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
• Number: Lebombo b 29;37m, Wolf 11x5+2;30m, Ishango 22m
• Pre-Socratic philosophers: substance, number, power, infinity,
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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”
Eratostenes (Ἐρατοσθένης)
Cirene, 276 BC - Alexandria, 194 BC
360º/7,2º= 50 x800 Km = 40.000 Km
El 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 pendulum
3. Law of uniformily accelerated movement
Mm
F G 2
r
E  mc
1
R   2 g 
2
8

G
R g    4 T 
c
tot  r  m  k  
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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
Henrietta Leavitt
Ed Pickering’s Harvard Harem
1913
… or
Harvard Computers
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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!
Some
of my
Books
K Kirsten
J Soto
E Gaztañaga
P Fosalba
S Gómez
S D Odintsov
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
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 a2 to get
When a increases, each
term on rhs decreases:
the universe should be
.
decelerating (a decreases)
Sean Carroll, Caltech
SSI 2009
a
Big Bang
t
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
Je-An Gu 顧哲安
F(R) Modified Gravity
arXiv:1009.3488
1
4




Sg 
d
x

g
R

f
R

16G
Goals
Explain
cosmic acceleration
as demanded by obs cosmology,
needs to pass
Model (parametrize)
deviations from GR
as a theory of modified gravity,
needs to pass
Cosmological Tests
Cosmic Expansion
Cosmic Structure
Local Tests
Solar-System Test
1
4
Sg 
d
x  g R  f R 

16G
F(R) Gravity
Cosmological Tests
Cosmic Expansion
Local Tests
Cosmic Structure
Solar-System Test
For a given expansion history H(t),
one can reconstruct f(R)
”designer f(R)”
generates the required H(t).
APPROACH
Consider the expansion H(t)
parametrized via
the CPL weff(z):
wCPL z   w 0  w a z 1  z 
 construct
Current observational constraints
(WMAP7+BAO+SN):
(1) weff  constant  0.980  0.053
72
(2) w0  0.93  0.13, wa  0.4100..71
f R;w 0 ,w a , fini , q j 
fini : initial condition of f(R)
qj : other cosmological parameters
1
4
Sg 
d
x  g R  f R 

16G
F(R) Gravity
Cosmological Tests
Cosmic Expansion
Local Tests
Cosmic Structure
Solar-System Test
 Key quantities distinguishing GR & F(R)


 Perturbed metric:
Geff
G
defined in
ds  1 2 dt  a 1 2 ij dx dx
2
2
2
i
 Evolution eqn. of matter density perturbation:
m  2Hm  4Geff m m  0
late-time,
j