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/
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/Rwhere 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/GHoIf the object has a measured flux F then its luminosity is given by
L = 4*pi*F*(cz/Ho)2and 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)Ho2Note that <theta*cz> means the average value of this product, but since more distant objects have higher redshifts 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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