The power spectrum and bispectrum of the CMASS BOSS galaxieskomatsu/meetings/lss... · 2015. 8. 8. · CMASS BOSS Galaxies: LRGs. 0:43 z 0:70 ˘7 105 galaxies Volume of 6Gpc3 10.000

IntroductionMeasurements

ModellingCosmological Parameters

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,L. Samushia, M. Manera & C. Wagner

.2014JCAP...12..029G

2015MNRAS.451.5058GarXiv:1408:0027

Theoretical and Observational Progress on LSS of the UniverseHector Gil-Marın Theoretical and Observational Progress on LSS The power spectrum and bispectrum of the CMASS BOSS galaxies



Conclusions

The BOSS surveyHistoryStatistical moments

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

Hector Gil-Marın Theoretical and Observational Progress on LSS The power spectrum and bispectrum of the CMASS BOSS galaxies



Conclusions


Introduction: the BOSS survey

CMASS sample with zeff = 0.57.

Anderson et al. (2013)




Conclusions


Introduction: Historical bispectrum measurements

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]

2015 SDSS-III (DR11 BOSS-CMASS) (∼ 7 · 105 galaxies)[HGM et al. 2015a, 2015b ]




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

Power Spectrum MonopoleBispectrum MonopoleReduced Bispectrum

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⋅105

P(k

) [(

Mpc

/h)3 ]

HGM et al. 2015aHector Gil-Marın Theoretical and Observational Progress on LSS 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. 2015a




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. 2015a




Conclusions

Galaxy BiasRedshift Space distortionsEstimating the parameters

Galaxy Bias

Galaxies are a biased tracers of dark matter.We chose a non-linear and non-local biasmodel,

δg (x) = b1δ(x) + 12b2[δ(x)2] + 1

2bs2 [s(x)2]

We choose that the bias is local in Lagrangian space,

δg (x) = b1δ(x) +1

2b2[δ(x)2] +

1

2[4

7(1− b1)][s(x)2]

which is in agreement with the synthetic halo and galaxy catalogues forthe power spectrum and bispectrum.




Conclusions


RSD: Power Spectrum

The Kaiser (linear order) prediction for the power spectrum multipoles is,

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.In our analysis we use more complex model,

Real space matter power spectrum is modelled through 2-loop RPT(HGM et al. 2012)

Redsshift space distortions for the power spectrum are modelledusing TNS model (Taruya, Nishimichi & Saito 2010)




Conclusions


RSD: Bispectrum

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

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

B(s)g (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


RSD: Bispectrum

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


RSD: Bispectrum

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

D(0)SQ1 =

2(15 + 10β + β2 + 2β2x212)

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 + 2β2x212)

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


RSD: Bispectrum

Tree level (already very complex!) only provides an accurate descriptionat large scales and at high redshifts.Empirical improvement of this formula through effective kernels method(Scoccimarro & Couchman (2001))

F2 → F eff2 (HGM et al. 2012)

G2 → G eff2 (HGM et al. 2014)

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




Conclusions


Estimating the parameters

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

The bias parameters: b1, b2

Dark matter power spectrum amplitude, σ28

Growth rate of structure f = d log δd log a

Fingers of God damping functions: σPfog , σB

fog

Shot Noise term amplitude term, Anoise




Conclusions


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

Cosmological paramers: f vs. σ8Dependence with the minimum scaleBreaking f and σ8 degeneracyComparison with CMASS DR11 fσ8 measurements

Cosmological paramers: f vs. σ8

Power Spectrum Monopole + Bispectrum Monopole.600 Mocks based on PTHalos (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. 2015a




Conclusions



-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 ]




Conclusions



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

f0.43σ8Hector Gil-Marın Theoretical and Observational Progress on LSS The power spectrum and bispectrum of the CMASS BOSS galaxies



Conclusions


Measurements: Dependence with the scale

No significant dependence with minimum scale used up tokmax = 0.17 hMpc−1,

1.5 1.6 1.7 1.8 1.9

2 2.1

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 Conservative cutoff at

kmax = 0.17 hMpc−1,

f 0.43σ8|z=0.57 = 0.582± 0.084b1.401 σ8|z=0.57 = 1.672± 0.060b0.302 σ8|z=0.57 = 0.579± 0.082




Conclusions


Measurements: Breaking f and σ8 degeneracy

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

and B(0),

We combine “a posteriori” the measurements on f 0.43σ8 with f σ8measurements (Samushia et al. 2013)

0.4

0.5

0.6

0.7

0.8

0.9

1

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. 2015b

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


Comparison with other CMASS DR11 measurements

Assuming a f planck = 0.777, we can project f 0.43σ8 bispectrum resultinto f σ8 plane to compare with P(0) + P(2) DR11 CMASS results:[f σ8]est. ≡ [f 0.43σ8]f 0.57Planck

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Chuang et al. ’13

Beutler et al. ’14

Samushia et al. ’14

Sanchez et al ’14

Reid et al. ’14

Alam et al ’15

Gil-Marin ’15

fσ8

CMASS sample

[f σ8]est. = 0.504± 0.069Hector Gil-Marın Theoretical and Observational Progress on LSS The power spectrum and bispectrum of the CMASS BOSS galaxies



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 b1.401 σ8, b0.302 σ8and f 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

Future work for DR12 sample,

Include full covariance of P and B

Include power spectrum quadrupole and perform a full fitP(0) + P(2) + B(0) in order to constrain f and σ8.

Improve modelling for RSD in the bispectrum.




Conclusions

Conclusions

Thank you for your attention!


Documents

The power spectrum and bispectrum of the CMASS BOSS galaxieskomatsu/meetings/lss... · 2015. 8. 8. · CMASS BOSS Galaxies: LRGs. 0:43 z 0:70 ˘7 105 galaxies Volume of 6Gpc3 10.000