John D John D BarrowBarrow

The Maths Of Whole The Maths Of Whole UniversesUniverses

A New Piece of GravityA New Piece of Gravity

- the ‘cosmological constant’ – does it exist?- the ‘cosmological constant’ – does it exist?

Gab + gab = Tab where = 8GN/c4 Geometry = mass-energy

Einstein’s Static UniverseEinstein’s Static Universe

(1917)(1917)

De Sitter’s Accelerating UniverseDe Sitter’s Accelerating Universe

Always expandingAlways expandingexponential curve R = exp[texponential curve R = exp[t/3]/3]

No matter – only No matter – only It has no beginning and no endIt has no beginning and no end

(1917)(1917)Willem De SitterWillem De Sitter

1872-19341872-1934

Friedmann’s universesFriedmann’s universes

1922,19241888-1925

Einstein’s Static UniverseEinstein’s Static Universeis unstableis unstable

Georges LemaîtreGeorges Lemaître(1894-1966)(1894-1966)

Einstein: ‘the biggest blunder of my life’Einstein: ‘the biggest blunder of my life’

Lemaître's UniverseLemaître's Universe

The best description of The best description of the visible universethe visible universe todaytoday

1927

> 0> 0

Tolman’s Oscillating Tolman’s Oscillating UniverseUniverse

Richard Tolman and Einstein at Cal Tech

1932

Tolman includes theTolman includes the22ndnd Law of thermodynamics: Law of thermodynamics:

Oscillations growOscillations grow(1932)(1932)

JDB + Dabrowski include theJDB + Dabrowski include thecosmological constant:cosmological constant:Oscillations always end Oscillations always end

(1995)(1995)

Kasner’s Anisotropic UniverseKasner’s Anisotropic Universe

Edward KasnerEdward Kasner(1878-1955)(1878-1955)

19211921

Volume abc time

(a,b,c) = (t(a,b,c) = (tpp, t, tqq, t, trr)) p + q + r = 1 p + q + r = 1 andand p p22 + q + q22 + r + r22 = 1 = 1-1/3 -1/3 p p 0 0 q q 2/3 2/3 r r 1 1

How Kasner Gave Us GoogleHow Kasner Gave Us GoogleKasner’s 9-yr old nephew Milton Sirotta invented the names

Googol = 10Googol = 10100100

Googolplex = 10Googolplex = 10GoogolGoogol

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.comwhen 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 HQThe Googleplex HQ

expansionexpansion shearshear rotationrotation

The Ten UniversesThe Ten UniversesOfOf

Bianchi and TaubBianchi and Taub

Shear DistortionRotation

Gravitational Waves

But

Everyone sees the same historyEveryone sees the same historySpatially homogeneousSpatially homogeneous

Ordinary Differential EquationsOrdinary Differential Equations

space

time

The End of a closed Universe will not be SimultaneousThe End of a closed Universe will not be Simultaneous

Misner’s Mixmaster UniverseMisner’s Mixmaster Universe

An infinite number of things happen in a finite time as t An infinite number of things happen in a finite time as t 0 0

1969

x 1/x - [1/x] : 0 < x < 1 continued fraction map

y = x sin(1/x)

Gauss’s Probability DistributionGauss’s Probability Distribution

xxn+1 n+1 = 1/x= 1/xnn – [1/x – [1/xnn]]

As n the probability of outcome x tends to p(x) = 1/[(1+x)ln2] : p(x) = 1/[(1+x)ln2] : 00

11 p(x)dx = 1 p(x)dx = 1Error is < (0.7)n after n iterations

p(x)

x

In aLetter to Laplace

30th Jan 1812‘a curious problem’

that had occupied him for 12 years

Distribution of the fractional

parts

xxn+1n+1 = 1/x = 1/xnn – [1/x – [1/xnn] = T(x] = T(xnn))

T(x)

x

n stepsn steps = = initialinitial exp[ht]: where h = exp[ht]: where h = 22/[6(ln2)/[6(ln2)22] ] 3.45 3.45

ldT/dxl = 1/x2 > 1

as 0 < x < 1

T(x) =1/x – kT(x) =1/x – k

(1-k)(1-k)-1-1<x<k<x<k-1-1

k integerk integer

Gödel's Rotating UniverseGödel's Rotating Universe

Allows time travelAllows time travelto occur !to occur !

1949

The Big Bang UniversesThe Big Bang Universes

The Evidence of a Hot Early HistoryThe Evidence of a Hot Early History

A temporary lambdaA temporary lambda

The Inflationary UniverseThe Inflationary Universe

1981

Even Smaller BeginningsEven Smaller Beginnings

Chaotic InflationChaotic Inflation

Geography is more complicated than we thoughtGeography is more complicated than we thought

Eternal InflationEternal Inflation

More than 10More than 1010107777

by-universes from our patch aloneby-universes from our patch alone

What is the What is the Probability of someProbability of some‘‘universe’ arising in universe’ arising in

the Multiverse?the Multiverse?N(A)/N(B) = /Or – maybe you could take the limit? - But….Or – maybe you could take the limit? - But….

Fraction of even numbers in 1,2,3,4,5,6,7,8,… Fraction of even numbers in 1,2,3,4,5,6,7,8,… 1/2 1/2

Fraction of even numbers in 1,3,2,5,7,4,9,11,… Fraction of even numbers in 1,3,2,5,7,4,9,11,… 1/3 1/3

Answer depends on how you count the universesAnswer depends on how you count the universes

The Probability Measure ProblemThe Probability Measure Problem

Any Beginning in a Quantum Universe?Any Beginning in a Quantum Universe?

Lemaître's universe describes our visible universe to high precisionLemaître's universe describes our visible universe to high precision

The Universe is Accelerating AgainThe Universe is Accelerating Again

> 0

Dark Energy Dominates the UniverseDark Energy Dominates the Universe

The Return of Einstein’s Lambda ?The Return of Einstein’s Lambda ?