WORLD VIEWS OF THE UNIVERSE

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 = H_oD, is due to 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 . Einstein himself made models for the Universe (circa 1910-1920s). Unfortunately, he errored because he believed in a 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.

Sloan Great Wall

I. Modern View of the Universe

Outdated Types of Universes

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.

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.

(a) At slow speeds, velocities simply add (top panel). In the picture, the arrow hits the target at speed = 300 km per hour for the arrow fired from the train while the arrow hits the target at speed = 200 km per hour for the arrow fired the person on the ground. (b) At high speeds, the speed an observer sees is limited by the speed of light (bottom panel). In the picture, the beam of light hits the target traveling at speed c, the speed of light, for both the beam fired the train and the beam fired from the ground. This is odd as the person on the train sees the beam of light leave him traveling at the speed of light c! This happenstance occurs because moving clocks run more slowly than stationary clocks.

The star distorts the shape of specetime. The moving ball simply follows the shape of spacetime. This is the ball's natural motion, it feels no force. Matter curves spacetime; spacetime tells matter how to move.

Example: Black Hole

Modeling of a black hole requires Einstein's GR, however, the idea behind them is simple. Black holes were predicted in the 1700s by Michell and Laplace who envisioned objects referred to as Dark Stars.
A nonrotating black hole is shown to the left. The mass of the black hole resides in its center, the Singularity where matter is compressed to infinite density. The radius where the escape speed from the mass is equal to the speed of light, c, is marked by the Event Horizon. The Event Horizon is not a solid surface. This radius is the Schwarzschild Radius.
Clocks run more slowly in the vicinity of the Event Horizon. Inside of the Event Horizon, an object would have to travel faster than the speed of the light to escape → even light cannot escape and the name Black Hole.
Falling into a Schwarzshild Black Hole (for further information, see see A.J.S. Hamilton [JILA])
Some Things about Black Holes

Black Holes Have No Hair
A black hole is characterized by its mass, its spin, and its electric charge

Cosmic Censorship
Singularities are cloaked by Event Horizons (no naked singularities)

In terms of spacetime, the Friedman models are flat (Euclidean space, critical universe), positive 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).

II. The Fate of the Universe

A great deal of effort is now directed toward determining which of the above models is the correct one for our Universe. How can we go about this exercise. Well, 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 H_o. 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.

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 an 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 distance, we see the object as it was when it was younger and by the same token, we see the Universe as it was when it was younger (as it was in the past).
In this way, we may be able to differentiate between the different Friedman model (open, closed, and flat).

Comment: 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 Critical Density is the amount of material the Universe must contain in order to be flat. The critical density depends on how fast the Universe expands. The faster the Universe expands the larger is the critical density. For the currently accepted Hubble constant, H_o ~ 73 km per second per Mpc, the critical density is around 9x10^-27 kg per cubic meter or around 6 hydrogen atoms per cubic meter. This is tiny, the density of air in this room is around 1.2 kg per cubic meter.
As a measure of how close we are to this critical density, we define Ω (the Greek letter Omega),
Ω_o = (average density)/(critical density)_o
In a flat Universe, we then have
Ω_m+Ω_Λ = 1
where the density is composed of two parts, the matter density, the dark matter and the normal matter, and the Dark Energy component. The flat universe is the dividing line between open and closed universes and is an unstable solution. If the universe has Ω_m+Ω_Λ < 1, then the universe is open. If the universe has Ω_m+Ω_Λ > 1, then the universe is closed.

Currently, WMAP has made the best measurement of Ω finding Ω = 1.02±0.02, a result consistent with 1.
This result is puzzling because, if at its beginning, the Universe was nearly flat but not quite flat, it would have evolved quickly to either a rapidly expanding open universe or a rapidly collapsing closed universe. For our Universe to be as old as it is and so nearly flat means that in the beginning, the Universe must have had Ω = 1 to 60 digits. Why would the Universe find itself with this one precise condition from the plethora of other possibilities (anything other than 1!). This is the
Flatness Problem

Scale Factor, R(t)
Properties of the different Friedman solutions can be readily visualized using another type of plot, one where we plot what is referred to as the scale factor for the universe, R(t) as a function of time. 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 often times chosen to be the size of the Universe today.
In terms of R(t), Friedman solutions for the evolution of the universe are shown to the right. The Flat universe that contains only matter, the green curve, stops expanding after an infinite amount of time has passed. The Flat universe dominated by dark energy, the curve continues to expand even after an infinite amount of time has elapsed. Open universes also never stop expanding even if they do not contain dark energy. A closed universe, is shown by the bottom curve.

III. The Fate of the Universe

The determination of the ultimate fate of the Universe is a major goal of cosmology and will be addressed in later classes, Topic 4.

The methods used to determine the appropriate universe model fall into three categories:

Dynamical Tests where we look at how fast the Universe is expanding as measured by the Hubble constant, H_o, and compare it to the amount of mass, radiation, and dark energy in the Universe

Topological Tests where we try and determine the shape of the Universe directly

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).

Viable 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.

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.

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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