Reading:

*Chapter 26* (ZG4)

Notes:

*pages 109-116*

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. |

- 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"?

- At what redshift was the universe
- 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?

- What is the
- 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?

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":

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:

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:

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 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:

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.

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)

Time since 0 | Event | Description | Temperature |
---|---|---|---|

15 x 10^{9} yrs | Now | Galaxies, stars, planets, and us | 3 K |

10^{9} yrs ? | Galaxy formation | bulges and halos of normal galaxies form | 20 K |

10^{6} yrs | Microwave Background | recombination - transparent to photons | 3000 K |

3 min | Nucleosynthesis | light elements formed | 10^{9} K |

6 sec | Electron-Positron pairs | creation of electrons | 6 x 10^{9} K |

2 sec | Neutrinos decouple | creation of neutrino background | 10^{10} K |

2 x 10^{-6} sec | Proton-Antiproton pairs | creation of nucleons | 10^{13} K |

2 x 10^{-10} sec | Electroweak unification | E-M and weak force same | 10^{15} K |

10^{-35} sec ? | Inflation | universe
exponentially expands by 10^{26} | 10^{27} K |

10^{-35} sec | Grand Unification | E-M/Weak and Strong forces same | 10^{27} K |

10^{-44} sec | Quantum Gravity | Unification of all 4 forces | 10^{32} K |

< 10^{-44} sec | Planck Era | No concept of space or time? | > 10^{32} K |

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.

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.

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

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

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

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:

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

or

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

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.

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:

becomes, as before, when the W term dominates

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

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.

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):

(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.

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.

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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*smyers@nrao.edu*
*Steven T. Myers*