mathematics & cosmology -past, present & future emili elizalde national research council of...
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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
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