dark matter and dark energy the dark side of the universe anne ealet cppm marseille
TRANSCRIPT
Lectures
• Lecture I• Basis of cosmology
• An overview: the density budget and the concordance model
• An history of the universe
• Lecture II• The dark matter mystery
• Lecture III• The CMB
• Lecture IV• The dark energy
Each lecture will be followed by a discussion on a given subject
some basic references:
Books: Peebles Principle of physical cosmology (princeton)
Liddle An introduction to modern cosmology
Weinberg Gravitation and cosmology
Review papers:
Freedmann astro-ph/0202006
Trodden/Carroll astro-ph/0401547
Peebles astro-ph/9806201
Web sites:astron.berkeley.edu/~mwhite/ site indexww.damtp.cam.ac.uk/user/gr/public/cos_home.html
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The Standard model of Cosmology
© Hu & White Scientific American 2004
5
The Standard Cosmological model
At different stages of the evolution of the universe different physics interplay: particle physic
nuclear physicgeneral relativitylinear perturbation theory (and non-linear)hydrodynamic, boltzmann…
…and compare to observations…
Evolution of an homogeneous fluid composed of all kind of particles (relativistic and not relativistic) with small inhomogeneities in an expanding univers
6
This lecture
• The cosmological principle
• Hubble’s law and Friedmann equations
• Geometry of the universe
• Weighing the universe
• The current universe budget
• The story of the universe
7
Cosmological distances
Mean distance Earth-Sun1 Astronomical Unit (AU)~ 1011 meters
Nearest stars
Few light years
~ 1016 meters Milky Way galaxy size ~ 13 kpc~ 1020 meters~ 40 000 light years
Alternative unit : 1 parsec ~ 3.261 lyr
1 pc = 1 AU / tan(1 arcsecond)1 arcsecond = 360 degrees / 3600
8
Large-scale structure of the universe
galaxies clusters of galaxies LSS
closest similar galaxy : Andromeda 2 Millions of light years ~ 1 MpcAfter .. Use the redshift as an indicator (see definition later..)
9
Cosmological principle
At large scale (> ~ 50 Mpc) , the Universe is isotropic = look the same in all directionhomogeneous= look the same at each point
( we are not particular)was used to do calculationsIn GR nearly proved later with large scale structure and CMB.
Angular distributionof 31000 radio sources
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Observational CosmologyCosmic Background Radiation (CMB)
Large Scale Structure (LSS)
Supernovae Ia
11
Edwin Hubble1889 – 1953
Distance Mpc)Rec
essi
onal
Vel
ocit
y (k
m/s
ec)
Hubble’s Great Discovery - The Universe is Expanding!
• In 1929 Hubble measured the red shift of nearby galaxies and found that nearly all were moving away from us.
• He used Cepheid variables as “standard candles” to measure distances.
• Result: The faster they are moving, the farther away they are.
• The Universe is expanding! Einstein declares his “Biggest Blunder.”
velocity distance
v Hor
Historical Note: Hubble was off by a factor of 8 in his measurement of H0.
Lesson: Beware of systematic errors!
H0 : Hubble’s constant
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Methods and distance ladder
Hubble parameter from supernovae
Age ~ 1/H0 ~ 14 109 yrs
H0 = h 100 Kms-1Mpc-1
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Hubble’s law is just what one would expect for a homogeneous expanding universe; each point in the universe sees all objects rushing away from them with a velocity proportional to distance
Consequence:The universe was denser and hotter in the pastIt starts with a mechanism known as the big bangThe left over of this dense phase is a radiation background called the cosmic microwave background (CMB)
We conclude that in the distant past everything in the Universe was much closer together:
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CMB
16
The theory (framework)
Gravitation and then the dynamic of the Universe are governed by the Einstein equations of general relativity:
gTGgRR 82
1
R
g
G
T
Ricci Tensor -> space-time curvature
Metric tensor -> space time distances dxdxgds 2
Gravitational constant = 6.67 10-11 m3 s-2 kg-1
RgR Ricci scalar
Energy-momentum tensor -> mass,energy
Cosmological constant
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Objects appear to be in different positions
Space is curved by matter
gTGgRR 82
1
18
• This is a non linear theory
• This is a gauge theory
but the measurements not depend of the gauge choice
We can choose the referentiel and the metric as we want , results will be the same..
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Cosmology : an application of GR
General relativity + cosmological principleFriedmann-Lemaitre model
Homogeneity
The spatial curvature Can be different at different timesAnd symetric (3 solutions)
Isotropy
The space-time is the same in all directions. This curvature give the expansion
Define a metric with this propertiesRelate it to the matter density with Einstein equation(at the universe scale, describe it as a perfect fluid)
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The Robertson-Walker metric
r are co-moving coordinates x = a(t) r a(t) is a function of time and r is a constant coordinate. a(t) is known as the scale factor of the universe and it measures the universal expansion ratea(t0) = 1 where t0 is todayThe comoving coordinates preserve today distance
Spatial curvature
222222
2222 sin
1)( drdr
kr
drtadtds
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The k parameter: the curvature
•Closed: k=+1 (tot>1) spherical geometryRecollapse, Big Crunch
Flat: k=0 (tot=1)Euclidean geometry
•Open: k<0 (tot < 1) hyperbolic geometry
a+b+c> 180
a+b+c= 180
a+b+c < 180
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The tensor T
p
p
pT
000
000
000
000
is the total density of the fluidp is the total pressure
The Ricci tensor R is calculated from the metric g
Assuming aPerfect Homogeneousfluid
23
• From Einstein field equations:
3
82
2G
a
k
a
a
pG
a
a3
3
4
1st Friedmann equation
2d Friedmann equation
Acceleration equation
2
2
24
2
1
a
kGp
a
a
a
a
Relate the expansion rate to the total energy density of allcomponents in the universe
Definition: w=p/ equation of state parameter
24
Reduced density definition
• Critical density at any time
c
total
c
ii
G
Hc
8
3 2
a
aH
H is the Hubble parameter whichis time dependent ….
with
Be carefull..often defined as the density today with t=t0 and the 0 is dropped….I will use the same convention ..m will be the matter density TODAY otherwise specify (idem r, , tot)
22aH
ki
putting22aH
kk 1 i
25
Fluid equation
03
p
a
a
From energy-momentum conservation :
pgUUpT Uis the fluid four velocity
The expansion of the universecan lead to local changes in theenergy density
0 T
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The cosmological constant
g
GT
8
This is like a prefect fluid with G
8
p 1w
This a constant component through spacetimeIt is equivalent to a vacuum energy
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Some solutions of the fluid equation
• p = 0 (matter dominated)
• p = /3 (radiation dominated)
• p = w
3
1
am
4
1
ar
)1(3
1ww a
For the cosmological constant cste
03
p
a
a
28
(a)
a
a-3
a-4
const
Radiation - Matter - Vacuum (Dark Energy?)
todayEarly Universe
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What is dominant when?Matter dominated (w=0): a-3 Radiation dominated (w=1/3): ~ a-4
Dark energy (w~-1): ~constant
• Radiation density decreases the fastest with time – Must increase fastest on going back in time– Radiation must dominate early in the Universe
• Dark energy with w~-1 dominates last
Radiation domination
Matter domination
Dark energy domination
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Putting it all together
r = r0 a-4 = cr a-4 since / a-4 and a today is 1
– 0 ≡ radiation density today
– use r0 ≡ r0 / crit by definition (and 0 is dropped )
• sim m,
• Therefore for k=0 and an unknown energy X (X= is w=-1):
tot = c [ r a-2 + m a-3 + w a-3(1+w)]
tot = r + m + w )
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Evolution of the universe:
• Putting these together and using
(t) = c [ r a-2 + m a-3 + w a-3(1+w)]
Freidmann eqn1 for k=0
H2 = H02
( r a-2 + m a-3 + w a-3(1+w))
)(3
82
2 tG
a
aH
00
a
aH
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First summary
The Universe can be caracterized byThe Hubble parameter in function of time
The densities of each component
)(3
82
2 tG
a
aH
)()( )1(332
20
2w
wmr aaaH
aH
mrkT 1
General assumptions: r can be neglected and T = 1 (flatness)
c
ii
33
Acceleration
• Acceleration parameter is defined by
2a
aaq
Today we have:
i
i
wq
2
)2)1(3(0
For an universe with k=0 and a cosmological constant 20mq
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Time evolution
• first Friedmann equation ,
• assuming a(t) = tExpansion evolution depend of the density content:
• p=0 matter dominated a(t) t2/3
• p=/3 radiation dominated a(t) t1/2
• Generalisation w a(t) t2/3(1+w)
• Cosmological constant a(t) eHt
)(3
82
2 tG
a
aH
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Destiny…
m=1 and =0 Einstein de Sitter universe :
m=0 and =0 empty universe
m=0 and =1 )3/exp()( tta
a(t) = (3/2H0t)2/3
a(t) =H0t
t0 = 2/3H0
t0 = 1/H0
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+…flat or not flat….
• Without :– k=+1 expand from a singularity and recollapse later
(big Crunch)– k=-1 (=k=0) expand for ever
• With dominant: all solutions expand exponentially (de Sitter)
Experimental evidence of flatness with high accuracy (WMAP) (see lecture 3)
Interesting to be aware if it is not completely true
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Destiny
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Measurements in an expanding Universe
(or how photons are traveling in the expanding Universe)
39
Redshift and scale factor
ta
taz 00
0
1
t0 t
time t0, a(t0)=1radius R0
time tradius a(t)R0
Redshift z
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In PracticeAll radiation and all wavelengths are redshifted
• In a model, we can calculate
when the light has been emitted and the distance of the source
H is 656 nmhere
z = (750-656)/656 = 0.14
t ~ – 2 billions yearsD ~ 600 Mpc
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comoving distance
• Comoving distanceUsing dx= c dt = a dr
)(20000 aHa
daa
aa
daa
a
cdtadraDc
))1()1(()(
11
1
))'1()'1((
'
)'(
')(
)1(3320
2
02
0
0)1(33
002
1
wwm
T
z
wwm
z
C
zzH
zH
a
dz
a
da
za
a
zzH
dz
zH
dzzD
Luminosity and observed flux
is the intrinsic luminosity of a given object at a redshift z, on geodesic:
ds2=0 = cdt2 – a(t)2 dr2/(1-kr2)Propagation of light from the object to us
1
0
1
1
02/1201
200
0
01
)1(
)()()(
0:
r
t
t
r
kr
drax
zH
dz
aHa
daa
ta
cdtadrax
kcaseNBx1 is the present day distance of the object
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Observed flux
• Take into account• Spatial distance between photons increased with a
factor (1+z)
• Photons are distributed on a surface of 4x12
221 )1(4 zx
l
l is the observed flux at a redshift z, related to cosmologicalparameters through r1
Luminosity distance
• Definition
24 Ldl
dL can be measured directly from the observed flux …and is related to the cosmological parameter via l
dL = Dc(1+z)
we will see later an application with supernovae (lecture 4)
45
Angular distance
– Defined such that =r / dA
• r=physical size of object• = angular size on sky
)1(
)(
)1( 2 z
zD
z
dd CL
A
Application with the CMB (lecture 3)
46
The age of the Universe
• Integrate
H2 = H02
(m a-3 + w a-3(1+w)) (neglect
radiation, flatness )
• If w=-1 (so x=):
• If w - 1 (given k=0 assumption)
1
0)1(33
1
02
1
)()( wwmo aaH
da
aaH
dat
t= 2/ 3(1+w) H0
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Measurements of the age of universe
• Krauss + Chaboyer (2003)
• Fit globular cluster age with
a monte carlo approach to take uncertainties into account
– stars age = 12.6 Gyr
– t0>10.4 Gyr 95 %
From the density estimation with CMB data + flatness ->
t0~13.4 Gyr
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a
49
The current budget
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WMAPDE= 0.7, M= 0.3
for a flat universe
NewStandard Cosmology:
73±4% Dark Energy
27±4% Matter
0.5% Bright Stars
Matter:
22% CDM, 4.4% Baryons,
0.3% s
A Revolution in Cosmology
Weak lensing mass census Large scale structure measurements
M= 0.3
Baryon Density
B= 0.044+/-0.004
Flat universe
total= 1.02+/-0.02
Big Bang Nucleosynthesis
Inflation
M- = C
M+ = C
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cdm
Cold dark matterThat we dont know
Cosmological constantOr a dark energyWe don’t know
= standard model
h 0.7
tot = 1
= 2/3 m = 1/3
b 0.04
= 0.005
= 0.00005
+ Initial adiabaticperturbations with an Harrison Zeldovitchspectrum n=1 (inflation)
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The story of the Universe
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The universe : a question of temperature…
• Expansion => volume increases a(t)3
• => temperature decreases as a(t)
• The radiation density is T4 ( Stefan law)• Matter as 1/Volume = 1/a3 = T3
=> radiation dominates when T increases
T = 1010 K / √t (s)T = 1010 K / √t (s)T 1/a(t)T 1/a(t)
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• At 1 s, the temperature is 1010 K..only 1 Mev
•At early phases of creation, energy was sufficient to create huge numbers of MASSIVE particles and antiparticles, hence there was a lot of annihilation. (highly energetic photons).
Upshot is that the photons we see then were created in the first second after the Big Bang.
However, the photons actually trace the condition of the Universe at age 300,000 years. At this time we say matter decoupled from radiation.
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]exp[~ Hta2/1~ ta 2/3~ ta
Expansionscale
Phases of Thermalstory
Equ
ival
ence
, à e
VE
quiv
alen
ce, à
eV
1 TeV1 TeV 1 MeV1 MeVTemperature
Inflation, G
UT
Baryogenesis
electroweak
transit
ion
Neutrinos d
ecouplin
g
Nucleosy
nthesis
Photons
decouplin
g(CMB)
Stru
cture
s form
ation
redshift
z <10z=1100
Thermal story of the Universe
za 1/1??? ?? ? ?
10101212 GeV GeV
End of inflatio
n
reheatin
g
Present
T=2.72 K
z=0
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Symmetry breaking• antimatter disappear• particles become massive
Baryogenesis• creation of proton and neutron
Nucleosynthesis• light elements
Universe start• atoms form•
Stars • … and life
neutrinos decoupling
uu d
InflationPlanck area
Visible matter: first budget
• Free H and He 4%
• Visible light from stars 0.5 %
• Neutrinos 0.5 %
• Heavy element 0.003%
Nuclear processes to produce such abundance : a great success betweenTheory and observation…
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Start from the beginning..
• t < 10-43s Planck time physic law not applied
• 10-43s < t < 10-32 s plasma of quarks etc..;Expansion of a factor 1050 ..antimatter disappears
• 10-32<t<10-6 s breaking strong and electroweak interaction
• 10-6<t<1s formation of nuclei …p,n,e,
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Nucleosynthesis
Hpn 2
nHeHH 322
pHnHe 33
nHeHH 423
All neutrons end up in He and thus neutron/proton asymmetry determines primordial abundance of Helium and other light elements …direct predictions
p and n are interacting.As the Universe cools (expansion) (lower 1 MeV), they stopto interact : at this point, the number of neutrons relative to protons (neutrons are slightly heavier) is 1 n for 6 p. (Neutron have a life time of 10 mn and give n->p+e+n )
For a radiation dominated epoch,the relative abundance of light element depend only on the baryon densityThis number has been measured by WMAP precisely
60
Big Bang Nucleosynthesis Theory vs. Observations:
Remarkable agreement over 10 orders of magnitudein abundance variation
Concordance region:b h2 = 0.02For h=0.7, b = 0.04.
Deuterium: strongest constraint
4He
b
61
• 15 mn after the Big Bang H,He,Li7
• as T decreases, not possible to continue….
Heavier elements are produced later (bmillion of years..) from nuclear processes in stars as supernovae
Wait …300 000 years ….T allows photons to decouple….
62
Stellar nucleosynthesis
Stars start with H et He gaz
contraction
T increases up to 20 106 K
H burns
T continues to increase 100 106 K
He burns
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Next lecture
• Understanding the dark matter• What we now from observationnal point of view• What we expect from theory….
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Hubble radius and horizon
t0
The light cone gives our horizon
Big bang
A structure with a size greaterthan the Hubble radius is only affected by the metric If the size is smaller than the Hubble radius, the structure is affected by causality (physic)
Us…
There are events outside our horizon and events which are inside but was outside the Hubble radius at a given time
This is a paradoxe with CMB …we will explain it later …
recombinaison
The universe is expanding with a radius which is the Hubble radius
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Distances dans l’ univers local• expansion linéaire
Loi de Hubble distance lumineuse
= /4dL2 = flux mesuré et = flux émis
• dL= a0 r(1+z) = c(1+z)/H0
1+ z =obs/emis=a0/a(t)
• On utilise la relation distance-luminosité• m-M=5log(D/10pc)-5
• Distances d’une chandelle standard (M=const.)• m=5log(z)+b
• b = M+25+5log(c)-5log(H0)• logH0=log(z)+5+log(c)-0.2(m-M)
Normalisation absolueMagnitude
mesurée