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:
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No-one before Hubble imagined that the spiral nebulae
might be other galaxies.
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Hubble discovered the redshift of galaxies.
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George Gamow was the first to predict the existence
of the Cosmic Microwave Background.
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To get the velocity of recession of a galaxy from its redshift,
you should use the formula given by special relativity.
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The universe can't expand faster than the speed of light.
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The Big Bang happened at a point in space (where is it?)
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If the universe is spatially closed (finite), it will
eventually recollapse, and if it is open (infinite), it will expand for
ever.
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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.
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The observable universe might be the inside of a black hole.
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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".
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Pressure in the early universe is the driving force
for the expansion.
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Cold dark matter is dust.
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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.
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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.
Historical
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No-one before Hubble imagined that the spiral
nebulae might be other galaxies.
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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.
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Hubble discovered the redshift of galaxies.
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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).
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George Gamow was the first to predict the
existence of the Cosmic Microwave Background.
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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.
Redshifts
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To get the velocity of recession of a galaxy from
its redshift, you should use the formula given by special relativity.
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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.
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The universe can't expand faster than the speed
of light.
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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
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The Big Bang happened at a point in space (where
is it?)
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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.
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If the universe is spatially closed (finite), it
will eventually recollapse, and if it is open (infinite), it will expand
for ever.
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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.
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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.
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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.
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The observable universe might be the inside of a black
hole.
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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
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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".
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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.
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Pressure in the early universe is the driving
force for the expansion.
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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.
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Cold dark matter is dust.
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First of all, be very careful using the word "dust" in cosmology, as it
has two completely different meanings:
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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.
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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.
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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.
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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.
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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.
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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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