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:

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 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

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

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

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

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

to keep up with the matter density, and thus

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

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

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

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

(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

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":

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 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:

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). For a brief description of this work, and my current microwave background work, see this page.

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.

Steven T. Myers - Last revised 03May96