The Early Universe

The Summary:

 
We can, like Einstein, modify the energy equation:

v^2 = 2GM/R + 2E + WR^2

The constant W is the cosmological constant.

Note that the effect of the cosmological constant term grows with R ->
expansion rate increases with time!

The cosmological term will dominate the density term (ignoring E/m) when
W > 2GM/R^3 or W > (8Pi G/3) x density.

In this case, v is proportional to R, and we get exponential expansion:

R= R0 e^(Ht/t0)

Note that if W = -(8Pi G/3) x density, then if E=0 then v=0. This was
Einstein's cosmological constant that allows a static universe.

If W > 0, then the cosmological constant term makes a universe that is
older than a Universe with the same expansion rate with W=0.

If W is large enough, then the age problem can be remedied.

The Density of the Universe

The density of matter in the universe evolves as (1+z)^3
The density of radiation energy evolves as (1+z)^4

The density was higher in the past. Currently, the energy density in
matter is about 10^4 times greater than in radiation. Thus, our Universe
is currently matter dominated

At a redshift of around 10^4, there was equal energy density in matter
and in radiation. Earlier than this, the universe was radiation dominated.

The temperature of the Universe, as measured by the radiation
temperature, is also higher in the past:

T(z) = T0 ( 1 + z )

The Universe was hotter and denser in the past!

If we extrapolate this trend back far enough, we reach the point when it
was infintely dense and infinitely hot. This is the Big Bang.

Cosmic Nucleosynthesis

 
Gamow and Ralph Alpher in 1948 showed that when T was around a million
degrees, protons and neutrons could fuse to form helium.

Nuclear fusion in the early Universe is like an inside out star - the
Univserse stars hot and ends cooler.

Only light elements can be fused in the low temperatures from 10^8 to a
few million K.

Cosmic nucleosynthesis is the competition between fusing together and
breaking apart by high-energy photons.

Gaps in the stable elements at atomic numbers 5 and 8 restrict cosmic
nucleosynthesis to the elements hydrogen (deuterium), helium and lithium
almost exclusively.

The heavy elements that we and the Earth are made of must have been
formed in stars, not the big bang.

The relative abundances of the nuclides are controlled by the relative
ratios of protons, neutrons and photons at the time of nucleosynthesis.

Gamow and Alpher calculated the current temperature T0 to be about 5 K.
 Nucleosynthesis occured at a time of around 3 minutes after the "Big
Bang".
 

The Microwave Background

 
The 5 K "cosmic background" would be seen at radio wavelengths:
lambdamax = 3mm/T(K)

This microwave background was discovered in 1965 by two physicists at
Bell Labs, Penzias and Wilson.

Penzias and Wilson received the Nobel Prize in 1978 for their discovery.
They measured the microwave background temperature to be about 3 K and
was isotropic on the sky to better than 10%.

There is a dipole in the microwave background at the level of 10^-3 of T0
due to our velocity of 570 km/s.

Recently, anisotropy in the microwave background at the level of 10^-5 of
T0 has been discovered by ground, balloon, and satellite based
experiments. These variations were imprinted by density fluctuations at
the time of recombination (z=1000).

The COBE satellite was launched in 1989 and measured T0=2.726 K, and the
anisotropy on scales 10 degrees and above.

Particles and Antiparticles

You can turn energy in mass using E=mc^2

Two photons of sufficient energy can create pairs of particles and
anti-particles.

To conserve quantum numbers, you need to create both a particle and its
antiparticle (which has all quantum numbers like charge reversed)

The rest mass of the electron (and positron) is me = 9.11 x 10^-31 kg
and thus two photons of me c^2 each can create an electron-positron pair.

The average photon energy is related to the temperature by E=kT. Thus,
when kT=mc^2, particle-antiparticle pairs can be easily created (and
subsequently destroyed).

Electron-positron pairs are created at temperatures above about 10^10 K
(times earlier than a few seconds)

Proton-antiproton pairs are created at temperatures above about 10^13 K
(10^-4 seconds and earlier)
 
 
 

The detail:

 

 

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 = v^2/2 - GM/R

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

 

v^2 = 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

 

v^2 = 2GM/R - kc^2

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:

 

v^2 = 2GM/R - kc^2 + WR^2

The constant W is the cosmological constant. Note that the effect of W 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

 

W > 2GM/R^3

or W > (8Pi G/3) x density. In that case, the universe will expand with v
proportional to R, and thus (using calculus) you find that

 

R(t) = R0 e^(Ht/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 W = -(8Pi G/3) x density, 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 W is even slightly different than -(8Pi G/3) x density, 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 W=0. This
is because the Universe was expanding slower in the past when W was on the
order of (8Pi G/3) x density. Thus, if W 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 W) with a moderate
H0=82, but does require a rather high value of W. There are some measurements
being done now that could rule this high a value of W 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.

The Density of the Universe

We have from the previous lecture the scaling of lengths with redshift

 

R(z) = R0 / ( 1 + z )

The density of matter in a sphere of radius R is given by 3M/4Pi R^3, so

 

Density(z) = Density(now) x ( 1 + z )^3

The matter density in the Universe was higher in the past. Note however that
the critical density 3H^2/8PiG was also higher in the past, since H was higher
(expanding faster). It turns out that for a flat universe at the critical
density, the density always remains at the critical density (since it must
always be flat). Therefore, it must be that

 

H^2 = H0^2 ( 1 + z )^3

to keep up with the matter density, and thus

 

H(z) = H0 ( 1 + z )^(3/2)

Radiation (all the light in the Universe) also has a density that will increase
in number of photons per unit volume as (1+z)^3 just like the matter density.
However, the energy of the radiation is being redshifted away, and must have
been higher in the past by the factor

 

E(z) = E0 ( 1 + z )

where E = h f = h c / Lambda the wavelength. Thus, the energy density of the
radiation evolves as

 

Urad(z) = Urad(0) ( 1 + z )^4

Since the energy of matter goes as mc^2, the energy density of matter is

 

Umat(z) = Umat(0) ( 1 + z )^3

Thus, at some point in the past the two were equal, Umat=Urad. This occurs at
1 + z = Umat(0)/Urad(0). Currently, the energy of the Universe is matter
dominated with Umat/Urad around 10^4. Thus, at a redshift z of around 10^4, the
energy density in matter and in radiation were equal. Before this, at higher
redshifts, the Universe was radiation dominated with most of the energy in
photons!

The Stefan-Boltzmann law is essentially a statement about the energy density of
radiation. We can rephrase it as

 

Urad = a T^4

(Note: the Stefan-Boltzmann constant "sigma" is equal to ac/4.) Thus, the
temperature of the radiation must also be higher in the past by the factor

 

T(z) = T0 ( 1 + z )

The Universe was hotter and denser in the past. This statement turns out to
have profound consequences, and follows from the expansion of the Universe and
the theory of relativity. In fact, if we carry it to its extreme, there must
have been a point (as it turns out not infinitely far back in time) when the
universe was infinitely dense and infinitely hot - this is the point called the
"Big Bang". This fiducial beginning is the point from which the age of the
Universe is measured. Q: What does the current temperature of the radiation in
the Universe T0 represent? How could we measure it?
 

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 5K.

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 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:

 

lambdamax = 3mm / T(K)

In the early 1960, 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).
 

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 me = 9.11 x 10^-30 kg, and thus two a photons
of energy E = me c^2 = 8.20 x 10^-13 J (511 keV) each or greater can create an
electron-postitron pair! Likewise, the proton mass is mp = 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.