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)
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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)
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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
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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