Post on 01-Mar-2021
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
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
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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
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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,
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 ]
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: 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)
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: Tree-level
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)
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: Tree-level
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,
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:
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).
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
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).
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: 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”
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
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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. . .
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
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.
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies
IntroductionRedshift Space distortions
BispectrumMeasurements
Conclusions
Conclusions
Thank you for your attention!
Hector Gil-Marın Modern Cosmology, Benasque The power spectrum and bispectrum of the CMASS BOSS galaxies