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World Views of the Universe
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Cosmological Facts
- Redshift-distance relation, Hubble Law (interpreted as expansion of the Universe) → Big Bang
- Prediction: Cosmic Microwave Background
- Prediction: Hydrogen/Helium abundances
Models of the Universe
Based on Hubble's law and the CMB, our world view of the Universe is rather
firm. On the assumption that Hubble's Law, cz = HoD, is due
to an uniform expansion of the Universe and that the Cosmological
Principle is valid, models for the Universe have been constructed
based on Einstein's General Theory of Relativity
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Einstein himself made models for the Universe (circa
1910-1920s). Unfortunately, he erred because he
believed in an homogeneous, isotropic, and
static Universe (on philosphical grounds).
Hubble Ultra-Deep Field Image |
Wait, recall that the Cosmological Principle
says that the Universe is homogeneous and isotropic for any observer
in the Universe. The appearance of the Universe can change with time, but
all observers must see the same Universe for any given time.
How is this consistent with the observed structure in
the Universe (structures more than 1 billion light years in size!)?
Despite the impressive sizes of these features
(e.g., the
Voids, and the Sloan Great Wall
--1.37 Billion light years across), these
structures are still less than 10 % of the distances to the farthest QSOs
(z ~ 6 ==> more than 25 billion light years away)
and thus fill less than 1 % of the volume of the Universe.
The Universe appears fairly bland on scales larger than these
observed features and the Cosmological Principle seems to hold for our
Universe.
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Sloan Great Wall |
I. Modern View of the Universe
After Einstein developed his General Theory of Relativity,
Alexander Friedman and Georges Lemaitre developed
the first models for the structure
of universes based on Einstein's General Theory of
Relativity. Lemaitre was the first to suggest that the
Universe came from an initial dense state, the
primeval atom and is said to be
the father of the big bang. The term, Big Bang, was
coined later by Sir Fred Hoyle in an attempt to mock the idea.
Using General Relativity and assuming that
the Cosmological Principle
was valid for our Universe, Friedman found
three types of evolving universes.
To get a feel for Friedman's models, we will usually demonstrate properties
of Friedman universes using two-dimensional examples even
though we know that our Universe is actually at least
four-dimensional in nature. Four-dimensional?
We know length,
heighth, and width, that is, there are three-dimensions for things
in our Universe. What is the fourth
dimension? That is, what is the fourth thing we must specify
in order to describe locations in our Universe?
Spacetime:
In general, to specify
events in our Universe we must tell you: (1) where the event takes place
(the spatial location of the event);
and (2) when the event takes place (the temporal location of the event).
So, to specify an events in a universe
→ we must consider
universes to have four dimensions, the 3 spatial locations and
the time of an event. These define the spacetime coordinates of an event.
Note that in our Universe, although time is in a sense a simple coordinate
equivalent to position, time
seems to occupy a preferred place in our Universe in that we can
move arbitrarily through space but we cannot
seem to move arbitrarily through time
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Interestingly, the framework upon which we
place these spacetime events
for the Universe can have different shapes and that the shapes can be
modified by the presence of mass and energy. This is completely contrary
to Newton who thought that space was rigid and that there was a master
clock in the Universe which kept time for everyone.
These notions are interesting because the way objects move is determined by
the shape of spacetime, objects simply follow the lay of the land
as they move. The way mass and energy fit into this idea is that mass and
energy are able to distort the shape of spacetime. Mass and energy tell space
how to bend. Spacetime then tells mass and energy how to move.
In terms of spacetime, the Friedman models are
flat (Euclidean space, critical universe),
postive curvature (e.g.,
a sphere, the closed universe), and
negative curvature (e.g., a saddle, open universe)
universes. In two-dimensions, positive and negative curvature spacetimes
look like
positive curvature |
negative curvature |
Abstract as are these concepts, these shapes
(topologies) have important consequences for how our Universe evolves,
the ultimate fates of the Universe.
If we can determine the topology
(geometry or shape) and composition of our Universe
then we can infer what is going to
happen to our Universe
in the distant future. In the coming weeks, we explore how astronomers
deduce the shape (topology) and composition of our Universe and thus
determine the ultimate fate of our Universe.
| Properties of Geometric Objects in Different
Universes
The geometry (curvature) of each type of universe is different; the
spacetime
of each universe has different character. Simple measurable differences
between the universes can be found from properties of different geometric shapes
and features in each spacetime.
Triangles, parallel lines, circumferences of circles behave differently in
the different universes (see the left panel).
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Hubble's Law
In the context of Einstein's view of the universe,
How are Hubble's Law
and galactic redshifts interpreted?
The galaxies and other denizens of the Universe are simply carried along by the
expansion of the spacetime.
In this sense, they may be stationary in the expanding
Universe; the apparent recession of distant galaxies with respect to our
home galaxy, the Milky Way, occurs because the distances of the
galaxies from us, grow with
time as the Universe grows in size! Each galaxy sees the other galaxies
behave in the same manner.
No one galaxy sits at the center of the universe,
furthermore, there is no center of the expansion. This difficult
conceptual point follows from the notion that the Universe itself is growing
in size, it is not expanding into anything and
so has no edges and therefore no center. All points in the Universe
are simply moving apart with time.
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Cosmological Redshifts
Again, in an epxanding Universe, galaxies and the other denizens of the
Universe are carried along by the
expansion of the spacetime and, in this sense, are stationary. So,
How Do Galactic Redshifts Arise?
The apparent recession of distant galaxies with respect to our
home galaxy, the Milky Way, occurs because the distances of the
galaxies from us, grow with
time as the Universe grows in size! In addition, the light in the Universe
(see the inset to the right) also stretches as the Universe expands.
In this manner, the expansion of the Universe redshifts the light from distant
galaxies; the wavelength of the light when produced is smaller than today.
Such redshifts caused by the expansion of the
Universe are referred to as
Cosmological Redshifts
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Hubble Law and Topology of the Universe
How does the Hubble Law, that is, the relation between the redshift of
and distance to far-away galaxies behave and how is it interpreted
in the context of Einstein's General Theory of Relativity
and the different Friedman models.
The different Friedman models for universes can only be differentiated
by how the Hubble Law behaves at large distances.
When we see at object, we see the object as it
was in the past. We do not see the object it is now! If an object is
at a distance of 1 million light years, we see the object as it was
1 million years ago! So when we see an object at large disstance, we see the
Universe when it was younger (as it was in the past). |
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Scale Factor, R(t)
Properties of the different spacetime solutions can also be
readily visualized using what is referred to as
the scale factor
for the universe R(t). The scale factor, R(t),
tells us how much bigger or smaller the Universe is today than
it was yesterday and so on.
- Scale Factor → Size = R(t) x Size (in the past)
where Size (in the past) is usually the size of the Universe at some
specific point chosen in the past.
In terms of the scale factor R(t), the evolution of three solutions
for universes dominated by matter
are shown to the right. Universes dominated by dark
energy can behave differently at long times.
Collapsing universes correspond to the bottom curve.
Open universes correspond to the top curve.
Flat universes (critical universes) fall right between
closed and open universes.
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A great deal of effort is now directed toward determining which of
the above models is the correct one for our Universe.
Initially, the Universe was driven to expand by some unknown impetus. The
current rate at which the Universe expands is measured by the
Hubble
constant Ho. The expansion is slowed by the gravitational
attraction of material contained in the Universe. If the amount of material
is large enough then the expansion of the Universe will be halted
and the Universe will eventually stop expanding and start to contract!
In this sense, we can define
an escape speed for the Universe. If the Universe
does not exceed this escape speed,
it will reach a maximum size and collapse.
There is then some critical amount
of material in the Universe (which determines how strongly gravity slows the
expansion). For this precise amount of stuff,
the Universe assumes a critical form
where it is neither open nor closed,
but is
flat. This amount of material leads to the definition
of the Critical Density.
The ultimate fate of an expanding Universe
is determined by the interplay
of;
- the mutual gravitational attraction of
material in the Universe
- the repulsive nature of Dark Energy
The methods used
to determine the correct universe model fall into three
categories:
- Dynamical Tests
where we compare the rate at which the Universe expands,
as measured by the Hubble constant, Ho, and the
amount of mass, radiation, and dark energy in the Universe
- Topological Tests
where we try and measure the shape
of the Universe
- Passive Tests
where we match
our most promising models to detailed observations of the Universe.
In particular, we see which model best explains snapshost
of the Universe, for example, such as those
taken at the Epoch of Recombination when we view
the Cosmic Microwave Background (CMB).
Before we get on to the details, I need to fill in a couple of
things about the
CRITICAL DENSITY:
The critical density is given by
ρc= 3Ho2/(8πG)
where Ho is the Hubble constant and
G is the gravitational constant.
Note that the faster the expansion (the greater the Hubble constant
Ho, the larger the critical density.) For the currently
accepted Hubble constant value, Ho ~ 22 km per second per
million light years, the critical
density is around 9x10-30 grams per cubic centimeter. This
critical density which controls the evolution of the Universe is
very rarefied; the density of air in this room is around
10-6-10-5 grams per cubic centimeter.
We define
the quantity known as Ω (the
Greek letter Omega) as
Ωo = (average density)/(critical density)o
For a
flat Universe, the total material in the Universe
takes the form
Ωm+ΩΛ = 1
where the material density is composed of two parts, the matter density
composed of the dark matter and normal
matter, and the Dark Energy component.
Here, the $Omega;s are the fractions of the critical density contained
in the matter component and Dark Energy component, respectively.
For an open universe, the measured density is
less than the critical density and the sum of the Ωs is less than l and
for a closed universe, the measured density is
greater than the critical density and the Ωs are greater
than 1. The determination of Ho
and the Ωs are crucial for the determination of the ultimate fate
of the Universeand will be addressed in later Topics.
We will spend a fair amount of time on these methods (later). Let me
first touch on some aspects of the second method. We consider results of
other more definitive tests later.
Topological Tests for the Shape of the Universe
In principle, if we sat down and drew large triangles and measured their
interior angles, we could determine the shape of the Universe. As a
practical matter this is difficult. The other geometric properties of the
Universe are also difficult to measure (as well). Are there other tests we
can apply? Yes.
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Angles, Areas, & Volumes
Depending upon the geometry of the Universe, the number density of galaxies
at large redshift (→ large distance) should
depend upon the geometry of the Universe. Why? Well, just as the
area of a circle depends upon the geometry of the Universe, the volumes of
objects may also depend upon the geometry of the Universe. The difference in
the way volumes depend upon distance will affect the way densities depend
upon distance. Tests for this effect have been carried out. The results
are interesting, but not conclusive.
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Parallel Lines
Depending upon the geometry of the Universe,
the paths of parallel lines can cross
or diverge, A consequence of this is that if
measures the angular sizes of distant
galaxies, they do necessarily have to decrease as 1/distance (as they
would in a flat universe). This effect has also also been studied. The
results are suggestive but, again, not conclusive.
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