Cosmological Distances

Intro Cosmology Short Course

Lecture 3

Paul Stankus, ORNL

Freedman, et al. Astrophys. J. 553, 47 (2001)

Riess, et al. (High-Z) Astron. J. 116 (1998)

W. Freedman Canadian

Modern Hubble constant (2001)

A. Riess American

Supernovae cosmology (1998)

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Two Modern Hubble Diagrams

Generalize

velocity redshift

distance magnitude

Einstein Equivalence

Principle Any small patch of GR

reduces to SR

Albert Einstein German

General Theory of Relativity

(1915)

x

t

x

t

General Curved Space

€

dτ 2 = dxμ gμν dxν

μν

∑Minkowski Space

x’

t’

€

dτ 2 = dt '( )2

− dx' c( )2

€

dτ = dt' For x’ constant “Proper time”

€

ds = dx' ' For t’’ constant Proper distance

€

ds2 ≡ −c 2dτ 2 = − cdt' '( )2

+ dx ' '( )2

x’’

t’’

€

dτ 2 = dt ' '( )2

− dx' ' c( )2 ?

What was the distance to that event?

Photons

Ust

t now

€

dτ 2 = dt 2 − a(t)[ ]2dχ 2 c 2 if dτ 2 > 0

ds2 = −c 2dt 2 + a(t)[ ]2dχ 2 if dτ 2 < 0Robertson-Walker

Coordinates

Past Event

Coordinate Distance Co-Moving Distance

€

= ds = a(now)ΔχA

B

∫ = Δχ

A B Well-defined… but is it

meaningful?

E

€

ds = 0A

E

∫

NB: Distance along photon line

Galaxies at rest in the

Hubble flow

Three definitions of distance

Co-Moving/ Coordinate

Angular Luminosity

Photons

t

Emitted

a(t0)

Observed

Robertson-Walker

Coordinates

h

dAngular

Euclidean: hd

Define: dAngularh/

Euclidean: FI/4d2

Define: dLum2I/4F

t0

dLuminosity

Intensity IFlux F

t

Physical Radius a(t0)

Robertson-Walker

Coordinates

t

x

Physical Radius x

Newton/Minkowski

Coordinates

I

F

I

Ft0

€

F =I

4π (Δx)2

dLum ≡I

4πF= Δx

€

FEnergy/Area/TimeObserved =

IEnergy/TimeEmitted

4π a(t0)Δχ[ ]2

Area of sphere1 2 4 4 3 4 4

1

1+ z(t1, t0)Rate of photons1 2 4 3 4

1

1+ z(t1, t0)Redshift of photons1 2 4 3 4

dLum ≡IEmit

4πF Obs= a(t0)Δχ

Physical Radius

1 2 4 3 4 1+ z(t1, t0)( )Effect of

Cosmic Expansion

1 2 4 3 4

t1

Us Us

Flat

Astronomical Magnitudes

Rule 1: Apparent Magnitude m ~ log(FluxObserved a.k.a. Brightness)

Rule 2: More positive Dimmer

Rule 3: Change 5 magnitudes 100 change in brightness

€

mBolometric = −log1005 FEnergy/Area/Time

Observed( ) + Constant

= −5

2log10 FEnergy/Area/Time

Observed( ) + Constant

Rule 4: Absolute Magnitude M = Apparent Magnitude if the source were at a standard distance (10 parsec = 32.6 light-year)

€

M Bolometric = −5

2log10 IEnergy/Time

Emitted( ) + Constant

Some apparent magnitudes

0-5

-10 -20-15 -255

1015

2025

SunFull Moon

VenusSaturn

Ganymede

UranusNeptunePluto

MercuryMars

Jupiter

Sirius

CanopusVeg

a

Deneb

Gamma Cassiopeiae

Andromeda GalaxyBrightest

Quasar

Naked eye, daylight

Naked eye, urban

Naked eyeGround-based 8m telescope

x100 Brightness

(Source: Wikipedia)

€

mBolometric = −5

2log10 FEnergy/Area/Time

Observed( ) + C

M Bolometric = −5

2log10 IEnergy/Time

Emitted( ) + C'

€

m − M = −5

2log10 F Obs

( )+5

2log10 IEmit

( ) + C' '

= 5log10 IEmit F Obs( ) + C' '

= 5log10 dLuminosity( ) + C' ' '

€

If dLum ∝ z then

5log10 dLum( ) = 5log10 z( ) + C

m − M = 5log10 z( ) + C'

Extended Hubble Relation dLum z

Proxy for

log(dLum)

Magnitudes and Luminosity Distances

t0 Big Bang tt0 Now

€

a(t) = t t0( )2 3

Flat

€

H0 ≡˙ a (t0)

a(t0)=

2

3t0

Hubble constant

q0 ≡ −˙ ̇ a (t0)

a(t0)H02 =

1

2 Deceleration parameter

1+ z(t1, t0) = t02 3t1

−2 3 Redshift from t1 to t0

€

dχ γ (t)

dt= ±

c

a(t)

χ γ (t) = ±c

a(t')dt ' + Const∫

Recall from Lec 2

€

dLum (t1, t0) = a(t0)Δχ γ (t1, t0)(1+ z) =c

a(t)t1

t0∫ dt ⎛

⎝ ⎜

⎞

⎠ ⎟

Cosmology!1 2 4 3 4

(1+ z)

dLum (t1, t0) =c

H0

z + z2 4 + O(z3)[ ]

€

dLum (t1, t0) =c

H0

z +1

2(1− q0)z2 + O(z3)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

More generally (all cosmologies):

Weinberg 14.6.8

Today’s big question:

Slope = Hubble

constant

Generic

0

Extended Hubble dLz

Open

2.512 in brightness

Generic

0

Open

Generic

0

Open