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The Basic Theory of Cosmic Expansion

The Big Bang: not the best name for a theory, but its what we're stuck with. If we trace back the observed expansion of the universe, we find that the universe has a finite age, one defined by the point at which the universe has an infinite energy density. The theory we have for the expansion can only approach this point. As we go back in time, the conditions of the universe go beyond the limits of our knowledge; we can talk about a bang, but we can only study its aftermath.

The basic theory for cosmological expansion is quite simple: a uniform universe is expanding away from an initial state of infinite density. At a given time after the birth of the universe, every observer in the universe sees the objects in the universe rushing away at the same rate as for every other observer. So, at a given time, every observer also sees the same local density of matter.

To calculate a model for this doesn't require difficult physics, and in fact, it doesn't even require general relativity if one is content to model the universe over redshifts much less than unity. The forces that determine the expansion of the universe at any given time are the force of gravity and pressure. For most of the history of the universe, particularly now, pressure plays no role, so the problem reduces to the gravity slowing of the universe's expansion.

The Matter Dominated Universe

When the only forces within the universe are pressure and the gravitational forces from matter, then general relativity gives us the Friedmann cosmology. As just stated, for the recent epochs, pressure is unimportant in the expansion of the universe. The appropriate theory for the age of galaxies is the matter-dominated Friedmann cosmology.

In this theory, the universe can be in one of three states, similar to the three possible states for an object moving away from the Sun: the universe can be bound, so that expansion eventually halts and then contracts; the universe can be at the escape velocity, so that the rate of expansion of the universe is always decreasing, but the expansion never halts; and the universe can be unbound, so that it eventually reaches a point where it expands at a constant rate forever. The first type of universe is called a closed universe. The second and third types are called open universes. The second type of universe is also called an Einstein-deSitter universe.

The parameter that determines which of these solutions is in effect is the average density of matter. If its value equals the closure density ρc, then the universe is at the escape velocity, if its value is greater that ρc, the universe is bounded, and if its value is smaller than ρc, the universe is unbound. The closure density is given by

ρc = 3 H0/8 π G
= 1.88 × 10-29 h2 gm cm-3
= 3 × 10-7 h2 Ms pc-3

In the last-two equations, h is the Hubble constant divided by the value 100 km s-1 Mpc-1. The units for the last equation is solar masses per cubic parsec. The current best value from type 1a supernovae is H0 = 60 km s-1 Mpc-1, so h = 0.6, for a closure density of 6.77 × 10-30 gm cm-3. The universe appears to have a density that is at least a factor of one-third less than this; the observations suggest that the universe is open.

For an open universe, the age of the universe is constrained to be 2/H03 and 1/H0, where the first value corresponds to a universe at the closure density, and the second value to a coasting, zero-density universe. For a Hubble constant of 60 km s-1 Mpc-1, the age of the universe is between 10.8 and 16.3 billion years.

The Cosmological Constant

In developing this theory for the expansion of the universe, we have ignored an effect that may now be manifesting itself in the observations, and that is the effect of a gravitating vacuum. The force of a gravitating vacuum is expressed as a variable called the cosmological constant. This variable has been a source of speculation ever since Einstein wrongly introduced it—he wanted a repelling force to precisely counteract the gravitational force of matter in a static universe. Recent research on the luminosities of type 1a supernovae at high redshift has ignited interest in a cosmological constant that is just now coming into play.

The theory that now has cosmologists in its grip is that over the first billion years of expansion the universe decelerated through the gravitational effects of the matter, and the effects of a cosmological constant were neglibible, but in the current epoch, the density of matter in the universe has fallen to the point that the repelling force of the cosmological constant is slightly larger than the attractive force of the matter. This means that the galaxies today are being forces apart at an increasing rate, so that whatever deceleration occurred earlier in our history is now being counteracted.

The age of an open universe when there is a cosmological constant and no matter is given by

H0 t= ΩΛ-1/2 sinh-1 Λ/(1 - ΩΛ)]1/2,

where the parameter ΩΛ is a dimensionless parameter that gives the strength of the cosmological constant. The left side is always greater than unity, and it goes to infinity as ΩΛ goes to unity. When a cosmological constant is present, the universe can be older than the age of a coasting universe.

To make this theory work in our epoch, the value of the cosmological constant would have to be at just the right. If it were too large, then it would destroy the proportionality of distance with redshift that we see at small redshift, but if it were too small, then its effects on the expansion of the universe would be be negligible in the current epoch.

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