Bad Cosmology

The aim of this page is to highlight some common misconceptions in cosmology; either mistakes commonly made by students or subtly wrong ideas that sometimes find their way into textbooks.  I have no ambitions to discuss "alternative cosmologies" which reject significant amounts of what is supposed by most cosmologists to be well known. There is an excellent web site by Ned Wright which discusses some of these fads and fallacies in cosmology.

The following statements are all WRONG, or at least highly misleading:


No-one before Hubble imagined that the spiral nebulae might be other galaxies.
The idea that nebulae are other "Milky Ways" goes back to the very first suggestions that the Milky Way itself is a system of stars in which our solar system is embedded. Both ideas were first put forward by Thomas Wright of Durham, in his An Original Theory of the Universe (1750), although his models of the structure of the Milky Way were very different from our present one, and strongly influenced by his religious ideas. Immanuel Kant, the famous philosopher, read a slightly misleading review of Wright's work and this inspired him to produce a model very similar to our present idea of the universe in his Universal Natural History and Theory of the Heavens (1755). Independently, Johann Lambert suggested a similar theory in 1761. These ideas were debated by astronomers for over 170 years; Hubble's achievement in 1924 was to decisively end the debate by measuring the distance to the Andromeda Nebula and hence showing that it and the other spirals were outside our  own Milky Way.
Hubble discovered the redshift of galaxies.
The first person to measure the Doppler shift of a galaxy was Vesto M. Slipher, an astronomer working for Percival Lowell at the observatory in Flagstaff, Arizona which Lowell had set up mainly to observe the "canals" on Mars.  The first galaxy Slipher observed, in 1913, was the Andromeda Nebula (as it was then known), which turned out to have a huge blueshift of 300 km/s (at that time, the largest speed ever measured). Over the next few years Slipher measured many more spiral nebulae and found that they were nearly all redshifted.  At this time the debate about the nature of spiral nebulae was in full swing and Slipher's data was enough to convince some (e.g. Arthur Eddington) that the nebulae were other galaxies. Hubble himself hardly ever measured redshifts; he relied on the results of Slipher and of Hubble's colleague at the Mt. Wilson Observatory, Milton Humason.  Hubble's contribution was to find ways of measuring distances to the spirals that did not depend on the redshift, which allowed him to show that the redshift did increase linearly with distance (Hubble's law).
George Gamow was the first to predict the existence of the Cosmic Microwave Background.
Gamow contributed a great deal to our understanding of the Big Bang; in fact he was the first to seriously try to calculate the physics of the early universe.  But the first prediction that there should be relic radiation left over from the Big Bang was made by Ralph Alpher and Robert Herman, in 1948. At the time, Alpher was Gamow's Ph.D. student, and Herman was a close collaborator of the two of them, so most people assume that Gamow's name was on the paper, but it wasn't.


To get the velocity of recession of a galaxy from its redshift, you should use the formula given by special relativity.
This is frequently claimed, but it makes no sense.  The special relativistic Doppler formula allows you to calculate the velocity in your local inertial frame of a moving source of radiation. But in cosmology,  you cannot extend a local inertial frame from the observer to a distant galaxy: the whole point of curved space-time is that different local frames are needed around each event.  One has to be quite careful about how to define "velocity" in this case, but there is a perfectly sensible definition using the so-called metric distance, which is the distance you would read from a tape measure running between our Galaxy and the other. The relation between rate of change of metric distance and redshift depends on how the expansion of the universe is accelerating, but one thing to note is that the metric distance to very distant galaxies can certainly increase faster than the speed of light, whereas the special relativistic formula will never give this result.
The universe can't expand faster than the speed of light.
This statement doesn't really make sense: you have to specify some measure of the size of the universe before you can talk about the expansion, i.e. the rate at which the size is increasing.  If the universe is homogeneous, as astronomers believe, then Hubble's law applies in a strict form: v = H D, where D is the metric distance between two galaxies, H is the Hubble parameter (which is constant everywhere in space although it may change with time), and v is the speed, i.e. the rate of increase of D. If you choose a galaxy far enough away, then v is greater than the speed of light, no two ways about it.  We already know enough about our universe to say for certain that there are galaxies far enough away for this to be true; in fact the universe might well be infinite in size.  An alternative way of defining the size of the universe is to use the scale parameter R that appears in the Friedman equation. If the universe is closed,  we can choose to think of it as the surface of a 4-D hypersphere,  in which case the scale parameter is the "radius", so it makes a sensible measure of the size of the universe. If the universe is open, there is no such easy interpretation of R, The Friedman equation shows that R increases at a speed of order the speed of light, maybe faster, maybe slower, depending on the density of the universe.

The expansion of the universe

The Big Bang happened at a point in space (where is it?)
Many people want to know where the big bang actually happened. But the whole concept of the Big Bang is that the universe is homogeneous, the same at all points. This could not be true if the Big Bang happened at one particular place. In fact the Big Bang happened everywhere at once (causing serious problems for causality, but then again, having it happen at different times in different places would be even worse!).  In theory, all the points in the universe were at infinitesimal distances from each other at the very moment of  the Big Bang (that's what is meant by calling it a "singularity"), but  the whole concept of a singularity is unlikely to survive when we properly put quantum mechanics into the picture (which we don't understand how to do yet). This means there is no point trying to imagine what happened at time zero; but if you pick a moment or two after (say 10-40 seconds), then you are still in the earliest phase of the Big Bang, points are separated by finite distances (just much smaller distances than they are separated by today!), and the super-hot conditions are the same everywhere in the universe.
If the universe is spatially closed (finite), it will eventually recollapse, and if it is open (infinite), it will expand for ever.
This is a claim frequently made in textbooks and popular treatments of cosmology. It makes two unstated assumptions (i) that closed universes always have positive spatial curvature (ii) that there is no dark energy. Assumption (i) could be wrong if the universe had a `compact' topology such as a 3-Torus. These are closed but can have zero or negative spatial curvature. It would therefore have been better to say that a Universe with positive spatial curvature recollapses, while one with zero or negative curvature expands for ever. But this still makes assumption (ii), which is strongly contradicted by recent observations: dark energy is almost certainly present (in fact, it dominates the energy density of the Universe). Give dark energy, all four possibilities might occur (closed/recollapse, closed/expand forever, open/recollapse, open/expand forever).  For what it's worth, the current best guess, assuming that the dark energy is a simple cosmological constant, is that the universe will expand forever, but it is too close to call on whether the universe is open or closed.
Even if the universe is closed, you can never see all the way round it, because by the time a photon travels all the way round the universe, the universe will have recollapsed in the Big Crunch.
This is another result which depends on the two questionable assumptions discussed in the previous item. With a large cosmological constant, a photon could travel round the Universe several times, even infinitely many times in the limiting case of Einstein's static, finite universe. Obviously our Universe is no close to the Einstein model. The best current data suggest that it might just be closed (to put it another way, it is closed but the radius of curvature R0 is much larger than the radius of the visible Universe). In this case the best bet is that even though it will last for ever, the Universe will expand so fast that a photon will only be able to reach a small fraction of the galaxies, even though it travels for ever at the speed of light. (Technically, the universe may have both a particle and an event horizon). However, this still assumes the simplest, `spherical' topology. If the topology is compact, the total size of the Universe could be much smaller, and light could travel around the universe many times (although the path would only close in certain special directions, thereby revealing the global anisotropy of such models). Current data is also consistent with a slight negative curvature for our universe, i.e. very large R0. Although compact (closed) negatively curved spaces can exist, they cannot be made much smaller than R0, and so they would behave very much like the spherical geometry, containing both a particle and an event horizon.
The observable universe might be the inside of a black hole.
This is sometimes said, meaning that the universe has an event horizon (a maximum distance to which we can ever send a signal).  But in my view this stretches the sense of "black hole" beyond breaking point. In normal usage a black hole is a region of space-time described by something closely approaching a Kerr metric, which means there is a definite central singularity, and a definite event horizon within which all world-lines converge on the singularity.  This is nothing at all like our nearly isotropic and homogeneous universe, where each point has a different event horizon.

The Big Bang

The early universe went through a "hadron era" when protons and neutrons were as common as photons, before it cooled to the point that protons and neutrons mostly annihilated, after which came the "lepton era".
Most textbooks fudge heavily on this issue, because the details depend on somewhat controversial particle physics. But it seems that there was no such era.  Baryons such as protons and neutrons were only formed when the quarks are confined in triplets (similarly, mesons formed as quark-antiquark pairs at the same time).  Because of the high density of the early universe, quark confinement probably took place at surprisingly low temperatures, when typical particle energies were around 200 MeV.  This is well above the rest mass of individual up and down quarks, so these were abundant until quark confinement; but well below the rest mass of baryons and their antiparticles, so these began to annihilate as soon as they formed and quickly dropped to negligible densities. So the only hadrons which were common in the "hadron era" were the pions, and even these were outnumbered by the muons, electrons and neutrinos. Thus one could really say that the lepton era started right after the quark era.
Pressure in the early universe is the driving force for the expansion.
No No No! Take another look at the deceleration equation: it shows that the higher the pressure, the more deceleration you get; pressure slows down the expansion! This is a result straight out of general relativity and I don't know of any way to justify it without using GR.  It is by no means obvious that the universe needs a driving force; according to the Friedman equation it expands willy-nilly even if it is completely empty; arguably any further "cause" for the expansion would be superfluous (although this doesn't stop theorists trying to find one).  The expansion can be accelerated by a negative pressure, as provided by the cosmological constant.
Cold dark matter is dust.
First of all, be very careful using the word "dust" in cosmology, as it has two completely different meanings:
  1. To astronomers, "dust" consists of micron sized grains made of carbon and heavier elements (or their compounds) which float around in space and absorb light passing by, obscuring distant stars; dense clumps of dust. are visible as "dark nebulae". Sometimes you can also see this dust illuminated by nearby stars; for instance it appears as hazy bluish whisps around the stars of the Pleiades cluster.  Just because this dust makes its presence felt by obscuring distant stars (and galaxies), it is detectable, and not therefore "dark" as far as cosmologists are concerned. Also, the actual mass of interstellar dust is very small compared to be mass of stars in a galaxy, so we don't have to bother about it when working out the density of the universe.
  2. To cosmologists, "dust" means any form of matter which does not exert a pressure which is comparable to its energy density, or in other words any form of matter which is cool enough that its particles are not moving at relativistic speeds.  Most cosmologists think of entire galaxies as constituting the "grains" of this dust!
Now, in cosmology, "cold dark matter" has a very specific technical meaning: "dark" means undetected except via its gravitational effects, and "cold" means matter whose particles were not moving relativistically when it froze out of thermal equilibrium with the photon background.  This is very important because matter that is cold in this sense can form tight clumps of material under the action of gravity, comparable to the properties of the "dark haloes" of galaxies.  From this definition, cold dark matter is dust in the cosmologist's sense, but this is not a very informative description because many types of matter are dust in this sense.  In fact we know that if there is enough cold dark matter to make up a reasonable fraction of the total density of the universe, it must be non-baryonic, and usually this is implied when cosmologists talk about cold dark matter.
The weak reactions that convert neutrons to protons stops about 1 second after the Big Bang because the reaction requires about 0.8 MeV, and after 1 second the temperature is too low to supply this.
The freeze-out of the weak reaction is crucial because it determines the neutron-to-proton ratio, and hence the helium abundance produced by Big-Bang nucleosynthesis.  The above explanation for the freeze-out temperature is given in both Roos' and Liddle's textbooks; the reaction they discuss is:
n <--> p + e- + antineutrino + 0.8 MeV
This explanation is very misleading.  First of all, this reaction is not the main one that allows conversion of protons to neutrons before 1 sec. This is because the decay is very slow (with a time-constant of 887 s) and the reverse reaction requires an unlikely 3-body encounter.  The most important reaction is
n + neutrino <--> p + e- + 0.8 MeV.
Secondly, the explanation confuses two different concepts; the break-down of thermal equilibrium due to freeze-out, and the fact that in thermal equilibrium the abover reactions will "shift over to the proton side" when the temperature falls below kT = 0.8 MeV. Freeze-out happens when the cross-section for a reaction becomes so low that any given particle is likely to undergo the reaction less than once in a Hubble time. While the reaction rate depends on the temperature, the Hubble time depends on the density of the universe. It is a vital and rather surprising co-incidence that the weak reaction happens to freeze out at the point when the temperature falls to around 0.8 MeV. If the expansion of the universe were controlled by some other formula than the Friedman equation, we would have got a different proton:neutron ratio; for faster expansion, freeze-out would have occured when the ratio was still nearly unity, so baryonic matter would have been almost entirely made of helium; for slower expansion, the reactions would have remained in equilibrium for long enough that the balance would have completely shifted to the proton side, giving virtually no helium production.
Electrons and protons combined to form atoms when the temperature fell to the point that there was about one ionizing photon in the CMB per atom.
This is the explanation given by Liddle's textbook for the temperature of "decoupling"; he uses it to get 2500 K, roughly the right answer. However, this is a fudge on multiple levels. First of all there is an unjustified factor of 3 in his equation, otherwise he would get 7600 K.  Secondly, to get the fraction of CMB photons which can ionize hydrogen (i.e. hf > 13.6 eV), he just uses the Boltzmann factor rather than integrating the Planck distribution; doing this correctly gets us back to 5700 K. Thirdly the one-to-one ratio, though it sounds plausible, has nothing to do with the real physics of the situation. Over the range of temperatures at which the number of electrons bound to protons increases from 10% to 90%, the number of ionizing photons per atom falls by many orders of magnitude.  Furthermore, decoupling of the photons does not take place until nearly all the electrons are bound, at about 2900 K, because electrons are very good at scattering photons.  At this temperature there are actually about 10-11 ionizing photons per atom! In reality the decoupling temperature depends on the balance between several processes, and even if we include only the most important of these the calculation is rather complicated.

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