Introduction Redshift Space distortions Bispectrum Measurements Conclusions The power spectrum and bispectrum of the CMASS BOSS galaxies H´ ector Gil Mar´ ın (ICG, University of Portsmouth) . . In collaboration with: W. Percival, L. Verde, J. Nore˜ na, M. Manera & C. Wagner . arXiv:1407.1836, arxiv:1407.5668,& arxiv:1408:0027 Modern Cosmology, Benasque 13th August 2014 H´ ector Gil-Mar´ ın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
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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
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
Outline
i Introduction
ii Redshift-Space Distortions
iii Bispectrum
iv Conclusions
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
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
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
The BOSS surveyStatistical momentsGalaxy Bias
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
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
The BOSS surveyStatistical momentsGalaxy Bias
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 .
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
The BOSS surveyStatistical momentsGalaxy Bias
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
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
The BOSS surveyStatistical momentsGalaxy Bias
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
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
IntroductionLinear order
Redshift space distortions: Introduction
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
IntroductionLinear order
Redshift space distortions: Introduction
Redshift space Real space
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
IntroductionLinear order
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),
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
IntroductionLinear order
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)
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
IntroductionLinear order
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
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
HistoryTree-level bispectrumEstimating the parameters
Bispectrum: History
Previous measurements of the bispectrum or 3-PCF in spectroscopicgalaxy surveys,
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
HistoryTree-level bispectrumEstimating the parameters
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.
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
HistoryTree-level bispectrumEstimating the parameters
Bispectrum: Estimating the parameters
The PS and BS models we considered here have 7 free independentparameters:
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
HistoryTree-level bispectrumEstimating the parameters
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.
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
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
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
DegenerationsPower Spectrum MonopoleBispectrum MonopoleReduced BispectrumDependence with the minimum scaleBreaking f and σ8 degeneracy
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
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
DegenerationsPower Spectrum MonopoleBispectrum MonopoleReduced BispectrumDependence with the minimum scaleBreaking f and σ8 degeneracy
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]
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
DegenerationsPower Spectrum MonopoleBispectrum MonopoleReduced BispectrumDependence with the minimum scaleBreaking f and σ8 degeneracy
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]
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
DegenerationsPower Spectrum MonopoleBispectrum MonopoleReduced BispectrumDependence with the minimum scaleBreaking f and σ8 degeneracy