1. Cheng Zhao Supervisor: Charling Tao
Cosmological parameter constraints and large scale structure of the universe
Cheng Zhao
Supervisor: Charling Tao

2. Outline Introduction Probes Large scale structure Perturbation theory
References

3. Standard cosmological model
Introduction
Standard cosmological model
Hot Big Bang cosmology with primordial fluctuations that are adiabatic and Gaussian. (Consistent with CMB measurements)
Wikipedia.org

4. Outline Introduction Probes Large scale structure Perturbation theory
References

5. Probes Overview Cosmic microwave background (CMB) Supernova
Temperature anisotropy
Polarization
Supernova
Large scale structure (LSS)
Geometry
Structure growth …

6. Probes CMB Geometry (curvature) Age Composition Primordial fluctuation
Inflation …

7. Supernova – standard candle
Probes
Supernova – standard candle
Hubble parameter
Dark energy …
Wikipedia.org

8. Large scale galaxy survey
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 …

9. Gravitational lensing
Probes
Gravitational lensing
Mass power spectrum
Matter fluctuation/density …
Wikipedia.org

10. Probes
Combination
Supernova Cosmology Project

11. Probes Parameterization 13 basic parameters 𝜏
Reionization optical depth
𝐴 𝑠
Scalar fluctuation amplitude
𝜔 𝑏
Baryon density
𝑛 𝑠
Scalar spectral index
𝜔 𝑑
Dark matter density 𝛼
Running of spectral index
𝑓 𝜈
Dark matter neutrino fraction 𝑟
Tensor-to-scalar ratio
Ω Λ
Dark energy density
𝑛 𝑡
Tensor spectral index 𝑤
Dark energy equation of state 𝑏
Galaxy bias factor
Ω 𝑘
Spatial curvature
Tegmark et al. 2004

12. Derived parameters (CMB)
Probes
Parameterization
Derived parameters (CMB)
𝑧 ion
Reionization redshift (abrupt)
𝑡 0
Age of universe
𝜔 𝑚
Physical matter density
𝜎 8
Galaxy fluctuation amplitude
Ω 𝑚
Matter density/critical density 𝑍
CMB peak suppression factor
Ω tot
Total density/critical density
𝐴 𝑝
Amplitude on CMB peak scales
𝐴 𝑡
Tensor fluctuation amplitude
Θ 𝑠
Acoustic peak scale
𝑀 𝜈
Sum of neutrino masses
𝐻 2
2nd to 1st CMB peak ratio ℎ
Hubble parameter
𝐻 3
3rd to 1st CMB peak ratio 𝛽
Redshift distortion parameter
𝐴 ∗
Amplitude at pivot point
Tegmark et al. 2004

13. 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: { 𝜔 𝑏 , 𝜔 𝑚 , ℎ, 𝜏, 𝑛 𝑠 , 𝐴 𝑠 }.
No preference for extending models.
Planck Collaboration, 2013 & Costanzi et al. 2014

14. Probes SDSS-III BOSS DR11
Geometry ( 𝐷 𝐴 , 𝐻) and structure growth (𝑓∙ 𝜎 8 ).
Consistent with Planck prediction within ΛCDM.
Total neutrino mass 𝑀 𝜈 =0.36±0.10 eV, higher than 𝑀 𝜈 <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

15. 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

16. Outline Introduction Probes Large scale structure Perturbation theory
References

17. 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)

18. Large Scale Structure Observations Photometric Spectroscopic 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)

19. Large Scale Structure eBOSS
Transition from deceleration to acceleration (𝐻(𝑧))
Structure growth (test of GR-ΛCDM)
Neutrinos
QSO/galaxy science …

20. 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
𝜉= 1 𝑅𝑅 𝐷𝐷 𝑛 𝑅 𝑛 𝐷 2 −2𝐷𝑅 𝑛 𝑅 𝑛 𝐷 +𝑅𝑅
Power spectrum
𝜉 𝑟 = 𝑑 3 𝒌 𝑃 𝑘 exp (𝑖𝒌∙𝒓)
𝛿( 𝒌 1 )𝛿( 𝒌 2 ) 𝑐 = 𝛿 𝐷 𝒌 1 + 𝒌 2 𝑃( 𝒌 1 )

21. Large Scale Structure Statistics Higher order statistics
3-point correlation function & bispectrum
4-point correlation function & trispectrum …
Anisotropic statistics
𝑃 ℓ (𝑘)≡ 2ℓ+1 2 −1 1 𝑑𝜇 𝑃(𝑘,𝜇) 𝐿 ℓ (𝜇)

22. Large Scale Structure
BOSS result
Anderson et al, 2012

23. Baryon Acoustic Oscillation (BAO)
Large Scale Structure
Baryon Acoustic Oscillation (BAO)

24. Large Scale Structure
Redshift space
de Lapparent V. et al, 1986

25. Large Scale Structure Redshift space Kaiser Effect & Fingers of God
Hamilton 1998

26. Redshift space distortion
Large Scale Structure
Redshift space distortion
2dFGRS
BOSS
Peacock et al, 2001
Samushia et al, 2014

27. Large Scale Structure Galaxy bias

28. Large Scale Structure Galaxy bias
Relation between galaxy density and dark matter density:
𝛿 𝑔 =𝑓( 𝛿 DM )
Linear estimation:
𝑏= 𝛿 𝑔 / 𝛿 DM

29. Estimate of galaxy bias
Large Scale Structure
Estimate of galaxy bias
2-point correlation function of galaxies and dark matter (N-body simulation):
𝑏= 𝜉 𝑔 / 𝜉 DM 1/2
Ratio of 2-point and 3-point correlation functions, which have different dependencies on the bias. (Noisy)

30. N-body numerical simulation
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.)

31. Outline Introduction Probes Large scale structure Perturbation theory
References

32. Dynamics of gravitational instability
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

33. Dynamics of gravitational instability
Perturbation Theory
Dynamics of gravitational instability
Peculiar velocity 𝒖:
𝒗 𝒙,𝜏 ≡ℋ𝒙+𝒖(𝒙,𝜏)
𝒑=𝑎 𝑚 𝒖
Cosmological gravitational potential Φ:
𝐺 𝑑 3 𝒙 ′ 𝜌( 𝒙 ′ ) 𝒙 ′ −𝒙 =𝜙 𝒙,𝜏 ≡− 1 2 𝜕ℋ 𝜕𝜏 𝑥 2 +Φ(𝒙,𝜏)
Poisson equation:
𝛻 2 Φ 𝒙,𝜏 = 3 2 Ω 𝑚 𝜏 ℋ 2 𝜏 𝛿(𝒙,𝜏)

34. Perturbation Theory Vlasov equation
Particle number density in phase space 𝑓(𝒙,𝒑,𝜏):
𝑑𝑓 𝑑𝜏 = 𝜕𝑓 𝜕𝜏 + 𝒑 𝑚𝑎 ∙𝛻𝑓−𝑎 𝑚 𝛻Φ∙ 𝜕𝑓 𝜕𝒑 =0

35. Perturbation Theory Lagrangian dynamics
Initial position 𝒒 and displacement field 𝜳(𝒒,𝜏):
𝒙 𝜏 =𝒒+𝜳(𝒒,𝜏)
Equation of motion:
𝑑 2 𝒙 𝑑 𝜏 2 +ℋ 𝜏 𝑑𝒙 𝑑𝜏 =−𝛻Φ
Poisson equation:
𝛻 𝑥 2 Φ=−𝛻∙ 𝑑 2 𝜳 𝑑 𝜏 2 +ℋ 𝜏 𝑑𝜳 𝑑𝜏 = 3 2 Ω 𝑚 (𝜏) ℋ 2 (𝜏)𝛿(𝒙,𝜏)
𝛻 𝑥 𝑖 = 𝛿 𝑖𝑗 𝐾 +𝜕 𝜳 𝑖 /𝜕 𝒒 𝑗 −1 𝛻 𝑞 𝑗

36. Lagrangian perturbation theory
Zel’dovich Approximation (ZA)
𝜓 ZA ≡𝛻∙ 𝜳 ZA =− 𝐷 1 𝜏 𝛿 (1) (𝒒)
2-order Lagrangian PT (2LPT)
𝜓 2LPT ≡𝛻∙ 𝜳 2LPT =− 𝐷 1 𝜏 𝛿 1 𝒒 + 𝐷 2 𝜏 𝛿 2 𝒒
Spherical collapse (SC)
𝜓 SC ≡𝛻∙ 𝜳 SC =3 1− 2 3 𝐷 1 𝜏 𝛿 (1) (𝒒) 1/2 −1

37. Perturbation Theory Cosmic web Density field: Jacobian 𝐽:
𝜌 1+𝛿 𝒙 𝑑 3 𝑥= 𝜌 𝑑 3 𝑞
Jacobian 𝐽:
1+𝛿 𝒙,𝜏 = 1 Det( 𝛿 𝑖𝑗 𝐾 +𝜕 𝜳 𝑖 /𝜕 𝒒 𝑗 ) ≡ 1 𝐽(𝒒,𝜏)
Local density field (ZA):
1+𝛿 𝒙,𝜏 = 1 1− 𝜆 1 𝐷 1 (𝜏) 1− 𝜆 2 𝐷 1 (𝜏) 1− 𝜆 3 𝐷 1 (𝜏)
𝜆 1 , 𝜆 2 , 𝜆 3 : eigenvalues of tidal tensor 𝜳 𝑖,𝑗 .

38. 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

39. Perturbation Theory
Cosmic web

40. References Anderson L. et al., MNRAS. 427. 3435A, 2012
Bernardeau F. et al., PhR B, 2002
Beutler F. et al., MNRAS B, 2014
BICEP2 Collaboration, PRL. 112, 2014
Coil Alison L., Planets, Stars and Stellar Systems Vol. 6, 2013
Costanzi M. et al., arXiv , 2014
de Lapparent V. et al., ApJ. 302L, 1986
Hamilton A. J. S., ASSL H, 1998
Hu J-W. et al., JCAP H, 2014
Peacock J. A. et al., Nature P, 2001
Percival W. J., arXiv: , 2013
Planck Collaboration, 2013 results.
Samushia L. et al., MNRAS S, 2014
Sanchez A. et al., MNRAS S, 2014
Tegmark M. et al., PhysRevD , 2004
Viel M. et al., MNRAS. 339L. 39V, 2009