1. Entropy, Heat Death, and the like The answer is we don’t know. The book good point probably won’t “bounce” because we will have gained entropy (disorder) therefore be so “disordered”, the universe won’t start “afresh.”

2. But maybe at high temperatures and density, our “normal” definition of entropy fails! So really, we don’t know!

3. Difference between energy density and entropy Suppose start with 2 separate gases: Hydrogen and He. Low entropy, highly ordered HHe wall Wall removed begins to mix H He box

4. contract box (like contracting universe) energy per unit volume goes up But arrow of time goes forward => H and He continue to mix! System proceeds to higher entropy!

5. In a closed universe, can we see the back of our heads? For our standard,  = 0 universe, the universe will re-collapse by the time the light from the back of our heads reaches our eyes.

6. How does q vary for the case where k = 0? Answer: not at all!= 1/2! Comes from 1 + kc 2 /R 2 =  = 2q

7. See chapter 11 of book for equations and discussion

8. Before the “ ,” life was simple: k and   and q 0 and the fate of the universe were all uniquely linked page 305, table 11.1: k  0 q 0 fate  1 0 < < 1 0< < 1/2 expand forever 0 1 1/2 almost forever +1 1< 1/2 < big crunch __

9. Look at (R/R) 2 + kc 2 /R 2 = (G8  /3) x . As prologue to “inflation” (why “we” like it):

10. Write  as  0 (R 0 /R) 3, for matter (fixed number of particles, change volume) density term goes up faster (as must the R 2 /R 2 term) as 1/R 3 increases faster than the kc 2 /R 2 term! => k becomes negligible => becomes effectively 0 =>  tends to 1 as we go back Also, stays 1 if it started as 1. (R/R) 2 + kc 2 /R 2 = (G8  /3) x    R 0 /R) 3..

11. Einstein’s “biggest blunder” Einstein didn’t know the universe was expanding. Static (and no  ) means  = (3kc 2 /8  GR 2 ); set R = 0 and solve escape eq  c 2 = -3p (from dU = -pdV and previous eq )* For ordinary matter and light (radiation),  = Einstein added a “fudge factor.” * math given as an “appendix”.

12. Einstein called this the cosmological constant (sometimes written and sometimes  we’ll stick  to using  ) To avoid p negative first  =>  +  /8  G =  p => p  c 2  /8  G = p use new  and p in our equations => Then using  c 2 = -3p = 3kc 4 /(8  GR 2 ) for p = 0 (matter dominated) then kc 2 /R 2 =  and  =  /4  G  > 0,  > 0, what we want => means k > 0 or k = +1, p < 0     

13. Summary Einstein wanted static universe and ordinary matter density and pressure positive Couldn’t do this without adding in a fudge factor (cosmological constant,  ) For matter dominated era this ( ,ordinary > 0) drives  > 0 and then k = +1 and p < 0 ! 

14. But then dU = -pdV with p The cosmological constant gives us in modern models, and accelerating universe! (more later)  

15. But Einstein’s model is not stable!

16. R 2 /R 2 = (G8  0 (R 0 /R) 3 +   c 2 /R 2  R decreases, then density (PE) term takes over and “rules” over  c 2 /R 2 collapse occurs R increases, then density term drops, no longer “rules”,  c 2 /R 2 rules, since  c 2 /R 2 > 0 is “repulsive”, expansion rules as  /3 will rule over c 2 /R 2 as R increases => This situation is not stable. KE PE (attractive) Repulsive for  /3 > c 2 /R 2

17. This is OK, because we observe the universe expanding today and even accelerating

18. We have two Omegas  m,   and,    q =  m /2    ;  m +    k  q =  m /2     m /2     /2  (3/2)   k = + 1 can still expand forever See page 312, table 11.2 Also from can see that q 0 < 0 means R 0 > 0, accelerating universe From Lecture 11 slide 2, see that   will overtake  m and we’ll always progress to an accelerating universe for k = 0 q 0 =  R 0 R 0 /R 0 2...  /8  G  c

19. Games people play with  Lemaitre universe ^ Inflation in an “empty” universe (father of current day inflation models) Current day models of “accelerating universe”

20. => Lamaitre Universe => Allows universe to be (much) older than implied by “normal expansion” equations. This allows for “seeing” in the backs of our heads (or complementary images of distant objects 180 degrees away.) But this effect is not seen in nature=> we reject these models. ^

21. Lemaitre see book (page 311) for good plot t R ^  causes slow down  takes over Cooking phase

22. De Sitter Universe, father of Modern Inflation Assume q =  1 and flat (k = 0), =>  m = 0 ! = “empty universe” => From escape equation => R = Rx((sqrt(  /3)) or R = R i exp(tx[sqrt(  /3)]) Key points are: R grows exponentially with time AND at t = 0 R has a finite size = no BB “singularity” = R does not go to an actual “point” as t goes to 0.

23. R can gets so large it exceeds the speed of light (by a lot), more on this later..

24. Appendix: U =  c 2 R 3,dU =  c 2 3R 2 dR + d  (c 2 R 3 ) =-pdR 3 = -p3R 2 dR For our static universe,  = 3c 2 /8  GR 2 and d  c 2 R 3 = -(2dR/R 3 )(3/8  G)c 2 R 3 x(R 2 /R 2 ) = -2c 2  R 2 dR => -2c 2  R 2 dR+3R 2  c 2 dR = R 2  c 2 dR = -p3R 2 dR Or  c 2 = -3p