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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

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The Cosmic Microwave Background�

from WMAP year 5 data.

2dFGRS large scale galaxy structure north and south

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Homogeneity and Isotropy

homogeneous

not isotropic 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

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Density - Evolution - Geometry

Open

Flat

Closed ρρ > c

ρρ = c€

ρ < ρc

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Big Bang Nucleosynthesis

Results:

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The Robertson-Walker metric

€

ds2 = c 2dt 2 − R2(t) dr2

1− k r2+ r2 (dθ 2 + sin2θdφ 2)

Space-time interval/metric

Light travel distance

Expansion factor

Curvature Term (open, closed, flat)

Normal spatial polar coords

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

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Einstein Field Equations

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Gµν = 8π G Tµν +Λ gµν

gµν = spacetime metric ( ds2 = gµν dx

µ dxν )Gµν = Einstein tensor (spacetime curvature)G = Netwon's gravitational constantTµν = energy - momentum tensorΛ = cosmological constant

The Friedmann eqns:•  Adopt standard GR •  Assume

– Homogeneity and Isotropy, i.e., uniform fluid – Adiabatic expansion (lossless!)

> Friedmann eqns: €

Gµν = 8πGTµν + Λgµν

€

δE = −pδV

€

R•

R

=8πGρ3

−kc 2

R2+Λ3

R••

R= −

4πG3

ρ +3pc 2

+

Λc 2

3

Λ = Cosmological constant = vacuum energy = dark energy = “Greatest blunder” = Inflation = Useful Fudge factor?

Hubble constant or H(t)

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Newtonian Analogy

€

R• 2

=8π G ρ + Λ

3

R2 − k c 2

Friedmann equation:

€

E = m2R• 2

−G M mR

, M = 4π3R3ρ

R• 2

=8π G

3ρ R2 + 2E

m

€

same equation if ρ → ρ + Λ8π G

, 2Em

→−k c2

m

R

€

ρ

Newtonian Analogy

€

E = m2

˙ R 2 − G M mR

Vesc =2 G M

R

€

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

V∞ > 0V∞ = 0

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€

escape velocity :

Vesc2 =

2 G MR

=2 GR

4π R3 ρ3

=

8π G R2 ρ3

Hubble expansion :V = ˙ R = H0 Rcritical density :

VescV

2

=8π G ρ3 H0

2 ≡ρρc

ρc =3 H0

2

8π G, ΩM =

ρρc

=8πGρ

3H 2

Critical Density •  Derive using Newtonian analogy:

R

€

ρ

Subscript, o = today’s values

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.

€

dp = (1+ z)da =dl

(1+ z)Why?

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Proper distance •  co-moving radial distance

–  now ( when photon received ):

–  when photon emitted:

•  Proper distance (for flat universe): •  Angular diameter distance:

€

dz= 0 = R(t0) ro = R0 ro€

ro0

te

to

€

dz= 0

€

dz

€

dz = R(te ) ro =R(te ) R0 ro

R0=dz= 01+ z

(1+ z) = λobservedλlaboratory

=R(to)R(te )

€

dp = dz= 0 = Roro

€

da = Rero =dp(1+ z)

Angular diameter distance

Proper distance

Angular diameter distance

Angles frozen at emission. If structure bound need to Use distance at emission to convert angular sizes to physical sizes

θ

r

€

r = da tan(θ)

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

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Luminosity Distance –  Luminosity ( erg s-1 )

–  area of photon sphere

–  redshift:

–  time dilation: lower photon arrival rate

–  observed flux ( erg cm-2 s-1 )

•  Luminosity distance

€

F = N h ν oA Δto

=L

4π dP2 1+ z( )2

=L

4π dL2

€

Ld = Pd (1+ z)

€

A = 4π dP2

€

λo = λe (1+ z )νo = νe 1+ z( )

€

L = N h νeΔte

€

Δto = Δte (1+ z )

€

ro0

€

te€

t0

€

Δte€

Δto

Sources look fainter.

Surface Brightness

•  Solid angle

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

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

Σ≡FΩ

=L

4π DL2DA

2

A=

L4π A 1+ z( )4

ADA

DA A2/=Ω

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Angular Diameter Distance

€

ΩM€

ΩΛ

€

ΩΛ = 0€

ΩM +ΩΛ = 1

Evaluating the Proper Distance

€

dp = Roro = Ro dro

ro

∫ = RocdtRte

tr

∫ (From metric : cdt = Rdr)

Want dp in terms of z and H(1+z) (z dependent Hubble constant) :

ddt

x =1+ z = R0R

=

d Ro Rdt

= −RoR2

dRdt

= −(1+ z)H

i.e., dt = −dxx H(x)

dp = RocdxxH(x)1

1+z

∫

Now need expression for H in terms of (1+z)

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Evaluating the Proper Distance

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From Friedmann eqn :

H 2 = 8πGρM3

+8πGρR

3−kc 2

R2+Λ3

Set to todays values and divide by Ho2 :

1= 8πGρM ,o3Ho

2 +8πGρR ,o

3Ho2 −

kc 2

Ro2Ho

2 +Λ

3Ho2

Simplify by defining : ΩM ,o,ΩR ,o,ΩΛ ,o,Ωokc 2 = Ro

2Ho2 ΩM ,0 +ΩR ,0 +ΩΛ ,0 −1( ) = Ro2Ho2(Ω0 −1)

Sub for kc 2 term in Fr. eqn. and divide by Ho2 :

H 2

Ho2 =

8πGρM3Ho

2 +8πGρR

3Ho2 −

Ro2

R2(Ω−1) + Λ

3Ho2

Evaluating the Proper Distance

€

Matter scales with z as : ρz = ρo(1+ z)3 (think of a box)

Radiation scales withz as : ρz = ρo(1+ z)4 (think of a box +loss of energy)

Cosmological constant does not scale with z

Sub for densities and use Ω notation and recall (1+ z) = Ro R :

⇒ H 2 = Ho2 ΩM ,0 (1+ z)

3 +ΩR ,0 (1+ z)4 − (Ω0 −1)(1+ z)

2 +ΩΛ ,0( )

€

Proper distance :

dp = R0 ro =R0R(t)

c dtte

t0

∫ = c dxx H(x)1

1+z

∫ = cH0dx

ΩR ,0 x4 +ΩM ,0 x

3 +ΩΛ ,0 + (1−Ω0) x2

1

1+z

∫

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

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