Mathematics & Cosmology - Past , Present & Future

1. Mathematics & Cosmology-Past, Present & Future EmiliElizalde NationalResearch Council of Spain Institut de Ciències de l’Espai (ICE/CSIC) Institutd’EstudisEspacials de Catalunya (IEEC) Campus UniversitatAutònoma de Barcelona (UAB) Colloquium CRM: Mathematics and SpaceScience, 14 Dec 2011

2. Trying to Understand • Number: Lebombo b 29;37m, Wolf11x5+2;30m, Ishango22m • Pre-Socraticphilosophers:substance, number, power, infinity, movement, being, atom, space, time, ... Mathesis • Plato:Ta mathemata(pillars of knowledge) ”Maβ, ZahlundGewicht” M Folkerts, E Knobloch, K Reich • Pythagorasschool:”all thingsarenumbers” • Galileo Galilei:”thebook of nature is written in mathematicallanguage” • Emmanuel Kant:”theproblem is to makeinteligiblethe idea itselfthatourUniverse is inteligible” • Albert Einstein:”the eternal mystery of theworld is itscomprehensibility; thefactthattheUniverse is comprehensible is a miracle” • EugeneWigner:”theirrationaleffectivness of mathematics in all natural sciences”

3. Eratostenes (Ἐρατοσθένης) Cirene, 276 BC - Alexandria, 194 BC 360º/7,2º= 50 x800 Km = 40.000 Km

4. El Proyecto Eratóstenes

5. (Pisa, 15 febraio1564- Firenze, 8 genaio1642) Galileo Galilei 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

7. But... the Universe was static !! (…and very small) • Whatisλ ? Non-physical • At that time: UNIVERSE = MILKY WAY (+ nebulae) • Ptolemy 150 DC, Almagest:mentions5nebulae • Persian, Arabic, Chinese astronomers: more nebulae • Edmund Halley 6 in 1715, Charles Messier103 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(EdPickering’s Harvard Harem), period-luminosityrelationshipof Cepheid variable starsHertzsprung-RussellDiagramm

8. Henrietta Leavitt Ed Pickering’s Harvard Harem 1913 … or Harvard Computers

9. Karl Schwarzschild: Black Holesolution (22 Dec 1915) • Willem de Sitter: masslessuniverstatic sol (just cc,’17) • Alexander Friedmann: expandinguniverse sol (1922) • Georges Lemaître: expanding universe (MIT 1925, AF solution); visited VestoSlipher(Lowell Obs, Arizona, 1912 redshifts) and Edwin Hubble(Mount Wilson, Pasadena); Keeler-Slipher-Campbell, 1918 • Led to Big Bang theory (Fred Hoyle, BBC radio 3rdProgramme, 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 (anticipatedinflation) • Friedmann-Lemaître-Robertson-Walker (FLRW 1931-37) • A Riess B Schmidt ’98, SPerlmutter ’99: accelerexpan!

21. Someof myBooks

22. K Kirsten E Gaztañaga P Fosalba S Gómez J Soto S D Odintsov SomeformerStudents and PostDocs: Joan Soto (CU,UB): Heavy Quarks Enrique Gaztañaga(PI,CSIC): ObservationalCosmology (Sloan,DES) Sergi Gómez (TU,URV): EngineeringInformatics (NN) Pablo Fosalba(IT,CSIC): ObservationalCosmology (PLANCK) August Romeo (UB): CognigtiveNeuroscience, Casimir Effect Sergio Leseduarte(Jena, Munich): Corporal motiondynamics, µcircuits Sergi R Hildebrandt(SLAC): TheoreticalCosmology (PLANCK) Klaus Kirsten(Prof, Math D, Baylor Texas): Zeta Functions Sergei D Odintsov(ICREA, ICE): QuantiumGravity, Forbes 5th Russia

26. EL ALEPH JORGE LUIS BORGES O God, I could be bounded in anutshell 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 ...

27. ThankYouGraciasMercès

28. Multiply by a2 to get When a increases, each term on rhs decreases: the universe should be decelerating (a decreases) . Friedmann equation (matter,radiation,curvature) a Sean Carroll, Caltech SSI 2009 Big Bang t

29. Two groups looked for the ‘deceleration’ of the universe expansion, using type Ia supernovae as standardizable candles SN 1994d

30. 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

31. Je-An Gu顧哲安 F(R) Modified Gravity Explain cosmic acceleration Model (parametrize) deviations from GR Cosmic Expansion Cosmic Structure Solar-System Test arXiv:1009.3488 Goals as demanded by obs cosmology, needs to pass as a theory of modified gravity, needs to pass Cosmological Tests Local Tests

32. F(R) Gravity ”designer f(R)” APPROACH Current observational constraints (WMAP7+BAO+SN): Consider the expansion H(t) parametrized via the CPL weff(z): Cosmic Structure Cosmic Expansion  construct Solar-System Test qj : other cosmological parameters fini : initial condition of f(R) Cosmological Tests Local Tests For a given expansion history H(t), one can reconstruct f(R) generates the required H(t).

33. F(R) Gravity Key quantities distinguishing GR & F(R) defined in Perturbed metric: Evolution eqn. of matter density perturbation: Cosmic Structure Cosmic Expansion late-time, sub-horizon Solar-System Test Cosmological Tests Local Tests