Announcements

• Exam 4 is Monday May 4. Will cover Chapters 9, 10 & 11. The exam will be an all essay exam. Sample questions are posted

• Project Presentations will be next week, starting Monday. Plan on a ~12 minute presentation with an additional 3 – 5 minutes for questions. A written paper is also required, not just a print-out of your presentation.

Edwin Hubble looked for Variables in the Andromeda “Nebula”

Since the period-luminosity relationship for Cepheid's had been recently determined, their luminosity could be calculated.

Careful examination of photographic plates yielded

Cepheid variables

Edwin Hubble: late 1924The closest of the spiral nebulae, the Andromeda nebulae, is over 1 million lightyears away and is at least 100,000 light-years in diameter. It cannot be part of the Milky Way which Harlow Shapley had determined is less than 100,000 lightyears across.

The original Hubble Diagram

Note the most distant object is only 2,000,000 parsecs

Hubble’s mistake was a systematic error in determining distance

Hubble’s actual distances were off by over a factor of 2 but their relative distances were the same so his conclusions were still correct.

Hubble FlowThe redshift of galaxies is not due to a peculiar velocity but caused by the expansion of space itself. Nearby galaxies may have peculiar velocities larger than their Hubble Flow velocity.

Einstein’s “Greatest Blunder”• Application of equations of General Relativity to a

simplified model of the universe showed it cannot be static

• Prevailing view in the 1910’s was a static universe

• Full extent of the universe was not even known

SolutionAdd a non-zero constant of integration to the equations

The Cosmological Constant LMade the universe static but unstable. Like trying to balance a pin on its head. You may be able to get it to stand upright but any disturbance knocks it over.

The de Sitter ModelPublished in 1917, well before Hubble showed the universe to be expanding.

Using Einstein’s equations of general relativity to solve for the universe will require a few simplifications

• No matter…the universe is empty. No stars, white dwarfs, neutron stars, black holes, gas, dust, nothing

• Space-time & L• Result: Exponentially expanding space

• Not widely understoodThose that did understand it didn’t accept it as

even an approximation of reality

A metric for an expanding universe

22222 zyxtcs Ordinary flat space-time metric

Expanding space-time metric

222222 )( zyxtRtcs Where R(t) is the scale factor

The flow rate of time isn’t changing but space is getting bigger

Consequence of the scaling factor: co-moving coordinates

The physical distance between objects is increasing and the rate of increase depends on

the original separation distance

What types of scale factors R(t) are possible and which is closest to the

observed universe?

The Robertson-Walker Metric

222222

2222 sin

1)( rr

kr

rtRtcs

• Metric works for any geometry…flat, spherical or hyperbolic

• Spherical coordinates instead of Cartesian coordinates

• k = curvature constant or shape factork=0…flat k<0…hyperbolic k>0…spherical

• Time flow rate doesn’t change

A few colored card questions

ClassAction website Cosmology moduleHubble’s LawHubble Constant 2Hubble Constant StatementsUnits of the Hubble ConstantEffects of Expansion Options 1 & 2

Five Minute Essay

According to the Hubble Law and the Robertson-Walker metric space is expanding. Does this mean you are expanding? Why or why not? Is the solar system expanding? Why or why not? How about the Milky Way? Why or why not? At what size do we consider space to be expanding and why?

Cosmic Time

• Any clock at rest with respect to the average mass distribution in the universe.

• All clocks that keep cosmic time are unaffected by any time dilation. They all always read the same time as all other clocks keeping cosmic time.

• No “real” or peculiar motion between clocks keeping cosmic time so no special relativistic time dilation.

• All expansion effects in the Robertson-Walker metric are in the spatial part. The time part is unaffected by the expansion

The Hubble Constant is the inverse of the age of the universe

If the expansion rate has remained constant then the time since the big bang is the Hubble time given by

HtH

1

H is usually given in km/sec/Mpc so a unit conversion is required to get tH in appropriate units of time

H is the slope of the line in the Hubble Diagram

The Hubble Length gives the size of the observable universe

H

cctD HH

If H is in km/sec/Mpc and c is in km/s then DH will be in megaparsec

Again, this assumes a constant expansion rate

Cosmological redshift is a result of the change in R with time

nowobs source

then

R

R

R is a length scale. As the universe expands R gets bigger.

1 obs now

source then

Rz

R

so

Note that this does not tell us how the universe evolved between then and now, only how it was then and how it is now.

If we assume a scale factor of 1 now, the redshift will give the scale factor for when the galaxy (or what ever is observed) was.

Thus, cosmological redshift is a measure of the scale factor.

Modeling the Universe

Beyond the de Sitter ModelThe de Sitter model was a little too simple with only space-time and L. The real universe is extremely complex. The only hope is to make some simplifications

• Take all matter in the universe, visible and dark, grind it into a uniform powder and spread it evenly throughout the universe. This gives rmatter for the universe. The matter will only interact through gravity (all dark matter).

• Take all the energy (only photons) in the universe and distribute it uniformly throughout the universe.

• The cosmological constant is zero. L = 0

• Once you get good at it (and get a bigger computer) you can start adding complications.

The simplest case:the Newtonian Universe

• Uniform distribution of mass

• Infinite

• Doing calculations with an infinite size not possible so just consider a sphere of radius R

• Look at a particle on the surface of the sphere

Velocity, Gravitational Force, Acceleration and Escape Velocity

Rt

Rv

gmR

MmGF t

stg

2

2R

MGg s

R

mt

As the sphere expands, the particle has a velocity given by

All the mass inside the sphere exerts a gravitational force on the particle given byDividing by the mass of the test

particle gives the gravitational acceleration

Using the gravitational force we can determine the escape velocity

R

GMv sesc

2

R

mt

? v toequalor than less an,greater th R Is esc

Kinetic energy is energy of motion

221 vmKE t

constant2

2 2 R

GMRE s

ER 22

Divide by the mass and rearrange to get energy (E) per unit mass and use the expression for the escape velocity

Now let R go to infinity so mass term vanishes we get

So

ER

GMRE s 2

22 2

What does it mean?

escvR

• If E∞<0…expansion will end and sphere will collapse.

• If E∞>0…expansion continues forever at an ever decreasing rate

• If E∞=0…expansion continues forever with rate decreasing to zero at infinite time

escvR

escvR

Moving from the expanding sphere to the expanding universe

ER

GMR s 2

22 3

34 RM s

RMs

Rr

ERGR 22382

The total mass in the universe may be infinite so use density (mass divided by volume) instead. If we use the scale factor instead of the radius of the universe for R we get rid of all infinities problems.

Now add General Relativity

Rr

222

3

8kcR

GR

R is now the scale factor and k is the curvature constant of the Robertson-Walker metric

This equation is known as the Friedmann Equation

Standard Models• Average density includes both average density of

matter and average density of energy

• Matter includes luminous (ordinary) matter and non-luminous (dark) matter

• Energy density contains only “normal” energy from photons. Largest constituent is the energy of the cosmic background radiation.

• Overall average density changes in time

• Follows Robertson-Walker Metric

• NO COSMOLOGICAL CONSTANT!!!

The Hubble “Constant” is related to the scale factor R

R

R

l

vH

Curvature is determined by the mass-energy density

Rearrange the energy equation and evaluate “now” gives

20

020

2

3

8H

G

R

kc

The Critical Density, rc, is the density required for a flat universe

G

Hc

8

3 2

Density Parameter W

c

22

2

1RH

kc

when plugged into the Friedmann equation gives

If k=1, W>1

If k=-1, W<1

If k=0, W=1

The Standard ModelsModel Geometry k Omega qo Age Fate

Closed Spherical +1 >1 >½ to < 2/3 tH recollapse

Einstein- deSitter

Flat 0 =1 =½ to = 2/3 tH Expand forever

Open Hyperbolic -1 <1 <½ but >0

2/3tH< to< tH Expand forever

All standard models have L = 0 so only gravity acts on them

Adding a Cosmological Constant

33

4 RR

GR

334 R

M s

Since the first term on the right

is proportional to one over R2 while the second term increases with R. Eventually, the second term will dominate and the expansion rate will begin to accelerate.

A universe with a positive cosmological constant will

eventually be dominated by L regardless of the geometry

22

2382

3kc

RRGR

Density, r, contains 1/R3 (mass divided by volume), so the first term goes as 1/R which decreases as the universe gets bigger

Other Cosmological ModelsModel Geometry L q Fate

Einstein Spherical Lc 0 Unstable

de Sitter Flat >0 -1 Exponential expansion

Steady State Flat >0 -1 Exponential expansion

Lemaître Spherical >Lc <0 after hover Expand, hover, expand

Closed Spherical 0 >½ Big Crunch

Einstein-de Sitter

Flat 0 ½ Expand forever

Open Hyperbolic 0 0<q<½ Expand Forever

Negative L Any <0 >0 Big Crunch