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Announcements

Announcements. Exam 3 is scheduled for Thursday April 11 (one week from Thursday). Format will be 7 MC’s and 4 SA’s (out of 7). Covers chapters 7 – 11. Sample questions are posted. Additional sample essays for Chapter 11 will be posted later today. A metric for an expanding universe.

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Announcements

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  1. Announcements • Exam 3 is scheduled for Thursday April 11 (one week from Thursday). Format will be 7 MC’s and 4 SA’s (out of 7). Covers chapters 7 – 11. Sample questions are posted. Additional sample essays for Chapter 11 will be posted later today.

  2. 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

  3. 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

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

  5. 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…flat k<0…hyperbolic k>0…spherical • Time flow rate doesn’t change

  6. 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

  7. 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?

  8. 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

  9. 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 tH in appropriate units of time H is the slope of the line in the Hubble Diagram

  10. The Hubble Length gives the size of the observable universe 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

  11. 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.

  12. Modeling the universe • The de Sitter model was a little to simple with only space-time and L. Let’s try to make a model that is more realistic • The real universe is extremely complex. The only hope is to make some simplifications • Take all matter in the universe, visible and dark, and 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. • 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.

  13. 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

  14. Velocity, Gravitational Force, Acceleration and Escape Velocity As the sphere expands, the particle has a velocity given by mt R 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

  15. Using the gravitational force we can determine the escape velocity mt R

  16. 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

  17. 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

  18. Moving from the expanding sphere to the expanding universe R R r Ms 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.

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