Download - An overview of Cosmology
10-12 August 2005 An overview of cosmology 1
An overview of CosmologyAn overview of Cosmology
CERN Student Summer School10-12 August 2005
Julien Lesgourgues (LAPTH, Annecy)
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What is Cosmology?What is Cosmology?
Astrophysics detailed description of « small » structures
Cosmology Universe as a whole Is it static? Expanding ? Is it flat, open or closed ?What is it composed of ?What about its past and future ?
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Part I : the expanding UniverseHubble lawNewtonian gravityGeneral relativity Friedmann-Lemaître model
Part II : the standard cosmological modelHot Big bang scenarioCosmological perturbationsCosmological parameters Inflation & Quintessence
Geometry and abstraction …
Concrete predictions, results, observations !!
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Part I :Part I :The Expanding The Expanding
UniverseUniverse
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Part I : (1) - The Hubble LawPart I : (1) - The Hubble Law
First step in understanding the Universe… first telescopes: observation of nebulae 1750 : T. Wright : Milky way = thin plate of stars? 1752 : E. Kant : nebulae = other galaxies?
Galactic structure not tested before… 1923!
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1842 : Doppler effect for sound and light
1868 : Huggins finds redshift of star spectral lines
1868 – 1920 : observation of many redshift of stars and nebulae random distribution late observations : excess of z>0 for nebulae
redshift :
z = v . n / c
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1920’s : Leavitt & Shapley : cepheids period / absolute luminosity
relation
measurement of distances of stars inside the Milky Way ( ~ 80.000 lightyear)
absolute luminosity
Lapparent luminosity
l = dL/ds = L/(4r2)
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1923 : Edwin Hubble : 2,50 m telescope at Mount Wilson (CA) cepheids in Andromeda distance of nearest galaxy = 900.000 lyr
(in fact 2 Mlyr) first probe of galactic structure !!! so : excess of redshifted galaxies Universe expansion ???
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IN GENERAL : expansion center
Against « cosmological principle » (Milne): Universe homogeneous … no privileged point!
QUESTION : is any expansion a proof against homogeneity?
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ANSWER : not if v = H r linear expansion… like infinite rubber grid stretched in all directions …
Proof that linear expansion is the only possible homogeneous expansion :
vB/A = vC/B homogeneity vC/A = vC/B + vB/A = 2 vB/A linearity
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1929 : Hubble gives the first velocity / distance diagram :
H = v / r = 500 km.s-1.Mpc-1 for Hubble ( 70 km.s-1.Mpc-1 for us )1 Mpc = 3.106 lyr = 3.1022 m
1000 km.s-1
0 km.s-1
2 Mpc0 Mpc
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THE UNIVERSE IS IN HOMOGENEOUS EXPANSION
1929 : starting of modern cosmology …
Remark : what do we mean by« the Universe is homogeneous »
(cosmological principle) ?
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example of structure homogeneous after smoothing:
today : data on very large scales confirmation of homogeneity beyond ~ 30 – 40 Mpc
local inhomogeneities scatteringv
r
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Part I : (2) – Universe expansion Part I : (2) – Universe expansion from Newtonian gravityfrom Newtonian gravity
on cosmic scales, only gravitation Newton’s law = limit of General Relativity (GR)
Newton’s law should describe expansion at small distances with v = H r << c …
but historically, GR proposed the first predictions / explanations !!!
F = G m1 m2 / r2
when v << c
speed of objectspeed of liberationOR
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Newton: finite Universe infinite Universe
but how to deal with infinity?
Gauss theorem : r = - G Mr / r2
Mr = constant = (4/3) r3 mass
r2 = 2 G Mr / r - k
= (8/3) G mass r2 - kr(t)
..
.
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Newtonian expansion law : ( r / r )Newtonian expansion law : ( r / r )22 = (8 = (8GG/3) /3) massmass - k/r - k/r22
mass(t) r(t)-3
same motion as a two-body problem :
mass <
mass =
mass > k 0 non-homogeneous expansion ???
v = H r and v << c r < RH c / H
.
r(t) v
3 ( ( rr / r ) / r )22
8 8 GG
.
GR
AV
ITY
/ IN
ERTI
A
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Part I : (3) – General Relativity and Part I : (3) – General Relativity and the Friedmann-Lemaître Universethe Friedmann-Lemaître Universe
Newtonian gravity
invariant speed of light General Relativity (Einstein 1916)
no more grav ( matter distribution grav E = grav )
three basic principles :1) space-time (t,x,y,z) is curved 2) curvature matter3) free-falling bodies follow geodesics
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1) how can we define the curvature :
of a 2-D surface ? embedded in 3D stay in 2D, and use angles :
stay in 2D, and use a scaling law : dl(x1,x2)
ex: sphere projected on ellipse, dl()
plane sphere
hyperboloid
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of 3-D space ? embedded in 4D : x1
2 + x22 + x3
2 + x42 = R2
stay in 3D, but provide a scaling law, like on a planisphere : dl(x1,x2,x3)
of 4-D space-time ? one more dimension time different from space (special relativity : - + + + )
intuitive representations:
cuts (t,x), (x,y), etc., Euclidian coordinates embedded in 3D + scaling law
t x t x
y + scale(t,x,y,z)OR
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2) curvature matter :
mathematical formulation = Einstein equation
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3) free-falling bodies follow geodesics
feel only gravity given one point and one direction (not E.M., etc.) one single line such that :e.g. galaxies, light… A, B, [AB] = shortest trajectory
example 1 : geodesics a sphere :parallel great circle
NO YES
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example 2 : gravitational lensing :
A sees an image of C lensed by B :
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Newtonian gravity versus G.R. : two different theories of gravity, i.e. two ways of
describing how the presence of matter affects the trajectories of surrounding bodies…
Newton Einstein
curvaturetensor
gravitationalpotential
matter
trajectories
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applying G.R. to the Universe: some history 1916 : Einstein has formulated G.R. 1917 : Einstein, De Sitter try to build the first cosmological
models (PREJUDICE : STATIC / STATIONNARY UNIVERSE)
1922 : A. Friedmann (Ru) investigate most general 1927 : G. Lemaître (B) HOMOGENEOUS, ISOTROPIC, 1933 : Robertson, NON-STATIONNARY Walker (USA) solutions of G.R. equations
1929 : Hubble’s law (first confirmation) 1930-65 : accumulation of proofs in favour of FLRW 1965 : CMB discovery : full confirmation
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basic principles of G.R. FLRW solution
space-time is curved ???????
free-falling objects follow geodesics
???????
curvature caused by / related to
matter???????
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summary of the situation :summary of the situation :
NEWTON
matter distribution grav forces changing the trajectories
General Relativity (GR)
three basic principles :1) space-time (t,x,y,z) is curved 2) free-falling bodies follow geodesics 3) curvature matter
Friedmann-Lemaitre model = application of GR to homogeneous Universe
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1) the curvature of the FLRW Universe : Universe space-time (t,x,y,z) curved by its own
homogeneous matter density (t) HOMOGENEITY decomposition of curvature in :
1. spatial curvature of (x,y,z) at fixed t 3-D space is maximally symmetric :
2. 2-D space-time curvature of (t,x) (t,y) (t,z) accounts for the expansion
3-plane 3-sphere 3-hyperboloidFLAT CLOSED OPEN RC(t)
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scale as a function of coordinates ? COMOVING COORDINATES (t, r, ) for Euclidian space : dl2 = dr2 + r2 (d + sin2d )
for FLRW : dl2 = a2(t) + r2 (d + sin2d )
a(t) scale factor 2-D space-time curvature k spatial curvature
– k = 0 : FLAT– k > 0 : CLOSED, RC(t) = a(t) / k1/2, 0 r < 1/ k1/2
– k < 0 : OPEN , RC(t) = a(t) / (-k)1/2
dr2
(1-k r2)
O
dl(r, )
(r+dr, +d+d)
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2) the geodesics in the FLRW Universe
photons : v = c ULTRA-RELATIVISTIC
for ordinary matter : v << c NON-RELATIVISTIC
(e.g. galaxies)
1) non-relativistic matter:
dl = 0 (r, ) = constant galaxies are still in coordinate space … … but all distances are proportional to a(t) a(t) gives the expansion between galaxies (although they are still !!!) like an inflated rubber balloon with points drawn on its surface …
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2) Relativistic matter : straight line in 3-D space, dl = c dt
c2dt2 = a2(t) + r2 (d + sin2d )
EQUATION OF PROPAGATION OF LIGHT
So l = c t is WRONG :
« bending of light in the Universe »(one of the two most fundamental equations in cosmology)
various important consequences …
dr2
(1-k r2)
O x
y (r2 , t2)
(r1 , t1)
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definition of the past light-cone : in Euclidian space :
= constant
re = c (t0 – te)
in Friedmann universe : = constant
dr c (1-kr)1/2
dt a(t)=
te
t0
re e
we are here
approximatelylinear region
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observable consequences of propagation of light equation:
the redshift :
z = = 0 / e – 1
z = a(t0) / a(te) – 1
remark 1 : Newtonian : z = v / c 1 G.R. : no limit, as observed …
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remark 2 : at short distance, we can recover the Hubble law ( z = v / c = H r / c )
then :
so : Hubble parameter = ( H0 = 70 km.s-1.Mpc –1)
te
t0
re e
ZOOM
nearbygalaxy
t0 – te = dt = dl / c
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2) the angular diameter-redshift relation
Euclidian space :
dl = r d with r = v / H = z c / H
G.R. :
dl = a(te) re d with re from
if dl is known, measurement of (d, z) k, a(t)
dldr
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CLOSED UNIVERSE : objects seen under larger angle
OPEN UNIVERSE : objects seen under smaller angle
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3) the luminosity distance-redshift relation
Euclidian space : with r = z c / H
G.R. : with re from prop. of light
if L is known, measurement of (l, z) k, a(t)
L absolute luminosityrl apparent
luminosity
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3) relation between matter and curvature :
FRIEDMANN LAW
Remark : for non-relativistic matter, E = m c2 / c2 = mass
then Friedmann law looks similar to Newtonian expansion law, but CRUCIAL DIFFERENCES :
1) a(t) r(t) : very different interpretation
2) k 0 not in contradiction with homogeneity
3) accounts for non-relativistic and relativistic matter
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NON-RELATIVISTICv << c
E = m c2
sphere with fixed comobile r fixed particle number
EV = constant = EV / V a(t)-3
ULTRA-RELATIVISTICv = c
E = ħ = ħ c /
V = 4/3 r3 a(t)3
EV 1 / 1 / a(t)
= EV / V a(t)-4
r
FRIEDMANN LAW is the same but DILUTION RATE is different
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.
. . .
. ..
in fact, in G.R., curvature matter relation given by
EINSTEIN EQUATION G = 8G T
in the FLRW solution : Friedmann law
EINSTEIN EQUATION conservation equation : = - 3 (a / a) ( + p )
1) non-relativistic : v << c p 0 / = - 3 a / a a(t)-3
2) ultra-relativistic : v = c p = / 3 / = - 4 a / a a(t)-4
3) in QFT : vacuum with p = - = 0 = constant
« cosmological constant »
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summary of the situation :
basic principles of G.R. FLRW solution
space-time is curved2 types of curvatures : {a(t), k}
free-falling objects follow geodesics
• non-relativistic matter : dl = 0 • ultra-relativistic matter : dl = c dt BENDING OF LIGHT EQUATION
curvature caused by / related to
matter
relation {a(t), k} (t) FRIEDMANN LAW + conservation equations
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Part II :Part II :The Standard The Standard
Cosmological ModelCosmological Model
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decomposition of quantities: (t,r) = (t) + (t,r) p(t,r) = p(t) + p(t,r) curvature = {a(t), k} + {(t,r), etc.}
HOMOGENEOUS PERTURBATIONS BACKGROUND
THEORY OF LINEAR PERTURBATIONS
Initialconditions
INFLATIONpart II - 4
HOMOGENEOUS COSMOLOGYpart II - 1
LINEAR PERT.part II - 2
NON-LINEARPERT.
_ _
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Part II : (1) – Homogeneous cosmologyPart II : (1) – Homogeneous cosmology
the evolution of the Universe depends : on SPATIAL CURVATURE
on the density of :
RADIATION : ultra-relativistic particles p = / 3 a-4
( photons, massless ’s, … )
MATTER : non-relativistic bodies p = 0 a-3
( galaxies, gas clouds, … )
COSMOLOGICAL CONSTANT [t-2] p = - = constant = c2 / (8G) ( vacuum ? … ? )
…
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Friedmann law:
most « complete » scenario :
phases can be skipped, but order cannot change
RADIATION DOMINATION : a t1/2 H = 1 / 2 t MATTER DOMINATION : a t2/3 H = 2 / 3 t CURVATURE DOMINATION :
k < 0 (open) : a t H = 1 / t k > 0 (closed) : a 0, then a < 0 or
VACUUM DOMINATION : a exp(/3t)1/2 H constant
BIG BANG
. .
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Friedmann law:
most « complete » scenario :
phases can be skipped, but order cannot change
RADIATION DOMINATION : a t1/2 H = 1 / 2 t MATTER DOMINATION : a t2/3 H = 2 / 3 t CURVATURE DOMINATION :
k < 0 (open) : a t H = 1 / t k > 0 (closed) : a 0, then a < 0 or
VACUUM DOMINATION : a exp(/3t)1/2 H constant
BIG BANG
. .
Future of the Universe:if = 0 :
if k < 0 or k = 0 indefinite decelerated expansionif k > 0 recollapse (BIG CRUNCH)
if 0 : k indefinite accelerated expansion
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the matter budget :
if we can measure {R, M, k, } today,
we can extrapolate back …
today :
flatness condition : 0 R + M + = 1
then : R0 + M
0 + 0 c0 X = X / c
0
so far : COSMOLOGICAL 4 independent parameters SCENARIOS {R , M , , H0}
MATTER BUDGET EQUATION
3 H02 c2
8G
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COLD or HOT BIG BANG ??? 1929–65 : no decisive observation in favour of Friedmann model
(apart from accumulation of redshifts)
works in cosmology remain marginal
but spectacular progress in particle physics…
studies based on the most simple possible scenario: Universe contains only non-relativistic matter evolution under the laws of nuclear physics
between Big Bang and today
COLD BIG BANG SCENARIO
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COLD BIG BANG : H2 = (8G/3c2) M a-3 t-2
NUCLEOSYNTHESIS : ensemble of nuclear reactions
p Hn De- 3He , 4HeLi , etc.
freeze-out due to expansion
timeNUCLEOSYNTHESIS RECOMBINATION
e-
pn
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pioneering works on nucleosynthesis :
1940 : Gamow et al. (USSR USA)
1964 : Zel’dovitch et al. (USSR)
1965 : Hoyle & Taylor (UK)
1965 : Peebles et al. (USA)
COLD BIG BANG no hydrogen
need to change H(tnucleo)
add relativistic matter (photons) with R >> M
HOT BIG BANG
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HOT BIG BANG :
H2 = (8G/3c2) (R + M)
& recombination
pn
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photon spectrum : before recombination, thermal equilibriumn(E)
blackbody spectrum :
T a-1 <E> E=ħc/
after recombination, Planck spectrum frozen and redshifted
so T0 a0 = Tnucleo anucleo
Gamow, Peebles et al. :
nucleosynthesis T0 1-10 K 0 1-10 mm
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discovery of the Cosmic Microwave Background
A. Penzias & R. Wilson, 1964, Bell laboratories (1964)
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discovery of the Cosmic Microwave Background
A. Penzias J. Peebles (Bell) (Princeton) B. Burke Ken Turner (MIT) (MIT)
publications by Penzias & Wilson and Peebles Nobel Prize for Penzias & Wilson … confirmation of HOT BIG BANG !!!
CMB = 25% of TV set noise…
listen to it at http//:www.bell-labs.com/project/feature/archives/cosmology/
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thermal evolutionthermal evolution of the Universe of the Universet T ¼
Planck time inflation ? 10-36 s 1018 GeV
GUT 10-32 s 1016 GeV
EW bayogenesis ? 10-6 s 1015 K 100 GeV
quark-hadron 10-4 s 1012 K 100 MeV
nucleosynthesis 1-1000 s 109-10 K 0.1 - 1 MeV
equality 104 yr 104 K 1 eV
decoupling 105 yr 2500 K 0.1 eV
structure formation 105 yr t0
today curvature / ? t0 13 Gyr 3 K 3.10-4 eV
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is there a curvature / domination today ? structure formation long-enough M.D. m 0.2
if k 1 or 1, curvature / domination started recently :
RD MD
k ? equality today a 10-3 a0 a0
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is there a curvature / domination today ? HOW CAN WE KNOW ?
k change k apparent luminosity / z relation SNIa a(t) angular diameter / z relation CMB
age of the Universe (measured with redshift of quasars)
t
2 / 3t H0 ( / c)½
- 1 / t
equality today t 104 yr 9 Gyr to 15Gyr ??
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DARK MATTER : galaxy rotation curves :
DM halo other compelling evidences nature of DM : non-luminous baryons ?
Hot dark matter (neutrinos) ? CDM : WIMPS (neutralinos) ? axions ?
mass(r) = b I(r)
grav(r) = 8G mass
v2(r) = r (grav/r)
z(r) I(r)
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so far :COSMOGICAL
SCENARIO
5 INDEPENDENTPARAMETERS
{R , B , CDM , , H0}
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Part II : (2) – cosmological perturbationsPart II : (2) – cosmological perturbations Universe not completely homogeneous even on large
scales … description of matter inhomogeneities ??
(clusters of galaxies, superclusters …) description of CMB temperature anisotropies ??
information on cosmological scenario measurement of cosmological parameters
this section : overview of mathematical framework intuitive description of main phenomena
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linear perturbation theory : x, x(t, r) = x(t) + x(t, r)
HOMOGENEOUS PERTURBATION BACKGROUND
CMB temperature homogeneity perturbations linear at least for t < tdec
x(t, r) x(t, r) / x(t) << 1
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Einstein equations (matter curvature) background equations :
Friedman law conservation equations
{, , CDM , B} {a(t) , k}
perturbation equations :{, , CDM , B} {curvature perturbations}
t, r) grav inside RH
partial derivative equations Fourier transformation …
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comoving Fourier space
xk (t) dr3 e-i k.r x(t, r)
comoving Fourier wavenumber k comoving wavelength 2k physical wavelength (t) = 2a(t)k
independent perturbation equations :
{, , CDM, B
}
linear system of ordinary differential equations… even at equilibrium, each wavelength redshifted
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stochastic theory : random initial conditions : P( x
k (t0) ) evolution under differential equation :
(d2/dt2) xk + … (d/dt) x
k + … xk = … Y
k + …
Early Universe gaussian distributions linearity : shape P( x
k (t) ) is preserved differential equation = evolution of r.m.s.
e.g.: xk P( x
k (t) ) x
k
t t
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intuitive description of the evolution : definition of the HORIZON :
Propagation of light :
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during radiation domination : a(t) t1/2 , RH = 2 c t
during matter domination : a(t) t2/3 , RH = 3/2 c t
physical process starting during RD, MD cannot affect(t) RH(t)
without violating causality…
RH(t) causal horizon for RD / MD
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evolution of wavelengths versus RH :
(t) = RH(t) k = 2a(t) / RH(t)
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evolution of wavelengths versus RH :
(t) = RH(t) k = 2a(t) / RH(t)
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PHOTON PERTURBATIONS : Planck spectrum (t, r) T/T(t, r)
1) before recombination ( decoupling equality) :
photons baryons grav. pot.E.M. gravity
tighly coupled fluid
gravity « acoustic oscillations » pressure
2) after recombination : photon decoupled « free streaming »
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PHOTON PERTURBATIONS : last scattering surface :
t
xy
x
y
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PHOTON PERTURBATIONS : last scattering surface mapped by COBE DMR (1994) :
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PHOTON PERTURBATIONS : evolution of perturbations :
primordial spectrum kn-1
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PHOTON PERTURBATIONS : spectrum of CMB anisotropies:
observation of CMB
T/T map of last scattering surface
Fourier spectrum with accoustic peaks
amplitude of the peaks B , CDM , n … !!!
position of the peaks angle under which RH(tdec) is seen
angular diameter – redshift relation
spatial curvature k !!!
RH(tdec)
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PHOTON PERTURBATIONS : main observations : COBE DMR (1994)
resolution 10º
(t) RH(tdec)
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PHOTON PERTURBATIONS : main observations : Boomerang (2000)
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PHOTON PERTURBATIONS : main observations : WMAP (February 2003)
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MATTER PERTURBATIONS : CDM : gravitationnal collapse efficient during MD (RD : grav follows baryons : follow during RD CDM during MD
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MATTER PERTURBATIONS : non-linear evolution : CDM
k (t) , Bk (t) 1 first for large k / small
hierarchical structure formation :
time
linear theory recovered by smoothing
(today: over 30 Mpc)
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MATTER PERTURBATIONS : observations : 2dF redshift survey
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Part II : (3) – cosmological parametersPart II : (3) – cosmological parameters
COSMOGICALSCENARIO
7 INDEPENDENTPARAMETERS …at least !!!{R , B , CDM , , H0 , A , n} H0
100 km.s-1.Mpc-1h
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1) nucleosynthesis B H, D, He, Li Bh2 = 0.020 0.002 , R 3 ’s
2) CMB anisotropies position : 0 = M + = 1.03 0.05 amplitude : Bh2 = 0.024 0.001 , n = 0.99 0.04
3) age quasars of age 11 Gyr open or
4) supernovae M = 0.5 0.5
A short selection of cosmological tests :
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A short selection of cosmological tests :1) nucleosynthesis
B H, D, He, Li Bh2 = 0.020 0.002 , R 3 ’s
2) CMB anisotropies position : 0 = M + = 1.03 0.05 amplitude : Bh2 = 0.024 0.001 , n = 0.99 0.04
3) age quasars of age 11 Gyr open or
4) supernovae M = 0.5 0.5
combined : B 0.044, CDM 0.23, 0.73 , h 0.71
5) large scale structure : perfect agreement
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Part II : (4) – Inflation & QuintessencePart II : (4) – Inflation & Quintessence « early problems » in the Hot Big Bang scenario :
flatness problem :
k grows like t (RD) or t2/3 (MD) k 0.1 at t0 k 10-60 at tP
horizon problem : causal horizon on CMB maps 1° 103 causally disconnected regions
origin of fluctuations : initially, RH …
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INFLATION : (Guth 79; Starobinsky 79)
defined as an initial accelerated expansion stage :
a
tINFLATION RD MD
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INFLATION : solves flatness problem :
log k
t
1
10-60
INFLATION RD MD
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INFLATION : solves horizon problem :
dH (t1 ,t2 ) RH (t2 ) for t1 << t2 and t1 RD, MD
dH (t1 ,t2 ) >> RH (t2 ) for t1 << t2 and t1 INFLATION
x
y
dH with inflation
dH without inflation
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INFLATION : solves generation of fluctuations :
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INFLATION : requires :
Friedman law a(t) > 0 p < 0conservation equation
candidates : p = -2 p < 0 but inflation forever …
slow-rolling scalar field : V = V
p = V
. .
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INFLATION with slow-rolling scalar field :
2 BONUS !!!
mechanism for generation of cosmological perturbations
quantum fluctuations wavelength stochastic background ofof scalar field amplification curvature perturbations
predictions : coherent, gaussian, VALIDATED BY adiabatic, scale-invariant OBSERVATIONS
mechanism for generation of first particles : PREHEATING
end of inflation oscillations of scalar field particle production
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« late problems » in the Hot Big Bang scenario :
magnitude of ¼ 10-3 eV !!!
problem for particle physicists : killing ¼ MeV / TeV …
problem for cosmologists : generating such small number
with respect to P¼ 1018 GeV …
« Cosmic coincidence problem » : why domination today ?
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many… unappealing proposals for Dark Energy, e.g. : slow-rolling scalar field : « quintessence »
= Vp = V
tracking solutionR
M
..
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« late problems » in the Hot Big Bang scenario :
magnitude of ¼ 10-3 eV !!!
problem for particle physicists : killing ¼ MeV / TeV …
problem for cosmologists : generating such small number
with respect to P¼ 1018 GeV …
« Cosmic coincidence problem » : why domination today ?
unsolved
improved
solved but … m 10-33 eV
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CONCLUSION 1CONCLUSION 1
INFLATION is a convincing, predictive theory… but need to relate to particle physics models (GUT, Susy, strings…)
no convincing theory for « DARK ENERGY »
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CONCLUSION 2CONCLUSION 2
cosmology has remarkable control on : cosmological parameters nucleosynthesis decoupling structure formation, lensing, etc., etc.
…but 23 + 73 = 96 % remains MYSTERIOUS !!!
10-12 August 2005 An overview of cosmology 94
dl2 = a2(t) + r2 (d + sin2d )
Remark 1: if { k = 0 AND a(t) constant }, r = a r Euclidian space Newton
dr2
(1-k r2)
__
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Remark 2 :
Euclidian : dl2 = dr2 + r2 (d + sin2d ) = dx2 + dy2 + dz2
does not depend on the choice of origin …
FLRW : k 0 dl seems to depend on the choice of origin …
do solutions with k 0 violate the assumption of homogeneity ??
NO, k 0 respects homogeneity !!!
r changes, but also dr, d and d : in fact dl is the same
no particular point is priviledged
dl2 = a2(t) + r2 (d + sin2d )dr2
(1-k r2)
O O’r r’
dl
10-12 August 2005 An overview of cosmology 96
Remark 2 : analogy with the 2-D mapping of the earth by axial projection:
on each map, thescale is a function of r
but all points on the sphere are equivalent
… the FLRW model is completely homogeneous !!!
dl2 = a2(t) + r2 (d + sin2d )dr2
(1-k r2)
earth
green map blue map
dlO
O’
10-12 August 2005 An overview of cosmology 97
summary of the situation :
basic principles of G.R. FLRW solution
space-time is curved2 types of curvatures : {a(t), k}
free-falling objects follow geodesics
??????????
curvature caused by / related to
matter??????????