Plot distance vs. velocity. We find:• Velocity proportional to distance• H0 is called Hubble’s constant

• Best fit today (includes many assumptions):

0v H d

0 70.4 1.4 km/s/MpcH Uncertainties:• Until recently, we didn’t know H0

very well, so we would write• h = 0.704 0.014• I’ll probably avoid h.

• Many published documentswrite answers in terms of h so that we remove the uncertainty in H0

0 100 km/s/MpcH h

2 0.02260 0.00053bh

• For almost any bright object, you can get red shift z:• You can then get the velocity from • Then use Hubble’s Law:

Distances from Hubble’s Law:

0

1 z

11 1

1

v c vz

v c c

0v H d

A spectral line from a galaxy normally at 415.0 nm appears instead at 423.0 nm. (a) How fast is the galaxy moving away from us?(b) What is the distance to the galaxy?

0

1 z

423.01.0193

415.0 1

v

c 0.0193v c 5790 km/sv

0

vd

H 5790 km/s

70.4 km/s/Mpc 82 Mpcd

Hubble’s Law isn’t perfect: low z:• Not everything expands• Galaxies, galaxy clusters, and smaller objects are not necessarily expanding• Galaxy superclusters expand more slowly than Hubble’s Law implies

• Objects have extra velocities, called peculiar velocities• Because gravity changes velocities• Typically of order 500 km/s or so

• This causes errors, usually random of vp /H0 in the distance• Typically 7 Mpc or so• Poor distance indicator for distances under 50 Mpc or so.

• We also have our own peculiar velocity that needs to be compensated for• We know how to do this

Peculiar Velocities

Failure of Hubble’s Law at High z:Another reason it fails: high z:• As z gets large (> 0.2) objects are moving relativistically• Relativistic corrections• Distance gets confusing as well

• We are also looking back into the past• Bad: Expansion might not be constant• Good: We can study the universe in the past (!)

• Still, higher z corresponds to greater distance• If we can accurately measure distance to some high z objects, we can set up a

calibration scale

The Cosmic Distance Ladder:

pc kpc Mpc Gpc

1 10 100 1 10 100 1 10 100 1 10

Radar

Parallax

1 AU

Moving Cluster

Light Echo

Cepheid Variables

Type Ia Supernovae

Spectroscopic Parallax

Cluster Fitting

Hubble’s Law

Planetary Nebulas

Hubble’s Law can be thought of two ways:• All galaxies are flying apart from each other• The space between the galaxies is expanding

Hubble Expansion

There is no special place in the universe• It is

meaningless to ask “where is it expanding from”• All observers

see the same thing

Age of the Universe: The naïve view:• Hubble’s Law tells us relation between distance and velocity• We can therefore figure out how long ago all this stuff was here:• Call t0 the time when everything left here:

0v H d

0d vt 0 0d H dt0 0 1H t

10 0t H

• This time is called the inverse Hubble time:• This is incorrect because the velocity of other

galaxies is probably not constant

10 13.9 0.3 GyrH

Assuming gravity slows things down, should the actual age of the universe be grater than the inverse Hubble time, or less?

• Speeds are slowing down• Speeds used to be greater• Higher speed Less time to get to where we are now

The Friedman Equation (1)• Assume the universe is homogenous and isotropic• We can treat any point (us), as center of the universe• We can use spherical symmetry around us

• Let a be the distance to any specific distant galaxy• And let its mass be m

• By Gauss’s Law, gravitational force is easy to find• Spherical symmetry tells us force is towards center• And caused by only mass closer than that galaxy

us distant galaxy

a

M

2

GMmF

a

2

2

d am

dt

• Because expansion of the universe is uniform, stuff closer than galaxy always closer• M is constant

• Multiply both sides by 2da/dt and integrate over time

2

2 2

d a GM

dt a

2

2 2

22

d a da GM dadt dt

dt dt a dt

22da GM

dt a

Don’t forget the constant of integration!• We could call it k, for example• For technical/historical reasons, it is called – kc2

• This makes k dimensionless

2kc

The Friedmann Equation (2)

us distant galaxy

a

• Mathematical notation: Time derivatives

M

daa

dt

• Divide both sides by a2

• Rewrite first term on right in terms of mass density

222da GM

kcdt a

2 22GM

a kca

2 2

2 3 2

2a GM kc

a a a

M V 343 a

2 2832 2

a kcG

a a

The resulting equation is called the First Friedman Equation• The derivation was flawed• M does not necessarily remain constant• In relativity, other contributions to gravity

• The equation is correct, including relativity

• The left side is the square of Hubble’s constant• Left side and first term on right independent of which galaxy you use• We can pick a particular galaxy to make k bigger or smaller• Normally chosen so that k = 0, +1, or -1

a Ha

aH

a

The Scale Factor

• The distance a is called the scale factor• All distances increase with the same ratio• This applies to anything that is not bound together• Even to waves of light!

2 2832 2

a kcG

a a

aH

a

1,0, or 1k

0 0

a d

a d

aEarly Universe

a0Now

Red shift can be thought of as a stretching of universe• Or is it Doppler shift?• Either view is correct, there isn’t a right answer

• This view does give us another relationship:

then

0 now

a

a

1

1 z

0 0

1

1

a d

a z d

• Most common way to label something in the distant past: z• Bigger z means earlier

• We’d like to know which value of k is right• Define the ratio:• If we know how big is, we’ll know k

• It is common to break into pieces• Ordinary atoms and stuff: b

• Dark Matter: d

• Matter: m = d + b

• Radiation: r

• Etc.• changes with time•When referring to values today, we add the subscript 0• Sometimes

2 2832 2

a kcG

a a

aH

a

1,0, or 1k

83

2

G

H

Dens. Curv. NameW < 1 k = -1 OpenW = 1 k = 0 FlatW > 1 k = +1

Closed83

2i

i

G

H

ii

2 22

kH H

a 2 2 1k a H

0 0t

Curvature Dens. Curv. NameW < 1 k = -1 OpenW = 1 k = 0 FlatW > 1 k = +1

Closed

• What’s the circumference of a circle of radius r?• According to Einstein, Space time can be curved!• Shape of space depends on density parameter:

•If = 1 then universe is flat and

•If > 1 then universe is closed and

•If < 1 then universe is open and

2C r

2 sinC a r a

2 sinhC a r a•Surface area of a sphere:

2

2 2

2 2

4 1,

4 sin 1,

4 sinh 1.

r

A a r a

a r a

•A closed universe is finite

So, What is ?Stuff Cont. to Stars 0.011Gas, Dust 0.035Neutrinos 10-4 – 10-2

Dark Matter 0.227 0.014Light ~ 10-5

Total ~ 0.273

b = 0.0456 0.0016

As we will see, we have missed a substantial fraction of

Age of the Universe, Round 2 (page 1)2 2

832 2

a kcG

a a

• What’s the current density of stuff in the universe?83

2

G

H

280 0 03 G H

• For ordinary matter, density 1/Volume

2

20 02

0

1kc

Ha

3

08 803 3

aG G

a

32 00

aH

a

22 20

2 20

akc kc

a a a

2

2 00 1

aH

a

• Substitute in: 3 22

2 20 00 02

1a aa

H Ha a a

Age of the Universe, Round 2 (page 2)

• Let x = a/a0 : 2

2 3 2 20 02

1x

H x H xx

3 22

2 20 00 02

1a aa

H Ha a a

2 10 1x H x

0 1x H x

10

1

1

dtH

dx x

11

0 0

0 1

dxt H

x

• If = 0.273, then• t0 = 11.4 0.3 Gyr

• If = 1.000, then• t0 = 9.3 0.2 Gyr

• These are younger than the oldest stars

The Age Problem