Lecture 26 - The Early Universe (4/22/99)

Quasars --- | --- Final Exam!

Reading:
Chapter 26 (ZG4)

Notes:

pages 109-116

Fluctuations
in the Cosmic Microwave Background radiation, as seen by the COBE satellite.
(Courtesy COBE/NASA)

Key Question:

What evidence is there to support
or refute the Hot Big Bang model?

Key Principle:

Big-Bang Nucleosynthesis

Key Problem:

Find out where the microwave
background and helium came from.

Investigations:

Distances

What are roles of distance in the flux-luminosity, angular diameter and
velocity-time relations? Are these the same distances?

What do we mean by proper distance or comoving distance?

What is the comoving distance at redshift z for a flat universe?

What is the horizon distance at z of infinity?

What is the angular diameter distance d_A for a given comoving
distance?

What is the luminosity distance d_L?

Why are these different?

How does surface brightness vary with redshift?

The Radiation Era

At what redshift was the universe radiation dominated?

How does the density evolve in the radiation era?

How does the scale factor R evolve with time t?

What is the relationship T(t) in this era?

What time t after the big bang was the universe at temperature T?

What do we mean by the "first three minutes"?

Big-Bang Nucleosynthesis

Why can the universe be considered as an inside-out star?

What happens when the universe was about 10^9 K? What redshift was this?

Why do only light elements get formed in primordial nucleosynthesis?

Why does the production of different isotopes of hydrogen, helium,
lithium, beryllium and boron depend on the ratio of photons to nucleons?

Why is the production of primodial helium at 25% one of the great
acheivements of the big bang model?

What is the current estimate of the baryon to photon ratio, assuming the
Omega in baryons were 1? How does this compare to big-bang nucleosynthesis
values?

If the temperature for nucleosynthesis were 10^9 K and the density for
nucleosynthesis were 10^-2, what is the redshift of nucleosynthesis?

The Cosmic Microwave Background

What should be the temperature of the universe now if it was 10^9 at
nucleosynthesis?

At what wavelengths should this radiation be seen?

Who got the Nobel Prize for discovering the Cosmic Microwave Background
radiation?

What is the current best value of T_0 as determined by the COBE
satellite with FIRAS?

To what level is the CMBR isotropic on large scales?

Where is the CMB dipole generated? What is our peculiar velocity with
respect to the CMB frame?

What level of fluctuations in the CMB did the COBE DMR experiment find
on large angular scales? What is our best model for the generation of these?

What can measurements of cosmic background anisotropies tell us about
the values of cosmological parameters?

How does the discovery and subsequent measurements of the CMBR support
the big-bang model?

How can a CMB be accomodated into a steady-state model?

Timescales and Temperatures

What is the characteristic temperature for the pair production of
particles of mass m?

At what time and temperature were electron-positron pairs able to be
produced? (511 keV/c^2)

At what time and temperature were proton-antiproton pairs able to be
produced? (938 Mev/c^2)

At what time and temperature were Z-anti Z pairs able to be produced?
(90 Gev/c^2)

At what time and temperature did the universe correspond the the most
energetic forseeable particle collider? (100 TeV/c^2)

What time and temperature corresponds to the Planck mass? (10^28 eV/c^2)

Why do space and time break down in the Planck era?

What do we mean by unification of forces? Electroweak
unification? Grand Unification?

Horizons and Problems

How far can light have traveled at time t after the big bang?

Why is this called the horizon?

What is the horizon size now?

What was the horizon size at nucleosynthesis?

What was the horizon size at last scattering?

If the horizon at nucleosynthesis was about 1 AU, how did the universe
become so uniform?

Since opposite sides of the sky were never in causal contact in a
straight FRW model, why is the CMB so isotropic?

What is the horizon problem?

What is the flatness problem? Is the universe flat?

What is the monopole problem? Are there magnetic monopoles?

What is the age problem? Is there an age problem?

Inflation

What is the cosmological constant and what does it do to the
dynamical equations?

Why can you not make a static universe without a cosmological constant?

How can a cosmological constant term drive exponential expansion?

How can a cosmological constant solve the age problem?

What is inflation?

How can inflation solve the horizon, flatness, and monopole problems?

What is a phase transition?

What is vacuum energy? How is it related to inflation?

What phase transitions are candidates for inflation?

What are cosmological defects such as cosmic strings? What can
they do?

Open Questions

Is the FRW model correct? Can we dismiss a steady-state Universe?

What is H_0? Omega_0? q_0? Lambda_0? t_0? Omega_B?

What is the universe made of? What is dark matter?

How did structure form? Are there defects?

Is inflation correct? If not, what set initial conditions?

What is outside our horizon? Are there other universes? Other physics?

What is the fate of the Universe? Heat death or big crunch?

Are quasars supermassive black holes? How and why did they form so
early?

What were the first generation of stars like?

How do galaxies form? Spirals vs. ellipticals?

How do stars form? What sets the mass function?

How do planets form? Are planetary systems common?

How does life form? Are we alone?

How little do we really know about the universe?

Cosmic Nucleosynthesis

There must have been a time in the past when the Universe was hot enough
(like the core of a star) for nuclear fusion to occur. Astrophysicists George
Gamow and Ralph Alpher in 1948 showed that when T was around a million degrees,
protons and neutrons could fuse to form helium. However, nuclear fusion in the
early Universe is like an inside out star - the Univserse stars hot and ends
cooler. Fusion in a star stars cool as they contract onto the main sequence,
then gets progressively hotter as heavier and heavier nuclei are "burned".
Because there are initially no heavy elements in the hotter earlier phases of
the universe, fusion can only occur near the end when the temperatures are low
enough to combine protons and neutrons to make deuterium, a nucleus of
"heavy hydrogen":

p + n -> 2H

You should recognize this as being similar to the first step in the
proton-proton chain (though there are still free neutrons around then,
unlike in stars, so we need not wait until the weak reaction turning protons
into neutrons which takes billions of years happens). Thus, only fusion to light
elements can occur in the short time that the Universe is cool enough so that
the deuterium formed is not broken apart by the high energy photons in the
radiation bath of the universe, yet not so cool that the proton-proton chain
will not work.

The cosmic nucleosynthesis is a competition between the fusing together and
breaking apart by high-energy photons:

Fusion <--> Destruction

At high temperatures, the destruction wins out. Thus, the building of the
elements heavier than hydrogen, known as nucleosynthesis, takes place in
the temperature range from about 10^8 down to a few times 10^6 K.

Because nuclei in this chain are built by progressive addition of neutrons
and protons, it is much like a ladder:

1H -> 2H -> 3He ->
4He -> 5? -> 6Li -> 7Li
-> 8? -> 9Be

This ladder is built by alternating the addition of protons (which change the
nuclear charge by +1 and thus the element) and neutrons (which only change the
atomic number or mass and thus the isotope of the given element). However, you
will notice that the nuclides with atomic numbers 5 and 8 are marked by "?" -
they should be isotopes of lithium (Li) and beryllium (Be) respectively but they
are unstable and decay back into what they were made from. Thus to bridge
these "gaps" to the heavier elements, you need to add deuterons (the
deuterium nuclei pn), not just single protons or neutrons. This is harder to do,
and thus there is much less lithium than hydrogen, deuterium and helium-3 and
helium-4, and there is essentially no nuclides heavier than lithium-7. All
nuclei heavier than these therefore must be formed in the centers of stars. That
is why we said earlier that we and the Earth are made from the stuff processed
in the centers of stars and blown out by supernove - "star stuff" indeed!

The relative abundances of the nuclides of hydrogen, deuterium, helium-3 and
helium-4, and lithium-6 and lithium-7 are controlled by the relative ratios of
protons, neutrons and photons at the time of nucleosynthesis. The ratio of
baryons (protons and neutrons) to photons at this time means a higher
fraction of helium (and deuterium and lithium) in the Universe today. More
neutrons relative to protons changes the relative abundances of helium-3 versus
helium-4, and deuterium and lithium-7. Using the observed abundances today
(though finding a cloud of gas that hasn't been contaminated by supernovae
ejecta is difficult) can pinpoint the time of nucleosynthesis. For example, the
fact that the mass fraction of helium-4 is around 24% is an important clue.

Gamow and Alpher used the best data at that time and deduced the redshift and
temperature of nucleosynthesis. They showed that a consistent picture for early
universe nucleosynthesis could be constructed, and furthermore that the heavy
elements must have been processed in stars. Finally, they calculated that the
current temperature T0 of radiation in the Universe should be about 5 K.

Nucleosynthesis occured approximately 3 minutes after the "Big Bang",
or projected point of infinite density and temperature. In the final lecture, we
will discuss what likely occured during the first three minutes of the Universe.
The Nobel-winning physicist Steven Weinberg wrote a wonderful book called
"The First Three Minutes" which I strongly recommend.

The Cosmic Microwave Background

The 5 K "cosmic background" predicted by Gamow and Alpher should be seen as
microwaves, since from the blackbody formula for the wavelength of maximum
emission:

max = 3
mm / ( T / 1K )

In the early 1960s, a team of Princeton astrophysicists prepared to look for
this microwave background using the latest in radio technology. However,
they were beaten to the punch by two physicists at Bell Labs looking for a
source of excess noise found in transatlantic radio communications. Penzias and
Wilson found that no matter which direction they pointed their antenna, they
found around 3 K of "excess thermal noise" in their system (even when they
cleaned all the bird droppings from their antenna). Discussions with the
Princeton scientists alerted them to the correct explanation, that they had
found the cosmic microwave background of Gamow and Alpher, purely by accident!
Penzias and Wilson would receive the 1978 Nobel Prize for their serendipitous
discovery.

As measured by Penzias and Wilson, the microwave background temperature T0 is
nearly 3 K, and was seen to be isotropic to better than 10% over the sky. Later
measurements improved on this: the COBE satellite
launched in 1989 has measured T0 = 2.726 K, and found intrinsic anisotropy only
at one part in 10^5! The spectrum of the microwave background is a perfect
blackbody to the precision of the measurement. The discovery of the microwave
background in 1965 was a vindication of the Big Bang model in that there almost
certainly had to be a time when the Universe was hot, dense, and opaque to
photons at the redshift z ~ 1000. The competing model at that time, the
so-called steady-state universe where the expansion was compensated for
by the continuous creation of matter so the same density could be maintained
indefinitely (and an infinitely old universe), was thus discredited, though some
stubborn adherents still try and tweak it so that it seems to work (though it
smells of epicycles).

One of the prime missions of the COBE satellite was to measure the
anisotropy of the cosmic microwave background radiation - in other words,
whether the temperature (brightness) of the radiation in different directions on
the sky is different from the 2.726 K average. Changes in brightness of the
radiation reflect changes in the density of the Universe at the time of
recombination at z = 1000. At this point the Universe was only about a
million years old, and there were only very small variations in the matter and
radiation density that would grow gravitationally over the course of 15 billion
years to form all the galaxies and varied structures we see today! Here are some
maps of the sky (at a wavelength of about 3mm) made by the COBE satellite in its
4-year lifetime:

The upper panel shows a strong variation in the temperature (coded from blue
to red, with blue cooler than the average and red hotter than the average) from
one side of the sky to the other. This pattern is called a dipole and is
simply due to the Doppler effect from our galaxy's (and thus the Earth's)
velocity of around 570 km/s (caused by the gravitational pull of the Virgo
cluster and the so-called "Great Attractor"). The magnitude of this dipole is
10^-3 of the T0. Q: How does this relate to v/c of the Earth's
velocity?

The middle panel shows the variations after subtracting the dipole. The
bright band across the center is radio emission from our galaxy. Note that the
map is in galactic coordinates with the galactic center in the middle!

The bottom panel shows the microwave background with the dipole and the
galaxy subtracted. Alot of the hot and cold spots are just instrumental noise,
but some of these (the signal-to-noise ratio is about 2) are real fluctuation in
the 2.7K background caused by very small density fluctuations at z=1000
when the radiation was last scattered by the ionized Universe! The level of the
fluctuations are around 10^-5 of the 2.726 K average background, very tiny
indeed. Here is a better map of the cosmic microwave anisotropies from the COBE
map:

The resolution of COBE was only around 7 degrees. This map has been smoothed
to 10 degrees. For more on the results from COBE see the COBE Home Page at
NASA.

On smaller angular scales, maps of the microwave background can be made from
ground-based telescopes. My PhD thesis at Caltech was on observations looking
for anisotropy on angular scales of 2' to 7' (arcmintues). For a brief
description of this work, and my current microwave background work, see this page.

Measuring the Cosmic Microwave Background

North and South galactic projections of the COBE DMR 4yr data.
Courtesy NASA/GSFC

Some microwave background experiments:

The COsmic
Background Explorer (NASA/GSFC)

The Mobile
Anisotropy Telescope (M. Devlin/UPenn)

The OVRO Cosmic
Microwave Background project (S.Myers/UPenn)

Microwave Anisotropy Probe
(NASA/GSFC)

Timeline of the Universe - Standard Model

Time since 0

Event

Description

Temperature

15 x 109 yrs

Now

Galaxies, stars, planets, and us

3 K

109 yrs ?

Galaxy formation

bulges and halos of normal galaxies form

20 K

106 yrs

Microwave Background

recombination - transparent to photons

3000 K

3 min

Nucleosynthesis

light elements formed

109 K

6 sec

Electron-Positron pairs

creation of electrons

6 x 109 K

2 sec

Neutrinos decouple

creation of neutrino background

1010 K

2 x 10-6 sec

Proton-Antiproton pairs

creation of nucleons

1013 K

2 x 10-10 sec

Electroweak unification

E-M and weak force same

1015 K

10-35 sec ?

Inflation

universe exponentially expands by 1026

1027 K

10-35 sec

Grand Unification

E-M/Weak and Strong forces same

1027 K

10-44 sec

Quantum Gravity

Unification of all 4 forces

1032 K

< 10-44 sec

Planck Era

No concept of space or time?

> 1032 K

Particles and Antiparticles

We used Einstein's equation from special relativity E=mc^2 to calculate the
energy converted from mass in nuclear reactions. It is allowed to convert energy
into mass also! Two photons of sufficient energy E can interact to create a
particle and antiparticle of mass m=E/c^2 or less. Since photons have no charge
and are not normal particles, conservation of quantum numbers (like charge,
baryon number, and others more obscure) require that particles be created in
pairs, along with an antiparticle (which has all quantum numbers reversed
compared with the particle). You need two photons for momentum conservation.

The rest mass of the electron is m_e = 9.11 x 10^-30 kg, and thus two a
photons of energy E = m_e c^2 = 8.20 x 10^-13 J (511 keV) each or greater can
create an electron-postitron pair! Likewise, the proton mass is m_p = 1.67 x
10^-27 kg, so two photons of energy 1.5 x 10^-10 J (938 MeV) each or more can
created pairs of protons and antiprotons. Note that the particle and
antiparticle will usually come back together in a short time, or encounter
another of its anti-partners, and annhilate turning into energy
(photons). (Note: particle - antiparticle annhiliation is a possible source of
energy, assuming you can find antiparticles sitting around somewhere, and is
perfectly efficient in the sense of E=mc^2. This should be familiar from "Star
Trek" as their stated source of energy for the starship.)

The average energy of photons in a thermal radiation bath is about E=kT,
where k is Boltzmann's constant (k = 1.38 x 10^-23 J/K). Thus, when the
temperature reaches T=mc^2/k then particle-antiparticles pair with masses m can
be created (and destroyed) at will. For electrons, this will occur at
temperatures of just under 10^10 K, which occur in the universe at a few seconds
after the big bang. At times earlier than 1 second, therefore, the Universe was
a sea of electrons and positrons being created and annhiliation spontaneously.
Likewise, when the Universe was just above 10^13 K, protons and antiprotons
could be created from the radiation, and thus earlier than 10^-4 seconds after
the big bang the Universe also had a sea of protons and antiprotons being
continuously created and destroyed.

Unification of Forces

There are four known forces in nature: strong nuclear, electromagnetic, weak
nuclear, and gravity (in decreasing order of strength with ratios of
1:10^-12:10^-14:10^-40 with ranges of 10^15m, infinity, 10^-17m, infinity
respectively). In the late 19th century, James Clerk Maxwell unified the
electric and magnetic foreces as aspects of the same phenomenon,
electromagnetism. His four equations describe the electric and magnetic
forces experienced by charged bodies. Can this be done with all the forces? It
is certainly philosophically appealing to think that all the forces of nature
are aspects of a single underlying theory!

In our current paradigm for physics, the standard model, as well as
most viable alternatives, the forces become unified as the energy of particles
undergoing the interactions increases. At high energies, you cannot tell
the difference between any of the forces, and it is only at low energies where
we experience the universe that the four fundamental forces become distinct.
This idea is related to the concept of symmetry in modern physics, since
a perfectly "symmetric" universe is one where everying is interchangeable,
particles and forces for example. Thus, the forces and particles become
differentiated through the mechanism of spontaneous symmetry breaking,
which gives particles their different masses (why is the proton more massive
than the electron?) and forces their different strengths and couplings to
particles (why is the strong force stronger than the weak force?).

The weak force is intermediated by the W and Z particles. The W and Z have
measured masses of around 90 GeV (1 GeV = 10^9 electron volts energy equivalent
mc^2. Thus, the proton mass is 0.94 GeV.) The neutral Z is slightly heavier than
the charged W+ and W-. Thus, at T of about 10^15 K, W and Z bosons can be
created and destroyed out of photons, and thus we might expect the
electromagnetic force carried by photons and the weak force carried by the W and
Z to be indistinguishable. Indeed, at this temperature and higher the
electromagnetic and weak forces are unified into the electroweak force.
The electroweak theory and unification was verified about 15 years ago in
particle accelerator experiments. The electroweak unification occured
approximately 10^-12 seconds after the big bang in our model.

Given the success of the electroweak unification theory, we might confidently
expect to be able to unify the electroweak and strong nuclear forces at even
higher energies. In fact, Einstein himself spend his last years looking for such
a grand unified theory or GUT. From the strength and nature of the strong
force, we expect that this should occur at energies of 10^14 GeV (10^23 eV)
corresponding to temperatures of 10^27 K. Unfortunately, this energy is well
above the capabilities of current and projected future accelerators (around
10^12 eV), but there some lower energy aspects of GUTs that can be tested even
at these energies. We expect that at temperatures above 10^27 K and higher,
which occurs 10^35 seconds and earlier after the big bang, the
electro-weak-strong forces are unified, and only gravity stands alone. Also, at
this so-called GUT transition when the strong force split off, some theories
state that there was a phase in the evolution of the Universe where a
cosmological constant dominated - this is the era of inflation described
below.

The Cosmological Constant

We now revisit Einstein's cosmological constant by amending the energy
equation. Consider first the standard energy equation with kinetic and potential
terms:

E/m = v2/2 - GM/R

It is easiest to rearrange this to solve for the velocity:

v2 = 2GM/R + 2E/m

What about the energy term E/m? It seems sensible that it be related to the
relativistic energy E=mc^2. If we write E = -k mc^2, where k is some constant,
then E/m = -kc^2. Then

v2 = 2GM/R - kc2

For a flat Universe, k=0. It turns out that k=+1 for a closed universe and
k=-1 for an open universe. Other values of k are not allowed by the equations of
general relativity. Note that we can see if k=+1, then you cannot have R >
2GM/c^2 or you will get negative v^2 - this is the closed universe.

Now, we can add the cosmological constant term, not as a constant (kc^2 is
already a constant in this equation) but with an R^2 dependence so it will
behave as a density:

v2 = 2GM/R - kc2 + R2

The constant (Lambda) is the
cosmological constant.

Note that the effect of Lambda grows with R, and thus the expansion
rate of the Universe increases with time. Note that the cosmological
constant term dominates the evolution of the right-hand side of the equation
when

>
2GM/R3

or

>   8G/3.

In that case, the universe will expand with v proportional to R, and thus
(using calculus) you find that

R(t) = R0 eHt/t0

This is exponential expansion. We will use this in the very early
universe to drive inflation.

Let us explore the possibilities in this modified energy equation. Note that
Einstein used it to make a static universe by noting that if we ignore the
curvature term kc^2 and if Lambda = -(8Pi G rho/3), then v=0 is the solution,
and a static universe (with constant density) is possible. This is what he later
called his greatest blunder since v was found to not be zero. Note also that if
Lambda is even slightly different than -(8Pi G rho/3), then we get a nonzero
v.

The implications of having a cosmological constant that is now greater than
the density term means that the Unverse is older than if the Universe had
Lambda=0. This is because the Universe was expanding slower in the past when
Lambda was on the order of (8Pi G Rho/3). Thus, if Lambda is large enough, then
the age problem can be remedied. This one of the proposed solutions to the age
problem, and allows a flat universe (k=0 determines the topology, not Lambda)
with a moderate H_0=82 km/s/Mpc, but does require a rather high value of Lambda.
There are some measurements being done now that could rule this high a value of
Lambda out (these are based on the counts of things like gravitational lenses
with redshift, looking for a pile-up when the universe was just changing from
density to cosmological constant dominated evolution.

Inflation

As we stated when we began our investigation into cosmology, by and large we
seem to live in a Universe that is homogeneous and isotropic. In particular, we
look in two very different directions on the sky, we find that the structure of
the Universe appears the same. For example, except for the dipole due to our own
velocity through space, the cosmic microwave background radiation appears to be
isotropic to the level of 1 part in 10^5 or so, even on opposite sides of the
sky. The crucial question is: How did the Universe know to make the cosmic
background 2.726 K in every direction that we see? This may seem like an
unimportant question, or some sort of metaphysics, except for the fact that in
our model for the expanding universe, those parts of the universe could never
have communicated with each other at the speed of light! Recombination
occured about a million years after the big bang, which was some 15 billion
years ago - that's how long it took light to reach us from when the microwave
background was last scattered and its temperature and fluctuation level
imprinted. Thus, it would have taken twice that, or a couple of million years
less than twice the present age of the Universe for any information,
forces, photons or particles to have interacted between these regions. One of
the cornerstones of physics is the concept of causality, that for
something to "cause" something else to happen, some interaction must have taken
place which occurs at the speed of light at a maximum. Thus, two regions of
space have been separated by a distance equal to or less than the light travel
time between for the age of the Universe are said to have been in causal
contact. Patches of the sky separated by an angular distance of more than 2
degrees were never in causal contact at the time that the microwave background
was generated. In general, it is difficult to see how the Universe seems
homogeneous and isotropic on scales that were never in causal contact given our
expanding universe model.

Of course, it could be solved by resorting to special initial
conditions. The subsequent evolution of a system (like the Universe) under
physical law is given by the inital conditions to say what we started with and
the laws of physics to tell us what happend to them - you should end up with the
Universe as we observe it now if you have the right model. Thus, we could put
the isotropy of the Universe as we see it down to the assumption that it started
out very homogeneous to begin with. However, it is much more satisfying if we
have some physical reason that the Universe should appear isotropic, like
it were mixed by heat and convection like the atmosphere of a star. But this
requires causal contact. The maximum distance which two points in causal contact
can be is called the horizon. This is a different sort of horizon from,
but related to, the event horizon we encountered when discussing black holes.
The problem of causality between two distant part of the observable universe is
called the horizon problem.

The solution is to modify the evolution of the Universe to make regions now
far apart even closer than they would be under the normal expanding model at
very early times. Thus, we need to change the energy equation to allow faster
than normal expansion early on. When we discussed the effect of a positive
cosmological constant in Einstein's equation, we noted that if the
cosmological constant term were large enough, you would get exponential
expansion, or inflation of the Universe:

v2 = 2GM/R - kc2 + R2

becomes, as before, when the W term dominates

v2 = R2

Using v=HR, we find that H is constant H^2 = W, and thus (using calculus)
that if R=R0 at t=t0 at some point during inflation, then

R(t) = R0 eHt/t0

For the universe to be as isotropic as we see it and yet purely causal, it
turns out that the universe must have inflated by a factor of e^60 or
more (10^26 or more).

If you think a bit, however, you should notice a problem in a cosmological
constant to drive inflation: how do you stop it? The exponential inflation with
H^2=W will go on forever unless we get rid of the W term after the universe has
inflated sufficiently. If for some reason W were to drop to zero (or at least
much smaller than 2GM/R^3), then the normal Hubble expansion as determined by
the density would take over. It turns out that using our standard model for
particle physics, there are indeed mechanisms for causing inflation, expanding
the Universe by 10^50 or so, then safely turning off inflation. There are some
theoretical indications in our theory that this occured at the time of the GUT
transition at 10^27 K. The mechanism of inflation is widely, though not
universally, accepted among cosmologists as having occured in the very early
Universe.

One of the consequences of inflation is that after expanding by such a large
amount, the curvature of the Universe at the end of inflation should be very
nearly flat. Most cosmologists who calculate inflationary models would say that
inflation predicts a Universe that is flat, and thus very nearly the
critical density. This solves another problem in cosmology, called the
flatness problem. It appears observationally that the universe is at
least within a factor of 10 of the critical density, that is that the observed
density is 10% or more of the critical density. In the expansion (after
inflation) the Universe diverges from the critical density, thus if it is within
a factor of 10 now, it had to be nearly the critical density much earlier on.
This is another worrisome case that would require fine-tuning of the initial
conditions (causality and all that). Inflation leads naturally to a very nearly
flat universe. The only problem is that most models of inflation naturally lead
to a Universe much closer to the critical density than a factor of 10. Thus,
inflation may do too good a job of solving the flatness problem if observations
indicating that the current density is only 10%-40% of the critical density. The
jury is still out on what our density really is, but this may be a potential
problem with the inflationary theory.

Also, more work needs to be done on the details of the exact physical
mechanism involved in the inflation (what particles are involved, how long it
went on, whether gravity waves were generated). It is interesting that in order
to get rid of the cosmological constant W at the end of inflation, it energy
which was driving the expansion of space must be turned into something. It turns
out it is converted into radiation and particles (thus making the matter-energy
density the critical density), and that all the matter and radiation in the
universe today was generated from the cosmological constant energy at the end of
inflation. All the matter and energy density that was present before inflation
was "inflated away" to low densities by the exponential expansion. Kind of cool,
eh?

Paul Steinhardt, professor of Physics and Astronomy at Penn, has been a
pioneer in the development of the inflationary theory in cosmology. See also the
Cosmology and
Astrophysics at Penn page for a description of this and other cosmological
research at the University of Pennsylvania.

The Planck Era

To unify the final (?) force, we need to go back even farther to temperatures
of 10^32 K at a time 10^-43 seconds after the "big bang"! The energy of this
full unification can be estimated by considering what it might mean for photons
and gravity to be equivalent. For example, is there an energy where a photon can
create a relativistic massive particle that is its own black hole? If we
remember what we've done before, we can see that this will occur when the
wavelength (h/mc) is equal to the Schwarzschild radius (2Gm/c^2):

mpl2 = hc / 2G

(where I have put in the correct factor of Pi to make it come out right).
This mass is called the Planck mass, and is equal to 10^19 GeV (or 10^32
K). Q: Calculate the Planck mass in kilograms, and the equivalent energy in
Joules and electron volts, and temperature in Kelvin..

At these energies and higher, there will be the creation and destruction of
these black hole -like particles. Since we learned from general relativity that
gravity means the curvature of space-time, then at these energies and higher,
the concept of space and time breaks down! (Is this another sort of unification
and symmetry?) In fact, the theory that must describe this will have to put
gravity on the quantum mechancial level (since the wavelength of the black hole
is equal to its size) - this is called quantum gravity, and this earliest
(?) phase of the Universe is known as the Planck Era. Since we have no
workable theory of quantum gravity (though there are some intriguing
possibilities), we really do not understand the Universe at times earlier than
this (of course, it is likely that time as we know it is not relevant in the
Planck era). This can in some sense be considered to be the "Big Bang", where in
our space-time continuum came into being.

Cosmology in a Computer

The NCSA has put together a
fascinating exposition Cosmos in a
Computer featuring some of the latest state-of-the-art simulations of our
Universe. Be sure to try the exhibit map to
navigate the site.

Some Final Thoughts

The picture I have given you of the Universe should be taken seriously not
because it is written in a textbook, but because we can calculate the relevant
quantities using the laws of physics, which were in turn built upon a sequence
of observations, hypotheses, theories, then further observations.

I hope you enjoyed this class, and feel that you learned something about this
Universe we inhabit. I have pushed you pretty hard to be able to calculate some
pretty important things about the Universe, and you should feel pretty good
about your performance - you have been astrophysicists for this semester!

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[email-masked] Steven T.
Myers