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

4

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

10

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

12

ww

w.a

stro

.gla

.ac.

uk/

user

s/ke

nton

/C18

5/

Methods and distance ladder

Hubble parameter from supernovae

Age ~ 1/H0 ~ 14 109 yrs

H0 = h 100 Kms-1Mpc-1

14

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:

15

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

17

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..

19

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)

20

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

21

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

22

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

26

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

27

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

29

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

30

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 )

31

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

32

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

34

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

35

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

36

+…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

37

Destiny

38

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

40

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

41

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

43

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

47

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

48

a

49

The current budget

50

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

51

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)

52

The story of the Universe

53

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)

54

• 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.

55

]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

56

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…

58

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,

59

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

63

Next lecture

• Understanding the dark matter• What we now from observationnal point of view• What we expect from theory….

64

65

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

66

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