1. John D Barrow The Maths Of Whole Universes

3. A New Piece of Gravity  - the ‘cosmological constant’ – does it exist? G ab +  g ab =  T ab where  = 8  G N /c 4 Geometry = mass-energy

4. Einstein’s Static Universe (1917)

5. De Sitter’s Accelerating Universe Always expanding exponential curve R = exp[t  /3] No matter – only  It has no beginning and no end (1917) Willem De Sitter 1872-1934

6. Friedmann’s universes 1922,1924 1888-1925

7. Einstein’s Static Universe is unstable Georges Lemaître (1894-1966) Einstein: ‘the biggest blunder of my life’

8. Lemaître's Universe The best description of the visible universetoday the visible universe today 1927  > 0

9. Tolman’s Oscillating Universe Richard Tolman and Einstein at Cal Tech 1932

10. Tolman includes the 2 nd Law of thermodynamics: Oscillations grow (1932) JDB + Dabrowski include the cosmological constant: Oscillations always end (1995)

11. Kasner’s Anisotropic Universe Edward Kasner (1878-1955)1921 Volume  abc  time (a,b,c) = (t p, t q, t r ) p + q + r = 1 and p 2 + q 2 + r 2 = 1 p + q + r = 1 and p 2 + q 2 + r 2 = 1 -1/3  p  0  q  2/3  r  1

12. How Kasner Gave Us Google Kasner’s 9-yr old nephew Milton Sirotta invented the names Googol = 10 100 Googolplex = 10 Googol In 1996 Larry Page and Sergey Brin called their new internet search engine ‘BackRub’. Sean Anderson suggested Googolplex. Shortened it to Googol but Anderson mistyped Google.com when he searched to see if Googol.com was available. It was, Brin liked it better, so Google.com was registered on 15 Sept 1997 The Googleplex HQ

13. expansionshear rotation

14. The Ten Universes Of Bianchi and Taub Shear Distortion Rotation Gravitational Waves But Everyone sees the same history Spatially homogeneous Ordinary Differential Equations space time

15. The End of a closed Universe will not be Simultaneous

16. Misner’s Mixmaster Universe An infinite number of things happen in a finite time as t  0 1969 x  1/x - [1/x] : 0 < x < 1 continued fraction map y = x sin(1/x)

17. Gauss’s Probability Distribution x n+1 = 1/x n – [1/x n ] As n   the probability of outcome x tends to p(x) = 1/[(1+x)ln2] :  0 1 p(x)dx = 1 Error is < (0.7) n after n iterations p(x) x In a Letter to Laplace 30 th Jan 1812 ‘a curious problem’ that had occupied him for 12 years Distribution of the fractional parts

18. x n+1 = 1/x n – [1/x n ] = T(x n ) T(x) x  n steps =  initial  exp[ht]: where h =  2 /[6(ln2) 2 ]  3.45 ldT/dxl = 1/x 2 > 1 as 0 < x < 1 T(x) =1/x – k (1-k) -1 <x<k -1 k integer

19. Gödel's Rotating Universe Allows time travel to occur ! 1949

20. The Big Bang Universes

21. The Evidence of a Hot Early History

22. A temporary lambda The Inflationary Universe 1981

23. Even Smaller Beginnings

25. Chaotic Inflation Geography is more complicated than we thought

26. Eternal Inflation More than 10 10 77 by-universes from our patch alone

27. What is the Probability of some ‘universe’ arising in the Multiverse? N(A)/N(B) =  /  Or – maybe you could take the limit? - But…. Fraction of even numbers in 1,2,3,4,5,6,7,8,…  1/2 Fraction of even numbers in 1,3,2,5,7,4,9,11,…  1/3 Answer depends on how you count the universes The Probability Measure Problem

28. Any Beginning in a Quantum Universe?

29. Lemaître's universe describes our visible universe to high precision The Universe is Accelerating Again  > 0

30. Dark Energy Dominates the Universe The Return of Einstein’s Lambda ?