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

Outline Introduction Probes Large scale structure Perturbation theory References

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

Outline Introduction Probes Large scale structure Perturbation theory References

Probes Overview Cosmic microwave background (CMB) Supernova 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

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

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 … http://www.sdss3.org

Gravitational lensing 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 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

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

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

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

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 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)

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 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 )

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)

Large Scale Structure BOSS result Anderson et al, 2012

Baryon Acoustic Oscillation (BAO) 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

Redshift space distortion Large Scale Structure Redshift space distortion 2dFGRS BOSS Peacock et al, 2001 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: 𝛿 𝑔 =𝑓( 𝛿 DM ) Linear estimation: 𝑏= 𝛿 𝑔 / 𝛿 DM

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)

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.)

Outline Introduction Probes Large scale structure Perturbation theory References

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

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)

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

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

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

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 .

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

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