Cosmology primer

The Cosmological principle

The standard model: (The Robertson-Walker metric & General Relativity)

Newtonian derivation of Friedmann eqns

Proper, angular, and luminosity distances

Derivation of the proper distance

The Cosmological principle

• Axiom: “The Universe is both homogeneous and isotropic”

Same in all locations Same in all directions Laws of Physics Universal

• Supporting evidence:

Cosmic Microwave Background Deep galaxy counts Large scale galaxy surveys

Page 1 The Cosmic Microwave Background from WMAP year 5 data.

2dFGRS large scale galaxy structure north and south

Page 2 Homogeneity and Isotropy

homogeneous isotropic not isotropic not homogeneous

Universal expansion is the only motion allowed by Cosmological Principle

Standard Cosmological Model

A metric system for defining the geometry = spacetime-metric

A census of the key contents= vacuum energy, matter, radiation

A model for how the contents modify the geometry = gen. rel.

Observational Cosmology:

Can measure geometry and use model to constrain contents: - Standard candles - Baryon Acoustic Oscillations - CMB Anisotropy Peak

Can obtain direct measurements of contents (matter, radiation): - Mass density via nearby studies - Radiation density via CMB

Can use orthogonal evidence: - Age of galaxy, stars, Earth < Age of Universe - Abundances via Big Bang Nucleosynthesis

Page 3 Density - Evolution - Geometry

ρ < ρc Open

ρ =ρ € c Flat

Closed ρ >ρ c

Page 4 Big Bang Nucleosynthesis

Results:

Page 5 The Robertson-Walker metric

Space-time interval/metric Normal spatial polar coords

2 2 2 2 2  dr 2 2 2 2  ds = c dt − R (t)  2 + r (dθ + sin θdφ )  1− k r 

Light travel distance Curvature Term (open, closed, flat)

Expansion factor € This metric is forced by the Cosmological Principle (universal expansion)

Spacetime interval for a photon = 0 (from special relativity)

If we consider photons only on radial paths in a flat Universe (k=0): c 2dt 2 = R2(t)dr2

€ Einstein’s General Relativity

• Spacetime geometry tells matter how to move

– gravity = effect of curved spacetime – free particles follow geodessic trajectories – ds2 > 0 v < c time-like massive particles – ds2 = 0 v = c null massless particles (photons) – ds2 < 0 v > c space-like tachyons (not observed)

• Matter (+energy) tells spacetime how to curve

– Einstein field equations – nonlinear – second-order derivatives of metric with respect to space/time coordinates

Page 6 Einstein Field Equations

Gµν = 8π G Tµν + Λ gµν

2 µ ν gµν = spacetime metric ( ds = gµν dx dx )

Gµν = Einstein tensor (spacetime curvature) G = Netwon's gravitational constant

Tµν = energy - momentum tensor Λ = cosmological constant

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The Friedmann eqns: • Adopt standard GR G = 8πGT + Λg • Assume µν µν µν – Homogeneity and Isotropy, i.e., uniform fluid – Adiabatic expansion (lossless!) δE = −pδV

€ Hubble constant or H(t) > Friedmann eqns:  •  R 8πGρ kc 2 Λ = Cosmological constant   Λ = − 2 + = vacuum energy €  R 3 R 3 = dark energy   = “Greatest blunder” •• R 4πG  3p Λc 2 = Inflation = −  ρ +  + = Useful Fudge factor? R 3  c 2  3

€ Page 7 Newtonian Analogy

m • 2 G M m 4π E = R − , M = R3ρ 2 R 3 m • 2 8π G 2E R R = ρ R2 + 3 m ρ

Friedmann equation:

2 € € •  8π G ρ + Λ R =   R2 − k c 2  3  Λ 2E same equation if ρ → ρ + , → −k c2 8π G m €

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Newtonian Analogy

m G M m E = R˙ 2 − 2 R 2 G M V = esc R

E > 0 V > Vesc R → ∞ V∞ > 0 € E = 0 V = Vesc R → ∞ V∞ = 0

E < 0 V < Vesc R → 0

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Page 8 Critical Density • Derive using Newtonian analogy: escape velocity : 3 2 2 2 G M 2 G  4π R ρ 8π G R ρ Vesc = =   = R R  3  3 Hubble expansion : V = R˙ = H R R 0 ρ critical density : 2  Vesc  8π G ρ ρ   = 2 ≡  V  3 H ρc 0 € 2 8πGρ 3 H0 ρ Subscript, o ρc = , ΩM = = 2 = today’s values 8π G ρc 3H

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Cosmological Distances

• Proper distance (dp) – Actual distance to object today (if one could stop time and measure it with a ruler).

• Angular diameter distance (da) – Proper distance at time of photon emission.

• Luminosity distance (dl) – Distance the photon feels it has travelled. d d = (1+ z)d = l p a (1+ z) Why?

€ Page 9 Proper distance

• co-moving radial distance 0 ro – now ( when photon received ):

dz= 0 = R(t0 ) ro = R0 ro – when photon emitted: dz= 0 € to R(t ) R r d d = R(t ) r = e 0 o = z= 0 z e o R 1 z € 0 + te € dz  λ R(t ) ( 1+ z) = observed = o   λlaboratory R(te )

• Proper distance (for flat universe):€ dp = dz= 0 = Roro • Angular diameter distance: € d d = R r = p a e o (1+ z) €

€ Angular diameter distance Angles frozen at emission. If structure bound need to Use distance at emission to

convert angular sizes to r physical sizes θ

r = da tan(θ)

Proper distance

Angular diameter distance

€ For non bound objects use proper distances to work out tangential sizes

Page 10 Luminosity Distance

N h νe – Luminosity ( erg s-1 ) L = 0 ro Δte – area of photon sphere 2 A = 4π dP t0 – redshift: Δto λo = λe (1+ z) € € νo = νe (1+ z) te Δte – time dilation:€ lower photon arrival rate € € Δto = Δte (1+ z) – observed€ flux ( erg cm-2 s-1 ) € N h ν L € L F = o = = A Δt 4 d 2 1 z 2 4π d 2 € o π P ( + ) L • Luminosity distance = (1+ z) Sources look d L d P fainter. €

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Surface Brightness

2 • Solid angle Ω = A/ D A

• Surface brightness D A A – Flux per solid angle (erg s-1 cm-2 arcsec-2)

2 F L DA L Σ ≡ = 2 = 4 Ω 4π DL A 4π A (1+ z)

– decreases very rapidly with z because: – expansion spreads out the photons – decreases their energy € – decreases their arrival rate

Page 11 Angular Diameter Distance

ΩΛ

ΩM € ΩM +ΩΛ = 1

€ ΩΛ = 0 €

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Evaluating the Proper Distance

ro tr cdt d = R r = R dr = R (From metric : cdt = Rdr) p o o o ∫ o ∫ R o te

Want dp in terms of z and H(1+z) (z dependent Hubble constant) : R d  R  d o R dR  x =1+ z = 0  = R = − o = −(1+ z)H dt  R  dt R2 dt −dx i.e., dt = x H(x) 1+z cdx dp = Ro ∫ 1 xH(x) Now need expression for H in terms of (1+z)

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Page 12 Evaluating the Proper Distance From Friedmann eqn : 8πGρ 8πGρ kc 2 Λ H 2 = M + R − + 3 3 R2 3 2 Set to todays values and divide by Ho : 2 8πGρM ,o 8πGρR,o kc Λ 1= 2 + 2 − 2 2 + 2 3Ho 3Ho Ro Ho 3Ho

Simplify by defining : ΩM ,o,ΩR,o,ΩΛ,o,Ωo 2 2 2 2 2 kc = Ro Ho (ΩM ,0 + ΩR,0 + ΩΛ,0 −1) = Ro Ho (Ω0 −1) 2 2 Sub for kc term in Fr. eqn. and divide by Ho : 2 2 H 8πGρM 8πGρR Ro Λ 2 = 2 + 2 − 2 (Ω −1) + 2 Ho 3Ho 3Ho R 3Ho

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Evaluating the Proper Distance

3 Matter scales with z as: ρz = ρo (1+ z) (think of a box) 4 Radiation scales withz as: ρz = ρo(1+ z) (think of a box +loss of energy) Cosmological constant does not scale with z

Ro Sub for densities and use Ω notation and recall (1+ z) = R : 2 2 3 4 2 ⇒ H = Ho (ΩM ,0 (1+ z) + ΩR,0 (1+ z) − (Ω0 −1)(1+ z) + ΩΛ,0)

Proper distance :

t0 1+z 1+z R0 c dx c dx € dp = R0 ro = c dt = = ∫ R(t) ∫ x H(x) H ∫ 4 3 2 te 1 0 1 ΩR,0 x + ΩM ,0 x + ΩΛ,0 + (1− Ω0 ) x

Depends only on redshift plus Ho plus contents (Ω’s)

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