1. The Cosmological Redshift DEBATE TIME!!!

2. The spectral features of virtually all galaxies are redshifted  They’re all moving away from us

3. Edwin Hubble, using Cepheids as standard candles, was one of the first to measure distances to other galaxies By measuring distances to galaxies, Hubble found that redshift and distance are related in a special way

5. y = mx +b velocity = (H 0 )  (distance) + 0 The Hubble Law H 0 is the ‘Hubble constant’

7. velocity = (H 0 )  (distance) The Hubble Law:

8. Redshift of a galaxy tells us its distance through Hubble’s Law: distance = velocity H 0 redshift (z) is defined:

9. Remember? Special relativity: We can re-write it:

10. And with a little bit of algebra we can get to this: The so-called Doppler factor. So the redshift can be written: And in the non-relativistic limit (v<<c)

11. Homework Fill in the “little bit of algebra” and show the last step

12. Homework Hubble originally got 250 km/s Mpc for the Hubble constant. How old is the universe with that value of H 0 ? If a galaxy has a redshift of 0.02, how fast is it traveling away from us? How far away is the galaxy?

13. Redshift as Doppler Shift

14. Cosmological Redshift A more correct approach is to note that the wavelengths of photons expandwiththeuniverse: The two approaches areactuallyequivalent

15. Propagation of Light Our view of the Universe depends upon the propagation of light through the curved space. To understand this, we to consider the paths of null geodesics. A null geodesic has no length and in metric notation is ds = 0 Light travels on null geodesics …. So lets look at light traveling on a radial line from an observer at r=0 need

16. Suppose a light ray is emitted at and is received today (r=0) at t 0 And set c=1 Remember that a(t) and R(t) are essentially the same thing… a time t 1 at a distance r 1

17. Propagation of Light Remembering that for a comoving source at distance the coordinate is fixed, then r,r, Hence, in an expanding universe there will be a redshift

18. Hubble ' s Law If weconsideranearbysource,thenwecan write then Hence, we can see we can derive Hubble ‘s l aw from Robertson-Walker metric. Hubble ' s constant H(t) gives the instantaneous expansion rate. the

19. Is Energy Conserved in an Expanding (or Contracting) Universe? Consider energies of photons Consider potential energies of unbound systems

20. Deriving the Friedmann Equation Can derive the evolution of R(t) using mostly Newtonian mechanics, provided we accept two results from General Relativity: Birkhoff ' s system, the determined 1) theorem: for an infinite gravitating force due to gravity at radius r is only bythe mass interior to that radius. 2) Energy contributes which equals: to the gravitating mass density, energy density (ergs cm -3 ) of radiation and relativistic particles density ofmatter

21. Deriving the Friedmann Equation Consider an isolated sphere of radius R s and mass M s sitting in an isotropic expansion (Hubble flow), then the gravitational acceleration of the outer edge of the sphere is

22. We define “now” with a 0 subscript, set R 0 = 1, and since density is proportional to R -3, we get which we substitute into the equation of motion to get Note that if is nonzero, the Universe must be expanding or contracting. It cannot be static. Deriving the Friedmann Equation

23. How do we integratethis?Multiply both sidesboth sides byby toget And rememberthat So thatSo that Now, also remember:

24. So that we have Which means that the Expression inside the Brackets must be constant. We arbitrarily scale the constant with c 2 Replacing with and dividing by R 2,

25. What does this mean? critical flat universe If k=0, is always positive, and the expansion continues at an ever slowing pace since is dropping this is called a critical or flat universe. closed universe If k>0, is initially positive, but will reach a point where it changes sign. Expansion turns into contraction. This is a closed universe. open universe If k<0, is always positive, and never goes to zero – expansion always continues. This is an open universe.

26. The Friedmann Equation A more complete derivation,includingthecosmological constantterm,gives: The Friedmann Eqn. is effectively theequation of motion for a relativistic, homogeneous universe. In order to derive cosmological models from it, we also need to specify the equation of state of the cosmological fluid which fills the universe.

27. Cosmological Parameters