1. 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.

2. 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.

3. Careful examination of photographic plates yielded Cepheid variables

4. Edwin Hubble: late 1924 The 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.

5. The original Hubble Diagram Note the most distant object is only 2,000,000 parsecs

6. 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.

7. Hubble Flow The 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.

8. 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 Solution Add a non-zero constant of integration to the equations The Cosmological Constant  Made 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.

9. The de Sitter Model Published 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 &  Result: Exponentially expanding space Not widely understood Those that did understand it didn’t accept it as even an approximation of reality

10. A metric for an expanding universe Ordinary flat space-time metric Expanding space-time metric Where R(t) is the scale factor The flow rate of time isn’t changing but space is getting bigger

11. 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

12. What types of scale factors R(t) are possible and which is closest to the observed universe?

13. The Robertson-Walker Metric Metric works for any geometry…flat, spherical or hyperbolic Spherical coordinates instead of Cartesian coordinates k = curvature constant or shape factor k=0…flatk 0…spherical Time flow rate doesn’t change

14. A few colored card questions ClassAction website Cosmology module Hubble’s Law Hubble Constant 2 Hubble Constant Statements Units of the Hubble Constant Effects of Expansion Options 1 & 2

15. 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?

16. 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

17. 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 H is usually given in km/sec/Mpc so a unit conversion is required to get t H in appropriate units of time H is the slope of the line in the Hubble Diagram

18. The Hubble Length gives the size of the observable universe If H is in km/sec/Mpc and c is in km/s then D H will be in megaparsec Again, this assumes a constant expansion rate

19. Cosmological redshift is a result of the change in R with time R is a length scale. As the universe expands R gets bigger. 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.

20. Modeling the Universe

21. Beyond the de Sitter Model The de Sitter model was a little too simple with only space- time and . 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  matter 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.  = 0 Once you get good at it (and get a bigger computer) you can start adding complications.

22. 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

23. Velocity, Gravitational Force, Acceleration and Escape Velocity R mtmt 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 by Dividing by the mass of the test particle gives the gravitational acceleration

24. Using the gravitational force we can determine the escape velocity R mtmt

25. Kinetic energy is energy of motion 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

26. What does it mean? 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

27. Moving from the expanding sphere to the expanding universe R MsMs R  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.

28. Now add General Relativity R  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

29. 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!!!

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

31. Curvature is determined by the mass-energy density Rearrange the energy equation and evaluate “now” gives

32. The Critical Density,  c, is the density required for a flat universe

33. Density Parameter  when plugged into the Friedmann equation gives If k=1,  >1 If k=-1,  <1 If k=0,  =1

34. The Standard Models ModelGeometrykOmega qoqoqoqoAgeFate ClosedSpherical+1>1>½ t o < 2/3 t H recollapse Einstein- deSitter Flat0=1=½ t o = 2/3 t H Expand forever OpenHyperbolic<1 0 0 2/3t H < t o < t H Expand forever All standard models have  = 0 so only gravity acts on them

35. Adding a Cosmological Constant Sincethe first term on the right is proportional to one over R 2 while the second term increases with R. Eventually, the second term will dominate and the expansion rate will begin to accelerate.

36. A universe with a positive cosmological constant will eventually be dominated by  regardless of the geometry Density, , contains 1/R 3 (mass divided by volume), so the first term goes as 1/R which decreases as the universe gets bigger

37. Other Cosmological Models ModelGeometryqFate EinsteinSpherical cccc0Unstable de Sitter Flat>0 Exponential expansion Steady State Flat>0 Exponential expansion LemaîtreSpherical >c>c>c>c <0 after hover Expand, hover, expand ClosedSpherical0>½ Big Crunch Einstein-de Sitter Flat0½ Expand forever OpenHyperbolic00<q<½ Expand Forever Negative  Any<0>0 Big Crunch