We now come to address Inflation. Which is the factor, the standard theory of
Quantum Cosmology today. It really started in 1980, where Alan
Guth physicist at MIT came up with what he called a spectacular realization.
This is a page from his journaled time. some of these ideas have been already
discussed by others but not known to Guth.
And since then many other people have improved the theory and added more to it.
Well, the reason why everybody got so excited and why inflation has achieved
such a great acceptance is that it solves three key cosmological problems which had
no obvious solution before. The first one of those is so called
flatness problem, why is the universe so close to being exactly flat.
The second is the horizon problem, why was cosmic microwaves so uniform? We'll
explain both of these in a moment. The third one is less obvious and this
is, where there are no magnetic monopoles observable today? Whereas many Big Bang
theories predicted there should be copious amounts of them created in the
early universe. Inflation also accounts for the observed
power spectrum of galaxy clustering and those are extensively fluctuations left
over from an earlier phase of the universe.
It also predicts the existence of random cosmic gravitational wave background, but
we're very far from being able to detect such a thing.
And implies that our universe is just a small part of a much, much, much bigger
universe. Let's tackle the flatness problem first.
In Friedmann Lemaitre models, omega always evolves away from unity.
So if the universe was slightly negative curvature, it'll become more so as it
evolves. If it was slightly positive curvature
it'll become more so as it evolves. A great demonstration of that is shown in
this diagram [UNKNOWN] my right. It shows the density of the universe, one
nanosecond after the big bang and it was 400 something sextillion 16 grams per
cubic centimeter. Add one more gram per cubic centimeter,
universe becomes closed and will collapse.
Subtract one, and universe goes to expansion forever.
So this is extremely sensitive, even the slightest deviation of omega equal one
from the early universe would map into a much bigger deviation today.
And yet we know the universe is very close to flat, so we must have started
really really close to flat, if not absolutely flat.
The horizon problem can be stated this way at any given time.
you can receive signals from points that are within the time elapse from the big
bang. And the particle-horizon distance is
three times speed of the light times, the age of the universe at the time.
Now remember that early on scale factor goes as time to the 2/3 power, and so it
means that the horizon expands faster than the universe, and as time goes on
you get to see more and more objects come into it.
Now look at cosmic micro background. It originates from roughly [UNKNOWN]
1000, and so it's temperature back then was about 10^4.5 higher than it is today.
At that time the horizon distance was given by the formula that I just
mentioned above, and since then it's expanded by a factor of 1000.
However, our horizon distance is bigger than that.
And therefore, there should be many disconnected, causally disconnected
regions in the sky, which were not in a causal contact at the time with when
microwave background was released. So, even though we can see that there is
one sky now, back then, patches of micro background scalar that are more than say
two degrees apart were not in a causal contact.
And yet, cosmic micro background is uniform to few parts in a million.
So how did these independent pieces of universe know to have same temperature
within a part in a million universes with which they could not have had physical
contact. The monopole problem is somewhat generic
to the particle physics, and prediction is that copious numbers of those will be
generated during the grand unified theory transition, the face transition.
Yet none have been found despite many intensive searches.
Not only that, but their masses are supposed to be so high that coupled with
density, they would totally dominate the dynamics of the universe.
A generic expectation is that future theory of everything that unifies gravity
with other interactions, all interactions are unified into one around planck time
or there abouts. The universe undergoes a phase
transition. Then the universe undergoes a phase where
gravity splits from the outer forces, and things remain roughly the same on the
logorythmic time axis by almost 10 more powers of 10 when there is another phase
transition. What is postulated here, is that physical
vacuum was not in its actual ground state at the time.
But it was at higher state, just like in atoms there is a ground state and then
electrons can go to higher orbits, and have excited states.
Also supposedly the universe, the vac, physical vacuum itself,
could've been in such a higher state. Somehow, in mechanisms for this are very
complex and beyond the scope of this class.
The vacuum undergoes a phase transition going to the actual ground state, the
true vac and in that process, vast amounts of energy are released and that
energy is used to drive exponential expansion.
Here is one schematic way to show it. The potential energy of some scale or
field Pi is plotted against potentially itself.
And if universe is a center of a high plateau zero this is a metastable state.
It will eventually roll down into the true vacuum, which is at some finite
value of Pi, and may even slosh around a little bit.
So in this schematic, the universe rolls down the scalar field potential.
So the decay of the field reheats the universe from that excess energy and all
of the matter/energy content of the universe can be created in that process.
So the universe undergoes phase transition and releases this latent heat
inflating exponentially. This could be just one of the many many
bubbles in a much larger universe that expand and maybe collide which is process
called reheating. However, this is something that has never
been observed yet. Now the energy density of a physical
vacuum can be described as a cosmological constant.
This is not the cosmological constant of today, but some much larger value from
the earlier universe. In that case the Friedmann equation is
give fairly simply and has an obvious solution, an exponential expansion.
It turns out that these inflationary models.
This exponential expansion phase goes for about 100 e folding times or 40 orders of
magnitude in size. And since the deviation of the density
parameter from unity also goes from exponentially.
That means, there was incredibly finely tuned to be close to unity to begin with.
So here is a schematic expansion diagram, it's plotted as 1 plus redshift rather
than 1 over the quantity, which would be the scale factor.
So there is a rapid initial expansion, and then universe enters into a
traditional Friedmann Lemaitre phase. So how does this solve the flatness
problem? Well think about, this as follows.
You can start with the region on a sphere, or this is now a 2 dimensional
equivalent of 4 dimensional space time. If you inflate this sphere by a large
amount, that region will appear spatially flat, and by that same token this great
expansion of the universe essentially flattens the local curvature which might
have been more substantial. So the density parameter differs from
unity by an epsilon, a very tiny number At the same time to solves the horizon
problem. Space can expand faster than the speed of
light, so the regions that were constantly disjoint at the time of the
recombination might have been spatially close to each other at the beginning of
the inflation. They might be really adjacent to each
other and that, that's why they have same energy density or same temperature of the
micro background today. The inflation carries them apart, and so
by the time, by the end of the inflation they were no longer in a casual contact.
But nevertheless, prior to that, they were nicely termalized.
Inflation also tackled the origin of the large scale structure we see.
In quantum physics, the vacuum is not empty but it's populated by virtual
particle-antiparticle pairs that appear and dissappears.
subject to the uncertainty principle. They cause essentially quantum
fluctuations of energy density in the early universe.
Now inflation blows up these minute quantum fluctuations of physical vacuum
to enormous size. Where in fact, where they can be really
seeds of the large skill structure that we observed today.
Is an remarkable prediction, that time quantum fluctuations can result in larger
structures that they're observed. And there is a specific prediction of the
functional form of the power spectrum of these fluctuations, which turns out
corresponse fairly closely to what's observed.
Note that this is not approve of inflation because one can come up with
different ways to reach the same observed state if large scale structure.
But the consistency is cert, certainly very encouraging.
Next time, we'll talk about even earlier universe.