The Hot Big Bang model predicts that the universe is filled with
  black-body radiation.  Penzias and Wilson discovered the cosmic
  microwave background (CMB), as Gamow had predicted.  Moreover, the
  spectrum of this radiation follows *exactly* (at least to the
  precision of present-day observations) the predicted black-body
  form.  A beautiful confirmation of the Big Bang model. 

The Cosmic Microwave Background is uniform to a part in 100,000,
consistent with Einstein's hypothesis, the "Cosmological Principle",
that the universe is homogeneous on the largest scales.  But on second
thought, this is quite surprising: the universe was only about 400,000
years old when the CMB photons were emitted, thus each point was in
contact with other parts no further than 400,000 light years away.
Yet the photons we see in the CMB coming from opposite directions are
separated by *much* larger distances than that.  How could they all
have "known" to have been at the same temperature?

Alan Guth (1981) found a solution, based on Einstein's cosmological
constant.  Before describing Guth's insight, let's talk about the
cosmological constant some more.  

  Soon after Einstein developed General Relativity, Willem de Sitter
said, 'What if there is a universe with *just* a cosmological
constant?'  That is, no attractive matter, but nothing but
cosmological constant.  Another way to phrase this is in terms of the
vacuum having a negative pressure (i.e., the universe is filled with
'dark energy'); the gravitational effect of negative pressure is to
give a *repulsion* effect to gravity.  In this case, the expansion of
the universe *accelerates*.

  As the universe expands, the energy density of vacuum energy (i.e.,
that cosmological constant stuff) stays the same.  Notice how
different this is from regular matter, whose density thins out as the
universe expands.  As we'll see below, our universe does appear to
have a vacuum energy density.  Thus in the far future, the vacuum energy
will dominate, and the universe will follow de Sitter's model, and
expand exponentially.

  So Alan Guth's idea was to imagine that the universe expands very
rapidly, exponentially, due to the dominance of some sort of negative
pressure/vacuum energy stuff right in the beginning (i.e., *really*
early, say when the universe was only 10^{-35} seconds old), and then
have the vacuum energy go away, so that our current, more stately
expansion could take over.  This causes the present-day universe to
have expanded from a tiny volume, and in a sense got the universe's
expansion started.  This idea is called "inflation", because the
present-day observable universe inflated from a very tiny volume.
(Remember throughout that the universe is infinite in extent; the part
that is the tiny volume is the currently *observable* part of the
universe).

The inflation idea solves several problems and makes a few predictions:

   -It gives an explanation of why the universe is expanding in the
first place.  It gets the expansion started. 
   -It gives an explanation of why the universe appears so smooth.  If
   the present-day observable universe inflated from a tiny volume,
   then that initial tiny volume would have had a chance to
   equilibrate and all be smooth and uniform. 
This is simply not explained in the standard Big Bang model.  Again,
the CMB radiation was produced 400,000 years after the Big Bang, so
light (or any other signal) could have traveled only 400,000 light
years in that time.  Thus no region much larger than that could have
come into equilibrium, and thus we would have expected the universe to
be very inhomogeneous on larger scales.  Regions of the CMB on
opposite sides of the sky are separated by many times this 400,000
light years, so we would expect it to be very inhomogeneous.  And yet
it is not.  Inflation naturally explains this.
   -The initial tiny volume would not be *perfectly* smooth; quantum
fluctuations (think Heisenberg's Uncertainty Principle) would have
imprinted tiny ripples.  Those get hugely magnified by inflation,
giving a natural explanation of the ripples (on scale of millions of
light years) that we observe in the cosmic microwave background.   
   From maps of the CMB, you can make a plot of the strength of the
 very subtle large-scale fluctuations you see as a function of the
 angular size of the fluctuations.  This has been exquisitely measured
 by the Wilkinson Microwave Anisotropy Probe (WMAP, named after the
 late Dave  Wilkinson of Princeton).  The results are in beautiful,
 detailed agreement with the prediction of the inflationary model.  

   -Any initial space curvature of the universe would become
negligible upon inflation.  Think of a balloon; when it is small, its
curvature is very noticeable.  Blow it up to as large on the Earth.
If you stand on it, it will look pretty flat, and you'd have to go a
very long ways on its surface before noticing that it's curved.
Indeed, the inflation model predicts that the observable universe
should accurately follow Euclidean geometry, i.e., the space part
would be flat.

  A bit more about those all-important fluctuations: consider a region
  that is slightly more dense in the early universe.  It has a greater
  mass than its surroundings, and therefore will pull material to it,
  and grow further in mass.  As the universe continues to expand,
  then,  initially dense regions continue to grow in density,
  eventually leading to the very structured universe we see around us
  today. 

  One important parameter in the Friedmann Universe is called Omega_matter: 
   Omega_matter = 8 pi G rho/(3 H^2)
where rho is the average density of matter in the universe, and H is
the Hubble Constant.  If Omega > 1, then the self-gravity of the
universe is enough to eventally halt the expansion, and cause it to
recollapse.   

   There is another important parameter, Omega_lambda, which is the
similar quantity, except using the density of vacuum energy (the "dark
energy" stuff).

   If Omega_matter + Omega_Lambda = 1, then the universe has a flat space
   geometry.  What does this mean?  

   In a closed universe, the angles of a triangle add up to > 180
degrees, and a circle has a circumference < 2 pi r.   We say that it
has positive curvature.   This happens if Omega_matter + Omega_Lambda
> 1. 

   There are two other logical possibilities: a flat universe, in
which the space part of the space-time is flat, like Euclid: the
angles of triangles add up to 180 degrees, and circles have a
circumference of 2 pi r.  We say that it has zero curvature; this is
the flat space case.  (Omega_matter + Omega_lambda = 1).  

   And an open universe (a saddle-shape) with negative curvature:
angles of a triangle add up to < 180 degrees, and a circle has a
circumference > 2 pi r.  Here Omega_matter + Omega_lambda < 1.  

Escher explored these geometrical ideas in his famous etchings, in his
angel/devil figures.

  Einstein thought that the cosmological constant idea was his
"biggest blunder", and so most astronomers didn't take seriously the
idea that there may be a cosmological constant in the present-day
universe.  However, about a decade ago, astronomers were carrying out
measurements of distant supernovae.  Using the inverse square law,
they could measure the distance to each.  Comparing the redshift and
the distance gives the Hubble Constant.  But we see the most distant
supernovae when the universe was younger than it is today, so they in
effect could see how Hubble's law has changed with time.  To their
great surprise, the expansion of the universe is speeding up, just as
one would expect if the cosmological constant/dark energy idea is
causing a repulsion between the galaxies.  (Without a cosmological
constant, one would expect the mutual gravity of all the galaxies to
counteract the expansion, causing it to slow down).  So Einstein's
Biggest Blunder turned out to be correct after all (note that Einstein
wanted the cosmological constant for all the wrong reasons: he wanted
it to keep the universe from expanding.  That was wrong.  We're now
using it in a different context.).

  Thus we are finding ourselves referring to the cosmological
constant in two different contexts: Once, in the very early universe,
when it causes inflation and gets the expansion of the universe
starts, and again, at the present, where it is causing the expansion
of the universe to accelerate. 

Measurements of the CMB fluctuations tie this all together.  The
  results are: 
    Omega_lambda = 0.73
    Omega_matter = 0.27
Note that they add up quite accurately to 1.  So we live in a flat
universe.  Perhaps more precisely, any curvature of the universe is on
scales much larger than we can measure.  (In the same sense that when
we look at the surface of the Earth, it looks flat, unless we look on
large enough scales to see its curvature).  

  Here's another way to think of this: 
  It turns out that you can get a handle on the geometry of the
universe by measuring the angular size of the fluctuations in the
Cosmic Microwave Background.  We know how large these fluctuations
should be (in light years), and we can measure their angular size.  In
essence, we can use this to *test* the small-angle formula.  
This experiment was done by several groups, most definitively by WMAP,
 They found, definitively, that the small-angle formula
holds as we've phrased it, i.e., Euclid was right and the universe is
flat (or very close to it), just as inflation predicted.  More
accurately, we know the true extent of the universe is *enormous*, and
therefore looks very close to flat.

  So what does this mean for the future of the universe? 

  As the universe expands, the energy density of vacuum energy (i.e.,
that cosmological constant stuff) stays the same.  Notice how
different this is from regular matter, whose density thins out as the
universe expands.  Thus in the far future, the vacuum energy will
dominate, and the universe will follow de Sitter's model.  Galaxies that
can see each other now will eventually fall out of causal contact;
light will not be able to travel between them any more.  So there will
be an event horizon, in analogy to what we saw with the black holes;
galaxies we will no longer be able to receive photons from.  

  So *why* did the universe expand briefly in its early history, as if
there was a cosmological constant, and *why* did this phase end?  The
idea is that at the very high energies associated with the high
temperatures of the early universe (and we're talking very early here,
of order 10^{-35} seconds after the Big Bang), quantum fluctuations
could put us into a state in which the universe has negative pressure,
like a cosmological constant.  Then, by a process called 'quantum
tunnelling', the universe could make a transition to a more ordinary
state, with ordinary pressure, and therefore no more repulsive
gravity, thus ending the inflationary state.

 Quantum tunnelling is an effect of Heisenberg's uncertainty
principle: If a particle finds itself with a small amount of energy,
not enough to get past some barrier, it won't get out, by our normal
ideas.  However, its energy is somewhat uncertain, according to
Heisenberg, and so it can very briefly have a larger energy than we
thought, and get out, "tunnelling" its way out.  It turns out to be
used also in the explanation of radioactive decay, and is also related
to the discussion of thermonuclear fusion and the de Broglie
wavelength.


  If this 'quantum tunnelling' seems magical, don't worry.  Quantum
mechanics makes all sorts of non-intuitive predictions, and this is
certainly one of the weirdest.  The great physicist Richard Feynman
(Princeton graduate, and student of John Wheeler) once said that if
you claim to really understand quantum mechanics, that probably means
that you just don't appreciate how weird it is!

  Quantum tunnelling is also invoked to imagine that the universe got
started by quantum tunnelling out of nothing at all.  But this idea
doesn't work; after all, how does *nothing at all* have any laws of
physics to work with?

  Moreover, if the universe 'quantum tunnelled' from an inflationary
state to the more "ordinary" expansion we observe today, why did this
happen only once?  Indeed, it has been suggested by Andrei Linde and
others that this has happened multiple times, even an infinite number
of times: new universes are tunnelling into existence all the time, so
our universe is only one of an infinite branching chain of universes.
(A "multiverse").

  We *still* need a way to get this whole process started.  Li and
Gott idea: have the universe create *itself* via time travel; one of
these branches loops around in time to create itself.  If we do this,
there is no earliest event, no initial singularity, and no edge in
time.

  We can also address the question of the arrow of time; if time could
go in reverse, radiation from an event today could go back to that
initial time machine, which would (it turns out) destroy it.  That
would be bad!  In particular, it would be an inconsistent model.  So
this is why time goes forward; if it went backward, the universe
couldn't exist in the first place.
  
  Is this model right?  It isn't clear yet.  In all this, we're
working in ignorance of the correct model of the physics of matter
under the extreme densities and temperatures of the very early
universe.  To have a complete understanding of this will require a
complete 'theory of everything'.  Perhaps string theory, when it is
all worked out, will turn out to be consistent with the Gott-Li model.

 
Notes for Twenty-third Lecture 

© Copyright 2009 J. Richard Gott and Michael A. Strauss