Large scale structure of the Universe

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Cosmology

• 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)

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The large scale structure of the Universe
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.

• Age of the Universe 15 billion years. Evidence
  from dynamics of universe expansion (model) AND
  age of oldest stars.
• Size of the Universe more complicated question.

• Units in astronomy
• Astronomical Unit AU 150 millions km
  (Earth/Sun distance)
• Parsec 3.26 light years (ly)
• Light Year 9.46 x 1015 m

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Size of Solar System (Plutos orbit) about 6
light hours. Size of Milky Way 105 ly x 103
ly Galaxies bunches of stars (in evolution),
with typically 1011 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
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How distances are measured?

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

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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!
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The Hot Big Bang Model

• Model for the large scale structure and evolution
  of the Universe.
• 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
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From a distant galaxy
In laboratory
Red shift
1929 Hubble discovered the empirical relation
From the nonrelativistic Doppler formula
A relation between the Galaxy velocity (away from
us) and its distance
Birth of Modern Cosmology!
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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)

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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)
The expansion looks the same as seen from A or
from B
H 70 7 km/sec Mpc
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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!)
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Homogeneous and isotropic expansion the shape of
the triangle must be preserved. Therefore
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1
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Seen from patch 1
Seen from patch 2
In any universe undertaking homogeneous and
isotropic expansion, the velocity/distance
relation must have the form
Now we see that
a(t) scale parameter
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Elements of a naïve thermal history of the
Universe

• Going backward in time means
• No structures (No stars, galaxies..) Only
  Matter and Radiation
• Higher densities and higher temperatures

e
Matter
Radiation
p
When E(?) gt 13.6 eV radiation and matter are
coupled. This took place at cosmic time 400,000
yrs. Radiations is in equilibrium with atoms.
(photodissociation)
(radiative recombination)
Before this era, let us imagine nuclei,
electrons and radiation, at some T.
Energy kT. Electrons streaming freely at this
point.
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Then by going backward some more in time energy
increases to
This took place at about T1010 K and cosmic
time 100 sec
Electrons cannot free stream.
Nucleosynthesis already taking place at that time
(from 1 sec to 300 sec).
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 1010 K (and cosmic time 0.1 sec).
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To summarize, a timeline of important events

• Tgt1010 K, Egt10 MeV, tlt0.1 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.
• T109 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)
• T5000 K, E10 eV, t400,000 years. No more
  radiation,e,p equilibrium. Atoms formation
  (hydrogen, helium). Photons decouple ? CMB

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Primordial Nucleosynthesis
Gamow, 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
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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.
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CMB temperature fluctuations (COBE)
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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)

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Concepts of General Relativity
Classical Physics concepts Special Relativity
concepts Spacetime of Classical Physics and
Special Relativity

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

Spacetime must be curved !!
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• 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

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Gravitation, a peculiar force field
Electric field F qE F m(i)a qE m(i)a a E
q/m(i) Depending on particle charge

• Gravity field
• P m(g) g
• P m(i) a
• m(g)g m(i)a
• a g m(g)/m(i)
• a g
• One for all bodies

If gravitation does not depend on the
characteristics of a body then it can be ascribed
to spacetime. It is a spacetime property.
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Equivalence between inertial mass and
gravitational mass
Free fall in gravitational field (apple from a
tree) cannot be distinguished from acceleration
(the rocket)

• Free fall the same for every body ? geometric
  theory of gravitation
• Gravitation equivalent to non-inertial frames (EP)

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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.
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The concept of elementary interaction
Newton
Action at a distance
Faraday Maxwell
Field concept
Quantum Fields (field quanta exchange)
Gravity (spacetime curvature)
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Spacetime geometry
Geometry 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.
P
L

• 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)

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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)
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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!
Einsteins 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)
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Newtonian, Minkowski, General Relativity
geometries
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.
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Einstein Tensor
Energy-Momentum Tensor T
Ricci Tensor
Ricci Scalar R
Riemann Tensor
Spacetime curvature
Momentum/Energy
Mass Density
Gravitational potential
(Classical Physics)
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A picture of the Universe expansion can be drawn
by using the surface of the sphere analogy
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
In the surface of the sphere analogy, the
geometry is non-Euclidean (but locally Euclidean)
and the space has a positive curvature.
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
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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
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Expansion of a classical distributed mass
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Friedmann equation in Newtonian form
General form of the solutions
We can calculate A using the present-day values
(H, R, density)
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Qualitative comments
Going in the past the first term dominates (R was
smaller)
There was a time when
(the beginning of the Big Bang)
What is the future o the Universe?
Let us define
Critical density
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If current density greater than critical density
the second term is negative and then there will
exist a time in which
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.
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Dark 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.
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The presence of Dark Matter is deduced by using
its gravitational effect on luminous matter. At
different scales!
Galaxy scale
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
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Galaxy Cluster 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 cant see.
Gravitational lensing effect. The light from a
far away source is deflected by Dark Matter
distortions and multiple images (Einstein Rings)
Cosmological scale
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Gravity Optics ! !
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So, what is dark matter ?
Some Einstein rings observed by Hubble Space
Telescope

• MACHOs (Massive Compact Halo Objects) Black
  Holes, Planets, Dead Stars
• Non-baryonic Matter (particle physics
  explanation) WIMP (Weakly Interacting Massive
  Particles) like neutralinos, neutrinos

Dark Matter is about 90 of the total matter in
the Universe ! !
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The Friedmann Model strikes back
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 do the full model
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Friedmann equation in Newtonian form

• Why the Newtonian Universe is not a good
  representation?
• Not homogeneuos and isotropic
• It is Euclidean

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)

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Einstein equation with the Cosmological constant
Homogeneity and Isotropy (Friedmann, Robertson,
Walker metric)
Scale parameter and geometry of the Universe
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
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The cosmological constant
The cosmological constant is an extra term in
Einsteins 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.
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After some simplification the result is
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
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)
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Our Universe is located at about
These slides at http//pcgiammarchi.mi.infn.it
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Two suggestions for further reading

• B. F. Schutz, A first course in general
  relativity, Cambridge University press.
• B. Ryden, Introduction to cosmology, Addison
  Wesley