The power spectrum and bispectrum of the CMASS BOSS galaxiesbenasque.org/2014cosmology/talks_contr/136_Gil.pdf · 2014. 8. 18. · CMASS BOSS galaxies H ector Gil Mar n (ICG, University

IntroductionRedshift Space distortions

BispectrumMeasurements

Conclusions

The power spectrum and bispectrum of theCMASS BOSS galaxies

Hector Gil Marın (ICG, University of Portsmouth)..

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

.arXiv:1407.1836, arxiv:1407.5668, & arxiv:1408:0027

Modern Cosmology, Benasque 13th August 2014

Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies



Conclusions

Outline

i Introduction

ii Redshift-Space Distortions

iii Bispectrum

iv Conclusions




Conclusions

The BOSS surveyStatistical momentsGalaxy Bias

Introduction: the BOSS survey

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

CMASS BOSS Galaxies: LRGs.

0.43 ≤ z ≤ 0.70

∼ 7 · 105 galaxies

Volume of 6Gpc3

10.000 deg2 area




Conclusions


Introduction: the BOSS survey

CMASS sample with zeff = 0.57.

Anderson et al. (2013)Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies



Conclusions


Introduction: Statistical moments

1 The power spectrum is the Fourier transform of the 2-pointfunction.

〈δk1δk2〉 = (2π)3P(k1)δD(k1 + k2)

It contains information about the clustering.

2 The bispectrum is the Fourier transform of the 3-point function.

〈δk1δk2δk3〉 = (2π)3B(k1, k2)δD(k1 + k2 + k3)

It essentially contains information about the non-Gaussianities:primordial + gravitationally inducedSince is gravitationally sensible → Test of GRIt is essential to break the typical degeneracies between biasparameters, σ8 and f .




Conclusions


Introduction: Galaxy Bias

Galaxies are a biased tracers of dark matter.

Eulerian linear bias model,

δg (x) = b1δ(x)

Eulerian non-linear, local bias model,

δg (x) =∑i

bii !

(δi (x)− σi )

Eulerian non-local bias model,

δg (x) = b1δ(x) +1

2b2[δ(x)2] +

1

2bs2 [s(x)2]

where s(x) is the tidal tensorHector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies



Conclusions


Introduction: Galaxy Bias

Galaxies are a biased tracers of dark matter.

Eulerian linear bias model,

δg (x) = b1δ(x)

Eulerian non-linear, local bias model,

δg (x) =∑i

bii !

(δi (x)− σi )

Eulerian non-local bias model (local in Lagrangian space),

δg (x) = b1δ(x) +1

2b2[δ(x)2] +

1

2[4

7(1− b1)][s(x)2]

where s(x) is the tidal tensorHector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies



Conclusions

IntroductionLinear order

Redshift space distortions: Introduction




Conclusions


Redshift space distortions: Introduction

Redshift space Real space




Conclusions


Redshift space distortions: Linear order

For the power spectrum, the linear term can be modelled analytically(Kaiser 1984),

P(s)g (k , µ) =

[b1 + f µ2

]2σ28Plin(k)

P(s)(k, µ) can be expressed in the Legendre polynomial basis, P(`),defined as,

P(`)g (k) =

(2`+ 1)

2

∫ 1

−1

dµP(s)g (k , µ)L`(µ),




Conclusions


Redshift space distortions: Linear order

where L` are the Legendre polynomials of order `.

P(0)g (k) = Plin(k)σ2

8

(b21 +

2

3fb1 +

1

5f 2)

Monopole

P(2)g (k) = Plin(k)σ2

8

(4

3fb1 +

4

5f 2)

Quadrupole

Measuring the amplitude of P(0)g and P

(2)g at large scales respect to Plin,

b1σ8 and f σ8 can be inferred.——————Following this idea, but with more complex modelling for non-linearscales, f σ8 has been measured from the DR11 CMASS BOSS galaxies:Beutler et al. (2013), Chuang et al. (2013), Sanchez et al. (2013),Samushia et al. (2013)




Conclusions


But 1− 2σ tension with the CMB...

Macaulay, Wehus & Eriksen (2013)Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies



Conclusions

HistoryTree-level bispectrumEstimating the parameters

Bispectrum: History

Previous measurements of the bispectrum or 3-PCF in spectroscopicgalaxy surveys,

1982, CfA Redshift Survey (∼ 1, 000 galaxies)[Baumgart & Fry (1991)]

1995, APM survey (∼ 1.3 · 106 galaxies)[Frieman & Gaztanaga (1999)]

1995, IRAS - PSCz (∼ 15, 000 galaxies)[Feldman et al. (2001), Scoccimarro et al. (2001)]

2002, 2dFGRS (∼ 1.3 · 105 galaxies)[Verde el al. (2002)]

2013, WiggleZ (∼ 2 · 105 galaxies)[Marın et al. 2013]

2014 SDSS-III (DR11 BOSS-CMASS) (∼ 7 · 105 galaxies)[HGM et al. 2014b, 2014c ]




Conclusions


Bispectrum: Tree-level

We can model the bias using perturbation theory.In real space

Bg (k1, k2) = σ48b

41

2Plin(k1) Plin(k2)

[1

b1F2(k1, k2) +

b22b21

+2

7b21(1− b1)S2(k1, k2)

]+ cyc.

,

and in redshift space

Bsg (k1, k2) = σ4

8 [2Plin(k1)Z1(k1)Plin(k2)Z1(k2)Z2(k1, k2) + cyc.] .

Z1(ki ) ≡ (b1 + f µ2i )

Z2(k1, k2) ≡ b1

[F2(k1, k2) +

f µk

2

(µ1

k1+µ2

k2

)]+ f µ2G2(k1, k2) +

+f 3µk

2µ1µ2

(µ2

k1+µ1

k2

)+

b22

+2

7(1− b1)S2(k1, k2)




Conclusions



Bispectrum monopole,

B(0)g (k1, k2) =

∫dµ1dµ2B

(s)g (k1, k2) .

B(0)g (k1, k2) = Plin(k1)Plin(k2)b41σ

48

1

b1F2(k1, k2, cos θ12)D(0)

SQ1

+1

b1G2(k1, k2, cos θ12)D(0)

SQ2

+

[b2b21

+bs2

b21S2(k1, k2)

]D(0)

NLB +D(0)FoG

+ cyc.

Scoccimarro et al. (1999)




Conclusions



Bispectrum monopole (β ≡ f /b1; x12 ≡ cos(θ12); y12 ≡ k1/k2),

D(0)SQ1 =

2(15 + 10β + β2 + 2β2x2)

15, (1)

D(0)SQ2 = 2β

(35y2

12 + 28βy212 + 3β2y2

12 + 35 + 28β+ (2)

+ 3β2 + 70y12x12 + 84βy12x12 + 18β2y12x12 + 14βy212x

212 + 12β2y2

12x212 +

+ +14βx212 + 12β2x212 + 12β2y12x312

)/[105(1 + y2

12 + 2x12y12)],

D(0)NLB =

(15 + 10β + β2 + 2β2x2)

15, (3)

D(0)FoG = β

(210 + 210β + 54β2 + 6β3 + 105y12x + 189βy12x12+ (4)

+ 99β2y12x12 + 15β3y12x12 + 105y−112 x12 + 189βy−1

12 x + 99β2y−112 x12 + 15β3y−1

12 x12 +

+ 168βx212 + 216β2x212 + 48β3x212 + 36β2y12x312 + 20β3y−1

12 x312 +

+ 36β2y−112 x312 + 20β3y12x

312 + 16β3x412

)/315,




Conclusions


Bispectrum: Beyond tree-level

Tree level only provides an accurate description at large scales and athigh redshifts.Empirical improvement of this formula through effective kernels method(Scoccimarro & Couchman (2001))

F2 → F eff2 (HGM et al. 2012) [arXiv:1111.4477]

G2 → G eff2 (HGM et al. 2014a) [arXiv:1407.1836]

9 free parameters each kernel to be fitted from dark matter N-bodysimulations. Independent of scale or redshift, weakly dependent withcosmology.




Conclusions


Bispectrum: Estimating the parameters

The PS and BS models we considered here have 7 free independentparameters:

The bias parameters: b1, b2 (local Lagrangian bias,bs2 = −4/7[b1 − 1]),

Dark matter power spectrum amplitude, σ28 : Plin(k)→ σ2

8Plin(k)

Growth rate of structure f = d log δd log a

Fingers of God damping functions: σPfog , σB

fog

Shot Noise term amplitude term, Anoise: Anoise > 0 (sub-Poisson);Anoise < 0 (super-Poisson).




Conclusions


Bispectrum Estimating the parameters

Estimation of the best-fit parameters, Ψ, and their error.

χ2diag.(Ψ) =

∑k−bins

[Pmeas.(i) (k)− Pmodel(k ,Ψ; Ω)

]2σP(k)2

+

+∑

triangles

[Bmeas.(i) (k1, k2, k3)− Bmodel(k1, k2, k3,Ψ; Ω)

]2σB(k1, k2, k3)2

,

〈Ψi〉 is a non-optimal and unbiased estimator of Ψtrue, (see Verde etal. 2001)

Ψtrue ' 〈Ψi〉 ±√〈Ψ2

i 〉 − 〈Ψi〉2

1σ-error is given by the dispersion of mocks around to their mean.




Conclusions

DegenerationsPower Spectrum MonopoleBispectrum MonopoleReduced BispectrumDependence with the minimum scaleBreaking f and σ8 degeneracy

Measurements: Degenerations

Power Spectrum Monopole + Bispectrum Monopole.600 Mocks based on pt-haloes (Manera et al. 2013) at zeff = 0.57.1σ contours from the mocks density of pointsData from NGC CMASS BOSS galaxies

-1

-0.5

0

0.5

Log 1

0[ f

]

-0.4

-0.2

0

Log 1

0[ σ

8 ]

-3

-2

-1

0

1

0.1 0.2 0.3 0.4 0.5

Log 1

0[ b

2 ]

Log10[ b1 ]

kmax=0.17 h/Mpc

-1 -0.5 0 0.5Log10[ f ]

-0.4 -0.2 0Log10[ σ8 ]

0.3

0.4

0.5

0.6

0.7

0.8

f0.43

σ 8

kmax=0.17 h/Mpc

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.3 1.4 1.5 1.6 1.7 1.8 1.9

b 20.

30σ 8

b11.40σ8

0.3 0.4 0.5 0.6 0.7 0.8

f0.43σ8

HGM et al. 2014b [arxiv:1407.5668]Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies



Conclusions


Measurements: Power Spectrum Monopole

0.85 0.9

0.95 1

1.05 1.1

1.15

0.01 0.1 0.2

Pda

ta /

Pm

odel

k [h/Mpc]

3.5

4

4.5

5

5.5

P /

Pnw

1⋅104

1⋅105P

(k)

[(M

pc/h

)3 ]

HGM et al. 2014b [arxiv:1407.5668]Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies



Conclusions


Measurements: Bispectrum Monopole

0⋅100

2⋅109

4⋅109

6⋅109

0.04 0.06 0.08 0.10

B(k

3) [(

Mpc

/h)6 ]

k3 [h/Mpc]

k1=0.051 h/Mpc k2=k1

0⋅100

1⋅109

2⋅109

3⋅109

4⋅109

0.04 0.06 0.08 0.10 0.12 0.14k3 [h/Mpc]

k1=0.0745 h/Mpc k2=k1

0⋅100

1⋅109

2⋅109

0.04 0.08 0.12 0.16k3 [h/Mpc]

k1=0.09 h/Mpc k2=k1

0⋅100

1⋅109

2⋅109

3⋅109

0.06 0.08 0.10 0.12 0.14

B(k

3) [(

Mpc

/h)6 ]

k3 [h/Mpc]

k1=0.051 h/Mpc k2=2k1

0.0⋅100

5.0⋅108

1.0⋅109

1.5⋅109

0.10 0.14 0.18 0.22k3 [h/Mpc]

k1=0.0745 h/Mpc k2=2k1

0⋅100

3⋅108

6⋅108

9⋅108

0.10 0.14 0.18 0.22 0.26k3 [h/Mpc]

k1=0.09 h/Mpc k2=2k1

HGM et al. 2014b [arxiv:1407.5668]




Conclusions


Measurements: Reduced Bispectrum

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

Q(θ

12)

θ12 / π

k1=0.051 h/Mpc k2=k1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1θ12 / π

k1=0.0745 h/Mpc k2=k1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1θ12 / π

k1=0.09 h/Mpc k2=k1

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

Q(θ

12)

θ12 / π

k1=0.051 h/Mpc k2=2k1

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1θ12 / π

k1=0.0745 h/Mpc k2=2k1

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1θ12 / π

k1=0.09 h/Mpc k2=2k1

HGM et al. 2014b [arxiv:1407.5668]




Conclusions


Measurements: Dependence with the scale

NGC, SGC, NGC+SGC

1.4 1.5 1.6 1.7 1.8 1.9

2

0.14 0.16 0.18 0.20

b 11.

40 σ

8

kmax [h/Mpc]

0.4

0.5

0.6

0.7

0.8

b 20.

30 σ

8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

f0.43

σ8

0 2 4 6 8

10 12

0.14 0.16 0.18 0.20

σ FoG

P

kmax [h/Mpc]

0 10 20 30 40 50

σ FoG

B

-0.8-0.6-0.4-0.2

0 0.2 0.4 0.6 0.8

Ano

ise

f 0.43σ8|z=0.57 = 0.582± 0.084 (14%) (kmax = 0.17 hMpc−1).f 0.43σ8|z=0.57 = 0.584± 0.051 (9%) (kmax = 0.20 hMpc−1).




Conclusions


Measurements: Breaking f and σ8 degeneracy

For constraining f and σ8 alone we need information from P(0), P(2)

and B(0),

Naively we combine “a posteriori” the measurements on f 0.43σ8 withf σ8 measurements (Samushia et al. 2013)

0.2

0.4

0.6

0.8

1

1.2

1.4

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

σ 8(z

eff)

f(zeff)

mocks

data

HGM et al. 2014c [arxiv:1408:0027]

f σ8|z=0.57 = 0.447± 0.028

f 0.43σ8|z=0.57 = 0.582± 0.084

—————————————–

f (z = 0.57) = 0.63± 0.16 (25%)

σ8(z = 0.57) = 0.710±0.086 (12%)

Results to be improved when thecombination is “a priori”




Conclusions

Conclusions

This methodology is non-optimal. The errors can be reduced by,

Including covariance will automatically shrink the error-bars

Including more triangles will also reduce the error-bars.

Therefore, there is still space for improvement. . .




Conclusions

Conclusions

We have combined the power spectrum monopole with thebispectrum monopole to set constrains in the cosmologicalparameters.

Using the galaxy mocks we have determined that b1.40σ8 andf 0.43σ8 are the parameters less affected by degenerations.

The results on f 0.43σ8 are robust under changes in the minimumscale used for the fit.

Combining f 0.43σ8 measurements with f σ8 measurements from thesame galaxy sample, f and σ8 can be estimated separately.




Conclusions

Conclusions

Thank you for your attention!


Documents

The power spectrum and bispectrum of the CMASS BOSS galaxiesbenasque.org/2014cosmology/talks_contr/136_Gil.pdf · 2014. 8. 18. · CMASS BOSS galaxies H ector Gil Mar n (ICG, University