Mathematics & Mathematics & Cosmology Cosmology

--Past, Present & Past, Present & FutureFuture

Emili ElizaldeEmili ElizaldeNational Research Council of Spain National Research Council of Spain

Institut de Ciències de l’Espai (ICE/CSIC)Institut de Ciències de l’Espai (ICE/CSIC)Institut d’Estudis Espacials de Catalunya (IEEC)Institut d’Estudis Espacials de Catalunya (IEEC)

Campus Universitat Autònoma de Barcelona (UAB)Campus Universitat Autònoma de Barcelona (UAB)

Colloquium CRM: Mathematics and Space Science, 14 DecColloquium CRM: Mathematics and Space Science, 14 Dec 20112011

Trying to UnderstandTrying to Understand• Number: Number: Lebombo b Lebombo b 29;37m, 29;37m, WolfWolf 11x5+2;30m, 11x5+2;30m, Ishango Ishango

22m22m

• Pre-Socratic philosophers:Pre-Socratic philosophers: substance, number, power, substance, number, power, infinity, movement, being, atom, space, time, ... infinity, movement, being, atom, space, time, ... Mathesis Mathesis

• Plato:Plato: Ta mathemata Ta mathemata (pillars of knowledge) (pillars of knowledge) ”Ma”Maββ, , Zahl und Gewicht” Zahl und Gewicht” M Folkerts, E Knobloch, K ReichM Folkerts, E Knobloch, K Reich

• Pythagoras school:Pythagoras school: ”all things are numbers””all things are numbers”

• Galileo Galilei:Galileo Galilei: ”the book of nature is written in ”the book of nature is written in mathematical mathematical language”language”

• Emmanuel Kant:Emmanuel Kant: ”the problem is to make inteligible the idea ”the problem is to make inteligible the idea itself that our Universe is inteligible”itself that our Universe is inteligible”

• Albert Einstein:Albert Einstein: ”the eternal mystery of the world is its ”the eternal mystery of the world is its comprehensibility; the fact that the comprehensibility; the fact that the

Universe Universe is comprehensible is a miracle”is comprehensible is a miracle”

• Eugene Wigner:Eugene Wigner: ”the irrational effectivness of mathematics ”the irrational effectivness of mathematics in all natural sciences”in all natural sciences”

EratostenesEratostenes ( (Ἐρατοσθένης) Ἐρατοσθένης) Cirene, , 276 BC - - Alexandria, , 194 BC

360º/7,2º= 50 x800 Km = 40.000 Km

El Proyecto EratóstenesEl Proyecto Eratóstenes

Galileo Galilei

(Pisa, 15 febraio 1564- Firenze, 8 genaio 1642)

Astronomer, Philosopher, Mathematician and Physicist: Il libro della natura è scritto in lingua matematica o oscuro labirinto? ... Il Saggiatore

*Albert Einstein: “Galileo is the father of modern science”

*Stephen Hawking: “Galileo is probably, more than anybody else, the real responsible of the birth of modern science”

LAWS:1. First law of movement (of inertia)2. Law of isochrony of the pendulum3. Law of uniformily accelerated movement

2mcE

kmrtot

TcGgRgR

42

1 8

2r

MmGF

• But... But... the Universe was static !! (…and very small)the Universe was static !! (…and very small)

• What is What is λλ ? ? Non-physicalNon-physical

• At that time: At that time: UNIVERSE = MILKY WAY (+ nebulae)UNIVERSE = MILKY WAY (+ nebulae)

• PtolemyPtolemy 150 DC, 150 DC, Almagest:Almagest: mentions mentions 55 nebulae nebulae

• Persian, Arabic, Chinese Persian, Arabic, Chinese astronomers: astronomers: moremore nebulae nebulae

• Edmund Halley Edmund Halley 66 in 1715, in 1715, Charles Messier Charles Messier 103103 in in 17811781

WilliamWilliam and and Caroline Herschel Caroline Herschel 3 catalogs, 1786-3 catalogs, 1786-1802, a total of 1802, a total of 2510 2510 nebulaenebulae

• And then, And then, Edwin Hubble, Edwin Hubble, in 1922-1924, saw in 1922-1924, saw Andromeda nebulae (.8 Mly) was Andromeda nebulae (.8 Mly) was far beyond our far beyond our Milky Way: Milky Way: nebulae were nebulae were other galaxies!other galaxies!

• Henrietta S Leavitt Henrietta S Leavitt (Ed Pickering’s Harvard Harem), (Ed Pickering’s Harvard Harem), period-luminosity relationshipperiod-luminosity relationship of Cepheid variable of Cepheid variable stars stars Hertzsprung-RussellHertzsprung-Russell Diagramm Diagramm

Henrietta Leavitt

Ed Pickering’s Harvard Harem

… or Harvard Computers

1913

• Karl Schwarzschild: Karl Schwarzschild: Black Hole solution (22 Dec 1915)Black Hole solution (22 Dec 1915)

• Willem de Sitter: Willem de Sitter: massless univer static sol (just cc,’17)massless univer static sol (just cc,’17)

• Alexander Friedmann: Alexander Friedmann: expanding universe sol (1922)expanding universe sol (1922)

• Georges Lemaître: Georges Lemaître: expanding universe (MIT 1925, AF expanding universe (MIT 1925, AF solution); visited solution); visited Vesto Slipher Vesto Slipher (Lowell Obs, Arizona, 1912 (Lowell Obs, Arizona, 1912 redshifts) and redshifts) and Edwin Edwin Hubble Hubble (Mount Wilson, Pasadena); (Mount Wilson, Pasadena); Keeler-Slipher-Campbell,Keeler-Slipher-Campbell, 1918 1918

• Led to Led to Big Bang theory Big Bang theory (Fred Hoyle,(Fred Hoyle, BBC radio 3 BBC radio 3rdrd Programme, 18:30 GMT, 28 March 1949)Programme, 18:30 GMT, 28 March 1949)

• Fred Hoyle, Thomas Gold, Hermann Bondi:Fred Hoyle, Thomas Gold, Hermann Bondi: SteadyState SteadyState ’48, “C-field” with a negative pressure, to be consistent ’48, “C-field” with a negative pressure, to be consistent with energy conservation with energy conservation (anticipated inflation)(anticipated inflation)

• Friedmann-Lemaître-Robertson-WalkerFriedmann-Lemaître-Robertson-Walker (FLRW 1931-37) (FLRW 1931-37)

• A Riess B SchmidtA Riess B Schmidt ’98, ’98, SS PerlmutterPerlmutter ’99: ’99: acceler expan!acceler expan!

SomeSomeof myof myBooksBooks

J SotoE Gaztañaga P Fosalba

Some former Students and PostDocs:Joan Soto (CU,UB): Heavy QuarksEnrique Gaztañaga (PI,CSIC): Observational Cosmology (Sloan,DES)Sergi Gómez (TU,URV): Engineering Informatics (NN)Pablo Fosalba (IT,CSIC): Observational Cosmology (PLANCK)August Romeo (UB): Cognigtive Neuroscience, Casimir Effect Sergio Leseduarte (Jena, Munich): Corporal motion dynamics, µcircuitsSergi R Hildebrandt (SLAC): Theoretical Cosmology (PLANCK)Klaus Kirsten (Prof, Math D, Baylor Texas): Zeta FunctionsSergei D Odintsov (ICREA, ICE): Quantium Gravity, Forbes 5th Russia

K Kirsten

S D Odintsov

S Gómez

EL ALEPH JORGE LUIS BORGES O God, I could be bounded in a nutshell and count myself a King of infinite space. Hamlet, II, 2.… En la parte inferior del escalón, hacia

la derecha, vi una pequeña esfera tornasolada, de casi intolerable fulgor. Al principio la creí giratoria; luego comprendí que ese movimiento era una ilusión producida por los vertiginosos espectáculos que encerraba. El diámetro del Aleph sería de dos o tres centímetros, pero el espacio cósmico estaba ahí, sin disminución de tamaño. Cada cosa (la luna del espejo, digamos) era infinitas cosas, porque yo claramente la veía desde todos los puntos del universo ...

Thank You Thank You

GraciasGracias

MercèsMercès

Friedmann equation (matter,radiation,curvature)

Multiply by a2 to get

When a increases, each term on rhs decreases:the universe should be decelerating (a decreases)

.

a

tBig Bang Sean Carroll, CaltechSSI 2009

Two groups looked for the ‘deceleration’ of the universe expansion, using type Ia

supernovae as standardizable candles

SN 1994d

Result: supernovae are

dimmer than expected

The universe is not decelerating

It is accelerating

Cannot be explained by

matter+radiation (see before)Riess, Schmidt et al. ‘98 Perlmutter et al.

‘99

F(R) Modified Gravity RfRgxd

GSg

4

16

1

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Tests Local Tests

as demanded by obs cosmology, needs to pass

Goals

as a theory of modified gravity, needs to pass

Explain cosmic acceleration

Model (parametrize) deviations from GR

Je-An Gu 顧哲安arXiv:1009.3488

F(R) Gravity RfRgxdG

Sg4

16

1

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Tests Local Tests

For a given expansion history H(t),one can reconstruct f(R)

generates the required H(t).”designer f(R)”

APPROACHConsider the expansion

H(t) parametrized viathe CPL weff(z): zzwwzw aCPL 10

Current observational constraints (WMAP7+BAO+SN):

72.071.00 41.0 ,13.093.0 (2)

053.0980.0constant (1)

a

eff

ww

w

jinia qfwwRf ,,,; 0 construct

qj : other cosmological parameters

fini : initial condition of f(R)

F(R) Gravity RfRgxdG

Sg4

16

1

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Tests Local Tests

Key quantities distinguishing GR & F(R)

G

Geff

Perturbed metric: ji

ij dxdxadtds 2121 222

Evolution eqn. of matter density perturbation:

defined in

042 mmeffmm GH late-time,sub-horizon