Chapter 2
Baryons, Cosmology, Dark Matter and Energy
2.1 Hubble expansion
We are all aware that at the present time the universe is expanding. However, what will
be its ultimate fate? Will it continue to expand forever, or will the expansion slow and
finally reverse? In order to see what role the constituent matter and energy – baryons,
photons, neutrinos, and other stuff not yet identified – of our universe may play in answering
this question, we explore their effects in an expanding homogeneous and isotropic universe.
Consider a small test mass m which sits on the surface of a spherical chunk of this universe
having radius R. If the mean energy density of the universe is ρ, then the mass contained
inside the spherical volume is
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M (R) = πR3 ρ
3
The potential energy of the test mass, as seen by an observer at the center of the sphere, is
U = −G
M (R)m
R
while its kinetic energy is
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1
dR
T = mv 2 = m
2
2
dt
!2
By Hubble’s Law the expansion velocity is given by
v = HR
where H = R1 dR
is the Hubble constant. Although the While the size of H has been debated
dt
in the past, recent determinations give a rather precise value of 71 ± 4 km/s/Mpc. (One
parsec = 3.262 light years.) The total energy of the test particle is then
1
8
Etot = T + U = mR2 (H 2 − πρG)
2
3
and the fate of the universe depends on the sign of this number, or equivalently with the
relation of the density to a critical value
ρcrit
3H 2
∼ 1.88 × 10−29 h2 g/cm3
=
8πG
where h ∼ 0.71 ± 0.04 is (today’s) Hubble constant in units of 100 km/s/Mpc. This means
ρ<
∼ ρcrit ⇒ continued expansion
ρ>
∼ ρcrit ⇒ ultimate contraction
2.2 Photon, baryon, and neutrino contributions to mass/energy density
So how does the measured mass/energy density of the universe match up to ρcrit ? We can
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certainly do one immediate calculation, for photons. You are probably aware that photons remained in thermal equilibrium with the matter as long as there were free protons
and electrons. But just as we calculated the n + p ↔ d + γ equilibrium, we can evaluate
the p + e− ↔ H + γ equilibrium, where H denotes the hydrogen atom. Given the ionization potential of H of 13.6 eV, one can calculate when the photons cool to the point that
photocapture can no longer efficiently break up newly formed atoms. One can show this
corresponds to a temperature of about 1 eV and to a time about 380,000 years after the
Big Bang. After this point, the photons decouple from the matter as they no longer see free
charges to scatter off. This decoupled background of photons is now redshifted to microwave
energies.
For the photon number density
nγ = 2
Z
1
d3 q
= 2ζ(3)Tγ3 /π 2 ∼ 408/cm3
3
(2π) exp(q/Tγ ) − 1
where ζ(3) ∼ 1.20206 is the Riemann zeta function and Tγ the today’s cosmic microwave
background temperature, measured (with great accuracy) to be about 2.73 K. Similarly for
the energy density in photons
ργ = 2
Z
d3 q
q
= π 2 Tγ4 /15 ∼ 4.6 × 10−34 g/cm3
3
(2π) exp(q/Tγ ) − 1
It follows that photons contribute only 0.0000485 of the closure density.
Now what we did in BBN allows us to estimate the baryonic (or nucleonic) contribution
to the ρ as well. The baryon to photon number density is η, which either BBN or cosmic
microwave background studies finds to be
ηBBN = (5.9 ± 0.8) × 10−10
ηCM B = (6.14 ± 0.25) × 1010
So these values are in great agreement. Using the CMB value, we then find
nnucleons = ηCM B nγ = 2.51 × 10−7 /cm3
and thus multiplying by the average nucleon mass (a detail – but we know the n/p ratio is
1/7 for doing this average)
ρb = 4.19 × 10−31 g/cm3 ∼ 0.0442ρcrit
That is, baryons provide only 4.4% of the closure mass. Clearly the electron contribution
to ρ, ρe ∼ (6me /7mN )ρb , is then neglible, about 2 ×10−5 of ρcrit , comparable to the photon
contribution.
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One can count the “visible” nucleons, by integrating over all of the luminous matter in stars
and gas clouds. Interestingly this yields
ρvis ∼ 0.02ρcrit
Thus one concludes roughly half of the baryons are not visible. Presumably these nucleons
are some place – perhaps nonluminous gas clouds – because we believe BBN, and because
the BBN prediction for η is now confirmed by CMB results. This problem is sometimes
called the dark baryons problem – though there are even more intriguing “dark” problems.
A second dark problem has to do with large-scale gravitational interactions of galaxies,
galaxy clusters, etc. For some time it has been clear that the total ρ is much larger than
that coming from photons and baryons (and electrons). For example, Doppler studies of
the rotation rates of spiral galaxies indicate that these systems are much more massive than
their luminosities seem to suggest
ρrot ∼ 20ρvis
This is too large a discrepancy to attribute just to the dark baryons. The origin of the “dark
matter” responsible for this discrepancy is a matter of current study: there are several possibilities. But regardless of the origin of the dark matter, it appears that the matter/energy
density of our universe is a lot closer to ρcrit than one would guess from our calculations of
ρb and ργ .
Just as we have a CMB, there will be a relic neutrino spectrum left over from the big bang.
These neutrinos would have decoupled when temperatures were slightly above 1 MeV. Since
that first second of the big bang, no further interactions have occurred. If we had some
means to detect these neutrinos, they would tell us about conditions at that very early time,
e.g., their temperature fluctuations (probably exceeding tiny!) over the sky would tell us
about the structure of the universe at 1 sec.
We do the calculation of the neutrino contribution to ρ making two assumptions. First is
the assumption that we have three flavors (thus 6 neutrinos in all), as the standard model
tells us, all of which are light. We will see that this is know from both cosmology, and from
a combination of tritium β decay and recent discoveries of neutrino oscillations. The upper
bound on the masses of the light neutrinos is about 1 eV.
With this assumption neutrinos are relativistic when they decoupled. It follows that each
neutrino flavor (e.g., νe and ν̄e ) contributes:
nν = 2
It thus follows
Z
1
d3 q
= 3ζ(3)Tν3 /(2π 2 )
(2π)3 exp(q/Tν ) + 1
3 Tν
nν = ( )3 nγ
4 Tγ
3
What about Tν ? In the very early universe electrons, positrons, neutrinos, and photons
would all be relativistic and in equilibrium, characterized by a single temperature. Then
there is an epoch around 1 MeV when the neutrinos have decoupled, but the electrons are
relativistic and in equilibrium with the photons. Let the temperature of this epoch be called
T . Still sometime later the positrons and electrons annihilate into two γs. This will clearly
heat the photons. It can be shown the constancy of the entropy then relates the new Tγ
(after annihilation) to Tν by
ρTγ
Tν
=( T
)1/3
Tγ
ργ + ρTe+ + ρTe−
where the superscripts T tell us to evaluate this at temperature T, when the electrons were
relativistic and in equilibrium with the photons. Thus
ρe− = 2
It follows that
Z
q
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d3 q
= ργ
3
(2π) exp(q/Tγ ) + 1
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Tν
4
= ( )1/3
Tγ
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Thus today’s cosmic neutrino background temperature is about 1.92 K. It follows
nν =
3
3 4
nγ = nγ
4 11
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Now consider today, when the temperatures are low. Nothing has occurred to change nν .
If neutrinos were massless, they would contribute very little to the mass energy, clearly. So
lets assume they have a mass. We assume that mass is large compared to today’s kinetic
energy – that’s the only way to make them important. On the other hand, they cannot be
so massive to invalidate our assumptions of relativistic neutrinos on decoupling. (We know
this is true experimentally.) It follows that their contribution to the mass/energy is their
number density times their mass. Summing over three flavors
ρν =
X
ρcrit X
3
nγ
mν (i) = 0.0106 2
mν (i)
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h
i
i
We will see later that the maximum of the sum over neutrino masses, using only laboratory
and neutrino oscillations, is 6.6 eV. And the minimum (from the neutrino mass difference
measured with atmospheric neutrinos) is 0.055 eV. Using h=0.71 we find
0.0011 <
∼ ρν /ρcrit <
∼ 0.14
So two things are important about this. First, there is neutrino dark matter. Second, based
on laboratory data only, it could be significant, though never more than 1/7 the closure
density.
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It turns out that the large-scale structure of our universe is sensitive to neutrino mass: a topic
that will likely come up next quarter. Current analyses that take into account large-scale
structure surveys, the CMB, and other cosmological tests appear to require that the sum of
neutrino masses cannot exceed 1 eV. Thus, while more model dependent, one concludes
0.0011 <
∼ ρν /ρcrit <
∼ 0.026
That is, neutrinos could be about as important as the visible baryons in the universe’s
mass/energy budget, but not more.
2.3 Dark matter
We have gone through the calculation of the cosmological density of baryonic matter from
the theory of BBN and the measurements of light element abundances in reasonable detail.
We have also mentioned the CMB constraint on η, the baryon-to-photon ratio. This comes
from an analysis of temperature fluctuations in the Cosmic Microwave Background blackbody spectrum – which Eric Agol will likely discuss in detail in his cosmology course, but
we will also summarize below. Temperature anisotropies are found at the level of 1 part in
105 and involve a typical angular size of about one degree.
One can understand the general physics relatively simply. First, the observers very carefully
measure the temperature of the black body radiation as a function of solid angle, plotting
the very small variations in this temperature as a function of multipolarity. In a Legendre
expansion the ` of the multipole maps into distance: the higher the ` (for a peak in the
power spectrum), the more rapid the variation with change of solid angle.
The picture of structure formation is that dark matter seeds – areas of higher density –
form the gravitational potential into which ordinary matter falls. This picture presumes
that there is some spectrum of density fluctuations associated with early cosmology. Ordinary (or baryonic) matter acts differently from the dark matter because it not only responds
to gravity, but also interacts with radiation. Gravity causes ordinary matter to flow into
potential wells; radiation pressure increases in regions of higher density and thus acts to
resist strong compression of ordinary matter. The result are acoustic oscillations of the ordinary matter that reflect the time scale – the time matter has had to flow since the Big Bang.
There are a couple of processes that connect temperature variations in the CMB to density
fluctuations, and thus to the structure of the universe at recombination. The most important
physics, at least on smaller scales of most interest to us, is the heating and cooling associated
with the interactions between ordinary matter and radiation, as that matter is acoustically
compressed or rarefied. If matter flows into a gravitational potential and achieves a higher
density, some of that kinetic energy associated with the inflow will be converted into heating
of the plasma. Thus a hot spot in the CMB at small scales indicates a high density region,
while a cold spot indicates a rarefied region.
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There are a couple of other effects that can also alter the temperature. One, the SachsWolfe, effect has to do with the gravitational red shift. If a photon comes out of a region of
high density – and thus from deeper in the gravitational well – it will loose more energy –
opposite of the effect describe above. This effect becomes more effective on larger scales, as a
large-scale overdensity generates a stronger gravitational potential and a larger gravitational
red shift. Thus it has a different signature.
Such temperature fluctuations, and their connections to density fluctuations, probe the dynamic processes that govern structure formation. The kinetic energy of inflow is transfered
to the plasma by processes like Compton scattering. This provides a radiation pressure that
resists matter flow, and can halt that flow. Likewise, if a flow is reversed, motion of matter
outward in a gravitational well must lead to a cooling of the plasma, by energy conservation.
The timescale for possible acoustic oscillations – compression and rarefaction – is governed
by the age of the universe at recombination, 380,000 years. This limits the size scale of
fluctuations: if the scale is too large, there is insufficient time for matter at that scale to
fully condense.
It is relatively easy to appreciate intuitively that the largest structures that can be seen
must correspond to the largest area that can condense over the lifetime of the universe. By
condense here we mean reach the density where the radiation pressure just halts the flow.
It is helpful to think of the process as an oscillator, with gravity working to compress the
spring, and with radiation resisting the compression (and becoming more effective as the
spring is compressed). When a spring oscillates, at the points of maximum compression and
maximum rarefaction, the spring is at rest. If one were to ”sample” the spring during its
motion, therefore, the ”power” would collect at these extremes.
At recombination, of course, the sampling time is fixed at 380,000. What varies are the
springs – the variety of density fluctuations that presumably follow some characteristic spectrum. A special spring – a special size scale – are the fluctuations that, over 380,000 years,
allow matter to reach the point of maximum compression. Power will collect in this mode.
Another special mode corresponds to a size scale about half of this. There the matter has
time to reach the point of maximum compression, be forced outward by the radiation pressure, and then again come to rest as gravity once again overcomes the diminishing radiation
pressure. These are the π and 2π peaks in the power spectrum. One can continue, forming
a second compression (3π) etc. Figure 2 shows – with somewhat diminishing clarity – the
first (compression), second (rarefaction), and third (compression) power peaks in the CMB
temperature fluctuations, as measure by WMAP and other CMB probes. The first peak
corresponds to an ` of about 200 – an angular scale of about one degree. The second corresponds to an angular size of about half a degree. Peaks in the power spectrum are not
seen for `s much smaller than 200 – there has not been enough time for large-scale regions
to compress to high density.
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Figure 1: The NASA/WMAP plot of the temperature of the CMB radiation, showing variations on angular scales of about one degree. The bluer regions are slightly cooler, and the
red slightly hotter. This reflects density fluctuations at the time of last scattering, which
influence the time of recombination and alter the energy loss of radiation as photons emerge
from regions of overdensity .
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Figure 2: The measured power spectrum for CMB temperature fluctuations. From Wayne
Hu’s web page.
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Figure 3: This shows how the CMB power spectrum is influenced by variations in the baryon
density. From Wayne Hu’s web site.
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Figure 4: This shows how the CMB power spectrum is influenced by variations in the total
matter density. Fits to the WMAP data require that there is about five times more cold
dark matter than baryonic matter. From Wayne Hu’s web site.
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As the mechanism for the rarefaction is the interaction of radiation with ordinary matter
interactions, the peak structure must be sensitive to the baryon to photon ratio η that we
introduced in our BBN discussion. The baryons act as a source of inertia in the compression and rarefaction. It should be clear that if one increases the number of baryons, then
the amplitudes of the oscillations should increase: there is more inertia on infall that the
radiation has to resist, overcome, and reverse. This is shown in the third figure. Low baryon
density tends to reduce the radio of the first two peaks (corresponding to compression and
rarefaction). The result is in good agreement with the BBN determination, as we noted
earlier, favoring just slightly larger values of η. In terms of a closure density, it corresponds
to a ρbaryons of about 0.0442.
Both the CMB and the BBN calculations – based on radically different physics governing
the universe at very different times – give similar results. We noted before this implies
that about half of the baryonic matter is nonluminous. Among the possible hiding places
are MACHOS – massive compact halo objects being probed in gravitational microlensing
searches – and matter hidden in nonluminous gas clouds.
There are also a couple of reliable determinations of the total matter density. The height
of the first acoustic peak in the CMB spectrum is quite sensitive to the matter density, as
shown in the fourth figure. The position of this peak requires
ρM ∼ 0.268 ± 0.018
(fraction of the critical density). Red-shift surveys measurements of the shape of the power
spectrum for large-scale matter inhomogeneities also probe this quantity, giving
ρM ∼ 0.40 ± 0.06.
These results are in reasonable agreement with each other, as well as with the values derived
by combining the known baryon density with the baryon-to-total-mass density ratio in clusters.
Distance Type Ia supernovae can be used as standard candles – even at large distances and
thus at past times – to probe the Hubble expansion. These indicate that in addition to dark
matter, space (the vacuum) is characterized by some dark energy. This dark energy is a sort
of negative pressure working against gravity, causing the universe to expand more rapidly
than it would due to matter along. They find
ρΛ − ρM ∼ 0.4 .
Again, this sensitivity is physically very plausible. Matter retards expansion, dark energy
accelerates it. Thus the expansion rate should test the difference. Actually, one can appreciate that things are actually richer, as cold matter and dark energy evolve differently
cosmologically. Since we know how the former evolves as the universe stretches, careful
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measurements can determine the equation of state of the dark energy. Distant supernovae
are the tool for probing the condition of the universe at earlier times.
Combined cosmological analyses also give
ρ ∼ 1.0 ± 0.04
That is, the universe is close to critical density. Combined with the above results, one deduces that ρM ∼ 0.27, again with 0.044 of this being baryons (visible and otherwise) and
the rest something beyond the standard model (like the lightest stable supersymmetric particle). The remainder is the dark energy, ρΛ ∼ 0.73 – whose nature is simply not understood.
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