• Large scale structure of the Universe

• Hot Big Bang Theory

• Concepts of General Relativity

• Geometry of Space/Time

• The Friedmann Model

• Dark Matter

• (Cosmological Constant)

Cosmology

The large scale structure of the UniverseThe large scale structure of the Universe

Age of the Universe: 15 billion years. Evidence from dynamics of Age of the Universe: 15 billion years. Evidence from dynamics of universe expansion (model) AND age of oldest stars.universe expansion (model) AND age of oldest stars.

Size of the Universe: more complicated question.Size of the Universe: more complicated question.

Cosmology is an evolutionary science (at least in principle) which does not allow controlled repetition of the system. (We cannot build a universe in a laboratory). Analogy with archaeology, geology, paleo-biology.

Units in astronomy:

• Astronomical Unit AU = 150 millions km (Earth/Sun distance)

• Parsec = 3.26 light years (ly)

• Light Year = 9.46 x 10^15 m

Size of Solar System (Pluto’s orbit) : about 6 light hours.

Size of Milky Way: 10^5 ly x 10^3 ly

Galaxies: bunches of stars (in evolution), with typically 10^11 stars.

Galaxies agglomerate in clusters with size of a few Mpc (e.g. Local Group)

Galaxy Clusters agglomerate in Superclusters with size: 200 Mpc

Dominant interaction in the Universe: Gravitation

How distances are measured?How distances are measured?

The “ Cosmic distance ladder “The “ Cosmic distance ladder “ Parallax methodsParallax methods Main-sequence fitting (HR plot)Main-sequence fitting (HR plot) Variable (Cepheid) starsVariable (Cepheid) stars Supernovae, cosmological methods Supernovae, cosmological methods

The Universe as seen by us is strongly dishomogeneus and anisotropic.

This statement holds true also on the galactic scales (kpc distances)

….and remains true also on the scale of galaxy clusters (Mpc distances)

However, if seen from distances of 100 Mpc or more, the universe gets homogeneus and isotropic. This is homogeneity and isotropy at large scales!

The Hot Big Bang ModelThe Hot Big Bang Model

Model for the large scale structure and evolution of the Universe.Model for the large scale structure and evolution of the Universe. Based on important experimental observations.Based on important experimental observations.

Cosmological Red Shift

Radiation is emitted from stars and other celestial bodies

This radiation has the same physical origin of the radiation we study in terrestrial laboratories (e.g. atom absorption and emission).

Stellar evolution and many other branches of astrophysics are based on such evidence. E.g. chemical composition of star surfaces are well known.

The radiation emitted by any source can be affected by the Doppler effect if there is a relative motion between the source and the receiver

'

z

Hz d

c

vz

c

Red shift

In laboratory

From a distant galaxy

1929: Hubble discovered the empirical relation

Birth of Modern Cosmology!

From the nonrelativistic Doppler formula:

v H dA relation between the Galaxy velocity (away from us) and its distance

Since our position in the Universe is hardly a privileged one, galaxy superclusters recede from each other with the cosmological Hubble law. Universe is expanding!

Two immediate consequences:

• In the far past all matter was lumped in very little space (the Big Bang)

• The timescale for this is roughly 1/H (assuming the expansion law was the same all over, which is not really the case)

v H d

The Universe is expanding into what ?

It is the space itself that is expanding? Yes.

Are rulers expanding? No, only gravitationally independent systems participate in the expansion!

The Hubble law is a linear expansion law which generates an homologous expansion (it is the same as seen from every Galaxy)

H = 70 ± 7 km/sec Mpc

The expansion looks the same as seen from A or from B

Naïve expansion model (assuming H = const) = Patch of size 100 Mpc

What we see in our Patch is consistent with isotropic and homogeneous expansion plus the “Cosmological Principle” (no privileged place in Universe!)

11

2

3

12 1 2r r r 23 2 3r r r

13 1 3r r r

Homogeneous and isotropic expansion: the shape of the triangle must be preserved. Therefore

13 13 0( ) ( ) ( )r t a t r t12 12 0( ) ( ) ( )r t a t r t 23 23 0( ) ( ) ( )r t a t r t

1212 12 0 12( ) ( ) ( )

dr av t a r t r t

dt a

13

13 13 13( ) ( ) ( )dr a

v t a r t r tdt a

Seen from patch 1:

Seen from patch 2: 2121 21 0 21( ) ( ) ( )

dr av t a r t r t

dt a

23

23 23 0 23( ) ( ) ( )dr a

v t a r t r tdt a

In any universe undertaking homogeneous and isotropic expansion, the velocity/distance relation must have the form

Now we see that:

( ) ( )a

v t r ta

( )a

H ta

a(t): scale parameter

Elements of a naïve thermal history of the UniverseElements of a naïve thermal history of the Universe

13.6bE eV

H e p

Going backward in time means:

• No structures (No stars, galaxies…..) Only Matter and Radiation

• Higher densities and higher temperatures

Matter Radiatione

p

When E(γ) > 13.6 eV radiation and matter are coupled. This took place at cosmic time 400,000 yrs. Radiations is in equilibrium with atoms.

Before this era, let us imagine: nuclei, electrons and radiation, at some T.

Energy ~ kT. Electrons streaming freely at this point.

e p H (photodissociation) (radiative recombination)

0.5kT MeV

e e

Nucleosynthesis already taking place at that time (from 1 sec to 300 sec).

Electrons cannot free stream.

Then by going backward some more in time energy increases to:

Then by going backward some more in time energy increases to give a mean energy 10 MeV. Therefore the reactions

became possible. These reactions mix p and n together making nucleosynthesis impossible. This is T around 10^10 K (and cosmic time 0.1 sec).

e en p e n e p

This took place at about T=10^10 K and cosmic time 100 sec

To summarize, a timeline of important events:

• T>10^10 K, E>10 MeV, t<0.1 sec . Neutrons and protons kept into equilibrium by weak interactions. Neutrinos and photons in equilibrium.

• t = 1 sec. No more p/n equilibrium. Beginning of nucleosyntesis. Neutrinos decoupling from matter.

• T=10^9 K ,E =1 MeV, t= 100 sec. Positrons and electrons annihilate into photons

• t = 300 sec nucleosysnthesis finished because of low energy available and no more free neutrons around Low mass nuclei abundance fixed

• Protons, photons, electrons, neutrinos (decoupled)

• T=5000 K, E=10 eV, t=400,000 years. No more radiation,e,p equilibrium. Atoms formation (hydrogen, helium). Photons decouple CMB

Primordial NucleosynthesisGamow, Alpher and Herman proposed that in the very early Universe, temperature was so hot as to allow fusion of nuclei, the production of light elements (up to Li), through a chain of reactions that took place during the first 3 min after the Big Bang.

The elemental abundances of light elements predicted by the theory agree with observations.

Y ~ 24% Helium mass abundance in the Universe

Cosmic Microwave Background

Probably the most striking evidence that something like the Big Bang really happened is the all pervading Cosmic Background predicted by G. Gamow in 1948 and discovered by Penzias and Wilson in 1965.

This blackbody gamma radiation originated in the hot early Universe.

As the Universe expanded and cooled the radiation cooled down.

CMB temperature fluctuations (COBE)

By way of summary, the 3 experimental evidences for Big Bang:

• Red shift (Cosmic Expansion)

• Primordial Nucleosynthesis

• Cosmic Microwave Background

Key concepts of the Hot Big Bang Model:

• General Relativity as a theory of Gravitation

• (Inflation)

Concepts of General RelativityConcepts of General Relativity

General Relativity: a theory of Gravitation in General Relativity: a theory of Gravitation in agreement with the Equivalence Principleagreement with the Equivalence Principle

Classical Physics concepts

Special Relativity concepts

Spacetime of Classical Physics and Special Relativity

Spacetime must be curved !!

Classical Physics

• Existence of Inertial Reference Frames (IRF)

• Relativity Principle (Hey man, physics gotta be the same in any IRF!)

• Invariance of length and time intervals

Special Relativity

• Existence of Inertial Reference Frames (IRF)

• Relativity Principle (Hey man, physics gotta be the same in any IRF!)

• Invariance of c

'

'

x x v t

t t

'

' 2

( )

( / )

x x vt

t t vx c

Gravitation, a peculiar force fieldGravitation, a peculiar force field

Gravity fieldGravity field P = m(g) gP = m(g) g P = m(i) aP = m(i) a m(g)g = m(i)am(g)g = m(i)a a = g m(g)/m(i)a = g m(g)/m(i)

a = ga = g One for all bodiesOne for all bodies

Electric fieldElectric field

F = qEF = qE

F = m(i)aF = m(i)a

qE = m(i)aqE = m(i)a

a = E q/m(i)a = E q/m(i)

Depending on Depending on

particle chargeparticle charge

If gravitation does not depend on the characteristics of a body then it can be ascribed to spacetime. It is a spacetime property.

Equivalence between inertial mass and gravitational mass

Free fall in gravitational field (apple from a tree) cannot be distinguished from acceleration (the rocket)

1. Free fall the same for every body geometric theory of gravitation

2. Gravitation equivalent to non-inertial frames (EP)

Einstein replaced the idea of force with the idea of geometry. To him the space through which objects move has an inherent shape to it and the objects are just travelling along the straightest lines that are possible given this shape (J. Allday).

Understanding gravitation requires understanding space-time geometry.

The concept of elementary interaction

Newton

Faraday

Maxwell

Action at a distance

Field concept

Quantum Fields

(field quanta exchange)

Gravity

(spacetime curvature)

Spacetime geometrySpacetime geometryGeometry: study of the properties of space.

Euclidean geometry: based on postulates

- example: given an infinitely long line L and a point P, which is not on the line, there is only one infinitely long line that can be drawn through P that is not crossing L at any other point.

L

P

Some consequences:

• The angles in a triangle when added together sum up to 180°

• The circumference of a circle divided by its diameter is a fixed number :

• In a right angled triangle the lengths of the sides are related by

(Pythagoras Theorem)

2 2 2c a b

Euclid geometry is a description of our common sense (= classical physics) three-dimensional space

However there are spaces that do not obey Euclid axioms. Spaces having a non-Euclidean geometry. We will consider the (2-dimensional) example of the surface of a sphere.

What are the “straight lines” on the sphere surface? They are the great cirlces! (the shortest path between two point is an element of a great circle).

Now, suppose we choose A as a point and we draw from B the parallel to A.

They meet at the North Pole!

(Euclid axiom does not hold)

Another consequence: the sum of the angles of a triangle is higher than 180°

With the example of a bidimensional space (the sphere surface) we have shown the existence of non-Euclidean (Riemannian) spaces. In this case parallel axiom does no hold true!

Einstein’s theory replaced gravity as a force with the notion that space can have a different geometry from the Euclidean. It is a curved space.

The sphere surface is 2-d and is a curved space when seen from “outside” (3-d)

We live in a 4-d curved (by gravity) spacetime

Three kind of geometry are in general possible (depending on energy content of Universe)

2

2 2 2 2 2

2

1 0 0

0 1 0

0 0 1

x

s y x y z

z

2 2

22 2 2 2 2 2

2

2

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

c t

xs c t x y z

y

z

Newtonian physics spacetime.

Length of a rules is invariant (as well as time interval dt)

Special Relativity spacetime: the 4-interval is invariant

g Matrix (spacetime metric)

General Relativity Spacetime: similar in structure to Special Relativity spacetime but now the gravity field makes the metric spacetime dependent.

( )g x

Newtonian, Minkowski, General Relativity Newtonian, Minkowski, General Relativity geometriesgeometries

8G GT

1

2G R g R R g R

Einstein Tensor Energy-Momentum Tensor T

Ricci Tensor Ricci Scalar R

, , , , ,

1

2R g g g g

( ) 8f g T

R R

Riemann Tensor

Spacetime curvature Momentum/Energy

Gravitational potential Mass Density2 4 G (Classical Physics)

A picture of the Universe expansion can be drawn by using the surface of the sphere analogy

In the surface of the sphere analogy, the geometry is non-Euclidean (but locally Euclidean) and the space has a positive curvature.

The “center of the Universe” lies outisde of the Universe

The Big Bang takes place everywhere!

The evidence is recession of other parts of the Universe from us

This space is closed (one can go all the way around and get back to the same place)

Geometry locally Euclidean means neglecting the curvature (or neglecting gravity). It is Minkowski space.

Our Universe 4-d expansion is in analogy with this toy-model

Friedmann Models (Newtonian version)Friedmann Models (Newtonian version)

Friedmann Models are models of the Universe as (large-scale) systems that are governed by (General Relativity) physical laws.

Recipe:

- Use the General Relativity Law (means selecting the equations)

- Assume the Universe in Homogeneus and Isotropic (means selecting a metric)

- Assume an energy content for the universe (means selecting E-p tensor)

Result: Equations for the evolution of the Universe (Friedmann, LeMaitre)

Let us start with a Newtonian model

, ( )S SM R t

2 ( )S

S

GM mF

R t

2

2 2 ( )S S

S

d R GM

d t R t

m

34( ) ( )

3S SM t R t ( ) ( )S SR t a t r

2 2 2 21 4( ) ( )

2 3S Sr a G r t a t U

Expansion of a classical distributed mass

21

2 ( )s S

S

dR GMU

dt R t

Friedmann equation in Newtonian formFriedmann equation in Newtonian form

2

2 2

8 2 1( )

3 ( )S

a G Ut

a r a t

General form of the solutions

21

2s S

s

dR GMA

dt R

We can calculate A using the present-day values (H, R, density)

32 2 0 00 0

41

2 3

RA H R G

2 3 220 0 00 0

38 8

3 3 8S

s

dR G R HGR

dt R G

Qualitative comments

Going in the past the first term dominates (R was smaller)

There was a time when0s

s

R

dR

dt

(the beginning of the Big Bang)

What is the future o’ the Universe?

Let us define:20

0

3

8 c

c

H

G

Critical density

2 3

20 00 0

8 8

3 3S

cs

dR G RGR

dt R

If current density greater than critical density the second term is negative and then there will exist a time in which

0sdR

dt

The expansion will then stop and the Universe will collapse back to the initial state

If current density smaller than critical density the second term is positive and the derivative will never get down to zero. Expansion will go on forever.

Dark MatterDark Matter

This is a golden era for cosmology:

• Measurements of the CMB (and its anisotropies)

• Existence of Dark Matter

• Existence of Cosmological Constant

In this section we discuss evidence for Dark Matter

Popular wisdom: that the matter in the Universe is made of ordinary baryons (the so called barionic matter). This matter has the property of emitting radiation (being mostly concentrated in stars).

However, there seems to be more matter than the one which is visible.

The presence of Dark Matter is deduced by using its gravitational effect on luminous matter. At different scales!

For instance, let us consider a spiral galaxy and plot the velocity of matter in the galaxy as a function of the distance from the center:

Keplerian rotation: due to gravitational force

This can be explained by the existence of a halo of matter surrounding the galaxy

Galaxy scale

In clusters of Galaxies: a galaxy cluster is a group of galaxies held together by their own gravity. However, when we measure the speed with which each galaxy moves, it appears a lot more gravity is required to hold the cluster together than can be explained by the stars we can see. Therefore, there must be a lot of dark matter that we can’t see.

Cosmological scale Gravitational lensing effect.

The light from a far away source is deflected by Dark Matter: distortions and multiple images (Einstein Rings)

Galaxy Cluster scale

Gravity Optics ! !

So, what is dark matter ?So, what is dark matter ?

• MACHO’s (Massive Compact Halo Objects): Black Holes, Planets, Dead Stars

• Non-baryonic Matter (particle physics explanation): WIMP (Weakly Interacting Massive Particles) like neutralinos, neutrinos

Some Einstein rings observed by Hubble Space Telescope

Dark Matter is about 90% of the total matter in the Universe ! !

The Friedmann Model strikes backThe Friedmann Model strikes back

Friedmann Models are models of the Universe as (large-scale) systems that are governed by (General Relativity) physical laws.

Result: Equations for the evolution of the Universe (Friedmann, LeMaitre)

Let us do the “full” model

Recipe:

- Use the General Relativity Law (means selecting the equations)

- Assume the Universe in Homogeneus and Isotropic (means selecting a metric)

- Assume an energy content for the universe (means selecting E-p tensor)

Friedmann equation in Friedmann equation in Newtonian formNewtonian form

Why the Newtonian Universe is not a good representation?

1. Not homogeneuos and isotropic

2. It is Euclidean

2

2 2

8 2 1( )

3 ( )S

a G Ut

a r a t

What is then the correct equation for the scale parameter?

Recipe:

- Use the General Relativity Law (means selecting the equations)

- Assume the Universe in Homogeneus and Isotropic (means selecting a metric)

- Assume an energy content for the universe (means selecting E-p tensor)

2

2 2

8 2 1( )

3 ( )S

a G Ut

a r a t

8G GT g Einstein equation with the Cosmological constant

2

2 2 2 2 2 2 22

( ) sin1

drds dt a t r d d

kr

Homogeneity and Isotropy (Friedmann, Robertson, Walker metric)

( )

( 1 , 0 , 1 )

a t scale parameter

k space curvature open flat closed Scale parameter and geometry of the Universe

( )T p g p U U Energy-momentum tensor of a perfect fluid

The result of all this produces two changes (with respect to the Newtonian form) in the scale parameter equation:

2

2 20

( ) ( )

2

S

t t energy density

U kcgeometrical

r R

The cosmological constant

The cosmological constant is an extra term in Einstein’s equation of General Relativity which physically represents the possibility that there is a density and pressure associated with “empty” space. This term acts as a “negative” pressure.

After some simplification the result is:

2

2 20

8

3 3M R

a G k

a a

Data indicates that the dynamics of the Universe is dominated by Dark Matter and Cosmological Constant. And that the Universe is geometrically flat.

In other words the equation is effectively

2

2 20

8

3 3DarkMatter

a G k

a a

The energetics of the Universe is mostly determined by the Matter component (30% of total energy, which is 95% Dark Matter) and the Cosmological Constant (70% of total energy content)

These slides at

http://pcgiammarchi.mi.infn.it

Our Universe is located at about

0.7

0.3DM

Two suggestions for further readingTwo suggestions for further reading

B. F. Schutz, A first course in general relativity, Cambridge B. F. Schutz, A first course in general relativity, Cambridge University press. University press.

B. Ryden, Introduction to cosmology, Addison WesleyB. Ryden, Introduction to cosmology, Addison Wesley