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Physics of the Large Scale Structure


Pengjie Zhang Department of Astronomy Shanghai Jiao Tong University The observed galaxy distribution of the nearby universe

0.7 billion lys

Observer
The observed galaxy distribution of the nearby universe

SDSS redshift survey Sloan great wall

1.4 billion lys

CfA2 redshift survey CfA2 great wall
The observed galaxy distribution of the nearby universe

Cosmic web can be described statistically 4 billion lys

SDSS redshift survey
What is the large scale structure?
Large scale structure
Intrinsic inhomogeneities
Not illusion of observation
beyond randomness
Not fluke of randomness
At >~ Mpc scale
Not internal structure of galaxies
Not distribution at specific region (since we do not know the initial condition)
Ensemble average > volume average
The large scale structure of the universe

Fundamental physics
Precision modeling Precision measurement Galaxy clusteringweak LSS
lensingclusters, void, SZ, ISW, peculiar velocities, etc.

Dark Baryons matter
The large scale structure of the universe
Part 1: • Deciphering the large scale structure (LSS) • With statistics and physics Part 2: • Tracers of LSS • Broadband power spectrum, BAO, redshift distortion, weak lensing, SZ effect, etc. Part 3 • Synergies of LSS tracers • Probe DM, DE, MG, neutrino, etc. • Reduce statistical errors • Control systematic errors
How to describe the large scale structure?

With statistics!  N-point correlation functions and their Fourier transforms – 2-point correlation– power spectrum  N-point joint PDF  Peak analysis  Topological descriptions: Minkowski functionals (and genus in particular), etc. Twopoint correlation function
3 3 P12 =(¯n1d x1)(¯n2d x2)[1+ξ(r)] x 2  P12: pairs of galaxies  Correlation function.: r what beyond average. Depends on the pair x 1 separation vector r.

 In real measurement, we have to eliminate observational effect
Correlation function: correlated feild

3 3 P12 =(¯n1d x1)(¯n2d x2)[1+ξ(r)]

n(x) −n The overdensity δ(x) ≡ field
n

ξ(r) ≡δ(x)δ(x + r)x The observed galaxy correlation function

Smaller scale measurement: Larger scale measurement: requires high number denisty
requires large volume
ξ(r) The z=0 correlation function: • 
a featureless power-law • Correlation length: ~5 Mpc/h 1

Nonlinear Linear regime r 1
10
100 Mpc/h
Features in the galaxy correlation function
  r ∼−1.8 ξ(r)  r0

ξ(r)

1
Bump at 100 Mpc/h eg. Zehavi+, 2004
eg. Eisenstein+, 2005
r 1
10
100

Type dependence. Deviation from power-law eg. Zehavi+, 2004
eg. Zehavi+, 2003
Closer look at the correlation function

x 2 r 3 3 P12 =(¯n1d x1)(¯n2d x2)[1+ξ(r)] x 1

3 3 P12(x1,x2)=(¯n1d x1)(¯nd x2)[1 + ξ(r ≡ x2 − x1,x1)]

Our universe should be homogeneous
ξ(r ≡ x2 − x1,x1) → ξ(r ≡ x2 − x1)

Our universe should be isotropic
ξ(r ≡ x2 − x1) → ξ(r) More features: anisotropies

Line of sight
ξ(r)=ξ(σ, π) x r 2 π σ x1

Observer
e.g. Peacock+, 2001, with 141,000 2dF galaxies
More features: anisotropies
ξ(r)=ξ(σ, π)

e.g. Peacock+, 2001, with 141,000 2dF galaxies
2010+: larger scale coverage, higher accuracy e.g. Li+, 2016, with 0.5M BOSS galaxies
Some anisotropies are so prominent that we can simply see by eyes in 1980s!

Observer
Correlation function –> Power spectrum

n(x) −n δ(x) ≡ n  δ(k) ≡ d3xδ(x)eik·x

homogeneity
 3  δ(k)δ(k ) =(2π) δ3D(k + k )P (k)  P (k)= d3rξ(r)eik·r Rich features in the power spectrum

Tegmark
Rich features in the power spectrum

P (k) ≡ P (|k)|) P (k)

5.0 0.4

4.5 0.2

0.0 4.0 [h/Mpc] || k

−0.2 3.5

−0.4

3.0 −0.4 −0.2 0.0 0.2 0.4 k [h/Mpc] ⊥ DR11 e.g. Li+ 2016
Even more features: “abnormal” correlation at horizon scales

0.04 17 < g < 19 19 < g < 19.5

0.02

Separate by ~3000 Mpc/h
0.00 -0.020.04 19.5 < g < 20 20 < g < 20.5

) 0.02 θ ( GQ

w 0.00 dN/dz -0.020.04 0.01 0.1 1 20.5 < g < 21 θ (degrees) 0.02

0.00

-0.02 0 0.5 1 1.5 2 0.01 0.1 1 z θ (degrees) Fig. 1.— Galaxy redshift distribution from applying our 17 < r<21 magnitude limit to the CNOC2 luminosity function and quasar redshift distribution inferred from quasar photometric red- shifts (solid lines). The fitted redshift distributions from Equa- Non-vanishing correlation! tion 8 are shown with dashed lines. In all cases, the amplitude scaling is arbitrary. First detected by Scranton+, 2005 at ~10sigma and then by other data
Even more features: “abnormal” correlation at horizon scales

Galaxies at z~1
CMB at z~1100
Separate by ~6000 Mpc/h

Non-vanishing correlation detected at ~4sigma, by NVSS/SDSS/WISE +WMAP/Planck Understanding LSS with physics

Initial conditions set at early time

Evolution under laws of physics

LSS we observe at late time Understanding LSS with physics

Initial fluctuations Fluctuations at z ∼ 100 Linear perturbation P (k) ∝ k0.96 at k → 0 0.96 Boltzmann equation
P (k) ∝ k P (k) ∝ k0.96−4 at k →∞

• High order perturbation theory • Halo model, Effective field theory of LSS • N-body simulations • Hydrodynamic simulations • Semi-analytical models • Halo occupation distribution, Conditional luminosity function
Movies of simulation: The Millennium The Elucid project

Complexities to understand LSS

 Task: evolve the universe from z~100 to z~0  Complexities – Dark matter, baryons, photons, neutrinos, dark energy – Gravity: nonlinear evolution at ~10 Mpc/h under gravity GR effect at >~103 Mpc/h – Non-gravitational forces: gastrophysics, galaxy formation and feedback – Mapping underlying matter/energy into observable signals (galaxy distribution, galaxies shapes, secondary CMB, etc.) – Eliminating observational contaminations in the desired cosmological signals An example of simplified treatment
 Set neutrinos=0, photons=0.  Neglect gastrophysics of baryons. Then only gravity. Baryons behave the same as DM  Assume smooth dark energy. Then DE only affects the expansion. Assume DE=Lambda  Assume flatness

The universe to evolve  Dark matter (=DM+baryons), Lambda  Flat  gravity Two steps

 Step 1: evolution of the dark matter field – Large scales: linear perturbation (GR effect can be included too) – Intermediate scales: spherical collapse – Small scales (and actually all scales): N-body simulations

 Step 2: identify virialized regions (dark matter halos) and put galaxies in – Halo mass function, halo bias/spatial clustering – Halo occupation distribution/Conditional luminosity function (the number of galaxies as a function of halo mass)


i, 2 + dx χδ hh
i equation
αβ  )

χδ φ ν

μ gg ∂∂ ) perturbation terms G 2 αβ δ differential ∂ =0) (1 + 2 2 1 2 evolution a P − ( ρ (1 + ν + Nonlinear μ Th =¯ ν 2 linear =  U ρ dt μ Neglect high order ) ν αβ

μ the background Perturb around ρU ν TgG μ

Step 1 ν αβ μ g = G = hgg ∂ ∂ )( += (1 + 2 ψ + ν μν )(

μ T −

ν

FRW μ ν FRW μ =

ν gG μ 2

ν ν μ ds μ Step 1linear evolution: Lambda: sub-horizon
2 2 −3 H (a)=H0 (Ωma +ΩΛ)   d2δ dδ dH/da 3 3 H2Ω δ + + − 0 m =0 da2 da H a 2 H2a3 a2

Linear  growth factor
a da δ(x, a) ∝ H δ(x, a =0) 3 3 Heath 1977
0 H a

Carroll, Press & Turner (1992) From sub-horizon to super-horizon
2 2 2 0 2


ðr þ 3KÞ 3a H ð a þ Þ¼4G ma m; GR effect

0 2 aHð a þ Þ¼4G ma W;

r2W 0 ¼ þ 30: m a2H GR effect


Z a ~ / da ½1 þ ð Þ m H 3 3 C k; a : 0 H a

3 2 2 0 1 3 2 ð Þ¼ a H H a þ R =H a C k;a 2 a 3 3 k H 0 da=H a 2ð Þ2 0 1 3 2 ¼ a H=H0 H a þ R =H a 3 1 2 a 3 3 : 3ðk 10 h MpcÞ H 0 da=H a ZPJ 2011
From linear to nonlinear scales
Halo model, halofit, EFT, etc.
high order pt
simulations

Linear
Nonlinear

N-body z=0 simulations today non-linear 10-104 Mpc/h k3P (k) evolution z=1 2 109-12particles Δ (k) ≡ z=2 2π2 2-4 linear z=3 ~10 cpus growth z=4  z=5 days-months

Credit: Percival’s lectures
Can it explain the observed galaxy clustering?
 Some but not all! r ∼−1.8 ξ(r)  r0

ξ(r)

1
Bump at 100 Mpc/h eg. Zehavi+, 2004
eg. Eisenstein+, 2005
r 1
10
100

Type dependence. Deviation from power-law eg. Zehavi+, 2004
eg. Zehavi+, 2003
The power-law correlation at ~1 Mpc/h
  k3P (k) 1 ∼−1.8 Δ2(k) ≡ ∼ ξ r ∼ ⇒ ξ(r) ∼ r 2π2 k z=0   non-linear r ∼−1 ξ(r)  evolution z=1 r0 z=2 linear z=3 growth z=4 z=5 eg. Zehavi+, 2004

Deviation from power-law eg. Zehavi+, 2003
What about the bump at ~100 Mpc/h? Baryon acoustic oscillations
Sound horizon cat decoupling
c  √ s 3

varying the baryon fraction

Bump at 100 Mpc/h eg. Eisenstein+, 2005
A missing ingredient

Type dependence. eg. Zehavi+, 2004

Larger DM halos More/larger galaxies

Smaller DM halos Less/smaller galaxies
Bias

2 ξhalo = b (M)ξDM

Type dependence. eg. Zehavi+, 2004
Jing 1998
Rich cosmological information


Large scale small scale

GR effect BAO, RSD-> DE, gravity z=0 PNG
non-linear evolution z=1 z=2 linear z=3 growth z=4 z=5

Neutrinos, DM

Cosmic magnification (weak lensing)