Mathematics & Cosmology -Past, Present & Future Emili Elizalde National Research Council of Spain Institut de Ciències de l’Espai (ICE/CSIC) Institut d’Estudis Espacials de Catalunya (IEEC) Campus Universitat Autònoma de Barcelona (UAB) Colloquium CRM: Mathematics and Space Science, 14 Dec 2011

Trying to Understand Trying to Understand Number: Lebombo b 29;37m, Wolf 11x5+2;30m, Ishango 22m Number: Lebombo b 29;37m, Wolf 11x5+2;30m, Ishango 22m Pre-Socratic philosophers: substance, number, power, infinity, movement, being, atom, space, time,... Mathesis 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 Plato: Ta mathemata (pillars of knowledge) ”Maβ, Zahl und Gewicht” M Folkerts, E Knobloch, K Reich Pythagoras school: ”all things are numbers” Pythagoras school: ”all things are numbers” Galileo Galilei: ”the book of nature is written in mathematical language” 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” 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” 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” Eugene Wigner: ”the irrational effectivness of mathematics in all natural sciences”

Eratostenes ( Ἐ ρατοσθένης) Cirene, 276 BC - Alexandria, 194 BC Cirene276 BCAlexandria194 BC Cirene276 BCAlexandria194 BC 360º/7,2º= 50 x800 Km = Km

El Proyecto Eratóstenes

Galileo Galilei (Pisa, 15 febraio Firenze, 8 genaio 1642)Pisa15 febraio1564Firenze8 genaio1642 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 pendulum 3. Law of uniformily accelerated movement

But... the Universe was static !! (…and very small) But... the Universe was static !! (…and very small) What is λ ? Non-physical What is λ ? Non-physical At that time: UNIVERSE = MILKY WAY (+ nebulae) At that time: UNIVERSE = MILKY WAY (+ nebulae) Ptolemy 150 DC, Almagest: mentions 5 nebulae Ptolemy 150 DC, Almagest: mentions 5 nebulae Persian, Arabic, Chinese astronomers: more nebulae Persian, Arabic, Chinese astronomers: more nebulae Edmund Halley 6 in 1715, Charles Messier 103 in 1781 Edmund Halley 6 in 1715, Charles Messier 103 in 1781 William and Caroline Herschel 3 catalogs, , a total of 2510 nebulae William and Caroline Herschel 3 catalogs, , a total of 2510 nebulae And then, Edwin Hubble, in , saw Andromeda nebulae (.8 Mly) was far beyond our Milky Way: nebulae were other galaxies! And then, Edwin Hubble, in , 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 Henrietta S Leavitt (Ed Pickering’s Harvard Harem), period-luminosity relationship of Cepheid variable stars Hertzsprung-Russell Diagramm

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

Karl Schwarzschild: Black Hole solution (22 Dec 1915) Karl Schwarzschild: Black Hole solution (22 Dec 1915) Willem de Sitter: massless univer static sol (just cc,’17) Willem de Sitter: massless univer static sol (just cc,’17) Alexander Friedmann: expanding universe sol (1922) 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 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 3 rd Programme, 18:30 GMT, 28 March 1949) Led to Big Bang theory (Fred Hoyle, BBC radio 3 rd 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) 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 ) Friedmann-Lemaître-Robertson-Walker (FLRW ) A Riess B Schmidt ’98, S Perlmutter ’99: acceler expan! A Riess B Schmidt ’98, S Perlmutter ’99: acceler expan!

Some of my Books

J Soto E GaztañagaP 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 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 Gracias Mercès

Friedmann equation (matter,radiation,curvature) Multiply by a 2 to get When a increases, each term on rhs decreases: the universe should be decelerating (a decreases). a t Big Bang Sean Carroll, Caltech SSI 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 Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Tes ts 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:

F(R) Gravity Cosmic ExpansionCosmic StructureSolar-System Test Cosmological TestsLocal 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 w eff (z): Current observational constraints (WMAP7+BAO+SN):  construct q j : other cosmological parameters f ini : initial condition of f(R)

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