1. Lecture 22 Cosmological Models ASTR 340 Fall 2006 Dennis Papadopoulos Chapter 11

2. Newtonian Universe Send r-> , k is twice the kinetic energy per unit mass remaining when the sphere expanded to infinite size

3. Fates of Expanding Universe FINITE SPHERE 1.E  <0, negative energy per unit mass; expansion stops and re-collapses 2.E  =0, zero net energy; exactly the velocity required to expand forever but velocity tends to zero as t and R go to infinity 3.E  > 0, positive energy per unit mass; keeps expanding forever; reaches infinity with some velocity to spare BIG LEAP -> CONSIDER SPHERE THE UNIVERSE What happens when R->  ? Explore R-> 

4. Standard Model From Newtonian to GR The Friedmann Equation Robertson Walker (RW) metric : k=0, +1, -1 In Friedmann’s equation R is the scale factor rather than the radius of an arbitrary sphere Gravity of mass and energy of the Universe acts on space time scale factor much as the gravity of mass inside a uniform sphere acts on its radius and E  replaced by curvature constant. Term retains significance as an energy at infinity but it is tied to the overall geometry of space

5. Standard Model Simplifications To solve we need to know how mass-energy density changes with time. If only mass  R 3 =constant. Now need relativistic equation of mass-energy conservation and equation of state i.e  E =f  m ). Notice that here  m R 3 =constant but  E R 4 =constant. Why the extra R? Mainly photons left out of Big-Bang. Red-shifting due to expansion reduces energy density per unit volume faster than 1/R 3 Photons dominant early in Universe are negligible source of space time curvature compared with mass to day. All models decelerate Also now dR/dt>0 expansion. For all models R=0 at some time. R=0 at t=0. Density -> infinity, and kc 2 term negligible at early times. Great simplification.

6. Fate of Universe-Standard Model While early time independent of curvature factor ultimate fate critically dependant on value of k, since mass-energy term decreases as 1/R. Fate of Universe in Newtonian form depended on value of E . In Friedmann Universe it depends on value of curvature k. All models begin with a BANG but only the spherical ends with BANG while the other two end with a whimper.

7. Theoretical Observables Friedmann equation describes evolution of scale factor R(t) in the Robertson-Walker metric. i.e. universe isotropic and homogeneous. Solution for a choice of  and k is a model of the Universe and gives R(t) We cannot observe R(t) directly. What else can we observe to check whether model predictions fit observations? Need to find observable quantities derived from R(t). Enter Hubble Since R and its rate are functions of time H function of time. NOT CONSTANT. Constant only at a particular time. Now given symbol H 0 Time evolution equation for H(t) Replaces scale factor R by measurable quantities H,  and spatial geometry

8. Observing Standard Model Average mass density critical parameter why? Measurement of H 0 and   give curvature constant k Explore equation: 1.Empty universe  k negative hyperbolic universe, expand forever 2.Flat or require matter or energy. 3.k=0 -> critical density

9. Critical Density If H 0 100 km s -1 Mpc -1 critical density is 2x10 -26 kg/m 3 or 10 Hydrogen atoms per cubic meter of space Scales as H 2 50 km/sec Mpc gives ¼ density Current value of 72 km/sec Mpc gives critical density 10 -26 kg/m 3  M =1 gives boundary between open hyperbolic universes and closed, finite, spherical universe In a flat universe  is constant otherwise it changes with cosmic time

10. Deceleration Parameter q Deceleration Parameter. Now q 0. All standard models decelerate q.0. Need cosmological constant to change it For standard models specification of q determines geometry of space and therefore specific model

12. Summary - Definitions