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IntroductionPower Spectrum
BispectrumConclusions
Constraining cosmological parameters using BOSSdata: Power Spectrum and Bispectrum
Hector Gil Marın
Laboratoire de Physique Nucleaire et de Hautes Energies
Rencontres de Moriond 20th March 2016
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Outline
1 Introduction
2 Constrains from the BOSS DR12 Power Spectrum multipoles[arXiv:1509.06386] and [arXiv:1509:06373]
3 Constrains from the BOSS DR12 Bispectrum(work in progress)
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
The BOSS surveyAnisotropy
Introduction: the BOSS survey
Apache Point Observatory (APO) 2.5-m telescope for fiveyears from 2009-2014.
Part of SDSS-III project. BOSS: Baryon OscillationSpectroscopic Survey
Total coverage area about 10,000 square degrees
Two samples with different target selection: LOWZ andCMASS
Two patches in the Northern and Southern Hemisphere
BOSS Galaxies: LRGs.
0.15 ≤ z ≤ 0.43 LOWZ
0.43 ≤ z ≤ 0.70 CMASS
∼ 7 · 105 galaxies in 6Gpc3
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
The BOSS surveyAnisotropy
LOWZ and CMASS samples
0
1
2
3
4
5
6
7
0.2 0.3 0.4 0.5 0.6 0.7
n(z)
x 1
04 [Mpc
h-1]3
z
LOWZ sample
CMASS sample
The two redshift bins havedifferent target selection.
We treat them as independentgalaxy population with differentgalaxy bias properties.
CMASS has ∼ 3 times the volume that LOWZ
For simplicity we only present the results for CMASS.
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
The BOSS surveyAnisotropy
Anisotropies: RSD + AP
General picture tell us (at sufficiently large scales) there is not aprivileged observer (Copernican principle). Therefore, the Universewe observe from Earth (or any other random point) must look likeisotrope and homogeneous at large scales.However there are two phenomenon that ’break’ this principle inthe observed data. Both of them relies on how the co-movingdistances are estimated:
1 Redshift space distortions (RSD)
2 Alcock-Paczynski (AP)
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
The BOSS surveyAnisotropy
Anisotropies: RSD
Kaiser approximation (only for large scales)
P(0) = P(k)[(bσ8)2 +2
3(f σ8)(bσ8) +
1
5(f σ8)2]
P(2) = P(k)[4
3(bσ8)(f σ8) +
4
7(f σ8)2]
but for smaller scales, more complex model is needed.Kaiser (1987)
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
The BOSS surveyAnisotropy
Anisotropies: AP
An extra source of anisotropy comes from AP-effect, Theco-moving distance to an observed galaxy is given by,
dcomov(z) =
∫ z
0
cdz ′
H(z ′)
where, H(z) = H0
√Ωm(1 + z)3 − Ωm + 1.
By assuming a wrong cosmological model (Ωm) we change theline-of-sight clustering respect to the angular clustering, creating ameasurable anisotropy: constrains H(z) and DA(z)→BAO!Alcock & Paczynski (1979)
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
ParametrizationMeasurements
RSD Constrains from the Power Spectrum
Goal
To get constrains on f σ8, DA, and H using the shape and theamplitude of P(0) and P(2) in redshift space.
Issues
Galaxy bias may be non-local, non-linear and stochastic
RSD small scale modelling
Dark matter small scales modelling: matter and velocity-fields
Need to include AP parameter to account for inaccuracieswhen assuming Ωm
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
ParametrizationMeasurements
Galaxy Bias
Galaxies are a biased tracers of DM.
The bias model can be as complex as we want: linear bias,non-linear bias, non-local bias,....
Eulerian non-local bias model (local in Lagrangian space),
δg (x) = b1δ(x)︸ ︷︷ ︸linear Eulerian
+
non-linear Eulerian︷ ︸︸ ︷1
2b2[δ(x)2] +
1
2[
47(1−b1)︷︸︸︷bs2 ][
tidal tensor︷ ︸︸ ︷s(x)2 ]︸ ︷︷ ︸
non-local Eulerian
McDonald & Roy 2009
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
ParametrizationMeasurements
Power Spectrum model
This scheme is applied the real space galaxy power spectrum(Beutler et al 2014)
We use the TNS model for describing the redshift space powerspectrum (Taruya, Nishimichi & Saito 2010),
P(s)g (k , µ) = DP
FoG(k , µ, σPFoG[z ])[Pg ,δδ(k) + 2f µ2Pg ,δθ(k)
+f 2µ4Pθθ(k) + b31A(k, µ, f /b1) + b41B(k, µ, f /b1)]
Pδδ, Pδθ, Pθθ → Dark Matter non-linear models (2-loop RPT)DPFoG → 1-parameter Lorentzian damping term
A,B → TNS functions
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
ParametrizationMeasurements
Parametrization
The total number of free parameters of the model is,
Bias Parameters: b1, b2
Cosmology Parameters: f , σ8
AP parameters: α‖, α⊥
Fingers-of-God: σFoG
Shot noise amplitude ANoise
8 free parameters.
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
ParametrizationMeasurements
Parametrization
The total number of free parameters of the model is,
Bias Parameters: b1, b2
Cosmology Parameters: f , σ8
AP parameters: α‖, α⊥→ DA/rs(zd) and H(z)rs(zd)
Fingers-of-God: σFoG
Shot noise amplitude ANoise
8 free parameters → 4 cosmological, 4 “nuissance”
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
ParametrizationMeasurements
Measurements
0.7 0.8 0.9
1 1.1 1.2
0.01 0.1 0.2
Pda
ta /
Pm
odel
k [h/Mpc]
103
104
105
P(l)
(k)
[(M
pc/h
)3 ]
CMASS (zeff=0.57)
Monopole
Quadrupole
8.7
9
9.3
9.6
9.9
CMASS sample (z=0.57)
DA(z
)/r s
(zd)
P(0)+P(2); kmax=0.23 hMpc-1
12
12.5
13
13.5
14
14.5
15
0.3 0.4 0.5H
(z)r
s(z d
) [1
03 km
s-1]
f(z)σ8(z) 8.7 9 9.3 9.6 9.9
DA(z)/rs(zd)
f σ8(zcmass) = 0.444± 0.038 [f σ8|no−AP = 0.438± 0.021]
H(zcmass)rs(zd) = (13.92± 0.44)× 103kms−1
DA(zcmass)/rs(zd) = 9.42± 0.15
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
ParametrizationMeasurements
Measurements
We compare the results with other BOSS papers / other redshiftsurveys
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
This work
fσ8
CMASS sample
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
fσ8
z
6DFGS
SDSS MGS
SDSS LRG
WiggleZLOWZ
CMASS
VIPERS
Planck + γ=0.420Planck + γ=0.545Planck + γ=0.680
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
ParametrizationMeasurements
What’s next?
But....
The 2-pt function / Power spectrum is not the end of thestory....
Gravitational evolution of galaxies generates a non-Gaussiancomponent. The full information is spread along higher ordercorrelation function: 3-point, 4-point, ... correlation functions.
Using the 3-pt function / bispectrum in combination with thepower spectrum we can improve the constrains on theseparameters.
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Statistical momentsMeasurementConstrains
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-pointfunction.
〈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 .
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Statistical momentsMeasurementConstrains
Measurement
-4-3-2-1 0 1 2 3 4
0 50 100 150 200 250 300
∆B /
σ B
Triangle Index
0.4 0.6 0.8
1 1.2 1.4 1.6
Bda
ta/B
mod
el
108
109
B [M
pc/h
]6
CMASS sample (zeff=0.57)
Equilateral
Isosceles
Scalene Bins of ∆k = 6kf
' 350 triangles fork < 0.17h/Mpc
' 600 triangles fork < 0.22h/Mpc
Full Covariance from mocks
SPT + eff Kernels (G-M et al2014) for Bispectrum modelling
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Statistical momentsMeasurementConstrains
Comparing P vs. P + B
How much statistical signal do we gain by adding the Bispectrum?The answer depends on the scales.
Large Scales: few triangle allowed, good modelling
Small Scales: lots of triangles, poor modelling.
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23
σ fσ 8
[%]
kmax [hMpc-1]
CMASS sample (zeff=0.57)
P(0) + P(2)
P(0) + P(2) + B(0)
Up to kmax = 0.16 no gain on f σ8.
For kmax = 0.22 hMpc−1 Gain of 2
Systematics of the model grow with kmax.
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Statistical momentsMeasurementConstrains
Constrains at k = 0.17 hMpc−1
8.7
9
9.3
9.6
DA(z
)/r s
(zd)
CMASS sample (z=0.57)
12 12.5
13 13.5
14 14.5
H(z
)rs(
z d)
[103 k
ms-1
]
0.5
0.6
0.7
0.8
0.4 0.6 0.8 1
σ 8(z
)
f(z)
8.7 9 9.3 9.6DA(z)/rs(zd)
12 13 14
H(z)rs(zd) [103 kms-1]
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Statistical momentsMeasurementConstrains
Comparing P vs. P + B
8.7
9
9.3
9.6
9.9
CMASS sample (z=0.57)
DA(z
)/r s
(zd)
P(0)+P(2); kmax=0.23 hMpc-1
P(0)+P(2)+B(0); kmax=0.17 hMpc-1
12
12.5
13
13.5
14
14.5
15
0.3 0.4 0.5
H(z
)rs(
z d)
[103 k
ms-1
]
f(z)σ8(z) 8.7 9 9.3 9.6 9.9
DA(z)/rs(zd)
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
σ 8(z
eff)
f(zeff)
CMASS sample
DR11 data
DR12 data
Planck15 data
Agreement between P and P+B
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Statistical momentsMeasurementConstrains
Comparing P vs. P + B
8.7
9
9.3
9.6
9.9
CMASS sample (z=0.57)
DA(z
)/r s
(zd)
P(0)+P(2); kmax=0.23 hMpc-1
P(0)+P(2)+B(0); kmax=0.22 hMpc-1
12
12.5
13
13.5
14
14.5
15
0.3 0.4 0.5
H(z
)rs(
z d)
[103 k
ms-1
]
f(z)σ8(z) 8.7 9 9.3 9.6 9.9
DA(z)/rs(zd)
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
σ 8(z
eff)
f(zeff)
CMASS sample
DR11 data
DR12 data
Planck15 data
Agreement between P and P+B
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Conclusions
We have measured the power spectrum monopole (isotropic)and quadrupole (anisotropic) signal from the BOSS DR12galaxies.
We have constrained cosmological parameters: the growth ofstructure of the Universe, f σ8, Hubble Parameter H(z) andthe angular diameter distance DA.
We have shown how by adding the bispectrum signal on thetop of the power spectrum multipoles, we can reducesignificantly the statistical error bars and break the degeneracybetween f and σ8.
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Extra slides
0.85
0.9
0.95
1
1.05
1.1
1.15
0 100 200 300 400 500 600 700 800
B /
Bno
sys
Triangle index
CMASS sample
ki<0.03 hMpc-1
ki>0.17 hMpc-1
k1>0.03 hMpc-1 & k3<0.17 hMpc-1
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Extra slides
[rij]CMASS
0 50 100 150 200 250 300 350
0
50
100
150
200
250
300
350
-1
-0.5
0
0.5
1
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Extra slides
-2-1 0 1 2 3 4 5
0 50 100 150 200 250 300
∆B /
σ B
Triangle Index
108
109
B [M
pc/h
]6
CMASS sample (zeff=0.57)
MD-Patchy Mocks
Data
MD-Patchy mean
QPM mean
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Extra slides
0.1
1
10
50
0 50 100 150 200 250 300
σ B /
B(0
) data
[%]
Triangle Index
CMASS sample
statistical errorssystematic correction
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Extra slides
0.35
0.4
0.45
0.5
0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23
fσ8
kmax [hMpc-1]
0.92 0.94 0.96 0.98
1 1.02 1.04 1.06
α ||
0.94 0.96 0.98
1 1.02 1.04 1.06
α ⊥
CMASS data (zeff=0.57)
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data
IntroductionPower Spectrum
BispectrumConclusions
Extra slides
0.94 0.96 0.98
1 1.02 1.04
0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23
fσ8
/ [fσ
8]tr
ue
kmax [hMpc-1]
0.96
0.98
1
1.02
1.04
α || /
[α||]
true
0.96 0.98
1 1.02 1.04 1.06 1.08 1.1
α ⊥ /
[α⊥]tr
ue
CMASS mocks
MD-Patchy
Inter-bias Haloes
High-bias Haloes
Low-bias Haloes
Rencontres de Moriond 20th March 2016 Constraining cosmological parameters using BOSS data