## Static Universe

With the discovery in the early 20th century that spiral-shaped 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).

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.

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.

## 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 expanding. To see how this produces a Doppler effect, consider a simple 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) or what are called the co-moving coordinates, an angle 'q' 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 '2pR' 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 'q' 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.

There are basically three possible shapes to the Universe; a flat Universe (Euclidean or zero curvature), a spherical Universe (positive curvature) or a hyperbolic 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 Universe can have positive curvature, a low mass Universe might have negative curvature.

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

never meeting (flat or Euclidean)

must cross (spherical)

always divergent (hyperbolic)

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 space-time 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 aside). Although this surface cannot exist within our three-dimensional space, a distorted version can be built by taping together top and bottom (see 2) and scrunching the resulting cylinder into a ring (see 3). 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. Topologically, the octagonal space is equivalent to a two-holed pretzel. 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), or turn-around and collapse (closed). Note that flat is the specific case of expansion to ever slower speeds approaching zero velocity.

One of the key factors 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 future of the Universe is usually dipicted in a radius versus time diagram. Here the 'radius,' or scale factor, of the whole Universe is shown with respect to time. Regardless of the overall curvature of the Universe, we can charactersize its size with a scale factor in order to measure the change in spacetime as the Universe expands. A closed Universe will expand to some maximum size than contract back to zero. A flat or open Universe will increase in size forever.

Our position in this diagram must be near the origin since it is difficult to directly determine which type of Universe we currently live in (at later times it becomes easier). Notice that since the Universe is currently 13.8 billion years old, the time axis is measured in very long timescales. Hubble's constant measures the expanision rate of the Universe as it is today, thus is the slope of the radius-time diagram for time=today. If the Universe were empty, i.e. no matter, then the expansion rate would remain unchanged with time. The existence of matter, and therefore gravity, causes all of the possible Universes to slow down with time.

The parameter used to measure the rate of change in the expansion rate of the Universe is called 'qo', the deaccelration parameter (although, under the right conditions, 'qo' can be positive and the Universe acceraltes). Knowledge of 'Ho' and 'qo' are the true goals of observational cosmology, for with both parameters it is possible to extract distances and ages of all things in the observable Universe.

## Cosmological Models

In modern cosmology, the different classes of Universes (open, flat or closed) are known as Friedmann universes and described by a simple equation:

In this equation, 'R' represents the scale factor of the Universe (think of it as the radius of the Universe in 4D space-time), and 'H' is Hubble's constant, how fast the Universe is expanding. Everything in this equation is a constant, i.e. to be determined from observations. These observables can be broken down into three parts gravity (matter density), curvature and pressure or negative energy given by the cosmological constant.

Historically, we assumed that gravity was the only important force in the Universe, and that the cosmological constant was zero. Thus, if we measure the density of matter, then we could extract the curvature of the Universe (and its future history) as a solution to the equation. New data has indicated that a negative pressure, or dark energy, does exist and we no longer assume that the cosmological constant is zero.

Each of these parameters can close the Universe in terms of turn-around and collapse. Instead of thinking about the various constants in real numbers, we perfer to consider the ratio of the parameter to the value that matches the critical value between open and closed Universes. For example, the density of matter exceeds the critical value, the Universe is closed. We refer to these ratios as Omega (subscript 'M' for matter, 'k' for curvature, Lambda for the cosmological constant). For various reasons due to the physics of the Big Bang, the sum of the various Omegas must equal one. And for reasons we will see later, the curvature Omega is expected to be zero, allowing the rest to be shared between matter and the cosmological constant.

The search for the value of matter density is a much more difficult undertaking. The luminous mass of the Universe is tied up in stars. Stars are what we see when we look at a galaxy and it's fairly easy to estimate the amount of mass tied up in stars, gas, planets and assorted rocks. This contains an estimate of what is called the baryonic mass of the Universe, i.e. all the stuff made of baryons = protons and neutrons. When these numbers are calculated it is found that 'W' for baryons is only 0.02, a very open Universe. However, when we examine motion of objects in the Universe, we quickly realize that most of the mass of the Universe is not seen, i.e. dark matter, which makes this estimate of 'W' to be much too low. So we must account for this dark matter in our estimate.