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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