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Mathematics & Cosmology �-Past, Present & Future

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Emili Elizalde

National Research Council of Spain

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Institut de Ciències de l’Espai (ICE/CSIC)

Institut d’Estudis Espacials de Catalunya (IEEC)

Campus Universitat Autònoma de Barcelona (UAB)

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Colloquium CRM: Mathematics and Space Science, 14 Dec 2011

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Trying to Understand

  • Number: Lebombo b 29;37m, Wolf 11x5+2;30m, Ishango 22m
  • Pre-Socratic philosophers: substance, number, power, infinity, movement, being, atom, space, time, ... Mathesis
  • Plato: Ta mathemata (pillars of knowledge) ”Maβ, Zahl und Gewicht” M Folkerts, E Knobloch, K Reich
  • Pythagoras school: ”all things are numbers”
  • Galileo Galilei: ”the book of nature is written in mathematical language”
  • Emmanuel Kant: ”the problem is to make inteligible the idea itself that our Universe is inteligible”
  • Albert Einstein: ”the eternal mystery of the world is its comprehensibility; the fact that the Universe is comprehensible is a miracle”
  • Eugene Wigner: ”the irrational effectivness of mathematics in all natural sciences”

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Eratostenes (Ἐρατοσθένης) �CireneCirene, 276 BCCirene, 276 BC - AlexandriaCirene, 276 BC - Alexandria, 194 BC

360º/7,2º= 50 x800 Km = 40.000 Km

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El Proyecto Eratóstenes

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Galileo Galilei

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

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

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*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 pendulum�3. Law of uniformily accelerated movement

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  • But... the Universe was static !! (…and very small)
  • What is λ ? Non-physical
  • At that time: UNIVERSE = MILKY WAY (+ nebulae)
  • Ptolemy 150 DC, Almagest: mentions 5 nebulae
  • Persian, Arabic, Chinese astronomers: more nebulae
  • Edmund Halley 6 in 1715, Charles Messier 103 in 1781

William and Caroline Herschel 3 catalogs, 1786-1802, a total of 2510 nebulae

  • And then, Edwin Hubble, in 1922-1924, saw Andromeda nebulae (.8 Mly) was far beyond our Milky Way: nebulae were other galaxies!
  • Henrietta S Leavitt (Ed Pickering’s Harvard Harem), period-luminosity relationship of Cepheid variable stars Hertzsprung-Russell Diagramm

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Henrietta Leavitt

Ed Pickering’s Harvard Harem

… or Harvard Computers

1913

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  • Karl Schwarzschild: Black Hole solution (22 Dec 1915)
  • Willem de Sitter: massless univer static sol (just cc,’17)
  • Alexander Friedmann: expanding universe sol (1922)
  • Georges Lemaître: expanding universe (MIT 1925, AF solution); visited Vesto Slipher (Lowell Obs, Arizona, 1912 redshifts) and Edwin Hubble (Mount Wilson, Pasadena); Keeler-Slipher-Campbell, 1918
  • Led to Big Bang theory (Fred Hoyle, BBC radio 3rd Programme, 18:30 GMT, 28 March 1949)
  • Fred Hoyle, Thomas Gold, Hermann Bondi: SteadyState ’48, “C-field” with a negative pressure, to be consistent with energy conservation (anticipated inflation)
  • Friedmann-Lemaître-Robertson-Walker (FLRW 1931-37)
  • A Riess B Schmidt ’98, S Perlmutter ’99: acceler expan!

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��Some�of my�Books��

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J Soto

E Gaztañaga

P Fosalba

Some former Students and PostDocs:

Joan Soto (CU,UB): Heavy Quarks

Enrique 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, µcircuits

Sergi R Hildebrandt (SLAC): Theoretical Cosmology (PLANCK)

Klaus Kirsten (Prof, Math D, Baylor Texas): Zeta Functions

Sergei D Odintsov (ICREA, ICE): Quantium Gravity, Forbes 5th Russia

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K Kirsten

S D Odintsov

S Gómez

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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 ...

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Thank You ��Gracias��Mercès

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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)

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a

t

Big Bang

Sean Carroll, Caltech

SSI 2009

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

as standardizable candles

SN 1994d

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Result: supernovae are

dimmer than expected

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The universe is not

decelerating

It is accelerating

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Cannot be explained by matter+radiation (see before)

Riess, Schmidt et al. ‘98 Perlmutter et al. ‘99

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F(R) Modified Gravity

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

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F(R) Gravity

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)”

APPROACH

Consider the expansion H(t)

parametrized via

the CPL weff(z):

Current observational constraints

(WMAP7+BAO+SN):

⇒ construct

qj : other cosmological parameters

fini : initial condition of f(R)

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F(R) Gravity

Cosmic Expansion

Cosmic Structure

Solar-System Test

Cosmological Tests

Local Tests

❖ Key quantities distinguishing GR & F(R)

∙ Perturbed metric:

∙ Evolution eqn. of matter density perturbation:

defined in

late-time,

sub-horizon