

Eff. kernel 2LPTH

SPT kernel 2LPTH

2

2.02

2.04

2.06

2.08

2.1

2.12

2.14

0 0.05 0.1 0.15 0.2 0.25 0.3

b eff

k [h/Mpc]

R=13 Mpc/hkmax=0.16 h/Mpc

R=14.5 Mpc/hkmax=0.14 h/Mpc

SPT: b1 ' 1.92 b2 ' 0.7Eff. : b1 ' 1.92 b2 ' 0.55

b2eff(k) = b21 +b1b2P(k)

∫dqB(k,q)

B(k1, k2) = 2P(k1)P(k2)F2(k1, k2) + cyc.

Smoothing scale problem.






Galaxy Mocks

Following a HOD prescription galaxies are added into the haloes,

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0.01 0.1

b2 eff /

b2 1*

k [h/Mpc]

mocks NGCmocks SGC

2LPT Haloes

0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

0.9 0.95 1 1.05 1.1

b 2/b

2*

b1/b1*

k3<0.3 h/Mpc

Main effects,

beff is reducedb1 is reduced but b2 increases!





Non-linear bias in z-space

Map the real space kernels into z-space kernels based on perturbationtheory,

b1F1 → Z1(k) ≡ (b1 + f µ2)

b1F2(k1, k2) +b22→ Z2(k1, k2) ≡ b1F2(k1, k2) + f µ2G2(k1, k2) +

+f µk

2

[µ1

k1(b1 + f µ2

2) +µ2

k2(b1 + f µ2

1)

]+

b22.

Bshhh(k1, k2) = 2P(k1)Z1(k1)P(k2)Z1(k2)Z2(k1, k2) + cyc.

Work in progress...





Conclusions

1 The bispectrum is the next natural step after the power spectrum toextract information from the large scale galaxy distribution

2 The effects of the mask have to be modeled in order to avoidspurious signals at large scales.

3 2LPT mocks seem to show good behaviour up to small scales,k . 0.3

4 Since b2 is not small, we need to assume an underlying Pmm toextract b1 / b2.

5 Effective kernel need to be used to avoid spurious signals.

6 Stay tuned for upcoming data results!


Documents

Measuring the CMASS galaxy bias using the bispectrum techniqueindico.ictp.it/event/a12214/session/12/contribution/8/... · 2014. 5. 5. · Measuring the CMASS galaxy bias using the