Transcript Cosmological parameter constraints

Cosmological parameter constraints
and
large scale structure of the universe
Cheng Zhao
Supervisor: Charling Tao
Outline
• Introduction
• Probes
• Large scale structure
• Perturbation theory
• References
Introduction
Standard cosmological model
Hot Big Bang cosmology with primordial fluctuations
that are adiabatic and Gaussian. (Consistent with
CMB measurements)
Wikipedia.org
Outline
• Introduction
• Probes
• Large scale structure
• Perturbation theory
• References
Probes
Overview
• Cosmic microwave background (CMB)
• Temperature anisotropy
• Polarization
• Supernova
• Large scale structure (LSS)
• Geometry
• Structure growth
•…
Probes
CMB
• Geometry (curvature)
• Age
• Composition
• Primordial fluctuation
• Inflation
•…
http://www.esa.int/spaceinimages/Images
Probes
Supernova – standard candle
• Hubble parameter
• Dark energy
•…
Wikipedia.org
Probes
Large scale galaxy survey
• Baryon Acoustic Oscillations (BAO, standard ruler)
• Redshift space distortion (RSD)
• Matter fluctuations/density
• CDM or WDM?
• Galaxy bias
• Dark energy
• Neutrino mass
• Non-Gaussianity
•…
http://www.sdss3.org
Probes
Gravitational lensing
• Mass power spectrum
• Matter fluctuation/density
•…
Wikipedia.org
Probes
Combination
Supernova Cosmology Project
Probes
Parameterization
13 basic parameters
𝜏
Reionization optical depth
𝐴𝑠
Scalar fluctuation amplitude
𝜔𝑏
Baryon density
𝑛𝑠
Scalar spectral index
𝜔𝑑
Dark matter density
𝛼
Running of spectral index
𝑓𝜈
Dark matter neutrino fraction
𝑟
Tensor-to-scalar ratio
ΩΛ
Dark energy density
𝑛𝑡
Tensor spectral index
𝑤
Dark energy equation of state
𝑏
Galaxy bias factor
Ω𝑘
Spatial curvature
Tegmark et al. 2004
Probes
Parameterization
Derived parameters (CMB)
𝑧ion
Reionization redshift (abrupt)
𝑡0
Age of universe
𝜔𝑚
Physical matter density
𝜎8
Galaxy fluctuation amplitude
Ω𝑚
Matter density/critical density
𝑍
CMB peak suppression factor
Ωtot
Total density/critical density
𝐴𝑝
Amplitude on CMB peak scales
𝐴𝑡
Tensor fluctuation amplitude
Θ𝑠
Acoustic peak scale
𝑀𝜈
Sum of neutrino masses
𝐻2
2nd to 1st CMB peak ratio
ℎ
Hubble parameter
𝐻3
3rd to 1st CMB peak ratio
𝛽
Redshift distortion parameter
𝐴∗
Amplitude at pivot point
Tegmark et al. 2004
Probes
Planck 2013 data release
In combination with WMAP polarized CMB data at
low ℓs, and CMB data from ACT and SPT at high ℓs:
• Matches well with minimal ΛCDM model, with
“vanilla” 6 parameters: {𝜔𝑏 , 𝜔𝑚 , ℎ, 𝜏, 𝑛𝑠 , 𝐴𝑠 }.
• No preference for extending models.
Planck Collaboration, 2013 & Costanzi et al. 2014
Probes
SDSS-III BOSS DR11
• Geometry (𝐷𝐴 , 𝐻) and structure growth (𝑓 ∙ 𝜎8 ).
• Consistent with Planck prediction within ΛCDM.
• Total neutrino mass 𝑀𝜈 = 0.36 ± 0.10 eV, higher
than 𝑀𝜈 < 0.23 eV of Planck.
• 2𝜎 tension of growth index 𝛾 from ΛCDM-GR
prediction, in combination with Planck/WMAP9.
Beutler et al, 2014 & Sanchez et al. 2014
Probes
BICEP2
• Measured 𝑟 = 0.2 ± 0.06 (Tensor-to-scalar ratio),
higher than 𝑟 < 0.11 without running spectral
index of Planck.
• B-mode polarization or dust polatization?
BICEP2 Collaboration, 2014
Outline
• Introduction
• Probes
• Large scale structure
• Perturbation theory
• References
Large Scale Structure
Probes
• Large scale galaxy survey
• Angular galaxy survey (model dependent)
• Gravitational lensing
• Lyman-α forest
• Hydrogen 21 cm emission line
• Quasar Clustering
Useful tool: N-body numerical simulation (covariance
matrix & galaxy bias)
Large Scale Structure
Observations
• Photometric
•
•
•
•
Spitzer
Dark Energy Survey (DES)
The VLT FIRST survey
Large Synoptic Survey Telescope (LSST)
• Spectroscopic
•
•
•
•
•
2/6-degree Field Galaxy Redshift Survey
CfA redshift survey
DEEP2 redshift survey
European Southern Observatory Slice Project (ESP)
Sloan Digital Sky Survey (SDSS)
Large Scale Structure
eBOSS
• Transition from deceleration to acceleration (𝐻(𝑧))
• Structure growth (test of GR-ΛCDM)
• Neutrinos
• QSO/galaxy science
•…
http://www.sdss3.org/future/eboss.php
Large Scale Structure
Statistics
• 2-point correlation function
𝑑𝑃 = 𝑛 1 + 𝜉 𝑟 𝑑𝑉
𝜉 𝑟 = 𝛿 𝒙 𝛿(𝒙 + 𝒓) , 𝛿 = 𝜌/𝜌 − 1
2
1
𝑛𝑅
𝑛𝑅
𝜉=
𝐷𝐷
− 2𝐷𝑅
+ 𝑅𝑅
𝑅𝑅
𝑛𝐷
𝑛𝐷
• Power spectrum
𝜉 𝑟 =
𝛿(𝒌1 )𝛿(𝒌2 )
𝑑 3 𝒌 𝑃 𝑘 exp(𝑖𝒌 ∙ 𝒓)
𝑐
= 𝛿 𝐷 𝒌1 + 𝒌2 𝑃(𝒌1 )
Large Scale Structure
Statistics
• Higher order statistics
• 3-point correlation function & bispectrum
• 4-point correlation function & trispectrum
•…
• Anisotropic statistics
2ℓ + 1
𝑃ℓ (𝑘) ≡
2
1
−1
𝑑𝜇 𝑃(𝑘, 𝜇)𝐿ℓ (𝜇)
Large Scale Structure
BOSS result
Anderson et al, 2012
Large Scale Structure
Baryon Acoustic Oscillation (BAO)
https://www.cfa.harvard.edu/~deisenst/acousticpeak/acoustic_physics.html
Large Scale Structure
Redshift space
de Lapparent V. et al, 1986
Large Scale Structure
Redshift space
• Kaiser Effect & Fingers of God
Hamilton 1998
Large Scale Structure
Redshift space distortion
2dFGRS
Peacock et al, 2001
BOSS
Samushia et al, 2014
Large Scale Structure
Galaxy bias
http://www.astr.ua.edu/keel/galaxies/largescale.html
Large Scale Structure
Galaxy bias
• Relation between galaxy density and dark matter
density:
𝛿𝑔 = 𝑓(𝛿DM )
• Linear estimation:
𝑏 = 𝛿𝑔 /𝛿DM
Large Scale Structure
Estimate of galaxy bias
• 2-point correlation function of galaxies and dark
matter (N-body simulation):
1/2
𝑏 = 𝜉𝑔 /𝜉DM
• Ratio of 2-point and 3-point correlation functions,
which have different dependencies on the bias.
(Noisy)
Large Scale Structure
N-body numerical simulation
• Trace dark matter particles (and baryons).
• Input:
• Linear power spectrum from CMB
• Gaussian random primordial fluctuation
• Cosmological parameters
• Initial conditions from perturbation theory
• Dynamics:
• Tree algorithm
• Particle-Mesh (PM) algorithm
• Hybrid methods (AP3M, AMR, etc.)
Outline
• Introduction
• Probes
• Large scale structure
• Perturbation theory
• References
Perturbation Theory
Dynamics of gravitational instability
• Assumption: collisionless cold dark matter (CDM).
• Discrete effects such as 2-body relaxation are negligible
• Non-relativistic
• Comoving coordinates 𝒙, conformal time 𝜏, and
conformal expansion rate ℋ:
𝒓=𝑎 𝜏 𝒙
𝑑𝑡 = 𝑎 𝜏 𝑑𝜏
ℋ ≡ 𝑑 ln 𝑎 /𝑑𝜏 = 𝐻𝑎
Perturbation Theory
Dynamics of gravitational instability
• Peculiar velocity 𝒖:
𝒗 𝒙, 𝜏 ≡ ℋ𝒙 + 𝒖(𝒙, 𝜏)
𝒑=𝑎𝑚𝒖
• Cosmological gravitational potential Φ:
′)
𝜌(𝒙
1 𝜕ℋ 2
3
′
𝐺 𝑑 𝒙
= 𝜙 𝒙, 𝜏 ≡ −
𝑥 + Φ(𝒙, 𝜏)
′
𝒙 −𝒙
2 𝜕𝜏
• Poisson equation:
𝛻2Φ
3
𝒙, 𝜏 = Ω𝑚 𝜏 ℋ 2 𝜏 𝛿(𝒙, 𝜏)
2
Perturbation Theory
Vlasov equation
• Particle number density in phase space 𝑓(𝒙, 𝒑, 𝜏):
𝑑𝑓 𝜕𝑓
𝒑
𝜕𝑓
=
+
∙ 𝛻𝑓 − 𝑎 𝑚 𝛻Φ ∙
=0
𝑑𝜏 𝜕𝜏 𝑚𝑎
𝜕𝒑
Perturbation Theory
Lagrangian dynamics
• Initial position 𝒒 and displacement field 𝜳(𝒒, 𝜏):
𝒙 𝜏 = 𝒒 + 𝜳(𝒒, 𝜏)
• Equation of motion:
𝑑2 𝒙
𝑑𝒙
+ℋ 𝜏
= −𝛻Φ
2
𝑑𝜏
𝑑𝜏
• Poisson equation:
2𝜳
𝑑
𝑑𝜳
3
2
2 (𝜏)𝛿(𝒙, 𝜏)
𝛻𝑥 Φ = −𝛻 ∙
+
ℋ
𝜏
=
Ω
(𝜏)ℋ
𝑑𝜏 2
𝑑𝜏
2 𝑚
𝛻𝑥𝑖 =
𝐾
𝛿𝑖𝑗
+ 𝜕𝜳𝑖 /𝜕𝒒𝑗
−1
𝛻𝑞𝑗
Perturbation Theory
Lagrangian perturbation theory
• Zel’dovich Approximation (ZA)
𝜓ZA ≡ 𝛻 ∙ 𝜳ZA = −𝐷1 𝜏 𝛿 (1) (𝒒)
• 2-order Lagrangian PT (2LPT)
𝜓2LPT ≡ 𝛻 ∙ 𝜳2LPT = −𝐷1 𝜏 𝛿
1
𝒒 + 𝐷2 𝜏 𝛿
• Spherical collapse (SC)
𝜓SC ≡ 𝛻 ∙ 𝜳SC = 3
2
1 − 𝐷1 𝜏 𝛿 (1) (𝒒)
3
2
𝒒
1/2
−1
Perturbation Theory
Cosmic web
• Density field:
𝜌 1+𝛿 𝒙
𝑑3 𝑥 = 𝜌 𝑑3𝑞
• Jacobian 𝐽:
1
1
1 + 𝛿 𝒙, 𝜏 =
≡
𝐾
Det(𝛿𝑖𝑗 + 𝜕𝜳𝑖 /𝜕𝒒𝑗 ) 𝐽(𝒒, 𝜏)
• Local density field (ZA):
1
1 + 𝛿 𝒙, 𝜏 =
1 − 𝜆1 𝐷1 (𝜏) 1 − 𝜆2 𝐷1 (𝜏) 1 − 𝜆3 𝐷1 (𝜏)
• 𝜆1 , 𝜆2 , 𝜆3 : eigenvalues of tidal tensor 𝜳𝑖,𝑗 .
Perturbation theory
Cosmic web
• All positive: knot
• 2 positive and 1 negative: sheet
• 1 positive and 2 negative: filament
• All negative: void
Knot
Sheet/Filament
Void
Perturbation Theory
Cosmic web
References
• Anderson L. et al., MNRAS. 427. 3435A, 2012
• Bernardeau F. et al., PhR. 367. 1B, 2002
• Beutler F. et al., MNRAS. 443. 1065B, 2014
• BICEP2 Collaboration, PRL. 112, 2014
• Coil Alison L., Planets, Stars and Stellar Systems Vol. 6, 2013
• Costanzi M. et al., arXiv1407.8338, 2014
• de Lapparent V. et al., ApJ. 302L, 1986
• Hamilton A. J. S., ASSL. 231. 185H, 1998
• Hu J-W. et al., JCAP. 05. 020H, 2014
• Peacock J. A. et al., Nature 410. 169P, 2001
• Percival W. J., arXiv:1312.5490, 2013
• Planck Collaboration, 2013 results.
• Samushia L. et al., MNRAS. 439. 3504S, 2014
• Sanchez A. et al., MNRAS. 440. 2692S, 2014
• Tegmark M. et al., PhysRevD. 69. 103501, 2004
• Viel M. et al., MNRAS. 339L. 39V, 2009