Cosmology

1. Cosmology Friedmann solution L = 0 L not necessarily = 0 Ask questions! If you are curious or don’t understand, it’s unlikely you’re the only one. and it makes class much more interesting! office hours: 10:45 - 11:45 Tuesdays and Thursdays any time I’m in my office by appointment via e-mail Cosmology January 31, 2006

2. Cosmology • The study of the universe • large-scale properties of the universe as a whole • origin, evolution, fate • Cosmological principle • the universe is isotropic and homogeneous • the same everywhere and the same in all directions • at a particular time • the universe can evolve

3. Big Bang Cosmology • Theoretical basis is • general relativity • cosmological principle • Basic idea is that the part of the universe we can see today was extremely small billions of years ago and has since expanded from a hot dense state to the huge and much cooler region we inhabit • this does not say that the universe started as a point! if the universe is infinite in extent today, it was infinite then

4. 8pGr Lc2 kc2 H2 = - + 3 R2 3 GR solutions for the universe • Alexander Friedmann (Russia) developed a dynamic equation for the expanding universe in 1922 • and independently by Abbes Georges Lemaitre in 1927 • actually a relativistic equation in the framework of general relativity, but a simplified, non-relativistic version is easy to derive and gives a good idea of what’s happening • Friedmann equation • with Einstein’s cosmological constant included

5. 8pGr Lc2 kc2 - + H2 = 3 R2 3 ( ) H(t) = R dR dR dt dt [ ] H0 = t=t0=now • Recall the scale factor R(t) • so the distance between, say, two galaxies changes as r = r0·R(t) and r0 is the distance at some arbitrary time (e.g. now!) • note - R is dimensionless, it is not a distance itself • Hubble’s law is actually v = H(t)·r and H(t) is the rate of change of thescale factor divided by the scale factor • if R(t=now)=1, the Hubble “constant” is the value of H now • G is Newton’s gravitational constant • The density of the universe is r(t) • changes with time, as the universe expands • proportional to 1/R3(t) • c is, of course, the speed of light

6. 8pGr Lc2 kc2 - + H2 = 3 R2 3 • k is the curvature parameter • units of inverse length squared • classically, it is related to the total amount of energy in the universe • in GR, energy (mass) causes space-time to curve, so k is called the curvature parameter • constant • L is the cosmological constant • units of inverse length squared • added by Einstein to stop gravity from changing the universe • on philosophical grounds, he believed in a homogeneous, isotropic and unchanging universe • after Hubble’s discovery, he called it his “biggest blunder” • L contributes to the change in the expansion rate • it may be positive, negative, or zero

7. 8pGr Lc2 kc2 - + H2 = 3 R2 3 • Interpretation of L is that vacuum fluctuations affect the geometry of space-time on a distance scale of 1/ÖL • Rohlf’s book says that L can be measured using the volume density of distant galaxies and that |L| < 3 x 10-52 m-2, so that vacuum fluctuations don’t affect space-time geometry over distances of less than 1026 m (about 1010 LY) • but he gives no reference and I haven’t found one • Making the equation correct relativistically requires only one change • rather than r(t) being the matter density of the universe, it is the total energy density in both matter and radiation • which have different time dependencies!

8. k = 0 – Einstein-de Sitter universe the energy density is such that the universe expands forever, with a decreasing expansion rate then where H is Hubble’s parameter at some time and r is the energy density at that time the energy density for which this equation holds is the critical energy density, rc now is as good a time as any, so about 6 protons/m3 If the density is less than or equal to the critical density, the universe expands forever if r < rc, k < 0 H(t) > 0 – always (if L = 0) an open universe If the density is greater than the critical density, the expansion will stop and the universe will collapse if r > rc, k > 0 H(t’) = 0 and H(t>t’) < 0 (if L = 0) for H<0 (collapsing), the right-hand side is still positive because r is increasing and R is decreasing a closed universe 8pGr Lc2 kc2 - + H2 = 3 R2 3 8pGr H2 = 3 3H02 ≈ 10-26 kg/m3 rc = 8pG k – curvature parameter assume L=0

9. 8pGr Lc2 kc2 H2 = - + 3 R2 3 Friedmann globally • Look at the solution more “globally” • how are the terms related? • how do they change with time? • L not necessarily 0

10. Contribution to the expansion from the mass/energy of the universe r is the energy density of the universe – mass and radiation energy everything we know about, including dark matter the mass part is µ 1/R3 the mass stays the same and the volume increases the radiation part is µ 1/R4 the equivalent matter density is 1/R3 from the volume increasing 1/R from the wavelength stretching if R doubles,photon energy is halved Which was more important at the beginning? will be at the end? 8pGr Lc2 kc2 H2 = - + 3 R2 3 Ephoton x nphoton c2 Friedmann globally ave

11. Contribution to the expansion from the geometry of the universe µ 1/R2 Contribution to the expansion due to a cosmological constant constant! can be positive or negative Matter contribution µ 1/R3 Radiation contribution µ 1/R4 Which contribution was most important at the beginning?in the middle?at the end? Rewrite the equation So the geometry of the universe is flat if WM + WL = 1 Open if WM + WL < 1 Closed if WM + WL > 1 8pGr Lc2 kc2 - + 1 = 3H2 R2H2 3H2 8pGr Lc2 kc2 H2 = - + 3 R2 3 Friedmann details = WM + Wk + WL and then WM + WL = 1 - Wk Note that all change with time

12. WM + Wk + WL = 1 empty universe Note -- even if the universe is closed and finite, it can still expand forever flat universe

13. WM + Wk + WL = 1 – Data Lots more detail on this after the exam!

14. L