Physics 321
Astrophysics II:  Lecture #24
Prof. Dale E. Gary
NJIT

Cosmology

The Big Bang--A Starting Point

The universe is expanding.  We have wonderfully precise measurements and firm evidence for this expansion: The expansion means that the universe was once much smaller, and running the clock backward to zero, we might imagine that the universe may have started at a single point, in the Big Bang.  If so, we can ask what things would have been like in the ultra compact, ultra high-energy environment of this point-like early universe.  We can go back to the point where our knowledge of physics breaks down, which is, indeed, incredibly far back, to much less than 1 s after the Big Bang began.
The Shape of the Universe
Note: This part of the lecture follows the treatment of Taylor and Wheeler in their recent text, Black Holes--Introduction to General Relativity, Addison Wesley Longman, 2000.
Einstein showed, with his theory of General Relativity, that we can interpret mass as a curvature in space-time.  In regions far from any mass, space-time is flat, and motions of objects proceed in straight lines in the absence of forces.  In regions near a mass, however, space-time is curved and motions proceed in curved paths (i.e. orbiting planets follow curved paths) in the absence of forces.  Thus, Einstein interprets the influence of the nearby mass not as causing a force, but as causing a curvature in space-time.   Objects (even massless photons) merely follow their geodesics or inertial paths in curves rather than straight lines.

If space-time can be curved due to the presence of mass, then the question arises as to what is the curvature of the Universe as a whole.  To understand curved space-time, let's look first at an analogy:  a one-dimensional world with a single coordinate, x, where length can be measured in only a single quantity dx.  A creature in such a world can only move in this one dimension, backwards or forwards, and would have no knowledge of any other dimensions, or directions of movement.  However, us 3-dimensional beings can view his universe from outside, and might see that his line of motion is curved into a circle, as in the figure below.

The creature would measure a distance dl along his (curved) line, and outside we would interpret his lengths as having a 1-dimensional length dx that depends on his position, because his motion is also changing its y coordinate.  We can write a metric for this space as
dl2 = dx2 + dy2
but since there is a relationship between the radius of curvature R and the coordinates x and y, R2 = x2 + y2, we can write y2 = R2 - x2.  Taking the differential, we have 2y dy = -2x dx, so that dy = -x dx/y = -xdx/(R2-x2)1/2, to eliminate dy and write
dl2 = dx2 + x2dx2/(R2-x2)
which can be written
dl2 = dx2/(1-x2/R2).
Analogy to 3-d Space
If we want to describe curvature in 3-dimensional space, we have the usual x, yandz coordinates, but we have to describe it in a 4-dimensional hyperspace, so we will introduce another spatial coordinate w.  Since we are 3-dimensional creatures, we cannot experience this fourth spatial dimension, but we can imagine a 4-dimensional being viewing our universe from "outside"  in order to visualize the curvature.  The curvature of our universe will again we described by a radius of curvature (or hyperradius) R.  Then we have the relationship
R2 = x2 + y2 + z2 + w2.
In an obvious analogy to the 1-d case, we can eliminate w to obtain the spatial, 4-dimensional line element with a length given by
dl2 = dx2 + dy2 + dz2 + r2dr2/(R2-r2)
where we have introduced the usual expression for a 3-dimensional spherical coordinate, r2 = x2 + y2 + z2.  Note that in such a spherical coordinate system, the square of an element of length is
dl2 = dr2 + r2sin2qdq2 + r2df2,
where we use dl for 3-d length to distinguish it from 4-d length dl.  Then we have
dl2 = dr2 + r2 sin2qdq2 + r2df2 + r2dr2/(R2-r2).
We can combine the first and last terms to get
dl2 = dr2/(1 -r2/R2) + r2 sin2qdq2 + r2df2 .
Finally, we introduce scaled distances, u = r/R and du = dr/R, to find the space part of the space-time metric:
dl2 = R2(t)[du2/(1 -ku2) + u2sin2qdq2 + u2df2].
Here we have introduced a variable k  and allow for a radius of curvature that can change with time, to make our result more general, as we will explain in a moment.  As we said, this is the space part of the metric, and expresses the curvature of space at a single moment of time.  To include the time part, for the full expression of space-time, we simply write c2dt = c2dt - dl2, or
c2dt2 = c2dt2- R2(t)[du2/(1 -ku2) + u2sin2qdq2 + u2df2].
You will find that this is the same expression as equation 25-6 in the text, which is called the Robertson-Walker metric.

Now we again run into trouble trying to interpret this equation for 3-dimensional space (4-dimensional space-time), because we have to step outside our universe and view it in another, hyperspatial dimension that we cannot visualize.  So let us instead consider only 2-dimensional space, with z = dz = 0, so that we can visualize the curvature as a third dimension.  We can then use the balloon analogy as follows:

Closed, Flat, or Open Universe?
Let's examine the closed universe model first.  A simplified treatment is the Friedmann Model, which corresponds to conditions of uniform dust distribution of mass in the universe, with zero pressure.  In this case, we can determine the way that the curvature should change with time, which is entirely determined by the average density of the universe:
(dR/dt)2 + c2 = (8p/3) rGR2
It is interesting to write this in terms of the total mass of the universe, again assuming uniform dust:
M = (4p/3) R3r
so that the above equation becomes:
(dR/dt)2 + c2 = 2GM/R2.
Consider this description of a closed universe, which will eventually stop expanding, eventually to collapse again.  In this case, dR/dt will go to zero, at which time the curvature of the universe will be the same as the Schwarzschild Radius for the universe--the universe is like one giant black hole!

It is also instructive to multiply this equation through by m/2, which gives

1/2 m (dR/dt)2-GMm/R2 = -mc2/2.
The first term is the kinetic energy of the universe, the second term is the potential energy of the universe, and the term on the right is the total energy.  It is negative for a closed universe, showing that the universe does not have enough energy to expand forever.

For a flat universe, the total energy is zero, while for an open universe it is positive.  What governs whether the universe will be open, closed, or flat?  Clearly it has to do with the total mass of the universe.  If the total mass is greater than some critical mass, Mcrit, then the universe will be closed.  We refer to the mass of the universe in terms of the ratio W = M / Mcrit.  The closed, flat, and open universes correspond to W < 1, W = 1, andW > 1, respectively.

The Visible Universe
Going back to the balloon analogy, imagine that we are at some point on the surface of the expanding balloon.  How much of our universe can we see?  It depends only on the rate of expansion.  Imagine galaxies as dots on the surface of the balloon.  As we look at more and more distant dots (galaxies), we see them receding at higher and higher velocity.  There is some point beyond which they recede at greater than the speed of light, and so obviously we cannot see them.  So there is a limit to the size of the visible universe, but there is an "outside" that could exist but not be visible.  Note that in all of the possible universes we have mentioned, open, closed, or flat, the expansion is expected to slow down, so that recession velocities will decrease with time.  In this case, new galaxies will become visible and the size of the visible universe will increase.  What is extremely puzzling is that there is now evidence, from studies of supernovae, that the universal expansion is not slowing down, but rather is speeding up!

Two studies of the brightness (or distance modulus) of supernovas as a function of redshift.  If the points lie along a straight line, the universal expansion rate is constant.  However, the more distant supernovas appear to be fainter (have a higher magnitude, or a larger distance modulus) than expected from their redshifts.  This means they are moving away more slowly, and since we are seeing them as they were long ago, it implies that the universe is speeding up in its expansion.  Is the effect real, or are the data too uncertain?  The authors think it is a real effect.
The Big Bang Fireball
If the universe did start in the Big Bang, it would have followed this scenario:
How Can We Know the Big Bang is Real?
All we can do is observe the present state of the universe and try to test various predictions of this scenario.  Two striking agreements are found: However, two striking disagreements are also found: An addition to the Big Bang theory, advanced to solve these problems, is the inflationary hypothesis put forth by Alan Guth, of MIT.  In this hypothesis, at a time about 10-35 s after the start of the Big Bang, the strong nuclear force separated from the electroweak forces.  This symmetry breaking was generated by the Higgs particle.   At a critical temperature of this time, a new "phase" grew out of this symmetry breaking, forming bubbles that grew exponentially with time--much faster than the previous powerlaw expansion.  Each bubble may represent a separate universe, and our universe is one of these regions of symmetry breaking.  During the inflation period, the universe doubled in size about every 10-35 s, until time t = 10-32 s, during which time it expanded a size smaller than a proton to the size of a grapefruit, about 10 cm.  This expansion was much faster than the speed of light, and left its mark in the two observations above--the flatness of the universe and the lack of lumpiness of the cosmic background radiation.  As a result of inflation, areas of our universe expanded beyond our horizon (to a place we cannot reach due to the limitation of the speed of light).  If the universe were closed, the expansion would slow down and part of our universe would slow and reenter our horizon.  Such a universe has positive curvature and would eventually fall back to a Big Crunch.  However, it appears that expansion is accelerating!  This means that the universe is open, and has negative curvature.  In this case, our universe will expand forever, and part of our currently visible universe will cross beyond the rim (to use a Babylon 5 phrase).  The next generation microwave anisotropy investigations will seek to determine the cosmological parameters for our universe to much better precision than the current values.
Some Questions for Discussion