Subject: Density Determination
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Date: Wed, 17 Jan 2007 13:36:12 -0600

How do Astronomer's Measure the Density of the Universe?

In order to measure the density of the Universe, it is necessary to sample a
region that is larger than the scale on which the Universe becomes approximately
homogeneous. The volume of this region must then be measured, as well as the
mass of the matter it contains. The ratio of mass to volume then gives the
density.

The volume of the surveyed region obviously depends on the distance of the
objects at the edge of the region. The volume of a sphere of radius R
is given by V = (4*pi/3)R3. The radius is the distance
Dmax of the most distant object in the sample. Since the
region has to be rather large, this distance will be determined using the Hubble
Law. If zmax is the maximum redshift at the edge of
the region, then Dmax = czmax/ Ho
is the radius of the surveyed region, and the volume is
V =
(4*pi/3)*(czmax/Ho)3 if the whole sky is
surveyed.

The mass of an object can be derived from the orbital velocities of the
particles within the object. Consider a mass on a string that you swing around.
As you make the mass go faster, the pull you need to exert on the string gets
larger and larger. If you replace the string by a central mass, this means that
the central mass must be larger if the orbital velocities are larger. When
applied to clusters of galaxies we find a mass to luminosity ratio that implies
a density of the Universe that is about 0.3 times the critical density.

A moderately technical further discussion of these idea follows below, with
links to more detailed derivations.

The mass of the objects in the surveyed region is determined using the virial
theorem or one of its variants, which states that
v2 = GM/R

where v is a typical velocity of an particle within the object
relative to the center of mass of the object, R is an effective radius
of the object, M is the mass, and G is Newton's gravitational
constant. In words, the virial theorem states that twice the kinetic energy is
equal to the magnitude of the potential energy. [A more technical derivation can
be found in the lecture notes from
my undergraduate course.] For particles in circular orbits, v is the
circular orbital velocity and R is the radius of the orbit. For hot
X-ray gas in clusters of galaxies, v is the typical thermal velocity
and is determined by the temperature of the gas. But the radius R is
determined using an angular size theta and the distance, D =
cz/Ho, so R = theta*D = theta*cz/Ho.
Therefore the mass is given by
M = R*v2/G = theta*cz*v2/GHo

If the object has a measured flux F
then its luminosity is given by
L = 4*pi*F*(cz/Ho)2

and thus the mass-to-luminosity ratio of the object is
M/L = (Ho/cz)*theta*v2/(4*pi*F*G)

If N objects of this mass are seen in the surveyed volume, then the
derived density of the Universe is given by
rho = N*M/V = N*(3/4*pi)*<theta*cz>/(czmax)3*(v2/G)Ho2

Note that <theta*cz> means the average value of this
product, but since more distant objects have higher redhifts and lower angular
sizes, the product should be fairly independent of distance. This density has
the same dependence on the Hubble constant as the critical density, so we
determine the ratio of the density to the critical density, Omega,
directly. When applied to clusters of galaxies these techniques give
approximately Omega = 0.3.

We can also compute the luminosity density of the sampled region:
N*L/V = N*<F*(cz)2>*Ho/[3*(czmax)3]

The luminosity density of the Universe is about 190 million solar luminosities
per cubic Megaparsec
(Mpc) at visual wavelengths for Ho = 71. Since the
critical density is 140 billion solar masses per cubic Mpc, the
mass-to-luminosity ratio of the Universe needs to be 700 solar if the Universe
has the critical density. But based on the masses of clusters of galaxies, the
ratio is only 300 solar. But the mass-to-luminosity ratio of the solar
neighborhood is only 3 solar! Thus the Universe has a mass that is 100 times
larger than the mass of the stars that we see. There is a large amount of dark matter that
we see only through its gravitational effects.

If we use the virial theorem on galaxies instead of clusters of galaxies then
we get a mass-to-luminosity ratio that is about 30. Thus the mass-to-luminosity
ratio appears to vary with the size of the region measured, from 3 in the solar
neighborhood to 30 in galaxies to 300 in clusters of galaxies. Is there a
possibility that for even larger objects the ratio could reach the critical
value of 700? For such large regions we cannot use the virial theorem because
these regions are still expanding with the Hubble flow. However, we can compute
the gravitational acceleration due to the large density contrasts in the nearby
superclusters. The density contrast, d(rho)/rho, can be measured by
counting galaxies. The gravitational acceleration is proportional to
d(rho) which is the measured density contrast times the unknown
density. The gravitational acceleration times the age of the Universe gives our
peculiar velocity relative to the CMB, which can be
determined from the dipole anisotropy
of the CMB. Different groups have reached different conclusions about whether
the resulting Omega could reach the critical value of 1. But it
definitely appears that the dark matter fraction
increases with the size of objects at least up to clusters of galaxies (1 Mpc
radius).

We can use the observed abundances of hydrogren, helium and lithium isotopes
to estimate the total density of baryons in the Universe. This gives a value of
4.4% of the critical density for Ho = 71. Since the
Omega = 0.27 from clusters of galaxies is much larger than 0.044, most
of the matter in the Universe must be non-baryonic.

One proposed way out of the need for dark matter is to modify the law of
gravity. This is done in the Modification Of Newtonian
Dynamics [MOND] theory of Milgrom. There are many papers about MOND on the
astro-ph preprint server, just in the 1993-1998 time period: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. Many papers continue to
published on MOND. While the MOND alternative to non-baryonic dark matter is not
widely accepted, it is not suppressed by the establishment, as proved by these
citations.

MOND proposes that the acceleration of gravity g is not always given
by the Newtonian formula gN = GM/r2. If
gN < ao, where ao is a new
constant of nature, then the acceleration is modified to be g =
sqrt(aogN) in the original version of MOND. Thus for
large distances the acceleration goes like 1/r instead of
1/r2, leading to constant circular velocities at large radii
for galaxies.

Aguirre, Schaye &
Quataert (2001) find that while MOND works well on galaxy scales, it fails
for Lyman alpha clouds and clusters of galaxies. The basic problem with MOND on
cluster scales is that a point mass in MOND gives the observed isothermal
profile in clusters, but most of the mass in clusters is due to a diffuse cloud
of hot gas. Thus the observed gas density profile generates a gravity field
under MOND that is not consistent with the observed gas temperature and density
profile. Sanders (2002),
written by a long time supporter of MOND, recognizes this problem and finds that
dark matter is needed in clusters even under MOND.

A long standing problem with MOND was that it did not have a relativistic
version. Bekenstein & Sanders (2005) have proposed a solution to this problem. But
Zhao et al. (2005)
find that this model requires different values for the universal constant
ao in different gravitationally lensing clusters of galaxies, so the
problem of not having a consistent relativistic version of MOND remains.

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