The basic idea of the post is that while people often talk about superclusters, they don’t really exist as structures in the universe. Much like constellations, they might be a convenient way to talk about what we see in the sky but have no existence outside our own minds. This is because superclusters are so large that in the context of an accelerating, dark energy dominated universe, they probably can’t be gravitationally bound.
Continuing my discussion of dark matter, I will now explain another piece of evidence supporting its existence: galactic rotation curves.
How rotation curves work is easily understood using basic Newtonian gravitation and assuming spherical symmetry. In a spherically symmetric system (where the properties of the system depend only on the distance from the center (the radius) and not the direction.
Due to the inverse square force law, the force on an object in a spherically symmetric system is equivalent to the force of a point mass at the origin with the total mass at radii less than that of the object. So,
where m is the mass of the object, ρ(r) is the mass density at radius r, and M(r) is the mass of the galaxy held within a sphere of radius r. For a circular orbit, this requires that
as well. Combining these results, we see that the mass is related to the rotational velocity v by
We can use measurements of the rotational velocities of objects around the centers of galaxies to estimate the mass and compare these to other methods. In particular, we see that galaxies appear to have a fairly well-defined size. Above some radius there aren’t any more stars. We might then expect the mass to cease increasing, so the velocity will then fall with 1/r.
In the 1970s, Rubin and Ford (and a few other colleagues) made a number of measurements of the velocity curves for a number of spiral galaxies. Some example papers can be found here and here. These galaxies appear to have something more akin to cylindrical symmetry than spherical symmetry, so the above equations won’t be exactly correct, but similar behavior will be expected. The velocity curves were measured by carefully taking spectrographic measurements at different positions in the galaxies. The positions of absorption and emission lines will change slightly depending on the velocities with respect to us, so the positions of these lines allow astronomers to extract the velocities from the data.
Rather than finding this 1/r behavior far from the centers of galaxies, they found that the velocity curves actually leveled off or in some cases increased even far beyond where stars appeared to be. This kind of rotation curve suggests that the total mass is approximately proportional to the radius. Alternatively, this means that the density is proportional to 1/r2.
This gradual reduction in the mass density means that much of the mass of galaxies seems to be spread over a much larger volume than the gas and stars. This mass is not emitting or absorbing light, since otherwise it would be easily detected with photodetectors. A simple explanation of this phenomenon is that the masses of galaxies are dominated by diffuse clouds of nonluminous “dark” matter. This dark matter provides most of the gravitational effects of galaxies except right near the center where normal matter is concentrated.
In this last part of this set of posts, I will discuss the thermal history of the universe. Also, note that if you find this subject interesting, this site has a nice poster on cosmology, as well as more posters on particle physics, nuclear physics, and plasma physics/fusion. The posters were designed by LBL (Lawrence Berkeley National Lab) and would make nice classroom decorations if you’re a physics teacher.
Anyway, when I say “thermal history of the universe” what I really mean is the evolution of smaller scale structures from subatomic physics to galactic clusters rather than the evolution of the global properties of the universe.
Conditions of the Early Universe
Cosmological measurements tell us that the early universe must have been very dense and very hot. It would be so dense and hot in fact, that all the various particles of the Standard Model would exist at thermal equilibrium. The temperature would be high enough (i.e. the kinetic energy of individual particles is high enough) that any known particle could be created in collisions between other particles, as long as that interaction is allowed. The high density means that each type of particle is readily created and destroyed in collisions, so the overall density reaches some equilibrium value.
The universe is assumed to expand in an adiabatic manner. That is, no energy (heat) is introduced into the system. So, if we consider a typical volume and follow it as it expands (this is a comoving volume: it expands along with the expansion of the universe), the average energy in the expanded volume does not change (other than small fluctuations since the universe is not completely uniform). If this expansion occurs in a reversible manner (a reasonable assumption), then the expansion is also isentropic.
In order to maintain equilibrium during this adiabatic expansion, the temperature must decrease and the number density of particles must also fall as the universe expands. So, the universe cools and gets less dense. At some point, the temperature gets low enough that very few particles will have enough energy to create the heaviest particles in interactions. Without any ability to generate new particles, these heavy particles must fall out of thermal equilibrium. From here, two things can happen:
- Heavy particle is unstable: The particles may continue annihilating, but even if they don’t they will eventually all decay away.
- Heavy particle is stable: At some point, the density of these particles gets so low that the probability of annihilating is basically zero. In other words, the mean free path of annihilation becomes larger than the observable universe. At this point, the particles can no longer be created in significant numbers (no production) but can also no longer annihilate in noticeable amounts. The density can only change due to the expansion of the universe. Thus, the comoving density (number of particles in a “unit” volume that expands with the universe) remains constant for the rest of time. We say that such particles have frozen out.
Particles don’t necessarily need to be heavy to undergo some sort of freeze out problem. As long as the density and cross sections are low enough for production and annihilation of a stable particle to cease, there should be some primordial signal from the early universe.
Evolution of Particles and Nuclei
- We start with all the particles of the Standard Model at equilibrium in a radiation-dominated expanding universe.
- As the temperature falls well below the electroweak scale (100 GeV – 1015 Kelvin!), production of top quarks, Higgs bosons, and W and Z bosons ceases and these unstable particles disappear except for briefly appearing in rare high-energy interactions.
- The universe cools even further, leading to the disappearance of tau mesons, bottom quarks and charm quarks (T around 1 GeV), muons and strange quarks, as well as all mesons and baryons except for protons and neutrons (T below 100 MeV). At all these scales, there is slightly more matter than antimatter, so even after annihilation of all antimatter, there is some remaining matter.
- At a temperature of a few MeV, the density drops enough that neutrinos can no longer be created in large numbers from electron-positron annihilation. The neutrinos then freeze out since they don’t annihilate very easily and don’t decay, leading to the theorized Cosmic Neutrino Background of “relic neutrinos.” Th
- The MeV scale is also interesting because it is also where nuclei first appear. The typical binding energy of a nucleus is a few MeV per nucleon, so nuclei would immediately be torn apart at higher energies. In the process of Big Bang Nucleosynthesis (BBN), the lightest few isotopes are created as the temperature falls enough for them to freeze out. The amounts of the lightest stable isotopes (hydrogen, helium, lithium) are set by BBN. The model explains observational data very well, giving us strong evidence for our model of the early universe.
- Below the MeV scale, the universe contains nuclei, electrons, neutrinos (noninteracting) and photons. At eV (12,000 Kelvin) to keV temperatures, the nuclei can start capturing electrons. This is the epoch of recombination, where neutral atoms are formed. The universe then contains mostly neutral particles, so photons freeze out as well. The universe goes from opaque to transparent. The photons left over from this form the Cosmic Microwave Background (CMB). The CMB is one of the most important tools for studying the early universe and actually reportedly contributes some of the static seen in broadcast TV (that is how it was discovered by Penzias and Wilson).
- At this point, the universe is around 400,000 years old and basically everything has frozen out. The universe is a pretty uniform fluid of noninteracting atoms, neutrinos, and photons.
After the epoch of recombination, not much seems to be happening. There is very little to measure from this era, so it is called the “Dark Ages”. One possible way to see what is happening is to look at the hydrogen 21 cm hyperfine transition line (something a number of people I know are working on). This is due to a spin transition of the electron in neutral hydrogen relative to the proton spin.
In general, what’s happening is that fluctuations in the density create gravitational potentials that attract even more matter. At some point in the first billion years or so, enough gas collects in a small enough area to start nuclear fusion, creating the first stars and heating up most gas enough to reionize it. This is the epoch of reionization. Larger and larger clumps of matter form and eventually they get big enough to start collapsing due to gravitational forces, creating the earliest galaxies and then galactic clusters as larger structures form. From here, galaxies and stars continue to form and evolve leading to the creation of the Milky Way and then our own solar system as well as countless other stars and galaxies.
There is one important thing I didn’t mention yet. It turns out that normal matter (i.e. atoms) is not sufficient to explain the structure formation we actually see. We need something else to provide the large gravitational potential wells in which galaxies and clusters formed in the early universe. This matter can’t interact much outside gravity (hence “dark matter:” it doesn’t interact with light). So, the matter component of the universe is actually composed of interacting matter (the Standard Model particles) and noninteracting matter (dark matter). Dark matter is the dominant form of matter in the universe and drives the formation of large scale structures. The “Standard Model of Cosmology” consisting of a universe composed primarily of dark energy, dark matter, and regular matter is typically called the ΛCDM (Λ(dark energy) Cold Dark Matter) model.
And that’s the end of my brief discussion of cosmology. Again, there will probably be more on this subject later. I plan to start discussing dark matter in much more detail in the near future.