The oldest cosmological paradox concerns the fact that the night sky should not appear dark in a very large (or infinite), ageless Universe. It should glow with the brightness of a stellar surface.

There are many possible explanations which have been considered. Here are a few:

- There's too much dust to see the distant stars.
- The Universe has only a finite number of stars.
- The distribution of stars is not uniform. So, for example, there could be an infinity of stars, but they hide behind one another so that only a finite angular area is subtended by them.
- The Universe is expanding, so distant stars are red-shifted into obscurity.
- The Universe is young. Distant light hasn't even reached us yet.

The premise of the second explanation may technically be correct. But the number of stars, finite as it might be, is still large enough to light up the entire sky, i.e., the total amount of luminous matter in the Universe is too large to allow this escape. The number of stars is close enough to infinite for the purpose of lighting up the sky. The third explanation might be partially correct. We just don't know. If the stars are distributed fractally, then there could be large patches of empty space, and the sky could appear dark except in small areas.

But the final two possibilities are surely each correct and partly responsible. There are numerical arguments that suggest that the effect of the finite age of the Universe is the larger effect. We live inside a spherical shell of "Observable Universe" which has radius equal to the lifetime of the Universe. Objects more than about 15 billion years old are too far away for their light ever to reach us.

The resolution of Olber's paradox is found in the combined observation that 1) the speed of light is finite (although a very high velocity) and 2) the Universe has a finite age, i.e. we only see the light from parts of the Universe less than 15 billion light years away.

__Lookback Time__:

The large size of the Universe, combined with the finite speed for light, produces the phenomenon known as lookback time. Lookback time means that the farther away an object is from the Earth, the longer it takes for its light to reach us. Thus, we are looking back in time as we look farther away.

The galaxies we see at large distances are younger than the galaxies we see nearby. This allows us to study galaxies as they evolve. Note that we don't see the individuals evolve, but we can compare spirals nearby with spirals far away to see how the typical spiral has changed with time.

__Cosmic Edge__:

One of the first cosmic riddles is `Is there an edge to the Universe?' This question illuminates one of the common problems in dealing with cosmological issues. By definition, all discussion of the characteristics of the Universe must face the fact that the Universe has to contain the properties of everything. Thus, the term `edge' of the Universe assumes that there exists something that is not contained in the Universe. Invoking an outside property the the Universe (an edge or outside to the Universe) is logically inconsistent since, by definition, the Universe must contain everything.

A corollary to this point is that the Universe must be boundless. This does not necessary mean that the Universe is infinite, although this is the simplest solution. Notice also that space is not a receptacle for the Universe, space is physical and is contained with the Universe. Lastly, if the Universe contains everything, the it must contain its own origin mechanism, a bootstrap program.

__Static Universe__:

With the discovery that spiral nebula were, in fact, other galaxies external to our own, our concept of a Universe became one of in a Newtonian universe of infinite size and mass, galaxies spread out in infinite space. However, there is a problem with a uniform, static Universe, any density enhancements would become unstable to gravitational collapse. Thus, the whole Universe should have collapsed (or be collapsing) into a giant black hole.

In the 1930's, Edwin Hubble discovered that all galaxies have a positive redshift. In other words, all galaxies were receding from the Milky Way. By the Copernican principle (we are not at a special place in the Universe), we deduce that all galaxies are receding from each other, or we live in a dynamic, expanding Universe. This solves the problem for gravitational collapse, only small regions will collapse to form galaxies. The rest of space keeps expanding.

The expansion of the Universe is described by a very simple equation called Hubble's law; the velocity of the recession of a galaxy (determined from its redshift, see below) is equal to a constant times its distance (v=Hd). Where the constant is called Hubble's constant and relates distance to velocity in units of megaparsecs (millions of parsecs).

__Hubble's Constant__:

The velocity of a galaxy is measured by the Doppler effect, the fact that light emitted from a source is shifted in wavelength by the motion of the source. The change in wavelength, with respect to the source at rest, is called the redshift (if moving away, blueshift if moving towards the observer) and is denoted by the letter z. Redshift, z, is proportional to the velocity of the galaxy divided by the speed of light. Since all galaxies display a redshift, i.e. moving away from us, this is referred to as recession velocity.

Of course, a key parameter in understanding the distance-redshift relation is the calibration of the whole system. This is know as the problem of the extragalactic distance scale, an ongoing research project for the last 30 years. The primary goal of the distance scale project is to compare the redshift, or recession velocity, of a galaxy with some independent measure of its distance. This is, of course, much more difficult than one would natively think since we can not travel to nearby galaxies, and they are much too far away to observe their motion or parallax.

As a result, distance scale work uses a chain of distance indicators working outward from nearby stars to star clusters in our own Galaxy to stars in nearby galaxies. Unusually bright stars, such as variable stars and supernovae, complete the distance ladder out to cosmological distances. The latest results from the Hubble Space Telescope are shown above, a plot of recession velocity with distance (in megaparsecs, millions of light-years). The straight, linear correlation indicates that the Universe is currently expanding at a rate of 72 km per sec for every Mpc. The rate, known as Hubble's constant, may change with time (see next lecture).

__Expanding Universe__:

A common question in cosmology is "why are all the galaxies receding from each other?" In other words, the cosmological principle requires that we not be at a special place in the Universe. Since all the galaxies are moving away from us, then they must all be moving away from each other. This is explained if the Universe, as a whole, is expanding.

In a real sense, Hubble's law, the recession velocity of galaxies, is an illusion. The galaxies are not moving, the space between them is literally expanded. To see how this produces a Doppler effect, consider a simply Universe that is a circle. To the observers in this type of Universe, they believe they live in a 1D structure. But, in fact, they live in a 2D structure, a circle. The position of the galaxies can be measured by the distance between them (S, see diagram below) or what are called the co-moving coordinates, an angle θ between the galaxies.

The radius of the Universe is given by R, notice that R is a quantity only seen in 2D space, not measured directly by the inhabitants of the 1D circle unless they measure 2πR by walking around the Universe. Now, we let the Universe expand by a factor of 2, R becomes 2R. The distance between the galaxies becomes 2S, but the co-moving coordinate, angle θ remains unchanged. Since the distance between the galaxies has increased, then the galaxies will appear to have moved apart by S/time of expansion. When, in fact, the galaxies have not moved at all, the space between them has increased.

Expanding spacetime also explains the redshift of galaxies, which is interpreted as Doppler motion. Since space expands, any photons traveling through that space (from distant galaxies to us) must also expand, i.e. the photons are `stretched' as they travel across the Universe.

So the redshift we see for distant galaxies is really an effect of spacetime expanding, not real motion. This is good because some of the redshifts for the most distant galaxies have recessional velocities in excess to the speed of light. But this is not a contradiction for special relativity since the space is expanding, not true motion. We will also see that photons created as gamma rays in the early Universe are now redshifted to the microwave region of the spectrum to make up what is called the cosmic microwave background (CMB).

__ Geometry of the Universe __:

Can the Universe be finite in size? If so, what is ``outside'' the Universe? The answer to both these questions involves a discussion of the intrinsic geometry of the Universe.

At this point it is important to remember the distinction between the curvature of space (negative, positive or flat) and the toplogy of the Universe (what is its shape = how is it connected). It is possible to different curvatures in different shapes. For example, a torus (donut) has a negative curvature on the inside edge even though it is a finite toplogy. All types of topologies are possible such as spherical universes, cyclindrical universes, cubical universes with opposited edges identified or more complicated permutations of the identifications including twists and inversions or not opposite sides. It could be that the topology of the Universe is very complicated if quantum gravity and tunneling were important in the early epochs. We will first consider the three most basic types.

There are basically three possible shapes to the Universe; a flat Universe (Euclidean or zero curvature), a spherical or closed Universe (positive curvature) or a hyperbolic or open Universe (negative curvature). Note that this curvature is similar to spacetime curvature due to stellar masses except that the entire mass of the Universe determines the curvature. So a high mass/high energy Universe has positive curvature, a low mass/low energy Universe has negative curvature.

All three geometries are classes of what is called Riemannian geometry, based on three possible states for parallel lines

or one can think of triangles where for a flat Universe the angles of a triangle sum to 180 degrees, in a closed Universe the sum must be greater than 180, in an open Universe the sum must be less than 180.

Standard cosmological observations do not say anything about how those volumes fit together to give the universe its overall shape--its topology. The three plausible cosmic geometries are consistent with many different topologies. For example, relativity would describe both a torus (a doughnutlike shape) and a plane with the same equations, even though the torus is finite and the plane is infinite. Determining the topology requires some physical understanding beyond relativity.

Like a hall of mirrors, the apparently endless universe might be deluding us. The cosmos could, in fact, be finite. The illusion of infinity would come about as light wrapped all the way around space, perhaps more than once--creating multiple images of each galaxy. A mirror box evokes a finite cosmos that looks endless. The box contains only three balls, yet the mirrors that line its walls produce an infinite number of images. Of course, in the real universe there is no boundary from which light can reflect. Instead a multiplicity of images could arise as light rays wrap around the universe over and over again. From the pattern of repeated images, one could deduce the universe's true size and shape.

Topology shows that a flat piece of spacetime can be folded into a torus when the edges touch. In a similar manner, a flat strip of paper can be twisted to form a Moebius Strip.

The 3D version of a moebius strip is a Klein Bottle, where spacetime is distorted so there is no inside or outside, only one surface.

The usual assumption is that the universe is, like a plane, "simply connected," which means there is only one direct path for light to travel from a source to an observer. A simply connected Euclidean or hyperbolic universe would indeed be infinite. But the universe might instead be "multiply connected," like a torus, in which case there are many different such paths. An observer would see multiple images of each galaxy and could easily misinterpret them as distinct galaxies in an endless space, much as a visitor to a mirrored room has the illusion of seeing a huge crowd.

One possible finite geometry is donutspace or more properly known as the Euclidean 2-torus, is a flat square whose opposite sides are connected. Anything crossing one edge reenters from the opposite edge (like a video game see 1 above). Although this surface cannot exist within our three-dimensional space, a distorted version can be built by taping together top and bottom (see 2 above) and scrunching the resulting cylinder into a ring (see 3 above). For observers in the pictured red galaxy, space seems infinite because their line of sight never ends (below). Light from the yellow galaxy can reach them along several different paths, so they see more than one image of it. A Euclidean 3-torus is built from a cube rather than a square.

A finite hyperbolic space is formed by an octagon whose opposite sides are connected, so that anything crossing one edge reenters from the opposite edge (top left). Topologically, the octagonal space is equivalent to a two-holed pretzel (top right). Observers who lived on the surface would see an infinite octagonal grid of galaxies. Such a grid can be drawn only on a hyperbolic manifold--a strange floppy surface where every point has the geometry of a saddle (bottom).

Its important to remember that the above images are 2D shadows of 4D space, it is impossible to draw the geometry of the Universe on a piece of paper, it can only be described by mathematics. All possible Universes are finite since there is only a finite age and, therefore, a limiting horizon. The geometry may be flat or open, and therefore infinite in possible size (it continues to grow forever), but the amount of mass and time in our Universe is finite.

__Measuring Curvature__:

Measuring the curvature of the Universe is doable because of ability to see great distances with our new technology. On the Earth, it is difficult to see that we live on a sphere. One stands on a tall mountain, but the world still looks flat. One can see a ship come over the horizon, but that was thought to be atmospheric refraction for a long time.

Our current technology allows us to see over 80% of the size of the Universe, sufficient to measure curvature. Any method to measure distance and curvature requires a standard `yardstick', some physical characteristic that is identifiable at great distances and does not change with lookback time.

The three primary methods to measure curvature are luminosity, scale length and number. Luminosity requires an observer to find some standard `candle', such as the brightest quasars, and follow them out to high redshifts. Scale length requires that some standard size be used, such as the size of the largest galaxies. Lastly, number counts are used where one counts the number of galaxies in a box as a function of distance.

To date all these methods have been inconclusive because the brightest, size and number of galaxies changes with time in a ways that we have not figured out. So far, the measurements are consistent with a flat Universe, which is popular for aesthetic reasons.

__ Density of the Universe__:

There are two possible futures for our Universe, continual expansion (open and flat), turn-around and collapse (closed). Note that flat is the specific case of expansion to zero velocity.

The key factor that determines which history is correct is the amount of mass/gravity for the Universe as a whole. If there is sufficient mass, then the expansion of the Universe will be slowed to the point of stopping, then retraction to collapse. If there is not a sufficient amount of mass, then the Universe will expand forever without stopping. The flat Universe is one where there is exactly the balance of mass to slow the expansion to zero, but not for collapse.

The parameter that is used to measure the mass of the Universe is the critical density, Ω. The cosmic density parameter, Ω, is usually expressed as the ratio of the mean density observed to that of the density in a flat Universe.

Given all the range of values for the mean density of the Universe, it is strangely close to the density of a flat Universe. And our theories of the early Universe (see inflation) strongly suggest the value of Ω should be exactly equal to one. If so our measurements of the density by galaxy counts or dynamics are grossly in error and remains one of the key problems for modern astrophysics.

__Dark Matter__:

From comparing the mass estimates to the observed amount of light from galaxies, and from the abundance of light elements, that there is a problem with the fraction of the mass of the Universe that is in normal matter or baryons. The fraction of light elements indicates that the density of the Universe in baryons is only 2 to 4% what we measure as the observed density. The rest of the mass appears to be `missing', meaning unobserved or dark.

Exactly how much of the Universe is in the form of dark matter is a mystery and difficult to determine, obviously because its not visible. It has to be inferred by its gravitational effects on the luminous matter in the Universe (stars and gas) and is usually expressed as the mass-to-luminosity ratio (M/L). A high M/L indicates lots of dark matter, a low M/L indicates that most of the matter is in the form of baryonic matter, stars and stellar remnants plus gas.

A important point to the study of dark matter is how it is distributed. If it is distributed like the luminous matter in the Universe, that most of it is in galaxies. However, studies of M/L for a range of scales shows that dark matter becomes more dominate on larger scales.

Most importantly, on very large scales of 100 Mpc's (Mpc = megaparsec, one million parsecs and kpc = 1000 parsecs) the amount of dark matter inferred is near the value needed to close the Universe. Thus, it is for two reasons that the dark matter problem is important, one to determine what is the nature of dark matter, is it a new form of undiscovered matter?, the second is the determine if the amount of dark matter is sufficient to close the Universe.

__Dark Energy__:

The current observations and estimates of dark matter is that 20% of dark
matter is probably in the form of massive neutrinos, even though that mass
is uncertain. The another 5% to 10% is in the form of stellar remnants
and low mass, brown dwarfs. The rest of dark matter is called CDM (cold
dark matter) of unknown origin, but probably cold and heavy. The
combination of all these mixtures only makes 20 to 30% the amount mass
necessary to close the Universe. Thus, the Universe appears to be open,
i.e. Ω_{M} is 0.3.

With the convergence of our measurement of Hubble's constant and
Ω_{M}, the end appeared in site for the
determination of the geometry and age of our Universe. However, all was
throw into turmoil recently with the discovery of dark energy. Dark
energy is implied by the fact that the Universe appears to be
accelerating, rather than decelerating, as measured by distant
supernovae.

This new observation implies that something else is missing from our understanding of the dynamics of the Universe, in math terms this means that something is missing from Friedmann's equation. That missing something is the cosmological constant, Λ. Einstein first introduced Λ to produce a static Universe in his original equations. However, until the supernova data, there was no data to support its existence in other than a mathematical way.

The implication here is that there is some sort of pressure in the fabric
of the Universe that is pushing the expansion faster. A pressure is
usually associated with some sort of energy, we have named dark energy.
Like dark matter, we do not know its origin or characteristics. Only that
is produces a contribution of 0.7 to Ω, called Ω_{Λ} , so that
matter plus dark energy equals an &Omega of 1, a flat Universe.

With a cosmological constant, the possible types of Universes is very open. Almost any kind of massive or light, open or closed curvature, open or closed history is possible. Also, with high Λ's, the Universe could race away.

Fortunately, observations, such as the SN data and measurements of Ω allow us to constraint the possible models for the Universe. In terms of Ω for k (curvature), M (mass) and Λ (where the critical values are Ω=1), the new cosmology is given by the following diagram.

SN data gives Ω_{&Lambda}=0.7 and Ω_{M}=0.3.
This results in Ω_{k}=0, or a flat curvature. This is
sometimes referred to as the Benchmark Model which gives an age of the
Universe of 12.5 billion years.